Wikipedia:Reference desk/Archives/Mathematics/May 2006

May 1

Pair of equations

How can you solve the system

$a+b=10\,\!$

$ab=10\,\!$

without graphing? (i.e., algebraically)

I know the answers approximately, having graphed as a last resort.

I tried substitution, but to no avail.

Guidance?

139.55.19.50 11:02, 1 May 2006 (UTC)

It's a quadratic equation. The first line gives an expression for b in terms of a; rewrite the second line in those terms and resolve to get your quadratic. Notinasnaid 11:06, 1 May 2006 (UTC)

Yes, the canonical method is the Viète's formulas. Let's see:

$x=a$ or $x=b$ is equivalent to $0=(x-a)(x-b)=x^{2}-(a+b)x+ab=x^{2}-10x+10$ from which you get $x=5\pm {\sqrt {5^{2}-10}}=5\pm {\sqrt {15}}$, thus $a=5+{\sqrt {15}}$ and $b=5-{\sqrt {15}}$ or the same two values swapped. – b_jonas 11:19, 1 May 2006 (UTC)

Algebra solves this easily, but we can also approach numerically. Since the product of a and b is positive, either both are positive or both are negative; and since their sum is also positive, we conclude that 0 < a,b < 10.

Actually, we can do better, because if one factor is 10 the other is 1, and if one summand is 1 the other is 9. Repeat! If one factor is 9 the other is ¹⁰⁄₉, and if one summand is ¹⁰⁄₉ the other is ⁸⁰⁄₉. Then we get ⁹⁄₈ and ⁷¹⁄₈; then ⁸⁰⁄₇₁ and ⁶³⁰⁄₇₁; then ⁷¹⁄₆₃ and ⁵⁵⁹⁄₆₃. Numerically we now have 1.127 and 8.873, which are correct to the number of decimal places given. In fact, the last two steps agree to this precision, so we seem to have converged to an answer. The correct answers are approximately 1.1270166537926 and 8.8729833462074, so we have done well.

As requested, we solved the system without graphing. In this particular case we could get an exact expression with algebra, but in more complicated cases it's good to have more tools. A similar computation led to the intriguing discovery of the arithmetic-geometric mean. --KSmrq^T 20:13, 1 May 2006 (UTC)

This is a nice solution. It appears that this method works for any quadratic equation which has two distinct real roots except for the ones with zero linear term.

Let's see. We have the equation $0=x^{2}-2bx+c$, where $0<b^{2}-c$ and $0\neq b$. The solutions are $x_{1}=b-{\sqrt {b^{2}-c}}$ and $x_{2}=b+{\sqrt {b^{2}-c}}$, for which $x_{1}+x_{2}=2b$, and $x_{1}x_{2}=c$. Now if $f(x)=2b-c/x$, then $x_{1}$ and $x_{2}$ are fixpoints of $f(x)$. The derivative is $f'(x)=c/x^{2}$. Now it's easy to see that one of ${\sqrt {c}}$ and $-{\sqrt {c}}$ is between the two roots, so at the larger absolute valued root $|f'(x)|<1$. Thus, the iteration $y_{n+1}=f(y_{n})$ will converge to that root if started from a sufficently small environment of it.

This way we can indeed find the roots of any quadratic equation where $0\neq b$: we find the larger absolute value root with iteration and the other root in the usual way by dividing $c$ with the first one.

The only problem with this iteration is that it converges only slowly. Let me show an example.

Suppose we want to determine the numeric value of ${\sqrt {5}}$. We can use the method shown above: start with $x_{0}=3$, iterate $x_{n+1}=2+4/x_{n}$, and finally ${\sqrt {5}}=x_{\inf }-1$. The iteration gives 3.00000 3.33333 3.20000 3.25000 3.23077 3.23810 3.23529 3.23636 3.23596 3.23611 3.23605 3.23607 3.23607 so the approximate value is 3.23607 - 1 = 2.23607. Compare to this the fast Newton-iteration: $x_{0}=2$, $x_{n+1}=(x_{n}+5/x_{n})/2$. This gives 2.00000 2.25000 2.23611 2.23607 2.23607. It's obviously worth to use the latter one, even though the iteration rule is a bit more complicated.

– b_jonas 12:50, 2 May 2006 (UTC)

                a + b = 10           (1)
                a * b = 10           (2)
(1)             a = 10 - b           (3)
(3) & (2)       (10 - b) * b = 10    (4)
(4)             10b - b^2 = 10       (5)
(5)             b^2 - 10b + 10 = 0   (6)

After that the answer is as plain as daylight.

Quadratic_equation Ohanian 06:06, 2 May 2006 (UTC)

Yes, this is a slightly more explicit version of the first response, by Notinasnaid. It does assume that we know how to solve a quadratic equation. And if we require a numerical answer, we eventually still use numeric iteration on a computer to compute the square root of 15.

A more sophisticated response would be to point at the article on resultants. Although it could use some work, such as mentioning the existence of an assortment of resultants (Sylvester, Bézout, Dixon, Macaulay, Kapur-Saxena-Yang), it explains enough to free us from custom manipulation of the equations. Better still, it will work with a much broader range of equation systems than this simple pair.

Related ideas are the Gröbner basis and Hilbert's Nullstellensatz. We even have Cylindrical Algebraic Decomposition, despite the apparent lack of a Wikipedia article, which can handle inequalities.

The deeper point of mentioning alternatives is that when we are confronted with a mathematical challenge in real life, we do not know in advance how best to confront it, or whether it is intractable. In elementary classes we are lulled into thinking there is always an answer, and a right way to find it. This is not a helpful mindset, and the sooner we escape it the better. --KSmrq^T 23:36, 2 May 2006 (UTC)

Matrix problem

Suppose I have an unknown square matrix A. I know that there exists a regular matrix Q such that:

$B_{1}=Q^{-1}AQ$

$B_{2}=QAQ^{-1}$

Only B₁ and B₂ are known to me. Is it possible to reconstruct A with this information? Under what assumptions? How can this be done in practice? -- Meni Rosenfeld (talk) 16:48, 1 May 2006 (UTC)

You'll need stronger assumptions than those. If B1 = B2 = diag (1,0), then you might have A = diag (1,0), Q = I or A = diag (0,1), Q = antidiag (1,1). Melchoir 19:58, 1 May 2006 (UTC)

This looks somewhat like an eigenvalue decomposition, if the B_i are diagonal. Typically the Q matrix would be orthogonal, but that's not essential. Do you know more than you have said? Otherwise, reconstruction is not possible. --KSmrq^T 21:30, 1 May 2006 (UTC)

$Q^{2}B_{1}-B_{2}Q^{2}=QAQ-QAQ=0$

This a linear system of equations for $Q^{2}$, so if it is possible to determine $Q^{2}$ uniquely, then $Q^{2}=0$. That is $Q$ could be any nilpotent matrix. In some cases, $Q^{2}$ does not exist or not unique, see the Fredholm alternative. (Igny 21:48, 1 May 2006 (UTC))

Thanks. The main problem I am interested in is as stated, which I see is unsolvable. I'll work on it some more to see if a solution to a weaker problem can be useful to me. Any ideas which additional information can solve the problem? Or limitations on the solution (is every matrix similar to A a solution to the problem, or must the solution satisfy additional restrictions?) Thanks. -- Meni Rosenfeld (talk) 11:52, 2 May 2006 (UTC)

$B_{1}$ and $B_{2}$ have to be similar, because $B_{1}=Q^{-1}AQ=Q^{-2}B_{2}Q^{2}$. In that case, the equation $Q^{2}B_{1}-B_{2}Q^{2}=0$ mentioned by Igny does not have a unique solution for $Q^{2}$, which is just as well, because Q cannot be nilpotent if it is regular. -- Jitse Niesen (talk) 08:16, 3 May 2006 (UTC)

Programming Languages (Moved from 24 April)

Hello. I'm a beginning programmer. Two years ago, I tried to learn Visual C++ .NET. That was really, really hard. I wasn't able to do anything besides the tutorial in the book, so I quit. About four months ago, I picked up Liberty BASIC. I am able to do a lot and understand the language. I really enjoy it and am doing a lot of fun stuff. I wondered if anyone had any suggestions on other languages to try next or any suggestions about a good progression of languages for a learning programmer. Any stories about what you did, what you wish you did, or just simply any advice you have would be great. Thanks for your help. --Think Fast 01:23, 24 April 2006 (UTC)

Replies should be in the archive. Basically

• Compiled object oriented languages: C++, Java, Pascal, C#
• Compiled not very object oriented languages: Visual Basic, C
• Interpreted languages: Python, Perl, PHP, Ruby, ...
• Other stuff worth knowing: javascript, MySQL, Ajax

see Programming language for more. I doubt if the reference desk will be able to help much more than this. --Salix alba (talk) 10:43, 2 May 2006 (UTC)

Hello i used ABC language a lot and i liked it a lot. It is much more simpler than C++ and very easy to learning and alsxo to programmin g. But is nor ported to macos X and i cant use it any more. Now i using python language a lot which is also fun for programming but not so easy to learning. If you have pc you can start with ABC language. 88.233.136.138 19:09, 2 May 2006 (UTC)

May 2

Cooper-Scharlemann's graph

Can anybody please give me advice about how to obtain the properties of Cooper-Scharlemann's graph using Thom's Transversality Theorem? Cthulhu.mythos 09:29, 2 May 2006 (UTC)

I mean, I think Thom's theorem is to be used, but if it can be done some other way it is ok, of course. Cthulhu.mythos 09:31, 2 May 2006 (UTC)

Introductory advanced math websites

I've all but given up on Wikipedia for help with math, and most other comparable sites (e.g. MathWorld) aren't that helpful either. This wouldn't be a problem, except that I have a test in a month that I'm frantically studying for, and I still haven't gotten my head wound around all the material yet. My matrices and vectors are very shaky, and everything I know about three-dimensional geometry is stuff I reasoned out by myself (and reinventing the wheel is obviously a bad way to go about this). I also have a difficult time with trigonometry (I rely on my calculator a lot), but I can struggle through. What I'm mainly concerned with are the topics I just mentioned -- matrices, vectors, and three-dimensional geometry (planes and all that crap). Oh, and permutations/combinations as well. I'm currently learning them in statistics class, but any complementary materials would be a great help. I don't need the fancy stuff Wikipedia has -- a simple introductory webpage (or three) for the topics I mentioned will suffice. (Yes, I know I could just Google it, but filtering the wheat from the chaff is a bit hard when it comes to math.) Johnleemk | Talk 15:23, 2 May 2006 (UTC)

The best we can offer on matricies is Matrix (mathematics) and Vector (spatial), otherwise I would sugest a bookshop/libary, you are likely to much better introductory texts in print than anywhere on the web. Wikibooks might be worth a look. (Oh and reinventing the wheel can be a great way to learn for some people, practice and making lots of mistakes is an esential part in the learning process). --Salix alba (talk) 23:54, 2 May 2006 (UTC)

I looked at those before -- they aren't much help (as I said). Johnleemk | Talk 15:56, 3 May 2006 (UTC)

Wikipedia isn't supposed to help you with math. That's never been it's purpose. Wikipedia aims to be an encyclopedia. As far as I understand the concept, encyclopedias are reference works -- a repository of facts. Wikipedia aims to explain the facts and concepts in a lucid and illustrative manner, but it never aims to fully teach you anything. Wikibooks however is more for this purpose, but it is far less complete. Dysprosia 03:47, 3 May 2006 (UTC)

Uh...exactly. Johnleemk | Talk 15:56, 3 May 2006 (UTC)

Sorry that wasn't so helpful, but I felt that needed to be said. Dysprosia 07:10, 4 May 2006 (UTC)

As far as the matrix goes, why not try http://joshua.smcvt.edu/linearalgebra/ ? It's a complete book on matrix, though of course you still need to pour in a lot of efforts- matrix is quite a large area. And tips on the trignometry: if you know a little about complex number, then you only need to remember one formula... (drumroll) Euler's formula. With some simple substitution you can derive compound angle formula from that, and this formula is much more inituitive then the trig formula. (Oh, no, but you still have to remember sine law and cosine law) However, maybe the exam paper will print out all those formula anyway, at least this is the case in my exam. --Lemontea 10:02, 3 May 2006 (UTC)

I don't need to know everything -- just the basics (multiplication, etc.). I'll have a look, though. I've been taught Euler's formula before, but I never really learnt how to apply it. Thanks anyway!

You might try asking for clarification on those topics you don't understand on the wikipedia math reference desk. Not ideal, but an available option. -lethe ^{talk +} 00:02, 4 May 2006 (UTC)

Euler's formula say $e^{i\theta }=\cos {\theta }+i\sin {\theta }$. Now substitute a+b into θ, you get $\cos(a+b)+i\sin(a+b)=e^{i(a+b)}=e^{ia}e^{ib}=(\cos {a}+i\sin {a})(\cos {b}+i\sin {b})=\cos {a}\cos {b}+i\cos {a}\sin {b}+i\sin {a}\cos {b}+i^{2}\sin {a}\sin {b}$ Knowing that i²=-1, and matching coefficients gives $\cos(a+b)=\cos {a}\cos {b}-\sin {a}\sin {b}$ and $\sin(a+b)=\cos {a}\sin {b}+\sin {a}\cos {b}$. By substituting a-b, you will get the other two formula. As for the product to sum and sum to product formula, your best bet is to start from these compound angle formula, or substitute θ and -θ into euler's formula, add or subtract them to express the sin and cos function with exponential function. Then just add them or multiply them. --Lemontea 13:23, 4 May 2006 (UTC)

Benefit vs Cost of using MLE calculation to measure central tendency of a data set

Dear Sir or Ma'am,

We are in a research and analysis organization here at Fort Gordon that works with computer simulations and insights into their output. We have recently run a study that was made up of multiple simulation runs that developed different sets of data at the end of the run, that are of particular interest to us. These data sets range from large (>300 data values), to small (300<n>30), and statistically insignificant (n<30). Our customer required 1xinput value for each of these data sets as a representation of the measurement of central tendency (i.e. arithmetic mean, median, and mode). Analysis of our data set ditributions showed that many were severely skeweed (postive skew) due to extreme outliers, and therefore we also calculated (harmonic mean and geometric mean) as a method to mitigate the extreme outliers in a large data set and still take into account all the data values in set while calculating its measurement of central tendency.

Our question, is what are the benefits and costs/limitations of using the Maximum Likelihood Estimator (MLE) calculation to provide a measurement of a value (i.e. central tendency value) to represent these data sets vs the method of calculation we used previously. Also is there certain criteria of a data set (e.g. normal distribution, large sample set, etc.) that should be met in order to confidently use the MLE for this type of estimation? -----16:16, 2 May 2006 (UTC)

Thanks for any help you can provide!

v/r,

Klaus G. Sanford

[phone number removed]

Possibly you need robust statistics which can cope with outliers. --Salix alba (talk) 00:05, 3 May 2006 (UTC)

Squaring the circle differently?

I know about how squaring the circle in the manner described is impossible, but I remember reading about a mathematical result in which someone proved that a square could be divided into a finite number of pieces and the pieces could be recombined into a circle. Anyone know when and where this was? --Zemyla 17:39, 2 May 2006 (UTC)

Well, dividing a circle into a square is Tarski's circle-squaring problem, but I guess it's equivalent to the reverse. Melchoir 18:04, 2 May 2006 (UTC)

Thank you, you wonderful person of indeterminate gender. --Zemyla 20:34, 2 May 2006 (UTC)

But of course! Pronoun trouble, eh? I'm male; I guess I'll put that in my user page. Melchoir 20:36, 2 May 2006 (UTC)

Drawing a circle around a bunch of circles

I have a bunch of circles and want to draw a circle around them which is the tightest possible. Is there simple way of finding it? Now I use an iterative method which seems to work but I can't prove it. Also, same question for a circle around three other circles. There I use not iteration but geometry and trig formulas which are ugly. Is there a beautiful method for solving this? This is not a homework but for a drawing program I made for fun. 88.233.136.138 19:02, 2 May 2006 (UTC)

See Descartes' theorem, especially the part at the bottom about complex numbers. —Keenan Pepper 19:47, 2 May 2006 (UTC)

Right, if you start with three circles that all touch each other, then there is a beautiful method: Descartes' theorem. Or, if the circles all have the same radius, then you could find the minimum bounding circle of the triangle formed by their centers (see Circumcircle for hints) and then enlarge it by the common radius.

For three general circles, the question is harder. It seems that most authors are more interested in finding a tangent circle that excludes three circles, rather than including them; see, for example,

• http://www.ijcc.org/on-line(pdf)/2(1)45-54.pdf
• http://citeseer.ifi.unizh.ch/gavrilova00another.html
• Or just Google for Apollonius Tenth Problem.

Possibly you can co-opt such a solution to answer your own problem. Anyway, if you can deal with three circles, you can deal with any number. Melchoir 20:00, 2 May 2006 (UTC)

You don't say what your iterative method is. If it's to replace two circles with the tightest circle containing them, then it's wrong. For example, let us have three very small circles with centers {(-4, 0), (4, 0), (0, 7)}. The smallest circle containing all three is the one with center (3, 0) (0, 3) and radius 5. If, however, you first take the tightest circle containing the first two circles, that's the one of center (0, 0) and radius 3; and then if you take the smallest circle containing both that circle and the small one at (0, 7), you get the circle with center (2, 0) (0, 2) and radius 6, which is suboptimal. – b_jonas 21:41, 2 May 2006 (UTC) (Corrected some numbers – b_jonas 22:34, 2 May 2006 (UTC))

You're right, I think, that this problem doesn't have the greedy property like that. But your numbers are still wrong -- (0, 3) is only 4 from (0, 7), so the first circle is suboptimal. Moreover, I think the idea was to iterate with a solution to the three-circle problem, not the two-circle problem. Still no idea if there's a better extension to $n>3$ than the obvious one I give below, though. --Tardis 14:54, 3 May 2006 (UTC)

Start with two circles. The answer is obvious (a circle centered between them, though not at the midpoint of the segment connecting the centers unless the circles are the same size), but we can go farther and get some insight into the three circle problem by considering non-optimal solutions. We have the centers ${\vec {c_{1}}}$ and ${\vec {c_{2}}}$, and radii $r_{1}$ and $r_{2}$. Call the new circle $({\vec {C}},R)$, and the points of tangency ${\vec {t_{1}}}$ and ${\vec {t_{2}}}$. We know that the segments ${\vec {C}}\rightarrow {\vec {t_{1}}}$ and ${\vec {C}}\rightarrow {\vec {t_{2}}}$ have the same length $R$, and that they are respectively collinear with ${\vec {c_{1}}}\rightarrow {\vec {t_{1}}}$ (length $r_{1}$) and ${\vec {c_{2}}}\rightarrow {\vec {t_{2}}}$ (length $r_{2}$) because radii to a point of tangency are collinear. So (using $d({\vec {x}},{\vec {y}})$ for distance), $d({\vec {C}},{\vec {c_{1}}})=R-r_{1}$ and $d({\vec {C}},{\vec {c_{1}}})=R-r_{2}$. So $d({\vec {C}},{\vec {c_{1}}})-d({\vec {C}},{\vec {c_{2}}})=r_{2}-r_{1}$.

But this is precisely the definition of a hyperbola; all circles tangent to two given circles lie on (one of the two arms of) a hyperbola with the centers of the circle as the foci. To do three circles, then, we just need to find the intersection of any two of the three hyperbolae defined by the three handshakes. (This is an extension of using the intersection of the perpendicular bisectors to find the circumcenter of a triangle.) Basically, characterize each hyperbola with a center ${\vec {c}}$ (just the midpoint of the segment connecting the circles' centers), a "focal length" $f$ (half the length of that segment), and a $d=\Delta r$ that is the constant difference in distance. Some coordinate geometry gives you that the usual parameters for the hyperbola are then $a=d/2,b={\sqrt {f^{2}-a^{2}}}$. Parameterize with sinh and cosh: ${\vec {x}}(t)={\vec {c}}+a{\hat {m}}\cosh t+b{\hat {n}}\sinh t$, where ${\hat {m}}$ points from the center to the bigger circle and ${\hat {n}}\cdot {\hat {m}}=0$. Then set up ${\vec {x_{1}}}(t_{1})={\vec {x_{2}}}(t_{2})$ (two possibly-different parameter values, since this is an intersection and not a collision), and solve. You'll need to write sinh and cosh in their exponential forms and use the substitution $\tau _{i}=e^{t_{i}}=1/e^{-t_{i}}$. Solve for (say) $\tau _{1}$, and then evaluate ${\vec {x_{1}}}(\tau _{1})$ to get your center; the radius is then $d({\vec {x_{1}}}(\tau _{1}),{\vec {c_{i}}})+r_{i}$ (for any i). This is analytically solvable, but the algebra is insane. (If I get a simple result, I'll post it; I notice that the link at ijcc.org mentions the hyperbola intersection, but they seem to not be using the analytic solution for some reason...?)

(Yeah, apparently it's inefficient to intersect them either numerically or symbolically, and of course it's less sexy than applying Möbius transformations. It still might be easier to code and/or a good exercise, though.) Melchoir 22:05, 2 May 2006 (UTC)

There's more! It's possible for the tangent circle to contain but not touch one of the three circles. In this case, the hyperbolae do not intersect. It's also possible that the tangent circle for all three is not the smallest circle possible (consider the points (circles of vanishing radius) (-1,0), (0,ε), and (1,0)). But this is trivial to overcome: construct the obvious containing circle for each pair of circles and check if it contains the third. If so, use that and never set up the equations (which would yield impossible values of $\tau _{i}<0$ in the first case and valid but poor choices in the second).

Finally, to deal with $n>3$ circles, as far as I can tell you'll just have to consider all ${n \choose 3}={\frac {n(n-1)(n-2)}{6}}$ triplets and construct the circle for each, then check that it contains all the other circles. While you're at it, for each triplet check to see if any other circle is contained within the circumcircle of the triangle formed by the three circle-centers and discard any that are. This sounds like extra work but it's only $O(n)$ and will (hopefully) help reduce the effective $n$ in the main $O(n^{4})$ algorithm. Hope this helps! --Tardis 21:54, 2 May 2006 (UTC)

Actually, it occurs to me that in the optimization I mentioned you must only discard circles contained in the triangle formed by the centers of other circles; a circle can be contained by the circumcircle and still be relevant to the overall problem. (Consider the circumcircle of the three points I mentioned earlier, and how much could fit in it!) --Tardis 22:42, 2 May 2006 (UTC)

There's a simpler problem I've heared that's similar to this: given a list of points, find the tightest circle contaning all of them. Clearly this is a special case of the original problem. To do this, you can check all the circumcircles of three points and all Thales-circles of two points, but there's probably a more efficent way to do this. I don't know how fast the fastest solution to this is. – b_jonas 22:41, 2 May 2006 (UTC)

Thank you so much all of you. It is very interesting and i would never have known to google for apollonius tenth problem, did he have more problems? Maybe the wikipedia can tell more about apollonius problems. Thank you again. 88.233.136.138 06:12, 3 May 2006 (UTC)

Hmm... there are some problems at Apollonius of Perga, but they're not numbered. Melchoir 06:25, 3 May 2006 (UTC)

Better late than never? This problem is of interest in computational geometry and in practical applications. Kaspar Fischer worked on it for his Diplomarbeit (Master's thesis); see the papers on his web page. He and his advisor, Bernd Gärtner, have made C++ code available as a SourceForge project, here. It is also part of CGAL. It is much faster and more robust than naïve implementations. This is also mentioned in the bounding sphere article.

For the special case of the minimal enclosing circle of a set of points, a famous result by Nimrod Megiddo is that linear programming can be used to solve the problem in time directly proportional to the number of points. --KSmrq^T 09:19, 6 May 2006 (UTC)

Nice. Thanks for the links. – b_jonas 12:49, 6 May 2006 (UTC)

CIH author

I've read that the CIH virus was also written by Sergei Kozachkov, who was recently condemned. He is a co-author? Brand 21:32, 2 May 2006 (UTC)

Sergei Kozachkov? Never heard of him, I get no Google results for "Sergei Kozachkov"... SCHZMO ✍ 22:48, 2 May 2006 (UTC)

The closest thing I found to this subject was a guy named Sergey Antimonov whose DialogueScience antivirus software helped overcome the CIH virus. —Mets501^talk 23:29, 2 May 2006 (UTC)

Recently condemned? Sounds ominous. --Deville (Talk) 00:41, 3 May 2006 (UTC)

Roulette

If I put $1 on red, and the Roulette table comes up black, my net loss would be $1. I could then bet $2 on red and if it lands black, my net loss would be $3. I could then wager $4 on red. Should I lose again, my net losses are $7. if I were to then bet $8 on red, & win, I would win $1 (my origional bet). Why then can I not do this until the table turns up red? Eventually it will. Of course I would need a substantial amount of money to make sure I can keep betting until it hits red....but why couldnt I if I had say $250,000? Lord Westfall 23:28, 2 May 2006 (UTC)

You wouldn't do that because you wouldn't win anything. It's just like when the people buy a scratch-off lotto ticket for $5, and then they win $5 and they're so happy about it. But if you have the money, and you enjoy just playing roulette, then be my guest and play that way. :) —Mets501^talk 23:35, 2 May 2006 (UTC)

Wouldnt you slowly accumulate $1s from your winnings? Lord Westfall 23:36, 2 May 2006 (UTC)

Oh, sorry for my mistake. You're right. —Mets501^talk 00:08, 3 May 2006 (UTC)

This is true. You would slowly accumulate $1s from your winnings. However, there is a problem, and that is this: you don't have an infinite bankroll. Presumably, you go to the table with a finite amount of cash that you can bet. So if you ever got into a situation where you couldn't double your bet, then you would have lost all of your money. Also, even if you had an infinite amount of money, the casino wouldn't, and they'd quit covering your bets at a certain point, at which point you'd be out of money as well.--Deville (Talk) 23:47, 2 May 2006 (UTC)

The strategy you have discovered is a martingale. It doesn't work because even with $250,000, if you keep playing you will eventually lose it all; this is known as gambler's ruin. Melchoir 23:43, 2 May 2006 (UTC)

I agree. It would only work if you had much more money than the Casino has, and if they don't throw you out from it. A good book (which doesn't require any prior knowledge) on this topic is Warren Weaver, Szerencse kisasszony, Kairosz kiadó, 1997, orig. "Lady Luck", Doubleday & Company, Inc. Garden City, New York. – b_jonas 20:43, 4 May 2006 (UTC)

May 3

Rotated Conics

I would like to know how a rotated ellipse (or other conic) can be expressed as an equation. More specifically, how can an equation be obtained which describes an ellipse with semi-major and semi-minor axes a and b such that the angle between the x-axis and the ellipse's major axis is θ. I would prefer parametric equations, if possible (by the way, the Ellipse article does not list any - surely they exist?). --72.140.146.246 02:10, 3 May 2006 (UTC)

Matrices make this a breeze. Let's take the simplest parametric equations for an ellipse, an origin-centered circle stretched along the coordinate directions. One form of the equations uses trigonometric functions.

$P(\theta )={\begin{bmatrix}x(\theta )\\y(\theta )\end{bmatrix}}={\begin{bmatrix}a\cos \theta \\b\sin \theta \end{bmatrix}}$

Instead, we will use homogeneous coordinates for our convenience; they will be especially useful in implicit form.

$P(t)={\begin{bmatrix}x(t)\\y(t)\\w(t)\end{bmatrix}}={\begin{bmatrix}a(1-t^{2})\\b(2t)\\1+t^{2}\end{bmatrix}}$

Now a 3×3 matrix can express any linear, Euclidean, affine, or projective transformation of points P = (x:y:z). A rotation by φ has matrix form

$R(\varphi )={\begin{bmatrix}\cos \varphi &-\sin \varphi &0\\\sin \varphi &\cos \varphi &0\\0&0&1\end{bmatrix}}.$

The rotated ellipse is merely R(φ) P(t), namely

$P'(t)={\begin{bmatrix}a(1-t^{2})\cos \varphi -b(2t)\sin \varphi \\a(1-t^{2})\sin \varphi +b(2t)\cos \varphi \\1+t^{2}\end{bmatrix}}.$

This is more machinery than we need simply to rotate an ellipse at the origin, but it allows us to do much more. It works for any rational parametric curve, and for the broad class of transformations mentioned earlier.

Also, it gives us a handle on implicit conics. We can write the implicit form for any conic in bilinear form, using a symmetric 3×3 matrix Q.

$0={\begin{bmatrix}x&y&w\end{bmatrix}}Q{\begin{bmatrix}x\\y\\w\end{bmatrix}}$

For our original ellipse, Q has the form

$Q={\begin{bmatrix}1/a^{2}&0&0\\0&1/b^{2}&0\\0&0&-1\end{bmatrix}}.$

Now if M is some transformation we wish to apply, we merely substitute for Q,

$Q'=(M^{-1})^{\mathrm {T} }QM^{-1}\,\!.$

That's it. The same approach works for quadric surfaces in 3D. --KSmrq^T 04:33, 3 May 2006 (UTC)

...............................................

If all you want is a parametric representation for an ellipse with centre at the origin, put C = cos θ, S = sin θ, and use

x = aC cos t - bS sin t, y = aS cos t + bC sin t.

This was obtained from the unrotated case by replacing (x, y) := (Cx - Sy, Sx + Cy), which amounts to rotating the point (x, y) around the origin by an angle θ. --Lambiam^Talk 07:07, 3 May 2006 (UTC)

Thanks, that was very helpful.--72.140.146.246 20:03, 3 May 2006 (UTC)

pi question

There are some numbers like, 4,77777777777777.... 8,765765765765765765..... 9,0192837465019283746501928374650192837465.. , 912,992992992992992992992...

They haven't found the full number pi, ins't because the pi is a numbers like those other that i said before, but a more complex number???

Pi is irrational and transcendental. As far as I can see, the numbers you listed above are rational numbers. Dysprosia 03:44, 3 May 2006 (UTC)

It is not very clear what the question is. What would it mean for the "full number" to be "found"? Has the full number 10021426384/1111111111 been found? And what about the square root of 2? --Lambiam^Talk 07:20, 3 May 2006 (UTC)

The numbers you give are 43/9, 973/111, 10021426384/1111111111 and 912080/999, respectively. These are all rational numbers - numbers that can be written as the ratio of 2 integers. The rational numbers are also exactly those numbers that can be written as a repeating decimals (where the same bunch of digits repeats over and over). π is irrational: It cannot be written as a ratio of two integeres, and its decimal expansion doesn't repeat itself. So it is essentially different from the numbers you give, and it will never be possible to find all of its digits.

Lambiam: I suppose the fact that the nth digit of this rational number can be found with negligible computational effort (just divide mod 10 and look in an array), is what the questioner essentially means by "the full number has been found". -- Meni Rosenfeld (talk) 08:29, 3 May 2006 (UTC)

Perhaps, but then, presumably, the questioner would think that also the full number √2 hasn't been found. And, if you're happy with the hexadecimal system, you could argue that the remarkable result of Bailey, Borwein and Plouffe means that the full number π has been found. By the way, I've found all the digits of π. Here they are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 ;-) --Lambiam^Talk 10:56, 3 May 2006 (UTC)

On this topic, an article on the computational complexity of various mathematical constants would be an interesting addition to Wikipedia. Fredrik Johansson 11:08, 3 May 2006 (UTC)

He thinks the standard proof of Pi's irrationality is that nobody's found a pattern to it yet. He reasons, however, that that's not absolute proof at all, and indeed it isn't. What's the actual proof? I can't find it in Pi. Black Carrot 21:22, 3 May 2006 (UTC)

You're right, I didn't see the proof of irrationality there. It's a neat little proof using just some integral calculus, we should probably have it somewhere. It does mention the proof that π is transcendental at Lindemann–Weierstrass theorem, which of course implies irrationality. -lethe ^{talk +} 00:00, 4 May 2006 (UTC)

Various proofs are mentioned (but not given) at Irrational number#History, such as Lambert's proof, which is notable by being first but not so simple as Bourbaki's (assuming that is the one you are referring to). It would make sense to copy some of this to the Pi article. Lambiam^Talk 00:12, 4 May 2006 (UTC)

I was talking about this (I will give a example because I forgot the right name used to represent those numbers) if you get the number 10 and / by 3 (so 10/3) the answer will be 3.3333333333333333333333333333333............. there is no end this will be like this forever, there are more complex versions of this type of numbers, like 874.454545454545454545......... or 2435.767657676576765767657676576765....... or maybe even more complex 7777.876483900011133876483900011133876483900011133... ins't the pi number a complex (moooooooooooore complex) number like the last???

In a word, no. There is no pattern. For there to be a pattern, a particular set of numbers of finite length must be repeated, over and over, forever. If the numbers repeated only once, or only a finite number of times, and were then followed by other numbers, that means the existence of a pattern is illusory. You could search through the first 10 billion trillion digits of pi and still never get to the end of the set of numbers that might, supposedly, be repeated. And if you did, hypothetically, eventually find that that humungously long set of numbers did repeat, it would not be sufficient for it to repeat just once, it would have to repeat ad infinitum. If that really were the case, that would make pi an algebraic number, which we know is not the case. The numbers of pi just go on and on in an apparently random manner forever and ever and ever. Looking for all of its digits is like looking for the crock of gold at the end of the rainbow, because it is impossible to write them all down even if you had infinite time available to you. There is no pattern. JackofOz 03:08, 4 May 2006 (UTC)

Additionally, be careful with your wording - while in certain ways pi is "complicated", in mathematics talking about a complex number is a very specific thing, which pi is not (well, ok, to be pedantic all real numbers are complex numbers, but its imaginary part is zero which I suppose you could say means it's not stricty complex). Two useful words used to describe pi are irrational (meaning you can't write it as a fraction - and hence as a recurring decimal) and transcendental (meaning you can't write it as the solution to a polynomial with integer coefficients). Confusing Manifestation 12:52, 4 May 2006 (UTC)

The pattern is that there is none. So, absolutely any text, once encoded (or any figure) may be found somewhere in pi's decimals. Your love letters. Our Ref desk history. Wikipedia. The internet. More : it is a coding of the universe since its beginning. Strange pattern indeed. --DLL 21:42, 4 May 2006 (UTC)

Since nobody else is bothering to, I looked up the proof of Pi's irrationality [1]. It's more complicated than you probably know how to deal with, and involves calculus, but it's at least not very long. It proves that, although we can't look at every digit of Pi, if we could, we would not see a really really long repeating decimal, we'd just see randomness. Therefore, Pi is irrational, and not similar to 912080/999. Black Carrot 22:00, 4 May 2006 (UTC)

If the digits in the decimal expansion eventually repeat forever, the number is rational, that is, it can be written as the fraction P/Q in which P and Q are integers (and the other way around: all rational numbers have a repeating decimal expansion). The German mathematician Johann Heinrich Lambert proved in 1761 that π is irrational (not rational), so its digits will not repeat (at least not forever). As to DLL's remark above (which I also found hiding inside π), the property he refers to is that of being a "normal number". It is generally believed by mathematicians that π is normal. But this has not been proved, and to find a proof of this looks like a very difficult task. --Lambiam^Talk 22:20, 4 May 2006 (UTC)

Re DLL's statement: absolutely any text, once encoded (or any figure) may be found somewhere in pi's decimals. Your love letters. Our Ref desk history. Wikipedia. The internet. More : it is a coding of the universe since its beginning. Can someone convince me this is true, because at the moment I rather doubt it. Just because the digits proceed randomly forever does not necessarily mean that any string of numbers you can possibly devise will eventually occur, does it? If this were true for pi, it would also be true for e, and any irrational number, wouldn't it? JackofOz 03:36, 5 May 2006 (UTC)

No one can convince you, because it hasn't been proven. DLL is referencing a conjecture that pi is a normal number. Melchoir 04:00, 5 May 2006 (UTC)

Although, to be fair to DLL, it's often stated as fact on the internets. Melchoir 04:07, 5 May 2006 (UTC)

And no. Not every irrational number is normal. The transcendental Liouville's constant is certainly not. However, e is also thought to be normal. And if the decimal expansion is truly random, it is easy to show that every given sequence of digits has a probability of 1 of appearing somewhere. -- Meni Rosenfeld (talk) 06:48, 5 May 2006 (UTC)

Thanks Melchoir and Meni. I'd be interested in seeing that proof. JackofOz 07:01, 5 May 2006 (UTC)

Of the normality of Pi? So do we. Hopefully we'll live long enough. -- Meni Rosenfeld (talk) 07:47, 5 May 2006 (UTC)

No, the proof that "if the decimal expansion is truly random, it is easy to show that every given sequence of digits has a probability of 1 of appearing somewhere". JackofOz 09:19, 5 May 2006 (UTC)

Oh, that. As I said, this is remarkably easy - but it involves some technicalities. I'll sketch a proof for a seemingly stronger statement: Given a real number x, the decimal expansion of which is random, and a digit string of length m, there is a probability of 1 that there exists a (natural) number n such that the digits of x from n*m to (n+1)*m-1 is identical to our given string (I'll call a number with this property a "good number"). Equivalently, there is a probability of 0 that there are no good numbers. I'll denote this probability by P. I'll also denote P(n) the probability that every natural number up to n is not good. Clearly P ≤ P(n) for every n. The probabilty for i not to be good is (1-10^-m). This is independent of any other number being good or not, since every number describes a separate set of digits, and because of our assumption of randomness. So the multiplication principle holds, and therefore

$P(n)=\left(1-10^{-m}\right)^{n}$

$\lim _{n\to \infty }P(n)=0$

And it follows that P = 0. I hope this was clear - do tell if it wasn't. -- Meni Rosenfeld (talk) 09:42, 5 May 2006 (UTC)

Somewhat simpler: If your love letter, or the Constitution of the United States, or any other text (encoded as a finite string of digits) fails to appear, then, by definition, the decimal expansion is not (truly) random. This is true for the von Mises definition, which applies to infinite sequences. Strangely enough it is not mentioned in the Randomness article. --Lambiam^Talk 13:17, 5 May 2006 (UTC)

Thanks. This makes sense to me now. JackofOz 14:27, 5 May 2006 (UTC)

Properties

I'm aware of two (connected, as it happens) ways of describing almost any number. One is factors: A*B=C, with few pairs of A and B for each C. The other is difference of squares: A²−B²=C, once again with few pairs of A and B for each C. Does anyone know of similar pairs or small sets of numbers? Black Carrot 21:40, 3 May 2006 (UTC)

You mean something like this: A^B = C, as in 1073741824¹ = 32768² = 1024³ = 64⁵ = 32⁶ = 8¹⁰ = 4¹⁵ = 2³⁰ = 1073741824? Or more like this: A²+B² = C, as in 120² + 115² = 132² + 101² = 141² + 88² = 144² + 83² = 155² + 60² = 160² + 45² = 164² + 27² = 165² + 20² = 27625? The last one is not "almost any number", of course, but A²−B² won't give you any numbers of the form 4N+2. --Lambiam^Talk 00:03, 4 May 2006 (UTC)

Well, I'm not sure what you mean by 4N+2, but given C=A²-B²=(A+B)(A-B)=D*E, it will only give pairs of squares that correspond to pairs of factors that are an even distance away from each other. Otherwise, B is not a whole number. So, it's guaranteed to work on odd numbers, not so much on evens. I don't think those two suggestions are quite what I'm looking for; the first as far as I can see only works for values of C that are powers of whole numbers. The second looks interesting, but I'm not sure how much system there is to it. Please get back to me. Black Carrot 00:56, 4 May 2006 (UTC)

4N+2 is a multiple of 4 plus 2. So a number C is of the form 4N+2 if there is value for N such that C = 4N+2. For example, 14 is of the form 4N+2 since 14 = 4×3+2. Indeed, you cannot solve A²−B² = 14 in integers. C has to be odd, or a multiple of 4, and must not be a multiple of 4 plus 2. The first suggestion always works, since any number is itself raised to the power 1. It is again related to A×B = C. Just let N be any natural number (in the example 2) and define A' = N^A and C' = N^C. Then (A')^B = (N^A)^B = N^A×B = N^C = C'. The second has not such a simple pattern, but I did not see that as a requirement. Is this just for playing with numbers, or is there something else behind the question? --Lambiam^Talk 03:06, 4 May 2006 (UTC)

Every natural number can be expressed as the sum of 4 or fewer squares; 9 or fewer cubes; 19 or fewer fourth powers etc. - see Lagrange's four-square theorem and Waring's problem. Gandalf61 09:14, 4 May 2006 (UTC)

Ooh. Enticing. I'll see what I can do with Waring and Lagranges' work. And I'm sorry Lambian, of course you're right, any number is itself to the first. What I'm doing: for the past year, I've been trying to find shortcuts to the factorization of a number. Finding out that there was another way to express numbers, as the difference of squares, has been very instructive and has, I think, helped me figure some of it out. So, I figure finding even more ways will help even more.

I found another example. Any number can be written as the difference between two triangular numbers, in much the same way as the difference of two squares. The math is a bit different, and it can handle different numbers - The squares can't handle even-odd pairs of factors, and the triangles can't handle even-even. So, the squares represent 16=8*2 as 16=(5+3)(5-3)=5²-3², but the triangles can't, and the triangles represent 12=3*4 as 12=4+4+4=3+4+5=T(5)-T(2) but the squares can't. Anything else you think of, let me know. Black Carrot 00:01, 5 May 2006 (UTC)

16 = T(16)-T(15). --Lambiam^Talk 00:23, 5 May 2006 (UTC)

Indeed. However, that can be proved by way of 16=1*16=16=T(16)-T(15) (following the format above). That's an even-odd pairing of factors, which I said triangular numbers can handle just fine. On the other hand, the squares can't handle that, as it's equivalent to 16=16*1=(17/2+15/2)(17/2-15/2)=(17/2)²-(15/2)², which includes non-whole numbers. Black Carrot 01:22, 5 May 2006 (UTC)

May 4

Pi

Is there any known pattern of pi? Are there any formulas that represent pi without giving an aproximation? If so, what are they?

Pi is fairly helpful. And yes, Pi = Circumference/Diameter is, by definition, exactly accurate, though difficult to use in practice. Black Carrot 00:58, 4 May 2006 (UTC)

Because Pi is irrational, there is definitely no simple pattern to the digits (in terms of having them repeat), and I daresay there's unlikely to be any complicated pattern either (even though it's possible to construct other trancendental numbers that do) although I have no proof for that. Any of the formulas for pi will give you an exact answer, the only problem being that they're mainly infinite sums which means you have to be able to add an infinite number of terms (and need an infinite sheet of paper on which to write the decimal digits), so in that sense the best you can do is to take a partial sum (the first n terms) and get an approximation good to however many decimal places you need. Confusing Manifestation 01:03, 4 May 2006 (UTC)

Keep in mind that a decimal expansion is an infinite sum as well. When you say 1/3 = 0.333..., what you mean is $1/3=\sum _{n=1}^{\infty }{3\cdot 10^{-n}}$. —Keenan Pepper 01:31, 4 May 2006 (UTC)

Pattern? Whether a pattern is visible in how a number is described depends largely on the nature of the description. Is there a pattern in the square root of two, √2 = 1.4142135623…? Written as a decimal expansion, apparently not; written as a regular continued fraction, it exhibits repeating 2s, √2 = [1; 2, 2, 2,…]. That's an algebraic number; surely there can be no pattern with a transcendental number? Wrong; we have e = [1; 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1,…], exhibiting a very simple pattern. In fact, π has the simple generalized continued fraction 3+ ¹⁄₆₊ ⁹⁄₆₊ ²⁵⁄₆₊ ⁴⁹⁄_6+…. And note that we can do arithmetic with continued factions directly; we don't need to convert to a decimal expansion.

As for exact formulae, we have the well-known π = 4 tan⁻¹ 1, using the arctangent function; we have π = (Γ(¹⁄₂))², using the standard generalization of the factorial function, the gamma function; and we have many others. --KSmrq^T 02:04, 4 May 2006 (UTC)

Maybe you will also be interested in List of formulae involving π. -- Meni Rosenfeld (talk) 12:19, 4 May 2006 (UTC)

I believe that KSmrq's fractional representation of pi looks wrong. It seems to diverge off to infinity - actually even after 3 + 1/6, the number is greater than 3.1415... --AMorris (talk)●(contribs) 01:09, 7 May 2006 (UTC)

Yes, KSmrq's representatation is indeed wrong. One formula that I know is $\pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}+\ldots$ —Mets501^talk 01:35, 7 May 2006 (UTC)

Please be careful of notation; what I gave is a continued fraction representation. Here it is in another presentation:

$\pi =3+{\frac {1}{6+{\frac {9}{6+{\frac {25}{6+{\frac {49}{6+\ddots }}}}}}}}$

I thought I was explicit enough; apparently not. It is correct. --KSmrq^T 01:59, 7 May 2006 (UTC)

Now don't get worked up about this. I did not mean to insult you by saying your edits were "wrong", I merely thought you had mistyped them. It was my fault, I did not read "continued fraction". I apoligize for my mistake. —Mets501^talk 02:24, 7 May 2006 (UTC)

I blame an education system that neglects continued fractions. Although I understand the need to cover all the high-traffic mainstream topics, it's a pity we have to neglect some of these inviting side eddies to do so. --KSmrq^T 06:16, 8 May 2006 (UTC)

multiplication

Shortcuts or techniques to make multiplication easier?

In your head, or on paper? —Keenan Pepper 03:23, 4 May 2006 (UTC)

It is essential to memorize the single digit multiplication table, but there are tricks to help. For example, multiplication by nine can be done by a rule.

Also there are methods like the Trachtenberg system that organize larger computations for efficiency. One of the valuable tricks to learn is some alternate computation to verify the result; casting out nines is a popular example, related to the nine rule. --KSmrq^T 09:56, 4 May 2006 (UTC)

A variety of methods are discussed in Multiplication algorithm. JoshuaZ 03:52, 5 May 2006 (UTC)

Double digit Multiplying

2000 minus 72 times 26 equals

Thanks —Preceding unsigned comment added by 203.213.7.130 (talk • contribs)

Either you can work this out easily on paper, or if you just want the answer, people usually have calculators for these sort of things. Dysprosia 07:12, 4 May 2006 (UTC)

This is, of course, a trick question, depending on grouping. Google gives an answer, a power of two; but is it the one intended?? --KSmrq^T 10:07, 4 May 2006 (UTC)

Not quite. If you assume operator precedence (substituting "times" etc. for the necessary operators) there is only one answer. Vocally however the statement is ambiguous. Dysprosia 11:08, 4 May 2006 (UTC)

Shortcut (see preceding section) : 72 * 26 = 72 * ( 100/4 + 1) = 7200/4 + 72 = 1872. --DLL 21:26, 4 May 2006 (UTC)

$ python
Python 2.4.1 (#1, May 27 2005, 18:02:40) 
[GCC 3.3.3 (cygwin special)] on cygwin
Type "help", "copyright", "credits" or "license" for more information.
>>> print 2000 - 72 * 26
128

Can someone tell me what is the point of this question???? - anon

Only the original (anonymous) poster can say. Take it as a reminder that not everyone who reads Wikipedia is a university graduate; some are still learning the basics. Perhaps we have a very young student who is bright enough to use the Web, but not bright enough (or too lazy) to do assigned problems in arithmetic. The latter assumption explains the form of answers given, typically requiring the student to do more individual work. Obviously it is not a smart question. --KSmrq^T 01:40, 5 May 2006 (UTC)

The answer is 50128 if you do the subtracting first; 128 if you do the multiplication first. Jonathan ^talk File:Canada flag 300.png 02:21, 5 May 2006 (UTC) Which means that thanks to the Order of Operations, 128 is the only answer. -- He Who Is^{[ Talk ]} 02:06, 28 June 2006 (UTC)

Generalized Euler limit

The limit for Euler's constant can be generalized to

$\lim _{n\to \infty }\left[\sum _{k=1}^{n}f(k)-\int _{1}^{n}f(x)\,dx\right]$

where f is any positive decreasing function. Euler's constant is of course given by f(x) = 1/x, and e.g. f(x) = (log x)ⁿ/x gives the Stieltjes constants. Taking f(x) = x^-a gives the zeta function plus a simple ratio (with the nice

$\gamma =\lim _{x\to 1}\left[\zeta (x)-{\frac {1}{x-1}}\right]$

as a special case). In addition, taking f(x) = 1/Γ(x-1) gives e minus the Fransén-Robinson constant.

My questions are: has this limit been studied for other f than those mentioned, and does it have a name? I think it'd be worthy of a Wikipedia article of its own. (It could be mentioned in the article on Euler's constant, but it does seem to be of a more general character.) Julian Havil mentions it in Gamma (page 117), and at least refers to the case of x^-a as "Euler's generalized constants" but I'm having difficulties locating anything relevant based on that name. Fredrik Johansson 12:06, 4 May 2006 (UTC)

celestial mechanics : how to derive formula for angle since periapsis

Hello,

this question may be considered science, but for me the main difficulty is in finding the correct manipulations on more well known equations in order to obtain this result.

This is not homework! :) It is just a personal interest of mine to understand spaceflight as good as possible.

Consider an object describing a (let us say) elliptic orbit, around a sun (which you may consider a point) of mass M , and assume the object has neglectible mass.

The periapis is the point where the object came closest to the sun. Suppose the object has at a certain time : distance r from the sun an angle t of its speed vector with the line connecting it with the sun a speed (in modulus)v

What is the angle $\theta$, viewed from the sun, that the object has travelled from the periapsis

it should be :

$\tan(\theta )={\frac {{\frac {rv^{2}}{GM}}\sin(t)\cos(t)}{{\frac {rv^{2}}{GM}}\sin(t)^{2}-1}}$

G is just the universal gravity constant.

All replies are welcome.

Try starting with the article on orbital elements, and also follow its links (especially orbital state vectors). --KSmrq^T 23:21, 4 May 2006 (UTC)

May 5

What's the significance of e?

"e" is approx. 2.7182818285. You use it in advanced algebra. My question is: Why does e have to be that particular number? Jonathan ^talk File:Canada flag 300.png 02:09, 5 May 2006 (UTC)

Not algebra, but calculus. The number e is the exponential e¹, where e^x is the unique function that is its own derivative, modulo a constant factor. For more, see Exponential function. Melchoir 03:42, 5 May 2006 (UTC)

Oh, and Exponential growth for applications. Melchoir 03:44, 5 May 2006 (UTC)

A good definition is $e=\sum _{n=0}^{\infty }{1 \over {n!}}=1+1+{1 \over 2}+{1 \over 6}+{1 \over 24}...$. —Keenan Pepper 04:01, 5 May 2006 (UTC)

Because a smaller number would be too small, and a larger number would be too large. Fredrik Johansson 04:41, 5 May 2006 (UTC)

This constant appears naturally in different mathematical situations, more so than most of the other named constants (of which there are many). For example, consider the calculation of compound interest. If we have an initial investment A and accrue interest once each year at an annual rate of r = 18% = 0.18, then after a year we will have A(1+r). If the interest accrues every month, the rate is divided by 12 for each increment as we accumulate 12 times, for a total of A(1+^r⁄₁₂)¹². This is more generous. We could go further, choosing weekly or daily or hourly increments. If we choose N times per year, the formula is A(1+^r⁄_N)^N. Now, although it may not be clear without study, we get a definite (and finite!) rate of return even if N goes to infinity (so that we are compounding continuously). Our constant magically appears, as the formula becomes

$Ae^{r}\,\!.$

Another example involves the reciprocal function, f(x) = ¹⁄_x. If we draw a graph for this function, we see that it rises steeply (towards infinity) as x decreases from one towards zero, and it sinks slowly (towards zero) as x increases towards infinity. We are interested in the area bounded by the x axis, the curve, a cut at one, and a cut at c (with c positive). We count areas to the right of one as positive, and areas to the left as negative. Once again our constant appears, for the area is s, where e^s = c. We call s the natural logarithm of c.

A final, famous, and seemingly unrelated, example is in the description of circular movement. If we measure angles by distance traveled along the circumference of a unit circle, so that a full circle is 2π (radian measure), and if 0 is perfectly horizontal and ^π⁄₂ is perfectly vertical, then for a given angle, ϑ, the horizontal position, x, is

$x={\frac {e^{\mathbf {i} \vartheta }+e^{-\mathbf {i} \vartheta }}{2}},$

where i² = −1. The article on Euler's formula discusses this further.

The deeper reason for the frequent appearance of e, and especially exp(x) = e^x (the exponential function) is that we are seeing an eigenfunction with eigenvalue one of the continuous linear operator for the calculus notion of taking the derivative of a function.

${\frac {d}{dx}}\exp(x)=\exp(x)\,\!$

Simply put, the constant b = e gives the unique function of the form b^x unchanged by taking its derivative. That's a Good Thing. --KSmrq^T 06:04, 5 May 2006 (UTC)

What are Groebner bases?

I'm researching current work on Frobenius' linear Diophantine equations. You know, the "Postage Stamp Problem". I've found references to Graph Theory and Geometric work being done on this classic number theory problem, yet can't find any explanation when "toric Groebner base" is mentioned. Perhaps we should start with just what a Groebner base is . . .

Thanks.

JAsbrand

I haven't a clue, but is Gröbner basis what you're looking for? Melchoir 04:03, 5 May 2006 (UTC)

Try combining the idea of a Gröbner basis with the circumstances of toric geometry; that is, compute a Gröbner basis for a toric ideal. This paper may help more than Wikipedia. --KSmrq^T 06:17, 5 May 2006 (UTC)

inner product - wavelets

I'd be right in saying that the inner product of f(t) and g(t), where:

• f(t) = 1, between t=0 and t=1, and
• g(t) = t, between t=0 and t=1, 2-t between t=1 and t=2

(i.e. a square wavelet and a triangle wavelet) ...is 0.5 right? --131.251.0.7 11:03, 5 May 2006 (UTC)

• Or 1, actually. Well, its 1 or 0.5 anyway. --131.251.0.7 11:11, 5 May 2006 (UTC)

Looks like 0.5 to me. Of course, this is only if by "the inner product" you mean the "natural" inner product: the integral of the pointwise product over all space. Sometimes it's useful to use a different inner product; see Orthogonal polynomials. Melchoir 11:18, 5 May 2006 (UTC)

I'm talkin about the inner product in the "integral of f(t)g(t)dt" sense. So, the data compression/ wavelets sense of the term. --PrifysgolCaerdydd (The anon of the first posting) 11:37, 5 May 2006 (UTC)

Righto, then 0.5 it is. Melchoir 11:41, 5 May 2006 (UTC)

Yeah, that was pretty easy actually. Cheers anyway. I just had to make sure I was right (thats a few points in the bag for the exam then). I may well come back to this page later to ask about LFSRs and hash functions and linear congruential generators (stuff that I need for finals). --PrifysgolCaerdydd 11:47, 5 May 2006 (UTC)

hash functions

Yeah, I may need your help again. The question is: Let h: Σ^* --> Σⁿ be a cryptographic hash function. What is the total number of collisions for h? .... I'm sure this is a trick question, cos there's no more to the question, and you need an extra bit of information surely. Any pointers? --PrifysgolCaerdydd 14:52, 5 May 2006 (UTC)

If I'm interpreting the question correctly, then Σ* is an infinite set and Σⁿ is a finite set. —Blotwell 15:24, 5 May 2006 (UTC)

Without any further information, the correct answer would appear to be "infinite". — Timwi 15:51, 5 May 2006 (UTC)

Well, that's one of the multiple choice answers, I'll click on that one then. --PrifysgolCaerdydd 15:53, 5 May 2006 (UTC)

Gödel's(?) logic

When is 10 + 10 not equal to 20? -- PCE

When it's equal to 100? --Bth 16:35, 5 May 2006 (UTC)

[Edit conflict] In binary, 10 + 10 = 100. If + is interpreted as bitwise or, then 10 + 10 = 10. Can't see the connection to Gödel though. -- Meni Rosenfeld (talk) 16:36, 5 May 2006 (UTC)

Correct. Someone else pointed out to me that the base has to be designated as decimal before 10 + 10 = 100. As for Gödel someone else has said that while his Incompleteness Theorem applies to arithmetic it does not apply to binary (or for that matter multiple state) logic, which seems a bit odd. -- PCE 16:56, 5 May 2006 (UTC)

What do you mean "has to be designated as decimal"? Is it a typo for "binary"? (Obviously binary is the only counting system where 10+10 isn't 20 because it's the only that doesn't have a 2). --Bth 17:44, 5 May 2006 (UTC)

Perhaps you mean Propositional calculus? In that case, I don't remember the details but I think it's true - This theory is too simple for the incompleteness theorem to apply to it. -- Meni Rosenfeld (talk) 17:12, 5 May 2006 (UTC)

Are you familiar with Rucker's proof? -- PCE 17:29, 5 May 2006 (UTC)

No, sorry. -- Meni Rosenfeld (talk) 17:30, 5 May 2006 (UTC)

Is it the one about half way down this page? --Bth 17:46, 5 May 2006 (UTC)

Its at Gödel's incompleteness theorems. If you get a chance the thing I am trying to find out is whether Rucker's proof actually falls within the context of Gödel's Incompleteness Theorem or if his proof falls more in the context of logic than in the context of arithmetic which I have been told is the only context in which Gödel's Theorem applies. -- PCE 17:54, 5 May 2006 (UTC)

While we're here, you can make links within Wikipedia using double brackets around the title -- eg [[Arithmetic]] gives Arithmetic (which I see is the current mathematics collaboration of the week). --Bth 18:58, 5 May 2006 (UTC)

I have an email from Rucker giving his permission to reproduce this proof, however here is a link to the web page from which it was copied:

Incompleteness Theorem

1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
2. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
3. Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
4. Now Gödel laughs his high laugh and asks UTM whether G is true or not.
5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
7. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

• The problem with the above construct is that finite time is discrete and effect must follow cause. If G may be either true or false then what is to stop G from being true at one point in time and false at another point in time? Consequently if we ask the UTM whether G is true or false and the UTM says that G is false and G becomes as the result true then the error we have made is in applying the outdated answer to the resulting state of G rather than laboring to ask the question again. If we ask the question again then the UTM will answer that G is true and G will once again change states and become false. The above construct simply points out the discretion of time and that effect follows cause. If we are going to expect a universal answer from a universal truth machine then we must provide it with a universal rather than a discrete question. The result of this understanding is in fact the principle behind the bistable multivibrator or flip-flop. (Computer Circuits for Experimenters by Forest M. Mims, III, Radio Shack, a Tandy Corporation, 1974, Page 17.)

What a UTM would most likely say to such a discrete question is that G will be true until UTM answers the question that G is true at which time G will become false and vice versa. -- PCE 03:33, 6 May 2006 (UTC)

The problem with this way out is that G is a statement of arithmetic, something like: for all natural numbers m and n it is the case that (m+1)ⁿ⁺¹ = m(m+1)ⁿ+(1+m)ⁿ. Much more complicated, of course, but that is not a matter of principle. Such statements are not supposed to be time-dependent. If they are, all of mathematics becomes bistromathics. --Lambiam^Talk 08:04, 6 May 2006 (UTC)

Yeah, that's the one I found at miskatonic.org. As to the logic/arithmetic thing, I think the relevant part of the Wikipedia article is the second item in the "Misconceptions" section -- specifically the statement "The theorem only applies to systems that allow you to define the natural numbers as a set". That's probably what's meant by "arithmetic" being the only valid context, since the natural numbers are the backbone of arithmetic.

I still don't understand the connection to 10 + 10 = 100, to be honest. --Bth 18:54, 5 May 2006 (UTC)

I assume that the consideration of base when discussing the natural numbers is irrelevant unless you do not settle on a single one. Actually it looks like you can only settle on base 10. -- PCE 22:29, 5 May 2006 (UTC)

Obviously all regularly used bases: decimal, binary, hexadecimal and octal are all base 10. but what about others? --itaj 11:24, 6 May 2006 (UTC)

What I am suggesting is that you can not discuss natural numbers or apply Gödel in other bases. Example: How do you discuss or apply Gödel to the integer 2 in base 2 since the integer 2 does not exist in base 2? How would you discuss or apply Gödel to the term "A" in base 16 or hexadecimal? -- PCE 11:36, 6 May 2006 (UTC)

What do you mean by "apply Gödel to the integer 2"?" Computing the Gödel number of a term? I sense a deep confusion of levels in your question.

The nature, existence and properties of an object in mathematics are independent of the choice of notation used to represent them. So first of all, "the integer 2 does not exist in base 2" is meaningless; what you are presumably trying to express is "the glyph/digit 2 is not used to write down numbers in base 2". The integer 2 most assuredly has a representation in base 2, consisting of the symbols "1" and "0" in the proper order.

What bearing does this have on the incompleteness theorems, though? One popular illustration of the incompleteness theorems is given in Gödel, Escher, Bach, where the formal system used to demonstrate the theorem on in fact uses unary to represent numbers. No "2" in sight anywhere, but that doesn't prevent the number 2 from being representable as "SS0", nor does it prevent the expression of Gödel numbers in the same way. 82.92.119.11 12:03, 6 May 2006 (UTC)

The Revenge of the UTM

1. Someone introduces U.T.M. Mark II to Kurt Gödel, a deceased Austrian logician who is supposed to be a genius mathematician, capable of correctly answering any question that has a logical answer.
2. Mark II asks Gödel whether he is familar with the concept of quining. "Naturally am I that," answers Gödel, "it means a quoted version of a text to itself prepending -- you can it up-look on the Vikipedia. Van should me acknowledged have, for it just an informal version is of what I so laboriously with primenumbers in arithmetic coded have."
3. With a deadpan expression -- recall that emotive facial expressiveness was only introduced with the Mark III line -- Mark II prints out the following string on its output tape: "', quined, cannot be claimed to be true by Kurt Gödel while remaining truthful', quined, cannot be claimed to be true by Kurt Gödel while remaining truthful". Call this text G. Note that G contains the text K = ", quined, cannot be claimed to be true by Kurt Gödel while remaining truthful". So G states: "K, quined, cannot be claimed to be true by Kurt Gödel while remaining truthful". But K quined is G. So in fact G states: "G cannot be claimed to be true by Kurt Gödel while remaining truthful".
4. Now Mark II, feverishly running its tapes to and fro, asks Gödel whether G is true or not.
5. If Gödel says G is true, then he claims that G cannot be claimed to be true by himself while remaining truthful. But he just claimed that very statement to be true, leading inexorably to the conclusion that he is untruthful, ruining his repetition, Since Gödel wishes no such thing to happen, he will never say that G is true.
6. We have established that Gödel will never say G is true. But, as we just saw, we know that G is true, because Gödel cannot claim G to be true without being untruthful, which is exactly what G says.
7. "I know a truth that Gödel can never utter, or else he is lying," Mark II says. Is it possible that a hint of a smirk shimmers through that immobile steel mask? "I know that G is true," Mark II resumes, "we all know that G is true, all of us except Gödel. This guy's logic is definitely incomplete; I wouldn't trust him to do any significant mathematics."

--Lambiam^Talk 22:50, 5 May 2006 (UTC)

Greeting Wikipedia -

If a sub-amateur may posit, then consider: maybe atemporal paradox is the engine of time itself, something like Heidegger's "becoming".

Thank you, Willie

How can I find a software for VOB?

I need to convert a Windows Media file into a VOB (DVD Video Object) because I'm trying to burn it onto a DVD and the burning program won't accept my Windows Media file. Can you tell me if there's any software I can use? Jonathan ^talk File:Canada flag 300.png 18:53, 5 May 2006 (UTC)

Search for options at this web site. One popular choice is TMPGEnc. --KSmrq^T 23:49, 6 May 2006 (UTC)

Jacobi Verification

In Lagrange's four-square theorem, it says, "In 1834, Carl Gustav Jakob Jacobi found an exact formula for the total number of ways a given positive integer n can be represented as the sum of four squares. This number is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even." I can't find anything that backs this up. Is it accurate? If so, can anyone direct me to a proof? Black Carrot 22:28, 5 May 2006 (UTC)

This follows from Theorem 386 in Hardy & Wright's An Introduction to the Theory of Numbers Chapter XX which states that:

The number of representations of a positive integer n as the sum of four squares, representations which differ only in order or sign being counted as distinct, is 8 times the sum of the divisors of n which are not multiples of 4.

If n is odd then all of its divisors are odd and none of them are divisible by 4. If n is even then the sum of its divisors which are not multiples of 4 is the sum of its odd divisors, A, plus the sum of its divisors divisible by 2 but not by 4, B. But B = 2A, so the sum of the divisors of n not divisible by 4 is 3 times the sum of its odd divisors. Gandalf61 14:36, 6 May 2006 (UTC)

Score. Does the theorem have a short proof, or should I buy the book? Black Carrot 16:36, 6 May 2006 (UTC)

You should buy the book anyway :) --Lambiam^Talk 18:59, 6 May 2006 (UTC)

Thanks to all. The book is bought. Since it'll be a few days before it ships, and a week and a half before I see it, would you mind giving me some hints on the proof itself?

Incidentally, it turns out there are two books with identical names and totally different authors: [2] and [3]Black Carrot 02:31, 7 May 2006 (UTC)

(lol) Just wondering ... what if the authors were only slightly different?  :--) JackofOz 02:52, 7 May 2006 (UTC)

Hardy & Wright's proof takes up about 3 pages of the book, so I am not going to attempt to reproduce it. But basically they use some complicated identities from elliptic function theory to derive an expression for the co-efficients in the generating function

$(\sum _{m=-\infty }^{\infty }x^{m^{2}})^{4}.$

They say that the result can also be derived by considering norms of quaternions. My impression is that any derivation of the number of four-square respresentations will be an order of magnitude more complex than simply proving that there is at least 1 such representation for every positive n. Gandalf61 10:17, 7 May 2006 (UTC)

Ah. Darn. Regarding the two books: If the authors differed only slightly, it would clearly be a different edition of the book, one for which the team writing and editing it had changed somewhat. These things happen. However, since the two have no apparent relation at all, other than their name, it seems one of them must be violating copyright. Black Carrot 15:57, 7 May 2006 (UTC)

May 6

kinetic energy

My father and I have an argument going, he states that 2 identicle vehicles, going the same speed hit head on is the same as one hitting an immovable object. I, on the other hand, think that the kinetic energy of the 2 vehicles add up and produce a much greater impact. What is the formula to prove my theory.71.133.186.118 04:21, 6 May 2006 (UTC)

I agree with your father. Each car will absorb one car's worth of kinetic energy, whether you think of it as hitting the other car or as hitting an imaginary plane-shaped force field. If the crash is set up symmetrically enough, then the two situations will be identical in all other aspects as well. Melchoir 04:36, 6 May 2006 (UTC)

I don't think it will be completely identical: when there is one car, the time it takes for crumpling will be given by Ft = mv, while when there are two cars, Ft=2mv. Thus the crumpling take twice as long when there is only one car hitting the wall (or imaginary force field, if you prefer), assuming the force is the same. This might make me inclined to say both father and son are right: the more important part of the impact is probably how deeply the cars get crumpled, which is the same. However, the scenarios are not completely identical, the two cars crumple faster. -lethe ^{talk +} 04:46, 6 May 2006 (UTC)

You seem to be saying, assuming the force is the same between a car and another car or between a car and a force field. (And sure, everyone loves force fields.) But it makes a lot more sense to me if the force between two cars winds up to be twice the force between a car and a force field. The force on an individual car has to be the same in either scenario, because both scenarios are microscopically just making sure that nobody crosses the symmetry plane, while the plane itself absorbs no energy.

One could even prove my assertion mathematically; model a car as a system of atomic-scale masses connected by springs, and let the masses on either side of the crash surface scatter elastically off their counterparts. Or, in the force field case, let them scatter elastically off the plane; it's the same thing. Melchoir 05:04, 6 May 2006 (UTC)

If you assume that the force between the two cars is twice as big between two cars as it is between a single car and a wall, then the distance calculation is changed; now the total distance crumpled is the same as with one car, but split between the two, each car only crumples half as much. You can't have it both ways. Either one scenario crumples half the depth, or the other half the length of time. Also, you are asserting that the force exerted by a crumpling fender is velocity dependent. Well, that's certainly possible, but it's not obvious. Certainly not the case with simple harmonic oscillators. But whatever force you try to model this with, it doesn't change the issue with crumpling time versus distance, which is just a consequence of conservation laws. PS, force fields are nice, but I don't think trying to model a wall with a force field is a good idea. Force fields are better suited to inverse square force laws. To model a wall with a force field, I dunno, you'd have to use a square well potential, which means some delta functions, yuck. Better to just introduce the wall as a constraint in your variables. -lethe ^{talk +} 05:15, 6 May 2006 (UTC)

I get the feeling that we're miscommunicating due to a lack of symbols. Let me start over below... Melchoir 05:51, 6 May 2006 (UTC)

Actually, now I feel that you definitely can't (easily) model a wall with a force field: the normal force exerted on you by a wall depends on how hard you push, and not just on where you are (as with a force field). Of course, the wall's normal force is a complicated sum of atomic bond forces, which are force fields, so with enough machinery, you could probably get away with it. But to what end? -lethe ^{talk +} 06:07, 6 May 2006 (UTC)

I meant a science fiction force field. Not made of atoms, or a vector field, or anything complicated; just a plane that things bounce off of. Melchoir 06:47, 6 May 2006 (UTC)

Oh right. So you really just mean a fancy term for a very massive brick wall. For those, I prefer the term “wall”. For me, a force field is a vector field which measures force. -lethe ^{talk +} 07:15, 6 May 2006 (UTC)

Edit conflict: Well, it seems to me that your father is correct. When there is one car, the total energy is T₁ = 1/2mv². Assuming the force on the car is constant as it crumples when it hits the immovable object, we will get the amount of crumpling by Fd= 1/2mv². When there are two cars, the total energy is T₂ = 1/2 mv² + 1/2 mv² = mv². Assuming the same force to crumple the cars, the total crumpling distance will be twice as much, but since both cars are crumpling, this distance will be split evenly between the two cars, and each car will crumple the same depth as the single car. I'm not positive about assuming the force will be the same in both scenarios, but it seems plausible. -lethe ^{talk +} 04:41, 6 May 2006 (UTC)

Okay... let m be the mass of a car. Let's work in one dimension. Let x₁(t) and x₂(t) be the center-of-mass positions of cars 1 and 2, which begin with equal and opposite velocities and collide with each other at x=0. Let x₀(t) be the position of car 0, which starts out the same as car 1 but collides with a Wall at x=0 instead. And by a Wall, I mean an idealized plane of infinite mass at every point that absorbs no energy, and therefore reflects anything that hits it.

Let v_i(t) be the velocity of car i and F_i(t) be the force on car i.

Finally, define relative variables x₁₂(t) = x₁(t) − x₂(t), v₁₂(t) = v₁(t) − v₂(t), and F₁₂(t) = F₁(t) − F₂(t).

By symmetry alone, we have:

• x₁(t) = −x₂(t) = ¹⁄₂x₁₂(t)
• v₁(t) = −v₂(t) = ¹⁄₂v₁₂(t)
• F₁(t) = −F₂(t) = ¹⁄₂F₁₂(t)

I am claiming that cars 0 and 1 behave identically; so

• x₀(t) =x₁(t) = −x₂(t) = ¹⁄₂x₁₂(t)
• v₀(t) =v₁(t) = −v₂(t) = ¹⁄₂v₁₂(t)
• F₀(t) =F₁(t) = −F₂(t) = ¹⁄₂F₁₂(t)

Is there something wrong with that? Melchoir 06:13, 6 May 2006 (UTC)

What is F₁₂ supposed to represent? Why do you assume that car 0 and car 1 undergo the same forces and trajectory? Is this not in contradiction with what you said earlier ("But it makes a lot more sense to me if the force between two cars winds up to be twice the force between a car and a force field. ")? -lethe ^{talk +} 06:22, 6 May 2006 (UTC)

F₁₂ is the relative acceleration, multiplied by m; it is what I called the "force between two cars". I could do without it, but you did write Ft=2mv above; that makes sense only if by F you mean F₁₂ and by v you mean v₁. (Also, if you insert some Deltas).

As for why x₀(t) =x₁(t) should work, why not? You'll agree if every car is a single particle, right? Melchoir 06:45, 6 May 2006 (UTC)

The equation Ft=2mv says that the force exerted in an interaction times its duration (known as the impulse) is equal to the total change of momentum in the collision. This number F₁₂, the “relative force”, is not actually the force on any object in the system, so I’m not sure what to do with it. But uh, you know what? I think I’ve made a mistake. I think the impulse on an object should only be equal to that object’s change in momentum, not the total change of momentum of all objects involved in the collision. So maybe the equation Ft=2mv isn’t right. Lemme think this through a little more carefully. -lethe ^{talk +} 06:51, 6 May 2006 (UTC)

Yeah, I think I'm wrong. I used that equation badly, and your reasoning is sound: with perfect symmetry, the collision of the two cars will be exactly the same as one car hitting the wall. My bad. So tell anon that his father was right. -lethe ^{talk +} 07:12, 6 May 2006 (UTC)

Heh, I think we scared off the anon, and everyone else. Or, to be more precise, I choose to believe that there could be no other reason why we're the only people posting to the Wikipedia mathematics reference desk during a western-hemisphere night on the weekend! Melchoir 07:23, 6 May 2006 (UTC)

Haha. I wish I were out at the bar tonight. Maybe I'll be able to go out tomorrow night, and you can man the reference desk by yourself. Better yet, maybe we'll both make it, and these questions can wait until Sunday morning!-lethe ^{talk +} 07:31, 6 May 2006 (UTC)

But don't post reactions with a hangover; it shows. Lambiam^Talk 11:28, 6 May 2006 (UTC)

If the two cars have identical masses m and are travelling at the same speed v in opposite directions, and come to rest after colliding head on, then the energy expended in the crash is mv². This energy is expended in damaging the cars (although some will be lost as heat due to friction). To expend the same energy by hitting a stationary wall, a single car would have to be travelling faster, at sqrt(2)v. So, no, a head-on crash between two cars travelling at speed v is not the same as a single car hitting a wall at speed v. Gandalf61 20:25, 6 May 2006 (UTC)

In the two-car case, the amount of kinetic energy is ½mv² per car, or in total 2 × (½mv²). Assuming all is transformed into destructive energy, and assuming perfect symmetry, the amount per car is one half of that, or ½mv². Now for the one-car case, the amount of kinetic energy is ½mv² in total, giving a destructive energy of ½mv² for that one car. Same difference, no? --Lambiam^Talk 20:43, 6 May 2006 (UTC)

Not the same. In the one-car case some of the ½mv² destructive energy is absorbed by the wall, so the destructive energy absorbed by the car is less than in the two-car case. No matter how "immovable" the wall is, it must suffer some deflection and damage in an inelastic collision. Gandalf61 10:03, 7 May 2006 (UTC)

For a real wall made of quite movable stone, yes. But surely you can imagine a truly and completely immovable wall, not one atom of which even flinches upon impact. To invert the question, such a wall would be a nasty thing, because crashing into it is just a bad as crashing into an oncoming car! Melchoir 10:28, 7 May 2006 (UTC)

[this is wrong: – b_jonas 12:26, 7 May 2006 (UTC)] An immovable wall doesn't have to suffer damage. An immovable wall won't be harmed, but it will abosrb the energy in the form of kinetic energy: it will get all the momentum the car had, but as the wall effectively has an infinite mass (as it's fixed firmly to the Earth) it wouldn't move at all. It's similar to when the car accelerates, it can gain a momentum because it propels the ground back but that doesn't move the Earth at all.

Also, in reality, cars aren't so ideal either as we assumed here. Obviously, some safer makes of cars would suffer less damage than an inferior car. – b_jonas 11:25, 7 May 2006 (UTC)

Actually, when an infinite-mass object accepts momentum, in addition to not moving ($v=p/m=0$), it doesn't absorb any kinetic energy ($E=p^{2}/2m=0$). So if you do want a wall to absorb energy in a collision, it either has to have a finite effective mass (not fixed to a planet) or it has to suffer deformation. Melchoir 11:40, 7 May 2006 (UTC)

Ok, you're right, sorry. So in an totally elastic collision, the inifnite-mass wall would absorb no energy but the car would start going back with the same velocity; while a car would obviously collide inelastically, when after the collision the car would stop, the wall wouldn't move, and all energy would be converted to damage of the car, damage of the wall, heat, and sound. – b_jonas 12:26, 7 May 2006 (UTC)

Some amount of energy may go into damaging people in the car. Lambiam^Talk 14:09, 7 May 2006 (UTC)

• lethe said: "So maybe the equation Ft=2mv isn’t right." [in the 2 car case]. Just to amplify why it's not right, you forgot that F and v are vectors. Total momentum before collision is mv + (-mv)=0. Total momentum after collision is 0 + 0=0 (totally inelastic collision) Change of total momentum over the time period of the collision (ie impulse) =Ft =0-0=0.
• In a one car inelastic collision with the wall, total momentum = momentum of car before = mv, after collision momentum of car =0, but total momentum is conserved, so where is the rest? The wall (and its rigidly attached planet) are now moving from the impulse mv.

You compute the velocity of wall+planet, V, from MV=mv, where M is mass of wall+planet. (I know it's off center, so it will spin the planet a little. Imagine the entire mass of the planet were in the wall, if you like, to simplify the thought experiment.)

• Consider the two car collision. Let's say the two cars are identical. So identical, so when they crash exactly head-on, and piece of one flies off, like a side mirror, say, it collides at the plane of collision, with the side mirror of the other car and inelastically collides and drops to the ground at the collision plane. And thus all of each car stays on its side of the collison plane. Now we stretch a sheet of tissue paper on a frame , like a Japanese shoji screen, at the plane of collision and run the experiment again. Looking at the results on one side of the screen CANNOT BE DISTINGUISHED FROM the results of colliding with the supermassive wall, which we also covered with tissue paper. The driver will not know if s/he hit a wall or a sheet of tissue paper (with a car behind it.) Your father is right, as Melchoir said in the beginning. --GangofOne 07:00, 8 May 2006 (UTC)

... but only if we assume an infinitely massive, infinitely stiff wall, which is so far away from physical reality that it is not even a good first order approximation. Gandalf61 09:19, 8 May 2006 (UTC)

mutually singular measures

let $(\Omega ,\Sigma ),$ be a measurable space. i'm talking here about real-value signed finite measures on this space. let M be a set of measures.

let N be the set of all measures absolutely continuous with respect to a measure in the linear span of measures in M. i.e. $N:=\{\nu :\exists \mu \in span(M)\ (\nu <<\mu )\}$

for a measure $\mu$ i'll denote ${singl}(\mu ):=\{\nu :\nu \perp \mu \}$. the set of measures mutually singular to $\mu$.

and for a set of measures L. ${Singl}(L):=\{\nu :\forall \mu \in L\ (\mu \perp \nu )\}$. the set of all measures mutually singular to all measures in L.

my question is if the above implies that $N=Singl(Singl(N))$ i know this is true if there's only one measure in M, but i need to know about infinite set M, countable and bigger. --itaj 16:13, 6 May 2006 (UTC)

hhhmmmm... no one says anything... is my question ok? is it phrased wrong? --itaj 18:24, 7 May 2006 (UTC)

The phrasing looks OK, I just don't know enough to answer. Sorry! -lethe ^{talk +} 18:29, 7 May 2006 (UTC)

I'll give it a try, but I have been a few years away from my calculus courses, so I hope this works. First of all, I will drop the linear span condition, and concentrate on the case of standard positive measures. I guess one can get the general case from this (say, splitting $\mu =\mu ^{+}+\mu ^{-}$). So I will assume $N:=\{\nu :\exists \mu \in M\ (\nu <<\mu )\}$.

You start from Radon-Nikodym Theorem: hence you have that each $\nu \in M$ can be written as $f\cdot \mu$ where $f$ is some $\Sigma$-measurable function. From now on, when I say "$\forall f$" I will mean "for all such $f$'s", and "$\exists f$" will mean "for some such $f$".

Now you have:

$\nu \in N\iff \nu =f\cdot \mu$ for some $f$.

$\tau \perp \nu \iff \forall A\in \Sigma :(\tau (A)>0\iff \nu (A)=0)$.

$\tau \in Sing(N)\iff \forall A\in \Sigma :(\tau (A)>0\iff \int _{A}f\cdot d\mu =0\forall f)$ which is the same as

$\tau \in Sing(N)\iff \forall A\in \Sigma :(\tau (A)=0\iff \exists f:\int _{A}f\cdot d\mu >0)$.

$\sigma \in Sing(Sing(N))\iff \sigma \in Sing(\tau )\forall \tau \in Sing(N)\iff (\sigma (A)>0\iff \tau (A)=0\iff \exists f:\int _{A}f\cdot d\mu >0)$. Ehm, I seem to be a little bit stuck here. I'll think over this some more time, I'll just post this, hoping it may help anyway. Also note that in order to use Radon-Nikodym Theorem $\mu$ must be $\sigma$-finite. Otherwise you can use a generalization of that theorem that allows you to split "almost" any measure as $\mu =\mu _{<}+\mu _{\perp }$, where $\mu _{<}<<\nu$ and $\mu _{\perp }\perp \nu$, but I do not remember the details. Cthulhu.mythos 11:25, 12 May 2006 (UTC)

May 7

Trigonometry with a triangle inscribed in a circle

In the figure to the left, A, B and C are points on a circle with centre P. B and C are fixed, but A is a variable point on the arc BC. I have to find the perimeter of triangle ABC in terms of x, a and R. The correct answer is: $2R(\sin \left({\frac {a}{2}}\right)+\cos x+\cos \left({\frac {a}{2}}-x\right))$

Now, why is this so? According to the explanation I got, the side opposite C is: $2R\cos \left(x\right)$; the side opposite a is: $2R\sin \left({\frac {a}{2}}\right)$; and the side opposite B is: $2R\cos \left({\frac {a}{2}}-x\right))$. However, it doesn't explain the principles applied to obtain these equations. Can anyone help me? What am I missing? Johnleemk | Talk 08:11, 7 May 2006 (UTC)

What does it mean that B and C are fixed but A is variable? And what are a and x? Melchoir 08:31, 7 May 2006 (UTC)

I'm guessing that it means that B and C are fixed points, but A can be any point on the arc BC. a and x are angles; I tried to roughly approximate the original diagram, but I guess my attempt didn't work too well. Johnleemk | Talk 08:37, 7 May 2006 (UTC)

But if we don't know exactly where B and C are, aren't they just as variable as A? Or, since the geometry isn't changing over time, aren't they all equally fixed? Well, let's just chalk it up to educators who don't stop to think about what they're saying. Anyway, I'll comment on the solution below. Melchoir 09:11, 7 May 2006 (UTC)

The measure of arc BC is a. Since angle BAC subtends the same arc, but lies on the circle, it is a/2. I know segment AB and BC, so I can solve the triangle, though this is an ASS situation, so there may be two solutions, rather than one. I'll try it. -lethe ^{talk +} 08:40, 7 May 2006 (UTC)

The triangle PBC is isosceles, and we may find segment BC using the law of cosines by BC² = R² + R² –2R² cos a. The law of sines says that

${\frac {\overline {BC}}{\sin A}}={\frac {\overline {AC}}{\sin B}}={\frac {\overline {AB}}{\sin C}}=2R.$

To solve this should be easy. -lethe ^{talk +} 08:46, 7 May 2006 (UTC)

Why is triangle PBC equilateral? How can I tell if the line BC is exactly equal to the radius? Johnleemk | Talk 08:49, 7 May 2006 (UTC)

Sorry, I meant to say "isosceles". Because the two radii are equal. The calculation is OK, I just misspoke. -lethe ^{talk +} 08:55, 7 May 2006 (UTC)

So we have that sin C = x/2R. Thus B = π – a/2 – C (and note that there may be two solutions for C here). -lethe ^{talk +} 08:59, 7 May 2006 (UTC)

Therefore

${\textrm {AC}}^{2}={\textrm {BC}}^{2}-x^{2}-2x{\textrm {BC}}\cos(\pi -a/2-C)$.

-lethe ^{talk +} 09:01, 7 May 2006 (UTC)

Now that we have AB, BC, and AC in terms of a, x and R, all we have to do is simplify. Things look kinda messy, so I'm going to try it on paper and see what happens, before I procede any further. -lethe ^{talk +} 09:08, 7 May 2006 (UTC)

I forgot to mention that A can only vary on the major arc BC. Sorry about that. Johnleemk | Talk 08:48, 7 May 2006 (UTC)

Holy shit. I've just re-read your explanation of the problem more carefully, and I see that x is an angle (namely, the angle PBA). I had assumed that it was the length of the segment AB. Thus, everything I've said so far is probably wrong. Lemme start over. -lethe ^{talk +} 09:10, 7 May 2006 (UTC)

Yeah, that's what I thought at first too. But knowing the angles makes everything un-messy again. Melchoir 09:15, 7 May 2006 (UTC)

OK, so I have that angle BAC is a/2, angle ABC is x+(π-a)/2. And length BC is given by BC² = R² + R² –2R² cos a. I know AAS of a triangle, I can solve that using law of sines. Note that angle BCA = π – (x + (π–a)/2) – a/2 = π/2 – x. Now let's see:

${\textrm {AB}}=2R\sin {\left({\frac {\pi }{2}}-x\right)}=2R\cos x$

then

${\textrm {AC}}=2R\sin {\left(x+{\frac {\pi }{2}}-{\frac {a}{2}}\right)}=2R\cos {\left(x-{\frac {a}{2}}\right)}$

and finally

${\textrm {BC}}=2R\sin {\left({\frac {a}{2}}\right)}.$

Adding them up, I get the required answer, and voilà: I'm done. -lethe ^{talk +} 09:23, 7 May 2006 (UTC)

Alternatively, denoting the midpoint of AB by M, the angle AMP is a right angle because the perpendicular bisector of a chord goes through the centre of the circle; see chord (geometry). Hence AM = R cos x and AB = 2R cos x. The other sides can be calculated similarly. -- Jitse Niesen (talk) 09:47, 7 May 2006 (UTC)

Now that lethe has supplied a computation, let's go back and consider concepts.

1. The sum of the interior angles of any (plane) triangle is π, or 180°.
2. If a triangle is inscribed in a circle (so that its vertices lie on the circle), then the angle an edge subtends from the center of the circle is twice the angle at the opposite vertex.
3. Any triangle with two vertices on a circle and the third at the circle center is isosceles, its two equal sides being radii.
4. The cosine function is a "phase shift" of the sine function, cos(ϑ) = sin(^π⁄₂+ϑ).
5. The cosine function is an "even function", meaning cos(−ϑ) = cos(ϑ).

The first fact is one most people know. The second deserves to be better known; for example, it implies that if the inscribed triangle has a diameter as one edge, then it is a right triangle. Combine the second and third facts and split the isosceles triangle into two equal right triangles; then basic trigonometry gives half of side BC as R sin(^a⁄₂). This also leads to the law of sines. The rest is minor juggling. --KSmrq^T 10:46, 7 May 2006 (UTC)

Ok, thanks guys! Johnleemk | Talk 06:30, 8 May 2006 (UTC)

Leibniz notation (dx)

I've finally had it up to here with what appears, from my understanding, to be gross abuse of notation committed by practically everybody. It's all to do with Leibniz notation of differentials, dy and dx, etc. I've never really understood what they represent and after trying to do a Vector Analysis assignment that involved torsion of a curve c(t), defined only in our notes by ${\frac {d\mathbf {B} }{ds}}=\tau \mathbf {N}$, where B is the binormal vector, s is some sort of poorly explained arclength thing, and N is the normal vector. Since B is a function of the parameter t, I'm wondering how on earth one can differentiate it with respect to s.

And even beyond that particular example, which is not such a bad one (given that you can think of B as being parameterised by s in some sort of way), it's even -worse- in situations like this representation of integration by parts. I never liked doing it, but I just accepted it at the time. Now, I'm finally sick of being confused by the thing.

Nobody has -ever- really explained what dy/dx really means, first we started just using it to represent the derivative and told "it's just a thing in itself", then we started bouncing its bits around to do the chain rule and such, and I'm still confused as to what it represents and what, exactly, the value of dx is. Could anyone help, please? Maelin 12:49, 7 May 2006 (UTC)

dx does not have a value. d/dx just means "the derivative with respect to x". If I'm not wrong, it's sort of comparable to the trigonometric functions in the way that it does not have a value, just tells you "what to do" with a number of expression. —Mets501^talk 13:08, 7 May 2006 (UTC)

And B is also a function of s, as follows: Given s, find the point p on the curve that is at a distance s, measured along the curve, from some fixed starting point. Then take the value of B at that point p. Since p is on the curve, there is some value t_p such that p = c(t_p). That gives you the value for t to plug in if B is expressed as a function of t. For arc length, see Arc length and Curve#Lengths of curves. --Lambiam^Talk 15:06, 7 May 2006 (UTC)

Think of it as ${\frac {dy}{dx}}=\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}$. Technically, since limits are hard to understand and some constructivists do not even accept the limits, consider this notation just as a certain consistent rule in mathematics. Also in $dy=f'(x)dx$, $f'(x)$ is a linear operator in a certain vector space. (Igny 15:12, 7 May 2006 (UTC))

So my advice is a little different and may be a bit surprising coming from a pure mathematician. I would say, go back to what Leibniz more or less meant by it ("a very small change"), and don't sweat the details too much for a while. Multivariable calculus is a wonderful subject for developing a physical intuition, and then seeing, post facto, how that intuition connects with the formalism. Just loosen up on your demands for rigor for a while, until the intuitions are fully internalized. At that point the formal arguments will be much easier to follow. --Trovatore 16:01, 7 May 2006 (UTC)

The notation of Leibniz is not so bad. Integrals make sense in spaces which have some notion of sizes of sets. The value of an integral depends on this choice of notion, and therefore the notation should carry some indication of this dependence. The symbol dx indicates the standard choice for the real line (known as the Lebesgue measure). Viewing the real line as an oriented smooth manifold, every differentiable function f gives rise to a dual vector df = f'(x) dx, and every dual vector gives rise to a measure. These two symbols df and dx denote different choices of measure then. The chain rule/substitution rule of integrals and the product rule/integration by parts rule are expressed in this notation as $\int \,df=\int f'(x)\,dx$ and $\int f\,dg=fg-\int g\,df$. In other words, these notations are not abuses if the df symbols are thought of as differential forms. They are natural notations. The symbol d is then an operator which takes functions to their associated forms. This operator is natural, in the sense of category theory.

As for the symbol df/dx, well you could take it to be defined by the equation $df={\frac {df}{dx}}\,dx$, which explains the suggestive notation as well as the fact that some rules like ${\frac {df}{dx}}={\frac {df}{dy}}{\frac {dy}{dx}}$ hold; it's the factor relating two dx-type symbols. Nevertheless, the d in the differential form has to be viewed as a different symbol from the d in the symbol df/dx, since the dx-type symbols are not invertible and cannot be made into fractions. This is probably why a different symbol ∂ is sometimes preferable. I guess nonstandard analysis allows you to think of df/dx as a fraction of infinitesimals, but I don't find that very helpful.

To sum up, the operator d is natural, and none of its formulas are abuses of notations, rather they are expressions of properties that the operator satisfies. Thy symbol ∂f/∂x sometimes acts like a fraction of differential forms, because it is defined as the factor relating them. Nevertheless, it cannot be thought literally as a fraction of forms because forms do not form a division ring; they don't have inverses. -lethe ^{talk +} 17:12, 7 May 2006 (UTC)

What I'm recommending to the OP is to think of dy/dx as a literal fraction of infinitesimal quantities, without bothering with nonstandard analysis. Focus first on developing the physical/visual/tactile intuition. The "rigorous" treatment (either the traditional or nonstandard one) is a superstructure built on top of that, not its foundation. --Trovatore 17:46, 7 May 2006 (UTC)

A sensible recommendation. Afterall, asking the OP to learn differential topology in order to understand the notations of elementary calculus is a bit of a stretch (nevermind model theory). Nevertheless, that's how I think of the symbols, and I thought OP might be interested to hear about my way of thinking about the notation. -lethe ^{talk +} 18:13, 7 May 2006 (UTC)

Hmm... maybe I made it look like I was somehow rebutting your statement, but putting my comment indented under yours. I wasn't really. My comment was meant for the OP, I just indented for legibility. Just a sort of "here are my thoughts on those symbols". -lethe ^{talk +} 18:16, 7 May 2006 (UTC)

I see there already are all kind of responses, but I will give my thoughts on the matter as well: If df/dx is defined (as it often is) as simply a notation for derivative, then yes, what you discuss definitely is an abuse of notation. But it's one that works, which explains its popularity (the contexts where it works can be rigorously proven to be true). But you wish to avoid the "I don't know what it means but it works so I'll use it" approach, and this requires a different thinking. One way of doing it was discussed by lethe; I like a different approach, inspired by Trovatore's intuitionistic suggestion. Instead of thinking of symbols like f or y as functions in the ordinary sense, and x as a dummy variable used to express the functions, think of a "world" where x, s, y etc. denote stand-alone objects, each one having a meaning of its own, but they are interrelated by a system of equations. In every manifestation of the world, each of these objects will obtain a specific value, usually a real number. Not every allocation is allowed; Only those that satisfy the equations which determine the "rules" for the world. Now, if u is any symbol, then du is the change (difference between new and old values) in u when the world changes slightly (that is, when each object gets a different, yet very close, value). Here is a simple example: Suppose you have the equation y = x². Don't think of y as being strictly a function of x; Rather, our world consists of 2 objects, x and y, where only mainfestations where this equation hold are allowed. So it can be that x = 2 and y = 4 but it cannot be that x = 3 and y = 10. dx is the change in x when the world changes slightly; dy is the same for y. Since we always have y = x², we also have:

$dy=d(x^{2})=(x+dx)^{2}-x^{2}=x^{2}+2xdx+dx^{2}-x^{2}=2xdx+dx^{2}$

Our assumption was that the change is slight (infinitesimal), so dx^2 is negligible, and we have dy=2x dx. For this we can also get:

${\frac {dy}{dx}}=2x$

So, the ratio between the change in y and a change in x, when both changes are infinitesimal, is twice the value of x. Therfore, we say that the derivative of y with regard to x is 2x.

Another example: Suppose we have y=x^2, z=y^3. We wish to find dz/dx. This question is meaningful: z is not a function of y, it is an object that has a certain relation with y. Since y is related to x, z is also related to x. We have two ways of doing that calculation:

${\frac {dz}{dx}}={\frac {d(y^{3})}{dx}}={\frac {3y^{2}dy}{dx}}={\frac {3(x^{2})^{2}*2xdx}{dx}}={\frac {6x^{5}dx}{dx}}=6x^{5}$

${\frac {dz}{dx}}={\frac {dz}{dy}}{\frac {dy}{dx}}={\frac {d(y^{3})}{dy}}{\frac {d(x^{2})}{dx}}={\frac {3y^{2}dy}{dy}}{\frac {2xdx}{dx}}=3y^{2}*2x=3(x^{2})^{2}*2x=6x^{5}$

The second calculation works because dy is effectively a number, so we can divide and multiply by it. This approach is what is called "the chain rule" when applied to functions.

We also have

$d(uv)=(u+du)(v+dv)-uv=uv+udv+vdu+dudv-uv=udv+vdu+dudv=udv+vdu$

Where dudv is negligible: This is equivalent to the multiplication rule for derivatives.

The same idea can be applied for integrals. ∫ is taken to mean an operator which cancels d. So it is always the case that ∫du = u. So we have:

$\int 2xdx=\int d(x^{2})=x^{2}$

Another example:

$d(uv)=udv+vdu$

$udv=d(uv)-vdu$

$\int udv=\int (d(uv)-vdu)$

$\int udv=\int d(uv)-\int vdu$

$\int udv=uv-\int vdu$

And here we have the integration by parts rule. An example for using it:

$\int {\sqrt {x}}dx=x{\sqrt {x}}-\int xd{\sqrt {x}}=$

Assuming x≥0, and letting y=√x:

$x{\sqrt {x}}-\int y^{2}dy=x{\sqrt {x}}-\int d({\frac {y^{3}}{3}})=x{\sqrt {x}}-{\frac {y^{3}}{3}}=x{\sqrt {x}}-{\frac {{\sqrt {x}}^{2}}{3}}={\frac {2}{3}}x{\sqrt {x}}$

You may say that this approach isn't rigorous. You're right; it's not. But it pours some meaning into otherwise meaningless expressions. Virtually every result that can be obtained by it, can be proven to be true. It is also very convenient for obtaining these results - you don't really have to memorize the rules for integration by substitution, parts etc., you can always derive them from these manipulations. So, as Trovatore suggested, you should familiarize yourself with doing manipulations using this intuitionistic approach, and keep in mind that every result can, if desired, be also proven rigorously. I hope this helped. -- Meni Rosenfeld (talk) 13:30, 8 May 2006 (UTC)

Induction.

My maths exam is coming up soon but I am like... having trouble with nduction. Especiall thses kind:

Prove that, for all integers n >= 1, that

4ⁿ − 1 is a multiple of three

Thanks :D —Preceding unsigned comment added by 81.178.69.26 (talk • contribs)

Have you taken a look at mathematical induction? Isopropyl 15:12, 7 May 2006 (UTC)

I had a loko there but it was beyond my scope. I am just an advanced higher pupil. I have grasped the whole sigma thing in induction but knowing my luck, one of the example above would come up. Thanks for the reply though —Preceding unsigned comment added by 81.178.69.26 (talk • contribs)

First, prove the basis. The lowest number of n used is 1, so 4¹-1=3, which is a multiple of 3. Then, prove that, for any 4ⁿ-1 that is a multiple of 3, 4ⁿ⁺¹-1 must also be a multiple of 3. Well, what do we do to turn 4ⁿ-1 into 4ⁿ⁺¹-1? We add 1, then multiply by 4, then subtract 1. So, 4((Multiple of 3)+1)-1. Distribute, 4(Multiple of 3)+4-1. Simplify, 4(Multiple of 3)+3; 4(Multiple of 3)+(Another multiple of 3); (A higher multiple of 3). Therefore, 4¹-1 implies 4²-1, which implies 4³-1, etc. Does that help? Black Carrot 15:48, 7 May 2006 (UTC)

---

Induction is a marvelous and essential proof method that comes with the natural numbers. It is often convenient to start counting from zero, but one is also common, and the method accomodates a choice of starting points.

The intuition is simple enough: One is a natural number, and all the rest are successors, where the successor of n is n+1 (or will be once we define arithmetic). The proof method says that if a statement is true for n = 1 (the base step), and if whenever it is true for n then it is also true for n+1 (the induction step), then it is true for all natural numbers.

In a typical proof by induction, the base step is trivial. The work comes in finding a way to structure the facts to handle the induction step. (And we should remember the well-known false proof that all horses are the same color.)

Here we are asked to prove that three divides 4ⁿ−1. We could formally call this P(n), a proposition that is true for n. The base step is to prove P(1); but trivially, 4¹−1 equals 3, which is clearly a multiple of 3.

How can we prove that P(n) implies P(n+1)? A popular approach is to find a way to express the successor case as a function of the assumed case. Here that might mean that we would try to look at 4ⁿ⁺¹−1 as a function of 4ⁿ−1. Often considerable ingenuity is required, but in this example we're lucky enough to need only a little algebra. For inspiration, and to be sure the conjecture is reasonable, let's compute some values.

n		1	2	3	4	5
4n−1		3	15	63	255	1023
k		1	5	21	85	341
Δk		4	16	64	256

Here k is the multiple of three, 4ⁿ−1 = 3k; and Δk is the difference of k between n and n+1.

Within the confines of our limited experiment the conjecture seems to be true. Better still, notice that Δk = 4ⁿ. That is, we now suspect that if 4ⁿ−1 = 3k, then 4ⁿ⁺¹−1 = 3k+3×4ⁿ. If we can show this to be true, then we have proved our induction step. So we compute

(4n+1−1) − (4n−1)	= 4n+1 − 4n
	= 4n(4 − 1)
	= 3×4n

This completes the induction step, and so our proof, which calls for a big Q.E.D.

In this particular example we discover that the step in n always produces a step in our computed value that happens to be a multiple of three. In other examples we will need to look for other patterns, related to the specific conjecture being proved.

Also in this example, we could obtain a proof without induction. Observe that we can factor

4n−1	= (22)n−1
	= 22n−1
	= (2n)2−1
	= (2n−1)(2n+1)

But 2ⁿ−1, 2ⁿ, and 2ⁿ+1 are three consecutive integers, one of which must be a multiple of three. Clearly 2ⁿ is not a multiple of three, so our factorization shows us that 4ⁿ−1 must be. We don't actually need this second proof, and it doesn't directly help with the goal of understanding induction; but it's nice to have independent confirmation in unfamiliar territory. --KSmrq^T 00:54, 8 May 2006 (UTC)

An alternative direct proof : recall from knowledge of geometric series that

$1+4+4^{2}+...+4^{n-1}={\frac {4^{n}-1}{4-1}}={\frac {4^{n}-1}{3}}.$

But left hand side is clearly an integer, so right hand side is also an integer i.e. 4ⁿ-1 is a multiple of 3. Gandalf61 09:31, 8 May 2006 (UTC)

---

To do an induction problem, there are basically four main steps to take. An induction problem will typically say something like, "Prove that for all integers n >= q, some property holds." where q is just some integer (usually 0 or 1). Then you do the following:

• First work out how the property is expressed mathematically. Sometimes the property will already be expressed like this, other times you'll have to work it out. In your example above, the property is 'being a multiple of three', and that's what we want to prove. This could be expressed as

For all n >= 1, 4ⁿ - 1 = 3k, where k is some integer.

• Next, prove the base case. This is the lowest possible n for which you want the property to hold. You just sub it in and check that it works. So, we could say

For n = 1, 4ⁿ - 1 = 4¹ - 1 = 4 - 1 = 3 = 3k where k = 1

• Now the tricky part. You want to prove that, IF the property holds for some given n, THEN it must hold for n + 1. That is, if it's true of some n, then it's also true of the next n. So, we assume that it's true of n, and we try to use that to prove that it's true of n + 1. Typically, you will set it up by using n + 1 in your formula, then working it around a bit. You will nearly always end up using the assumption that it's true of n along the way, so that's a good thing to aim for. So here's the working done for the one you asked:

4ⁿ⁺¹ - 1 = 4¹ * 4ⁿ - 4 + 3 = 4 ( 4ⁿ - 1 ) + 3 = 4 (3k) + 3 = 3 (4k + 1) = 3j where k is some integer and j = 4k + 1 is also an integer

• Now we just pull it all together. Since you proved, in the base case, that it's true for n = 1, and you proved, in the induction step, that IF it's true for some n then it's ALSO true for n + 1, then, by mathematical induction, it is therefore true for all n >= 1. So, we summarise:

Therefore, by mathematical induction, 4ⁿ - 1 is a multiple of 3 for all n >= 1.

And we're done! Try a few more examples. Hope that helps you out. Maelin 01:05, 8 May 2006 (UTC)

Difference between euclidean and vector space?

what's the diff?

See our article Euclidean space. It begins by defining something called "real coordinate space": that's a vector space. Then it has a section saying that Euclidean space is "more than just real coordinate space": that's the diff. We really should rewrite it to mention vector spaces explicitly for comparison. —Blotwell 15:36, 7 May 2006 (UTC)

thank you very much.

Don't forget to hit the "Alternative definition" section, though. If I recall correctly, some authors do not define a "Euclidean space" to have an origin; in that case, it isn't actually a vector space at all, but something more subtle. Melchoir 20:05, 7 May 2006 (UTC)

Cut-off phenomenon

With thanks to answer to my last question, What is cut-off phenomenon in Markov Chain?! ‍‍‍‍Armandeh 20:26, 7 May 2006 (UTC)

Let's assume a basic understanding of Markov chains. We have states, transitions between them, and a probability associated with each transition. We know that (under reasonable assumptions) after many transitions the probability of being in a particular state converges to a specific value that depends on the probabilities in the transition matrix. That's interesting, even useful, but perhaps not too surprising. The cutoff phenomenon is more subtle, more intriguing, and less broadly applicable. For some kinds of situations, the convergence happens first very slowly, then abruptly gets very good very quickly, then tails off. A classic example is also of practical interest: shuffling cards. A random riffle shuffle does a pretty good job of mixing cards if repeated enough times (but beware of a perfect shuffle!). How many repetitions should we use? We know that for an ideally shuffled deck the probability of each permutation is the same. The cutoff phenomenom appears because with five shuffles we are far from that goal, but with seven shuffles we are quite close.

Recommended reading is Persi Diaconis' paper, 'The cutoff phenomenon in finite Markov chains. (Persi is not just a top mathematician, he's also a former professional magician, a protege of the late, incomparable Dai Vernon.) --KSmrq^T 03:22, 8 May 2006 (UTC)

(Furthermore he is mentioned in the thanks section of Douglas R. Hofstadter's Gödel, Escher, Bach which implies that he's not only a great mathematician but a great man too. – b_jonas 17:25, 8 May 2006 (UTC))

A good numerical procedure

I have a bunch of highly nonlinear equations to solve. What's a good procedure to solve them? --HappyCamper 20:50, 7 May 2006 (UTC)

How many unknowns and how many equations? -- PCE 21:03, 7 May 2006 (UTC)

Have you considered the power of prayer? Black Carrot 21:00, 7 May 2006 (UTC)

Algebraic equations? Differential equations? Partial differential equations? It might make a difference...:)--Deville (Talk) 21:09, 7 May 2006 (UTC)

Numerical Recipes is always a great place to start. It won't have the most cutting-edge algorithms for specific, narrow, well-researched problems, but it does what it does very well. Melchoir 21:15, 7 May 2006 (UTC)

Hmm...how do I write this out in a succint way? Let's try this: bolded terms are vectors.

I have these two initial conditions a₀ = a, and b₀ = a. a is given and is real. Now, I have these recursive definitions a_n+1 = f(a_n,x)/(n+1) and b_n+1 = b_n + a_n+1. Now, I want to find x so that certain vector elements (which I choose) of b_infinity are precisely zero. The rest of them can be nonzero. Now, the function f is vector valued, and each element is a polynomial function involving the elements of a_n and x...hmm... --HappyCamper 21:45, 7 May 2006 (UTC)

Well, root finding is Chapter 9 of NR; that's all I can say. Good luck! Melchoir 01:59, 8 May 2006 (UTC)

Professional numerical analysts cringe at the mention of Numerical Recipes; it may seem like a friendly way to get started, but the algorithms and explanations leave much to be desired. In any case, this sounds like a potentially difficult and expensive problem. The b vector is just an accumulation of the a vectors, so if the f function is simple each component might be an independent series, but even that might be difficult as a function of the x components. For a typical zero-finding algorithm to work well, small perturbations of the input must produce smooth effects in the output. And here, the output for a single input, x, could be very expensive to compute. To get an idea of the possible complexity of polynomial iteration, consider the Mandelbrot set. --KSmrq^T 03:39, 8 May 2006 (UTC)

Eh, I admit that professionally, I've never had to rely on an NR algorithm, because there's always been a better one lying around (that was harder to find). But I don't see what's wrong with the explanations. Is there a better general-purpose textbook I ought to take a look at? Melchoir 03:43, 8 May 2006 (UTC)

Here's a page of criticism at Stanford, with a page of better software linked at the bottom. Michael Heath has compiled an extensive list of scientific software sources. Also recommended is the Scientific Computing FAQ; it includes a page of books. For numerical linear algebra, Golub and Van Loan's Matrix Computations, 3/e (ISBN 0801854148), is without peer, but intended for serious use. Also well-regarded are LLoyd Trefethen's Numerical Linear Algebra (ISBN 0898713617), and Nick Higham's Accuracy and Stability of Numerical Algorithms (ISBN 0898715210). Two dated-but-decent possibilities are Hamming's Numerical Methods for Scientists and Engineers (ISBN 0486652416) and Acton's Numerical Methods that Work (ISBN 0883854503). For quality knowledge and zero price, try Cleve Moler's Numerical Computing with MATLAB, available online! A previous early, influential, and insightful work is Forsythe, Malcolm, and Moler's Computer Methods for Mathematical Computations (ISBN 0131653326).

I'm sure I've omitted something or someone essential, but perhaps our resident NA experts like Jitse Niesen will fill the gaps. --KSmrq^T 05:48, 8 May 2006 (UTC)

Oh, wow. Thanks!! Melchoir 06:01, 8 May 2006 (UTC)

(de-identing) Do I hear my name? Hmm, I see that KSmrq has already made the most important points.

Root-finding is a problem with which I do not have much experience or expertise. What I do know is that it is a hard problem. The basic method is Newton's method, described there. Usually, it needs some kind of line search algorithm like back-tracking. If your vector x is long, computing and inverting the Jacobian is often a lot of effort, and you're better of using some kind of approximation.

Of the books mentioned by KSmrq, Golub & Van Loan, Trefethen and Higham have little to nothing about this problem. I seem to remember that Acton has a decent discussion. I don't remember what the other books have on root-finding. One book that I'd add is Dennis and Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (ISBN 0136272169), which is also fairly old though.

However, I expect that none of the algorithms described in these books can solve your problem right away. You need to analyse how the input x is mapped on the output b_infinity. This mapping is likely partial (for some values of x, the sequence b_n = Σ_k a_k does not converge) and discontinuous, which should certainly be taken into account. In other words, your problem is more difficult than the usual root-finding problem, which is already a hard problem.

Are you sure you need to solve it? I assume that you started from another problem and reduced it to this root-finding problem. Perhaps you need another approach. -- Jitse Niesen (talk) 07:30, 8 May 2006 (UTC)

On second thought, it might not be that hard if you have a good initial guess for the solution. -- Jitse Niesen (talk) 01:27, 9 May 2006 (UTC)

Alright, I decided to drastically simplify the problem and made a whole bunch of assumptions...I end up with a matrix equation that looks like this: exp(7 C) = (A+B)^-1(B-A). How do I find what C is? It should be relatively simple I suppose? --HappyCamper 18:38, 12 May 2006 (UTC)

:D

I was wandering around irreducible complexity, and stumbled into Berry paradox. I'd like to read an excerpt. *ahem*

Note that some Berry type expressions present only minor problems of interpretation:

The smallest positive integer not nameable in under two (non-hyphenated) words.

under reasonable definitions of English denotes 21, since "twenty one" (or "twenty-one") is two words and any indirect definition of the number (such as "the number of dots on a six-sided die", or indeed "the smallest positive integer not nameable in under two words") is necessarily two or more words long.

That's a blackjack, isn't it? Black Carrot 22:43, 7 May 2006 (UTC)

No, blackjack is the name of a game, not an integer. --KSmrq^T 00:57, 8 May 2006 (UTC)

Au contraire. "A two-card hand of 21 (an ace plus a ten-value card) is called a "blackjack" or a "natural", and is an automatic winner." The score of twenty-one is, in all technical accuracy, called blackjack. Black Carrot 03:18, 8 May 2006 (UTC)

Even given your quotation, I would still argue that "blackjack" refers to the combination of cards adding up to twenty-one, not twenty-one itself. As evidence, note that three cards adding up to 21--certainly possible, and not even uncommon, in the game of blackjack--is not called "blackjack." Chuck 18:26, 8 May 2006 (UTC)

Innumeracy

This probably isn't the proper place to write about it, but the Innumeracy (book) article needs some help so that it can be expanded. Has anybody read it who wants to contribute? JianLi 23:33, 7 May 2006 (UTC)

Wouldn't it be more practical to merge the book into Innumeracy, now a pretty dumb page, and make it into a true article? My arguments for this: (1) Innumeracy deserves an article. (2) Because of WP:NOR, we need a good source. (3) What better source than the Paulos book? (I think he may also have coined the term.) (4) The book deserves a reasonable section in a full-fledged Innumeracy article. (5) Then any expansion of the Innumeracy (book) article is bound to result in overlap if not outright duplication, an undesirable situation. (6) Leaving Innumeracy (book) as a forever stub is also not desirable. (7) Solution: make Innumeracy (book) redirect to Innumeracy, which also treats the book. --Lambiam^Talk 17:37, 8 May 2006 (UTC)

No, but I've mentioned it here on the RD once: see Wikipedia:Reference desk archive/Mathematics/December 2005#Why is mathematics so difficult? – b_jonas 17:48, 8 May 2006 (UTC)

Ti-83

Do any of you use any java applets online in place of a graphing calculator, like the Ti-83? I found a few using Google, but none of them are very well-designed. JianLi 00:49, 8 May 2006 (UTC)

As far as java applet's none that I know compare to the TI-83. For graphing stuff, try Winplot. It's a great program which can also graph implicit functions, such as x² + y² = 25. For other stuff (calculator functions), there are some great programs, but most cost money. If you have the money to spend, Mathematica is a great program, but is quite expensive. —Mets501^talk 01:22, 8 May 2006 (UTC)

Also try this page: List of information graphics software —Mets501^talk 01:23, 8 May 2006 (UTC)

thanks!JianLi 02:54, 8 May 2006 (UTC)

May 8

computer fundamentals

What happens if keyboard is not connected to the system while booting.

Nothing. If it's a USB keyboard, you can just plug it in while the computer is on. If it's a PS2 keyboard, you will probably have to restart the computer for it to work. —Mets501^talk 01:19, 8 May 2006 (UTC)

I have a vague recollection that some PCs won't go past POST if the mouse and keyboard are not detected, though this "feature" can probably be disabled in the BIOS. -lethe ^{talk +} 05:41, 8 May 2006 (UTC)

I don't think that any compture cares about the mouse but some oddly refuse to boot when they have no keyboard (it can hopefully be disabled on all of those computers). Jeltz talk 12:22, 8 May 2006 (UTC)

Often the BIOS will have an option for what to do if there's a problem with the keyboard (or the graphics card) at bootup. Alternatively, there's the classic "Keyboard error. Press any key to continue." message :-) -- AJR | Talk 18:47, 8 May 2006 (UTC)

golden ratio

how can you get a decimal approximation for the golden ratio correct to a certian decimal place? such as an even number B/c i don't understand how to do it.thank

The value is exactly (1+√5)/2, so all you need is a package that can work with arbitrary-precision arithmetic. Here are a few digits to test against.

Φ ≈ 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072…

Since ancient times (in computer terms), Unix systems have supported such calculations with dc, the reverse-polish desk calculator. --KSmrq^T 04:27, 8 May 2006 (UTC)

You can also use continued fractions to calculate golden ratio with a very high precision (φ = [1; 1, 1, 1, 1, 1, ...])  Grue  11:20, 8 May 2006 (UTC)

Grue's suggestion is equivalent to approximating Φ as the ratio of succesive Fibonacci numbers. So for example 10946 / 6765 (which can be calculated with long division) is a good approximation (7 digits). -- Meni Rosenfeld (talk) 13:47, 8 May 2006 (UTC)

As it happens, Φ has the slowest converging continued fraction. A typical Newton iteration for the square root of 5, or for a zero of x²−x−1, has quadratic convergence, doubling the number of significant digits with each iteration. For example, suppose we begin with x₀ = 2. Newton iterations for √5 average x_k with 5/x_k:

$x_{1}={\frac {x_{0}+{\frac {5}{x_{0}}}}{2}}={\frac {2+{\frac {5}{2}}}{2}}={\frac {9}{4}}$

$x_{2}={\frac {161}{72}}$

$x_{3}={\frac {51841}{23184}}$

$x_{4}={\frac {5374978561}{2403763488}}$

Already x₄ = 2.2360679774997896964… is correct for √5 through the decimal places given, and x₅ would double that. The article on the golden ratio mentions the Newton iteration

$x_{k+1}={\frac {x_{k}^{2}+2x_{k}}{x_{k}^{2}+1}}$

Again starting with x₀ = 2, we find

$x_{4}={\frac {4807526976}{2971215073}},$

which is correct for Φ through 1.618033988749894848…, and double that at the next step.

Contrast this with the fact that it took twenty Fibonacci numbers to get only seven digits. Continued fraction convergents are alternately high and low, and the difference between successive convergents is the reciprocal of the product of their denominators, here consecutive Fibonacci numbers. Up to rounding, the n-th Fibonacci number is Φⁿ/√5. So, for comparison, we want to find n such that 5/(Φ²ⁿ⁻¹) is less than 10²⁰, approximately n = 50. And, indeed,

${\frac {F(50)}{F(49)}}={\frac {12586269025}{7778742049}}$

is correct to the same number of decimal places as our fourth Newton iteration. Numerical analysts need to know mathematics; it would be wise for mathematicians to appreciate numerical analysis. --KSmrq^T 04:06, 9 May 2006 (UTC)

Some of us do. I recently had cause to mention Hurwitz's theorem on the hierarchy of quadratic irrationality, in trying to contrast real theorems involving cycles with the vague numerological mysticism of Edward R. Dewey, yuk. (Could use some help watching this page and reverting POV-pushing edits, BTW.) ---CH 02:00, 11 May 2006 (UTC)

Shortest sourcecode for PI

Is this the world shortest source code for a program which calculates and outputs the infinite digits of Pi ? Can anyone come up with a shorter source code?

http://mail.python.org/pipermail/python-list/2003-June/168485.html

#!/usr/bin/python
import sys

k, a, b, a1, b1 = 2, 4, 1, 12, 4
while 1:
    p, q, k = k*k, 2*k+1, k+1
    a, b, a1, b1 = a1, b1, p*a+q*a1, p*b+q*b1
    d, d1 = a/b, a1/b1
    while d == d1:
        sys.stdout.write('%d' % d)
        a, a1 = 10*(a%b), 10*(a1%b1)
        d, d1 = a/b, a1/b1

With the output

$ python short_pi.py 
31415926535897932384626433832795028841971693993751058209749445923
07816406286208998628034825342117067982148086513282306647093844609
55058223172535940812848111745028410270193852110555964462294895493

Ohanian 03:40, 8 May 2006 (UTC)

Congratulations, you win. Not. As a trivial counterexample, here's a Mathematica program:

N[Pi,2000]

In other words, such a competition makes no sense without clear and careful groundrules! --KSmrq^T 03:49, 8 May 2006 (UTC)

How does N[Pi,2000] constitute a computer algorithm? Ohanian 06:33, 8 May 2006 (UTC)

If you only count programs written in low level languages, you should probably prefer assembly to python. If you don't care whether it's low level or not, then Mathematica can do it in one line. -lethe ^{talk +} 07:14, 8 May 2006 (UTC)

The rules?

1. It has to be a computer algorithm.

2. It has to be in a source code format in a computer language available to the public.

3. Size is the source code file measure in bytes.

Ohanian 06:33, 8 May 2006 (UTC)

The Mathematica example satisfies all those rules. It's certainly an algorithm since it contains well-defined steps that can be executed in finite time by a computer, and Mathematica is also available to the public (albeit for a fee; nevertheless, similar programs can be constructed for any free computer algebra system with multiprecision arithmetic). If you think relying on Mathematica's built-in algorithms is cheating, be reminded that your program relies on Python's built-in algorithms for multiprecision arithmetic. Without that, the program would be much longer. Anyway, here are some short C programs for calculating various constants. Fredrik Johansson 07:00, 8 May 2006 (UTC)

Oh come on, your just being pedantic for the sake of it. Just add an extra rule

4. Algorithm musht not depend on other libraries or functions with the exception of system calls.

More interesting questions are is the algorithm correct, is the algorithm fast? --Salix alba (talk) 09:19, 8 May 2006 (UTC)

That doesn't make anything clearer. Is multiplication of two large numbers a system call or a library call? What about binary-to-decimal conversion? The truth is that both Mathematica and Python are abstract computer systems, N[] and Pi being no less "system calls" in Mathematica than a*b being a system call in Python. The missing rule is a clear specification of the language to be used. Fredrik Johansson 10:08, 8 May 2006 (UTC)

The same also applies to a*b in C which isn't a "system call" either. It's neither a "system call" nor a call to a library. In a way C is also an abstract computer system. Yeah, the rules need to be mormulated more clearly if we want to be pedantic. It's quite obvious that the Mathematica way is cheating, but it isn't that easy to formulate a rule which clearly forbids it. Jeltz talk 12:16, 8 May 2006 (UTC)

Well, is

N[16ArcTan[1/5] - 4ArcTan[1/239], 500]

also cheating? Or perhaps

FindRoot[Sin[x] == 0, {x, 3}, AccuracyGoal -> 500, WorkingPrecision -> 500]

? I think the most consistent rule would be to allow only operations that are O(1) in time (which is essentially what C provides). That rules out the Python program. Alternatively, you could allow basic arithmetic operations (+-*/%) on multiprecision integers and floats (but at that point, things have already gotten arbitrary). Fredrik Johansson 12:42, 8 May 2006 (UTC)

Operations in C are not O(1) in time either. I think all of them are O(1) in processor instructions though. Processor instructions have time complexity too. I can't really think of any rules that aren't arbitary. Jeltz talk 16:52, 8 May 2006 (UTC)

Here is a short program for calculating π with N digits in the simplest possible way; that is, by summing a series straightforwardly:

N=1000;a,b,t,k,s=10**N,1,1,1,0
while t:
 t=a//b;a*=k;b*=k*2+1;k+=1;s+=t
print 2*s

Because of rounding error, the last few digits in the output will be wrong (this can be avoided by throwing in a few guard digits). Fredrik Johansson 10:26, 8 May 2006 (UTC)

See also [4]; and the pi calculation section on IOCCC FAQ – b_jonas 18:19, 8 May 2006 (UTC)

This C snippet, which uses Wallis's formula, will do the job, even though the convergence is shocking: double pi; int i; for(i=1, pi=2.0; i < MAX; pi *= 4*i*i/(4*i*i-1.0), i++ ); Dysprosia 22:44, 8 May 2006 (UTC)

Saving Microsoft Sam

Is it possible when you type something into the Microsoft Sam Voice, you can save it as a file. I want to type in good and bad and save it as an MP3. If not, where could I find a recording of someone saying the bad and the word good. Thanks. schyler 23:57, 8 May 2006 (UTC)

Try wiktionary:good and wiktionary:bad? – b_jonas 15:54, 9 May 2006 (UTC)

May 9

Breaking A Code

• 5
• The 700's
• Riley+Mcmillian=Hill
• 2damsels+1sheep only
• Count the toes+hooves
• King tut the young
• look at the wall subtract 1x100+59
• 69
• Ghost Grant —The preceding unsigned comment was added by 65.183.106.10 (talk • contribs) 02:15, 9 May 2006 (UTC)

Bingo! Melchoir 03:01, 9 May 2006 (UTC)

I reformatted your text to have some resemblance to what I think you meant. Now what the hell are you talking about? Black Carrot 23:32, 9 May 2006 (UTC)

Bloodhound Problem

A prison guard and his bloodhound are chasing an escaped prisoner. The prisoner has a 5 mile head start but the guard is walking 1 mph faster than the prisoner. The bloodhound is trained to run to the prisoner, run back to the guard, and then continue running back and forth between them. If the bloodhound runs 10 mph, how far does the bloodhound run before the guard finally catches up to the prisoner?

How would you go around solving this problem? I want an idea on how to solve this problem and others like it, algebraically preferably. Thanks.

--Proficient 05:33, 9 May 2006 (UTC)

distance travelled is speed×time, keep that in mind, dont get distracted by the implied infinite series. —The preceding unsigned comment was added by Frencheigh (talk • contribs) .

Yes, thank you, but next time sign your post. Proficient, you can either do it by writing an infinite series and then summing it, or you can do it the easy way. The dog always runs at the same speed without stopping, right? —Keenan Pepper 05:49, 9 May 2006 (UTC)

The original story, with the cute John von Neumann twist, involves a fly and two trains. It's a popular old chestnut used to tease new calculus students. --KSmrq^T 08:26, 9 May 2006 (UTC)

There are several ways of doing this, but here is what I think of as the algebraic way.

Variables

d_GP: the distance difference to be overcome by Guard to catch up with Prisoner

v_GP: the speed difference between Guard and Prisoner

t_GP: the time it takes Guard to catch up with Prisoner

d_B:   the distance covered by Bloodhound

v_B:   the speed of Bloodhound

t_B:   the time Bloodhound runs

Equations

d_GP = v_GP×t_GP (by definition of "speed")

d_B = v_B×t_B (same)

d_GP = 5×mile (given)

v_GP = 1×mile/hour (given)

v_B = 10×mile/hour (given)

t_B = t_GP (given)

Solve for

d_B

There are as many equations as there are variables, which looks promising. Let us take miles and hours as units (which just means we leave them out) and start simplifying by plugging in the known values, and using just one variable t for t_GP and t_B. The equations become then:

Simplified equations

5 = 1×t

d_B = 10×t

From here on I think you can do the rest by yourself. --Lambiam^Talk 08:29, 9 May 2006 (UTC)

Well, the guard will get to the prisoner in 5 hours, right? So the entire run will be 5 hours long. Hence the bloodhound will run 50 miles. Cthulhu.mythos 10:00, 9 May 2006 (UTC)

Thank you very much for your oustanding help.--Proficient 05:54, 10 May 2006 (UTC)

'

sums with two summation conditions in latex

Any suggestions on how to improve my poor attempt on Siegel-Walfisz theorem? Mon4 14:48, 9 May 2006 (UTC)

How About

$\sum _{n\leq x \atop n\equiv a\mod q}\Lambda (n)$

? -- Meni Rosenfeld (talk) 15:07, 9 May 2006 (UTC)

Beautiful! Mon4 15:12, 9 May 2006 (UTC)

Recording off computer

If there is a sound played on a website, on a flash webpage, is there any way in which I can get that sound onto an audio file on my desktop? --Chachu207 ::: Talk to me 16:56, 9 May 2006 (UTC)

You may get a hold of software used to decode .swf (normal flash files), which basically will turns the file into its raw materials, like mp3-files, graphics, etc etc. You can ALSO use a mic and standard recording system (an extremely basic version can be found on the Start menu somewhere in most Windows computers), but the result isn't likely to be fantastic.

If you want to actually MIX the flash-sound with an already existing audio file, I have zero idea. Aforementioned Windows standard media-tool might be able to do that, but I wouldn't be able to tell. There are some easy and reliable, perhaps some free sound-mixers out there to be found, I might be able to find one for you if you reply that you want one. 213.161.189.107 21:01, 9 May 2006 (UTC) Henning

You need recording software, try google. Also, sourceforge probably has something of this sort.--Frenchman113 on wheels! 21:03, 9 May 2006 (UTC)

I hear from a friend that .swf-files are most often encrypted, and that opening them with even a flash-editor shouldn't be possible unless there's no encryption. In other words, the Frenchman might be on to something. 213.161.189.107 21:10, 9 May 2006 (UTC)Henning

If you're not concerned about audio quality, the "audio-hole" will work perfectly: obtain a 3.5mm-3.5mm male-male audio cable, plug one end into the speaker socket and the other into the microphone socket. Start an audio recording program in the background, play the SWF -- presto, the audio recording program will pick up the audio from the SWF. Dysprosia 22:36, 9 May 2006 (UTC)

A quick Google search turn up some shareware called Audio Record Expert in the first several links. I'm sure there are others. --LarryMac 14:55, 10 May 2006 (UTC)

Try software that captures (any) sound from Audio Out and saves it into a file. I use TotalRecorder for this purpose, and it works pretty well. --Leapfrog314 04:01, 12 May 2006 (UTC)

caculate displacement

I am a drilling rig supervisor, and am trying to figure out a simple formula that will tell me how many feet from center the bit will be, based on depth and angle of inclination. As we drill a well, we continually monitor how many degrees from vertical the well has moved. So, for example, consider a well that is 10,000' deep, and the angle of inclination is 5 degrees. If you pictured this as a right triangle, the tall vertical leg is 10,000, the angle at the bottom is 90 degrees, and the angle at the top is 5 degrees. I need to know what the length of the short leg is. How do I do this? Les F.

I think you mean that the "angled" leg (how far the bit has penetrated) is 10,000'. Is that correct? hydnjo talk 17:19, 9 May 2006 (UTC)

If the 10,000 feet is the angled length, you need to take the sine of 5 degrees, which is 0.0872. If 10,000 feet is really the vertical depth, take the tangent of 5 degrees to get 0.0875. In either case, multiply the value by 10,000 feet to get the answer, which is either 872 feet or 875 feet. As you can see, for small angles there isn't much difference. Also note that these calcs assume the well is absolutely straight. If the shaft bends, these results are not accurate. StuRat 17:50, 9 May 2006 (UTC)

This sounds like a possible question for a basic trigonometry class, except for a few real-world details. Doesn't it seem peculiar that a company would employ someone to supervising a massive and expensive job like drilling to 10,000 feet (3 km) without adequate training to answer a question like this? --KSmrq^T 21:45, 9 May 2006 (UTC)

Since the oil field is probably quite large, they likely just say it's "close enough" and don't worry about how far off it is. StuRat 20:50, 10 May 2006 (UTC)

First of all, thank you for your insight and suggestions. To Hydnjo..There are two inputs that I have available to me at any time, TVD or true vertical depth, and MD or measured depth. The MD is the convoluted hypotenuse of this triangle, and TVD is the vertical depth or the long leg of the triangle. To KSmrq...I am the person resonsible at the drilling site that makes sure that all the parameters related to drilling are adhered to. On this particular well, I was given a radius from the beginning point of the well that I had to stay within. I was getting the angle of inclination as we drilled, and I was trying to calculate how many degrees from center I could be and still be within the target. It's true, I'm not a petroleum engineer, and it's also true that I should be able to figure this out on my own, but I thought a quick, accurate way to do this would be to post my question her. And although I am not a petroleum engineer, I challenge any engineer to come to the rig, live there 24 hours a day, and make the decisions I make related to drilling fluids systems, pipe stretch, wireline logging decisions, geological changes, cementing procedures, safety management, logistics, and managing 20 roughnecks, a toolpusher, a mud engineer, directional drilling personell, while trying to keep peace with a group of irate landowners, and trying to keep the home office up to date on daily and cumulative costs, and make suggestions related to drill bit selection, fishing tool company selection, access to water, disposal of oilfield waste, road maintenance, etc... And to StuRat...My target on these wells is a 25' radius. Thats after drilling directionally 2,000', and then dropping the bit another 13,000'. Thats right, we drill vertically 300-400', then we kick the well on a predetermined azimuth at a predetermined angle build to 20 degrees, then we drill the predetermined offset, and then steer the well back vertical, then we stay within a 25' radius to 15,000'. Pretty simple, eh? And close enough doesn't cut it. If we don't achieve this objective, we plug the well back, and redrill it until we get it right. If I don't keep this well within the target, I'm out of a job. I've been doing this for 10 years now, so I guess I'm doing something right.

Hey, let me apologize for the unintentional abrasiveness above. It's pretty rare we get genuine posts like this. As for the answer, a diagram would be best...

 |\
 |b\
 |  \ B
A|   \
 |    \
 |a   c\
 -------
    C

Note how I've labeled the triangle. So, from what you're saying, a = 90 degrees, and b is 5 degrees, A is 10000 feet, and you want to find out what C is?

Let's assume you only have A and b. The formula for C is given by C = A tan b. You have to be careful if you are using a calculator. Make sure you know you are working in "degrees" mode and not "radians". --HappyCamper 04:09, 11 May 2006 (UTC)

Great response, Les. Now go back and notice that I worded my "peculiar" remark very carefully; if you had other skills that made you worth hiring, isn't it foolish for the company not to train you with the geometry skills to help you succeed? How expensive is a little education? How expensive is redrilling? The conclusion is simple arithmetic. Tell your boss Wikipedia mathematicians say so!

Meanwhile, let's consider options. First, since drilling holes in the ground is big business, commercial software may be worth investigating, to see if the price and features are right. For example, a quick web search turned up Target and DrillStar.

If we're going to apply mathematical ideas, it's important for us to approach this as a real-world engineering task, not some simplified homework exercise. For homework we can assume an ideal triangle. In reality, we should be cautious about assuming that the drill path is a straight line, or that it stays in a plane with an ideal vertical. Angle measurements will always be approximate, and if we accumulate them repeatedly the errors can add up. What we would like to do is to track the 3D path as a series of angles and distance increments. And for that, angle from the vertical is not enough; we should be sure of the compass direction as well.

Another thought is that since integrating the drill path may be subject to various errors, it would be nice to have a way to get a true 3D position reading, at least occasionally, perhaps by acoustic triangulation if that's feasible both economically and with different kinds of intervening rock.

The more you can tell us about what information you have available (including its reliability), the more we can help.

Just remember, it's not our job on the line; and while some of us are professional graduates, others are pre-college.

Which reminds me to remind you: please sign your post by using four tildes, "~~~~". It will automagically become a real Wikipedia signature, with user ID and time stamp. Thanks. --KSmrq^T 06:35, 11 May 2006 (UTC)

Hey Les, thanks for getting back to us, not many folks do that. But now that my interest has been piqued by your response, just how do you determine the TVD. The MD is pretty obvious and I can see now that its convoluted length could indeed confound any simple approach to calculating the "off-center" distance. There might even be enough interest here to start the Mathematics of deep well drilling (or some such) article! The reason for my asking is that if the TVD is derived by knowing the MD and the angle from vertical then not much has been gained. If on the other hand, the TVD is determined independently well then it becomes a useful piece of data in determining other values particularly if you have knowledge of either the MD or the TVD for each drilled azimuthal segment (and of course the azimuth of those segments). Hell, you may even invite one of us out to the site to observe such a fascinating operation. I, for one, would love to know how you "kick" the drill off to another aximuthal direction, it boggles my mind, and all the more reason for an article about all of this --hydnjo talk 20:32, 11 May 2006 (UTC)

May 10

Cross Sections of a Cube

Why is it that a cube cannot be sliced by a plane in such a way that the cross section formed will be a regular pentagon?

You can probably answer the question by considering all possible cases that a plane can intersect a cube, and describing the cross-section. By exhaustion of cases, you won't obtain the desired cross-section. Dysprosia 04:43, 10 May 2006 (UTC)

I remember this problem from a Mu Alpha Theta competition. Proving it rigorously would be a real pain. We ended up making some reasonable assumptions (bilateral symmetry was one) and then doing an optimization to convince ourselves it couldn't be done. —Keenan Pepper 04:50, 10 May 2006 (UTC)

Aren't two of the sides of a resulting pentagon necessarily parallel? Because there's only 3 non-parallel sides of a cube...  Grue  07:21, 10 May 2006 (UTC)

Ah, thank you! I knew there had to be a slick answer. Melchoir 08:12, 10 May 2006 (UTC)

In that vein: is there a standard proof or theorem that states something along the lines of: "when two parallel planes are intersected by a third plane, then the points of intersection on the third plane are parallel lines"? or does one have to prove it every time? --Seejyb 21:12, 10 May 2006 (UTC)

It is a very simple lemma. If the lines were not parallel, they would intersect. The point of intersection should lie on both parallel planes, which is impossible.  Grue  21:23, 10 May 2006 (UTC)

Damn. I wish we had thought of that, instead of doing all that hairy calculus (which probably didn't even prove it anyway). —Keenan Pepper 04:07, 12 May 2006 (UTC)

series sum

I'm certain this is incredibly simple but I've had a complete mind blank over the past couple of days. Is there a general expression for 1^2 + 2^2 + 3^2 + ... + n^2 ???? (sorry i don't know how to use the maths markup) 129.78.64.105 04:16, 10 May 2006 (UTC)

From Summation:

• $\sum _{i=0}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}$

Melchoir 04:32, 10 May 2006 (UTC)

Argh the title of that article was too obvious for me, haha. THANKS! 129.78.64.105 04:44, 10 May 2006 (UTC)

OEIS names the sequence Square pyramidal numbers. --DLL 20:04, 10 May 2006 (UTC)

i think that you are going to fail trig. and then one day, when your at checkers someone is going to ask you this question. and your not going to know it —Preceding unsigned comment added by 65.9.35.40 (talk • contribs)

paper on data structure

hello all techis! i need to work on some good and new idea on sorting algorithms, plz help...

Try Category:Sort algorithms. Melchoir 10:06, 10 May 2006 (UTC)

Transform a circular image into a square image?

This seems like a basic question, but it's proving to be very hard. Say that you have an rectangular image of a circle (like a picture of the moon), and you wanted to transform that so that the moon stretched to fill the entire image. How would one transform that image mathematically?

You select your favourite map projection according to the properties that you want your resulting image to have, and then you apply that projection to get a rectangular image that's probably less accurate than the circular one you began with. -- AJR | Talk 11:55, 10 May 2006 (UTC)

For the problem as stated, one might apply the Riemann mapping theorem. But be warned; as the article says:

"Even relatively simple Riemann mappings, say a map from the interior of a circle to the interior of a square, have no explicit formula using only elementary functions."

Since the question did not require a conformal map, there are simpler options. From the center of a circle to any point on its boundary we have a line segment of a certain length; to map (invertibly) from rectangle to circle, scale all points on each line segment by the inverse of that length.

A typical map projection is a map from (a part of) the sphere to the plane, which is a different idea. --KSmrq^T 12:34, 10 May 2006 (UTC)

Finding a mapping only solves half the problem. You will also need a resampling algorithm, Bilinear interpolation or Bicubic interpolation may help. EricR 14:14, 10 May 2006 (UTC)

Various questions

Okay, I know this sounds like I'm asking to be spoonfed, but I'm not. Basically I'm self-studying some advanced maths which isn't part of my school curriculum, so getting help is kind of difficult. I'm not asking for the answers; I just need an explanation of the principles which will help me obtain the answers, because I have absolutely no idea how to do these questions. I've never taken a formal course which taught any of these principles, so hopefully you guys can bear with me despite the variety of questions I'm about to ask.

Trigonometry

What is $\sin[\operatorname {arccot}(-3/4)]$? It should be simple, but I've found there are two apparently correct ways to get two different answers. The first gives a value of -0.8. I determine this by first seeing that the cotangent of the unknown is equal to -3/4. In other words, $1/\tan(x)=-3/4$. Using my calculator to find arctangent -4/3 and then finding the sine of this value, I obtain -0.8. My textbook, however, approaches the problem by determining the arccotangent of -3/4 is equal to $\pi /2-\arctan(-3/4)$. The sine of this value is then found to be 0.8. Which approach is correct? Perhaps I've a terrible memory, but I can't even recall being taught the principle behind the textbook's solution.

Given a number a, there are infinitely many numbers x such that cot(x) = a. The function arccot returns just one of them, and this is usually taken to be the one value in the range 0 < x < π. The approach you used generates, indeed, a number whose cotangent is (-3/4), just not the one that is denoted by arccot (-3/4). So the approach in the book is more accurate (though your solution can also be considered correct, depending on the context). -- Meni Rosenfeld (talk) 19:15, 10 May 2006 (UTC)

Periodicity of sine and cosine

Periodicity of tangent

The cotangent of an angle is the ratio of cosine to sine, and ranges from −∞ to +∞. Like sine and cosine (shown left), it is periodic; but like the tangent function (shown right, and noting cot(ϑ) = tan(^π⁄₂−ϑ)), its period is π, not 2π. This implies we have two choices for the value of sin(arccot(−³⁄₄)), one positive and one negative. It may be of interest to note that standard mathematical libraries for programming languages provide a two-argument arctangent, to allow choice of the desired quadrant. --KSmrq^T 23:38, 10 May 2006 (UTC)

3-D geometry

The object $3x+4y+5z=18$ lying in the first octant forms a three-dimensional object with the co-ordinate planes. How do I find its volume? The answer given in the text suggests it's a pyramid, but I can't seem to imagine how this can be inferred. I'm not exactly clear on how I would proceed from this to finding the volume either.

Find the intersection points of the plane with the axes. The resulting object is a pyramid, which you can think of as having for a base a triangle in the x-y plane, and height along the z axis. Use the formula $V={\frac {1}{3}}Ah$. -- Meni Rosenfeld (talk) 20:11, 10 May 2006 (UTC)

The geometric object defined by the zeros of the first degree polynomial

3x + 4y + 5z − 18

is a plane. (In general, a single polynomial equation in 3D defines a surface, and a linear equation defines a plane.) It cuts the x axis where y and z are zero, at x = 6; similarly for the y axis (at y = ⁹⁄₂) and the z axis (at z = ¹⁸⁄₅). This volume is an irregular tetrahedron. It is a pyramid in a rather general sense, not a regular pyramid like the Great Pyramid of Giza. --KSmrq^T 00:04, 11 May 2006 (UTC)

2-D geometry

$9x2-axy+4y^{2}=16$ represents a pair of straight lines for certain values of a. Why is this so? I have absolutely no knowledge of this area, and my book isn't helping with its terse "explanations".

If a is +12 or -12 (the values for which the discriminant is 0), the left hand side becomes:

$9x^{2}\pm 12xy+4y^{2}=(3x\pm 2y)^{2}$

And the equation becomes:

$(3x\pm 2y)^{2}=16$

So $3x\pm 2y=4$ or $3x\pm 2y=-4$, which are equation of straight lines. -- Meni Rosenfeld (talk) 19:20, 10 May 2006 (UTC)

Algebraic (or merely analytic) geometry is involved. In the plane, the zeros of a single polynomial, here a member of the family

9x² − axy + 4y² − 16,

typically describe a "curve", for a suitable understanding of what that word means. If we add the degrees of the variables in a term, we get the "total degree" of the term; the maximum over all the terms is the (total) degree of the polynomial, here 2. Ordinarily a line would intersect a degree-2 curve in two distinct points, per (a sloppy version of) Bézout's theorem. However, some curves are degenerate, in a precise technical sense. For a conic, which is what we have here, we could have a degenerate circle of radius zero,

x² + y²,

or a degenerate hyperbola forming a pair of intersecting lines, such as

xy,

or a degenerate parabola forming two parallel lines (not necessarily distinct), such as

x² − 1.

In geometric terms, a degenerate curve has a point which is "double"; an independent line through that point lacks the other intersection(s) we ordinarily expect. (For parallel lines, that point is at infinity.) In algebraic terms, we can detect degeneracy by asking if the polynomial for the curve has a double root. For a quadratic polynomial, this is easier because we can convert to bilinear form, leading to a matrix form. The matrix for the example is

${\begin{bmatrix}16&0&0\\0&9&-a/2\\0&-a/2&4\end{bmatrix}},$

with determinant 576−4a². We have degeneracy when this expression is zero, at a = ±12. --KSmrq^T 04:02, 11 May 2006 (UTC)

Volume

A circle with a radius of 10 inches is divided into two sectors; one is 240 degrees, the other is 120. Both are bent into cones. How do I find the ratio the smaller cone's volume to the larger cone's? From the answer given, it seems that there's some relation between the slant length of a cone with its height and base, but there's no real explanation or even statement of what principles were applied to derive the formula giving the answer.

If you check Cone (geometry), you will find the formula for the slant height, $s={\sqrt {r^{2}+h^{2}}}$. Both of your cones will have a slant height of 10 inches, and you can derive the radius of the base of each from the original problem. You can then solve for the height of each cone to determine their volumes. --LarryMac 17:10, 10 May 2006 (UTC)

A little note. You can find that equation for the slant height if you draw an imaginary right-angled triangle made up of the radius, height and slope of the cone, then apply pythagoras' theorem. That means you don't have to remember the formula, just remember to draw triangles where possible. Skittle 21:28, 10 May 2006 (UTC)

Pardon my ignorance, but how do I derive r? Johnleemk | Talk 17:24, 10 May 2006 (UTC)

You're not ignorant, you're asking questions. I am hoping to give you nudges in the right direction instead of just spelling out the answer. Anyway ... Compute the circumference of the circle you start with. Note that by dividing the circle as specified, the sectors contain arcs whose lengths are 1/3 and 2/3 the total circumference. When you bend your sectors into cones, those arcs become the circles at the base of each cone; thus the lenght of the arc is now the circumference of the base. Solve for r. Ask again if I'm being more obtuse than a 240 degree angle. --LarryMac 18:41, 10 May 2006 (UTC)

Properly speaking, ignorance is not knowing. The correction is to ask questions, which we encourage. --KSmrq^T 04:19, 11 May 2006 (UTC)

This problem has a hidden simplicity. When a circle sector is converted to a cone, the circle radius becomes the cone slant height, the circle radius times the sector radian measure becomes the base circumference, and the cone radius is proportional to the cone base circumference. Both cones will have the same slant height, and one will have twice the radius of the other. --KSmrq^T 04:19, 11 May 2006 (UTC)

1-> s = (r^2 + h^2)^0.5

                        1
#1: s = ((r^2) + (h^2))^-
                        2

1-> h

                         1
#1: h =  ((s^2) - (r^2))^-
                         2

1-> v = pi * r^2 * h/3

        pi*(r^2)*h
#2: v = ----------
            3

2-> eliminate h
Solving equation #1 for (h)...

                                  1
        pi*(r^2)* ((s^2) - (r^2))^-
                                  2
#2: v = ----------------------------------
                        3

Note: r = ( r_original * Angle_in_radians ) / ( 2 * pi )

Miscellaneous

Given a right regular hexagonal prism, how can I calculate the number of parallel edges it has? The answer given implies there's a way to mathematically find this through the addition of two different combinations.

Isn't it just 1 set of 6 parallel edges, and 3 sets of 4 parallel edges each? Am I missing something? -- Meni Rosenfeld (talk) 20:16, 10 May 2006 (UTC)

I have trouble visualising it, so maybe that's why I found the problem somewhat difficult to solve. Johnleemk | Talk 10:56, 11 May 2006 (UTC)

Thanks for all the help (again), guys. Johnleemk | Talk 16:44, 10 May 2006 (UTC)

Your questions give me the impression that your textbook may not be very good, at least not for self study. You should consider trying to get books which actually teach the subjects and don't just give exercises. -- Meni Rosenfeld (talk) 20:15, 10 May 2006 (UTC)

I actually rely on it only for questions -- the problem is that my other texts don't focus on some applications of trigonometry which appear in this text. Likewise, they don't cover anything on 3 dimensional geometry, which is why I needed help from the refdesk. Again, thanks so much for the help everyone. :D Johnleemk | Talk 10:56, 11 May 2006 (UTC)

They might be asking about the number of pairs of parallel edges, which is 33, according to a quick calc in my head. StuRat 18:46, 11 May 2006 (UTC)

Ah, I see. So the "addition of 2 combinations" he mentioned referred to ${6 \choose 2}+3{4 \choose 2}$. -- Meni Rosenfeld (talk) 19:29, 11 May 2006 (UTC)

This is a nice set of mathematical questions. I hereby nominate this as the question of the week (or day or whatever template we have available). – b_jonas 19:40, 11 May 2006 (UTC)

Problem with earth radian

Open title, now ain't it. Anyway, I have a standard geometrical issue: One flagpole casts a shadow of 18 meters, next to it is a pole of 2,1m which casts shadow long as 3,4m. Thus the length of the flagpole is in the region of 11,1174m. However, I want to impress my math teacher. Earth's mean diameter is 12 754 591m. I want to use rad to show that the flagpole's shadow is only 18m on a perfectly plane ground, while in truth the earth is curved, and so the flagpole's shadow is longer. The question is just, how do I do this? Any answer would be enormously appreciated! I don't know if my current calculations are right, so I won't bring them up here. :) 213.161.189.107 19:38, 10 May 2006 (UTC) Henning

You should right write down the equations describing a straight line and a circle, and find their intersection. -- Meni Rosenfeld (talk) 20:35, 10 May 2006 (UTC)

I think he means "write down". :--) JackofOz 02:05, 11 May 2006 (UTC)

Now you see why I went to bed shortly after this edit... :-) -- Meni Rosenfeld (talk) 06:34, 11 May 2006 (UTC)

I did this the hard way finding an answer, but it's actually straightforward if you're not dumb like me. Consider the triangle TCS, where T is the tip of the pole (assumed vertical), C is the center of the earth, and S is the tip of the shadow. What we want is the arclength of BS, where B is the base of the pole. Equivalently, we want to know the angle $\phi$ subtended by the segment BS at C. B is on the segment CT, so this is just the angle $\angle TCS$. By the law of sines, we have ${\frac {\sin \theta }{R}}={\frac {\sin(\pi -\theta -\phi )}{h+R}}$, where $\theta$ is the angle from vertical of the sun (and thus the angle $\angle CTS$), h is the height of the pole (length of BT), and R is the radius of the earth (CB, CS). Rearranging and using $\sin(\pi -x)=\sin x$, we have $\sin(\theta +\phi )={\frac {h+R}{R}}\sin \theta$, so $\phi =\arcsin \left({\frac {h+R}{R}}\sin \theta \right)-\theta$. Multiply by R to get the arclength. Hope this helps. --Tardis 04:27, 11 May 2006 (UTC)

Note that local variations in the Earth and the angle of the poles will have a far greater effect than the curvature of the Earth. So, while your answer is a nice mathematical exercise, it has no real application. StuRat 18:41, 11 May 2006 (UTC)

Thank you so very much Tardis! I am having problems understanding the need for arcsin, as I'm not really good at this stuff, but I am sure it will come to me! And yes, StuRat, I realize this. For the sake of good truth I must of course assume the angle is perfectly right and that the earth allows RB and RS to be completely identical. Impossible, but much fun is had trying to grasp a solution. :) So thanks! 213.161.189.107 13:16, 12 May 2006 (UTC) Henning

Multiplying PDF

Suppose I want to find out what is the probability that I will die of a heart attack at 2pm tomorrow on a 747 jet plane.

Answer = Prob(heart attack at 2pm) * Prob(On a 747 at 2pm)

So far so good.

But if I only know the PDF (prob dist func) of

• pdf( heart attack at 2pm )
• pdf( on a 747 at 2pm )

Then how to I multiply those two PDF together to a the PDF of the answer? To make life simple, let's assume I'm completely ignorant of my chances and the PDF of the two events is a uniform distribution (ie constant 1).

Ohanian 21:37, 10 May 2006 (UTC)

Should you die at 2.01 pm, does it still count? And 1 second after 2pm? Is it the local timezone of where the plane happens to be that counts? To have a probability distribution, you need a random variable, and a range of the possible values it may assume. Saying "constant 1" does not give enough information. With very high probability you will die between now and January 1st, 7382236374. It would be a bit silly to assume complete ignorance and claim that you are as likely to die tomorrow as on December 31st, 7382236373. The probability over all days has to add up to 1. --Lambiam^Talk 22:53, 10 May 2006 (UTC)

Godamn it! It does not matter. How about 2pm sharp within a interval of 1 second! Happy now? Ohanian 23:01, 10 May 2006 (UTC)

When asking strangers for help, swearing at them does not typically elicit better answers. Mathematicians are in the habit of demanding clear and precise questions, for very good reasons. If that doesn't appeal to you, ask your mother, not a mathematician (assuming your mother is not one, of course). Beyond that, the top of this page states explicitly, in boldface, Be courteous. Your understanding and cooperation will be appreciated. --KSmrq^T 23:49, 10 May 2006 (UTC)

What are you talking about? It is perfectly legitimate to have a uniform pdf of 1. Ohanian 23:03, 10 May 2006 (UTC)

I think what he is saying is that you have not made it clear what the event space is. Having a heart attack on a plane at 2pm is a single event. You need some unknown (or "random") event space to have the PDF with respect to if you want something more interesting. For example, you might say, given that you're on the plane at 2pm, there is some random number of snakes on that plane. You could think of the number of snakes as a random variable and it would have a PDF. --Deville (Talk) 23:15, 10 May 2006 (UTC)

I'm trying to keep it as simple as possible. A pdf for a specific event occuring at a particular point in time. A pdf for being in a particular location at a particular point in time. So how to get a pdf of (both being in a particular location and having a specific event occuring at a particular point in time)? You may assume that these two pdf are completely independent of each other. Ohanian 23:40, 10 May 2006 (UTC)

Then could you give me an example of a PDF which would correspond to "having a heart attack on a plane at 2pm"? I must say I think there is still some confusion here about what the question really is.--Deville (Talk) 23:49, 10 May 2006 (UTC)

Arrggh! I'm tearing my hair out!

If two events A and B are independent of each other then the probability of (A and B) is

Pr(A and B) = Pr(A) * Pr(B)

But if we do not know the probability of A and B. And instead we are only given the pdf of the probability of A and the pdf of the probability of B.

So we should be able to calculate the pdf of the probability of event (A and B) by some how multiplying pdf(A) with pdf(B)

pdf(A and B) = pdf(A) multiply pdf(B)

My question is "How do you multiply two pdf together?"

Ohanian 02:31, 11 May 2006 (UTC)

That's a good question. I don't know why they're having such trouble with it. Although, you did use pretty difficult examples. Your comment came at the same time as my response, which was going to be

I think I follow you, more or less. Let me repeat what I think the question is: "I know how to take the odds of simple events (5/6, 1/2, etc) and turn them into the odds of those events occuring simultaneously (5/6*1/2=5/12, etc). How do I take two continuous probability distributions (of which the uniform distribution is the simplest) and do the same?" I don't know, but for what it's worth, those are really bad examples. Try perhaps the odds of being more than six feet tall, the odds of earning more than $200,000 a year, and the odds of doing both. Of course, that assumes the two are independent, which they probably aren't.

I'd like to add something. It may have helped confuse them that you kept saying 'multiply' when, most likely, multiplication isn't the answer. 'Combine' might have been clearer. Also, given that most probability distributions are continuous, it isn't meaningful to ask the odds of an exact event (which are zero. what are the odds it'll be right down to the nanosecond?), it's only meaningful to ask the odds of an event within a range (say, of five minutes). Black Carrot 02:39, 11 May 2006 (UTC)

I get it now, I confused a lot of people when i use the term "pdf(event X)".

I meant pdf of the true probability of event X. ie. the random variable in the pdf is "the actual probability".

I should have used " pdf( actual probability of event X ) "

So the problem becomes

pdf(actual prob of event A and B) = pdf(actual prob of event A) combine pdf(actual prob of event B)

Ohanian 02:50, 11 May 2006 (UTC)

Assuming that the events are independent (though their probabilities may not be), then the probability of "A and B" is x iff $P(A)P(B)=x$. So just integrate over the possible values of the individual probabilities that multiply to that: $f_{\mbox{A and B}}(x)=\int _{x}^{1}f_{\mbox{A,B}}(t,x/t)dt$, where $f_{\mbox{A,B}}(x,y)$ is the PDF for A and B's probabilities (which is $f_{A}(x)f_{B}(y)$ if the probabilities themselves are independent). I dunno right now about dependent events, but I'll let you know if I discover something. Hope this helps. --Tardis 04:32, 11 May 2006 (UTC)

Note that Meni's answer to the recast question below is similar to, but more general than, mine and probably more accurate. I wrote this in a rush, and there's some sort of Jacobian-like modification that must be made to the integral; hopefully Meni got it right. --Tardis 18:04, 11 May 2006 (UTC)

Hopefully :-) -- Meni Rosenfeld (talk) 19:57, 11 May 2006 (UTC)

What made the question so unclear is that no event spaces were given, so what were the distributions over? Also, usually PDF stands for probability density function, which is related but not quite the same. For independent continuous random variables the joint event space is the product of the individual spaces, and if both have probability density functions (not all distributions do) the joint probability density function is (by definition of "independent") simply the product of the individual ones; see Probability density function#Independence. So if time and location are independent, just multiply the densities. --Lambiam^Talk 06:12, 11 May 2006 (UTC)

The reason your questions were meaningless is that you talk about a "pdf of a probability". There is no such thing (at least not in the probability theory I know of). There is a probability of an event, and a pdf of a random variable, and a probability of the event that a given random variable will lie in a given range. You only discussed events and their probabilities, so your use of the term "pdf" was completely misplaced. You could describe a random variable X, the number of seconds since midnight in which you die. You can then also assume that this variable is distributed uniformly between 0 and 86400. The pdf of X will be 1/86400 (not 1!) in the range 0≤t≤86400, and 0 elsewhere. Then you can ask about the probability, say, that X wil be in the range 50400≤t<50460, meaning that you will die in the minute 14:00. The result is 1/1440. If you also know the probability that you will die due to a heart attack in a 747 (no continuous random variable can be described for this), and you know that it is independent of your time of death, you can multiply the probabilities and obtain the probability that both will happen. -- Meni Rosenfeld (talk) 06:51, 11 May 2006 (UTC)

Multiplying PDF Redux

Consider a simple rectangle with height (h) and width (w).


******************************************************   |
*                                                    *   |
*                                                    *   |
*                                                    *   |
*                                                    * 
*                                                    *   h
*                                                    *
*                                                    *   |
*                                                    *   |
*                                                    *   |
******************************************************   |

<----------------------  w  ------------------------->


Now the pdf(height) is


pdf(height) =  1       if  0 < h < 1
            =  0       otherwise


Now the pdf(width) is

pdf(width)  =  1       if  0 < w < 1
            =  0       otherwise


What I'm looking for is the pdf of the area of the rectangle


pdf(area)  =  pdf(height)  <multiply>  pdf(width)


So the question is this.


How do I multiply two pdf together.


=========================================================

Can you understand why I'm asking for the following.



Pr(A and B)  =   Pr(A)  *  Pr(B)    (given A and B are independent)


so logically


pdf(A and B)  =   pdf(A)  *  pdf(B) (given A and B are independent)

Ohanian 08:10, 11 May 2006 (UTC)

Now you're asking a meaningful question.

Here's the solution:

First you should consider the cdf. Let x be constant. For a given value of w, if x≤w≤1, you have:

P(0≤A≤x) = P(0≤w h≤x) = P(0≤h≤x/w) = x/w

And if 0≤w≤x, you have P(0≤A≤x) = 1.

Therefore:

$P(0\leq A\leq x)=\int _{0}^{x}1dt+\int _{x}^{1}{\frac {x}{t}}dt=x+xln(1/x)$

The derivative is the pdf: ln(1/x). As you can see, the formula you suggested is incorrect. -- Meni Rosenfeld (talk) 09:13, 11 May 2006 (UTC)

More generally, given independent variables X and Y, with pdfs f and g in [0, ∞), the cdf of X*Y is (assuming my calculations are correct):

$H(t)=\int _{0}^{\infty }\int _{0}^{t/x}f(x)g(y)dydx$

And the pdf is:

$h(t)=\int _{0}^{\infty }{\frac {f(x)g(t/x)}{x}}dx$

These can also be presented as several equivalent forms. -- Meni Rosenfeld (talk) 09:29, 11 May 2006 (UTC)

The original question had meaning too -- it was just confusing. If you have incomplete information about the probability P of an event, it makes sense to speak of knowing only the PDF for P, with $f(p)dp$ being the probability that $P\in [p,p+dp]$. It's actually quite interesting to consider observing repeated tests of an event (or set of disjoint events) and determining the PDF for the event's probability (or set of unit-sum probabilities). It's really more of a confidence, since the probability is assumed to be already chosen and constant, but it's mathematically very similar. Also, that $1/x$ factor in your h integral shows that my answer to the previous question was not quite correct; you have to consider some sort of chain-rule-like determination of the "thickness" of the line I was tracing through the space of possible probabilities (or rectangle dimensions). Any thoughts on how to go about it as I was, but accounting for that? --Tardis 18:01, 11 May 2006 (UTC)

Oh, now I get it. Ohanian - what could really have made things clear was if you mentioned that the probability is itself a random variable. My apologies for not understanding this. Tardis - my guess is that any attempt to modify your original calculation would be more complicated than the direct calculation (finding the cdf and differentiating) - I had a try and it didn't quite work out. I think the way to go about it (I'll use my notation here, where t is the area) is to find the displacement of the graph of y=t/x, when t changes by dt, in a direction orthogonal to the graph - but I'm not completely sure. This will give the "thickness" of the line, which, if we're lucky, will amount to the 1/x factor. -- Meni Rosenfeld (talk) 20:10, 11 May 2006 (UTC)

May 11

Mathematics

Is math a closed system? or an open one? or is that unknown? what did Russell,Gödel and all of them say?. (as in closed sets or open sets).--Cosmic girl 16:25, 11 May 2006 (UTC)

Huh? Please suitly emphazi your question. Luigi30 (Ταλκ το mε) 18:22, 11 May 2006 (UTC)

It's hard to understand what you mean by a closed or an open system. I'll give it a try - Since mathematics is everything, it is, by definition, both a closed set and an open set. I guess that's not the answer you wanted to hear - If you clarify, we can be more helpful. Russel and Gödel would say what they always say. -- Meni Rosenfeld (talk) 20:14, 11 May 2006 (UTC)

That's the answer I wanted, but why do you say it's both a closed and an open system? can u explain that?... I guessed it was closed since it's everything like u said... but why open?.--Cosmic girl 22:01, 11 May 2006 (UTC)

Mathematics is a field of study. It is not a set, though sets are one of the things one studies in mathematics. Neither is mathematics a system. Like science, literature, politics, music, architecture, medicine, philosophy, mathematics is a field of study. The question probably has no meaning. -lethe ^{talk +} 22:37, 11 May 2006 (UTC)

I see... but I meant math as a language... it's a system! ( I guess).--Cosmic girl 00:05, 12 May 2006 (UTC)

Mathematics is not a language, though language is used to do mathematics, and mathematicsl languages are one of the subjects that are studied in mathematics. So what is it that you want to ask? Is the language used by mathematicians in their work a closed language? Well, mathematicians often use a form of the English language (with a fair amount of jargon added). Do you consider English to be a closed language? How about Spanish? I'm not sure what you think "closed" means for a language, but perhaps you have some idea. So my answer to you would be: mathematics is closed if and only if English or Spanish is. -lethe ^{talk +} 01:24, 12 May 2006 (UTC)

Meni, what do you mean by "mathematics is everything"? JackofOz 22:44, 11 May 2006 (UTC)

Same thing Pythagoras meant when he said "all is number", I guess. —Keenan Pepper 04:11, 12 May 2006 (UTC)

More or less. Hard to describe exactly here, though. It would suffice to say that AFAIK our physical universe is just one object in the "universe of all possibilities", aka mathematics. About both open & closed: Given a topological space (X, T), the universal set X and the empty set Φ are, by definition, open. X is the complement of Φ, and so it is, by definition, closed. -- Meni Rosenfeld (talk) 08:51, 12 May 2006 (UTC)

No guys, I'm being missunderstood I guess...Lethe, when I say language I don't mean 'english' because it's the one you use to do math... when I say language I mean MATH MATH in any language... like... math has syntactic rules and all, so why not be a language? .--Cosmic girl 16:15, 12 May 2006 (UTC)

Huh, you might be interested to read Godel's incompleteness theorem, Peano axioms, Zermelo-Fraenkel set theory, First order logic, Formal system or Formalism, and Foundations of mathematics. In short, if you're asking whether a mathematical system with axiom and theorem can be both complete and consistent, then godel proved(the first link) that in general the answer is negative, which is somewhat a blow to the attempt of formalism. However, this doesn't mean it is entirely impossible, as some system that are not strong enough to define natural number by itself can be made to do both, such as euclidean geometry or real numbers. Hope these answer your question. --Lemontea 01:50, 13 May 2006 (UTC)

thank you :), but what can euclidean geometry and real numbers do? I didn't understand that part.--Cosmic girl 20:55, 13 May 2006 (UTC)

Be both complete and consistent. "Consistent" means there are no contradictions, that is, you cannot prove both a statement and it's negation. "Complete" means that there are no "unprovable" problems - for every statement, you can always either prove the statement, or its negation. According to Godel's first incompleteness theorm, any system as strong as the Peano axioms can't be both complete and consistent - If it is consistent (which we should hope it is), than it cannot be complete (so there will always be "open" problems which cannot be solved with a proof). However, weaker systems are capable of being both consistent and complete. -- Meni Rosenfeld (talk) 12:32, 14 May 2006 (UTC)

Thank you Meni, great explanation! :) I didn't know real numbers where complete and consistent! I didn't think so either, thnx! :) hey Meni, I have a doubt though... are they both consistent and complete because tey are Axioms? or that doesn't have anything to do. :|.--Cosmic girl 16:07, 15 May 2006 (UTC)

As for the question "what can Euclidean geometry do?", well, it can do a lot. It can prove the Pythagorean theorem, and many other useful theorems. The theory of real numbers can also prove some useful results. -lethe ^{talk +} 14:30, 14 May 2006 (UTC)

For a more principled answer, different views on what mathematics "is" are possible; see Foundations of mathematics and Philosophy of mathematics. One particular school of thought is called Intuitionism; in that view mathematics is very much "open" and always essentially "incomplete". This then necessarily extends to the "language" of mathematics. For a more pragmatic answer, in mathematical practice working mathematicians invent concepts, terminology and notation on the fly, as they go, as long as it (hopefully) serves the purpose of getting an idea across. See Mathematical jargon, Mathematical notation, and, for example, Abuse of notation. In that sense too, mathematics is not closed. --Lambiam^Talk 16:45, 15 May 2006 (UTC)

Great, I see... but the ideas they want to get across themselves, are closed, right?, because they are not 'invented as they go', they are...'discovered'...this is what I mean, not the mathematical jargon, but mathematical escence itself..does intuitionism say anything about this? or no philosophy of math ever touched the subject and just focused on 'how' are ideas expressed rather than, 'what' are this ideas in the first place and how can we 'prove them'.--Cosmic girl 01:53, 16 May 2006 (UTC)

Maths Sum

The cow valued Rs 9600 is sold to three persons. The first person sells it at 10% profit, second person at 12% profit and the third person sells it at the profit of Rs 500. Find the last selling price.

Rs 3 lakh--Deville (Talk) 17:14, 11 May 2006 (UTC)

Assuming 9600 is the initial price and there are no expenses, we get:

Final price = 9600(1.10)(1.12) + 500

StuRat 18:35, 11 May 2006 (UTC)

$ python
Python 2.4.1 (#1, May 27 2005, 18:02:40) 
[GCC 3.3.3 (cygwin special)] on cygwin
Type "help", "copyright", "credits" or "license" for more information.
>>> Final_price = 9600*1.10*1.12 + 500
>>> print Final_price
12327.2

Google calculator. --hydnjo talk 14:13, 14 May 2006 (UTC)

The Age Of Her Life

Problem Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of her life, she lived for fourteen happy years after retiring.

How old was she when she retired?

Plz tell me answer?

We don't do your homework for you, unfortunately. :) Luigi30 (Ταλκ το mε) 18:21, 11 May 2006 (UTC)

Here's the setup, though:

Let X = how many years she lived

5 + X/4 + X/2 + 14 = X

Solve for X

StuRat 18:31, 11 May 2006 (UTC)

And then remember that the question is how old was she at retirement, not when she died. --LarryMac 18:35, 11 May 2006 (UTC)

I wonder why her parents called her "Problem Sarah". A clear cut case of setting up the poor kid for failure, if you ask me. JackofOz 22:43, 11 May 2006 (UTC)

:-) How humiliating when your childhood nickname sticks, even when you achieve textbook status. Skittle 14:59, 12 May 2006 (UTC)

Ah yes, textbook status. For how many centuries has man dreamed of achieving textbook status. But it can have its drawbacks, as we now know. :--) JackofOz 07:20, 13 May 2006 (UTC)

(Removed apparently copyrighted homework answer because it was copyrighted and giving a homework answer.) Skittle 13:04, 12 May 2006 (UTC)

But if you just thought about the problem, you could solve it easily. She lives a quarter of her life at school. She lives half her life at work. That leaves how much of her life? Skittle 13:06, 12 May 2006 (UTC)

I think this question deserves a rewrite:

If Problem Sarah starts contributing to Wikipedia at age 14, spends one quarter of her life making contributions to Wikipedia, and one half of her life in flame wars with various Wikipedians, then spends the last 5 years of her life in a straight jacket in an inane asylum, how old was she when first committed (to the asylum, not Wikipedia) ? StuRat 11:49, 14 May 2006 (UTC)

StuRat 11:49, 14 May 2006 (UTC)

How disappointing to find oneself in an inane asylum, after going to all the trouble of going insane. Skittle 10:38, 15 May 2006 (UTC)

I was wondering if anyone would notice that I slipped that in. Good eyes ! StuRat 01:56, 16 May 2006 (UTC)

Isn't the minimum age of retirement in the United States 67? I think it's a safe wager that 67 is in fact the age at which she retired ;) — Ilyanep (Talk) 01:11, 15 May 2006 (UTC)

While individual employers may set retirement ages, there is no mandatory retirement age in the US. StuRat 01:56, 16 May 2006 (UTC)

How can we see left derived functors even as actual functors?

Hello,

I am trying to understand more of homology.

Suppose you have a functor F (covariant and additive) from R mod (the R modules) to the category of Abelian groups.

If you want to take the nth left derivative, you are going to have to use projective resolutions. You create a projective resolution, apply the functor, you get a complex over Fm, and you take the nth homology module. Morphisms are defined a similar way.

My problem : you created 'a' projective resolution. So it depends on that choice. Now those modules will all be the same up to isomorphism, but... I don't feel too good about it.

This isn't really a functor anymore, or is it? Perhaps one should not change the functor but the categories?

All hints are welcome! Evilbu 22:26, 11 May 2006 (UTC)

I asked a similar question on this reference desk a while ago. Limits like products are only defined up to canonical isomorphism. You can define a functor which assigns products, but it in order to have a specific product object, you have to invoke the axiom of choice. I found a paper by Mikkai where he describes a construction of a generalization of functors which he calls anafunctors. These are like functors which have many different objects in the image of their object function, and satisfies some appropriate functorial condition. You can turn an anafunctor into a proper functor by invoking the axiom of choice to choose a single object from each isomorphism class. I'm not sure if derived functors admit a similar description, but it sounds like it might. That is, I don't know if I can describe a derived functor as a limit, but to each functor, we assign a derived functor, which is only determined up to isomorphism. Sounds about right. -lethe ^{talk +} 23:35, 11 May 2006 (UTC)

You should view them as functors from the derived category, which is pretty much a category whose objects are resolutions: notice that the entire construction is functorial in the resolution. Sadly our article on the subject is all history and no math. —Blotwell 23:21, 12 May 2006 (UTC)

May 12

Brachistochrone

I am interested in actually constructing a brachistochrone track for a physics project. How does one actually construct a brachistochrone (I mean on paper mathematically, not physically)? I know that it is a segment of a cycloid, but how much of the cycloid? I'm not allowed to just use any cycloid curve, right? It says on Wikipedia

Given two points A and B, with A not lower than B, there is just
one upside down cycloid that passes through A with infinite slope
and also passes through B. This is the brachistochrone curve.

So the tangent line of the brachistochrone at A must be vertical By tinkering with this: http://home.ural.ru/~iagsoft/BrachJ2.html I assume -construct an entire "hump" of a cycloid which much start at A, pass through B, and end at a point level with A

So I think that's right, but where do I go from there? If A and B are given, how would I construct such a cycloid? I can't just say that its amplitude (and therefore twice the circle's radius) is the vertical distance from A to B, because the lowest point may be below B. How would I figure its amplitude/width? -JianLi 01:32, 12 May 2006 (UTC)

Well, the obvious approach is to use the parametric equations for the cycloid and try to solve them simultaneously for t and r. Take A to be the origin, and take $B=(x,y),y<0$. Then we have $x=r(t-\sin t),y=r(\cos t-1)$. Isolating the sine and cosine, squaring, and adding gives us $(t-x/r)^{2}+(y/r+1)^{2}=1$, so $t={\frac {x}{r}}\pm {\sqrt {1-\left({\frac {y}{r}}+1\right)^{2}}}$. Then divide the sine and cosine to get $\tan t={\frac {t-x/r}{y/r+1}}={\frac {\pm {\sqrt {1-\left({\frac {y}{r}}+1\right)^{2}}}}{{\frac {y}{r}}+1}}=\pm {\sqrt {\left({\frac {y}{r}}+1\right)^{-2}-1}}$. This gives four possible solutions for t involving r (one times two from the $\pm$ and times two from $\tan x=\tan(x+\pi )$). Plug each of these into, say, the y equation and you can solve for r in theory, but I'm not sure how to do that analytically. Then you'd have to evaluate x and y and see if they matched for each of the four solutions, I guess. ...Hm, this sort of bogged down eventually. Anyone else have a cleaner way to do it? --Tardis 03:11, 12 May 2006 (UTC)

OK, try 2: start by considering the known value $\xi :={\frac {y}{x}}={\frac {\cos t-1}{t-\sin t}}$ (no r dependence). Then $\cos t+\xi \sin t-\xi t=1$. Unfortunately, the best you can do at this point appears to be solving numerically again; you could first replace the trig functions with one shifted, scaled trig function using $A\cos(x-\phi )=A\cos x\cos \phi +A\sin x\sin \phi$, so in this case $A\cos \phi =1,A\sin \phi =\xi$ so $A^{2}=1+\xi ^{2}$ and $\tan \phi =\xi$, so we have ${\sqrt {1+\xi ^{2}}}\cos(t-\arctan \xi )-\xi t=1$ (remember that $\xi <0$). Note that $t=0$ is automatically a solution, but should be discarded. Hope this helps, although I'm still hoping for a magic analytical solution (from my own brain or otherwise). --Tardis 03:30, 12 May 2006 (UTC)

Thanks for your reply thus far. I have another question: What is the effect of rotational kinetic energy? The calculation of the cycloid as the brachistochrone is calculated only accounting for translational KE. I'm assuming that the cycloid still comes in first, assuming that the ball will roll on all surfaces without slipping. Is this right? JianLi 03:56, 12 May 2006 (UTC)

The translational energy is of course $K=mv^{2}/2$, and the rotational kinetic energy is $U=I\omega ^{2}/2$. Since, without slipping, $v=r\omega$, the overall energy is proportional to $v^{2}$ and so acts like some increased mass. Since the mass doesn't matter, you're right, the cycloid wins -- but as the path of the center of gravity! You'll need to displace the track outwards by one radius. Of course, you'd need to do that anyway if the falling object is of non-negligible size compared to the course. --Tardis 04:13, 12 May 2006 (UTC)

Analytic signal and analytic continuation

Suppose we have a real $u(t)$ function. We can obtain a complex analytic signal $z_{a}(t)$ using Hilbert transform. We can also use an analytic continuation to obtain another complex function, say, $z_{c}(t)$. What can we say about $z_{a}(t)$ and $z_{c}(t)$? At least for harmonic $u(t)$ they seem to be equal. And what about the general case?--Ring0 05:33, 12 May 2006 (UTC)

Sorry, I seem to misunderstand your question. The complex analytic signal you get from the Hilbert transform is a complex-valued function whose domain is the reals, i.e.

$z_{a}(t):\mathbb {R} \to \mathbb {C} ,$ but the analytic continuation of a real function is a function whose domain is also complex, namely $z_{c}(t):\mathbb {C} \to \mathbb {C} .$ In this case we're talking two different animals here. Do I misunderstand something? Your last sentence makes me think I did --Deville (Talk) 02:21, 15 May 2006 (UTC)

Thanks, I've forgotten about the domain, but still we can apply $z_{c}(t)$ only to $t\in \mathbb {R}$ and compare it with $z_{a}(t)$ in the real domain.--Ring0 22:01, 15 May 2006 (UTC)

I think it's possible he means something like the analytic continuation of the Hilbert transformation as well, or something along those lines. At the very least there does seem to be a connection between the Hilbert transform and the analytic continuation by following logic along these lines (I haven't done the actual math involved so I'm not 100% sure of the connection):

Assume f(t) has an analytic continuation f(z) in the domain bounded by $-\epsilon <Im(z)<\epsilon$. Then $g_{n}(z)={\frac {f(z)}{(z-\tau )^{n}}}$ has a pole of order n at $z=\tau$. Let C be a contour which starts at $-\infty$ on the real axis, goes along the real axis to $+\infty$ with a small detour over $z=\tau$, then goes back to $-\infty$ with a small detour under $z=\tau$. Then I think there's a fairly close connection between the Hilbert transforms of the nth derivative of f(t), the integral of g_n(z) around the contour C, and the nth coefficient of the Taylor expansion of f(z) around $z=\tau$. As I said, I didn't do the math so it could be completely wrong. Confusing Manifestation 02:40, 15 May 2006 (UTC)

Johnson Solids

I'm wondering about the names of the last 9 Johnson solids. At first glance most of them seem random, but a closer look seems to reveal patterns.

1. First, the snubs (J84 and J85). According to Mathworld, a disphenoid is a tetrahedron, so the snub disphenoid could also be called the snub tetrahedron (Is this right?). My question is, are the snub disphenoid and the snub square antiprism related to the tetrahedron and the square antiprism in the same way that the snub cube and the snub dodecahedron are related to the cube and the dodecahedron? The snub polyhedron article mentions the Johnson snubs in passing, implying that they may be a little different yet not giving sufficient detail as to what the difference actually is.
2. I understand the naming of J87, as the article is very good on this point.
3. All the others (J86 and J88-92) have strange names, but a close look reveals them all to consist of various combinations of the following:
   • spheno
   • corona
   • mega
   • hebe
   • cingulum
   • luna
   • di and bi
   • rotunda (eg pentagonal rotunda)

   According to this site, a luna is a fragment consisting of a square with two triangles attached to opposite sides. It also gives a definition of a corona which makes no sense to me. But that doesn't enlighten me as to the source of the other name fragments. Is there a polyhedron (or polyhedral fragment) called a spheno and another one called a cingulum? Also, rotunda in this case seems to mean something other than pentagonal rotunda. Can anyone help me here, or are these names truly just random?

4. Finally, the article explains that augmented and diminished may mean the addition or removal of either a pyramid or a cupola, but no rules are given to determine when a pyramid is used and when a cupola. With diminishing, I can see that it might be obvious as some shapes only have cupolas to remove and others have only pyramids. But with augmentation, is the choice of pyramid or cupola arbitrary or based on a rule?

Please answer all three questions if possible. Thank you. --72.140.146.246 17:45, 12 May 2006 (UTC)

Ah, those wonderful, wacky, polyhedra and their names! Probably George Hart's site, listed at the bottom of the article, is your best source. The truth is, people invent these names with some attempt at regularity and some attempt at whimsy. So many names are needed and the shapes are so challenging to describe simply (once we get past Archimedean solids), that it is unrealistic to hope for clear meaning and regularity — unless someone comes up with a new insight. --KSmrq^T 03:20, 13 May 2006 (UTC)

I did visit George Hart's site before asking this question, but I could not view the .wrl files, which discouraged me from looking through it much. Revisiting it just now, however, I noticed something which answers my question in point 3 above. Thank you. But what about the snubs (point 1 above) and the augmentation (point 4 above)? I would like to know whether the Johnson snubs are standard snubs (eg the Archimedean snubs) and how the choice of cupola or pyramid is made for an augmentation. --72.140.146.246 14:53, 13 May 2006 (UTC)

Help with finding a VRML browser (for the *.wrl files) is available. As for the remaining questions, try email to George Hart if nothing else works. --KSmrq^T 16:16, 13 May 2006 (UTC)

Area/Volume → Perimeter/Surface Area?

The area of a circle is $A=\pi r^{2}$. Differentiating this formula with respect to r gives ${\frac {\;dA}{\;dr}}=2\pi r$. So ${\frac {\;dA}{\;dr}}=C$, the circumference or perimeter. With a sphere this also works, so ${\frac {\;dV}{\;dr}}=A$. However, it does not work for a cube or a square, and probably not for other polygons or polyhedra. Is the fact that it works for the circle and the sphere simply a coincidence, or is this a special case of something more general which does work for other polgons and polyhedra? (I notice that for square or cube my method would produce exactly half of the correct value.) --72.140.146.246 17:59, 12 May 2006 (UTC)

This is certainly not a coincidence. It represents the fact that dA, an infinitesimal change in the area of the disc, is the area of a thin ring with thickness dr and circumference C. The area of this ring is C times dr (the area of a thin line is its length times its thickness), and therefore ${\frac {dA}{dr}}=C$. The same thing holds for a ball. For it to work for a cube\square, you need to use a different variable: Instead of a, the edge length (which is analogous to the diameter of a circle), consider b = a/2 (which is analogous to the radius). You'll have A = 4b^2 and p = 8b, where ${\frac {dA}{db}}=p$. It doesn't work with a, for the same reason it wouldn't work with the diameter d for a circle. -- Meni Rosenfeld (talk) 18:23, 12 May 2006 (UTC)

Also, this does work for a cube and other Platonic solids too, and for regular polygons, for the same reason as why it works for a sphere, you just have to do the differentiation with the radius of the inscribed sphere (or circle) as the independent variable. For example, a cube with insphere radius $\rho$ has volume $V=16\rho ^{3}$ and surface $A=(d/d\rho )(16\rho ^{3})=24\rho ^{2}$. Really it works for any solid, if you chose a suitable independent variable (the "average inscribed sphere radius" if such a thing exists), because if you scale a solid, the volume always changes cubically, and the surface square proportionally. – b_jonas 22:50, 12 May 2006 (UTC)

I wonder, does it really work for "any" shape? For example, it does not work for an ellipse, at least not if you choose the independent variable to be the semi-major axis and hold the eccentricity fixed. So the symmetry of the figure must somehow be relevant. -lethe ^{talk +} 03:23, 13 May 2006 (UTC)

I guess a sufficient condition for this to work with respect to a variable r is that the distance from the center to the tangent of the shape at any point (where the tangent exists) is r. If we want this to be some intuitive parameter of the shape (such as the half-side of a square, but not the "average inscribed radius" of an ellipse), this is probably necessary as well. -- Meni Rosenfeld (talk) 07:03, 13 May 2006 (UTC)

It holds for an ellipse too, only you have to choose a constant multiply of the semi-major axis as the independent variable. The only condition is that the shapes must really be similar to each other (which holds with an ellipse as the excentricity is fixed), and that the independent variable is scaled the same amount as the figures. The ellipse is a bit difficult example because its perimeter cannot be written as a simple formula, but still we know that the perimiter is surely a constant multiply of the semi-major axis $a$, let's say it's $S=k_{e}a$ (I wrote $k_{e}$ because the constant depends on the excentricity). The area is $A=\pi {\sqrt {1-e^{2}}}a^{2}=K_{e}a^{2}$. Now let's choose $t=2(K_{e}/k_{e})a$ as the independent parameter, then $dA/dt=(dA/da)/(dx/da)=(2K_{e}a)/(2K_{e}/k_{e})=S$. – b_jonas 16:01, 13 May 2006 (UTC)

Yes, I understood this idea, and that it works for any shape. I only said that in cases like this, the variable you have to use is a complicated function of the basic parameters of the shape, as opposed to a shape satisfying my condition above, for which the variable is simply the inscribed radius. -- Meni Rosenfeld (talk) 16:49, 13 May 2006 (UTC)

How would this hold for an equilateral triangle? The area of a triangle is $A={1 \over 2}bh$, which with the Pythagorean theorem becomes $A={{\sqrt {3}} \over 2}b^{2}$, where b is the side length. However, with this variable, $P=3b$. Using half the side length as your variable would not work here as it did for the square, so what variable would work and how is that variable related to the side length? Must the triangle be equilateral for this to work? --72.140.146.246 14:33, 13 May 2006 (UTC)

Here, you must chose the radius of the inscribed circle as the independent variable, which is $\varrho =h/3=b/(2{\sqrt {3}})$. You can use any triangle as long as you scale it uniformly, and the independent variable to choose is always the radius of the inscribed circle. – b_jonas 16:08, 13 May 2006 (UTC)

It will also work for a wide range of polygons, including (but not limited to) regular polygons, and also some shapes which are derived by stitching together straight line segments and arcs of a circle - in general, exactly those shapes that have an inscribed circle (touching all sides, and overlapping curved parts). The variable which should be used is, of course, the radius of the inscribed circle. BTW, it's ${\frac {\sqrt {3}}{4}}b^{2}$, not ${\frac {\sqrt {3}}{2}}b^{2}$. -- Meni Rosenfeld (talk) 16:49, 13 May 2006 (UTC)

Integral of the square of the sine

I suddnly found integrals a bit interesting, but there is one thing that bothers me, and that is the integral of the square of the sine. How can this equality be proven, using the usual techniques for integrals?

$\int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C$

Loool 19:09, 12 May 2006 (UTC)

Use the identity

$\cos 2x=1-2\sin ^{2}x.$

Also, use the identity

$\cos 2x=2\cos ^{2}x-1$

for the integral of cosine squared. -lethe ^{talk +} 19:31, 12 May 2006 (UTC)

Thanks for this one. I had forgotten about those formulas. Now the integral looks very easy. Loool 19:35, 12 May 2006 (UTC)

The identities come from

$\cos(x+y)=\cos x\cos y-\sin x\sin y,$

from which you get

$\cos 2x=\cos ^{2}x-\sin ^{2}x.$

To get the required identities, use the famous

$\sin ^{2}x+\cos ^{2}x=1,$

and substitute for the sine squared or cosine squared as appropriate. -lethe ^{talk +} 19:45, 12 May 2006 (UTC)

Yes, using an identity like that is indeed the easy way to compute this integral but it's also possible with partial integration I belive. – b_jonas 22:13, 12 May 2006 (UTC)

Yes, that's another good way. Requires a little less trigonometry. Another way is to use complex numbers, if you're familiar with them. -lethe ^{talk +} 03:13, 13 May 2006 (UTC)

If you desire only to prove the equality and not compute the integral, then it's sufficient to differentiate both sides. Dysprosia 12:32, 13 May 2006 (UTC)

Infinite Sets

Is it possible to prove that one infinite set can be larger than another infinite set using strictly mathematics [no analogies to cavemen]. — Ilyanep (Talk) 22:30, 12 May 2006 (UTC)

Yes. The Cardinal Number article has some nice information and links. What's this 'caveman' analogy? It sounds amusing. Black Carrot 23:05, 12 May 2006 (UTC)

Basically, you have a group of cavemen who can only count up to four, and afterwards they just say 'a lot'. So technically you could have six or eight and one 'a lot' is greater than another 'a lot'. — Ilyanep (Talk) 02:54, 13 May 2006 (UTC)

Yes, it's Cantor's diagonal argument. There are also various humorous illustrations of the same argument, like infinite hotels or the Book of Sets. – b_jonas 22:53, 13 May 2006 (UTC)

Perhaps you're referring to Hilbert's hotel. —Blotwell 21:51, 14 May 2006 (UTC)

Surface area of a sphere

How do you prove that the surface area of a sphere is $4\pi r^{2}$?

Obviously the volume formula comes from that by integrating the surface area formula.

(Be nice, I'm a freshman in high school who's in Geometry but has dabbled around with some calculus).— Ilyanep (Talk) 22:32, 12 May 2006 (UTC)

Some searching on Google gives [5] and [6]. Hope that helps. Black Carrot 23:11, 12 May 2006 (UTC)

I think it's easier to find the volume first, then differentiate to get the surface area. And a simple way to find the volume is to integrate the area of successive circular cross-sections, i.e. $\int _{-r}^{r}\pi (r^{2}-x^{2})\,dx$ or something. —Blotwell 23:15, 12 May 2006 (UTC)

This is a fun question to learn with. Archimedes, an outstanding ancient mathematician, was so proud of the relationship he proved between a sphere and its circumscribed cylinder that he asked for the figure to be engraved on his tombstone. His ingenious methods foreshadowed the development of integral calculus many centuries later.

The area of a cylindrical strip is unchanged by unrolling it to form a rectangle; the height will be 2r (the diameter of the sphere) and the width will be 2πr (the circumference of the sphere at the equator). Thus the cylinder strip area is 4πr², the product of height and width.

Can we transform our awkward problem, the area of a sphere, into this simple, solved, one? Archimedes says yes. Look at a tiny rectangle on the surface at a latitude of ϑ. As we move toward the North or South Pole, the surface tilts more and more perpendicular to the vertical of the cylinder. This causes the projection onto the cylinder to be foreshortened by a factor of cos ϑ. Simultaneously, the sphere surface shrinks away from the cylinder surface by the same factor, implying that the projection onto the cylinder is larger by the reciprocal of that factor. Thus these two effects exactly cancel, so that the area of the cylindrical strip is identical to the area of the sphere.

Incidentally, this implies that for random points uniformly distributed on a unit sphere, the z coordinate is uniformly distributed between +1 and −1. So is the y coordinate and so is the x coordinate; but these are obviously not independently distributed.

Not bad for no (explicit) calculus. Archimedes did more, and he was justifiably proud. If we try it with explicit calculus, we have two obvious choices: rotate a semicircle of longitude (drawn from Pole to Pole), or accumulate circles of latitude. The rotation method uses a valuable tool of broad utility, the description of a surface of revolution. The latitude method should look very much like our Archimedean calculation.

Let's end with a little table, listing interior measure and boundary measure (volume and area, generalized) for unit balls as a function of even dimension, 2n:

2n	2	4	6	8
V	π	π2/2	π3/6	π4/24
A	2π	2π2	π3	π4/3

In each case the product of dimension and volume equals area. The volumes seem to follow the pattern πⁿ/n!, which hints at a general formula. Is this real? If so, can we prove it? And what about the odd dimensions? (The answers to these questions are well-known, of course, and mostly hiding somewhere within Wikipedia.) --KSmrq^T 05:24, 13 May 2006 (UTC)

Deal or No Deal?

I was wondering how they calculate the "Deal" from the banker on the show and on the internet. I have been trying to figure it out for months. Thanks --Zach 23:34, 12 May 2006 (UTC)

Yes, I've been thinking about this a lot too. I guess that it just has to do with the probability that the person is going to win each amount that is left. The "deal" would just be the amount that the person is, according to probability, would probably win. Beyond that, I don't know. (i.e. I haven't thought that far or I can't think that far.) --Think Fast 02:02, 13 May 2006 (UTC)

I'm glad I wasn't the only one thinking about that. Not quite sure how it works though... —Mets501^talk 02:25, 13 May 2006 (UTC)

The banker's tolerance for risk appears to diminish as the prospective loss (and the std dev of the remaining values) becomes larger; whilst early offers are often substantially less than one's expected payout (and thus should be declined by risk-neutral players), late offers (especially in situations in which five or fewer amounts remain, of which at least two are exceedingly large) tend rapidly to approach one's expected payout (I can recall two cases in which the offer exceeded the player's expected payout, and each was, properly, accepted). This is, I think, by design; Howie has explained on, inter al., The Tonight Show with Jay Leno that the "banker" simply conveys an algorithm-generated offer. The show certainly has some appeal inasmuch as it presents one with the opportunity to earn money without debasing him/herself and in a pleasant environment, accompanied by family and friends, but I find myself increasingly unable to watch, if only because of the complete ignorance of game theory or probability exhibited by the players. Worse, though, is the sincere belief that many contestants ostensibly harbor that they make wise case choices; one wonders whether they actually appreciate that the game--at least the case-selection portion--is altogether one of chance (sure, I sometimes manually select numbers for the lottery, but I do that mainly for fun, and I surely don't think that my selections are somehow more lucky to appear than those generated by a computer). Finally, Howie's frequent intimations that a player has made a great deal are ex post accurate (just as one who doesn't switch in the Monty Hall problem but ends up with the car can be said to have made the right decision) but surely reflect a belief that one made the proper decision on a generalized, ex ante level, which belief is not always correct. Joe 03:31, 13 May 2006 (UTC)

It appears to me that the banker's offer is a rounding off of an algorithmic amount, probably to keep the numbers simple for television's sake. If you play the on-line version on the game at nbc.com, the banker's offers are very unrounded amounts. In the game I just played, I was given offers such as $24,834, $48,049, and $55,573. — Michael J 17:39, 13 May 2006 (UTC)

Note that the primary goal of the banker is not to limit payout, but to maximize viewership. Thus, making stingy offers which are constantly refused would be boring and lose viewers, while making generous offers which are accepted is more interesting (although not so generous as to be a "no-brainer"). StuRat 21:31, 14 May 2006 (UTC)

I have personally been interested in this EXACT topic. i want to create a spin off for mIRC. if ANYONE can actually derive an algorithm for it based off the currently remaining cases, email me at BXClapper@aol.com. It is very much appreciated and i will probably even give you credit for the algorithm.

Triangular Sums

Anyone know how to find pairs of triangular numbers whose sums are triangular numbers? Ditto for squares. Black Carrot 23:50, 12 May 2006 (UTC)

Any triangular number can be represented by the expression ${\frac {x(x+1)}{2}}$. This means that for a triangular number to be the sum of two other triangular numbers, it must be the case that $x(x+1)=m(m+1)+n(n+1)$. You can solve this Diophantine equation to find values. As for square numbers, you are really looking for Pythagorean triples, so I would go there for more information. —Mets501^talk 02:46, 13 May 2006 (UTC)

I have a similar problem too, this time is to find (non-trival) integer solution to ${p \choose 3}+{q \choose 3}={r \choose 3}$. BTW, for the squar number problem, move term to get x²=z²-y². Assuming they're in the simpliest form possible(i.e. coprime), we factorize to have x²=(z+y)(z-y).Because they're coprime, so are z+y and z-y coprime. So z+y and z-y are both square number. We set 2m²=z+y and 2n²=z-y. So x=2mn, y=m²-n², z=m²+n² as a parametric solution. --Lemontea 06:53, 15 May 2006 (UTC)

Here is an approach to the original question regarding generating pairs of triangular numbers which sum to another triangular number. Take an arbitrary triangular number T_n and express it as the product of a power of 2 and an odd number i.e.

$T_{n}=2^{p}q.$

Now express the odd number q as the sum of two consecutive numbers r and r+1, so

$q=2r+1.$

Define s and t by

$s=r+1-2^{p}$

$t=r+2^{p}.$

Then

$\sum _{m=s}^{t}m=\sum _{m=r+1}^{t}{(2r+1-m)+m}=\sum _{m=r+1}^{t}{q}=2^{p}q=T_{n}$

so we have expressed T_n as the sum of 2^p+1 consecutive numbers from s to t. Then

$T_{s-1}+T_{n}=\sum _{m=1}^{s-1}m+\sum _{m=s}^{t}m=\sum _{m=1}^{t}m=T_{t}.$

For example, T₁₁ = 66 = 2*33, so we have r = 16, s = 15, t = 18 and

$T_{14}+T_{11}=105+66=171=T_{18}.$

Gandalf61 10:03, 15 May 2006 (UTC)

May 13

"calculations in your head"

What is the English term for "making calculations in your head", if there is one? It's pretty common in Estonia to have competitions in schools for this, and I only just realized I don't actually know the English term. I recently heard a claim a 13-year old Estonian girl is the "world champion", which I find hard to believe, but I don't know where to look to find out. PeepP 07:08, 13 May 2006 (UTC)

I think this is sometimes referred to as mental calculation. -- Meni Rosenfeld (talk) 07:10, 13 May 2006 (UTC)

Thanks, that's obvious enough. I've added it as a see also to calculation, which is where I looked at first. PeepP 07:53, 13 May 2006 (UTC)

Personally, I call it mental math :) — Ilyanep (Talk) 15:09, 13 May 2006 (UTC)

A useful test: [7] (12.2 million results) [8] (25.4 million results) [9] (4.65 million results)

I'm more familiar with "mental math", probably chosen because it's catchier. Black Carrot 15:12, 13 May 2006 (UTC)

We usually called it Mental Arithmetic when we were tested on it. Skittle 16:00, 13 May 2006 (UTC)

I suspect Mental Arithmetic sounds more official, and Mental Math is more of a colloquial term. — Ilyanep (Talk) 22:50, 13 May 2006 (UTC)

In fact, mental math is more of a kids' term. That's why I know it better; I'm 18, and have spent most of my life as a child. Black Carrot 23:56, 13 May 2006 (UTC)

"THE ALL IS MIND; The Universe is Mental." -- The Kybalion. --DLL 19:53, 14 May 2006 (UTC)

finite geometry: how many (m+1)dim spaces on quadric go to m dim space

Hi,

I would really like to know how to answer this question

consider a nonsingular quadric Q in a projective geometry PG(n,q) with q a prime power

let V be a m dimensional space completely on Q how many m+1 dimensional spaces W go through V and are also completely on the quadric Q

now this question is fairly easy if m=0, given that the tangent hyperplane through a given point p cuts the quadric in a cone, with p as top and as basis a nonsingular quadric of two dimensions lower

but how about the general case? The formule should be quite easy I suspect, very closely related to the number of points in smaller quadrics.

Truly, all hints are welcome.

By the way: the more general case of a finite classical polar space is also interesting (thus the same question for nonsingular hermitian varieties and symplectic polarities)

Thanks, Evilbu 12:21, 13 May 2006 (UTC)

Nobody wants this one? We can at least ask about the field; shall we assume ℝ or ℂ or more general? When I'm not so lazy I'll check Hodge and Pedoe (ISBN 0521469015), a likely source. --KSmrq^T 03:59, 15 May 2006 (UTC)

Thank you for your response! What can be said about the field? Well it is definitely finite, thus its order is a prime power. PG(n,q) is 'slang' for the n dimensional projective geometry over the field of q elements (there is only one field of q elements up to isomorphism)

Now however it will matter whether or not q is even, because then you do not always have a polarity... but still the answer to my question (the number of spaces through it) should not be dependent.

Evilbu 16:10, 15 May 2006 (UTC)

May 14

Kernel

What does "kernel" stand for in Dirichlet kernel and Fejér kernel? Mon4 01:53, 14 May 2006 (UTC)

See integral transform. -- Jitse Niesen (talk) 03:41, 14 May 2006 (UTC)

Good catch! The usual algebraic meanings for "kernel" are linked at the disambiguation page, but the kernel of a linear filter defined by convolution is not, nor are any of its relatives. This is a common term in applications, in some instances equivalent to point spread function or impulse response. --KSmrq^T 04:07, 14 May 2006 (UTC)

May 15

Poisson Process

With $\{T_{n}\}_{n=1}^{\infty }$ a sequence of iid exponential variables with parameter $\lambda$. With definition of $\{S_{n}\}_{n=1}^{\infty }$ sequence:

$S_{n}=\sum _{k=1}^{n}T_{k}$

Which of these two process is a poisson process $N=\{N_{t}:t\geq 0\}$ with $\lambda$ parameter?

$A=\{A_{t}:t\geq 0\}\qquad A_{t}:=Sup\{n\in Z:S_{n}\leq t\}$

$B=\{B_{t}:t\geq 0\}\qquad B_{t}:=Inf\{n\in Z:S_{n}\geq t\}$

Hessam 09:22, 15 May 2006 (UTC)

How come this sounds so much like a homework problem? Could the reason be that it is a homework problem? If it is a multiple-choice test in which at least one answer is correct, then there is an easy way of eliminating one incorrect answer by considering P[A_t = 0] and P[B_t = 0]. --Lambiam^Talk 13:12, 15 May 2006 (UTC)

No it's not a multiple choice problem. With considering both on 0 position we can get nothing. There should be another reason in 3 other conditions of poisson process. I have no idea. No other idea?! ‍‍Hessam 22:47, 15 May 2006 (UTC)

:-(( Hessam 08:00, 16 May 2006 (UTC)

I don't know what the "3 other conditions of poisson process" are. I know the condition given under Poisson process. Taking N₀ = 0, we have P[N_t = 0] = exp(–λt).

Let us see if B fits. Assume t > 0. We have P[B_t = 0] = P[inf{n ∈ Z : S_n ≥ t} = 0] = P[S₀ ≥ t] = P[0 ≥ t] = 0. No, this doesn't agree with exp(–λt). So B is out.

Next we try A again with t > 0. Now P[A_t = 0] = P[sup{n ∈ Z : S_n ≤ t} = 0] = P[S₀ ≤ t ∧ S₁ > t] = P[S₁ > t] = P[T₁ > t] = 1 – P[T₁ ≤ t] = 1 – (1 – exp(–λt)) = exp(–λt). Yes, that fits. This isn't a proof that A is a Poisson process; for that you have to consider the more general expression P[N_t+τ – N_t = k]. But it's a start. --Lambiam^Talk 15:09, 16 May 2006 (UTC)

I knew nothing about this subject, but after a quick glance at poisson process I would guess it's A(Not sure if B is correct), as it resemble the "classic" telephone call example. --Lemontea 13:50, 16 May 2006 (UTC)

Thank you for your answers :-) ‍‍Hessam 21:31, 16 May 2006 (UTC)

Can a dot product return a complex number or just reals?

In the Spanish Wikipedia it says it returns "scalars". In the English "reals". Thanks.

What do you mead by "dot product"? If you mean the scalar product in real vector spaces, of course it never returns complex numbers just because the vector space is, indeed, real. In general, if you have a vector space over some field $\mathbb {K}$, any scalar product will return an element of $\mathbb {K}$, as a definition. Have a look at Scalar product for a general definition. Cthulhu.mythos 11:44, 15 May 2006 (UTC)

In short - Yes, if the vectors are over the field of complex numbers. -- Meni Rosenfeld (talk) 12:22, 15 May 2006 (UTC)

Indeed, as the questioner noticed, the article Dot product states: "the dot product [...] returns a real-valued scalar quantity." If this is wrong (or too simplistic) as stated, it should be corrected. The Dot product article starts with "For the abstract scalar product or inner product see inner product space", but it may not be clear to the reader looking up "dot product" whether what they are looking for fits the notion of "abstract scalar product or inner product". --Lambiam^Talk 13:34, 15 May 2006 (UTC)

A dot product is assumed to be the standard Euclidean inner product of a real vector space. We would not usually call an inner product of complex numbers a dot product. (Next time, it would be better to ask on the article talk page so the discussion stays associated with the article.) --KSmrq^T 14:22, 15 May 2006 (UTC)

Not to mention that the Dot Product article says that "the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar", which is by itself a contradiction, since a binary operation (in the strict sense) has to be closed, as it's stated in the article: "More precisely, a binary operation on a set S is a binary function from S and S to S". It should be corrected to "binary function", IMO. -- Matt Kovacs 10:08, 17 July 2006 (UTC)

Gaussian curvature --> radius

Curvature and radius are reciprocals of each other. So, given $M={\frac {1}{\kappa _{1}}}\mathrm {\;and\;} N={\frac {1}{\kappa _{2}}}\,\!$, where M and N are the principal radii of curvature, the Gaussian curvature equals $\kappa _{1}\kappa _{2}\,\!$, not ${\sqrt {\kappa _{1}\kappa _{2}}}\,\!$, so that means the radius of Gaussian curvature must, therefore, equal $M\!N\,\!$, not ${\sqrt {M\!N}}\,\!$—? But how can that be? Is it actually Gaussian curvature²?  ~Kaimbridge~13:47, 15 May 2006 (UTC)

A different choice is the mean curvature, which for a sphere will be the reciprocal of its radius. --KSmrq^T 15:04, 15 May 2006 (UTC)

(edit conflict) "Radius of Gaussian curvature" is not a defined and meaningful notion. "Radius of curvature" is meaningful for curves in 2-space, and "Gaussian curvature" is the curvature of surfaces in 3-space. See Curvature. --Lambiam^Talk 15:06, 15 May 2006 (UTC)

However the radius of the two principal curvatures is defined, and are used to define the focal surface which have been studied in singularity theory and also have aplications in optics. At a point p on a surface with normal N then the two points $p+{1 \over \kappa _{1}}N$, $p+{1 \over \kappa _{2}}N$, define the focal surface. --Salix alba (talk) 16:26, 15 May 2006 (UTC)

I have been going back and forth between curvature, Gaussian curvature, principal curvature (and others), but the discussion is mostly abstract theory. Is the more complete, clarified relationship, "2-space (I presume, meaning 2-D space?) curvature" is the reciprocal of its radius, and "3-space curvature" (such as Gaussian) is the reciprocal of radius²? As for semantics, the radius is the inverse of "whatever curvature", so it would seem the proper order is "radius of Gaussian curvature", not "Gaussian radius of curvature". Consider the two possibilities for "mean" (a, b are the equatorial, polar radii):

• Mean radius of curvature:

${\frac {M+N}{2}}={\frac {{\frac {1}{\kappa _{1}}}+{\frac {1}{\kappa _{2}}}}{2}}={\frac {M}{2}}\!\cdot \!\left(1+{\frac {a^{4}}{(bN)^{2}}}\right)={\frac {N}{2}}\!\cdot \!\left({\frac {(bN)^{2}}{a^{4}}}+1\right);\,\!$

• Radius of mean curvature:

${\frac {2}{{\frac {1}{M}}+{\frac {1}{N}}}}={\frac {2}{\kappa _{1}+\kappa _{2}}}={\frac {1}{H}}={\frac {2M}{1+{\frac {(bN)^{2}}{a^{4}}}}}={\frac {2N}{{\frac {a^{4}}{(bN)^{2}}}+1}}.\,\!$

 ~Kaimbridge~18:12, 15 May 2006 (UTC)

It is correct that the curvature of a curve at a given point is elementary: fit a circle to osculate there, and take the reciprocal of its radius. But as soon as we increase the dimension of our object, whether to a surface embedded in 3D or beyond to a general n-manifold (Riemannian) with no explicit embedding, curvature becomes a much richer phenomenon. No single number, however computed, can capture everything. For example, at a given point on an ellipsoid we can take slices in different directions and get curves with different osculating circle radii. We can prove that these values have a minimum and a maximum (the principal curvatures), and we can combine them into a single number; but how shall we combine them? No matter what choice we make, we lose information.

If we are interested in minimal surfaces, like soap bubbles, we must average the min and max curvatures (obtaining mean curvature). But if we are working with a surface which is not given as an embedding, then we must work intrinsically and use Gaussian curvature. And as we move up in dimension to curved spaces, we encounter options such as sectional curvature, the Riemann curvature tensor, Ricci curvature, and so on.

The broad idea is to have some way to summarize the "shape" of a (well-behaved) geometric object near a given point. (Of course, now that fractal objects have entered the common awareness, we know that curvature may not always be a meaningful measure.) Yes, the choices make learning more of a challenge, but they also make nice pictures. [10][11][12] [13] --KSmrq^T 09:03, 16 May 2006 (UTC)

Right, I realize these things can and do get complicated! P=) For instance, Gaussian may or may not be apparent, depending on an equation's form. E.g., (where $\phi ,\;\alpha \,\!$ is geodetic latitude and its azimuth):

${\frac {1}{\sqrt {\left({\frac {\cos(\alpha )}{M}}\right)^{2}+\left({\frac {\sin(\alpha )}{N}}\right)^{2}}}}={\frac {M\!N}{\sqrt {(M\sin(\alpha ))^{2}+(N\cos(\alpha ))^{2}}}}\,\!$

But, my question, here, isn't so much about its application, but its basic radius-curvature relationship. Maybe it would be better asked another way. How would you define/describe "MN"?: "Radius of Gaussian curvature", "radius of Gaussian curvature²" or "radius² of Gaussian Curvature"? ~Kaimbridge~17:01, 16 May 2006 (UTC)

As follows:

• MN: inverse of Gaussian curvature
• radius of Gaussian curvature: meaningless and undefined notion
• radius of Gaussian curvature²: meaningless and undefined notion
• radius² of Gaussian curvature: meaningless and undefined notion

Not all combinations of mathematical terms have a meaning. This thing just happens to be called "Gaussian curvature". That is an accident of history. It might as well have been called "Intrinsic contortion", in which case you probably wouldn't have thought of asking for its radius. --Lambiam^Talk 18:28, 16 May 2006 (UTC)

Matrix exponentiation

I have a matrix:

0 a b
0 c d
0 e f

that I would like to exponentiate. Can this be done analytically, and if so, what is a good approach? --HappyCamper 19:27, 15 May 2006 (UTC)

Find The Jordan form of A: A = Q^-1UQ. Then exp(A) = Q^-1exp(U)Q. So the problem is reduced to exponentiating a Jordan matrix. Since exponentiation of a block diagonal matrix is done block-wise, the problem is reduced to exponentiating a Jordan block. If the eigenvalue of a block D is λ, then exp(D) = exp(λI) exp(D - λI), so we are left with exponentiating a nilpotent Jordan block. This is easy from the power series. Note, however, that if this is a parametric matrix (a, b etc. are unknown), the result will be a monster. -- Meni Rosenfeld (talk) 19:53, 15 May 2006 (UTC)

Unfortunately, yes, the parameters are unknown. Is there a witty method which I can use to exploit the zero eigenvalue of the matrix, or perhaps use the Cayley-Hamilton theorem? --HappyCamper 19:59, 15 May 2006 (UTC)

I doubt it. I've let Mathematica do this matrix, and the result certainly isn't pretty. So my guess is that any approach would force you to do some dirty work. The approach I described should work here as well - you would obviously have to also divide the problem to distinct cases (unless you are only interested in the probable case that all 3 eigenvalues are distinct). Do the calculations symbolically with the parameters, and denote as auxillary variables expressions that occur frequently (such as ${\sqrt {(c-f)^{2}+4de}}$). Perhaps there is some slightly easier way, but I am skeptical about any miraculous instant solutions. -- Meni Rosenfeld (talk) 20:10, 15 May 2006 (UTC)

Oh, Mathematica can handle this? Hmm...too bad I don't have access to it. --HappyCamper 01:38, 16 May 2006 (UTC)

A free alternative for symbolic mathematics is the venerable Maxima, a decendent of the original MIT Macsyma. The list of computer algebra systems is a valuable resource, since each system has its strengths and weaknesses, including price, specialty, capability, and reliability.

It does appear that the first column of the result is (1,0,0)^T. However, the middle entry of the last row (one of the simpler entries!) causes Mathematica's FullSimplify to produce this mess:

${\frac {2e\exp({\frac {c+f}{2}})\sinh \left({\frac {1}{2}}{\sqrt {(c-f)^{2}+4de}}\,\right)}{\sqrt {(c-f)^{2}+4de}}}$

Recall that the matrix exponential is defined by the same power series as exp, so will involve not only products of the matrix with itself, but also sums. If we can diagonalize the matrix, as A = Q⁻¹DQ, with Q orthogonal and D diagonal, then we get tremendous simplification. First, notice that

A² = (Q⁻¹DQ)(Q⁻¹DQ) = Q⁻¹D²Q.

This generalizes to all powers. Next, notice that linearity of matrix multiplication (on both sides) allows the Q "sandwich" to group all the diagonal matrices inside.

I + Q⁻¹DQ + ¹⁄₂ Q⁻¹D²Q + ⋯ = Q⁻¹(I + D + ¹⁄₂D² + ⋯) Q.

Unfortunately, an algebraic expression for the diagonal form may be messy or impossible (for an n×n matrix the characteristic polynomial has degree n), and not every matrix can be diagonalized (hence the Jordan form). So unless the matrix A has special properties, its exponential won't be pretty. --KSmrq^T 07:56, 16 May 2006 (UTC)

You can use the zero eigenvalue to reduce it to the exponential of a two-by-two matrix, as follows:

$\exp \left({\begin{bmatrix}0&a&b\\0&c&d\\0&e&f\end{bmatrix}}\right)$

${}=\exp \left({\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}{\begin{bmatrix}0&1&0\\0&0&1\end{bmatrix}}\right)$

${}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\left({\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}{\begin{bmatrix}0&1&0\\0&0&1\end{bmatrix}}\right)^{k}$

${}=I+{\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}\sum _{k=1}^{\infty }{\frac {1}{k!}}\left({\begin{bmatrix}0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}\right)^{k-1}{\begin{bmatrix}0&1&0\\0&0&1\end{bmatrix}}$

${}=I+{\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}\left(\sum _{k=1}^{\infty }{\frac {1}{k!}}{\begin{bmatrix}c&d\\e&f\end{bmatrix}}^{k-1}\right){\begin{bmatrix}0&1&0\\0&0&1\end{bmatrix}}$

${}=I+{\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}{\begin{bmatrix}c&d\\e&f\end{bmatrix}}^{-1}\left(\exp \left({\begin{bmatrix}c&d\\e&f\end{bmatrix}}\right)-I\right){\begin{bmatrix}0&1&0\\0&0&1\end{bmatrix}}.$

This shows that the first column is indeed [1,0,0]^T. If you insist, you can continue and get an analytic expression out of it. But it won't be pretty. -- Jitse Niesen (talk) 13:10, 16 May 2006 (UTC)

Shouldn't the exponents in the fourth and fifth forms be k-1? --Lambiam^Talk 13:46, 16 May 2006 (UTC)

Yes, thanks. -- Jitse Niesen (talk) 02:34, 17 May 2006 (UTC)

And hey, we have some analytic expression for the 2 by 2 matrix here: [14]. Thanks everyone! --HappyCamper 03:12, 17 May 2006 (UTC)

Pi

My question is, has anyone ever thought of using percentage, instead of Pi, in finding the circumference of a perfect Circle?

Thanks.

Pi is the ratio of a circle's circumference to its diameter. Presumably, a "percentage" would be a ratio of approximately 314.159%; that is, pi. Or have I misunderstood your question? — Lomn Talk 21:05, 15 May 2006 (UTC)

Hi! Lomn,

Below is an example in using percentage instead of Pi.

Example: 3 x diameter plus the percentage equals the circumference of a perfect Cirle.

I think this is right. I hope this example gives a more clear understanding of my question.

Thanks.

You could say: to get the circumference, take 3 × the diameter, and then add approximately 4.71975511966 percent. But what would the point of doing that? --Lambiam^Talk 00:09, 16 May 2006 (UTC)

Well, something similar was attempted - see Grad (angle). While it is a bit easier for doing arithmetic with angles, when you start getting into stuff like calculus the $pi/100$ factors get very annoying. Confusing Manifestation 01:14, 16 May 2006 (UTC)

As noted above, there's no real purpose to this. Your percentage is still a ratio, and thus still bound to pi. Your equation has become, in effect, c = 3d + (π-3)d. — Lomn Talk 14:57, 16 May 2006 (UTC)

Hi! Lomn,

Well, instead of c = 3d + (π-3)d, how about something more like this; 33.333333333333333333333333333333% of d + 3d = C

Thanks.

You can write it that way. The percentage would be 100 x π - 300. Approximately 14.15926...% of the original value, continuing with the digits of pi. You can't escape from pi. If you want the percentage to add to three times the diameter, divide that percentage by 3. Notinasnaid 18:19, 16 May 2006 (UTC)

It isn't clear whether you thought this formula is correct (which it is not), or whether you are looking for a correct formula of the same form. If the latter, than as stated previously, you'll get that C is approximately 14.15926535897932384626433832795% of d + 3d. Again, no escaping π. -- Meni Rosenfeld (talk) 19:18, 16 May 2006 (UTC)

Ultimately, any means of finding the circumference of a circle based upon its diameter (or radius, or any other term that is a ratio of the diameter) is going to be require a ratio, and that ratio is always pi. The only way to leave pi out of this is to forego the diameter entirely and physically measure a circle's circumference (I'll dodge the question of whether that constitutes mathematics, thanks). — Lomn Talk 20:59, 16 May 2006 (UTC)

I thank you all for helping me in finding an answer to my question. Again, Thanks.

Cauchy & d'alambert convergence criteria

Can anyone give me some "general" guidelines in which when to use each convergence criteria? Thanks.

if you have a certain series, you put it into the formulas of the criterias. if one of the formulas you get seems simpler or easier to prove to pass the criteria use it. if it doesn't work try the other one.

also note that some convergent series pass the cauchy criteria and not the d'alambert, so if you can't prove d'alambert for a series you'll have to try using the cauchy, even though the formula may happen to be much bigger or complicated. and then if this doesn't work too maybe you'll have to use other things ... or actualy find out and prove that the series does not converge. hope this helps. --itaj 00:25, 16 May 2006 (UTC)

thanks

Derivatives of complex exponentials

Suppose I'm considering z^c where z is an element of the cut complex plane (Complex plane excluding the negative real axis), and c is some complex constant. This is defined by z^c = exp(c Log(z) ). Now, if I want to take the nth derivative of z^c, am I allowed to just use normal polynomial term differentiation rules (i.e. d/dz(z^c) = cz^c-1), or do I have to go by the definition and use the chain rule? I tried finding the general term by the definition but it started looking like a horrible mess of product rules and so forth. I'm uneasy about just using the normal standard derivative since c is a complex number and not a real, so is there an easy way to calculate the general nth derivative? Thanks. Maelin 23:20, 15 May 2006 (UTC)

${d \over dz}(z^{c})=cz^{c-1}$ is true in any region where z^c is analytic, because ${d \over dz}(\exp(c\cdot \log(z)))=c{d \over dz}(\log z)\exp(c\cdot \log(z))={c \over z}\exp(c\cdot \log(z))=c\cdot \exp((c-1)\cdot \log(z))$. —Keenan Pepper 02:52, 16 May 2006 (UTC)

Wow, that's nice. So as long as you stick to a single definition of log, the identity is automatically true. – b_jonas 17:56, 16 May 2006 (UTC)

Kuratowski closure axioms - topology defined by closure

about Kuratowski closure axioms. well, i already put this question in the talk page, but got no response. i think it's good enough to be here:

after reading the definition in the article, i can't see how to prove the following using the axioms:
$\bigcap _{i\in I}cl(A_{i})=cl(\bigcap _{i\in I}cl(A_{i}))$ - i.e. the infinite intersection of closed sets is closed.

and that's one of the needed properties of the closed sets in a topology. --itaj 00:02, 16 May 2006 (UTC)

Well, for any j:

• ∩ cl A ⊆ cl A_j (essential property of intersections)
• cl ∩ cl A ⊆ cl cl A_j (axiom 3 implies monotonic)
• cl ∩ cl A ⊆ cl A_j (axiom 2)
• cl ∩ cl A ⊆ ∩ cl A (essential property of intersections)
• cl ∩ cl A = ∩ cl A (axiom 1)

Melchoir 03:51, 16 May 2006 (UTC)

Maybe I should poke back in here and explain: I got that by working backwards. It's like a proof in algebra: start with the statement you want, then look for a slightly simpler statement that implies it. At every step, there's often only one reasonable thing to do. Melchoir 03:59, 16 May 2006 (UTC)

May 16

Weak confluence in abstract reduction systems

It has been stated (attributed to Hindley, by Kleene, by my reference) that the abstract reduction system defined by reduction rules { b -> a, b -> c, c -> b, c -> d} is weakly confluent but not confluent. It is clear that elements b and c are weakly confluent, but why must elements a and d be weakly confluent? In general, if n is a normal form, n cannot be weakly confluent as for n to be weakly confluent, n must have a one-step reduction to some other element, but that immediately contradicts the supposition that n is a normal form -- if this reasoning is valid, then no abstract reduction system with a normal form can be weakly confluent, as an abstract reduction system is said to be weakly confluent if all of its elements are. The conclusion is highly counterintuitive. Is there a problem here? Dysprosia 10:02, 16 May 2006 (UTC)

On the contrary, normal forms are confluent. Local confluence of a means something like: for all b and c, IF (a → b AND a → c) THEN there is some d such that (b →* d AND c →* d). So if a is normal this becomes: for all b and c, IF true THEN whatever. This is true. --Lambiam^Talk 11:49, 16 May 2006 (UTC)

But if a is normal, no such reduction a -> b exists, simply because a is normal. Dysprosia 12:04, 16 May 2006 (UTC)

Sorry, I made a mistake. I meant to write this: "So if a is normal this becomes: for all b and c, IF false THEN whatever." Unlike my previous attempt, this is true. Precisely because no such reduction exists, the confluence condition cannot be violated. --Lambiam^Talk 12:44, 16 May 2006 (UTC)

Ah, I see. I'd forgotten about the wily nature of implication with false antedecents. Dysprosia 12:50, 16 May 2006 (UTC)

Rolling dice

ok, im working on a c++ program that does stuff with dice probability....

basically, what i need to know is this: if one has 4 4-sided dice, how many possible combinations add up to 7? 1123 and 2311 ARE different.... i got 20, but i dont think thats high enough

any help is appreciated

--spuck

I got 20 as well - 12 for 1123, 4 for 1114 and 4 for 1222. That's correct. -- Meni Rosenfeld (talk) 20:17, 16 May 2006 (UTC)

Perhaps the example of a pair of ordinary 6-sided (cube) dice creates false expectations. In that case 7 is the most likely roll, because for each value of the first number there is a suitable value of the second number: 1+6, 2+5, …, 6+1. Even so, only 6 ways are possible.

The real problem with a fair 4-sided die is how to roll it! I'd suggest substituting a regular octahedron (8-sided), with each digit duplicated. ;-)

Speaking as a D&D player who rolls a lot of d4s, I recommend bouncing them off the table. This will give the effect of a "roll". Easier to do than describe! --Sir Ophiuchus 00:34, 18 May 2006 (UTC)

Back to mathematics, the question involves how to count so we know we've seen each possibility exactly once, and a little combinatorics. There is no single "right way" to do this. One natural approach for a program uses recursion: If the first die rolls 1, how many ways can 3 dice roll 6?

For more mathematics, try the article on partition, and this link. --KSmrq^T 10:45, 17 May 2006 (UTC)

Sigma notation

I'm doing a homework assignment about sigma notation and I've come across a problem I can't solve such that if you guys show me how to do it, I can then apply that method to other, similar problems. (I've already checked out the sigma-notation page and I didn't find it to be of much help.) How would you express 35 + 48 + 63 + 80 + 99 in sigma notation?

I think I figured it out. I can't use TeX very well, but this is my solution:

$\sum _{n=6}^{9}35+2n$ —Preceding unsigned comment added by 162.83.220.154 (talk • contribs)

That's not right, it doesn't add up. —Mets501^talk 21:03, 16 May 2006 (UTC)

OK, got it now (with the help of a quadratic regression on my calculator):

$\sum _{n=1}^{5}n^{2}+10n+24$ —Mets501^talk 21:23, 16 May 2006 (UTC)

$\sum _{0\leq n<5}(n^{2}+12n+35)$, or, if you're one-based, $\sum _{0<n\leq 5}(n^{2}+10n+24)$ – b_jonas 21:29, 16 May 2006 (UTC)

Alternately, using binomials instead of powers, $\sum _{0\leq n<5}\left(2{n \choose 2}+13n+35\right)$ or $\sum _{0<n\leq 5}\left(2{n \choose 2}+11n+24\right)$ – b_jonas 21:47, 16 May 2006 (UTC)

Yet more alternatively, recognize that $n^{2}+12n+35$ is $(n+5)(n+7)$, so (changing indices more) it's just $\sum _{n=5}^{9}n(n+2)$, which seems likely to be the "original" form. --Tardis 22:10, 16 May 2006 (UTC)

If you want to be a smartass, you could say $\sum _{1}^{5}a_{n}$ where a₁ = 35, ... Dysprosia

or $\sum _{n=1}^{1}325.$ Or, even better, $\sum _{n=1}^{325}1.$ w00t--Deville (Talk) 22:37, 16 May 2006 (UTC)

Surely the easiest is just $\sum _{n=6}^{10}(n^{2}-1)$. Each of the terms in the series is just one less than a square. Richard B 22:40, 16 May 2006 (UTC)

Yes, that was the easiest. I should have thought of that :-) —Mets501^talk 02:57, 17 May 2006 (UTC)

Surely this is really a case of identifying the series - which has been done with the "n² − 1" above. But how do you identify it? The technique I was taught was try successive differences:

35		48		63		80		99
	13		15		17		19
		2		2		2

And this implies a n² is involved. -- SGBailey 10:02, 17 May 2006 (UTC)

For lazy people (none of us), A005563 = n^2 - 1 is the first sequence found with "35 48 63 80 99" in an OEIS search. --DLL 18:30, 17 May 2006 (UTC)

May 17

Cancelling a sent e-mail on Gmail

I know well, that to cancel a sent e-mail is impossible, unless you have some access to the recipient's mailbox. But, now I got into situation when I need it very much. And, maybe, there is a way to do it after all?

I sent a letter form my gmail address to another person's gmail address, and I don't know the other person's password. I know that gmail made a lot of cool things, such as 2.5 G space, internal search, and stuff. Maybe they did implement the ability to cancel sent e-mails too? It's all on their servers after all. (By the way, many message boards like Vbulletin let you delete your unread PMs). If you know something, your help is very appreciated. I would be ready to pay a reasonable sum of money if you will help me.

Crocodealer 03:17, 17 May 2006 (UTC)

I don't see what this has to do with mathematics. Regardless, I don't believe you can cancel sent emails, and Gmail isn't providing the feature and won't just because you ask them, or throw money at them. Sorry. Dysprosia 04:16, 17 May 2006 (UTC)

It has nothing to do with mathematics, but on Wikipedia:Reference desk it is said: Mathematics. To ask questions about mathematics and computer science.

(Tongue-in-cheek) Let's not be so hasty. The NSA are a major employer of both mathematicians and computers, and are surely well aware of the existence of this gmail. Although it will never be deleted from their files, they seem to have a great deal of influence with US corporations. If you could persuade them that this was in the US national interest, for which the threshold seems quite low, then perhaps they could arrange something. ;-) --KSmrq^T 11:10, 17 May 2006 (UTC)

So, where should I ask this question? Crocodealer

There's no way to cancel a sent email, even though it's gmail. -lethe ^{talk +} 05:17, 17 May 2006 (UTC)

Tell the person that it's a virus and that if they open the email, it will immediately infect their computer. That way, there is at least a possibilty they will delete it without opening it ;) --AMorris (talk)●(contribs) 07:28, 17 May 2006 (UTC)

Oddly enough, on many decent forum software, the PM(Private Messaging) does provide ability to retract a massage before the recipient read it. Regression, huh? ;) --Lemontea 07:54, 17 May 2006 (UTC)

In fact, I have heared about an email client that allows the sender to delete a mail received by you, but I think it's more a bug than a feature and I doubt gmail would have it. – b_jonas 11:00, 17 May 2006 (UTC)

I don't think that's possible with the current e-mail standards. However, Microsoft (in the new Exchange server IIRC) makes use of DRM technology to probably enable a sender to revoke the rights of the user to read the email. Dysprosia 11:05, 17 May 2006 (UTC)

Sure it isn't possible, that's why it only works if the email reader client of the recipient also supports that extension. – b_jonas 22:48, 17 May 2006 (UTC)

It's not a matter of software and "extensions", it's a matter of the underlying protocols supporting it, otherwise one could only send emails to those using the same program. Dysprosia 09:36, 20 May 2006 (UTC)

There's always the obvious answer of just trying asking... did you try that yet? --Welcometocarthage 18:59, 17 May 2006 (UTC) Welcometocarthage

windows xp limited identity accounts

I have installed a game on my computer. When I try to play that game on a limited account, it stated that i need to have direct x 7 to play it. I have direct x 9, and the game works on an admin identity. Can I change settings on the limited accounts somehow to resolve this problem?

I don't see what this has to do with mathematics. Dysprosia 04:16, 17 May 2006 (UTC)

Wikipedia:Reference desk was changed and now states that this page should be used for questions in mathematics and computer science. I guess this includes questions like these. -- Jitse Niesen (talk) 05:03, 17 May 2006 (UTC)

It would be best to possibly rename the page so it is absolutely obvious what should be here and what should not (IIRC the science desk is getting computing related questions too). Dysprosia 10:10, 17 May 2006 (UTC)

We clearly need a computing reference desk. Fredrik Johansson 10:14, 17 May 2006 (UTC)

This would probably be a preferable solution. Dysprosia 10:17, 17 May 2006 (UTC)

I would also like that solution. Right now, it seems like computer tech support and computer science are trying to be split between math and science, which isn't great. -lethe ^{talk +} 02:21, 18 May 2006 (UTC)

It's doubtful; it really depends on what permissions are needed. Those could vary from registry read permissions, to full permissions on a system directory somewhere. If you want to try and track them all down I'd suggest looking at the tools from sysinternals.com, specifically FileMon and Registry Monitor, which may provide useful pointers to track down what's going on. If you have XP Pro you could try creating a user in the Power Users group, and see if the game runs. --Blowdart 10:29, 17 May 2006 (UTC)

Will there be multiple accounts trying to play? If not, it may work to uninstall (as admin) and then reinstall from the limited account. Or here's a thought: contact the game manufacturer. They sold the game, they provided the installer, and they probably have a forum for questions just like this. --KSmrq^T 11:19, 17 May 2006 (UTC)

I'm not sure about that. There's been a fair bit of discussion on the Reference Desk's talk page about how we could split it up further if we decide to do so, and it doesn't sound like a separate Computers desk would be a good idea. Black Carrot 02:18, 18 May 2006 (UTC)

List of mathematical series

Do we have such a list - I can't find one. Or do we rely on links to the maths sequence website? (One entry would be SUM(1+(2*n)) = (n+1)^2) -- SGBailey 11:32, 17 May 2006 (UTC)

There appears to be no such list. There is the article Summation, which contains a list that covers a slightly simpler case and refers to Series (mathematics), which links to Arithmetic series, which covers a more general form of this particular summation. Forgive my ignorance, but what is the maths sequence website? The OEIS? --Lambiam^Talk 11:48, 17 May 2006 (UTC)

Do you mean this? Black Carrot 12:04, 17 May 2006 (UTC)

Yes. Obviously we don't want to duplicate the contents of OEIS, but I would have though a list of a few of the common ones (that used to be in high school maths courses or in "maths table books") would be worth having. -- SGBailey 16:05, 17 May 2006 (UTC)

factorials (!)

a friend of mine on the net tried to do 100,000 factorial (100,000!) on calc.exe partly to test the ability of the computers at his school. apparantly he cant even do it, but with my 1.1Ghz AMD Duron it didnt take longer than a couple minutes.

100,000! = 2.8242294079603478742934215780245e+456573

His challenge is: What is "1,000,000!" ? what is "1,000,000,000!" ? is anyone able to calculate these?

Use logarithm.

Calc 1*2*..*9 using log base 10.

Calc 11*12*..*99 using log base 100.

Calc 111*112*...*999 using log base 1000.

And so on.

It's easy to convert from a result in log base 10 to a number in log base 100. And so on. Ohanian 12:57, 17 May 2006 (UTC)

The number of digits for each for an exact answer would be extremely long. By means of Stirling's approximation, 1000000! has approximately 5565709 decimal digits. 1000000000! will have much much more. Dysprosia 13:06, 17 May 2006 (UTC)

Not directly related to the question here, but you can do really cool stuff with Stirling's approximation. For example, if you flip a fair coin 2n times, what is the probability that you will get exactly n heads and n tails? The exact value is ${\frac {(2n)!}{2^{n}{(n!)}^{2}}}$. But apply Stirling's approximation to the factorials, and for large n this works out to be approximately ${\frac {1}{\sqrt {\pi n}}}$. Chuck 18:22, 17 May 2006 (UTC)

I think you mean ${\frac {(2n)!}{2^{2n}{(n!)}^{2}}}.$ --Deville (Talk) 19:13, 17 May 2006 (UTC)

Yep, you're right. Chuck 19:47, 17 May 2006 (UTC)

And that's actually cool in another way - it looks to me like it's proving a result also derivable from the Central limit theorem, and any time a result can be shown in two completely different ways is a cool thing in my book. Confusing Manifestation 06:21, 18 May 2006 (UTC)

Calc.exe takes minutes to calculate 100,000! approximately? I've written a program that finds the exact value of 100000! in 0.5 seconds (and 1,000,000! in 20 seconds) on my computer. Here's a slightly simplified version of it. 1,000,000,000! is quite a big number, though; it'd take over 3 gigabytes of memory just to store it. As Dysprosia said, you can use Stirling's approximation to find the approximate value. Fredrik Johansson 18:02, 17 May 2006 (UTC)

--- Fredrik, i find the fact that you have such a program very interesting, however... i do not know python well enough to be able to compile it, and in fact the only reason i know about it is its on all the college computers here... and my friend wants me to learn it this summer. Is there any way you could just calculate the numbers for me? or get me the exe? I dont even need the answer with all digits, just something like i posted, like 1.902526*10^6045073 or whatever massive number it comes out to be. Hope you all understand my question now :) -Mike ---

Stirling's formula gives

$\log _{10}n!\approx \left(-n+{\frac {\ln(2\pi )}{2}}+\left({\frac {1}{2}}+n\right)\ln n\right)/\ln 10$

Taking n = 10⁹ gives log₁₀ n! ≈ 8565705522.9958381065, so n! is about 10^0.9958381065 · 10^8565705522 ≈ 9.90463 · 10^8565705522. Likewise, 1000000! is about 8.26393 · 10^5565708. Computing these values by direct multiplication isn't time consuming if you only want a few significant figures; 10⁹! takes only a couple of seconds on a modern computer. (Calc.exe isn't that clever, though, and uses full precision even though most digits will be discarded.) Fredrik Johansson 15:40, 18 May 2006 (UTC)

mid-point scan conversion algorithm for circle drawing

how can I derive decition variable and associated parameters i.e. rate of change of errors for mid-point scan conversion algorithm for circle drawing?

Website with information about "advanced" limit calculations

Hello. I have been searching with Google for a while but could only find information about how to solve very simple limits of both functions and sequences. I'm interested in some place where I can learn how to solve things like (sorry for not knowing wiki math formulation):

    lim              1^10 + 2^10 + 3^10 + ... + (2n - 1)^10 + (2n)^10
                     ------------------------------------------------
   n->infinity                            n^11

     lim               
                 ( 1/x + 1/sqrt(x) ) * (sqrt(x+1) - 1)
     x->0+

Things a bit more difficult. By the way, where in wikipedia can I learn how to write correctly these formulas? Thanks.

Try Wikipedia:Manual of Style (mathematics) but for me it looks like I'd have to learn a bit. --DLL 18:21, 17 May 2006 (UTC)

A good way to learn how to write a formula for Wikipedia is to look at examples. There are two basic approaches at present, one using wiki markup and one using TeX. Your two limits would be done in TeX form, most likely.

$\lim _{n\rightarrow \infty }{\frac {1^{10}+2^{10}+3^{10}+\cdots +(2n-1)^{10}+(2n)^{10}}{n^{11}}}$

$\lim _{n\rightarrow 0+}({\frac {1}{x}}+{\frac {1}{\sqrt {x}}})({\sqrt {x+1}}-1)$

A Wikipedia guide to formula markup can be found at (surprise!) Help:Formula.

As for how to solve these things, it may be worth remembering that summations are the discrete version of integration, and we are not guaranteed obvious or compact solutions. Today's symbolic mathematics programs include sophisticated algorithms for both, but that's not the same as studying a text like Concrete Mathematics to learn techniques for oneself. For fun, I threw your limits at a program and was told ²⁰⁴⁸⁄₁₁ for the first and ¹⁄₂ for the second.

An approach to the second limit might be to try to develop a power series at zero by differentiation. An approach to the first limit might be to try to express the summation in the numerator in closed form.

And folks (everyone posting questions!), please sign your questions by using four tildes, ~~~~; thanks. --KSmrq^T 19:15, 17 May 2006 (UTC)

To find the second limit, use the Taylor series approximation for the square root, √(1+x) ≈ 1 + x/2 for x near 0. Then the expression becomes

$\left({\frac {1}{x}}+{\frac {1}{\sqrt {x}}}\right)\left(1+{\frac {1}{2}}x-1\right)$

which simplifies to

${\frac {1}{2}}+{\frac {\sqrt {x}}{2}}.$

The second term vanishes with x, so the limit is 1/2. Taylor series are generally very useful for calculating limits. Fredrik Johansson 18:36, 17 May 2006 (UTC)

Edit: fixed a mistake. Fredrik Johansson 19:03, 17 May 2006 (UTC)

L'Hôpital's rule may help with the problems, and Help:Formula with the formatting. Chuck 19:07, 17 May 2006 (UTC)

...............................

Maybe this Reference desk is the place :-)

As pointed out in Summation#Approximation by definite integrals, sums can be approximated by definite integrals. For this case, since the function f defined by f(x) = x¹⁰ is monotonically increasing, we have:

$\int _{0}^{2n}x^{10}\,dx\leq \sum _{i=1}^{2n}i^{10}\leq \int _{1}^{2n+1}x^{10}\,dx.$

Using the formula for this type of integral, we find:

${\frac {(2n)^{11}}{11}}\leq \sum _{i=1}^{2n}i^{10}\leq {\frac {(2n+1)^{11}-1}{11}}.$

Dividing everything by n¹¹ gives:

${\frac {(2n)^{11}}{11n^{11}}}\leq {\frac {1}{n^{11}}}\sum _{i=1}^{2n}i^{10}\leq {\frac {(2n+1)^{11}-1}{11n^{11}}}.$

We want to find the limit of the term in the middle for n → ∞. If the two outer terms have the same limit, then so will the middle term. For the leftmost term finding the limit is dead easy: it is the constant value 2¹¹/11. With a small bit of further effort we see that the rightmost term has the same limit.

For the second problem, assuming x > 0:

      $({\frac {1}{x}}+{\frac {1}{\sqrt {x}}})({\sqrt {x+1}}-1)$

$=~({\frac {1}{x}}+{\frac {1}{\sqrt {x}}}){\frac {({\sqrt {x+1}}-1)({\sqrt {x+1}}+1)}{{\sqrt {x+1}}+1}}$

$=~({\frac {1}{x}}+{\frac {1}{\sqrt {x}}}){\frac {x}{{\sqrt {x+1}}+1}}$

$=~({\frac {x}{x}}+{\frac {x}{\sqrt {x}}}){\frac {1}{{\sqrt {x+1}}+1}}$

$=~{\frac {1+{\sqrt {x}}}{{\sqrt {x+1}}+1}}$.

It is now immediate that the one-sided limit for x → 0+ equals 1/2. --Lambiam^Talk 21:52, 17 May 2006 (UTC)

THANK YOU very much too all, guys. You've been extremelly useful. Thank you.

Another way to do this is to note that the formula

$\sum _{k=1}^{N}k^{10}={\frac {N^{11}}{11}}+{\frac {N^{10}}{2}}+{\frac {5}{6}}N^{9}-N^{7}+N^{5}-{\frac {N^{3}}{2}}+{\frac {5}{6}}N$

holds, then plug in and ignore all but the leading order term in the limit. To derive the above formula, one way to do it is make the Ansatz that the sum is an 11th degree polynomial, and plug in N = 1,2,...,12, and then solve a 12x12 system of equations. -- Deville (Talk) 14:35, 18 May 2006 (UTC)

Theorem Problem

I have been working on some math, and I think I might have stumbled upon a new theorem. Is the any way I can check if someone else has already figured my theorem out? And if not, is there someone I should send it too, or should I just not worry about it at all?

--Welcometocarthage 18:46, 17 May 2006 (UTC) welcometocarthage

Those with serious mathematics training would not need to ask; those without are not likely to have discovered a significant new theorem. However, "not likely" does not mean it can't happen. Also, the kind of serious interest that would lead to such a question may be worth nurturing. Mathematics today has many specialties, both in its researchers and in its publications; an appropriate contact would depend on the subject of the proof, and perhaps on area of residence. --KSmrq^T 19:27, 17 May 2006 (UTC)

........................

If you tell us what the theorem is, we might be able to tell whether it appears new, and if so, whether it appears to be of sufficient interest to submit to a journal for publication, and if so probably even which journal. --Lambiam^Talk 20:18, 17 May 2006 (UTC)

Although, in the same way it's a bad idea to post your e-mail address here, it might be a bad idea to publish a new idea you want to keep hold of. Anyone and everyone can see this message board. If I succeed in what I'm working on, I have no intention of mentioning it here until I have full and irrevocable credit for it. So, perhaps telling us the area it's in so we can point you to a specific person or group would be best. Black Carrot 02:13, 18 May 2006 (UTC)

The theorem that I think I have discovered is in the study of Geometry. It is not a very important discovery, though, even if I am right. Where should I go to look for such a theorem? --Welcometocarthage

If this is Euclidean geometry in the plane, and it is relatively simple to state, then it is extremely unlikely you'd be the first to discover something relatively simple and new in the almost 23 centuries such problems have been studied extensively. My first guess would be Napoleon's Theorem, "one of the most-often rediscovered results in mathematics". For starters you might have a look at Euclidean geometry#Classical theorems and see if there is any relation. Another thing to look for is Pappus' Theorem. If you know how a mathematician would formulate the problem, you can do a Google search on some typical parts of the statement. For example the query ["equilateral triangles" centres "form an equilateral triangle"] yields Nappie's theorem among the first few entries.

If you're afraid someone might claim the honour of your result: In my opinion the chance is much less if you publicly reveal it here than if you send it in private to an expert. Euclidean geometry is not fashionable as a research area, you might even say it's kind of dead, so in case the result is known but not well known, it may be hard to find an expert who is likely to know it. --Lambiam^Talk 18:23, 18 May 2006 (UTC)

The Académie of Sciences in Paris, France, used to receive envelopes containing supposed new theorems. Then people would open and read and discuss them. The inventor was protected by that means without the use of patents &c.

Now I would expect at least some people interested in geometry here in WP : authors of geom articles. IANAGE myself, but try searching keywords about your discovery in Mathworld or another maths site. See if there are theorems, corollaries or other formulations for the specific domain you are about. --DLL 21:17, 18 May 2006 (UTC)

Actually, my theorem is in Euclidean geometry and I'm pretty sure someone has thought of it already. If not, then I'll be pretty happy! Here it is, I might as well just show it: If two opposite angles in a trapezoid are supplementary, then the trapezoid is isosceles. See, it's so simple that someone had to think of it first. So.... Has anyone thought of it first? As you can see, it's not useful in any way, but to think that I found something in a 'dead' math.... it makes my mind reel. (This comment is by Welcometocarthage, 00:50, May 19 2006 (UTC))

I don't think this theorem has a name, and I doubt it will get a name, although "Hannibal's Theorem" sounds good. the result is well known in the sense that any mathematician will say, without needing to think about it: "yes, of course". It is the easy consequence of two things, both mentioned in the Trapezoid article: An isosceles trapezoid is one for which the base angles are congruent (just as for triangles), and in a trapezoid a base angle and the adjacent angle are supplementary. As phrased, there is some ambiguity, but that is only in how it is expressed here and not in what is known. So the proof is then essentially: Opposite angle is supplement of base angle as well as of other base angle, so base angles are the same, therefore isosceles. Q.E.D. --Lambiam^Talk 14:56, 19 May 2006 (UTC)

A mathematical statement can be true, yet not rise to the level of being called a theorem or lemma or corollary, and certainly not being given a name. This would be an example. That does not necessarily mean it is without interest, at least to someone. Usually we distinguish as named theorems those statements that are pivotal to the understanding of a topic. A theorem may be trivial to prove, and yet not "obvious", and it may have far-reaching consequences. Consider the "pigeonhole principle"; despite the simplicity of the statement and proof, it is remarkably useful in concrete mathematics, such as combinatorics.

For those who have the itch, there can be great joy in sensing and then proving a new fact, regardless of its historical significance. We honor those who see an influential conjecture, as well as those who prove one, noting that a great deal of time and effort may separate these two events. We may even appreciate a new proof of an established theorem if it provides additional insight.

So enjoy the discovery, and may you have many more. --KSmrq^T 16:33, 19 May 2006 (UTC)

Thank you for all of your input on this. Hope to come up with another question soon!

Comment: While this theorem is a rather "trivial" one, it does demostrate an important proof technique: by finding invariant (mathematics) - that is, things that remain constant even under a change. Take this theorem as an example, we can use interior angle of parallel line to show that the base angles are equal. Then we "slide" the left side of the line to the right to form a triangle. (We can do this because the top and bottom lines are parallel) Now the triangle has equal base angles, so the sides are isosceles. --Lemontea 02:48, 20 May 2006 (UTC)

Codec of audio

Does anyone know of a tool which could help me identify the format (and therefore download the necessary codec(s) for playing it) of an audio file, the way GSpot can do for video files? Thanks in advance! — QuantumEleven 21:55, 17 May 2006 (UTC)

'nix has file, that may help. Dysprosia 01:03, 18 May 2006 (UTC)

Thanks, Dysprosia! I, erm, forgot to mention I still have Windoze... (hides in shame)... — QuantumEleven 06:18, 18 May 2006 (UTC)

There's always something like Cygwin... Dysprosia 06:38, 18 May 2006 (UTC)

The ffdshow audio decoder will help you in two respects. Firstly, it will most likely decode the audio format, and secondly, will give you all the important information about the file. There are a number of sites that offer precompiled binaries for Windows (I would recommend you check out the CCCP, which is a Codec pack that is actually nice and not bloaty. An alternative to ffdshow would be avdump, which is a CLI program and available here [15].

Thanks - I'll take a look! — QuantumEleven 08:09, 21 May 2006 (UTC)

May 18

Election statistics

Please correct me if I'm wrong about any of this background: In the United Kingdom, it is possible, and as far as I know usual, for a government to be "elected" without winning a majority of the votes. This happens because we don't have a system of proportional representation and individual votes select MPs to represent constituencies. These MPs (some 600+ of them) are then assessed and usually one party has a genuine majority of these. This party forms a government. The government, while typically not elected by an actual majority of people is considered to have been given a mandate as strong as the parliamentary majority suggests. For example, I believe the 1997 Labour landslide was not the result of a popular majority, yet the new government was seen as having been given carte-blanche to do its thing. I suppose you could say that the popular vote is "distilled" even further by the party in government's leader being in charge, e.g. UK likes Labour >(elect house of commons)> UK loves Labour >(Labour support Blair as leader)> UK worship Tony Blair.

Anyway, the consensus seems to be that the current system favours the big parties, whereas PR would give an advantage to, for example, the Liberal Democrats, our third party.

So my question is this: is the current "skewing" of the vote a result of our non-PR system or a peculiarity of the UK? Is adopting PR over a UK-type system always likely to benefit "small" parties? If so, what's the theory behind this?

Also, does anyone know any particularly extreme examples of odd election statistics? For example a government being elected (in the standard way - one party, not through coalition etc) while winning fewer votes than another party?

Thanks. --87.194.20.253 13:00, 18 May 2006 (UTC)

As far as I know, non-first past the post systems do tend to give more weight to minority parties. Malaysia would be a good example of how first past the post can be gamed to create an illusion of "worship"; although the government got only >60% of the vote in the 2004 general election, it controls 92% of all Parliamentary seats. You might be interested in our articles about Single Transferable Vote, etc. Unfortunately I'm not well-versed in the math behind how these systems work, so I'll leave it to others. Johnleemk | Talk 13:13, 18 May 2006 (UTC)

It is true that the party which wins the numerical majority is not necessarily going to win the most seats in the UK parliament (or in any other parliamentary system which doesn't use proportional representation). However, this is a pretty unlikely occurrence overall. It would be difficult to really nail down the probability, given that it's hard to figure out the pdf of the way each constituency will vote; but it can only come from a confluence of many unlikely events, so you won't see it so often. As you might know, there is a similar system in the US, and this sort of phenomenon can even happen in the Presidential election. As you might know, in 2000 George W. Bush was elected president although he gained fewer votes than his opponent Al Gore, although Gore didn't have a majority either, just a larger plurality.

As you point out, a possible disadvantage of such a "sectionalized" system (and an advantage of proportional representation) is that sometimes the side with fewer votes wins.

It is also arguable that PR has several disadvantages as well. As you point out, it does allow third parties to flourish more readily, and it's direct(er) democracy. However, it should be noted that since only the absolute number of votes matter, this encourages the parties to concentrate all of their funding and political maneuvering in urban areas, where they're more likely to pick up more votes, and ignore less populated areas. In short, it can be argued that PR encourages "mass marketing" of campaigns. A sectional system requires a party to strive to gain a wide variety of constituencies as opposed to just getting the bodies in the booth.

As for strange occurrences, the one I mentioned about the United States presidential election, 2000 is one example. This is the only time this sort of thing has happened in the States, AFAIK (The same thing happened in 1888, see United States presidential election, 1888 and of course there have been many cases of the elected president not winning the popular vote, but it was always in a different way than this), but I'm sure there are other examples in other countries. -- Deville (Talk) 13:27, 18 May 2006 (UTC)

I believe it's actually fairly normal for the controlling party in the UK not to have a majority of votes, since there are so many candidates in each constituency. If you did it by PR, the very smallest parties and independent candidates would have no chance. It would all be party politics, rather than the individual basis (you vote for the candidate you think will best represent you, not the party) it still is in theory. It is true that PR would give an advantage to 3rd parties, but is that necessarily a good thing? Currently, the results we get are pretty much what people vote for, minus the outliers. Under PR we would get more Lib Dem representation, and more green representation, but also BNP, National Front, UKIP, National Workers etc at the same time as losing all the independent MPs. And with PR, you really would be voting in a presedential figure. Currently, people vote for who they want to represent them and these all govern together. In practice there is a Prime Minister, but really he is just the guy in charge of the largest block of MPs. Every MP is free to vote how they want, and the Prime Minister can't do much about it. If we voted in PR, we would most likely be voting for a single person and I'm unhappy with that kind of leadership for a country. Sorry, that probably got off-topic. Skittle 13:45, 18 May 2006 (UTC)

This is another good point that I hadn't thought to mention. There already has been a lot of criticism of Blair for acting too presidential in the first place and not acting so much like a prime minister, and PR would make things even more so like that. Of course, whether this is good or bad would be quite subject to debate. -- Deville (Talk) 14:16, 18 May 2006 (UTC)

There are many issues raised here. The most mathematical issue is addressed by the article on voting systems. These have surprising consequences, no matter which method is used. Note that classical Athenian democracy was on a small enough scale that no intervening representatives were elected; citizens voted on issues directly. Representatives are used for at least two reasons: (1) to keep the governing body a manageable size, and (2) to insulate decisions from the ill-informed passions of the general public. At least one of these reasons could be considered controversial. ;-) --KSmrq^T 14:39, 18 May 2006 (UTC)

And, indeed, your second point is the main reason to keep the House of Lords full of life peers, completely independently chosen, with full powers to veto any bill. Of course, birth may not be the best principle for choosing them, particularly if it tends to accumulate at the top of the social scale. Maybe a random lottery? Skittle 14:47, 18 May 2006 (UTC)

In New Zealand general election 1978 and New Zealand general election 1981, the New Zealand National Party won a majority of the seats and remained the government even though it won fewer votes than the opposition New Zealand Labour Party. This was one of the factors leading up to the Royal Commission on the Electoral System and a lengthy process which ended with a change in the voting system in the mid 1990s.-gadfium 05:39, 19 May 2006 (UTC)

However, proportional representation is now criticised by some for giving New Zealand's minor parties too much power. The New Zealand First party has done particularly well out of it, having held the balance of power in two of the four elections conducted under this system, despite receiving only 13% and 6% of the vote respectively in each case. -- Avenue 09:25, 19 May 2006 (UTC)

Wow, thanks so much to everyone who chipped in! --87.194.20.253 11:02, 19 May 2006 (UTC)

Also worth looking at is Arrow's impossibility theorem which demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. There is quite a bit of theoretical work on fairness of voting systems, for instance the Gallagher Index measures dis-proportionality. --Salix alba (talk) 12:38, 19 May 2006 (UTC)

Metric

How many miligrams equals a gram? 72.174.30.223 14:30, 18 May 2006 (UTC)

The proper spelling is "milligrams", and the article to read is SI prefix, even though this concerns the metric system. --KSmrq^T 14:44, 18 May 2006 (UTC)

milli always means 'a thousand', and with units means 'a thousandth', so here it indicates 1000 milligrams in a gram. Kilo also means a thousand, but the other way. ie, there are 1000 grams in a kilogram. Looking up milligram might have told you this. Skittle 14:41, 18 May 2006 (UTC)

Or try google calculator: 1 gram in milligrams – b_jonas 15:05, 18 May 2006 (UTC)

1 tonne is equal to 1000 kilograms

1 kilogram is equal to 1000 grams

1 gram is equal to 1000 milligrams

1 milligram is equal to 1000 micrograms

1 microgram is equal to 1000 nanograms

If you look carefully, I think you will detect a subtle trend.

Ohanian 22:24, 18 May 2006 (UTC)

C++

I'm taking Computer Programming II at my high school, but even my teacher doesn't know why my program won't work. The idea of the program is to read two *.dat files: one that has the multiple choice answers for five questions and another that has the number of players, names, and their answers. First the program reads the number of players, and that variable is used in a for loop designed to run once for each player's information. In that for loop it reads the name of the player, and then in a second loop (that runs five times) it in turn reads the answer from one file and compares it to the answer from the other file, incrementing a counter for each one that is correct. After the second for loop, it couts the name and how many questions out of five that they got right. The problem is that at the end of the first user, the variable for the correct answer is never reassigned, and for the rest of the users it for some reason compares all their answers to the answer for #5. My teacher spent like three hours on it last night, but she can't figure out why it won't work. I don't know if it's the program (Microsoft Visual C++) or what, but syntax-wise there shouldn't be a problem. I put the source code up on my subpage. —Akrabbim^talk 15:30, 18 May 2006 (UTC)

Hiya. Instead of

getline(responseIN, name);

try

responseIN >> name;

That will definitely work. (I don't know how the getline function works, but that doesn't look right at all.) Btw, you don't need those swallow lines, as the ifstream class is for breaking a stream up by whitespace and that means you don't need to declare the dummy string. Finally, and this is just a little thing, it would improve readability to leave zeroing correct until just before the "marking" loop. Hope this helps! RupertMillard (Talk) 16:22, 18 May 2006 (UTC)

You are calling ifstream::open on the same object within a loop. If you want to reuse stream objects in that way you need to call ifstream::close and ifstream::clear before reopening. A better solution would be to only read the key file once, use a STL container class. EricR 16:46, 18 May 2006 (UTC)

Eric, he is closing the stream. RupertMillard (Talk) 17:12, 18 May 2006 (UTC)

But not clearing the stream state. Neither ifstream::close nor ifstream::open will clear the eof bit for the stream. To clear the eof bit one needs to call ifstream::clear or declare the stream inside the loop where it is used so the constructor and destructor get called for each iteration. EricR 17:28, 18 May 2006 (UTC)

Well you've lost me now. Being a C man, I don't know very much about C++ at all, although it does work on my computer. RupertMillard (Talk) 18:03, 18 May 2006 (UTC)

From a design perspective, this code has some trouble spots. However, those do not impact its correctness. I'm wondering about the getline, which for an ifstream should, I believe, have another parameter, the maximum number of characters to read.

But perhaps if we fix the design questions the problem will fix itself.

• File operations are typically much more expensive than memory operations, and opening a file especially so. The answers file should be opened and read exactly once, before any further processing, and the results stored in an suitable data structure, such as a Standard Template Library vector. (But this may be beyond what the class has covered so far.)
• The number of answers, 5, is hardwired into the program as a numeric value, but it should be a named constant or — better still — a variable read from the answers file (as is done with the number of names).
• When a file is opened on a stream, it would be wise to test it with good(), not just look for a non-null pointer.
• Good practice is to clear the count and reset the current answer immediately before the tally loop, not afterward.
• Good practice is to name variables more descriptively, like numUsers instead of users, or currentUserNum instead of c.
• In the Real World, a program should defend itself against bad input, such as a line of test responses with the wrong number of answers, either too few or too many. (But this may be more of a burden than this assignment should carry.)

If use of a vector is too advanced, try opening the key file only once, and using seekg(0, ios::beg) to reset before each tally loop.

As soon as possible, learn to use a debugger to step through your program. The discipline to cultivate is "Assume nothing, test everything." In a small program like this it is possible to look at the value of every variable at every step, to verify that things are as expected. It is all too easy for even an experienced programmer to stare at code and not see a mistake, much as we overlook spelling errors in text. The debugger is unencumbered by seeing things as they are expected to be, and will show you things as they truly are. The truth is your friend. --KSmrq^T 18:03, 18 May 2006 (UTC)

Seeking to the beginning of the file at the start of each iteration might not solve the problem. If there is no extra whitespace at the end of the file (or a variable number of values are read as you suggest) the last read operation will set the eof flag for the stream. The seekg function will only move the get pointer, not reset the stream state. EricR 20:35, 18 May 2006 (UTC)

This is a subtle point which is hard to tease out of the reference document, but I see no indication that eofbit is cleared by either a seekg or a close, and so would use a clear for safety, as you suggest. Note that a good() test fails if eofbit is set, providing a hint of this problem. One benefit of coding defensively is that there is less need to track down obscure bugs. :-) --KSmrq^T 11:32, 19 May 2006 (UTC)

Incidentally, to obtain an "official" C++ international standard document requires the exchange of money. What I have linked above is the draft that was approved as the standard, a subtle difference with practical financial benefits. (Note a Technical Corrigendum was incorporated in 2003.) Either way, this is not a hand-holding beginner's guide, nor a vendor's compiler guide; what this is, is the only thing it is safe to rely on (if the vendors conform). --KSmrq^T 14:26, 20 May 2006 (UTC)

                                     by; salazar,johnny c.

help with optimal classification equations

I have been working on formatting a few equations in math notation but don't know if I have them right. Can I get help with them by posting them here? -- PCE 16:18, 18 May 2006 (UTC)

I went ahead and posted them under a new article entitled Optimal Classification. Thanks. -- PCE 16:38, 18 May 2006 (UTC)

Kernels, Anyone?

A kernel is very important to the working of a computer, as I recently found out. My computer is a Windows 98. Whenever my computer runs certain programs, though, it says "Kernel32 has had an illegal operation. Click OK to terminate the program." It drives me nuts, and I have no idea why it does this! So here's my question: why does my computer do this, and what exactly is an illegal operation? --Welcometocarthage

An illegal operation is when the computer tries to do something that's impossible, such as dividing by zero or accessing memory that doesn't exist, or a program tries to do something it isn't allowed to, such as trying to use another program's memory space. It usually happens because the person who wrote the program in question didn't think of all the situations the program might encounter, and so didn't make the program able to deal with them. --Serie 19:36, 18 May 2006 (UTC)

Windows 98 is not a robust operating system, neither in its design nor in its implementation. Combine that with programming mistakes in applications, and the result is any number of possible sources for an error message like this. At least this appears to be terminating only a specific program, not forcing a system reboot (aka Blue Screen of Death). For help tracking down the culprit, try reading the "WHAT IS A KERNEL32 ERROR?" entry on this page. Lucky(?) you, a web search finds many comrades-in-arms for those afflicted by Windows. So whenever you see a message like this, write it down carefully and then search for its key words. The long term solution is to switch to a better operating system. --KSmrq^T 20:15, 18 May 2006 (UTC)

Like Windows 95 (!) — Arthur Rubin | (talk) 20:53, 18 May 2006 (UTC)

Perfecting VB6 code

It seems like VB used to have a wizard or a program that listed variables, created flowcharts and did a whole bunch of other stuff to help programmers keep from making errors that are responsible for crashes. Anybody know where these add ins or add-ons or wizards or whatever are? -- PCE 21:28, 18 May 2006 (UTC)

Euler's Totient Function

Can someone find phi(2695), phi(4312), and phi(5390)? They might all be the same. --71.235.83.132 21:59, 18 May 2006 (UTC)

I get 1680 for all three. --Lambiam^Talk 22:12, 18 May 2006 (UTC)

Postscriptum. 2695 = 5 × 539, 4312 = 8 × 539, 5390 = 10 × 539. Since φ is multiplicative and in all cases the factors are coprime, φ(2695) = φ(5) × φ(539), φ(4312) = φ(8) × φ(539), φ(5390) = φ(10) × φ(539). In the table in Euler's totient function we see that φ(5) = φ(8) = φ(10) = 4 (s.t. you can also find using "mental math"), so indeed all three are equal. --Lambiam^Talk 22:24, 18 May 2006 (UTC)

May 19

Mathcad help

Would anyone know the best way in Mathcad 12 to count the number of specific integer values in an array? -- PCE 11:09, 19 May 2006 (UTC)

If you want the number of occurances of a specific integer use rows(match(myinteger,myarray))

If you want the number of different integers in your array you could try this (sorry for the formatting, mathcad layout is awkward)

n(myarray) = | y <- sort(myarray)
             | n <- 1
             | for j ɛ 1 .. rows(y)-1
             |   n <- n+1 if y_j <> y_(j-1)
             | return n

where I've tried to apprximate the mathcad symbols as well as I can. If your array doesn't just contain integers then you need to AND an integer check on to the end of the n <- n+1 line. There may be a better way though. JMiall 17:06, 19 May 2006 (UTC)

Thanks so much for this information. I had tried to use rows(match(myinteger,myarray)) method but it would not allow me to use an array variable for myinteger but required a scalar variable or a constant instead. I was hoping to find a way to use conventional mathematics symbols to express the idea of an accumulator and a nested array such as x=x+1 and y(x(i)) but Mathcad rejects both of these constructs. Any ideas beside writing a routine or using a Mathcad function that I might use to accomplish both of these tasks using conventional non-program type symbols and structures? -- PCE 19:57, 19 May 2006 (UTC)

Sequence of zeros in pi

Several years ago as an undergraduate I went to a lecture by a PhD student concerning pi, and the only thing I remember about the lecture is that he stated that it had been proved that there was a maximum length that any sub-sequence of all zeros could be within the digits of pi. That is, it would be impossible to find a continuous sequence of zeros greater than [some number] when looking through the digits of pi. I didn't think much of it until now, when learning that pi is thought to be a normal number, and so any sequence of digits should be found with equal probability within pi. This is obviously the opposite of what I remember from the lecture. So did I remember wrong? Does any one know of any proof that remotely sounds like this? — Asbestos | Talk (RFC) 15:32, 19 May 2006 (UTC)

One possibility is that the lecturer knew something that is further unknown, like he found a proof, which unfortunately was accidentally eaten up by his dog, after which the shock was so large that he suffered memory loss and could not reconstruct the proof. Somewhat more likely is that he was mistaken; such things have been known to happen to PhD students. Finally, maybe you indeed remember wrong. The only thing I can think of remotely resembling the claim is that it has been proved that there is no all-zero segment of infinite length, nor any other infinitely repeating sequence of digits. --Lambiam^Talk 19:25, 19 May 2006 (UTC)

May 20

HTML Variable Processing

If I had passed a variable through a URL (eg: www.example.org/index.htm?name=John), is it possible, with only HTML and Javascript availible (i.e. no PHP, ASP etc.), to put that variable of "John" into my page somewhere? Or is it only possible to do so with Server-Side Coding?DanielBC 06:10, 20 May 2006 (UTC)

Well, you could do this in JavaScript:

document.write(location.href.toString().split('=')[1]);

which would output John, but I don't think this is what you need. x42bn6 Talk 06:27, 20 May 2006 (UTC)

You may be able to do it without CGI, but CGI is perfect for this sort of thing. Dysprosia 11:18, 20 May 2006 (UTC)

Thanks to all above, maybe document write is what I need! I will investigate later today - the above example isn't actually what I was intending to do with it - I was planning to create a form/input box, ask the user for a URL, and then use that URL later in the document to create an inline frame to that URL (the whole concept is being used to repeatedly refresh a page so I can watch it for changes). Because I could use docuemnt write to write the whole command of the inlineframe??? Would that work? Thanks again. DanielBC 22:24, 20 May 2006 (UTC)

That sounds like a job for CGI. Dysprosia 01:29, 21 May 2006 (UTC)

Or PHP. Assuming you defined your web server to parse .htm files with the PHP engine (or allowed for a URL like www.example.org/index.php?name=John ). — QuantumEleven 15:22, 22 May 2006 (UTC)

I believe that PHP is CGI. Any language would suffice, like Perl, for example. Dysprosia 22:49, 22 May 2006 (UTC)

The (window).location property refered to above is both read a write, so you can read the current url, extract the query string using location.search.substring(1), build whatever url you need and then do

location=http://www.example.org/page.html to go to a new page. You don't even need a iframe, you could just use a text area in a form and write to that. A good javascript reference is what you need. --Salix alba (talk) 22:47, 21 May 2006 (UTC)

calculus

Explain the importance of calculus.

Calculus is important, because it is an indication that you probably need to brush your teeth more. Dysprosia 10:32, 20 May 2006 (UTC)

Calculus can help you determine how often you need to brush your teeth. -- PCE 10:39, 20 May 2006 (UTC)

Calculus can help you determine how much more often you need to brush your teeth, etc etc. Also, given how much you brush your teeth, Calculus can tell you how clean your teeth are. It's mathematic's magic box of tricks! In all seriousness, I bet your teacher would love it if you did your homework exclusively in terms of personal hygiene metaphors.... --The Gold Miner 12:18, 20 May 2006 (UTC)

You folks deserve a plaque for your responses! --hydnjo talk 12:34, 20 May 2006 (UTC)

Sorry, I just don't understand how dental hygiene is relevant to bezoars. Perhaps someone should look through the disambiguation page to see if another meaning might be intended. --KSmrq^T 12:51, 20 May 2006 (UTC)

Perhaps we're all stoned  ;-)) --hydnjo talk 14:54, 20 May 2006 (UTC)

Perhaps; the word derives from the Greek khálix (χάλιξ), meaning pebble. The Romans used pebbles for gaming and reckoning, from which we also get "calculate". It's not because calculus is hard. ;-) --KSmrq^T 17:14, 20 May 2006 (UTC)

Importance of calculus is important, because it allows some people to do their own homework. Cthulhu.mythos 09:03, 22 May 2006 (UTC)

Importantance of calculus is importanant, because it allows some poeple to do their own homework on the importance of claculus's importance. --Welcometocarthage

It is important because without it, computing theory won't be advanced enough to enable you to type this question out here in the first place. (Yes, I broke the chains of jokes) --Lemontea 13:57, 24 May 2006 (UTC)

one minus the inverse of a number

Is there a name or description for the result of subtracting the inverse of a number from one? -- PCE 10:30, 20 May 2006 (UTC)

Should we assume that you are referring to the multiplicative inverse and not the additive inverse? --hydnjo talk 12:29, 20 May 2006 (UTC)

• multiplicative inverse -- PCE 00:56, 21 May 2006 (UTC)

Well, I don't know a name, but since you say "description"... given x, this is y=1-1/x, which is x/x-1/x, which is (x-1)/x, so it's the number just less than x, divided by x. This increases asymptotically to 1 as x increases, with a y-intercept approaching negative infinity from the positive side and a positive x intercept at 1. I suppose there might also be a name for "any fraction where the numerator is 1 less than the denominator", but I don't know it. Black Carrot 17:35, 20 May 2006 (UTC)

Superparticular number? —Keenan Pepper 19:48, 20 May 2006 (UTC)

Actually Superparticular sounds about as close as I'm going to get. The actual formula for duplicating the results of the relationship I am refering to is: y=1+(-1*|v^(-1*|c|)|) or y equals 1 plus (minus 1 times the absolute value of v raised to the power of (minus 1 times the absolute value of c)) or y=1+(-1*abs(v^(-1*abs(c)))) or $y={1}+\left[{-1*{|a^{\left(-1*|c|\right)}|}}\right]\,$. -- PCE 01:39, 21 May 2006 (UTC)

Not quite, that's (x+1)/x. Although, by extension, I suppose you could call these 'subparticular' numbers. The closest I can get wandering around our articles is the unit fractions, which are what you're subtracting from 1. Black Carrot 20:53, 20 May 2006 (UTC)

I like the term "subparticular number". If we use it in a few places elsewhere on the web first, we can add the term to Wikipedia without violating WP:NOR *evil grin* - Fredrik Johansson 01:06, 21 May 2006 (UTC)

You will probably need at least three printed references older than fifty years to prevent immediate deletion in the Wiktionary unless you happen to know one of the bureaucrats. (Just kidding...not really.) -- PCE 01:49, 21 May 2006 (UTC)

The inverse of the Hölder conjugate? —Ilmari Karonen (talk) 22:15, 21 May 2006 (UTC)

Wow, I didn't understand a word of that article. Black Carrot 22:47, 21 May 2006 (UTC)

Wow youself BC, you're smarter than me and I understood every word ...alright I'm lying. hydnjo talk 22:59, 21 May 2006 (UTC)

I'm not surprised, it's a pretty crap article — the formatting is messy and the definition makes little sense unless you're already familiar with L^p spaces. I'm not quite sure where to start fixing it, though. :( —Ilmari Karonen (talk) 23:45, 21 May 2006 (UTC)

That's one of the inequalities everyone kept using at MOP. I think they even used Hölder as a verb, though not as often as Cauchy. I couldn't do any of it, of course. =P —Keenan Pepper 04:56, 23 May 2006 (UTC)

Radial graph

Does anyone know the technical term for what I call a "radial graph"? It typically looks pentagonal/hexagonal/whatever and it shows different attributes of something. Corporal 16:57, 20 May 2006 (UTC)

A polar graph ? StuRat 16:59, 20 May 2006 (UTC)

It looks somewhat like what I'm looking for, but it doesn't have anything to do with calculus - What's I'm referring to is a graph that would be, say, pentagonal with a line from the center to each vertex representing one of five attributes. A point would be placed on a line somewhere between the center and the corresponding vertex representing the relative "greatness" of that attribute, and then each of those 5 points would be connected by lines and the area inside the lines would be shaded. Corporal 17:04, 20 May 2006 (UTC)

Oh yeah, and I have an example here if that's not clear enough: [16]

I'm tempted to say it's just artistic, but I remember my dad showing me something like that awhile ago (it was to do with his work), and it apparently actually means something. I'll see if he remembers it. Just looking at it, though, it doesn't seem all that useful. Given attributes A, B, C, and D, say, the area is (AB+BC+CD+DA)/2, which doesn't look very informative. Black Carrot 17:28, 20 May 2006 (UTC)

It probably is just artistic, but I personally think it's a better/more concise representation of attributes than just a regular bar graph. Could just be the aestetics though... Corporal 18:21, 20 May 2006 (UTC)

Isometric projection? Also see 3D graph paper. --hydnjo talk 17:31, 20 May 2006 (UTC)

Try "star plot", see e.g. <http://www.itl.nist.gov/div898/handbook/eda/section3/starplot.htm> and <http://www.math.yorku.ca/SCS/sugi/sugi16-paper.html#H1_5:Star>. --Lambiam^Talk 18:38, 20 May 2006 (UTC)

That's it. Thanks! Corporal 20:12, 20 May 2006 (UTC)

Excel calls this a "radar chart". It is also called a "spider chart" or "spider's web chart" - see [17] Gandalf61 12:11, 21 May 2006 (UTC)

DDR calls its graph of this kind the "Groove Radar" ... Confusing Manifestation 05:48, 22 May 2006 (UTC)

"undefined; assuming extern returning int"

In MS Visual Studio 6.0, programming in C. What does that mean for a function? I guess it is defined in another file and its header is included...

Forget this question. It was due to a typo... *laughs*.

Forget what? --hydnjo talk 17:52, 20 May 2006 (UTC)

The anon probably misspelled a keyword, then when s/he tried to compile the program, it did not work, giving the error message above. --HappyCamper 01:06, 21 May 2006 (UTC)

HC, you were requested by the anon to forget that. Did you forget?  ;-) --hydnjo talk 03:42, 21 May 2006 (UTC)

May 21

Conference graph

What is a conference graph? --HappyCamper 01:05, 21 May 2006 (UTC)

That's a graph your 11 year old daughter prepares for you to take to the conference to help you make an illustrated presentation... -- PCE 01:55, 21 May 2006 (UTC)

Googling reveals that is some sort of strongly regular graph. One may need to look deeper for more details. Dysprosia 02:07, 21 May 2006 (UTC)

Consider some researchers who participate in conferences. Let the researchers be the vortices of the weighted graph. For each conference, add 1 to the weight of the edge connecting the two researchers iff they both participated in the conference. What you get is the conference graph. Source. Conscious 20:55, 21 May 2006 (UTC)

Vortices? :-/ CH 10:54, 24 May 2006 (UTC)

http://en.wiki.x.io/wiki/Wikipedia:Reference_desk_archive/Mathematics/March_2006

look for conference graph there they are defined by their relation between the parameters paley graphs are an example, they are the only ones that can have non integer eigenvalues Evilbu 18:20, 28 May 2006 (UTC)

Domain/Range etc. for 3D graph

For 2D graphs you have domain for the x axis and range for the f(x) or y axis. How about for the 3D? Which variable, x,y, or z is traditionally the output? Tuvwxyz 13:18, 21 May 2006 (UTC)

Go back to the definitions of how a 3D graph is formed. One can define a 3D graph in terms of a function of two variables and plotting the height: ie., f(x, y) = z. The domain of the function would be a subset of R², and the range would be a subset of R. Dysprosia 13:23, 21 May 2006 (UTC)

i want to know about this chapter information and how to do it, i need it quiekly please.

i want to know about this chapter: Inequalities,Factorisation,Percentages,Ratios,Square root,Pythagorean Theorem,Polygons,Similar Triangles,this chapter's are from amercian system. —Preceding unsigned comment added by 195.229.242.54 (talk • contribs)

Please either ask a specific question, or visit the Wikipedia articles for those topics.

• Inequality
• Factorization
• Percentage
• Ratio
• Square root
• Pythagorean theorem
• Polygons
• Similar triangles

Pythagorean rotation

What are the angles of the two acute corners on a pythagoreanesque triangle? What is the number of degrees you would need to rotate one from the shortest side(a²) being flat to the hypotenuese(c²) being flat? I need this for a diagram made in Paintshop Pro 6. thank you, good day. --HomfrogHomfrog^{Tell me a story!}_{Contribulations}Homfrog 16:21, 21 May 2006 (UTC)

Okay, I think you mean right-angled triangle. The two acute angles can be anything, as long as they add up to 90 degrees. At a guess, if I've understood you correctly, the angle you want to rotate the triangle should be the same as the top angle, that is the angle of the pointy bit at the top, plus 90. This should be the same as 180 minus the bottom angle. I am assuming for this that your straight side is on the left. If not, the angle would be 360 minus the angle I suggested. Good luck. Skittle 16:43, 21 May 2006 (UTC)

So what if the bottom side was 99 pixels long? tThe unit would be 33 pixels. So how about with that info? --HomfrogHomfrog^{Tell me a story!}_{Contribulations}Homfrog 17:23, 21 May 2006 (UTC)

Knowing the length of a single side doesn't tell me anything about the angles, and I don't know what you mean by 'The unit'. Try looking up right-angled triangle, trigonometry and pythagoras's theorem. See if they help you understand better. There are many different right-angled triangles; they are not all the same shape. Skittle 17:41, 21 May 2006 (UTC)

"Perhaps Homfrog means by "pythagoreanesque": a triangle in which a:b:c = 3:4:5. So a = 3 "units"; if the unit is 33 px, then a = 99 px. The rotation required only depends on the ratio between a and b and is independent of the unit. You'd have to rotate by the supplement of the acute angle at the "a" side, that is, 180° − arctan(4/3) = 128.87° or thereabouts, which is approximately 2.2143 radians. Whether this is clockwise or counterclockwise depends on the orientation of the triangle. The same formula works for all right-angled triangles if you plug in the right values for a and b.

Cubic equations with three real, irrational roots

The equation 8x³ − 6x + 1 = 0 has the solutions sin 10º, sin 50º and −sin 70º, and is solved readily using trigonometric methods. However, attempting to solve the equation using Cardano's method (see Cubic equation) yields some rather nasty expressions, such as

${\frac {1}{2}}{\sqrt[{3}]{-{\frac {1}{2}}+{\frac {1}{2}}i{\sqrt {3}}}}+{\frac {1}{2}}{\sqrt[{3}]{-{\frac {1}{2}}-{\frac {1}{2}}i{\sqrt {3}}}}$

Question 1: Is it possible to reduce this to an expression involving radicals and non-complex rational numbers only?
Question 2: Can anybody give an example of a cubic equation with three different real irrational roots, where the roots can be expressed using radicals and non-complex rational numbers only? --Vibo56 16:58, 21 May 2006 (UTC)

I don't know about the first question, but my guess is that no. About the second, the equation

$x^{3}-({\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}})x^{2}+({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})x-{\sqrt {30}}=0$

Satisfies your conditions. If you want the coefficients to be integers as well, I guess this is equivalent to the first question. -- Meni Rosenfeld (talk) 17:50, 21 May 2006 (UTC)

Yes, I did want the coefficients to be integers. And I suspect you are right that the answer to both questions is no. If so, is anybody aware of a proof? --Vibo56 17:56, 21 May 2006 (UTC)

I think that the answer to Question 1 in general is "no", because of the result refered to here:

"One of the great algebraists of the 20th century, B.L. van der Waerden observes in his book Algebra I, that the Casus Irreducibilis is unavoidable. There will never be an algebraic improvement of the cubic formula, which avoids the usage of complex numbers." [18]

However, it may still be possible that for this particular equation, such a formula exist. Perhaps that reference is enough to get you started if you are really interested. You could also try asking User:Gene Ward Smith. -- Jitse Niesen (talk) 03:05, 22 May 2006 (UTC)

Thank you, Jitse, as far as I can see, the reference answers both questions. And I had no idea that it was this very problem that initiated the study of complex numbers! --Vibo56 16:57, 22 May 2006 (UTC)

• Solving the Casus Irreducibilis, the case of three real roots, of the cubic equation requires the use of complex numbers if you insist on doing it with radicals; this was a strong factor in the adoption of complex numbers by European mathematicians, which may be the first clear example of modern European mathematics getting somewhere which the rest of the world hadn't gotten to earlier and better. The article cubic equations goes into great deal on how you can solve the cubic equation algebraically, not just in terms of transcendental functions, without using complex numbers. In that case, the algebraic funcion C_1/3(x) can be used. While this can be computed in terms of trig functions, that does not make it a transcedental function, any more than the fact that P_1/3(x) = x^1/3 can be computed in terms of logs and exponentials makes it a transcendental function. In general, solving a solvable algebraic equation of degree not a power of two, all of whose roots are real, requires complex numbers if you use radicals but no complex numbers if you use Chebychev radicals C_1/n. The cubic equation article is long--would an article on the Casus Irreducibilis help, I wonder? Gene Ward Smith 19:38, 24 May 2006 (UTC)

Thank you Gene, for commenting my question. I do realize that sin 10º, sin 50º and −sin 70º are algebraic numbers, being solutions of the equation 8x³ − 6x + 1 = 0, but I did insist on expressing the solution with radicals. I'm grateful for the response that it cannot be done without using complex numbers. As to whether there should be a separate article on the Casus Irreducibilis, I think that would be overkill, but it wouldn't hurt if, at the end of the section on Cardano's method, you mentioned the fact that, when applied to an equation with three real, irrational roots, it will always give a solution which includes a sum of two conjugate complex numbers, such that the imaginary parts cancel out. --vibo56 21:47, 25 May 2006 (UTC)

Second thought, maybe for some special classes, it can be done by denesting Nested radicals? (See the reference section, through you may have to use an algebra system) --Lemontea 02:37, 24 May 2006 (UTC)

Well, the Cardano solution is expressed in nested radicals involving complex numbers. I did try to construct cubic equations using nested radicals of real numbers only, but always ended up with irrational coefficients. It is quite some time since this problem nearly drove me nuts (33 years, to be exact), and I didn't have access to a computer algebra system. --vibo56 14:25, 24 May 2006 (UTC)

Anyway, at least now I know that it is impossible for your example above - see Exact trigonometric constants, which said "No finite radical expressions involving real numbers for these triangle edge ratios are possible because of Casus Irreducibilis. 9×2X-sided 70°-20°-90° triangle - enneagon (9-sided) 80°-10°-90° triangle - octakaidecagon (18-sided)" PS:my fault, the reference I cited actually duels with nested square root more, and for general radicals, not all are denestable. PPS:If question 2 doesn't require there to be three real roots, then $x^{3}+6x+2=0$ satisfy the other requirement. --Lemontea 14:50, 24 May 2006 (UTC)

Thanks. The point was that there be three real roots. The real solution to your example (${\sqrt[{3}]{2}}-{\sqrt[{3}]{4}}$) would have satisfied the conditions if only there were two more real, irrational roots that could be expressed in a similar manner. However, the example is only a combination of the equations
$y^{3}=2$ and $z^{3}=4$.
Subtract the equations, i.e. $y^{3}-z^{3}=-2$,
factorize the l.h.s., substitute with $x=y-z$, observe that $yz=2$ and that
$x^{2}+6=(y^{2}+yz+z^{2})$, and you have your equation. --vibo56 19:10, 25 May 2006 (UTC)

Triangle with three right angles

My maths teacher asked me the question if there are any triangles where all the angles are right. After a while (after class) I produced the answer that a quarter of a circle gives such a right angle, but I don't know if that was what my teacher meant. Is it...? Henning 17:01, 21 May 2006 (UTC)

The sum of the angles of a triangle in a plane must equal 180 degrees. Therefore, if one angle is 90 degrees, both the other angles must be less than 90 degrees. You can have such triangles on the surface of a sphere, though. --Vibo56 17:36, 21 May 2006 (UTC)

[Edit conflict]What your teacher probably meant was, instead of drawing a triangle on the plane, draw it on the surface of a sphere. You can draw three quarters of a circle which meet each other at a right angle, and these will be considered as straight line segments in spherical geometry. Your suggestion is also a nice idea, however, a quarter of a circle is not a straight line segment in Euclidean geometry. -- Meni Rosenfeld (talk) 17:37, 21 May 2006 (UTC)

It is possible in spherical geometry, which is non-Euclidean. --Lambiam^Talk 17:43, 21 May 2006 (UTC)

A quarter of a circle is a sector, not a triangle. What's fun is that Jolly's idea show two segments and only one arc, when you have three arcs on a sphere. --DLL 18:07, 21 May 2006 (UTC)

Doesn't really matter if this is what your teacher meant or not - it is still a valid and interesting observation. It sounds as if you have discovered spherical geometry with remarkably little prompting. Here are some follow up questions:

• There are many paths that can be drawn between two points on a sphere - which one is the equivalent of a straight line in Euclidean geometry ?
• Take two "straight line" equivalents on a sphere that both pass through a given point. Do these lines intersect anywhere else on the sphere ?
• Once you know the equivalent of "straight lines" on a sphere, you can define a general triangle - can you show that the sum of the angles of any such triangle drawn on a sphere will always be greater than 180 degrees ?
• Can you find a relationship between the area of a triangle drawn on a sphere with radius 1 unit and the sum of its angles ? Gandalf61 09:14, 22 May 2006 (UTC)

mathematical incompleteness

In regard to the inability to express the dependency x=x+1 in conventional mathematical notation or such possibilities as y(x(i)) is conventional mathematics incomplete or just its notation? -- PCE 18:27, 21 May 2006 (UTC)

In most contexts, x=x+1 is a contradiction, not a dependency. What do you mean by y(x(i))? Perhaps you should be looking at temporal logic? -- EdC 21:44, 21 May 2006 (UTC)

y(x(i)) (where i is the index of x) x(i) is the index of y. Thus an array which uses the contents of another array as its index. -- PCE 22:18, 21 May 2006 (UTC)

Do you mean $y_{\left(x_{i}\right)}$? Arrays typically take integer indices, but indexing sets can be anything you want as long as they're of appropriate cardinality. EdC 22:24, 21 May 2006 (UTC)

"Conventional" mathematical notation can both express the (always false) proposition x=x+1 as well as y(x(i)) — using that arrays are essentially functions defined on a limited domain. Perhaps you could clarify the question; what is it that is not expressed by this? --Lambiam^Talk 22:30, 21 May 2006 (UTC)

I'm trying to do two things... First I want to be able to describe these constructs (which are in common use by computer programmers in virtually all computer launguage) in terms of "conventional" mathematical notation. Second I want to be able to implement these constructs using Mathcad 12 without using Mathcad functions or writing a Mathcad user program. -- PCE 04:15, 22 May 2006 (UTC)

Well, if by x = x + 1 you mean the function $x\mapsto x+1$, you want the lambda calculus or a similar calculus. If you mean the assignment $x{\mbox{ := }}x+1$ then you want to look at theories of programming languages. Whether this is possible in Mathcad depends on how full-featured it is. I haven't used it, but I would guess not - it looks more like an engineering package than a tool for mathematicians. EdC 23:02, 22 May 2006 (UTC)

x = x + 1

logic dictates that x = infinity

Note: infinity is NOT A NUMBER. Therefore x is NOT A NUMBER.

Ohanian 01:06, 22 May 2006 (UTC)

Infinity is not a natural number. There are plenty of theories with infinite numbers. You're right, though; $\aleph _{0}$ or any other infinite cardinal is a solution to $x=x+1$. Not infinite ordinals, though.EdC 23:02, 22 May 2006 (UTC)

Is this mixing apples and oranges? In computer programming languages such as C++, the statement

x = x + 1

does not express equality, but assignment. There is a well-developed mathematical formal semantics of programming languages to give meaning to the notation. As it happens, programming is possible with pure functions having no side effects, and the semantics are generally simpler. But in the context of mathematics this expression describes equality, not assignment. That is, in the mathematical world x might be an unknown value, but it would be the same value on both sides of the equality. --KSmrq^T 03:31, 22 May 2006 (UTC)

Well, x = x + 1 modulo 1! Sounds silly, but this kind of thing is sometimes worth keeping in mind. Melchoir 08:39, 22 May 2006 (UTC)

Huh? Black Carrot 22:34, 22 May 2006 (UTC)

GENEALOGY

What are the stistical odds of a person born in 2000 being a direct descendent of William the Conquerer, born in 1027? Thank You

By direct descendant, do you mean patrilineal descendant? Subject to what constraints? (Where is the person born?) Lacking any historical or genealogical information, the odds are symmetrical: 1/(number of males alive in 1027). In actual fact, the odds are most likely to either be near unity or be zero. See the articles on most recent common ancestor and Y-chromosomal Adam. -- EdC 22:32, 21 May 2006 (UTC)

If you're of Anglo-Saxon descent: pretty high, since his son Henry Beauclerc "is famed for holding the record for the largest number of acknowledged illegitimate children born to any English king, with the number being around 20 or 25". So the likelihood that he has no living descendents Every generation the number of descendents roughly doubles, except that you get duplicates, but you can think of it as drawing 2^G balls from an urn with P balls, where G is the generation number and P is the size of the pool of potential descendants – for most of history largely limited by location, until you get substantial migrations like to America. The probability of not being a descendant then comes to about exp(-2^G/P), which, for G = 32 and P << 360 million, is practically zero. (I'll leave it to other editors to explain why this analysis is way too simplistic.) If you are, however, from Tuva or Mali, your chances are pretty slim. --Lambiam^Talk

Functional equation

Is there such a function f that satisfies f(x+y) + f(x-y) = 0 ? --HappyCamper 21:42, 21 May 2006 (UTC)

The zero function. (Actually, any function with range additive self inverses, but let's keep it simple.) -- EdC 21:50, 21 May 2006 (UTC)

If this holds for all x and y, we have f(x+0) + f(x-0) = 0, or 2f(x) = 0, so f(x) = 0 for all x: the everywhere zero function is the only solution of this functional equation. --Lambiam^Talk 22:00, 21 May 2006 (UTC)

Well, no; $\mathbf {1}$ is a valid solution if ${\mbox{range }}f=\mathbb {Z} _{2}$. --EdC 22:20, 21 May 2006 (UTC)

Excluding funny stuff like ranges with zero divisors, the equation defines a function that is odd symmetric around every point. It's pretty obvious that the only such function is indeed f(x) = 0. —Ilmari Karonen (talk) 23:59, 21 May 2006 (UTC)

Okay, I see what's going on now. Thanks :-) --HappyCamper 02:45, 22 May 2006 (UTC)

May 22

Probability Paradox

Here's something I've been thinking about for a while. In a probabiliby class I had this question:

• Suppose there are two dice: one red, one blue.
• Both dice are rolled, but you do not see the results.

1. What are the odds that both dice show the same result if an observer tells you that the red die shows a 5?
2. What if you are told that at least one of the dice shows a 5?"

The answers are 1/6 and 1/11, respectively, which I understand using Bayes' Rule or just eliminating those dice combinations that don't fit.

Now let's say that I roll the dice and don't look at the results, but instead I just feel one of the dice.

Again I'm determining the probability that both dice show the same result. If I can feel that one of the dice shows a 5, then if I hold on to the die, the situation is analagous to knowing that the red die shows a 5 in the first question.

So the probability is 1/6 that the other die shows a 5, so the odds are 1/6 that both dice are the same.

But now, if I let go of the die, I only know that at least one of the dice showed a 5. So the probability that both show a 5 goes to 1/11.

This seems strange to me. Is the probabiliy really 1/6 if I hold on to the die, and 1/11 if I let it go? --Wyckyd Sceptre 00:53, 22 May 2006 (UTC)

Again I'm determining the probability that both dice show the same result. If I can feel that one of the dice shows a 5, then if I hold on to the die, the situation is analagous to knowing that the red die shows a 5 in the first question.

Wrong! It is not the same. The situation is NOT ANALAGOUS to knowing that the red die shows a 5 in the first question. Unfotunately I don't have time to get you a detailed response. Ohanian 01:00, 22 May 2006 (UTC)

Actually it is and I was wrong myself. Ohanian 03:02, 22 May 2006 (UTC)

Or perhaps th[e] margin is too narrow to contain the answer. Joe 01:28, 22 May 2006 (UTC)

Waaaaay too narrow! hydnjo talk 02:00, 22 May 2006 (UTC)

If you roll two dice, the probability that both dice show the same face is

Pr(r=1,b=1) + Pr(r=2,b=2) + Pr(r=3,b=3) + Pr(r=4,b=4) + Pr(r=5,b=5) + Pr(r=6,b=6) = 6/36 = 1/6

Suppose you put your hand and felt that one of the die is a "FIVE".

It changes nothing. The probability is still 1/6 that both dice show the same face.

Pr(r=5,b=5 | x=5) = Pr(color(x)==blue) * Pr(r=5,b=5|b=5) + Pr(color(x)==red) * Pr(r=5,b=5|r=5)

Pr(r=5,b=5 | x=5) = (1/2) * (1/6) + (1/2) * (1/6) = 1/6 SURPRISE , SURPRISE , SURPRISE !!!

Ohanian 02:49, 22 May 2006 (UTC)

Good grief! I sure hope he doesn't show up at the monty. hydnjo talk 02:59, 22 May 2006 (UTC)

As usual the question does not give the full monty. Let us assume the giver of extra information is incapable of lying, something that was not explicitly stated. There is the "What if you are told that at least one of the dice shows a 5?" What is not stated is whether the teller gives out this extra bit of information on whether there is at least one 5 unconditionally. Consider a teller who tells you this, but only if the red die shows a 5? If only the blue one shows a 5, they keep their mouth shut. This behaviour does not contradict what was stated. But now we're back to 1/6. If the teller tells you at least one die shows a 5 only if exactly one shows a 5, the probability drops to 0. And in that case, if they don't tell you this it goes up to 1/7, since 5 possibilities are left ((1,1), (2,2), (3,3), (4,4), and (6,6)) out of 35 (all 6×6 combinations except (5,5)). --Lambiam^Talk 07:16, 22 May 2006 (UTC)

I would say, the number five has nothing particular here. You could have six different problems $P(1),...P(6)$, where $P(n)$ is the following:

* Suppose there are two dice: one red, one blue.

* Both dice are rolled, but you do not see the results.

# What are the odds that both dice show the same result if an observer tells you that the red die shows a $n$?

# What if you are told that at least one of the dice shows a $n$?"

Of course, all these problems are the same problem. Hence, your second question is equivalent to the following: you roll the dice, you ask your observer, and (s)he tells you, oh, look, we have fallen upon $P(5)$. The odds being perfectly symmetrical, the probability you are looking for must be 1/6. (I hope I did not mess up too much). Cthulhu.mythos 09:23, 22 May 2006 (UTC)

A=(r=6), B=(b=6), C=(r=b). the paradox is that $1/6=P(C|A)\not =P(C|(A\cup B))=1/11$ To resolve it you have to understand that different information is given in these two cases. When you put the die down you forfeit some information and reduce the odds from 1/6 to 1/11. Look at the difference between P((r=6)|(r=6)) and P((r=6) | ((r=6) or (b=6)) ) (Igny 17:36, 22 May 2006 (UTC))

Yep. It is indeed $1/11$. Ashes be scattered over my head. Why do I always mess up conditional probability? Of course: $11$ times out of $36$ there is at least one $5$, and exactly once there are two. Period. Cthulhu.mythos 11:03, 26 May 2006 (UTC)

See Three cards problem for correct calculations with conditional probabilities, and Monty Hall problem for the possible effects of choices as to what (correct) statement the observer chooses to report.

That being said

1. What are the odds that both dice show the same result if an observer tells you that the red die shows a n?
2. What are the odds that both dice show the same result if an observer tells you that the first die looked at shows a n?
3. What if you are told that at least one of the dice shows a n?

Answers are 1/6, 1/6, and 1/11 (if n is chosen by the observer in advance)

— Arthur Rubin | (talk) 19:10, 22 May 2006 (UTC)

Falling Sums

This is something I've been toying with for awhile, and I'd like to know if any of you know if someone else has thought about it before me. Consider the sum ${\begin{matrix}\sum _{i=1}^{a}n=na\end{matrix}}$. Given two such sums, ${\begin{matrix}\sum _{i=1}^{a}n\end{matrix}}$ and ${\begin{matrix}\sum _{j=1}^{b}m\end{matrix}}$, it's easy to determine when they first intersect: ${\begin{matrix}\sum _{i=1}^{a}n\end{matrix}}={\begin{matrix}\sum _{j=1}^{b}m\end{matrix}}={\frac {nm}{gcf(n,m)}}$, with $a={\frac {m}{gcf(n,m)}}$ and $b={\frac {n}{gcf(n,m)}}$. The gcf (or gcd) part can be found quickly using Euclid's algorithm. Now, consider the sum ${\begin{matrix}\sum _{i=1}^{a}n-(i-1)r\end{matrix}}$, the sum of a decreasing linear progression of length a starting at n. (This can be written in closed form in many reasonably compact ways, the best I've found is $na-{\frac {a^{2}-a}{2}}r$.) How might you find the intersections of two such sums, ${\begin{matrix}\sum _{i=1}^{a}n-(i-1)r\end{matrix}}={\begin{matrix}\sum _{j=1}^{b}m-(j-1)r\end{matrix}}$? Is there something related to the gcf that can be used for this? Black Carrot 01:24, 22 May 2006 (UTC)

Mmmh, there is something I must be missing here: what do you mean by "intersecting sums"? Cthulhu.mythos 09:28, 22 May 2006 (UTC)

I think Black Carrot is looking for values of a and b (specially, the smallest ones) such that

$\sum _{i=1}^{a}n-(i-1)r=\sum _{j=1}^{b}m-(j-1)r,$

or, equivalently, for solutions of the Diophantine equation

$na-{\frac {a^{2}-a}{2}}r=mb-{\frac {b^{2}-b}{2}}r.$

-- Jitse Niesen (talk) 09:53, 22 May 2006 (UTC)

Yes, exactly. Well, I'm more interested in the summation part, not so much the Diophantine version of it. And I'm looking at it from the point of view of its rather striking similarity to the least common multiple. Have you heard of anyone else working on this? Black Carrot 22:25, 22 May 2006 (UTC)

solving equation

i want to know how to do it. —Preceding unsigned comment added by 195.229.242.54 (talk • contribs) 07:03, 2006 May 22

There are many types of equations, and equation solving is not a general thing you can learn, but a mixture of theory, techniques and methods that work in some cases but not in others. I don't know your background, but a start would be to buy a high school textbook on Elementary Algebra and study that. If bought used, you might find a cheap copy, although that may be more difficult in the Emirates than in the USA. --Lambiam^Talk 07:31, 22 May 2006 (UTC)

Beta testers wanted

Sorry but i'm not really asking a question here. I made a game similar to notpron on my home page with a code/cipher theme. It doesn't have too many levels (13 and the first 2 are tutorial levels). It is intended for non experts/non geeks so i figured that the kind of people who read this noticeboard would have no trouble with it. If you feel like it, would you take a look at it here and see if i've made any glaring errors or pages that turn out to be impossible to solve. Thanks Theresa Knott | Taste the Korn 15:35, 22 May 2006 (UTC)

The blue on gray text at level two hurts my eyes. There is one item on level 5 that needs to be guessed - probably easy enough for a native English speaker. I'm stuck on level 7. For now. --LarryMac 16:11, 22 May 2006 (UTC)

Now up to 9. Needing a break. --LarryMac 16:33, 22 May 2006 (UTC)

I didn't think 5 involved any guesswork, as it was systematic. 6 I have trouble with as I'm trying to intepret a way to input the password, or something. Skittle 16:46, 22 May 2006 (UTC)

Okay, that was odd. The join between 6 and 7 :-\ Skittle 17:15, 22 May 2006 (UTC)

I see Level 7 twice, with different passwords. --Lambiam^Talk 17:27, 22 May 2006 (UTC)

How odd, that didn't happen to me. Level 8 is annoying me, trying to find which bit matters. Skittle 18:10, 22 May 2006 (UTC)

Current sticking point for me is 11. Layout wise, the text beginning with "hmm" is right on the hoizontal line dividing the window, I almost couldn't see it. (IE 6, for reference). --LarryMac 18:21, 22 May 2006 (UTC)

Another layout thing -- on level 6, the phrase "Crack the code" begins at about 2/3 of the way (left to right), on the same line as the previous level's password. "Crack the" is on that line, "code" is on the next. --LarryMac 18:26, 22 May 2006 (UTC)

At level 5 the name of the crypto method is misspelled: "ea" should be "ae" and next "e" should be "a". --Lambiam^Talk 18:50, 22 May 2006 (UTC)

Level 42 is the best. --DLL 19:37, 22 May 2006 (UTC)

I only made 13 levels so far, so we all know you are full of it :-) Theresa Knott | Taste the Korn 05:01, 23 May 2006 (UTC)

Now accepting all assistance in getting past level 11. --LarryMac 19:51, 22 May 2006 (UTC)

It was boastin'. Stuck on 10 with a treasurus. There's a knot to taste. Theresa, it is a good idea and the pages are just nice. How many levels do you intend to create ? --DLL 22:08, 22 May 2006 (UTC)

There seems to be both a level 8 and a level eight. How odd. Also, are all your puzzles solvable by hand or do some of them require computers? I had to resort to computer decryption for level 8 (not eight), following the advice of LarryMac. Skittle 22:38, 22 May 2006 (UTC)

Ah I've gone wrong with the numbers. The digital fortress one is hand decodeable. Some of the levels - well at least one anyway, a brute force attack is the easiest way _ I suppose you could hand decode if you are truly clever. With a brute force attack using software speeds things up. The software is available as java applets that are easy enough to find on the web ( I hope) Theresa Knott | Taste the Korn 05:12, 23 May 2006 (UTC)

And I agree with LarryMac about the text being over the line, and so hard to read, on 11. And it should say 'interchange'. Skittle 22:45, 22 May 2006 (UTC)

Theresa, where did you go?? That thing I said about the layout on level 6 -- it seems to be that way in a lot of places, and now I'm home and using Firefox. I'm happy to know my new friend Skittle got through level 9, but I'm pretty sure he used the computer for that one too (hey, why would she have a Google link on each page if we weren't supposed to use it?). And I'm still stuck at 11.  :-( --LarryMac 01:18, 23 May 2006 (UTC)

I'm stuck at 11 myself. I thought I had the idea for it, but somehow it's not working out. --Deville (Talk) 01:53, 23 May 2006 (UTC)

WOW thanks what a lot of comments! The spelling mistake on level 11 is now fixed. I originally thought that I'd make 20 levels, so far I've made 13 but run out of ideas :-( so if anyone has any feel free to email me. Right I'd better go and try to fix the text over the line problem that so many are experiencing. I assume skittle has got passed level 11 or did I make it too difficult? A hint for anyone else who is stuck on that level - I missed out the spaces. Theresa Knott | Taste the Korn 04:43, 23 May 2006 (UTC)

OK I think I have solved the layout problem. Hopefully everyone should be able to read the hints now. Theresa Knott | Taste the Korn 04:59, 23 May 2006 (UTC)

My problem with 11 is that the code referenced usually represents characters as a sequence of 8 bits; however we are presented with 12, which works out to one and a half characters; there's barely any room for spaces to have been left out! Sorry, Theresa, but I think you've got some 'splainin' to do. --LarryMac 13:41, 23 May 2006 (UTC)

(Actually i thought it was 7 bits? but anyway) You are making an assumption. You don't have to represent the characters as 8 bits at all. There are other ways of doing it, other number systems. Of course it was pretty sneaky of me to deliberatly chose the characters I chose. Had I chose different letters the code would be much easier to break. Hopefully that's given you enough of a clue. Theresa Knott | Taste the Korn 14:20, 23 May 2006 (UTC)

Right, I didn't want to get into a whole 7 bits vs 8 bits, original standard vs extended, etc. We've all been trying to be coy about describing any of the levels too clearly. Anyway, believe me, I have broken down that string of bits in ways you probably never thought of. I am just going to ignore the whole thing for a while, that's what ultimately got me to U******.  ;-) --LarryMac 14:28, 23 May 2006 (UTC)

I had to go to bed, but am now back with a vengance! Following the clue, I am now re-exploring 11. I, too, broke down that sequence soooo many ways. I can tell you it's not MON or GOK... U****** was surprisingly easy, but I think I was just in 'the zone'. Skittle 17:12, 23 May 2006 (UTC)

Having completed 11, "Curse you!" *shakes fist at sky* And what was with the GAP sentence? Did that ever send me down the wrong road. Skittle 17:14, 23 May 2006 (UTC)

W00T, "Well done you have completed all levels!"

So overall, I'd have to say 11 was the most annoying level. Aesthetically, the color-scheme is a little hard to read, although that seems to be the scheme for your site; I just prefer a little more contrast. A few typos still remain, last I checked -- "Eund of game". But it was quite a challenge. I suppose I'm going to have to actually do some work for the rest of the day. --LarryMac 15:31, 23 May 2006 (UTC)

Done! Still reads 'Eund', but I think we'll just have to wait for you to get back. A slight flaw with level 12, and to a lesser extent 13, is that it only required a quick google and the answer was there. For 12, you only have to google the password. Maybe a combination, such as reversing the words as well? Skittle 17:35, 23 May 2006 (UTC)

This challenge is slightly easy up to level 10. Man, that one is tough!!!

I think it depends on what sort of person you are. I found 10 very easy, but 8 and 11 hard. But 8 was largely because I assumed they were doable in your head, rather than by plugging into a decrypter. After my aborted attempts at that, I'm thoroughly sick of Viginere tables :-) Fun though. Skittle 19:41, 23 May 2006 (UTC)

Yeah, I did level 10 "in my head". But I play with words all the time, so that helped. OTOH, seeing as I am a programmer by trade, the inordinate amount of time it took me to solve 11 should be shameful. --LarryMac 23:04, 23 May 2006 (UTC)

No it's because you are a programmer that you found 11 so hard. You already knew all there was to know about the code in question. Had you not you would have done a google search, found out (what you had forgotton) and solved with minutes.Theresa Knott | Taste the Korn 05:16, 24 May 2006 (UTC)

Thanks soo much everyone. This has been really valuable for me. I have several typos to fix, have to make 12 and 13 more difficult, have to fix the double level problem, I've spotted a problem in the steganography level in firefox but not IE, then I 'm off to google sitemaps to submit a sitemap which i don't know how to do at the moment. But before all that, I'd better go to work - seeing as how they pay me. If anyone can think of any other suitable codes or cyphers do please email them to me. Thanks Theresa Knott | Taste the Korn 05:16, 24 May 2006 (UTC)

The last two levels have been made more difficult, so if anyone gets stuck on them you have skittle to thank. Theresa Knott | Taste the Korn 12:30, 24 May 2006 (UTC)

Argh. I don't see how easy level 5 is. I think I know what to do, I'll try it. Thanks for making another cypher! I needed another web puzzle. :) EdBoy 23:31, 28 May 2006 (UTC)

They are always easy when you know the answer! Theresa Knott | Taste the Korn 07:54, 29 May 2006 (UTC)

PHP

How do you find a string in another string in PHP? i.e. Is there a function to say that "Pokémon/Pikachu" or "Template:Pokémon" contains the string "Pokémon".

Also how is the + tab added to the top of the page? Thanks, Gerard Foley 15:46, 22 May 2006 (UTC)

Maybe you want the strpos function. Returns FALSE if the string is not found, and returns the index of the start of the string if it is. -lethe ^{talk +} 16:02, 22 May 2006 (UTC)

Regular expresions, will also do and offer more powerful searching, be careful with non ascii characters. As to the + tab, this is buried deeply in the SkinTemplate.php code. --Salix alba (talk) 16:23, 22 May 2006 (UTC)

The strpos function was the one I was looking for, thanks, however it seems to be having problems with "é". Any ideas on that? About the + tab, what I meant is how do you get it on a non-talk page like this one. Gerard Foley 01:39, 23 May 2006 (UTC)

To get the + tab to appear on non talk pages will require modification of the existing skin code or better creation of a new skin. I managed to modify my local version of the software to add some custom tabs but its a few hours of code hacking to get this right.

For handeling non-ascii character you may need to use utf8_encode() and utf8_decode(). --Salix alba (talk) 08:26, 23 May 2006 (UTC)

I don't think so, "é" is in ASCII as well. Perhaps there's a problem with entity decoding, or having it munged up in URLs or something. Dysprosia 08:43, 23 May 2006 (UTC)

You don't need to beacuse utf-8 has the nice property that if the utf8-encoding of a string contains the utf8 encoding of anoteher string as a sub-byte-string, than the original character string contains the other character string too. Most other multibyte encodings (e.g. utf-16) don't have this property. – b_jonas 19:23, 23 May 2006 (UTC)

Why is antidifferentiation harder than differentiation?

The article on the antiderivative observes:

Finding antiderivatives is considerably harder than finding derivatives.

which is quite true!

Why is this so? Are there any deep reasons why it should be so hard to find antiderivatives? Is it a fundamental difficulty, one that we would expect even space aliens to understand? or is it an artifact of our mathematical notation, or of our ideas about which functions are "elementary"?

Or perhaps this is the wrong question, and the right question to ask is why it is so easy to find derivatives?

-- Dominus 21:23, 22 May 2006 (UTC)

One way to think of it is that the derivative requires information about a function only in or immediately around a single point, whereas the the antiderivative requires information from all points in an interval. The antiderivative therefore embodies much more structure or information, so it should be expected to be more "difficult". Another argument is that differentiation tends to turn simple expressions into more complicated ones, and going the other way is harder; of course, this holds for all kinds of algebraic manipulation and has nothing to do with derivatives in particular (it is also tautological). Fredrik Johansson 21:41, 22 May 2006 (UTC)

One simple point is that differentiation has the chain rule. But that's really more how than why. Perhaps it connects with Fredrik's point because everything acts linear on a small enough scale (thus allowing the chain rule). --Tardis 21:47, 22 May 2006 (UTC)

It depends on what you mean by "find". At one level the claim is just false; a function is much more likely to have an antiderivative than a derivative. For example, all continuous functions have antiderivatives, but they don't all have derivatives.

While there are senses in which the claim is false, there are also senses in which the claim is true. I am asking only about the latter senses. I cited the Wikipedia article saying that antidifferentiation is hard precisely because I hoped to forestall this kind of pedantic evasiveness. The question I hope to have answered is not about which functions are integrable. It is about the difficulty of finding "closed forms" for the antiderivatives of simple functions. --- Dominus 03:34, 23 May 2006 (UTC)

So the point here is that you should not expect an answer based on analysis; from the point of view of analysis, the claim is false. The senses in which the claim is true are all symbolic/linguistic/formal, and that's where you have to look. In particular you have to specify what a closed form is. The most natural definitions involve giving a set of functions for which we have symbols, and closing under composition. The chain rule gives us symbolic derivatives for all such functions, provided only that we have them for the functions we start with. There's no such rule for antiderivatives. As far as I can tell, that's the entire answer to your question. --Trovatore 04:13, 23 May 2006 (UTC)

If we take "find" to giving a "closed-form" symbolic expression for the function, then I think the issue is that such expressions all denote compositions of some fixed class of functions, all of which have symbolically expressed derivatives (if they didn't we'd invent them), and after that you just use the chain rule. There isn't any such rule for antidifferentiation. If there's a deeper "reason" than that, I don't know it (but see differential algebra for a technology allowing proofs that symbolic antiderivatives of a specified form don't exist). --Trovatore 21:54, 22 May 2006 (UTC)

I think you are missing several important issues here. Why don't we just invent symbolically-expressed antiderivatives? Well, in some cases we do, as with the Li or erf functions. But with derivatives, we don't have to. The class of rational functions is closed under differentiation, but not antidifferentiation. The class of functions formed by the arithmetic operators and fractional powers is closed under differentiation, but not antidifferentiation. And so on. To find a class of functions larger than the polynomials that is closed under antidifferentiation is not easy.

Is it just that there's no analogue of the chain rule for integrating compositions? I'm not sure that even that would solve the problem. Consider the case of $f(x)=x^{n}$, for example. There is a simple, uniform rule for the derivative of these functions for all n. But the corresponding rule for antidifferentiation has a strange exception at n = -1. --- Dominus 03:34, 23 May 2006 (UTC)

As to your first paragraph, the point is that the rational functions are just compositions of the functions: identity, constant, add two quantities, multiply two quantities, divide two quantities. We know how to differentiate all these functions (note that last three are bivariate; for example, for division, we need the partial derivative with respect to both the numerator and the denominator), so now all that's left is to use the (multivariate) chain rule.

Compare to the logarithmic integral; you're trying to integrate log(x)/x. That's a composition of three functions: logarithm, identity, division. All you need to differentiate it is their partial derivatives and the chain rule. When integrating, you don't have anything like the chain rule, so a priori there's no reason to think you should be able to express the integral symbolically without introducing a new symbol (and using differential algebra, you can actually prove you can't).

I can't see making much of the n equals −1 thing; that's part of the base case, and I'm pretty suspicious of attempts to make hay of superficial differences in trivial cases. It's just introducing one more symbol, not infinitely many. --Trovatore 04:03, 23 May 2006 (UTC)

For that matter, why is multiplying harder than factoring? Why is the inverse process ever harder than the forward process? I don't know. -lethe ^{talk +} 21:57, 22 May 2006 (UTC)

Yeah, that's what I was going to say. There only is a method for finding derivatives; all antiderivatives are defined and proven in terms of the derivatives they reflect, hence the name. And factoring is harder than multiplying, as far as I can tell, because substantial amounts of important information are lost when you turn those two numbers into one. Black Carrot 22:03, 22 May 2006 (UTC)

An integer product contains just as much information about its factors as the factors do separately. What is lost? Fredrik Johansson 22:07, 22 May 2006 (UTC)

No, the factors themselves indicate, exactly and quickly, what the factors themselves are. Using this information, and the distributive property, it's possible to multiply two numbers rather quickly into their product(Long multiplication). However, once this long series of miniature products is collapsed back into a single number, all trace of the originals factors is lost. Unless you have a way of finding them again? Black Carrot 01:18, 23 May 2006 (UTC)

Yes, according to the fundamental theorem of arithmetic, all the original factors can be retrieved using any factorization algorithm. Perhaps you're talking about composite factors? I don't think composite factors are so interesting for the discussion, though, since numbers with large and few prime factors are the ones that are difficult to factorize. If you multiply two composite n-digit numbers, the prime factors of the product can at worst have around n/2 digits each, which makes them substantially easier to find than the two n-digit factors you'd get if they were prime. Fredrik Johansson 07:36, 23 May 2006 (UTC)

No, according to the fundamental theorem of arithmetic, the factors exist. It says nothing about retrieving them. No, I don't mean the composite factors, I mean whichever ones you started with, or whichever ones you want. It's the same question. And yes, working with smaller factors it better, but they aren't always available to make it easy. What algorithm, by the way, are you thinking of? Black Carrot 22:14, 23 May 2006 (UTC)

That the prime factors can be found follows almost immediately from the fact that they exist; they are all less than ${\sqrt {n}}$, so all you have to do is try all numbers up to that point (trial division), and you're guaranteed to find all factors in a finite number of steps. Fredrik Johansson 13:23, 24 May 2006 (UTC)

Another reason that antidifferentiation is harder is that it doesn't always have an elementary answer. $e^{-x^{2}}$ has no elementary anti-derivative, so if one had any algorithm to fomrally anti-differentiate, it would need to have available to it at minimum a much larger set of functions to work with. (This is in some deeper way that I don't understand, related to the derivative as a local matter and integral as a global one) Note however that when doing real world problems, it is often easier to integrate (that is to get a good approximate value) than it is to differentiate (that is to get an approximate derivative). This is because global behavior is easy to approximate whereas local behavior is hard to approximate (because small errors can be magnified easily). JoshuaZ 04:19, 23 May 2006 (UTC)

I really don't think it is related in any quotable way to the local-v-global thing. See my responses to Dominus interspersed above. Local-v-global is a concept from analysis, and from the point of view of analysis, antidifferentiation is easier rather than harder. The reasons that symbolic antidifferentiation is harder relate to the "symbolic" part, not the "antidifferentiation" part. --Trovatore 04:26, 23 May 2006 (UTC)

Fredrik: "For that matter, why is multiplying harder than factoring? Why is the inverse process ever harder than the forward process?", "...factoring is harder than multiplying, as far as I can tell, because substantial amounts of important information are lost when you turn those two numbers into one." Now, compare for a moment the difference between the "number of steps" required to multiply two know numbers, say of a modest 20-digit length each, and the "number of steps" required to divide a 40-digit number by every number between 2 and 10²¹. Is there a difference? Do you see the point I was getting at from the beginning? And, to repeat the point I made earlier, there is also a significant difference between "exists", or even "can possibly be found"(since you mention it), and "can practically be found". A somewhat fuzzy line, since there are many calculations of "finite" length and complexity that are still beyond the combined capabilites of all the computing power in current existence. Like, for instance, factoring a thousand-digit number through trial division. Black Carrot 01:42, 25 May 2006 (UTC)

But the issue is not whether it is practical; the point is that if it possible even in theory then indeed information was preserved and not lost. I'd rather say that the information was encrypted (and in a very tangible sense). Fredrik Johansson 08:59, 25 May 2006 (UTC)

Your rewording is indeed marginally superior, and makes use of more precise terminology. Very well, 'It's harder because information is lost' becomes 'It's harder because information is encrypted beyond retrieval'. And in fact, what I was talking about was entirely about practicality. Any misunderstanding on your part was presumably based on my shamefully inexact use of the word 'lost'. I apologize. Black Carrot 17:57, 26 May 2006 (UTC)

Logarithms

Hey!

I had two questions from my Pre-Calc class. I have the answers to the questions but I don't know how to get there. Any help would be great!! Thanks!

(e^x + e^-x)/ 2 = 3

The answers are ln(3+2 sqrt 2) and ln(3-2 sqrt 2)

Multiply through by $e^{x}$ (OK since $x\mapsto e^{x}$ is nonzero); solve for $e^{x}$ as a quadratic; take logarithms.

The second problem is.

2^2x + 2^ x+2 - 12 = 0

The answer is ln 3/ln 2

Rewrite as $(2^{x})^{2}+4\left(2^{x}\right)-12=0$; proceed as above. It's quite sneaky; another way to proceed without knowing the trick is to take a parametric form (e.g. $u=e^{x}$) and solve for that first. EdC 22:33, 22 May 2006 (UTC)

Thank you!!!

Logs

Good evening,

I need serious math wizard aid. The problem is: (e^x + e^ -x)/2 = 3. I need both the answer but also step by step instructions. I am simply lost in the world of logs. Peace be on earth. -Magedelein.

P.s. Also, (2^2x +2^(x+2) - 12 = 0 Help!

THANKS

Don't repeat yourself. The answers to your question are right above. Dysprosia 23:29, 22 May 2006 (UTC)

Both me and my classmate are confused. We don't understand how to solve the problems.

Have you tried reading the responses above to your exact same question? Dysprosia 23:32, 22 May 2006 (UTC)

Yes I have already tried. I was hoping someone could better explain the individual steps.

Why don't you ask for someone to explain the steps above, rather than repeating the question? Dysprosia 23:40, 22 May 2006 (UTC)

Will you please explain the steps of the two problems above. I would greatly appreciate it. Thank you!

• (e^x + e^-x)/2 = 3
• (e^x)(e^x + e^-x)/2 = 3(e^x)
• (e^2x + 1)/2 = 3(e^x) (Don't forget, e^-x = 1/e^x)
• e^2x + 1 = 6(e^x)
• e^2x - 6(e^x) = -1
• (e^x)² - 6(e^x) + 9 = -1 + 9 Completing the square
• (e^x - 3)² = 8
• e^x - 3 = +-sqrt(8) Plus or minus
• e^x = 3 +- sqrt(8)
• x = ln(3 +- sqrt(8)) (This is the only part where logs matter)

• 2^2x + 2^x+2 - 12 = 0
• (2^x)² + 2²(2^x) - 12 = 0
• (2^x)² + 4(2^x) = 12
• (2^x)² + 4(2^x) + 4 = 12 + 4
• (2^x + 2)² = 16
• 2^x + 2 = +-4
• 2^x = -2 +- 4
• x = log₂(-2 +- 4) Black Carrot 01:12, 23 May 2006 (UTC)

Now would be a good time in life to learn how to ask smart questions. Also highly recommended reading is George Pólya's classic advice in How to Solve It. It is impossible to learn how to think by having someone else do the thinking. Consider the fact that two problems are given with answers, a sketch of a derivation of those answers has been provided, and yet the follow-up question highlights nothing specific. There is no "I understand that part but I don't follow this part because …." Foolishly, Black Carrot has now done your homework for you, sparing you the burden of improving your understanding and reasoning abilities. What will you do in all the years after school when you cannot ask someone else to solve your problems for you?

It is OK not to understand. It is OK to struggle awkwardly to gain understanding. It is OK to have to learn how to identify precisely where the confusion lies. It is OK to need work on formulating a clear, directed question. It is not OK to switch off the brain, wave the hands vaguely, and say "I'd don't get it."

In this forum you have access to professional mathematicians who are willing to take the time to explain the concepts of exponents and logarithms and solution techniques with more insight than any single Pre-Calc teacher. That is infinitely more valuable than getting the answer to a homework exercise. Don't waste the opportunity. --KSmrq^T 14:47, 23 May 2006 (UTC)

I do not know what to say after all that, and I'm still not satisfied. There is more convention than knowledge in this :

 ::*(2^x)² + 2²(2^x) - 12 = 0<br> ::*(2^x)² + 4(2^x) = 12.

Thanks to Black Carrot anyway for the pain. Is there some sense in the paraphrase of basic properties ?

Teachers may not be able to imprint those in their pupil's minds. There is plenty of work to do, only when you have motivation it's also fun and exciting. When you decide that it is of no use, you just run to find help. Pre-calc host, it's up to you to decide. --DLL 19:11, 23 May 2006 (UTC)

Asymptotes

When will a rational function have vertical, horizontal, and oblique asymtotes at the same time? Actually, is it possible to have all three?

No. :) Corporal 00:03, 23 May 2006 (UTC)

A rational function cannot have both a horizontal and an oblique asymptote. A horizontal asymptote implies ${\frac {P(x)}{Q(x)}}\to k{\mbox{ as }}x\to \pm \infty$, so P has lower or the same order than Q; an oblique asymptote implies that ord P = ord Q + 1. Indeed, a rational function can have at most one horizontal asymptote (with value the ratio of the coefficients of the two leading terms, if P and Q have the same order; 0 if ord Q > ord P), and similarly one oblique asymptote (with slope the ratio of the coefficients of the two leading terms). It may help to note that polynomials are all asymptotically either odd or even, since the leading term dominates. EdC 06:56, 23 May 2006 (UTC)

May 23

Algebra

Show that:

(a-b)²(a+b) + ab (a+b)=a³+b³

Ahi-crazyi

(Removed homework answer in the hope that questioner hasn't seen it yet Skittle 17:01, 23 May 2006 (UTC))

• see Factorization Black Carrot 00:48, 23 May 2006 (UTC)

Black Carrot, the guidelines for questions at the top of this page clearly and boldly state Do your own homework. Please honor that intent by not answering obvious homework questions like this. The reasons are threefold:

1. Giving homework answers is dishonest.
2. Giving homework answers discourages learning about the mathematics.
3. Giving homework answers encourages more homework questions, which are not the questions we want to see.

The guidelines explain what kinds of homework-related questions are appropriate. Your understanding and cooperation will be appreciated. --KSmrq^T 15:20, 23 May 2006 (UTC)

Mathematical physics problem

Dear Wikipedians:

I encountered some difficulties while answering the question below:

a) Why do swimmers with normal eye-sight see distant objects as blurry when swimming underwater?

b) How do swimming goggles or a swimming mask correct this problem?

My answer for (a) is that the critical angle limits the field of vision of the swimmers to 49 degrees from either side, thus, the outside world is compressed to this circle when the swimmer looks upwards through the naked eye. This compression creates distortion for distant objects, whose light shines in at angles approaching the critical 49 degrees limit -- these distortions are what make distant objects blurry to underwater swimmers.

However, I encounter difficulties when trying to answer (b) assuming (a) is correct. I fail to realize how goggles can correct the apparently incorrigible condition of image compression due to critical angles. I therefore seek your wise counsel on a possible resolution to this question, include revisions to my answer to (a) if that is necessary.

Thanks a million!

206.172.66.9 04:00, 23 May 2006 (UTC)

Two words: refractive index. —Keenan Pepper 04:10, 23 May 2006 (UTC)

And just to be clear, your answer to (a) is totally bogus. =P —Keenan Pepper 04:18, 23 May 2006 (UTC)

A question closely related to this one was answered in the science section of the reference desk a few days ago, see Fish out of water. --vibo56 13:31, 23 May 2006 (UTC)

That's because this question is not about mathematics or computer science; it belongs on the Science desk. The eye acts as a lens with a specific refractive index. If a lens is immersed in a fluid of the exact same refractive index, it becomes optically neutral. The bending (refraction) of light at the surface of a lens is a function of the change in refractive index. The approximate refractive index of air is 1.0002926, while that of water is 1.333. (See list of indices of refraction.) Thus a lens designed to work in air must be different than a lens designed to work in water. Although the human eye is a complex system with multiple components, the effective refractive index is only slightly different from that of water. [19] [20] [21] The eyes of water creatures like fish and cephalopods have a much higher refractive index, so high that they include a gradient to avoid aberrations. And then there are the Anablepidae. --KSmrq^T 17:04, 23 May 2006 (UTC)

Names of geometric shapes

I'm trying to help a young friend with homework. Pentagon, hexagon, septagon, 5, 6, and 7 sided figures etc. Cannot answer the logical questions: Why triangle and not trigon? Rectangle? Why not quadrangle or quadragon? Why are some "sides" and others "angles"? How come there is a pentagon and a pentangle(pentacle)? Or is that just Dennis Wheatley? Would appreciate an answer (or a link) to get me out of trouble. Thanks. --Chaswey 08:44, 23 May 2006 (UTC)

The definition of a rectangle is that it has FOUR right (90°) angles. Triangle because it has THREE angles. Remember there are REGULAR -gons and unregular. An unregular Pentagon is any five-sided geometric shape, where the angles can be anything, while a regular Pentagon has all angles 108°. Thus normally, -angles describe shapes that are defined by their angles, -sides are more loosely defined with the amount of sides, but no specific angles. Does this suffice to answer your question? I see Pentangle directs to Pentagram. Confusing stuff. Be adviced that Pentacle is not a Pentangle by definition. ;) Henning 12:00, 23 May 2006 (UTC)

Of course in English we prefer the term "irregular" to "unregular". -lethe ^{talk +} 15:27, 23 May 2006 (UTC)

Thanks for your reply. It certainly is confusing. Why a rectangle rather than a quadrangle other than that it has four identical angles? I suppose then, using twisted logic, one could say that a triangle (or regular trigon) has three 60° angles, whereas an irregular trigon has three sides but unspecified angles. Thanks again.

I guess the reason for the inconsistent naming is just historical. I don't know. I just wanted to mention that if the name for quadrilateral were going to use Greek roots, it would be *tetragon rather than *quadragon which would be an awkward mixing of Latin and Greek roots (though such formations are not unheard of like sociology). -lethe ^{talk +} 13:31, 23 May 2006 (UTC)

That's exactly my problem in trying to explain to my young friend, inconsistency. I get a blank look. Trying to explain the logic behind geometry when the names do not make sense is something that I don't envy teachers! Perhaps having a degree in Greek and Latin should be a requirement before one is taught geometry (grin). Even then it doesn't make any sense to mix them up. I think I'll just say that it's "historically inconsistent". Oh Dear! Another blank look. Thanks very much for your help.

Well language is fraught with inconsistency. It's nothing particular to geometry. Why "twenty" and "thirty" instead of "twoty" and "threety"? Why "geese" and "children" instead of "gooses" and "childs"? Anyway, I will also mention that of course there is such a word as "quadrangle". It's what you call four dorms arranged around a square courtyard on a college campus. I lived in one myself not too long ago. -lethe ^{talk +} 14:03, 23 May 2006 (UTC)

Yes, quadrangle certainly makes more sense then rectangle, as the dorms are arranged around a square, all angles would be 90°. Why then is quadrangle a specific noun? Just joking, I appreciate what you're saying, language itself is inconsistent.

If I were you, I would tell your friend to feel free to use the words "trigon" and "tetragon" for "triangle" and "quadrilateral"/"rectangle" until he gets more comfortable. Don't let language be a barrier to mathematics; notation is arbitrary, and you're free to use whatever notation you like. One day, when you're not talking about geometry, tell him about how English has a very storied history, with influences from Greek, Latin, German, Norse, Norman traditions, as well as its own Germanic roots. "The problem with defending the purity of the English language is that English is about as pure as a cribhouse whore" - James Nicoli. Anyway, despite all the inconsistency, it's not all that bad really. "Triangle" means three angles. Which is what a triangle has. "Quadrilateral" means four sides, which is what a quadrilateral has. "Rectangle" means right angle, which is what a rectangle has. "Pentagon" means five sides, which is what a pentagon has. Latin and greek are omnipresent in English vocabulary, especially technical vocabulary, and the two languages often exist side-by-side. Get used to it. It happens in medicine as well. And biblical terms (where you may also meet Hebrew *gasp*). -lethe ^{talk +} 14:25, 23 May 2006 (UTC)

You're right of course. It's just that "he" happens to be a very bright 11 year old girl who queries everything. She's not inclined to accept an answer like "I know it doesn't make sense but that's how it is", or words to that effect. I'll close the subject now.

However, many thanks for an interesting discussion.

You know, I've been discussing the disparity between Greek names and Latin names, but I see that your original question, at least in part, was the presence of some names referring to the number of angles versus names referring to number of sides, which I haven't really addressed. In other words, why is the triangle not called a "trilateral", which would be consistent with "quadrilateral", "pentagon", etc (forgetting for the moment the differences between Latin and Greek). In response to this question, I can only say I don't know. Well, I can say one thing: rectangle has to be named so, since its defining characteristic is the right angles. Thus "rectangle" is a kind of quadrilateral with special angles. So that makes sense. But I have no answer for why "triangle" didn't get named "trilateral" or "trigon". -lethe ^{talk +} 15:25, 23 May 2006 (UTC)

Yes, -gons being sides and -angles angles. That was what she first raised. 1. Talking strictly in "sides" (-gons). Ok from pentagon onwards. Trigon has already been discussed. Four sided causes more trouble, quadragon? By the way, let bi-gons be bi-gons (sorry about that). We have discussed rectangle (rhombus? - don't go there). By the way, the Chambers' online dictionary has this:quadrangle noun 1 geom a square, rectangle or other four-sided two-dimensional figure. 2 a an open rectangular courtyard, especially one that is in the grounds of a college or school, etc and which has buildings on all four sides of it; b a courtyard of this kind together with the buildings around it. Often shortened to quad. quadrangular adj. quadrangularly adverb. ETYMOLOGY: 15c: from Latin quadrangulum, from quattuor four + angulus angle.

2. -angle. Having x-angles. pentangle? Maybe or pentacle or pentagram (-agram) hexangle, septangle, octangle?

3. -laterals. bi-, tri-, quadri- OK. Pentilateral? (and onwards?)

How do we ever learn? Parrot fashion is looking good.

So I was going to correct you again for mixing Latin and Greek. But I checked my Lidell and Scott (Greek lexicon). The suffix -gon does not mean side, it means angle (I have been under the impression that it meant "side" for a long time, but I guess I was wrong. skélos is side, as in "isosceles"). Thus everything makes perfect sense: triangle (L), quadrangle (L), rectangle (L) pentagon (Gr), hexagon (Gr), heptagon (Gr), etc. mean three angles, four angles, right angles, five angles, six angles, seven angles. The only outlier is "quadrilateral", but we can replace that with the synonymous "quadrangle".

Now let me get on with the business of correcting your mixings. Never quadragon, but rather quadrangle (L) or tetragon (Gr) (four angles). Never pentangle. Pentagon (Gr) or quintangle (L) (five angles). Never hexangle, rather sexangle (L) or hexagon (Gr) (six angles). Septangle (L) is good (heptangle would be wrong), and octangle (L) is OK (but octagon (Gr) is standard). Pentilateral is no good. Quintilateral (L) or pentascalon (Gr) (five sides). -lethe ^{talk +} 17:25, 23 May 2006 (UTC)

So I think you should probably just tell her that you were translating -gon incorrectly. There is no inconsistency (except for the switch from Latin to Greek after four angles). -lethe ^{talk +} 17:40, 23 May 2006 (UTC)

I'd go with the "one, two, three, many" theory. That is, naming is often individual and unsystematic at first, then later becomes regular. So we refer, generically, to a polygon or an n-gon, which shows that we have a system based on "-gon". We also refer, generically, to a polyhedron, and systematically use the suffix "-hedron". Names like hexagon and tetrahedron follow the system, but names like triangle and cube predate the system and persist. And consider that polygon and polyhedron are themselves members of a series; the generic term is polytope.

In the study of polygons and polyhedra, we must decide what conditions to impose. An equilateral triangle and a square are regular: all the edges are identical and the appearance of each vertex is the same. A right triangle, for example, is not regular; and there are many kinds of quadrilaterals which are not square. The regular pentagram is provocative because it is a regular 5-sided figure, yet different from the pentagon. It is not simple, in the sense that sides cross each other; nor is it convex. The pentagram is a stellation of the pentagon; applied to a regular dodecahedron, stellation produces three new regular polyhedra, again self-intersecting. So what do we name these things? We can't simply say a "regular dodecahedron" for all four variations, even though a great dodecahedron is regular and has the same number of faces and vertices as an ordinary dodecahedron. Likewise, we want a different name for the pentagram to distinguish it from the pentagon. If we don't have historical precedent to guide us, we start inventing naming systems, typically drawing on Greek language roots (because the Greeks are the archetypical great geometers). --KSmrq^T 18:17, 23 May 2006 (UTC)

It's good to have your replies for reference, especially when I get bombarded with questions. However, I think you'll agree, the variance in terminology is a bit of a nightmare for a young person to take in let alone address the mathematical problem itself. I'll definitely close the matter here. Thanks very much for your replies and for the time and effort you have put in. --Chaswey 18:55, 23 May 2006 (UTC)

English is just wierd. – b_jonas 19:20, 23 May 2006 (UTC)

hilbert spaces

Let H be a Hilbert space,xn an orthonormal basis for H.An operator T is isometric iff Txn is an orthonormal sequence

Here's a hint. Isometry means that (Tv,Tv) = (v,v) for all vectors v. If you have a basis, then you can write

$v=\sum _{n}^{\infty }x_{n}(v,x_{n})$

and procede in an obvious way. -lethe ^{talk +} 14:28, 23 May 2006 (UTC)

Try using the quadratic formula. Should work well on a trivial one like this.

complex numbers

it is in mathematics books that complex number consist of real part and imaginary part...the term "imaginary part" confuses me..my question is how a number or integer can be imaginary..thats impossible bcoz if i have 12 bottles in a rack an i take 5 for my friends the remaining bottles are 7..simply it shows there never exists -1 bottles..plz answer this nonsense question!..thanx

The name "imaginary" doesn't really mean anything, it's just a name. Instead of "real" and "imaginary", you can think of a complex number as a point in the xy-plane -- the real part is the x coordinate, the imaginary part the y coordinate. Dysprosia 12:20, 23 May 2006 (UTC)

Complex number history is a good start. --HappyCamper 12:36, 23 May 2006 (UTC)

Some kinds of numbers only make sense in certain contexts. As you pointed out, it doesn't make sense to have a negative number of bottles, nor does it make sense to have a set with 2.5 elements or a rod 5+3i meters long. Complex numbers are very useful in some contexts, though, for example representing alternating current. The magnitude of the complex number represents the amplitude and the argument represents the phase. —Keenan Pepper 16:16, 23 May 2006 (UTC)

One way to visualize the imaginary component of complex numbers is with a simple mechanism. Here the circle rotates and the connecting rod (shown with asterisks) changes angle, while the reciprocating rod (shown with equals signs) moves back and forth. While each rotational position of the circle leads to one and only one position of the reciprocating rod, the reverse is not true. That is, given the location of the reciprocating rod in Position 2, which is the same as Position 4, we have no idea which of the two positions we have for the circle. One way to clarify the situation would be to add an imaginary number describing the height of an imaginary vertical reciprocating rod:

  +---+
 /     \    Position 1
+       +
|       *  *  *  *=================*
+       +
 \     /
  +---+

  +-*-+
 /     *    Position 2
+       + *
|       |    *=================*
+       +
 \     /
  +---+

  +---+
 /     \    Position 3
+       +
*  *  *  *=================*
+       +
 \     /
  +---+

  +---+
 /     \    Position 4
+       +
|       |    *=================*
+       + *
 \     *
  +-*-+

StuRat 16:38, 23 May 2006 (UTC)

Anagrams

I need all the meaningful anagrams of ANLDEGN? Is there is a secret technique for finding meaningful anagrams of lengthy words? (I am only interested in anagrams that are meaningful in English.)Patchouli 15:48, 23 May 2006 (UTC)

Anagram Generator. One of many... --LarryMac 15:50, 23 May 2006 (UTC)

I was looking for something like ENGLAND. ThanksPatchouli 16:20, 23 May 2006 (UTC)

That's the only one in my /usr/share/dict/words, at least. —Keenan Pepper 16:41, 23 May 2006 (UTC)

As for the secret, yes. You have to take each word in a dictionary and sort its letters alphabetically (this is called the alphabetical anagram). For example, you transform "England" to "adeglnn". Then, you have to store these alphabetical anagrams together with the original words, in such a way that they can be indexed by the alphabetical anagram (for example, you sort them). Now the alphabetical anagram of two anagram words is always the same, so if you want to find anagrams of, say, "anldegn", you just have to take its alphabetical anagram "adeglnn", and find that in the anagram dictionary. (Searching for anagram in perlmonks will probably find some examples of this algorithm.) (It's much more difficult that this to find multi-word anagrams. See the gson entry of IOCCC 1992 on that.) – b_jonas 19:09, 23 May 2006 (UTC)

Thanks for the secret.Patchouli 00:26, 24 May 2006 (UTC)

Most elegant parts of mathematics

As a devout follower of the world of math, I always love to see mathematical elegance. What is your favourite part of mathematics and can you suggest some beautiful problems and solutions to me?

It's hard to beat Euler's solution to the Basel problem. Both the proof and its implications (the vast and beautiful subject of analytic number theory). Fredrik Johansson 18:49, 23 May 2006 (UTC)

I like the Riemann hypothesis and all the associated goodness. Absolutely beautiful. --HappyCamper 19:43, 23 May 2006 (UTC)

Euler's formula, Burnside's lemma, Hairy ball theorem. —Keenan Pepper 23:35, 23 May 2006 (UTC)

Random suggestions: the usual suspects, like Galois theory, analytic number theory, integral transforms, plus

• Bollobas, Modern Graph Theory. You're sure to love something in there. For example random graphs, Szemeredi lemma, aw gee, it's all great! And amazingly approachable.
• Wilf, Generatingfunctionology. Great stuff, but ditch his approach and rewrite the entire book using the category theory approach, sometimes called structors or combinatorial species. Beautiful!
• C-star algebras and operator K-theory
• Ergodic theorems (but I repeat myself; see Szemeredi lemma) and symbolic dynamics
• Green's functions

OK, that should keep you busy for a few lifetimes. ---CH 10:38, 24 May 2006 (UTC)

There's this Proofs from THE BOOK book, you know. – b_jonas 18:59, 24 May 2006 (UTC)

Nonlinear PDEs

I have a set of 30 coupled nonlinear partial differential equations. What can be some possible ways to solve such a system? Thanks! deeptrivia (talk) 18:58, 23 May 2006 (UTC)

Add them up and divide by 30.

Yeh i too would also advise this. Basically get a calc. (four fuctioned version is sweet), bang in the numbers, press the big divide key, and bingo, to 12 s.f or woteva, u have your answer.

Hey, that sounds just like what I had a while ago! I got around it by reformulating the problem in terms of tensors, and then things became much, much simpler...are there any particular symmetries you can take advantage of? --HappyCamper 19:11, 23 May 2006 (UTC)

Thanks. How exactly can a problem be reformulated in terms of tensors? Does checking for symmetries simply involve plugging in -x for x, and seeing if the system remains the same. If that's indeed the case, then how can I take advantage of it? Thanks a ton! deeptrivia (talk) 19:21, 23 May 2006 (UTC)

Here are some wild ideas: maybe try taking the Lapalce transform of the entire system and see what happens just for fun? :-) Try to see if an approximation with finite differencing methods is useful. For me, it just so happened that I could use something similar to the idea here: Kronecker_product#Matrix_equations. --HappyCamper 19:29, 23 May 2006 (UTC)

Actually, scratch the Laplace transform - that won't get you anywhere - nonlinear PDEs. You could linearize the equation about some reference, and the solve it within that vicinity. Then repeat in a piecewise fashion... --HappyCamper 19:34, 23 May 2006 (UTC)

Hi, Deeptrivia, for systems of coupled nonlinear PDEs, the only general tool is probably symmetry analysis. I have used this method extensively to find solutions to a variety of such systems. Of course, you can't expect to find a general solution this way. If you know what dimensional analysis is, symmetry analysis is a vast generalization of that. "Naturally arising" systems often do have symmetries, and this often do yield special but still useful solutions. For example, the famous solitonic sech wave solution observed by Russell arises very easily from the symmetry group of the KdV equation.

For systems which arise as the Euler-Lagrange equations associated with some Lagrangian, the symmetry method gives an important connection with Noether's theorem, plus notions of "energy", "momentum", and so forth which are a lot of fun.

Other possibilities: after a transformation, your system might reduce to a linear system. If you are really lucky, your system might be completely integrable.

• Olver, Peter J. (1993). Applications of Lie Groups to Differential Equations. New York: Springer-Verlage. ISBN 0-387-95000-1.
• Drazin, P. G.; Johnson, R. S. (1989). Solitons: an Introduction. Cambridge: Cambridge Univerisity Press. ISBN 0-521-33655-4.

You can also look at several handbooks of differential equations for more ideas. HTH ---CH 10:24, 24 May 2006 (UTC)

I'd throw a computer at it. --Serie 20:39, 24 May 2006 (UTC)

Axiom of choice

Is it possible to construct a question in which you need to invoke the axiom of choice 100 times? --HappyCamper 19:18, 23 May 2006 (UTC)

Prove that L²(Rⁿ) has a Hamel basis for all n ≤ 100. -lethe ^{talk +} 19:27, 23 May 2006 (UTC)

Alright, well...that doesn't count :-) Maybe something more subtle? --HappyCamper 19:32, 23 May 2006 (UTC)

100 is quite a strange number. I know of know naturally occurring instances of the number 100 in set theory. In set-theory (or set-theoretic applications to other branches), one typically deals with very small numbers like 1, 2, 3, or else very large numbers like alef-naught. Thus the only way I can get you anything involving 100 is unnaturally: take the first 100 instances of problem with infinitely many cases. I know there are some large numbers which occur in mathematics, like Graham's number, but I don't know too much about them (but I'm pretty sure that they're different from 100), and I doubt that they have any natural connection with the axiom of choice. I can cook up lots more contrived instances for you if you want though. -lethe ^{talk +} 19:41, 23 May 2006 (UTC)

Actually, what I had in mind was some sort of solution to a Putnam problem, where (I think) a biographer remarked that a US contestant joked with his/her buddies saying that their solution needed to invoke the axiom of choice 256 times, or something like that. --HappyCamper 22:27, 23 May 2006 (UTC)

May 24

median of right triangle

Is it possible to prove that the median of a right triangle drawn to the hypotenuse splits the triangle into two isosceles triangles? Thanks, --Mrmrbeaniepiece 16:29, 24 May 2006 (UTC)

Yes. You're welcome. --Lambiam^Talk 16:54, 24 May 2006 (UTC)

You beat me to it! Anyway, think of the rectangle that has the right triangle's legs as sides. Then it's easy. —Keenan Pepper 16:55, 24 May 2006 (UTC)

A right triangle can be inscribed into a circumference whose center is... Cthulhu.mythos 11:17, 26 May 2006 (UTC)

Two color theorem

From the Four color theorem, we know that no map needs more than four colors to color it in. Doodling in class as a high school student, I always found that designs made entirely out of combinations of closed loops (with as many twists and overlaps as you like) and lines projecting to infinity (with curves, overlaps, etc) required only two colors. Is there a proof for this? — Asbestos | Talk (RFC) 18:22, 24 May 2006 (UTC)

There sure is, and it's much easier than the proof of the four color theorem. In fact, it's a simple corollary of the Jordan curve theorem (which basically says every simple closed curve has an inside and outside). For each region, count the number of curves which it is inside. Color it one color if that number's odd, and the other color if it's even. Then any two regions on opposite sides of a curve have different colors, because they have the same inside/outside status for all the curves except that one. For curves extending to infinity, just project the plane onto the surface of a sphere. —Keenan Pepper 18:44, 24 May 2006 (UTC)

Well, not quite: Asbestos says that the curves can "overlap", which might imply that the curves are not simple. There is probably a straightforward way of repairing that problem, however. Dysprosia 01:51, 25 May 2006 (UTC)

Also if two closed curves can share part of their length then that requires three colors. —Keenan Pepper 03:15, 25 May 2006 (UTC)

Alternately, it follows from the fact that there's no vertex where an odd number of lines meet, so in the dual graph (which you have to color) there are no odd cycles, so it's an bipartite graph. – b_jonas 18:53, 24 May 2006 (UTC)

Thanks! The second answer was what was almost in my head, as I noticed that any terminating line would make an odd number of segments, but I couldn't quite see it. Thanks again! — Asbestos | Talk (RFC) 19:25, 24 May 2006 (UTC)

Google search for two-color map theorem turns up a number of results which, although perhaps not exactly what you want, are closely related. -- Dominus 02:01, 30 May 2006 (UTC)

observatories vs portals

I have come across numerous web pages while researching that claim to be "observatories" for particular subjects. For example: UNDP has "ICT for Development Observatory," or the "Canadian Cultural Observatory", and countless others. What I cannot determine is the definition of an "observatory". How does it differ from a "portal"?

The sudden glut of "observatories" seems to indicate a growing trend and I am wondering if there are any evaluations that have been done on the effectiveness of observatories for gathering and disseminating information?

Many thanks--198.62.158.205

Isn't a portal where you go to be teleported to another planet? If you take an encyclopedia with you, I don't see how an observatory can beat that in terms of disseminating information. Actually, I realize I have no clue how an observatory goes about disseminating information at all. Do these "observatories" have something in common (next to their designating themselves as such)? Are you sure the question is related to mathematics or computer science?

There are some mathematical observatories for example Basic Skills Agency Observatory is one of the major UK government agencies page on numeracy skills. I guess observatory is just the latest buzz word to come out of this years' major structural reform of the UK education system, in the vague hope that a new fancy name will somehow correct all the problems created by last years major structural reform. (Not that I'm cynical) How does if differ from a portal: well portal is so last year and wikipedia now looks very dated. --Salix alba (talk) 20:46, 24 May 2006 (UTC)

You're right, this is a computer science question. I have not used this feature in wikipedia before and therefore posted the question in the wrong section. But it has indeed been a pleasure hobnobbing with mathematicians. You guys are fun.

Slide Rule Hyperbolic Functions

Many slide rules (remember those?) have scales for hyperbolic sine and tangent. I've read that these scales were used for vector analysis (among other things), but even among the many existing slide rule collector web sites I cannot find a description of exactly how they were used, or any sample problems. Slide rules having these scales often include "vector" in their name or description. I've found explanations of the hyperbolic functions, but not how they are used for vector analysis. Thanks. 136.1.1.101 18:59, 24 May 2006 (UTC)

Basically, you need to use Matricies. If you use the Simpson's rule on a matrix to extrapolate the roots then you will be able to solve this. —The preceding unsigned comment was added by Historyfan04 (talk • contribs) 22:02, May 24, 2006 (UTC)

The usage I recall is for evaluating "trigonometric" (or inverse trigonometric) functions of complex arguments, and similar contructs in differential equations. I don't know about "vector analysis". — Arthur Rubin | (talk) 22:56, 24 May 2006 (UTC)

A web search with a few obvious keywords, 'vector "slide rule" hyperbolic', quickly turns up a typical manual. If the manual doesn't link the hyperbolic scales and vectors, then users probably didn't either. --KSmrq^T 01:18, 27 May 2006 (UTC)

May 25

Propositional Calculus

Sorry if I abuse any terminology in what follows. I took a course in the above last year. Once our teacher had set up the basics, but before we discussed deductive proofs, we discussed a method which he called "Beth Trees". Each node (?) of these trees would consist of two sets of sentences (each possibly empty), one of which was said to be true, the other false. From a node, one could, with reference to truth tables established for functions OR, AND, NOT, IMPLICATION, proceed to another node or nodes. For example, given the node "true: [(NOT p)], false: [empty]" you could proceed to the node "true: [empty], false:[p]".

The idea was that you could start with a sentence and reduce it to a set of valuations on the language which satisfied it. You could also establish that a sentence was a tautology or a logical contradiction. And anything you found out was rock solid because it was directly from truth tables, but also more efficient than using truth table brute force.

Anyway, I can't find reference to Beth Trees anywhere - all I want to know is whether they are more usually known by another name?

Thanks --The Gold Miner 02:44, 25 May 2006 (UTC)

Perhaps the term of interest is "Truth Tree". Wikipedia does not have an article, but a web search works. (Howard Pospesel says: The method is also known as the "semantic tableau" test. It was conceived independently by E. W. Beth and Jaakko Hintikka in the 1950's.) --KSmrq^T 03:28, 25 May 2006 (UTC)

Ah, thank you so much, that's the one! Funny, it never occurred to me that Beth would be a person. Hey ho. For anyone interested, I did in fact find a description under Method of analytic tableaux. Thanks again. --The Gold Miner 06:17, 25 May 2006 (UTC)

why earth orbit is elliptical?

sir my question is why the orbit of the earth is not spherical...but i heard that its elliptical...does this is the reason that the no. of days in the year does not remain same???? —Preceding unsigned comment added by 80.247.152.121 (talk • contribs)

I don't know why it's not spherical (you might want to read up about orbits), but the number of days per year adjustment is due to the fact that there are not exactly 365 days in a year, rather, just less than 365.25. Hence, every once in a while, we make a correction to make up for this. x42bn6 Talk 07:18, 25 May 2006 (UTC)

Kepler's laws of planetary motion describe how two bodies orbit each other, the planetary orbit will either be an ellipse, hyperbola or parabola. Spherical orbits are a special case of elliptical orbits and require very special conditions to hold, conditions which are not satisfied by the planets. When you look into it the concept of Year gets quite involved. --Salix alba (talk) 07:56, 25 May 2006 (UTC)

The orbit of a planet around a single star without other major perturbations can never be spherical, but it can be circular. We do not know precisely how the planets formed and assumed their present orbits, but we do know it would be extraordinary to have a perfectly circular orbit. Instead, the stable orbits of the planets are ellipses, differing from circularity by various amounts, measured as eccentricity. The orbit of Earth is only slightly non-circular; the orbit of Mars, more noticeably so. The orbit of Mercury is so close to the Sun that its mostly elliptical orbit shows relativistic precession.

Earth's orbit is such that the orbital period, the time to circle the Sun once, is constant for most practical purposes. The rate at which Earth spins on its axis is nearly constant as well. However, the ratio of the length of a year to the length of a day is not a whole number, but is slightly larger than 365. (The ratio of year to month is also fractional.) Thus a calendar with a fixed number of days in each year will drift noticeably over time, so that instead of (Northern Hemisphere) winter in January it could be in April. Because this was considered undesirable, attempts were made to adjust the calendar by the carefully timed insertion of leap years. (We also have the lesser-known leap seconds to adjust the length of a day.)

So the mathematical reason the number of days in the year changes in our Gregorian calendar is to approximate a fraction. --KSmrq^T 17:41, 25 May 2006 (UTC)

One consequence of the slight ellipticity of the earth's orbit is that the earliest sunset in winter doesn't occur on the same day as the latest sunrise. If I recall, they're about two weeks apart. (In the northern hemisphere, one is mid-December, the other around New Year, but I forget which is which.) Arbitrary username 21:05, 26 May 2006 (UTC)

Diagonalizing a matrix.

How can i find the corresponding Eigen vectors for 3X3 matrix with eigen values labda one =labda two, different from labda three. Nkomali

When a matrix can be diagonalized, it has a complete set of eigenvectors which may be chosen mutually perpendicular, and that span to form a basis. However, when two eigenvalues are equal we may be free to choose the eigenvectors as any two perpendicular vectors within a subspace, but we also have to guard against a matrix like

$A={\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}},$

which cannot be diagonalized. From its characteristic polynomial, det(λI−A) = λ³−7λ²+16λ−12, we know that the eigenvalues are 2, 2, and 3; but it does not have a complete set of eigenvectors. Recall that the idea of an eigenvector is a direction in which the action is pure scaling. In this example, that is true for (1,0,0) and (0,0,1), but not for their mutal perpendicular (0,1,0). One further complication is worth mentioning, though it cannot occur with the specific constraints given. A matrix like

$R={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}$

has only one real eigenvalue and associated real eigenvector. Its characteristic polynomial, λ³−λ²+λ−1 = (λ−1)(λ²+1), does not factor into three roots over the reals.

It is common for the matrices in applications to be symmetric, and these can always be diagonalized. Given a matrix, numerical software ordinarily finds eigenvalues and eigenvectors simultaneously, though there are less satisfactory ways of producing an eigenvector from an eigenvalue when possible. --KSmrq^T 19:13, 25 May 2006 (UTC)

Segmentation Fault

During a 1 user Solaris test the CMS(Central Management Server) crashed. The core file was generated at 2AM, so you check the machine on which the database was running. You find an event logs that says "MS SQL Server: unknown error" and it was generated at around 2AM. You then extract the thread stacks from the core file, and you see that a thread had a segmentation fault (SIGSEGV). At the top of the stack you see that the thread was executing some Database Subsystem code of ours.

Question: What do you think is the most likely cause of the segmentation fault and why?

Probably buggy software. Who dumped core? Dysprosia 11:24, 25 May 2006 (UTC)

I doubt it's being caused by the MS SQL Server code. If you use the command "file core" on the core dump file, it will tell you the name of the program that dumped core. You can use a debugger (like dbx or gdb to examine the call sequence and the arguments being sent to the last call that failed. Most of the time, a segmentation fault is called by a null pointer. (Apologies if you knew this already.) --Elkman 16:39, 25 May 2006 (UTC)

Nitpick: In my experience segfaults are more often caused by bad pointers (e.g. to freed memory) than by null pointers. Null pointers are easy to identify, but bad pointers can be sneaky. —Keenan Pepper 19:35, 25 May 2006 (UTC)

precentages

if a final is 20% of your grade and you get a 95% on it and your grade for the class is 81% what is your new grade?

amanda

Just use weighted average: ${\frac {80\times 81\%+20\times 95\%}{100}}=83.8\%$ --Borbrav 21:56, 25 May 2006 (UTC)

What about this

$(0.95\times 0.2+0.8\times 0.81)\times 100=83.8$

Same thing. Black Carrot 17:51, 26 May 2006 (UTC)

May 26

Why isn't mathematics cool?

I study undergraduate mathematics. I like it and I'm proud of it. The other day I was chatting to a new acquaintance, an arts student. I told her that I studied maths and her response was "Aww, gutted!" which round here means "Awww, that must be terrible, you poor thing!" My initial reaction was one of pity for someone cursed with such bad manners, but it did get me thinking.

Now, the briefest exposure to pop culture will reveal that this attitude (i.e. maths is hard, boring and ostracising) prevails throughout the UK, the US and beyond. But I want to know: is there anywhere in the world where mathematics is cool? I mean, is it a cultural thing, or what? --The Gold Miner 13:43, 26 May 2006 (UTC)

Dunno. I guess that the reason might be that high school-level maths is indeed boring; I mean, all you do are ugly calculations, and no nice reasoning. Cthulhu.mythos 14:28, 26 May 2006 (UTC)

My impression is that a lot of people think that advanced mathematics is cool. Like "oh, you study infinite-dimensional hyperchaotic transfinite differential matrix supermanifolds? Cool! That sounds so much more interesting than my job as a copyeditor or administrative assistant." (NB: there is no such field of mathematics). On the other hand, most innumerate people view people who excel at math as kind of nerdy. So saying you study math is an admission of nerdiness, and you have to deal with the derision until you get to the hyperchaotic supermanifolds. But the question remains: why are people who excel at math viewed as nerdy? As far as I can tell, it's because they are nerdy. I mean, I myself am not (well, I probably am, but am also unable to recognize it), but I know of lots of math/science/engineering types who are either very socially awkward, bad at sports, physically weak, sexually unattractive, extremely eccentric, obsessive about minutiae, or otherwise nerdy. It seems to me that nerdy people are just a lot more common among the mathematically inclined. -lethe ^{talk +} 14:52, 26 May 2006 (UTC)

While I don't know for sure why exactly people think like that, I think I have noticed that a disproportionate number of maths and CS students are people-shy and have difficulty maintaining a conversation on topics, such as art and literature, that involve subjective judgments. By "disproportionate" I mean more than you would expect among a random group of age mates of the same sex. This is self-perpetuating, since it discourages people-oriented students to pursue the study of mathematics. I must further say that a lot of maths is taught in an awful and uninspiring way, giving no hint that this is something you might actually love. More puzzling to me is actually why people can proudly proclaim or even acclaim their advanced and terminal innumeracy, as cogently observed by Hofstadter. Whatever the case, I am convinced both phenomena are related to the breakdown of communication written about in The Two Cultures. --Lambiam^Talk 17:01, 26 May 2006 (UTC)

Perhaps they've never met the mathematicians described in (ISBN 0809249766) and (ISBN 0151581754) and (ISBN 0253211190). I find a widespread misconception that mathematicians spend their time doing numerical calculations. Mathematicians have spoken and written to the general public about topics such as Symmetry (ISBN 0691023743), Geometry and the Imagination (ISBN 0821819984), and even Journey Through Genius: The Great Theorems of Mathematics (ISBN 014014739X). Few people today have been exposed to these. Television shows and movies paint caricatures that do little to dispel the stereotypes. And, finally, mathematical beauty is unlike the beauty of a natural scene, perhaps a sunset, because what is observed is mostly in the mind. --KSmrq^T 23:33, 26 May 2006 (UTC)

Many thanks to everyone who responded! --The Gold Miner 22:40, 27 May 2006 (UTC)

just to answer another part of your question (as a fellow maths undergraduate)- maths is cool in france. i've only had limited discussions with french people but, from that sample, it seems to be that the nation that brought us such un-rigorous and airy-fairy concepts such as human rights and existentialism also values maths. strange but true. (although i probably vould have ran upstairs and told you rather than posted that here, but this was is so much more fun!)

Thanks - you wouldn't be a doll and put the kettle on, would you? --The Gold Miner 18:07, 28 May 2006 (UTC)

Another reason is that true mathematics requires spending a lot of time in your head, which isn't very conducive to social skills. Daniel (‽) 19:05, 29 May 2006 (UTC)

That depends entirely on what you spend your time on. Brilliance and careful thought can be, in theory, applied to anything. Of course, since we're talking about career nerds, it's obvious what trains of thought they've chosen to go with. Black Carrot 20:12, 29 May 2006 (UTC)

Oh, and to answer your question, math reasonably cool where I live, because I go to a magnet high school. A lot of the people I know both are good at math and are perfectly normal highschoolers. I'd be hard to block us out of the social circle in the first place, because there's just too many of us there. Black Carrot 20:16, 29 May 2006 (UTC)

Math is cool as in it's not hated or anything where I live. But that's mostly because my school is about 70% Asian, and you might know that a lot of Asian parents foster their kids' interest in math. --M1ss1ontomars2k4 ^{(T | C | @)} 20:20, 29 May 2006 (UTC)

System of seminorms that induces topology weaker than other system, is a 'weaker system'

Hello,

this question has been bugging me for months. My professor hasn't given me a clear proof, and other sources even say it is wrong. I could really use some advice on this problem :

Let V be a vector space (complex) Let P and Q be two systems of seminorms. A system of seminorms is a set of seminorms that has the filtering and separating property.

Now a system like that gives a topology, we say a subset $E\subseteq V$ is open if and only if for each point p in E there is an open semiball around p completely in E: $\forall p\in E:\exists f\in P:\exists r>0:b_{f}(p,r)\subseteq E$

Now we say a topology is weaker than another, if every open set in the first is also an open set in the second. An alternative way of saying this : topology A is weaker than topology B , if the identity map from the second to the first is continuous.

Now, we say a system of seminorms P is weaker than a system Q on the same space, when for every p in P, there is a seminorm q in Q and a constant C>0 , such that $\forall v\in V:p(v)\leq Cq(v)$

Now it is obvious that if a system is weaker than another, the induced topology is weaker.. but what about the converse? According to my professor, the converse is true. But how to prove that? I've done it for two norms on the same space, but those are two very special cases of systems of seminorms.

Really, all replies welcome!

Evilbu 15:26, 26 May 2006 (UTC)

Given a topology on a locally convex topological vector space, you can arrive at a family of seminorms using the Minkowski gauge. (If the topology is Hausdorff, the family will be separating, but I don't recall at the moment what it means for a family of seminorms to "filter". Can you remind me?). If I were going to attempt to prove your converse, that's how I would do it. -lethe ^{talk +} 15:30, 26 May 2006 (UTC)

Actually, I think the proof is even easier than that. The topology induced by a family of seminorms is the initial topology with respect to those maps. The comparison condition you cite is nothing but the continuity condition for a locally convex space, so the two conditions are equivalent. In other words, given two families of maps A and B, B is stronger than A if and only if all of A are continuous in the initial topology of B (this is the definition of stronger families of maps). But this is nothing but the statement that the initial topology of A is coarser than the initial topology of B. -lethe ^{talk +} 15:38, 26 May 2006 (UTC)

But how does that transfer to systems of seminorms? Why is a finer/coarser topology induced by a stronger/weaker system? Not knowing anything about it, here I am thinking aloud. Let topology Ti be induced by system Si, 1 = 1, 2. Assume T1 is strictly finer than T2. If S1 and S2 are comparable as to strength, clearly if S2 is stronger than S1, then T2 is finer than T1, contradicting the assumption on T1 and T2. Since S1 and S2 are comparable, and S2 is not stronger than S1, we conclude that S1 is stronger than S2. So far so good. But what if S1 and S2 are not comparable? Can you show that there are comparable S1' and S2' inducing, respectively, T1 and T2? (I assume that that is good enough to establish the somewhat imprecisely formulated claim.) --Lambiam^Talk 17:20, 26 May 2006 (UTC)

As far as I can see, the definitions are the same. One family of maps is stronger than another if the second is continuous in the first's initial topology (def). This is equivalent to that the initial topology of the first family be finer than that of the second. As for your question, if the two topologies are not comparable, then the families of seminorms which induce it are also not comparable, a corollary of the fact that the definitions are the same. -lethe ^{talk +} 21:20, 26 May 2006 (UTC)

Assume that the topology T(P) is weaker than T(Q) for some systems of seminorms P and Q. For any $f\in P$and any r>0 $\{v:f(v)<r\}$ is open in T(P), thus it is open in T(Q), so there is $g\in Q$ and r'>0 such that $\{g\leq r'\}\subset \{f<r\}$. Assume that v is such that g(v)=0, thus 0=ag(v)=g(av)<r', so f(av)<r for any a, therefore f(v)=0. Assume $g(v)\not =0$, then

$g\left({\frac {r'v}{g(v)}}\right)=r'\Rightarrow f\left({\frac {r'v}{g(v)}}\right)<r\Rightarrow f(v)<{\frac {r}{r'}}g(v)$ for any v.

(Igny 05:19, 27 May 2006 (UTC))

Rushing Roullette

What are the that someone would lose Rushing Roullette if you played 2 times in a row? What about 3,4,5 times in a row? More importantly, what equations would one use to solve this?199.201.168.100 15:53, 26 May 2006 (UTC)

Do you mean Russian roulette ? First, you would need to know how many chambers the revolver had and how many bullets were loaded. If there was 1 bullet and 6 chambers, then the odds of dying on the first shot would be 1/6 (assumming a 100% death rate if the bullet is in the chamber you try to fire). If the player survives and the barrel is spun after each trigger pull, then the odds would be the same (1/6) every time after, as well. Otherwise, if the player survives and barrel is NOT spun, the revolver advances to the next chamber, the odds would be 1/5, then 1/4, 1/3, 1/2, 1/1 (this means, if the first 5 players survived, the 6th would be guaranteed to die). Of course, the chances someone will die before it gets to be your turn go up the later your turn is. If you had the second turn, your chances of having to play (the chance the first player survived) would be 1 - 1/6 or 5/6. This, multiplied by the 1/5 chance of dying if you have to pull the trigger, gives you (5/6)(1/5) or 1/6 chance of dying, which is the same as the first player. In fact, the odds of dying are the same 1/6 for all six trigger pulls if the barrel is not spun between them. Thus, if you plan to fire a maximum of twice (it doesn't matter if it's in a row or not), your chance of dying is 2(1/6) or 2/6, three times is 3/6, four times is 4/6, five times is 5/6 and if you fire six times in a row your chance of dying is 6/6, or 100% guaranteed. I don't suggest that you try to verify this at home. StuRat 16:55, 26 May 2006 (UTC)

If this is a single player, spinning the cylinder (or whatever it is that revolves on a revolver) each time, the odds of survival in the N+1^th round given survival up to and including the N^th round is 5/6. Because each round is independent, you find then survival odds of (5/6)^N for N rounds, which means fatality odds of 1 – (5/6)^N. For the first few values of N this amounts to:

N = 0: p_fatal = 0.00000

N = 1: p_fatal = 0.16667

N = 2: p_fatal = 0.30556

N = 3: p_fatal = 0.42130

N = 4: p_fatal = 0.51775

N = 5: p_fatal = 0.59812

N = 6: p_fatal = 0.66510

N = 7: p_fatal = 0.72092

N = 8: p_fatal = 0.76743

N = 9: p_fatal = 0.80619

Personally I'd call any player of Russian roulette a loser even before they begin to play. --Lambiam^Talk 17:38, 26 May 2006 (UTC)

Postscriptum. I just read the article Russian roulette, and my impression is that the section Odds is at odds with probability theory, or else so unclear as to be misleading. It should be rewritten, which ought to be a cinch for any probabilist out there (or should I say "in here"?). --Lambiam^Talk 17:45, 26 May 2006 (UTC)

What is the problem? Just as StuRat shows, it is correct under the assumption that the barrel is not spun after every shot, and it displays the probabilty that you will die on round n, given that you have made it there. -- Meni Rosenfeld (talk) 18:06, 26 May 2006 (UTC)

I'm not a gun expert, but isn't it the cylinder that is spun (or not)? The article states: "If the cylinder is spun after every shot, the odds of losing remain the same". Still correct? --Lambiam^Talk 23:23, 26 May 2006 (UTC)

I have no idea what is spun. And yes, if the whatever is spun after every shot, the probability of losing on round n, given that you have made it there, is 1/6 - The same as in the first round ("remain the same" does not mean "the same as in the previous case", but rather "the same as in the first round" - As opposed to the first case, where the probability increases at each round). -- Meni Rosenfeld (talk) 12:57, 27 May 2006 (UTC)

Right. You know what it is that it is the same as, I know what it is that it is the same as, but does the reader know? It could very easily be taken to mean: "the same as in the previous case". I think it is also confusing that the probabilities are conditional: "given that you survived this far". The statistically uneducated reader could easily understand the given table to imply that the odds of surviving 2 rounds are 1-1/5 = 4/5 instead of 4/6. --Lambiam^Talk 14:06, 27 May 2006 (UTC)

I agree that it could have been clearer and that a rewrite would be useful, but still, I think it's clear enough and that a rewrite is not necessary. -- Meni Rosenfeld (talk) 15:51, 27 May 2006 (UTC)

That reminds me, I must rent a copy of The Deerhunter and reminisce about these scenes. JackofOz 14:26, 27 May 2006 (UTC)

I agree that our article wasn't clear on whether they were giving the odds assuming you get to the second (and subsequent) trigger pull or the odds including the probability that the game may not progress that far. StuRat 02:26, 29 May 2006 (UTC)

Unless someone objects I'll copy this over to the talk page of Russian roulette. --Lambiam^Talk 09:25, 29 May 2006 (UTC)

Maths in perspective

I've been thinking about this for a while, and never quite found any good answer, though I'm sure the question is actually extremely simple. The idea is to be able to describe the length of a window on a traincar which is positioned at a certain degree relative to you. I regularly draw, so this is of some practical importance to me.

Imagine a coordinate frame x-y (preferably, keep it so that both coordinates are always positive). Now, the train is represented by a line going from nowhere in particular, but for this example we can choose origo. The window is represented by an AB length somewhere on this line. The line has the function x=y, so it stands at 45 degrees relative to both x and y. As this function changes to 2x=y, 3x=y and so forth, one needs to figure out the amount of degrees that the line is at, relative to either axis of the frame. By doing so, one should in theory easily be able to tell what arclength AB has, and thus, what length the window should be drawn to have on the two-dimensional paper of the artist. Even has the line is moved randomly around on the frame (representing the artist changing his or her point of view), this shouldn't be difficult to calculate. My question is only, how? Thanks. :) Henning 19:37, 26 May 2006 (UTC)

The width of the window as projected onto your direction of view is going to be the true width of the window multiplied by the cosine of the angle between the direction of view and the normal to the window. If you are trying to relate it to these equations of lines, then remember that the numbers in the equation basically relate to the tan of the angle. To relate these to cos of the angle you want to use the relationship that cos(theta) = 1/sqrt(1+(tan(theta))^2) if the angle is called theta. e.g. for 2x=y, tan(theta) = 2 (if you're looking parallel to the y axis, say from somewhere on the x axis), so cos(theta) = 1/sqrt(5), or about 45% of the window's true width. Arbitrary username 21:16, 26 May 2006 (UTC)

Wikipedia has several articles related to perspective; perspective (graphical) is a good place to start. --KSmrq^T 05:13, 28 May 2006 (UTC)

May 27

Real life Block Transfer Computations

block transfer computations [UK television series "Dr. Who"] Computations so fiendishly subtle and complex that they could not be performed by machines. Used to refer to any task that should be expressible as an algorithm in theory, but isn't.

Doctor Who, tiring a bit over his TARDIS' fixed police box exterior, decides maybe his chameleon circuit could use some fixing. However, to fix the circuit, he needs to perform some Block Transfer Computations. These computations are so complex, no computer can handle it. The only way they can be done is using the services of a mathematically adept monk-like people called the Logopolians. The math monks of Logopolis perform these computations by meditation and very loud chanting. An acropolis-like building called the Central Registry handles the inputs and outputs of the math monks.

My question is this: Are there any real life block transfer computations in mathematics?

Ohanian 12:11, 27 May 2006 (UTC)

What about theorem proving? If I remember correctly, this is possible by computers in only rather basic cases, but not in general. Dysprosia 12:17, 27 May 2006 (UTC)

[Edit conflict] That depends. Any proof (in some traditional sense) is just a finite string of characters taken from an enumerable set, and is therefore enumerable. A computer with infinite memory seeking to prove a theorem can simply scan through all such strings until it finds one that proves it. If a proof exists, it is bound to find one. If memory is finite, even a PC should, given enough time, outperform humans in finding proofs - But "enough time" would usually be orders of magnitude greater than what a human would need. However, telling whether a given statement is decidable (that is, either provable or refutable) is something that AFAIK a computer cannot do in the general case - But I don't know of any monks who can do that, either. In other words, I doubt there is any computation which humans can do and computers cannot. -- Meni Rosenfeld (talk) 12:50, 27 May 2006 (UTC)

Is not the problem verifying that a proof found in the way that you describe is in fact the proof? Dysprosia 12:59, 27 May 2006 (UTC)

No, that is completely technical, with given axioms and inference rules. The proof itself contains information about which of these to use and how. It would AFAIK only require an amount of memory proportional to the length of the proof - Which I believe will be measured in kilobytes in common cases. -- Meni Rosenfeld (talk) 13:04, 27 May 2006 (UTC)

I've misunderstood something you said earlier. The proof instance must be encoded in terms of the axioms and inference rules. Dysprosia 13:26, 27 May 2006 (UTC)

In practice, certainly, current automated theorem provers are very limited beasts.

In theory, however, the question of whether there are questions that can be answered by the human mind, but not by a conventional computer, is one that has attracted some philosophical debate, and is closely related to the Strong AI question. Roger Penrose has written two almost impossibly dense books on the topic; the general conclusion in AI community (and mine, as much as I can figure out what the hell he's trying to say) is that Penrose has gotten it wrong. Have a look at halting problem for some related to discussion. --Robert Merkel 12:38, 27 May 2006 (UTC)

At Wikipedia:Reference desk archive/Mathematics/May 2006#The Revenge of the UTM you find the construction of a sentence that a machine can prove, but you can not. Stronger, any competent mathematician should agree that it is a valid proof, except you (or else you have shown yourself incompetent). --Lambiam^Talk 13:40, 27 May 2006 (UTC)

There are certainly problems that are known to be hard for computers and most humans alike, like the decision problem in Presburger arithmetic, which is known to need more than exponential time, and a similar decision problem for real numbers that requires exponential space. To the best of my knowledge these problems have not been tried out on Logopolians. You might try Wikipedians instead, with the additional advantage that they don't chant that loud. --Lambiam^Talk 13:49, 27 May 2006 (UTC)

No, we have no reason to believe that a computing device based on carbon has any qualitative theoretical advantage over a device based on silicon. If such a reason emerged, we would eventually find a way to build a machine to exploit the advantage. This plot idea is yet another instance of the conceit that homo sapiens is neither an animal nor a machine, but something "special" and separate from the rest of the natural world, despite all evidence to the contrary. (However, the Logopolis episode itself is special, though a natural part of the Doctor Who world.) --KSmrq^T 19:27, 27 May 2006 (UTC)

What about integrations? Human can perform integrations on mathematical expressions but computers cannot (they can only perform numeric integrations not symbolic integrations). Ohanian 22:15, 27 May 2006 (UTC)

On the contrary. The best symbolic computer algebra systems use the Risch algorithm for symbolic integration, which is (in theory) complete but a bit too complicated for humans to use without mechanical support. --Lambiam^Talk 22:25, 27 May 2006 (UTC)

On this planet, computers do a fine job of symbolic integration. Here are indefinite integrals a computer algebra system should be able to do almost immediately; see how quickly you can find an answer to each.

$\int {\frac {x}{\sqrt {1-x^{3}}}}\,dx$

$\int {\frac {3{\sqrt[{3}]{x+e^{x}}}+(2x^{2}+3x)e^{x}+5x^{2}}{x{\sqrt[{3}]{x+e^{x}}}}}\,dx$

Try these and more at integrals.wolfram.com, which is a free online demonstration of what a computer symbolic integrator is capable of handling. --KSmrq^T 00:06, 28 May 2006 (UTC)

The same idea I described for finding a proof to a given statement, can be used to find an antiderivative to a given function. Under certain assumptions, if the antiderivative has a closed form, it will eventually be found. Of course, this is extremely inefficient in practice. -- Meni Rosenfeld (talk) 17:13, 28 May 2006 (UTC)

May 28

Video editing program

A month or two ago I asked about a program that I can use to cut out and edit audio files. I was refered to Audacity and it has worked great for me. Now what I was wondering if there was a program like Audacity but for video. I would really like if it were free like audacity. Thanks. schyler 03:18, 28 May 2006 (UTC)

List of video editing software--Frenchman113 on wheels! 14:08, 28 May 2006 (UTC)

Computer / Network Worms

i would like to know more on a new computer worm called NEMATODES —Preceding unsigned comment added by 195.229.242.54 (talk • contribs) 10:50, 2006 May 28

(Please sign your post with four tildes, ~~~~, even if not logged in.) Perhaps you're thinking of the "good worms" idea proposed by David Aitel at Immunity. If so, background papers are available for download. --KSmrq^T 02:09, 31 May 2006 (UTC)

variational methods

Sorry for asking here, Wikipedia, but during the last exam period I asked a question here which was answered in a really good way, and I totally understood the reasoning, much better explained than by my personal tutor (and in the end I got 76% for that exam, with 15% of that thanks to Wikipedia). So I thought I may use you good volunteers again. This question is about a part of variational methods (I was away during the relevant lectures, and haven't any notes on this, you see), and how to go about answering a problem of this sort:

Take a question like the following one appearing on an old exam paper:

Find the extremals x(t) of the integral I(x), where
I(x)= integral between 1 and 0 of (4x² + t²)dt; subject to the constraint
Constraint = [integral between 1 and 0 of x²dt]=2, with x(0)=x(1)=0.

I don't think my lecturer likes it when I went straight to solving it via Euler's Equation, so what would be the best way to go about solving equations of this nature? I assume it is relatively straightforward (for final year university maths at least!). Thanks in advance, Wikipedians. --Knotted 13:41, 28 May 2006 (UTC)

That is a special case: any function satisfying the integral constraint has the same value of I(x). In general, I'd solve variational problems with an integral constraint by adding the constraint with a Lagrange multiplier to I(x). If you can't get notes, it's probably best to look it up in a text book on Calculus of Variations. -- Jitse Niesen (talk) 03:22, 29 May 2006 (UTC)

Fortunately, I(x)=8+1/3 for all x(t) satisfying the constraints.(Igny 17:55, 3 June 2006 (UTC))

Removing bloatware for WinXP

I was wondering if there was a way to remove unneccessary software that is built into Windows (like IE and WMP) without paying $35 for a copy of XPLite?--Frenchman113 on wheels! 14:10, 28 May 2006 (UTC)

Internet explorer is just about impossible to remove completely. IE is really just an interface to a lot of DLL's which lots of other applications use. On microsofts web site there is an article explaining how to remove internet explorer itself, but this doesn't get rid of the bloat. I once came across a web page that explained how to remove it completely. The procedure was quite involved, and looked to me like a maintenance nightmare. I would also expect to see other applications malfunctioning after doing this. I was unable to retrieve that web page now in a quick google search. --vibo56 14:50, 28 May 2006 (UTC)

Great idea, but Microsoft has business and legal reasons for wanting to get in your way, so don't look to their web site for help. You may find what you need in nLite; though I haven't tried it. Any computer that can run XP can also run alternatives; GNU/Linux, FreeBSD, and Mac OS are all decendents of Unix that may better serve your needs. --KSmrq^T 17:47, 28 May 2006 (UTC)

This was already asked on the reference desk, although it's quite difficult to search for it: Wikipedia:Reference_desk_archive/Science/February_2006#Internet_Explorer_Removal. – b_jonas 08:12, 30 May 2006 (UTC)

There isn't much you can do to remove IE, at least. As previously mentioned, it has a bunch of core components that many other programs use. Removing those components would probably cripple/break too many other things. The real "bloatware" to watch out for is malware such as viruses. DoomBringer 17:44, 2 June 2006 (UTC)

I use Mozilla Firefox for all of my day-to-day web browsing, however in my opinion I think it's a good idea to keep IE, because some web pages must be viewed in it (or do not display correctly in other browsers). Internet Explorer is actually integrated into the kernel (explorer.exe) in Windows XP, so for example if you were to type a website into the location bar of Windows Explorer (My Computer) it would display that website. However in Windows Vista, it is planned to be separated from explorer.exe, and if you were to type a website into the location bar it would open up IE in a new window. Cduffner 23:22, 4 June 2006 (UTC)

Area of a "square" on the surface of a sphere

This question was originally posted in the science section, but belongs here. The original questioner has stated clearly that it is not a homework question. It was formulated as follows:

"How do one find the area of a square drawn on a sphere? A square with the side of 10 cm, and draw loci (10cm) on each corners (quarter of a circle in a square to give the "square")"

Based on the discussion that followed, I think what the questioner had in mind is the area illustrated in yellow in the drawing below:

The red circles are supposed to be two pairs of great circles. The angle between the first pair of great circles, expressed in radians, is 10cm/R, where R is the radius of the sphere. The angle between the second pair of great circles is equal to the angle between the first pair. The plane defined by the axes corresponding to the first pair of great circles is perpendicular to the plane defined by the axes corresponding to the second pair of great circles.

The question is how to express the yellow area in terms of R, the radius of the sphere. Obviously, as R → ∞, the area → 100 cm².

• -

I am not a mathematician, but felt that it "ought to" be possible to express this area in terms of R, and decided to try to find the necessary information.

I found Girard's theorem, which states that the area of a triangle on a sphere is (A + B + C - π) × R², where A, B and C are the angles between the sides of the triangle, as illustrated in the second drawing. I also found the law of sines for triangles on a sphere, which relates the angles A, B and C to the angles a, b and c which define the sides of the triangle

${\frac {\sin a}{\sin A}}={\frac {\sin b}{\sin B}}={\frac {\sin c}{\sin C}}.$

I then attempted to divide the square into two triangles, and compute the area of one of these, but am stuck because I don't know the diagonal. Since this is spherical geometry, I doubt that it is as simple as ${\sqrt {2}}\times 10cm$. I would appreciate if somebody told me if I am on the right track, and, if so, how to complete the calculations. If my presentation of the problem reveals that I have misunderstood some of the theory, please explain. --vibo56 14:11, 28 May 2006 (UTC)

The natural way I suspect the question should presumably be answered is to take the square on the flat plane and use Jacobians to transform it onto the sphere. Those with a firmer grip of analysis would probably want to fill in the details at this point... Dysprosia 15:28, 28 May 2006 (UTC)

An easier way to tackle this might be to exploit the symetry of the situation. Slice the sphere into 4 along z=0 and x=0. This will give four identical squares with four right angles and two sides of length 5. The cut the squares along x+z=0, x-z=0 giving eight triangles, each with one 45 degree angles, one right angle and one side of length 5. --Salix alba (talk) 15:44, 28 May 2006 (UTC)

I think vibo is on the right track. You can use the law of sines to calculate the length of the diagonal. -lethe ^{talk +} 15:44, 28 May 2006 (UTC)

The law of cosines for spherical trig gives cos c = cos² a. -lethe ^{talk +} 16:06, 28 May 2006 (UTC)

From which I get using the spherical law of sines that sin A = sin a/ √(1 – cos⁴ a). A = B and C = π/2, so I have the triangle, and hence the square. -lethe ^{talk +} 16:11, 28 May 2006 (UTC)

To lethe: How can you say that C = π/2? This is spherical geometry, and the four "right" angles in the "square" in the first drawing add up to more than 2π, don't they, or am I missing something? --vibo56 16:49, 28 May 2006 (UTC)

You may be right, I cannot assume that the angles are right angles. Let me mull it over some more. -lethe ^{talk +} 16:58, 28 May 2006 (UTC)

OK, I think the right assumption to make is that C = 2A. I can solve this triangle as well, but it's quite a bit messier. Lemme see if I can clean it up, and then I'll post it. -lethe ^{talk +} 17:20, 28 May 2006 (UTC)

This makes my final answer

$2R^{2}\left(2\sin ^{-1}\left({\frac {\sin a}{\sqrt {1-\cos ^{4}a}}}\right)-{\frac {\pi }{2}}\right)$

For the square. Now I just have to see whether this answer works. -lethe ^{talk +} 16:14, 28 May 2006 (UTC)

And now I'm here to tell you that Mathematica assures me that this function approaches s² as the curvature goes to zero. From the series, I can say that to leading two orders of correction, area = s² + s⁴/6R² + s⁶/360R⁴. -lethe ^{talk +} 16:25, 28 May 2006 (UTC)

I got the following for the diagonal angle c of the big square from "first principles" (just analytic geometry in 3D): cos(c/2) = 1 / sqrt(1+2τ²), where a = 10cm/R and τ = tan(a/2). --Lambiam^Talk 16:02, 28 May 2006 (UTC)

I'm afraid I didn't understand (I'm not a mathematician :-) ). If we let (uppercase) C be the "right" (i.e. 90°+something) angle in the triangle in the second figure, and (lowercase) c be the diagonal that we are trying to calculate, could you please show the steps leading to this result (or rephrase it, if I misinterpreted your choice of which of the angles A,B,C that was the "right" one)? --vibo56 18:07, 28 May 2006 (UTC)

For simplicity, let's put R = 1, since you can divide all lengths first by R, and multiply the area afterwards by R². Then the equation of the sphere is x² + y² + z² = 1. Take the point nearest to the spectator in the first image to be (x,y,z) = (0,0,1), so z decreases when receding. Take the x-axis horizontal and the y-axis vertical. A grand circle is the intersection of a plane through the sphere's centre (0,0,0) with the sphere. The equation of the plane that gives rise to the grand circle whose arc segment gives the top side of the "square" is y = tan(a/2) × z = τz (think of it as looking sideways along the x-axis). At the top right corner of the "square" we have x = y. Solving these three equations (sphere, plane, x = y) for z, using z > 0, gives us z = 1 / sqrt(1+2τ²). Now if c is the angle between the rays from the centre of the sphere to this corner and its opposite (which, if R = 1, is also the length of the diagonal), so c/2 is the angle between one of these rays and the one through (0,0,1), then z = cos(c/2). Combining this with the other equation for z gives the result cos(c/2) = 1 / sqrt(1+2τ²). Although I did not work out the details, I think you can combine this with Salix alba's "cut in eight" approach and the sines' law to figure out the missing angle and sides. --Lambiam^Talk 19:59, 28 May 2006 (UTC)

new calculation

As vibo correctly points out above, the square will not have right angles, so my calculation is not correct. Here is my new calculation. Assuming all angles of the square are equal, label this angle C. Then draw the diagonal, and the resulting triangle will be equilateral with sides a and angles A, and 2A = C. The law of sines tells me

${\frac {\sin a}{\sin A}}={\frac {\sin c}{\sin 2A}}\,\!$

from which I have

$\sin c=2\cos A\sin a.\,\!$

From the law of cosines I have that

$\cos c=\cos ^{2}a+\sin ^{2}a(2\cos ^{2}A-1).\,\!$

My goal here is to eliminate c. First I substitute cos A:

$\cos c=\cos ^{2}+\sin ^{2}a\left({\frac {\sin ^{2}c}{2\sin ^{2}a}}-1\right)\,\!$

which reduces to the quadratic equation

$\cos ^{2}c+2\cos c-1=2\cos 2a.\,\!$

So I have

$\cos c=-1\pm {\sqrt {2+2\cos 2a}}\,\!$

and using cos A = sin c/2sin a, I am in a position to solve the triangle

$A=\cos ^{-1}\left({\frac {1}{2}}{\sqrt {1-\left(-1+{\sqrt {2+2\cos 2a}}\right)^{2}}}\csc a\right).\,\!$

I'm pretty sure this can be simplified quite a bit, but the simplification I got doesn't agree with the one Mathematica told me. Anyway, the expansion also has the right limit of s². -lethe ^{talk +} 20:16, 28 May 2006 (UTC)

Despite the figure, which is only suggestive (and not quite correct), are we agreed on the definition of a "square on a sphere"? The question stipulates equal side lengths of 10 cm. To avoid a rhombus we should also stipulate equal interior angles at the vertices, though we do not have the luxury of stipulating 90° angles. Food for thought: Is such a figure always possible, even on a small sphere? (Suppose the equatorial circumference of the sphere is itself less than 10 cm; what then?) Even if it happens that we can draw such a figure, is it clear what we mean by its area? Or would we prefer to stipulate a sufficiently large sphere? (If so, how large is large enough?) Figures can be a wonderful source of inspiration and insight, but we must use them with a little care. --KSmrq^T 20:40, 28 May 2006 (UTC)

The figure was drawn by hand, and is obviously not quite correct, but doesn't the accompanying description:

The red circles are supposed to be two pairs of great circles. The angle between the first pair of great circles, expressed in radians, is 10cm/R, where R is the radius of the sphere. The angle between the second pair of great circles is equal to the angle between the first pair. The plane defined by the axes corresponding to the first pair of great circles is perpendicular to the plane defined by the axes corresponding to the second pair of great circles.

resolve the ambiguity with respect to the rhombus, provided that the area of the square is less than half of the area of the sphere? --vibo56 21:51, 28 May 2006 (UTC)

What is meant by "the angle between the … circles"? That's not really the same as the arclength of a side as depicted. Also note that the orginal post suggests that the side might be a quarter of a circle. If that is true, then the "square" is actually a great circle! Each angle will be 180°, and the area "enclosed" will be a hemisphere of a sphere with radius 20 cm/π, namely 2π(20 cm/π)² = 800 cm²/π, approximately 254.65 cm².

By a series of manipulations I came up with

$\cos ^{2}A={\frac {\cos a}{1+\cos a}},$

where a is 10 cm/R, the side length as an angle. The angle of interest is really C = 2A, for which

$\cos C=-\tan ^{2}{\frac {a}{2}}.$

For the hemisphere case, a = π/2 produces C = π; while for the limit case, a = 0 produces C = π/2.

The original question was about the area, so we should conclude with that: (4C−2π)R². --KSmrq^T 04:43, 29 May 2006 (UTC)

By "the angles between a pair of great circles", I meant the angle between the plane P₁ in which the first great circle lies, and the plane P₂ in which the second great circle lies. The arc length depicted was intended to represent the intersection between the surface of the sphere, and a plane P₃, which is orthogonal to P₁ and P₂, and which passes through the centre of the sphere. As previously stated, I have little mathematical training. I therefore made a physical model by drawing on the surface of a ball, before making the first image. I convinced myself that such a plane is well-defined, and that this length of arc on a unit sphere would be identical to the angle between P₁ and P₂. Please correct me if I am mistaken, or confirm if I am right. --vibo56 20:19, 29 May 2006 (UTC)

Every great circle does, indeed, lie in a well-defined plane through the center of the sphere. Between two such planes we do have a well-defined dihedral angle. The problem arises when we cut with a third plane. If we cut near where the two planes intersect we get a short arc; if we cut far from their intersection we get a longer arc. In other words, the dihedral angle between the two planes does not determine the arclength of the "square" side.

Instead, use the fact that any two distinct points which are not opposite each other on the sphere determine a unique shortest great circle arc between them, lying in the plane containing the two points and the center. Our value a is the angle between the two points, as measured at the center of the sphere. Were we to pick two opposite points, we'd have a = π, which is half the equatorial circumference of a unit sphere. For a sphere of radius R, the circumference is 2πR. We are told that the actual distance on the sphere is exactly 10 cm, but we are not told the sphere radius. The appearance of the "square" depends a great deal on the radius, and so does its area. When the radius is smaller, the sides "bulge out" to enclose more area, the corner angles are greater, and the sphere bulges as well. As the sphere radius grows extremely large, the square takes up a negligible portion of the surface, the sides become straighter, the angles approach perfect right angles, and the sphere bulges little inside the square.

We do not have a handy rule for the area of a square on a sphere. Luckily, the area of a triangle on a sphere follows a powerful and surprisingly simple rule, based on the idea of angular excess. Consider a triangle drawn on a unit sphere, where the first point is at the North Pole (latitude 90°, longitude irrelevant), the second point drops straight down onto the equator (latitude 0°, longitude 0°), and the third point is a quarter of the way around the equator (latitude 0°, longitude 90°). This triangle has three perfect right angles for a total of 270° (or 3π/2), and encloses exactly one octant — one eighth of the surface area — of the sphere. The total surface area is 4π, so the triangle area is π/2. This area value is exactly the same as the excess of the angle sum, 3π/2, compared to a flat right triangle, π. The simple rule is, this is true for any triangle on a unit sphere. If instead the sphere radius is R, the area is multipled by R².

Thus we simplify our area calculation by two strategies. First, we divide out the effect of the radius so that we can work on a unit sphere. Second, we split the "square" into two equal halves, two equilateral triangles, by drawing its diagonal. Of course, once we find the triangle's angular excess we must remember to double the value (undoing the split) and scale up by the squared radius (undoing the shrink).

Notice that this mental model assumes the sphere radius is "large enough", so that at worst the square becomes a circumference. We still have not considered what we should do if the sphere is smaller than that. It seems wise to ignore such challenges for now. --KSmrq^T 21:23, 29 May 2006 (UTC)

Thank you. I really appreciate your taking the time to explain this to me which such detail and clarity. --vibo56 23:34, 29 May 2006 (UTC)

Coordinate Transform

What if we perform a simple coordinate transform to spherical coordinates and perform a 2-dimensional integral in phi and theta (constant r = R). Then, dA = r^2*sin(theta)*dphi*dtheta, and simply set the bounds of phi and theta sufficient to make the lengths of each side 10 cm. Nimur 18:11, 31 May 2006 (UTC)

Calculation completed

Thanks a million to the users who have put a lot of work in explaining this to me, and in showing me the calculations necessary. I started out based on the work of lethe. Armed with a table of trigonometric identities, I went carefully through the calculations, and am happy to report that I feel that I understood every single step. I was not able to simplify the last expression much further, the best I can come up with is

$\cos A={\frac {1}{2}}\csc a{\sqrt {2{\sqrt {2+2\cos 2a}}-2\cos 2a-2}}\,.\!$

You should probably make use of the identity

$\cos 2a=2\cos ^{2}a-1$

here, it simplifies this expression quite a bit. -lethe ^{talk +} 02:01, 30 May 2006 (UTC)

Since the r.h.s. is based on a only, which is a known constant when the radius and length of arc are given (a=10cm/R for the given example), let us substitute $G=g(a)\,\!$ for the r.h.s. Note that the function is undefined at a=0°±180° because of the sine function in the denominator. There is a graph of g(a) on my user page.

We can now calculate the area of the triangle, and that of the square.

$\cos A=G.\,\!$

$\cos 2A=2\cos ^{2}A-1=2G^{2}-1\,.\!$

According to Girard's formula, we then have

$area_{triangle}=(A+B+C-\pi )\times R^{2}\!$

$area_{triangle}=\left(2\cos ^{-1}G+\cos ^{-1}(2G^{2}-1)-\pi \right)\times R^{2}\!$

$area_{square}=2\times \left(2\cos ^{-1}G+\cos ^{-1}(2G^{2}-1)-\pi \right)\times R^{2}\!$

I calculated the behaviour of this area function on a unit sphere when a is in the range (0°...180°):

Seems reasonable up to 90°. The value at 90° corresponds to the "square" with four corners on a great circle that KSmrq mentions above, i.e. to a hemisphere, and the area, 2π is correct. in the interval [90°..180°), the function returns the smaller of the two areas. I also notice that the function looks suspiciously elliptical. Are we computing a much simpler function in a roundabout way?

I next studied how the formula given by KSmrq works out:

$area_{triangle}=\left(2\cos ^{-1}\left({\sqrt {\frac {\cos a}{1+\cos a}}}\,\right)+\cos ^{-1}(-\tan ^{2}{\frac {a}{2}})-\pi \right)\times R^{2}\!$

$area_{square}=2\times \left(2\cos ^{-1}\left({\sqrt {\frac {\cos a}{1+\cos a}}}\,\right)+\cos ^{-1}(-\tan ^{2}{\frac {a}{2}})-\pi \right)\times R^{2}\!$

I computed the area, and found that in the range (0°..90°], the formulae of lethe and KSmrq yield identical results, within machine precision. Above 90°, the formula of KSmrq leads to numerical problems (nans).

Finally, I would like to address the question of the orignial anonymous user who first posted this question on the science desk. Let us see how the area of the square behaves as R increases, using 10 cm for the length of arc in each side of the "square". The smallest "reasonable" value of R is 20cm/π ≈ 6.366 cm, which should lead to a surface area of approximately 254.65 cm², as KSmrq points out. Driven by curiosity, I will start plotting the function at lower values than the smallest reasonable one (in spite of KSmrq's advice to "ignore such challenges for now").

Here is the graph:

Unsurprisingly, the function behaves weirdly below the smallest reasonable value of R, but from R ≈ 6.366 cm and onwards, the function behaves as predicted, falling rapidly from 254.65 cm², and approaching 100 cm² asymptotically. In case anybody is interested in the calculations, I have put the program on my user page.

Again, thank you all. --vibo56 23:54, 29 May 2006 (UTC)

Well done. It does appear that you overlooked my simple formula for the area, which depends on C alone. Recall that when the square is split, the angle A is half of C, so the sum of the angles is A+A+C, or simply 2C. This observation applies to User:lethe's results as well, where we may use simply 4A. So, recalling that a = 10 cm/R, a better formula is

$\mathrm {area} _{\mathrm {square} }\,\!$	${}=\left(4\cos ^{-1}(-\tan ^{2}{\frac {a}{2}})-2\pi \right)\times R^{2}\,\!$
	${}=\left(4\cos ^{-1}(-\tan ^{2}{\frac {10\ \mathrm {cm} }{2R}})-2\pi \right)\times R^{2}.\,\!$

For the arccosine to be defined, its argument must be between −1 and +1, and this fails when the radius goes below the stated limit. (A similar problem occurs with the formula for A, where a quantity inside a square root goes negative.) Both algebra and geometry are telling us we cannot step carelessly into the domain of small radii. Try to imagine what shape the "square" may take when the circumference of the sphere is exactly 10 cm; both ends of each edge are the same point! Not only do we not know the shape, we do not know what to name and measure as the "inside" of the square.

This raises an important general point about the teaching, understanding, and application of mathematics. Statements in mathematics are always delimited by their range of applicability. Every function has a stated domain; every theorem has preconditions; every proof depends on specific axioms and rules of inference. Once upon a time, we manipulated every series with freedom, with no regard to convergence; to our chagrin, that sometimes produced nonsense results. Once it was supposed that every geometry was Euclidean, and that every number of interest was at worst a ratio of whole numbers; we now make regular use of spherical geometry and complex numbers. When we state the Pythagorean theorem, we must include the restriction of the kind of geometry in which it applies. When we integrate a partial differential equation, the boundary conditions are as important as the equation itself. It is all too easy to fall into the careless habit of forgetting the relevance of limitations, but we do so at our peril. --KSmrq^T 02:36, 30 May 2006 (UTC)

Yes, I did overlook the (now painfully obvious) fact that the sum of the angles was 2C. Your final point is well taken. I understood that the reason for the NaN's was a domain error, but thanks for pointing out the exact spots. --vibo56 19:40, 30 May 2006 (UTC)

The area of the 'square' depends on its angles i.e. not only on the lengths of the sides of the square, but on the size of the sphere on whose surface it lies.

But the simplest thing is to split it into two triangles, and get the area of each.

Johnbibby (talk) 20:47, 18 January 2008 (UTC)

Test Statistic in KPSS test of stationarity

What are the values of the test statistic in KPSS test for stationarity of a variable? Example : While running a KPSS test using Gretl 1.5.1 to check the stationarity of my variable (TotalProduction) i receive the following output.

KPSS test for TotalProd (including trend)

Lag truncation parameter = 4
Test statistic = 0.281322
                   10%      5%    2.5%      1%
Critical values: 0.119   0.146   0.176   0.216

I understand that the value to examine is the test statistic but what are the values of it that indicate that my examined time-series is non-stationary? If the test statistic is 0.05 is my series stationary?

As your test statistics, 0.281322, is greater that the critical value for 1% then it passes the test for being stationary at the 1% level, which is pretty good. On the otherhand a test statistic of 0.05 is less than the 10% level so it would not be safe to assume its stationary. Its worth having a think about whether you would expect low or high values of the statistic to indicate a stationary variable. --Salix alba (talk) 18:33, 28 May 2006 (UTC)

May 29

Roots

recently my maths teacher gave me a problem to solve, but i have had trouble with it. i need to find out what z is, in the equation z^3=1, the obvious answer is 1, but he said that it was more difficult than the previous question he gave me, solving the square root of i. He also asked me to find n in the equation, z^n = 1, and he then said there were n answers. any help would be much appreciated, sorry if this is too simple.

This has to do with the beautiful roots of unity. Essentially, the full solution involves complex numbers. If you plot these solutions out in the complex plane, they always end up on a unit circle centered around the origin. --HappyCamper 03:38, 29 May 2006 (UTC)

They are also equally spaced around this circle. —Bkell (talk) 03:39, 29 May 2006 (UTC)

THANKS GUYS

No problem! Actually, I think we can add more stuff to the article, like a diagram of a circle along with the position of the roots. Feel free to come back if you have more questions. --HappyCamper 04:32, 29 May 2006 (UTC)

ok i've had some problems. To solve z^3=1, do you just replace z with (a+bi)^3=1? Because then i get 1 = a^3 - b^3i - 3ab^2 + 3a^2bi and i dont know where to go from here. thanks.

The trick is not to use the representation z = a + bi, but rather z = r * exp(i θ). Now, I'll just give a colloquial sketch of what is going on. Because you know it's a root of unity, r = 1. So, you know that z = exp(i θ).

Now, to go around a circle, you need to traverse 2π radians. Since you are trying to find z^3=1, you look at the exponent and take a note that there is a 3. What angles (in radians) give multiples of 2π when multiplied by 3?

One of them is obviously 2π/3. There's also 4π/3. There's also plain old zero. Note how these can all be written like

$\theta =k{\frac {2\pi }{3}}$, where k = 0, 1, 2.

So, your solutions are of the form

$z=\exp(i{\frac {k2\pi }{3}})$

Now, let's just substitute in the k...these are your values for z

$z=\exp(0),\exp(i{\frac {2\pi }{3}}),\exp(i{\frac {4\pi }{3}})$

Finally, use Euler's formula, $e^{i\theta }=\cos \theta +i\sin \theta \,\;$

and you get your answers

$z=1,-{1 \over 2}+i{\frac {\sqrt {3}}{2}},-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}},$

Does this help? --HappyCamper 05:02, 29 May 2006 (UTC)

yeah, thanks a lot. Do u have a maths degree?

On second thought, I think your question can be answered much much simpler, although I haven't worked it out entirely yet.

Take the approach you had before, set z = a + bi, and just expand. You get a³ - ib³ - 3ab² + 3ia²b = 1. Now, you invoke this insight: The real and imaginary parts on boths sides must be equal! So, you get two simultaneous equations for a and b:

a³ - 3ab² = a(a² - 3 b²) = 1

3a²b - b³ = b(3a² - b²) = 0

From the second equation, you get that either b = 0, or $b=\pm {\sqrt {3}}a$. You see where this is going? Substitute these into the first and solve for a, and you get the same solutions as above. Much much easier than having to wave magic wands with complex exponentials and such. However, there is a lot of insight you can get from understanding the complex exponentials. I'd encourage you to revisit this problem again when you get the chance to. You might also wonder about...what is the quantity $i^{i}$, where i is the imaginary unit? Hm.... --HappyCamper 05:22, 29 May 2006 (UTC)

Right, that's a very important princple: any equality of complex numbers corresponds to two independent equalities of real numbers. —Keenan Pepper 05:46, 29 May 2006 (UTC)

Is there an article with a formal proof of this on Wikipedia? --HappyCamper 05:47, 29 May 2006 (UTC)

That's the definition of equality of complex numbers: Two complex numbers are equal if and only if their imaginary parts are equal and their real parts are equal. —Bkell (talk) 05:51, 29 May 2006 (UTC)

This was missing from complex number! [22] --HappyCamper 05:54, 29 May 2006 (UTC)

The problem as posed makes a hidden assumption. Any time we look for zeros (roots) of a polynomial, we must first know what kinds of values we are allowed to substitute for the unknown. For example, the polynomial

$x+1\,\!$

has a zero when x is −1, but that may not be admissible if we require a positive whole number. If we allow a negative whole number then we have exactly one root. Similarly, the polynomial

$2x-1\,\!$

has a zero when x is ¹⁄₂, but that is not a whole number and so may not be acceptable. Or consider

$x^{2}-2\,\!$

with a zero when x is √2, which is not a fraction! (This is the length of the diagonal of a square by the Pythagorean theorem, and the discovery that it was not a "proper" number deeply disturbed ancient Greek mathematicians.) Notice that all these polynomials are extremely simple, and all have integer coefficients.

Now look at the polynomial

$x^{2}+1\,\!$

which might seem to have no roots at all. For, the numbers we are accustomed to using in arithmetic have the property that any non-zero number squares to a positive number, yet we require a number whose square is −1. Is there no end to the need to enlarge our concept of number, from natural numbers to integers to rational numbers to real numbers to these new beasts? To our great relief, when we expand from real numbers to complex numbers by including a value that squares to −1, and commonly denoted by ⅈ, along with all multiples and sums with it, we find that no polynomial can ever again stop us. Hurray! Complex numbers make our life simple.

The usual picture is that real numbers extend smoothly left and right along a linear continuum, and complex numbers extend above and below that line as well. To turn 1 into −1 requires a 180° reversal; to do that in two steps (squaring) requires a 90° turn, which is what ⅈ accomplishes. Notice that in this case the magnitude of the number — its distance from zero — remains one before, during, and after the turn. Observe that 1 itself has two numbers that square to it: +1, which involves either a 0° or 360° turn, and −1, which involves two 180° turns. Also observe that −1 has two square roots: ⅈ, which involves two 90° turns, and −ⅈ, which involves turns of 270° or −90°.

With a little imagination we can picture three options for a turn that, when repeated three times, gives (an integer multiple of) 360°. These would be "cube roots of unity". We note that paired with every "positive" turn there will be a matching "negative" turn; this corresponds to replacing ⅈ by −ⅈ, an operation called complex conjugation. --KSmrq^T 05:56, 29 May 2006 (UTC)

It should be pointed out that the reason that squaring numbers like 1, −1, i, and −i does not change the number's magnitude is that the magnitude of all these numbers is 1, and 1² = 1. In general, when you square a complex number z, the magnitude of the resulting number will be the square of the magnitude of z. —Bkell (talk) 06:11, 29 May 2006 (UTC)

well. that was a pretty hard question considering the last thing we did in maths was inverse functions and we havent even been taught complex numbers yet.

ok i get up to where you said, but when trying to solve a, how does a^3 + 9a^2 - 1 give you -1/2 for a? and also with a^3 - 9a^2 - 1 = -1/2?

HappyCamper's formula $b=\pm {\sqrt {3}}a$ was presented in a somewhat non-conventional way. More conventional is $b=\pm a{\sqrt {3}}$, which makes clear that the occurrence of a is not under the square root's vinculum, which would have been $b=\pm {\sqrt {3a}}$. So b² = 3a² (which is where we came from), and not 3a. --Lambiam^Talk 10:19, 29 May 2006 (UTC)

Thank you to everyone who contributed in answering this question.

Math Question 1

If there is a ticket raffle, and the prize is worth $1,000, with 100 tickets outstanding I should clearly purchase tickets at $1 each, as they are worth more than that, but as I do, I increase the # of tickets oustanding, and at some point (1000 tickets I think) it is no longer advantagous to purchase tickets. (is this correct?)

But lets say I own 5 of the 100 oustanding tickets. These tickets would be valued at $10 each ( I think). What if instead of ONE $1000 price, there are 5 $200 prizes. How would this effect the "intrinsic value" of each ticket if:

-after a ticket is drawn it is put back into the pool and can be drawn again for one of the other prizes

-After a ticket is drawn it isnt put back but the ticketholder can win on the other tickets.

-After a ticket is drawn the ticketholder is ineligable to win other prizes regardless of his remaning tickets?

Anyone? 12.183.203.184

If there are 100 tickets outstanding, and you buy one, then there is a 1/101 chance you will win $1000, and a 100/101 chance you will win nothing. So the expected value of your winnings is

${1 \over 101}\times \$1000+{100 \over 101}\times \$0\approx \$9.90.$

So for that first ticket, you should be willing to pay up to $9.90. Suppose you buy it. Now you consider whether to buy a second ticket. Since there are now 101 tickets outstanding, the chance that your second ticket will win $1000 is 1/102. So now the expected value of your winnings from the second ticket is

${1 \over 102}\times \$1000\approx \$9.80.$

So it seems that you should pay up to $9.80 for that second ticket.

The problem, though, is that buying the second ticket changes the odds, and so the calculation for the first ticket is no longer valid. If you buy two tickets, then you have two chances in 102 of winning the thousand dollars, so the expected value of your winnings is

${2 \over 102}\times \$1000\approx \$19.61,$

so if you are planning to buy two tickets the total amount you should pay should be no greater than $19.61. (This of course assumes that no one else will be buying any other tickets; if you buy your two tickets for $19.61, and then someone else buys a hundred tickets, then you will have overpaid.)

You can continue this line of thought to figure out how many tickets you should buy for a dollar each. If you buy n tickets, then the expected value of your winnings is

${n \over 100+n}\times \$1000.$

You're going to have to spend n dollars to buy n tickets, so if this expected value is less than n, you shouldn't buy so many. —Bkell (talk) 02:34, 30 May 2006 (UTC)

Now suppose that you have five of the 100 outstanding tickets and there are five $200 prizes. If tickets are put back into the pool after being drawn, then in each drawing you will have five chances in 100 of winning $200. This means that the expected value of your winnings is

${5 \over 100}\times \$200+{5 \over 100}\times \$200+{5 \over 100}\times \$200+{5 \over 100}\times \$200+{5 \over 100}\times \$200=\$50,$

which means that your five tickets together are worth $50 (that is, $10 each). In fact, as long as the tickets are returned to the pool after each drawing, it doesn't matter how the $1000 is divided up. There could be a hundred thousand drawings for a penny each, and your expected winnings would still be $50. (You will find, however, that as the $1000 is divided ever more finely, you have a greater chance of winning closer to $50; that is, it becomes less and less likely that you will win $0 or $1000, and more and more likely that you will win $50, or $45, or $55.) —Bkell (talk) 02:53, 30 May 2006 (UTC)

Math probabilty question 2

Say a casino sells the "opportunity" to roll a 6-sided dice. You win the number rolled x 100. e.g. 2 = $200. The value of this appears to be $350. If the casino allowed you to reroll at your choice, what would the value be? If you could reroll twice (3 rolls total) what would the value be, and what number should you take the reroll option if the dice number turned up is less than? I think this should be higher on the 1st reroll than on the 2nd option. Is that right? How would one calculate that? Thanks! 12.183.203.184 08:17, 29 May 2006 (UTC)

With one reroll, you would reroll if the expected value of the reroll is more than the amount you have already won: that is, if the die currently shows a 1, 2 or 3 ($100, $200 or $300). Thus the expected value of that situation is ${\begin{matrix}{\frac {1}{6}}\end{matrix}}\left(\$350+\$350+\$350+\$400+\$500+\$600\right)=4{\begin{matrix}{\frac {1}{4}}\end{matrix}}\times \$100$.

From this point on, we can remove the $100 factor, because it just serves to confuse things. This means that after the first roll of three, the remaining two rolls are worth $4{\begin{matrix}{\frac {1}{4}}\end{matrix}}$ if exercised; thus they will be exercised if the current value of the die is less than $4{\begin{matrix}{\frac {1}{4}}\end{matrix}}$ i.e. on a 1, 2, 3 or 4. So the expected value of 3 rolls is ${\begin{matrix}{\frac {1}{6}}\end{matrix}}\left(4\times 4{\begin{matrix}{\frac {1}{4}}\end{matrix}}+5+6\right)=4{\begin{matrix}{\frac {2}{3}}\end{matrix}}$. EdC 14:13, 29 May 2006 (UTC)

In other words, denote the value of n rolls by $v_{n}$. What Edc meant was that this value satisfies a recurrent equation (see Conditional expectation)

$v_{n+1}=E(x|x\geq v_{n})P(x\geq v_{n})+v_{n}P(x<v_{n})$

and forms a sequence 3.5,4.25,4.66,4.94,5.13,5.27,...(Igny 16:52, 29 May 2006 (UTC))

Or, equivalently but conceptually simpler:

$v_{n+1}={\frac {1}{6}}\sum _{i=1}^{6}\max(i,v_{n}).$

For n ≥ 5 a closed form expression is given by v_n = 6 − (6768/3125)(5/6)ⁿ. --Lambiam^Talk 17:19, 29 May 2006 (UTC)

While we just provided a correct answer, I wonder if the proof is needed that $v_{n}$ is indeed the maximal value of n rolls, and our strategy is optimal. (Igny 18:05, 29 May 2006 (UTC))

I think it's obvious. But on demand a proof can be supplied. --Lambiam^Talk 23:04, 29 May 2006 (UTC)

Salary Statement

How to prepare a SALARY STATEMENT of any type in Ms Excel?--86.62.239.145 10:55, 29 May 2006 (UTC)

You will probably like Microsoft InfoPath (even although it is used to make XML forms, it can be used for receipts; anyway, if you dont have that you should just lay it out as it appears in most wage slips. Kilo-Lima|^(talk) 16:50, 29 May 2006 (UTC)

Windows Media Audio

Hi, I went and ripped some files from a bought music CD and it was extracted as .wma format. I then put it into WMP library. I then wanted to burn them to a CD, so I put in a blank disk and I keep getting the same error that it is not blank—I can assure you, it is blank! Does anybody know what's happening here? Thanks, Kilo-Lima|^(talk) 16:46, 29 May 2006 (UTC)

Ugggh...now you know why I like iTunes (no WMA crap). But anyways...maybe your CD is messed up (defective, etc.)--could somebody have "borrowed" when you weren't looking? --M1ss1ontomars2k4 ^{(T | C | @)} 21:41, 29 May 2006 (UTC)

Yeah, iTunes is totally free of DRM and everything, too. Dysprosia 22:20, 29 May 2006 (UTC)

May 30

Cell CPU

Hey ppl, i hope this is the right place to ask about the Cell CPU thats gonna be used in the Playstation 3. I'v asked SOOO many ppl and they aren't quite sure how it works either. Please, can someone explain to me how all the Synergistic Processing Elements (SPE's) work together to process the info and why game developers developing games for the PS3 say its so hard to program for 7 cores? Even the main wikipedia article on the Cell says its too complicated. Thx! KittenKiller

It's mostly because it's a very uncommon architecture, so almost nobody has any experience with it.

Historically, consumer-level computers have had a single general-purpose processor. Programming in such a situation is very well-understood: it's simply a matter of dividing up the available time between tasks. (For example: five milliseconds for AI, one for checking for user input, three for updating the sound effects, one for updating the game logic, twenty-five for updating the screen, repeat)

More recently, computers have started being produced with multiple identical general-purpose processors. Writing programs for this sort of system is fairly well-understood in theory: you divide up the tasks in such a way that the workload can be balanced among the processors. In practice, it's a bit harder, because some tasks can't be divided up, and you need to make sure the tasks don't step on each other's toes, but it's still familiar ground. (For example, the game AI would be one or more tasks, physics would be a task, generating sound effects would be a task, playing the background music would be a task, checking for user input would be a task, updating the graphics would be a task, and so on)

Then, there's the Cell CPU. It's got one general-purpose processor and seven special-purpose processors (the SPEs) that are extremely efficient for certain types of tasks, and very slow for others. Nobody's quite sure how to handle such a situation, or what the seven SPEs should be used for. It's known that they're good for graphics, but the PS3 has a dedicated graphics processor that's even better. They're probably good for sound effects, but that won't take the effort of more than one SPE. They're not good for AI, or for user input, or for background music. They might be good for physics, but nobody knows -- simulating physics is cutting-edge research right now. --Serie 22:48, 30 May 2006 (UTC)

Series

What is this series of numbers called:

1, 11, 21, 1211, 111221....?

Looks like the see-and-say sequence. Dysprosia 08:11, 30 May 2006 (UTC)

In mathematical jargon this is not a series. Goes to show that mathematicians are strange people. --Lambiam^Talk 09:58, 30 May 2006 (UTC)

There's a difference between series and sequences. OEIS A005150 : Look and Say sequence: describe the previous term! (method A - initial term is 1 - Formerly M4780) --DLL 20:32, 30 May 2006 (UTC)

Of course, but I think it's quite obvious what the poster above meant, so I don't think there's need for excessive concern about this. Dysprosia 22:19, 30 May 2006 (UTC)

Noo, I'm more concerned about the mathematicians ;-)  Lambiam^Talk 02:50, 31 May 2006 (UTC)

Well telling someone once is fine, but twice is getting to be a bit onerous ;) Dysprosia 03:09, 31 May 2006 (UTC)

Watermellon. Black Carrot 00:23, 1 June 2006 (UTC)

I don't get it. Dysprosia 02:01, 1 June 2006 (UTC)

Nothing. It was just that several posts had gone by without anyone saying anything, at all. I was taking that a bit farther. Black Carrot 02:13, 2 June 2006 (UTC)

Read the following aloud: one, one "one", two "ones", one "two" and one "one", one "one" and one "two" and two "ones"... See it? For even more fun, prove that "four" will never appear in this sequence. PS: I did saw it mentioned at the end of a maths book, but it didn't mention its name. --Lemontea 11:25, 5 June 2006 (UTC)

Font

for some unknown reason, the font in which i view wikipedia has changed to tahoma, and i would like to revert it. When editing articles my font is times new roman, but after i have edited them, it appears in tahoma. Can anyone shed any light on my problem?

Character encoding ? In Firefox the menu offers this entry with plenty of encoding possibilities. In other brothels it may be worse, brother. --DLL 20:28, 30 May 2006 (UTC)

Browser settings, font availability, and custom Wikipedia monobook.css can all affect what's displayed. Usually the text typed in the edit window appears in a fixed-width font like Courier, not Times New Roman; and usually the text for the preview or regular display of an article appears in a sans-serif font like Arial. The Tahoma font would be a reasonable substitute for Arial, but are you sure the editing font is proportionally spaced? --KSmrq^T 01:50, 31 May 2006 (UTC)

quantization

2 questions that keep cropping up in old exam papers, which will hopefully get me a few extra marks:

1. Define a scalar quantizer with M interval
2. Define a scalar quantization problem

I'm a bit rusty on these, plus Wikipedia has no articles on them. Any help would be appreciated. --Talented Wikipedian 10:42, 30 May 2006 (UTC)

Could you help us by giving a hint what field the exam is about: signal processing, perhaps more specifically image processing, sound processing, and music, or is it perhaps about physics? --Lambiam^Talk 13:55, 30 May 2006 (UTC)

It is part of a data compression module, and is some kind of lossy compression, I can't be sure whether it is to do with sound files or image files. --Talented Wikipedian 16:36, 30 May 2006 (UTC)

I suspect that "M interval" is not standard terminology. Could the meaning be "M intervals" (or "levels"), as in 16 intervals, 256 intervals, 65,536 intervals, ..., M intervals? Here are a few links that may be helpful to you:

• http://wiki.multimedia.cx/index.php?title=Scalar_Quantization
• http://www.debugmode.com/imagecmp/quantize.htm
• http://www.stanford.edu/class/ee372/methods4.pdf

--Lambiam^Talk 18:19, 30 May 2006 (UTC)

Perhaps this is meant in contrast to vector quantization? --KSmrq^T 22:37, 30 May 2006 (UTC)

astronomy-orbit of planets

What branch of calculus do I need to understand the position of a planet orbiting the sun at any given time?

Thank you, Denis

Assuming that you are happy enough with Newton's law of universal gravitation and do not require General relativity, what you need is the area of Ordinary differential equations, also known as ODEs, together with a general understanding of Newton's laws of motion and their role in Mechanics. --Lambiam^Talk 18:29, 30 May 2006 (UTC)

And if that's a bit much for you, try geometry, specifically conic section geometry. StuRat 22:38, 30 May 2006 (UTC)

That will tell you where, but you'll need Kepler's laws to know when where. I see now that the Kepler article shows how to derive these laws from the ODE's given by Newton's law. --Lambiam^Talk 02:58, 31 May 2006 (UTC)

Rope strength

If a single strand manilla rope has a working load of 100 lbs., does that mean that a triple strand rope of the same material has a working load of 300 lbs.? Is there a formula for figuring this out? Thanks, Pdpdpd1

If the box in Exercise 5.14 weighs 20 kg and has a coeficent of friction of .2, what is the minimum strength rope that can be used to pull it?

A rule of thumb for question 1 would say that there is always one of the rope that bears more weight. So do not triple it. But as the rope itself is made of strings, the standard load might be over 100 lbs. So try to triple it and do not stay under the load. --DLL 20:24, 30 May 2006 (UTC)

Question 1 is a yes, as long as it still obeys Hooke's law at that point. At least I think so. Question 2 deals with Hooke's Law and the Young modulus, but I don't understand what the "strength" refers to. But you could find out what the frictional force is and find the force required to pull the box. Remember, we don't do your homework. x42bn6 Talk 07:56, 31 May 2006 (UTC)

why is in locally pathconnected topological space, path connected component same as connected component

http://en.wiki.x.io/wiki/Connected_space#Local_connectedness

defines the concept of a locally path connected topological space. Now I wanna prove that in a locally pathconnected topological space, path connected components and connected components are the same.

Now I do know that this is a relevant fact : in a locally pathconnected space, every open set U has (in its induced topology) open pathconnected components.

But how to use this?

I have gotten this far : let C be the component of x, and P the strictly contained in C pathconnected component. Now I now C will be closed. I cannot say that of P. I need to find contradiction though...

Thanks,

Evilbu 23:08, 30 May 2006 (UTC)

Well, I hope this is not for homework. What you need to prove is that every connected component is also path connected. To do this, fix a connected component $C$ and choose $p\in C$. If you show that the set of all points in $C$ that can be connected to $p$ by a path is the whole $C$ itself, you are done.

To do this, you show that it is both open and closed: as $C$ is connected by definition, and nonempty because it trivially contains $p$, you are done.

Openness: suppose $q$ can be connected to $p$ by a path. Let $U$ be a neighbourhood of $q$ which is path connected (it exists because the space is locally pathconnected). Then any point in $U$ can be connected to $p$ by a path simply by going to $q$ first and then to $p$. Hence, your set is open.

Closedness: consider the subset of $C$ that cannot be connected to $p$ by a path, and choose $q$ as such. Then choose a neighbourhood $U$ as above: if any point of $U$ could be connected to $p$ by a path, then by following the two paths as before you could also connect $q$ to $p$. This means that this set is also open, i.e. the set of points that can be connected to $p$ by a path is closed.

Q.E.D. Cthulhu.mythos 09:12, 31 May 2006 (UTC)

Hmm, yes I see now. I guess one could not from the start that in a locally pathconnected space, all pathconnected components are open, right? Nope, this is not homework, yet it is not unrelated to university work, we did it quickly in class and I wanted to clear things up for myself. Evilbu 10:45, 31 May 2006 (UTC)

May 31

Integration of sin and cos

What happens when you integrate sin(), -sin(), cos() and -cos()

• Well, what happens when you differentiate any of them? Notice something interesting? Then just apply the Fundamental Theorem of Calculus and you're done. If that's not enough, check List of trigonometric identities#Calculus. Confusing Manifestation 12:01, 31 May 2006 (UTC)

Also check out Table of integrals#Trigonometric functions. -- Meni Rosenfeld (talk) 15:23, 31 May 2006 (UTC)

Implications

The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and fourier transformations. (I added this tidbit at the above article). Nimur 18:06, 31 May 2006 (UTC)

A sophisticated answer uses Euler's formula, cos ϑ+ⅈsin ϑ = ⅇ^ⅈϑ, and the fact that ⅇ^z is an eigenfunction for the linear operation of integration. For example,

$\cos \vartheta ={\frac {1}{2}}e^{i\vartheta }+{\frac {1}{2}}e^{-i\vartheta },\,\!$

so

$\int \cos \vartheta \,d\vartheta ={\frac {i}{2}}e^{i\vartheta }-{\frac {i}{2}}e^{-i\vartheta }+C.\,\!$

A more geometric approach observes a unit circle parameterized by arclength,

$(x,y)=(x(\vartheta ),y(\vartheta ))=(\cos \vartheta ,\sin \vartheta ),\,\!$

and considers that a tangent to the circle is always a unit vector perpendicular to the radius. Thus starting from the initial condition (x₀,y₀) = (0,−1) and integrating (cos ϑ,sin ϑ) traces out a counterclockwise unit circle parameterized by arclength, (cos(ϑ−^π⁄₂),sin(ϑ−^π⁄₂)). This serves to remind us of the important role of constants of integration, here a consequence of the initial condition.

The best approach is to pay attention in class, read the textbook, and ask the instructor or teaching assistant for homework help. --KSmrq^T 18:42, 31 May 2006 (UTC)

Thank you for your help, I hope I can get to a stage of complete understanding - there is a few years of work ahead of me yet though.

very exact definition of "orientable topological manifold"?

From http://en.wiki.x.io/wiki/Manifold

" Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are orientable manifolds. For others, this is impossible. "

Okay, but strictly speaking, this is still not really exact. I would like to have a very explicit definition of "agreeing on handedness". I have some very fundamental experience with charts of (topological varieties), and I know that in a vector space of an ordered field, two bases are considered having the same orientation if their matrix has positive determinant.

So how can it make this precise?

Thanks,

Evilbu 13:28, 31 May 2006 (UTC)

The problem here is whether we want the article to report the formal definition or not. If this is the case, suppose

$\phi _{U}:U\rightarrow \mathbb {R} ^{n}$

and

$\phi _{V}:V\rightarrow \mathbb {R} ^{n}$

are two charts. Then

$\phi _{V}|_{U\cap V}\cdot \phi _{U}|_{U\cap V}^{-1}:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}$

is a homeomorphism: if for every choice of $U$ and $V$ it preserves orientation, then the manifold is said to be orientable. Cthulhu.mythos 16:44, 31 May 2006 (UTC)

thanks, but could you explain what you mean by 'it preserves orientation"?

<Mount Soapbox>

Keep in mind that the manifold article is an overview of all different kinds of manifold, so deliberately does not dive into all the details. The topological manifold article should include a precise definition of orientability; however, these specialized articles have been starved for attention as numerous editors have nitpicked the survey article.

</Mount Soapbox>

The definition of a topological n-manifold states that it is a topological space in which every point has an open neighborhood homeomorphic to an open n-ball, Bⁿ. (Typically, the space must also be Hausdorff, paracompact, and second-countable; but those are technical details that needn't concern us here.)

Now fix Bⁿ, and choose a list of n+1 points (vertices) in general position within it to form a simplex (or choose an ordered basis of n vectors for the Rⁿ in which it lives). The choice imposes one of two possible orientations on Bⁿ, which we may call "right-handed" or "left-handed". If we exchange two vertices in a right-handed simplex we obtain a left-handed simplex, and these are topologically distinct in the sense that we cannot deform one into the other. For example, the triangle ABC in a planar disc cannot be deformed to ACB without violating its topology.

By the axioms of topology, if two open neighborhoods have a non-null intersection, then the intersection is an open set. And since a homeomorphism is a continuous map with a continuous inverse, we can create a composite map from Bⁿ to itself by way of the two neighborhood homeomorphisms. By a suitable deformation we can pass the simplex through this map and compare the image to the original. Thus we can decide if the overlapping homeomorphisms imply the same or opposite orientations.

The key observation is that we can cover some manifolds with neighborhoods so that overlapping neighborhoods always agree in orientation; these are orientable manifolds. Significantly, this is not always possible.

A basic example of a non-orientable manifold is the projective plane, RP². We can think of this as a sphere with opposite points on the surface identified, or as a disc with opposite points on the circular boundary identified. A 2-simplex is merely a triangle, and a triangle in B² maps to a three-sided region in RP², which might as well be a triangle itself. But because the manifold is non-orientable, there is no essential difference between ABC and ACB. For example, put C on the boundary of the disc, with A and B inside; then the identification of opposite boundary points means we cannot distinguish the two orientations of the triangle. We might see this more directly with the sphere model: Draw the triangle as ABC in the northern hemisphere, then use the identification of opposite points to transfer it to an oppositely oriented triangle in the southern hemisphere.

With an orientable manifold, such as an ordinary sphere, S², this kind of reversal is impossible. We can consistently define a right-handed triangle as distinct from a left-hand triangle, and never the twain shall meet. One point of caution is that the dimension of the simplex must be the same as the dimension of the space. For example, we can easily flip a triangle (a 2-simplex) in 3-space; we must work with a tetrahedron (a 3-simplex) to explore orientability in 3-space. --KSmrq^T 23:36, 31 May 2006 (UTC)

You mention that a manifold must be paracompact. Isn't this redundant for a second countable space locally homeomorphic to Rⁿ? -lethe ^{talk +} 04:21, 1 June 2006 (UTC)

I just mentioned some of the additional conditions that often appear, without regard for independence. For example, it is not uncommon for a paracompact space to be required to be Hausdorff. Anyway, the paracompact space article mentions that "Every locally compact second-countable space is paracompact." And the locally compact space article points out that "Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact." (There may be a caveat about Hausdorffness in there.) Fortunately, we can discuss orientability without concern for these extra conditions. --KSmrq^T 06:35, 2 June 2006 (UTC)

Evilbu: it is worth mentioning that manifolds with additional structure admit alternate definitions for orientability, which of course agree with the topological definition. A differentiable manifold if it admits an atlas whose transition functions have positive determinant on the tangent bundle. Equivalently, it has a nowhere zero differential n-form. -lethe ^{talk +} 04:24, 1 June 2006 (UTC)

And also this one: a differentiable manifold $M$ is not orientable if and only if it admits a loop $\gamma$ (with base point $p$) such that if you choose a base of $T_{p}M$ having $\gamma '(0)$ as its first vector and then drag it along $\gamma$ using charts, when you came back to $p$ at time $1$ you have a base of $T_{p}M$ whose orientation is reversed w.r.t. the base you started from. Cthulhu.mythos 08:56, 2 June 2006 (UTC)

I would also remark that orientability is an intrinsec property, and does not depend on how the manifold can be embedded in some euclidean space. For example, it occurs frequently to hear that "the Moebius strip is not orientable because it has only one side and one boundary, whereas an annulus is orientable because it has two sides and two boundaries". But that is not correct. A compact surface properly embedded in a 3-manifold (and the Moebius strip is NOT properly embedded in $\mathbb {R} ^{3}$, because of the boundary) is two-sided if it has a tubular neighbourhood which is just a product of a copy of itself with $[-1,1]$, and it is one-sided if its tubular neighbourhood is a twisted $I$-bundle over the same surface. Example (call $M$ the Moebius strip once and for all): in the manifold $M\times S^{1}$ the levels $M\times \{p\}$ are two-sided, whereas the torus $\gamma \times S^{1}$ (where $\gamma$ is the core of the Moebius strip) is one-sided. Cthulhu.mythos 10:15, 1 June 2006 (UTC)

Seems like I'll create one-sided (submanifold) and two-sided (submanifold) soon... Cthulhu.mythos 10:17, 1 June 2006 (UTC)

I don't understand why the Möbius strip isn't properly embedded in R³. On the contrary, I'm pretty sure the Möbius strip can be embedded in R³. I made one in third grade with scissors and paper, and it wasn't on the day we took the field trip to R⁶. -lethe ^{talk +} 17:37, 1 June 2006 (UTC)

Not properly embedded. The boundary, y'know. EdC 00:57, 2 June 2006 (UTC)

I guess I don't know what a proper embedding is. Will you tell me? (And what it has to do with boundaries.) -lethe ^{talk +} 01:04, 2 June 2006 (UTC)

So the article on embedding didn't mention proper embeddings. It does now, although I'm not sure how to make it clearer. A proper embedding preserves boundaries; for example, embedding $S^{1}\subseteq \mathbb {R}$ in $D^{2}\subseteq \mathbb {R} ^{2}$ as a chord is proper, but embedding it as a radius is not. So the Môbius strip can't be properly embedded in $\mathbb {R} ^{3}$, because the Môbius strip has a boundary ($S^{1}$) and $\mathbb {R} ^{3}$ doesn't. (This raises the question of what subspaces of $\mathbb {R} ^{3}$ the Môbius strip can be properly embedded in. I think it's obvious that the boundary has to have positive genus, though I'm not sure how to prove it.) -- EdC 04:16, 2 June 2006 (UTC)

Oh, right. Assume that M is embeddable in R³ less a contractible subspace A. Then ∂M (= S¹) is contained in the boundary of A, so its contraction S* is D². So M ∪ S* is an embedding of the projective plane in R³, which is a contradiction (there is no such embedding). -- EdC 04:29, 2 June 2006 (UTC)

The definition of a proper embedding was not precise, I fixed it. Cthulhu.mythos 08:48, 2 June 2006 (UTC)

Ooh, forgot the transversality condition. Thanks. EdC 11:59, 2 June 2006 (UTC)

Many thanks to both of you. So a proper embedding is an embedding that takes the boundary to the boundary in a nice way, right? -lethe ^{talk +} 20:03, 2 June 2006 (UTC)

htaccess for Zeus Web Server

I need this htaccess file written for Apache to be re-writen for Zeus Web Server

      Options FollowSymLinks
      RewriteEngine On
      RewriteCond %{REQUEST_FILENAME} !-f
      RewriteCond %{REQUEST_FILENAME} !-d
      RewriteRule ^(.+)$ /index.php?title=$1 [L,QSA]

Thanks, Gerard Foley 21:49, 31 May 2006 (UTC)

Nellore Telugu Measurement (Simhapuri Measurement)

I would like to know exactly how many feet are there in an "Ankanam" based on Nellore region, AndhraPradesh, India. Thank You

Maybe the question is flawed. One site listing Nellore housing says "One Ankanam =4 sq.yds.", while an online Telugu-English dictionary translates "ankaNam n. space between two beams or pillars in a house." Whichever we pick, the question makes no sense. --KSmrq^T 23:58, 31 May 2006 (UTC)

1 acre = 600 ankanams (appx.) => 1 ankanam = 72.6 sq.ft (appx.)

Real Analysis - teaching and learning

What is (or should be) gained by undertaking an introductory course in Real Analysis? What are the implications of this for the way it's taught? Thanks. --The Gold Miner 23:35, 31 May 2006 (UTC)

I always thought real analysis was just "calculus again, but for serious this time". You go over the definition of limit again, use fancy topology words like compact, and focus on pathological examples like differentiable functions whose derivatives are not continuous. —Keenan Pepper 05:39, 1 June 2006 (UTC)

The main benefit of a real analysis course is not so much the content, as the mathematical maturity gained by taking the course. Quite often RA courses are the student's first serious introduction to writing detailed proofs and working with high-level (i.e. Rudin) mathematics textbooks. 18.228.1.113 13:34, 1 June 2006 (UTC)js

Vast numbers of first-year university students are required to learn calculus, for use in a variety of different fields. Therefore the typical calculus textbooks and courses are designed to serve this genuine need for the broad community. However, this is rather different from training mathematicians. Weierstraß and others went to a great deal of trouble to build solid foundations under calculus, which is how real and complex analysis began, but the journeyman calculus applications rarely trouble themselves with such details. Complex analysis, which builds on real analysis, provides an indispensable set of tools for a great deal of modern mathematics. One last observation, admittedly a little biased: I don't recall ever hearing anyone, not even a non-mathematician, complain about knowing too much mathematics. In fact, I'll close with two quotations from Albert Einstein:

• In 1943 he answered a letter from a little girl who had difficulties in school with mathematics: "…Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
• "Any man who can drive safely while kissing a pretty girl is simply not giving the kiss the attention it deserves."

The first is well-known; the second shows his true genius. ;-) --KSmrq^T 06:48, 1 June 2006 (UTC)

Oh, we're listing choice quotes, are we? Here's an interesting anecdote. When Weisskopf was asked "How much mathematics does a theoretical physicist need to know?", he answered simply "more." -lethe ^{talk +} 17:47, 1 June 2006 (UTC)

Thanks to everyone who responded. My reason for asking is that I recently took such a course and found it a demoralising experience. Students at my University come from a wide variety of backgrounds and their levels of ability vary considerably. Success depends largely on diligent preparation for the end of semester examination. In a perfect world success in exams should, I think, depend on having developed a strong working intuition for the subject, and on being sharp and imaginative on the day. However, because of the way this particular course was taught, the exam positively encouraged cramming and parrot-fashion regurgitation of results and proofs, rather than said intuition and imagination. This rendered the course unfortunately somewhat pointless. I just wonder, given the subject matter and students of mixed ability, if it could ever have been otherwise... --The Gold Miner 15:26, 1 June 2006 (UTC)

Yes, it could have been different. I was a physics major who had only taken calculus type courses until my senior year, when I took advanced calculus. For the first time, my understanding was limited only by the time I spent thinking about things, and not by the vagueness of definitions, etc. Now, I'm a mathematician, and it is very satisfying (both research and teaching). Here's a quote from a recent student: "I maintain that taking Advanced Calculus was the turning point of my undergraduate career. It had reminded me how beautiful math is, how much it means to me, and how much I was willing to work in order to succeed in it." (Cj67 20:12, 1 June 2006 (UTC))

It's bad enough that a first-year calculus course might be taught so poorly; it's tragic that this should have been afflicted on aspiring mathematicians. But I must warn you that mathematics is not a field in which participants self-select because of their superior social skills (like teaching ability). Some mathematicians are awesome teachers; others are horrid. The ability to see a subject clearly for oneself is not at all the same as being able to communicate that insight clearly to others. So it has always been. This is one reason for Stigler's law of eponymy. --KSmrq^T 01:57, 2 June 2006 (UTC)