mm-308 === Subject: : gamma functionSince there's not good way to put this in terms of ASCII, I'll use page references. How would one go from equation #2 to equation #3 on http://mathworld.wolfram.com/GammaFunction.html ? I keep thinking that it's either substitution or integration by parts, but I can't seem to find anything that works.Thank you,Alec Colvin === Subject: : Re: gamma functionColvin..> How would one go from equation #2 to equation #3 on> http://mathworld.wolfram.com/GammaFunction.html ? ...In #2, just write t = u^2, dt = 2u du. The result is #3.LH === Subject: : Re: Die Cantor Die> Certainly, the mind does have a computational> element, but there is a lot more to the human mind than> computation. > Well, Penrose would agree with you. But I've never seen> an argument for your view that I find remotely convincing.> Presumably the workings of the brain obey the laws of> quantum mechanics, which in turn could be modelled in a> large computer. So maybe you're saying the mind is more > than the workings of the brain.> I will not continue this line of discussion.Is that a promise? === Subject: : Re: Die Petry Die> There is a key concept here that you don't seem to understand: there is > a difference between mathematics and applying mathematics. > When trying to distinguish between productive lines of inquiry,> and mental masturbation, it's a good idea to consider the possible> applications of your inquiry.Well you have shown how to do the latter. It's a shame you haven't done more of the former.> we don't know in advance if a non-computational aspect of math will lead > to an advance in computational math. > There is some truth to this, although mostly you are arguing with> your own misinterpretation of what I'm saying. As I have pointed> out way too many times, the study of formal systems does fall within> the scope of mathematics as the study of phenomena observable in> the world of computation. === Subject: : Re: puzzle: GCDs of Infinite Set of Integer Pairs>Hey I went away for 3 weeks to NZ, and you guys have managed to turn this into>one of the longest threads in the history of usenet, but AFAICS you still>haven't resolved the basic point for me, even though the discussion has got back>to pretty well exactly the same place.>My contention is:>You can say 'select a real randomly from the range [0,1]' but you can't actually>do it. It's like saying 'Imagine you can travel faster than the speed of light'.>A fine thought experiment, and you can draw all sorts of conclusions about a>universe that allows it, but this is not that universe.>I contend that:>1. There is no physical way of doing it (spin a pencil etc fails due to the>quantum nature of reality)Spin a pencil, etc does NOT fail due to the quantum nature ofreality. Space itself is not quantized.Radioactive decay works too, because time isn't quantized either.However, that doesn't give you a uniform distribution. It still gives>2. There is no way to describe a finite process which results in selecting such>a number randomly>3. There is no way to accurately describe the number you selected. If you>describe the number precisely, I'm entitled to call you a liar, and assert with>certainty that you didn't choose randomly.With probability 1, but not certainty.>On this basis, I assert that it's overly optimistic at best to say choose a>real number at random in the interval [0,1]>Oh, and I can't agree with this obviously:>>Who says they won't be the same no matter how many times I try? That's>>the way to bet, but it's not a certainty.>If ever there was a certainty, this is it. You can't even choose a number *once*>let alone twice. Even if I am eventually persuaded that you can in fact choose a>number at all, you definitely can't pick it again.That's easily refutable. The two events are independent, so if I canchoose a number once, I can certainly choose it twice.-- Matthew T. Russotto mrussotto@speakeasy.netExtremism in defense of liberty is no vice, and moderation in pursuitof justice is no virtue. But extreme restriction of liberty in pursuit of a modicum of security is a very expensive vice. === Subject: : Re: puzzle: GCDs of Infinite Set of Integer Pairs) Spin a pencil, etc does NOT fail due to the quantum nature of) reality. Space itself is not quantized.You're going to have to measure it. There's your quantum nature.SaSW, Willem-- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you !#EOT === Subject: : Re: puzzle: GCDs of Infinite Set of Integer Pairs) My contention is:) You can say 'select a real randomly from the range [0,1]' but you can't) actually do it. It's like saying 'Imagine you can travel faster than the) speed of light'. A fine thought experiment, and you can draw all sorts of) conclusions about a universe that allows it, but this is not that universe.You may want to read up on the 'Axiom of Choice', which isn't quitethe same as this, but it does seem to imply it.And the fun bit is, some mathematicians accept it, and some don't.Mathematics is perfectly happy either with or without it.SaSW, Willem-- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you !#EOT === Subject: : Simple honest plottingI'm writing a program that must plot smooth but possibly highly oscillating functions (antenna array factors), based on its input. My program is purely numerical, so I can't do anything too fancy to decide when the plot is too coarse. My question is, are there any simple heuristics I could use to decide when the plot is adequate?The only one I've been able to come up with is to subdivide the function so that every three points are very nearly collinear, but I don't want to depend upon a constant to determine what very nearly collinear is.Alex-- The only math in the movie, The Matrix, is in the title.(paraphrased from a posting to alt.math.recreational) === Subject: : Re: Simple honest plotting> I'm writing a program that must plot smooth but possibly highly> oscillating functions (antenna array factors), based on its input. My> program is purely numerical, so I can't do anything too fancy to decide> when the plot is too coarse. My question is, are there any simple> heuristics I could use to decide when the plot is adequate?> The only one I've been able to come up with is to subdivide the function> so that every three points are very nearly collinear, but I don't want to> depend upon a constant to determine what very nearly collinear is.used the curvature of the function to determine when points neededto be closer. This worked for functions and also for contour maps.Given the location, slope, and curvature at each point (which definesa circle passing through it), if adjacent points couldn't be approximatedwithin a couple pixels, the difference needed split. Of course, I hadthe function derivatives to work with. If I didn't, I guess I would haveused sets of 3 or more points to estimate them.- Brian === Subject: : Well-order the reals (was GCDs of Infinite Set of Integer Pairs) > Which reminds me of something tangentially related to where>this mess has now drifted: I've heard that you can well-order>the set of real numbers [such that every set of real numbers>will have a unique least member under this ordering]. How?>(Or is this one of those things that has been proven deductively,>but never by example?)> I think that's one of those things implied by the Axiom of Choice. Well, yes. :) AIUI, the Axiom of Choice says (or implies)that *any* set can be well-ordered. And the set of real numbersis a set, so... At the same time, it might be that the AoC is sufficient, butnot necessary, to prove that the reals can be well-ordered. Idunno. Daniel W. Johnson suggested that it had been proved that therewas no finite way to construct the well-ordering, though, whichkind of rules out the yes, you do it as follows... sort ofanswer I was hoping for. :-/-Arthur === Subject: : finding a Maclaurin seriesCould someone show me how to find the Maclaurin series for somefunction, either sin x, cos x, or ln (1+x)? thanks === Subject: : Re: finding a Maclaurin series> Could someone show me how to find the Maclaurin series for some> function, either sin x, cos x, or ln (1+x)? thanksTry reading your text first. === Subject: : > Bush vs Kerry, the winner is SharonComments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL belowX-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact X-Mail2News-Contact: http://80.65.224.85/http://www.nationalvanguard.org/story.php? id=2371~~~~~~~~~~~~~~~~~~~~~If you consider the content of this post to be particularly offensive, disgusting or plain illegal,it is probably 'designer abuse', a message designed specifically to hurt the remailer's reputation/existence. http://groups.google.com/groups?selm=6THHPRAL38002.4374074074% 40anonymous&oe=UTF-8&output=gplainSome people hate this remailer so badly that, for example, they did not hesitate to celebrate the death of 148 French tourists in a plane crash.Those people seceded from the human race, so don't hesitate to report them directly to the police. http://groups.google.com/groups?selm=Ymx1ZWpheQ%3D%3D .19d787f018eb3019d6fd3faa2125547c%401073158846.cotse.net&oe= UTF-8&output=gplain http://groups.google.com/groups?selm=Pine.LNX .4.58.0401181826110.31463%40thetis.deor.org&oe=UTF-8&output= gplainMore about the subject will be available http://frogadmin.yi.org/HOS/ === Subject: : Re: Question on modular arithmetic ETAtAhUArMFEYAxTPZuWaVF7HiFPcSEwCdMCFAxEtg364CChZztNDu8NSFvD4qS r I assume you are referring to an arithmetic series translated over tomodular arithmetic. For example, 1+2+3+... trnslated into mod 7 becomes1+2+3+...+6+0+1+2+3+... . In that case there is an easy extension ofthe arithmetic-sum formula -- IF the modulus is odd.In an ordinary arithmetic series you have the sum formula:(n/2)*(a_1+a_n)where n is the number of terms and a_1 and a_n are the first and lastterms. Now in modular arithmetic, what you do is use that formula butwith the division by 2 replaced by a multipliation by the inverse of 2.In modulo 7 you have 2*4 == 1 so your formula becomes:4*n*(a_1+a_n)which may be easily figured. For any odd modulus M, the mltiplicativeinverse of 2 is nothing more than (M+1)/2. So your arithmetic sumformula in mod M, with M odd, is:(M+1)/2*n*(a_1+a_n)Let's do my example above: a_1 = 1 and the common difference is 1, andwe are working in modulo 7. Then we have the following partial sums:n a_n 4*n*(a_1+a_n) Actual sum1 1 4*1*2 == 1 12 2 4*2*3 == 3 1+2 == 33 3 4*3*4 == 6 3+3 == 64 4 4*4*5 == 3 6+4 == 35 5 4*5*6 == 1 3+5 == 16 6 4*6*0 == 0 1+6 == 07 7 == 0 4*0*1 == 0 0+0 == 0and of course after that, the series repeats in this case. The formulaagrees with the actual sum determined by direct recursion.This method does not work for even M because when M is even, there is nomultiplicative inverse for 2. (M+1)/2 is then not an integer, and youcan't multiply 2 by any integer in an even modulus to get 1. For evenmoduli you cannot express the sum in terms of just (the residues of) thefirst and last terms.--OL === Subject: : When Cayley transform is orthogonal ?Let A be a n x n real matrix such that det(I-A)=/=0 .The Cayley transformation of A is C(A)=(I-A)^{-1}(I+A).How we can describe all matrices A for which C(A) is orthogonal ?=== Subject: === Subject: : Re: Trapezoidal Rule & Simpson's Rule - Comparison> I'm looking for a simple and obvious example where the Trapezoidal Rule> would give a better approximation to a given definite integral than> Simpson's Rule. Assume the number of subintervals used is the same for both approximations.=== Subject:=== Subject:==Sorry for following correction : Proposition. The inequality === Subject:=== Subject:=== Subject:=== Subject: Subject:RT(f;a,b) <= RS(f;a,b) for all [a,b] , subset in [A,B] , if and only if f:[A,B]---> is a concave function in the sense of Jensen, i.e. (J)-concave on [A,B]. === Subject: : Re: Trapezoidal Rule & Simpson's Rule - Comparison>Let f(x) = x^(2/5) on the interval [-1,1] and use one subinterval.> I assume you mean two subintervals, i.e. you evaluate f at the three > points -1, 0 and 1, with Trap = 1/2 f(-1) + f(0) + 1/2 f(1) and> Simpson = 1/3 f(-1) + 4/3 f(0) + 1/3 f(1).> A singularity is not necessary, of course: try cos(pi x) which> the trapezoidal rule (for these points) happens to get exactly right > but Simpson doesn't. And this generalizes easily to give examples> with more subintervals.Right, ever since the OJ trial, I've been confused about Simpson's Rule. === Subject: : Re: Help with equationhr@gfxn.com (xillion) asked:> Can anyone tell me what the mathematical> equation is for the following> statement:> The whole is worth more than> the sum of its parts.That would be: ( __ ) ( )U(W) > U( /_ P_i ) ( ) ( i in I ) whereU(*) = utility functionW = WholeP_i = a part__/_ = sumI = set of parts(view in fixed-width font)(this is based on expected values, and ignores risk theory)HTH === Subject: : The characteristics of polynomial functions in fittingHello list,I want to fit a dataset {(x, y)} to a function like:y = f(P(x)), where P(x) is a polynomial function of x with a certainorder. I believe this data is like a function encapsulating thepolynomial function. Assume I have plenty of points, and the data isaccurate (say, they are exactly the output of y=f(P(x) ). We can alsoassume the points (x, y) are on grids, where x, y are only integers.The problem is that f() is unknown, and I don't know how to figure itout just from the statistics of the data points. May be this is atopic in the function space, is there anyone can help me about it?Please forgive my ignorance in this area.My practical problem is actually on multi-variate fitting, but here wejust discuss the simple case first.Steven === Subject: : Re: The characteristics of polynomial functions in fitting> Hello list,> I want to fit a dataset {(x, y)} to a function like:> y = f(P(x)), where P(x) is a polynomial function of x with a certain> order. I believe this data is like a function encapsulating the> polynomial function. Assume I have plenty of points, and the data is> accurate (say, they are exactly the output of y=f(P(x) ). We can also> assume the points (x, y) are on grids, where x, y are only integers.> The problem is that f() is unknown, and I don't know how to figure it> out just from the statistics of the data points. May be this is a> topic in the function space, is there anyone can help me about it?> Please forgive my ignorance in this area.> My practical problem is actually on multi-variate fitting, but here we> just discuss the simple case first.> StevenWhat do you know about P(). If no more than you have told us so far, there isn't much we can say about f(). === Subject: : Re: Catholic or Protestant background ?> Went to a bunch of schools, the military, behaved, (well, maybe)> till, still drunk, right after the doctoration ceremonies, my ex-high> school cronies at the bar told me that I would never catch up with> their earnings until I would be 55, if lucky....... Having just gone> thru 4 years of penniless academic slavery, the realization of this> negative $$-aspect-future made me so furiously mad that marched> right away over to my now ex-professor/advisor's office where I> placed my cum laude diploma carefully into my square tassel hat, took> a onto it and poured a half a quart of good vodka over it. Threw> a match at it and watched the beautiful faint blue flame evolve> into a yellow sodium fire that began to stink more & more.> Just as I was celebrating my achievement professor schmuckface,> as I called him, came waltzing in and asked me what I was doing.....> deja vue........---ex-communication, etc....... left town in a jiffy.> Never looked back .......went to see the world.......ahahahahaha.......This is the most believable thing you have ever posted. === Subject: : Re: Catholic or Protestant background ?I see that this garbage has been posted in sci.physics, sci.math and sci.chem and, apparently nowhere else.This it has been posted only in places where it is totally irrelevant and OT to the ultimate.If there actually is a just god, may he/she/it condemn the OP to a suitable level of hell.PLONK! === Subject: : Re: Catholic or Protestant background ? charset=Windows-1252> I see that this garbage has been posted in sci.physics, sci.math and > sci.chem and, apparently nowhere else.> This it has been posted only in places where it is totally irrelevant > and OT to the ultimate.> If there actually is a just god, may he/she/it condemn the OP to a > suitable level of hell.> PLONK!Jeff, here's some Jesus warrior who doesn't like you nor me.Would you please administer to his soul, or do you wantto send him to Ravencrag? ....Virgil is sooo, pious, AAHahahahahahaha..........AHahahahhaha..........Virgil, a kacksacking bible beater, cranked himself....Virgil, listen, a just god WILL give you a special stool in heavenfor you rallying for him........AHAHAHAHahhahaa...ahahahanson === Subject: : Re: How to setup an observer by Femlab?> I'm trying to construct an observer for a control system by Femlab.> The problem is basically like this.> 1. We need to simulate the diffusion and convection of a system. For> this model, every parameter is known. (I've done this)> 2. An observer is running in parallel to fit the first model. The> initial states of the observer are unknown. We are trying to find out> the initial states by observing some of the outputs of the first model> (don't know how to do)What you have is essentially an optimization problem; you need to set upthe criterion fcn measuring a difference between a desired and actualresponse, then find the initial conditions which minimize thosedifferences. afaik, Femlab may not be equipped for these kind ofproblems, but you should check that w/Comsol tech support.--Dr.B.Voh-------------------------------------------- ----------Applied Algorithms http://sdynamix.com === Subject: : Re: tensorsOk, cool, that makes a lot more sense now, thanks! === Subject: : Re: generalization of stokes' and divergence theorems?Cool, thanks for the help!