mm-294 I hope that you enjoyed this complimentary ASCII babe, brought to you> free as a gift from your friends at the mathematical advertising> society.> Your dear friend,> Nathaniel Deeth> Age 11> fuffyThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) Guardian, 1966 ==== Subject: Re: Sex and Mathmike_deeth@yahoo.com (Mike Deeth) made some kool advert, but Cantor isall about Bijections, (or lack of them), So I think we need to showa bijection between the two most prominent features of his pic.Hence I remove the coverings. (Here I use Taylor's uncovering lemma!) Anon ==== Subject: Re: Sex and MathIn sci.math, Bill Taylor< 716e06f5.0311122046.7f477f3f@posting.google.com>:> mike_deeth@yahoo.com (Mike Deeth) made some kool advert, but Cantor is> all about Bijections, (or lack of them), So I think we need to show> a bijection between the two most prominent features of his pic.> Hence I remove the coverings. (Here I use Taylor's uncovering lemma!)[Taylor's uncovering lemma used too many times ]> Hmmmm... there's a thought.... I wonder what another application > of the uncovering lemma would reveal??> -------------------------------- ==== ========================= ==== => AnonWell, most likely, the sequence would be:[-1] woman in bikini[0] naked woman[1] musculature[2] bones[3] nothing-- #191, ewill3@earthlink.netIt's still legal to go .sigless. ==== Subject: Create your png Magic Square via a free web applicationNow you can create your PNG Magic Square for free using the webapplication you can find at http://www.pivari.com/squaremaker.htmlIt is based on our perl module Math::MagicSquare you can find athttp://search.cpan.org/ , exactly athttp://search.cpan.org/~fpivari/Math-MagicSquare-2.02/If you want to give a similar service inside your web site (internetor intranet) to your students, surfers, ... we can sell you the cgiapplication (a simple executable file that doesn't requirecustomization) for Windows, Linux, Solaris, AIX, HP-UX, FreeBSD,Tru64, SGI, ...Contact us directly at mailto: info@pivari.com ==== Subject: asking for your opinion I hope this does not seem to out of place and I hope that you are able tohelp me. I am about to complete my second year of my maths degree. I am going to bedoing a major in mathematics next year and I a am thinking about doinghonours the year after next. I have been looking into some areas of maths that I might be interes indoing more study in and one of them is that of complex (chaotic) systems.Well pretty much anything to do with Differential Equations I could beinteres in I just need to find out what they are really about. What I am after is just some good books on the area so that I can get a bitof a taste of it before I decide it is what I want to do further study in.Any of the other areas of applied maths that are rather interesting or newand exciting that would be helpful as well. I mean really anything as Ireally just want to see what is out there just so I have plenty of optionsStephen ==== Subject: Re: asking for your opinion> I hope this does not seem to out of place and I hope that you are able to> help me.> I am about to complete my second year of my maths degree. I am going tobe> doing a major in mathematics next year and I a am thinking about doing> honours the year after next.> I have been looking into some areas of maths that I might be interesin> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anything to do with Differential Equations I could be> interes in I just need to find out what they are really about.> What I am after is just some good books on the area so that I can get abit> of a taste of it before I decide it is what I want to do further study in.> Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well. I mean really anything as I> really just want to see what is out there just so I have plenty of options> StephenFor a good book on just ODE's, check out Morris Tennenbaum' s book fromDover, Ordinary Differential equations. It is only about 15 USD! Whichmakes it particularly nice. It is pretty comprehensive too, for a Intro.book.Lurch== Subject: Re: asking for your opinion === Hi Sephen,Nature has a very limi ability for solving differential equations, shemainly works through itteraative processes. Chaos theory is about analsingthe possible outcomes of those itterative processes.It also links in with the biggest debate in science: Einsein's God does notplay dice. Einstien's world was deterministic obeying the laws of Newton(with relativitstic corrections). Quantum mechanics took the view thateverything works by chance. Chaos theory shows that complex systems work ina psudorandom fashion.My guess is that 21st century appied maths is going to be about applyingchaos theory to classical physics to propery understans physics.-- regards BruceBruce Harveybruce@bearsoft.co.ukThe Alternative Physics Sitehttp://users.powernet.co.uk/bearsoft> I hope this does not seem to out of place and I hope that you are able to> help me.> I am about to complete my second year of my maths degree. I am going tobe> doing a major in mathematics next year and I a am thinking about doing> honours the year after next.> I have been looking into some areas of maths that I might be interesin> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anything to do with Differential Equations I could be> interes in I just need to find out what they are really about.> What I am after is just some good books on the area so that I can get abit> of a taste of it before I decide it is what I want to do further study in.> Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well. I mean really anything as I> really just want to see what is out there just so I have plenty of options> Stephen ==== Subject: Re: asking for your opinion> I hope this does not seem to out of place and I hope that you are able to> help me.> I am about to complete my second year of my maths degree. I am going to be> doing a major in mathematics next year and I a am thinking about doing> honours the year after next.> I have been looking into some areas of maths that I might be interes in> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anything to do with Differential Equations I could be> interes in I just need to find out what they are really about.> What I am after is just some good books on the area so that I can get a bit> of a taste of it before I decide it is what I want to do further study in.(Table of contents at http://www.jesus.ox.ac.uk/~dacheson/calcon.html)Glendinning, P. Stability, Instability and Chaos: An Introduction to theTheory of Nonlinear Differential Equations, Cambridge University Press1994.Essentially the lecture notes of a third year Cambridge undergraduatecourse.> Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well.Acheson, D.J. Elementary Fluid Dynamics, Oxford University Press 1990(http://www.jesus.ox.ac.uk/~dacheson/efdcon.html)And, in the interests of equal time,Batchelor, G.K. An Introduction to Fluid Dynamics, Cambridge UniversityPress 1967 (reprin 2000)This, however, is far more dense and less readable than Acheson.-- ==== Subject: Re: asking for your opinionMy opinion is that if your Differential Equations teacher had been a f***ingmindkilling brainslayer with a psycho-like uncomprehendibility and optionalTotal-Encephalon-Annihilation ModeT like my teacher was, then you wouldn'tbe willing to study that field. Luckily yours, it seems, wasn't. Excuse mefor the rhetorics and for the unusefulness of my post, but you know... onone hand, people must know that being into maths doesn't mean that youcannot have a good command of the language, on the other hand, sometimespeople need to feel luckier than someone else to be happy ;-).Stephen ha scritto nel messaggio> I hope this does not seem to out of place and I hope that you are able to> help me.> I am about to complete my second year of my maths degree. I am going tobe> doing a major in mathematics next year and I a am thinking about doing> honours the year after next.> I have been looking into some areas of maths that I might be interesin> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anything to do with Differential Equations I could be> interes in I just need to find out what they are really about.> What I am after is just some good books on the area so that I can get abit> of a taste of it before I decide it is what I want to do further study in.> Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well. I mean really anything as I> really just want to see what is out there just so I have plenty of options> Stephen ==== Subject: Re: asking for your opinionI will be sure to tell my DE's teacher that thanks.> My opinion is that if your Differential Equations teacher had been af***ing> mindkilling brainslayer with a psycho-like uncomprehendibility andoptional> Total-Encephalon-Annihilation ModeT like my teacher was, then youwouldn't> be willing to study that field. Luckily yours, it seems, wasn't. Excuse me> for the rhetorics and for the unusefulness of my post, but you know... on> one hand, people must know that being into maths doesn't mean that you> cannot have a good command of the language, on the other hand, sometimes> people need to feel luckier than someone else to be happy ;-).> Stephen ha scritto nel messaggio> I hope this does not seem to out of place and I hope that you are ableto> help me. I am about to complete my second year of my maths degree. I am going to> be> doing a major in mathematics next year and I a am thinking about doing> honours the year after next. I have been looking into some areas of maths that I might be interes> in> doing more study in and one of them is that of complex (chaotic)systems.> Well pretty much anything to do with Differential Equations I could be> interes in I just need to find out what they are really about. What I am after is just some good books on the area so that I can get a> bit> of a taste of it before I decide it is what I want to do further studyin.Any of the other areas of applied maths that are rather interesting ornew> and exciting that would be helpful as well. I mean really anything as I> really just want to see what is out there just so I have plenty ofoptions>>Stephen>>== Subject: Re: differentiable...problem... ==== Subject: Re: differentiable...problem...>> if f is differentiable on (0, infinite)>> and lim [f(x) + f'(x)] = L (x->infinite)>> show that lim f(x) = L (x->infinite) and lim f'(x) = 0 (x->infinite)>| f e^x (f + f') e^x>| lim f + f' = L => lim f = lim ----- = lim ------------ = L>| x->oo x->oo x->oo e^x x->oo e^x>> Not even the existence of lim(x->oo) f(x) ? >L'Hospital's rule for the form lim f/g, lim g = oo >needs no hypotheses on existence or value of lim f>> [1] A. E. Taylor, L'Hospital's Rule>> Amer. Math. Monthly, Vol. 59, No. 1 (Jan., 1952), pp. 20-24..>> [2] A. M. Ostrowski, Note on the Bernoulli-L'Hospital Rule>> Amer. Math. Monthly, Vol. 83, No. 4 (Apr., 1976), pp. 239-242.. >Jstor requires a subscription (e.g. most major universities). >Alternatively, many public libraries subscribe to the Monthly.Oh, I'd have to ask for an inter-library copy be mailed.For math papers, four pages might not be short and simple.How demanding are they? Within the scope of 2nd year calculus?All the l'Hopital's rules I know come of Cauchy's mean value theorem.If f',g' exist on [a,b], g(a) /= g(b), then some z between a,b with f'(z)/g'(z) = (f(b)-f(a))/(g(b)-g(a))Does Taylor or Ostrowski have a similar approach?Is one to be recommended over the other?> However when f e^x -> k; then> f = fe^x e^-x -> 0> f e^-x = f e^x e^-2x -> 0L = lim f+f' - lim 2f = lim f'-f> 0 = lim f = lim f e^-x / e^-x = (f' - f)e^-x / -e^-x = lim f-f' = -L> 0 = lim f+f' - lim f = lim f'So if lim f = k and lim f' exists, then lim(x->oo) f+f' = k + lim f'> k = lim(x->oo) f = k + lim f'; lim f' = 0If lim f' doesn't exist, then for n >= 2> f_n(x) = sin(x^n)/x -> 0> (f_n)'(x) = nx^(n-1) cos (x^n)/x - (sin x^n)/x^2 -> oscillation---- ==== Subject: Re: differentiable...problem...> if f is differentialbe on (0, infinite)> and lim [f(x) +f'(x)] = L (x->infinite)> show that lim f(x) = L (x->infinite) and lim f'(x) = 0 (x->infinite)Everyone responds with L'hospital's rule, but that hardly helps onesee *why* it is true. Here's how I look at it (although this is NOTrigorous so dont use it in your answer).Let O be an extremely huge number and o an extremely tiny one. Precisely, let O be so large that f(w)+f'(w) is within a given, tinyreal epsilon of L for all w>O (which we know we can do by hypothesis). Then we should have something likef(O) + f'(O) = L +/- oIn other words, f(O) + f'(O) ~= L (~= means approximately equal to)We can approximately rewrite f(O) as L-x and approximately rewritef'(O) as x. Here x is conjured out of thin air by the fact that(L-x)+x=L. What we are asked to prove is basically that x is verysmall. So suppose that, on the contrary, x is not small at all. Butx is approximately f'(O), so if it is not small, we are forced toconclude that the rate of change of f at O is not small either. Butwe stipula that for all w>O, f(w)+f'(w) is within epsilon of L. Sowe must conclude that the rate of change very quickly balances outso as not to violate this stipulation. But how quickly must this beaccomplished? Since epsilon can be made arbitrarily small, no matterhow quickly the rate of change balances itself out, it just ain'tquick enough... so we're forced to conclude that x is, in fact, small,which is to say that f'(O) is approximately 0 and f(O) isapproximately L.(I must reemphasize this is completely nonrigorous and very handwavey. The point is to make it clear *why* the proposition is true- L'hospital's rule is calculus's version of induction, it gets the jobdone but doesn't explain much)P.S. I find it somewhat amusing how in your postings you try topresent yourself as some sort of stereotypical highschool-agedhot-girl. I'm guessing you are really a 40 year old male truckdriver from Texas? :)== Subject: Re: Linear independence === I found it! > == Subject: Re: linear independence/square roots/primes >>How to prove that the square roots of the >>primes are linearly independent over the rationals? >Let p1, p2, ..., p_i, ... be a sequence of different primes. >INDUCTION HYPOTHESIS: sqrt(p_n) does not belong to the field > E_(n-1)=Q[sqrt p1, sqrt p2, ..., sqrt p_(n-1)] >Proof n-1 => n: By ind. hyp., E_(n-1) has 2^(n-1) automorphisms >(sqrt p_i -> +- sqrt p_i). Suppose sqrt (p_n) belongs to E_(n-1). >Apply all automorphisms, and you see that in the Q-linear >decomposition of p_n there are AT MOST two nonzero summands. >More exactly, >(1) sqrt p_n = r+s*sqrt(q_1*q_2*...*q_i) , r,s from Q. where q_j >are some different primes from the sequence p1, ..., p_(n-1).How's that done? >But (1) is clearly impossible. QED---- ==== Subject: compact definition....book-topology def)A collection B of subsets of a space X is said to cover X,or to be a covering of X, if the union of the elements of B is equal to X.it is called an open covering of X if its elements are open subsets of X.def)A space X is said to be compactif every open covering B of X contains a finite subcollection that alsocovers X--------------------------------i think that meaning of compact is X = U(G_i_n) by upper definition.but in the other book,i saw that definition of compact is X C U(G_i_n) {C : inclusion sign}which of definiton is right??advice ...please.......... ==== Subject: Re: compact definition....> book-topology > def)> A collection B of subsets of a space X is said to cover X,> or to be a covering of X, if the union of the elements of B is equal to X.> it is called an open covering of X if its elements are open subsets of X.> def)> A space X is said to be compact> if every open covering B of X contains a finite subcollection that also> covers X> --------------------------------> i think that meaning of compact is X = U(G_i_n) by upper definition.> but in the other book,> i saw that definition of compact is X C U(G_i_n) {C : inclusion sign}> which of definiton is right??> advice ...please..........since the G_i are subsets of X, any (finite) union of them is, andthus the second implies the first ( U{G_i_n} C X C U{G_i_N}) andclearly the first implies the second. ==== Subject: Re: compact definition....hot-girl> def)> A collection B of subsets of a space X is said to cover X,> or to be a covering of X, if the union of the elements of B is equal to X.> it is called an open covering of X if its elements are open subsets of X.> def)> A space X is said to be compact> if every open covering B of X contains a finite subcollection that also> covers XThis definition is correct.> --------------------------------> i think that meaning of compact is X = U(G_i_n) by upper definition.> but in the other book,> i saw that definition of compact is X C U(G_i_n) {C : inclusion sign}> which of definiton is right??They are the same if the sets G_i_n are contained in X. In the other book,when he says X C U(G_i_n) {C : inclusion sign}the author is no doubt defining a compact _subspace_ X of some other space.Again the definition is okay.LH ==== Subject: Re: Marketing shift, core issues>>At least now I won't have to feel guilty if I decide to use tactics,>>as I've given fair warning.>And if this doesn't work, you can start pestering Congress to get your >theorem passed into law.He'll have to kick their cots to wake them up./BAHSubtract a hundred and four for e-mail. ==== Subject: Re: Marketing shift, core issues >>I've been thinking about my problems with getting any kind of>>admission that my math arguments showing the core error in mathematics>>are correct, so I've gone to marketing books. You know, in case you're curious, this sounds really really stupid.>> If your results were correct you'd be able to convince people of>> them by explaining the proofs carefully. But in fact they're wrong,>> people continually explain what the errors are, and tactics from>> marketing books are not going to change that. If his results were correct, the same people who point out errors to him >> with saintly patience would instead have provided all these explanations.>>Having spent some time now reading marketing tactics, I can clearly>see that both Ullrich and Bau are selling a viewpoint to the>readership.>>Here the assertion is that if I were right then people would>necessarily agree with me!!!>>Is that true in your experience?> It's certainly true in _my_ experience that when I'm right about> something and have a valid proof that I'm right then competent> mathematicians agree I'm right, after I've explained the proof,> yes. That includes cases where they were certain at first I was> wrong, by the way.Don't you agree, David, that from C1-C4--indeed, from C3,C4 alone--any*competent* mathematician could have deduced Ex~(x=x)?C1 AxAy[x=y -> Az(z in x <-> z in y)]C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] C3 EyAx[x in y <-> Et(x in t) & A] (with y not free inA)ClassificationC4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (WeakExtensionality)someone will point out the error.?Did your Homies correct you? Or did they *defend* your mistake--and you for having made it? --John ==== Subject: Re: Marketing shift, core issues>I've been thinking about my problems with getting any kind of>admission that my math arguments showing the core error in mathematics>are correct, so I've gone to marketing books.You know, in case you're curious, this sounds really really stupid.> If your results were correct you'd be able to convince people of> them by explaining the proofs carefully. But in fact they're wrong,> people continually explain what the errors are, and tactics from> marketing books are not going to change that.If his results were correct, the same people who point out errors to him > with saintly patience would instead have provided all these explanations.>>Having spent some time now reading marketing tactics, I can clearly>>see that both Ullrich and Bau are selling a viewpoint to the>>readership.>>Here the assertion is that if I were right then people would>>necessarily agree with me!!!>>Is that true in your experience?>> It's certainly true in _my_ experience that when I'm right about>> something and have a valid proof that I'm right then competent>> mathematicians agree I'm right, after I've explained the proof,>> yes. That includes cases where they were certain at first I was>> wrong, by the way.>Don't you agree, David, that from C1-C4--indeed, from C3,C4 alone--any>*competent* mathematician could have deduced Ex~(x=x)?Two comments, the second for anyone who missed your first few hundredrepetitions of this question:(i) your question has no relevance here - you seem to be confusinga statement and its converse.(ii) the original claim was that Ex~(x=x) followed _in_ standardset theory. Since Ax(x=x) is a theorem of standard set theory, ifthis follows then it also follows that C1-C4 are simply inconsistentwith standard set theory.>C1 AxAy[x=y -> Az(z in x <-> z in y)]>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in>A)Classification>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak>Extensionality)>someone will point out the error.?>Did your Homies correct you? Or did they *defend* your mistake-->and you for having made it? A few people recently have poin out that to conclude that whatI said was erroneous you need to take it out of context. Which isof course true.>--JohnGiggle: When you set up a new yahoo account with a silly namelike pzuffy the idea is to avoid using your real name in postsyou make under the silly name.(I'm assuming the point to pzuffy was to make it appear thatthere's someone other than you out there who agress with thethings you say. If the point was something other than thatthen never mind. Although it's hard to see what other pointthere could have been - if the point was to set up a newaccount because the old one was getting too clogged withspam, or for some other reason other than hiding youridentity, it's hard to see why you'd choose a name like'pzuffy'.) ==== Subject: Re: Marketing shift, core issues>Don't you agree, David, that from C1-C4--indeed, from C3,C4 alone--any>*competent* mathematician could have deduced Ex~(x=x)?> Two comments, the second for anyone who missed your first few hundred> repetitions of this question:> (i) your question has no relevance here - you seem to be confusing> a statement and its converse.> (ii) the original claim was that Ex~(x=x) followed _in_ standard> set theory. Since Ax(x=x) is a theorem of standard set theory, if> this follows then it also follows that C1-C4 are simply inconsistent> with standard set theory.>C1 AxAy[x=y -> Az(z in x <-> z in y)]>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in>A)Classification>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak>Extensionality)>someone will point out the error.?>>Did your Homies correct you? Or did they *defend* your mistake-->and you for having made it? > A few people recently have poin out that to conclude that what> I said was erroneous you need to take it out of context. Which is> of course true.Don't *all* of the following indicate that you had *no idea* (and*still* have no idea) wh Ex~(x=x) follows from C1-C4? ==== ========================================================= === When you said set theory I assumed you meantZF. That's what set theory with no qualificationmeans these days.If you meant NGB set theory then no, C1-C4 arenot inconsistent with set theory. It does _not_follow that C1-C4 give an example of somethingwhich is not equal to itself, or an example ofsomething which does not exist. [DON'T SNIP--J.C.]It is correct that I have no idea why Ex~(x=x)follows from C1-C4. This is because (assumingthat NBG is consistent) NBG has a model inFOL=. In that model everything is equal toitself.[DON'T SNIP--JC]C1 AxAy[x=y -> Az(z in x <-> z in y)]C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] C3 EyAx[x in y <-> Et(x in t) & A] (with y not free inA)ClassificationC4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (WeakExtensionality) ==== ===================================== ==== =>You have no idea why Ex~(x=x) follows from C1-C4>because you are brain-dead analysis teacher who>can only work with the routines he has memorized.Could be. Now show us why Ex~(x=x) _does_ followfrom C1-C4. ==== ===========================================Exhibit of proof of Ex~(x=x) from C1-C4 and someone will pointout the error.IT'S ULLRICH BAIL-OUT TIME! SNAP TO IT, MAGIDIN! ON THE DOUBLE,HUGHES!> Giggle: When you set up a new yahoo account with a silly name> like pzuffy the idea is to avoid using your real name in posts> you make under the silly name.> (I'm assuming the point to pzuffy was to make it appear that> there's someone other than you out there who agress with the> things you say. If the point was something other than that> then never mind. Although it's hard to see what other point> there could have been - if the point was to set up a new> account because the old one was getting too clogged with> spam, or for some other reason other than hiding your> identity, it's hard to see why you'd choose a name like> 'pzuffy'.)You've got your fuffy. I've got my pzuffy. --John ==== Subject: Re: Marketing shift, core issues>>I've been thinking about my problems with getting any kind of>>admission that my math arguments showing the core error in mathematics>>are correct, so I've gone to marketing books. You know, in case you're curious, this sounds really really stupid.>> If your results were correct you'd be able to convince people of>> them by explaining the proofs carefully. But in fact they're wrong,>> people continually explain what the errors are, and tactics from>> marketing books are not going to change that. If his results were correct, the same people who point out errors to him >> with saintly patience would instead have provided all these explanations.>>Having spent some time now reading marketing tactics, I can clearly>see that both Ullrich and Bau are selling a viewpoint to the>readership.>>Here the assertion is that if I were right then people would>necessarily agree with me!!!>>Is that true in your experience?It's certainly true in _my_ experience that when I'm right about> something and have a valid proof that I'm right then competent> mathematicians agree I'm right, after I've explained the proof,> yes. That includes cases where they were certain at first I was> wrong, by the way.> Don't you agree, David, that from C1-C4--indeed, from C3,C4 alone--any> *competent* mathematician could have deduced Ex~(x=x)?> C1 AxAy[x=y -> Az(z in x <-> z in y)]> C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] > C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in> A)Classification> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak> Extensionality)> someone will point out the error.?> Did your Homies correct you? Or did they *defend* your mistake--> and you for having made it? > --John> fuffyThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) Guardian, 1966 ==== Subject: Re: Mike Turner in NY Times Science 11/11/03>> I think we are so confused that we should keep an open mind to>> tinkering with gravity, said Dr. Michael Turner, a cosmologist at>> the University of Chicago.>> previously>> Q to Ed Witten: How can the cosmological constant be so close to>> zero but not zero?DL> Close to zero? WTF does close mean? Is there a maximum valueDL> we can compare it to to see how close it is?The current estimates are that the energy density contribu by thecosmological constant (or dark energy) is 0.7, in units of the closurephysics that make predictions about the cosmological constant. Thetypical value predic is 10^50 or so. -- Lt. Lazio, HTML police | e-mail: jlazio@patriot.netNo means no, stop rape. | http://patriot.net/%7Ejlazio/sci.astro FAQ at http://sciastro.astronomy.net/sci.astro.html ==== Subject: Re: Mike Turner in NY Times Science 11/11/03>> I think we are so confused that we should keep an open mind to>> tinkering with gravity, said Dr. Michael Turner, a cosmologist at>> the University of Chicago. previouslyQ to Ed Witten: How can the cosmological constant be so close to>> zero but not zero?> DL> Close to zero? WTF does close mean? Is there a maximum value> DL> we can compare it to to see how close it is?> The current estimates are that the energy density contribu by the> cosmological constant (or dark energy) is 0.7, in units of the closure> physics that make predictions about the cosmological constant. The> typical value predic is 10^50 or so.Excellent! (I'm going to pretend to understand that.) ;-)--Denis Loubetdloubet@io.comhttp://www.io.com/~dloubet ==== Subject: Re: Comprehensive Math tutorial software wan.> My school gets a lot of kids that are behind their nominal peer group> in math skills.> I'm looking for a self-contained math remediation system for junior> and senior high.See if ALEKShttp://www.aleks.com/will meet most of those needs. They offer UNLIMI numbers of 48-hour freetrials, so you'll have time to check it out. The 48-hour free trials don'tallow CONVENIENT tracking of a particular student's progress over time,which is how the company gets people to sign up for the real service.I don't work for the company; I've just used the free trial a few timeswhile checking my son's progress in other math courses.Hope this helps!-- Karl M. Bunday Christ has set us free. Galatians 5:1Learn in Freedom (TM) http://learninfreedom.org/kmbunday AT earthlink DOT net (preferred email address)-- tml ==== Subject: origin of homotopy and homologyhi,Does anyone know how Poincare came up with the idea of homotopy and homologyand how he defined them originally?Thanks!Sarah ==== Subject: Determining number of scenarios to get the final score of a hockey gameI use to know how to do this with my eyes closed - but it's been sometime and I forget the formulas. Can someone help me figure out thenumber of different combinations for the following.The Final score of the hockey game was Team A 13 Team B 4I went to sleep with the score tied 2-2.What I am trying to figure out is, how many different ways could thetwo teams have scored to arrive at the 13-4 final score.Example it could have went 3-2, 4-2, 4-3, 5-3,...11-3. 11-4...13-4or it could have went 3-2, 4-2....12-2, 13-2, 13-3, 13-4Thanks! ==== Subject: Re: Determining number of scenarios to get the final score of a hockey game> I use to know how to do this with my eyes closed - but it's been some> time and I forget the formulas. Can someone help me figure out the> number of different combinations for the following.> The Final score of the hockey game was Team A 13 Team B 4> I went to sleep with the score tied 2-2.> What I am trying to figure out is, how many different ways could the> two teams have scored to arrive at the 13-4 final score.There were 13 goals scored after you went to sleep, and 2 of them werescored by Team B. The number of combinations is the binaomialcoefficient 13C2 = 13! / (2! * 11!) = 78.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== Subject: Re: Determining number of scenarios to get the final score of a hockey gameI once got interes in the number of ways to get a particularbowling score (e.g., 1 way to get a 0 and 1 way to get a 300).Wrote a program to do this (try it, it was kind of challenging).paper on th topic that had already been written (in J Rec Math or Math Mag orsomewhere similar). As I recall, the score that can be attainedthe most number of ways is in the 70's.Chip Klostermeyer> I use to know how to do this with my eyes closed - but it's been some> time and I forget the formulas. Can someone help me figure out the> number of different combinations for the following.> The Final score of the hockey game was Team A 13 Team B 4> I went to sleep with the score tied 2-2.> What I am trying to figure out is, how many different ways could the> two teams have scored to arrive at the 13-4 final score.> There were 13 goals scored after you went to sleep, and 2 of them were> scored by Team B. The number of combinations is the binaomial> coefficient 13C2 = 13! / (2! * 11!) = 78.== Subject: Re: Determining number of scenarios to get the final score of a hockey game === Mathematics Magazine: Volume 63, Number 4, Pages: 239-243 (1990)A Generating Function for the Distribution of the Scores of all PossibleBowling GamesCooper, Curtis N. and Kennedy, Robert E.> I once got interes in the number of ways to get a particular> bowling score (e.g., 1 way to get a 0 and 1 way to get a 300).> Wrote a program to do this (try it, it was kind of challenging).> paper on th topic that had already been written (in J Rec Math or Math Magor> somewhere similar). As I recall, the score that can be attained> the most number of ways is in the 70's.> Chip Klostermeyer> I use to know how to do this with my eyes closed - but it's been some> time and I forget the formulas. Can someone help me figure out the> number of different combinations for the following.The Final score of the hockey game was Team A 13 Team B 4I went to sleep with the score tied 2-2.What I am trying to figure out is, how many different ways could the> two teams have scored to arrive at the 13-4 final score.There were 13 goals scored after you went to sleep, and 2 of them were> scored by Team B. The number of combinations is the binaomial> coefficient 13C2 = 13! / (2! * 11!) = 78. ==== Subject: Re: Cardinality of 2^n numbers?> Suppose we had the set of all integers which resembled an integer 2^n> where n is any integer. This would start from 1 in the case where n => 0 to indefinately high. Let's call this set Q.hope you don't mind if i rewrite it:let Xn be a sequence of sets with card(Xn)=n, let P(Xn) denote the set of all subsets/its power setcard(P(Xn))=2^n.then you claim that, aslim card(Xn) is alpeh-0 and lim card(P(Xn)) is aleph-0, you have a contradiction. what you haven't done is show: lim(P(Xn)) is the same as P(lim(Xn))indeed an element in lim(P(Xn)) (assuming X1 is contained in X2 iscontained in X3 etc which we may as well do) is in one of the P(Xn),and thus contains only a finite number of elements, and lim(P(Xn)) isnot therefore the set of all power sets of a set of countably infintecardinality. ==== Subject: Re: Cardinality of 2^n numbers?W. Dale Hall scribbled the following:> This is simply not so. If K is the set of powers of 2, it is just not> the case that 2^K is the set of all natural numbers. Since you're the> one making this assertion, I'll ask you to tell me how one associates> a natural number with an arbitrary subset of K, so that every natural> number is matched with a subset, and so that no two different subsets> are matched to the same natural number. According to Cantor's theorem> it cannot be done. Given any set X, its collection of all subsets has> more elements than X.OTOH, isn't N bijective with the set of the *sums* of the *finite*subsets of the powers of 2?-- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland ---------- http://www.helsinki.fi/~palaste --------------------- rules! --------/Life without ostriches is like coffee with milk. - Mika P. Nieminen ==== Subject: Re: Cardinality of 2^n numbers?> OTOH, isn't N bijective with the set of the *sums* of the *finite*> subsets of the powers of 2?Well N u {0} *is* the set of sums of the finite sets of powers of 2.I suspect this is whatshisorhername's conceptual problem. He/she/ithas forgotten that infinite sets have lots of infinite subsets.-- ==== Subject: Re: Tricky integral> berlin.de:>> oo ln(x)>> S --------------------- dx>> 0 (x^n) -1>> n=2,3,4,.....> Split the integral at x=1. In one piece, replace x by 1/x and> simplify. Then both integrals have the same range, so you can> recombine them. When you simplify, the things are a lot easier.If you follow this hint, things go like this. The above becomesint_0^1 ln(x)(x^{n-2}+1)/(x^n-1) dxSo that convergence issues are much easier to deal with. Now expand everything but ln(x) in a Maclauren series to getint_0^1 ln(x) (-1-x^{n-2} - x^{n} - x^{2n-2} - x^{2n} ...) dxThe series converges absolutely on [0,1), which helps. Integrateterm by term, fudging at the endpoints. int_0^1 x^m ln (x) dx = 1/(m+1)^2, so we are left with1+ sum_k=1^oo 1/(kn-1)^2 + 1/(kn+1)^2= 1+ psi_1(1-1/n) + psi_1(1+1/n), where psi_1(x) is the polygamma function.So this gives us some nice identities for the polygamma function.Let F(n) be the value of the integral for a specific value of n.F(2) = pi^2(Csc[pi/2])^2/2^2 = pi^2/4 on one hand and onthe other hand we have F(2) = 1+ psi_1(1/2) + psi_1(3/2)So, 1+ psi_1(1/2) + psi_1(3/2) = pi^2/4.Similarly, F(3) = 4pi^2/27 = 1+psi_1(2/3)+psi_1(4/3)Bart ==== Subject: Online calculation of statistical varianceCalculating the variance and mean of a sequence of numbers X1 to Xn is straightforward.If a new number x is added to the sequence the mean and variance changes. The mean can be recalcula without summing over all the values again like this: newmean = (oldmean*n + x) / (n+1).Can something similar be done to calculate the new variance without iterating over all the element over again ?The answer is yes if it is possible to calculate (X1+X2+...+Xn+x)^2by using the result from the expression (X1+X2+...+Xn)^2.This would increase the speed of my code since then I would not need to iterate over all the numbers for each time I add a new number to calculate the new variance. ==== Subject: Re: Online calculation of statistical variance> Calculating the variance and mean of a sequence of numbers X1 to Xn is > straightforward.> If a new number x is added to the sequence the mean and variance > changes. The mean can be recalcula without summing over all the > values again like this: newmean = (oldmean*n + x) / (n+1).> Can something similar be done to calculate the new variance without > iterating over all the element over again ?> The answer is yes if it is possible to calculate (X1+X2+...+Xn+x)^2> by using the result from the expression (X1+X2+...+Xn)^2.> This would increase the speed of my code since then I would not need to > iterate over all the numbers for each time I add a new number to > calculate the new variance.If you're the sort of person who worries about negative variances, youmight use something likeIF n > 0 THEN m = n+1 dev = newValue - mean meanAdjustment = dev/m mean = mean + meanAdjustment var = n*var/(n-1) + dev*meanAdjustment {guaranteed non-negative} n = mELSE mean = newValue var = -1 {or some other error indicator} n = 1ENDIFIan Smith ==== Subject: Re: Online calculation of statistical varianceJust keep track of sum(i,x(i)) and sum(i,x(i)**2).I.e.: procedure newobservation(x : double) begin sum := sum + x; sumsq := sumsq + x**2 n := n + 1; mean := sum/n; if (n>1) then var := [1/(n-1)]*(sumsq - (1/n)*sum**2); end; -------------------------------------------------------------- --Erwin Kalvelagenerwin@gams.com, http://www.gams.com/~erwin------------------------------------ ----------------------------> Calculating the variance and mean of a sequence of numbers X1 to Xn is > straightforward.> If a new number x is added to the sequence the mean and variance > changes. The mean can be recalcula without summing over all the > values again like this: newmean = (oldmean*n + x) / (n+1).> Can something similar be done to calculate the new variance without > iterating over all the element over again ?> The answer is yes if it is possible to calculate (X1+X2+...+Xn+x)^2> by using the result from the expression (X1+X2+...+Xn)^2.> This would increase the speed of my code since then I would not need to > iterate over all the numbers for each time I add a new number to > calculate the new variance. ==== Subject: Re: Complex numbers and 2x2 matrices (was Re: Complex numbers and 2x2 matrixes)> Also, can we have things such as e^(quat) or ln(quat) where quat is a> quaternion? Has anybody ever figured out how to do this? Also, what> about using lie algebras which work on quaternions?Any quaternion q can be written a+bu, where a,b are real and u^2 = -1.So the set of {x+yu:x,y real} for this fixed u is isomorphicto the complex numbers. Compute e^q and ln(q) in it just as in thecomplex numbers.Generalizing functions of two variables is where problems come, if youuse two quaternions that don't commute.-- http://www.math.ohio-state.edu/~edgar/ ==== Subject: Re: A neat computer program/*The following C program computes a short approximation of Pi ci byEric Raymond in The New Hacker's Dictionary:Note that it computes pi by computing its own area. To get amore accurate result, draw a bigger picture.*/#define _ F-->00 || F-OO--;int F=00,OO=00;main(){F_OO();printf(%1.3fn, 4.*-F/OO/OO);}F_OO(){ _-_-_-_ _-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_-__-_-_-_-_-_-_-_-_-_-_-_-_-_-_-__- _-_-_-_-_-_-_-_-_-_-_-_-_-_-__-_-_-_-_-_-_-_-_-_-_-_-_-_-_-__- _-_-_-_-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_-_-_-_ _-_-_-_-_-_-_-_-_ _-_-_-_}-- http://www.math.ohio-state.edu/~edgar/ ==== Subject: Re: What is the explicit expression for this series?>[series for n from 0 to N of] n*A^n>firstly with N as a finite number and then with N as infinite>A is a constant: 0Is there an explicit expression?Yes! Consider 1+x+x^2+...+x^N = [ x^(N+1) - 1]/(x-1)1) differentiate both sides2) multiply both sides by x3) Put x = A (here , A can be an real or complex number) N can't be infinite. But if you mean the infinite series instead ofthe finite one, then yes. Just take the limit as N->+oo.For this case you must have |A| <1 ==== Subject: Re: What is the explicit expression for this series?It needed several try because at first i used the wrong closed form for thegeometric serie, i used (1-A^N)/(1-A) instead of the right one(1-A^(N+1))/(1-A) but at last i found the right expression bydifferentiating and multiplying by A as you sugges.It should be (A-N*A^(N+1)+N*A(N+2)-A^(N+1))/((1-A)^2)It works with several test case, so i suppose it is right.Thank to everybody for the hint. ==== Subject: Re: Euler booksThe library of the CWI at Amsterdam (Centrum voor Wiskunde en Informatica)possesses a copy of Euleri Opera Omnia (in an English edition as far as I know).URL is http://www.cwi.nl/Best regards: Johan E. Mebius (mailto:j.e.mebiusremove@ewi.tudelft.nl) ==== reply to =====> where can I find english or german translations of some of euler's