mm-259 === Subject: Re: a/b+b/c+c/a=n Except for n=142 and n=177, we have explicit solutions.> The following is a complete list of the values of n under 200... except for the possibility that the values n=147 and n=177 must be added.> (We can know that as soon as we have explicit solutions for these two n.Is it 142, or is it 147?-- Gerry === Myerson (gerry@maths.mq.edi.ai) (i -> u for email)Subject: Re: Axiomization of Number Theory http://www.arxiv.org/html/cs.lo/000307 so we can link to it but even> better would be to state them with abstract and publication date and> journal to interest us in reading your arxiv.When http:// is prefixed to address news browser recognizes it has weblink, highlights it and will open URL if clicked upon.> I'm not sure what you mean, but anyone who is interested is invited to> click on www.arxiv.org/html/cs.lo/0003071Does you browser recognize that as a web address?Mine and others don't as you didn't include http://nothing appears to click on and it sits there as an inertascii text going nowhere, doing nothing, meriting === dismissal.Subject: Re: Bush unveils new economic plans based on the Theorems of CantorMiracle Bush Quadruples National Treasury Every Month.Amazing, how did he do it?He drove US into bankruptcy.But then the treasury would be broke.Exactly, four times nothing is nothing.> In a press conference today the honorable George W. Bush unveiled a> new economic plan which is based on the indisputable principals of> Cantor.> In this new plan, said the president, 99% of the weight of taxes> will fall upon the 20 poorest percent of the population. But wait!> he said, silencing a cry of outrage which arose. According to the> principals of the transfinitude, in the long run this is a fair> program as it will tax all citizens equally. You see, the set of all> natural numbers has the same magnitude as the set of all natural> multiples of 100. This shows that in the long run, the rich will end> up paying just as much as the poor and all will be fair and just.> The president then went on to justify his planned $6billion tax relief> cut for the next fiscal year. You see, said G.W., Cantor has shown> us that the magnitude of an infinite set is not affected by the> addition or removal of a finite number of elements. Therefor, in the> long run, this $6B tax refund will have no adverse effects> whatsoever.> The president then changed the topic to that of the war in Iraq. To> save on spending, we will no longer issue our soldiers firearms or> helmets. While I understand that this will increase their casualty> rate tremendously, please bear in mind that, if we assume we will> forever be at war somewhere or other, then again the theorems of> Cantor assure us that this will have no impact whatsoever on the> magnitude of the set of our casualties. For, if this increases the> death rate by a factor of 500, Cantor assures us that the set of> naturals has the same magnitude as the set of natural multiples of> === 500.Subject: Circles in complex space.In modern physics it's been explained that an object can rotate incomplex space and take 720 degrees to return back to its orginalposition. How is this possible, exactly? Obviously it's not the sameas taking the argument of a circle on a z plane for each point andthen figuring the area geometrically as if this were real space.What exactly is happening?Also, does this mean you can take a measure through complex space andget a negative or === imaginary distance?(...Starblade Riven Darksquall...)Subject: Re: Complex Numbers>David,>After reading your posts it occurred to me that perhaps all Riemann>surfaces(= 1 dimensional>analytic manifolds) are orientable. I went to the library and found that>yes this is true.>Furthermore the proof is very simple:>Definition: A differentiable manifold is orientiable if there is achoice>of atlas such that for>every pair of coordinate charts (f,U) and (g,V) in the atlas, theJacobian>of the map>fg^(-1) is positive.>Theorem. Every Riemann surface is orientable.>Proof: Let f and g be given as above with also fg^(-1) analytic per the>definition of Riemann surface.>Let fg^(-1) = u + iv. The Jacobian of fg^(-1) is det([u_x, u_y], [v_x,>v_y]). The Cauchy-Riemann equations>say that u_x = v_y and u_y = -v_x. Thus we have that the Jacobian is>(u_x)^2 +(u_y)^2 >= 0. We cannot have = 0 because h^(-1) exists. QED.>Thus not only is it the case that an appropriate atlas exists, but infact>_every_ atlas>shows that a Riemann surface is orientable!>Consider the case of the mobius strip. Intuitively if we followcoordinate>charts around the strip,>the orientation is reversed and if one uses analytic functions then(after>appropriate identifications)>following the coordinate charts around the mobius strip changes theidentity>function to>complex conjugation. Complex conjugation is NOT an analytic function asmay>be easily>verified from the definition of derivative.>Comments would be welcome as I am not an expert on Riemann surfaces.> Neither am I, but it sounds right to me. What I'm not sure about is> whether you're suggesting that this helps us decide which of the> two square roots of -1 is i and which is -i.>David C. Harris> ************************> David C. UllrichNo, I am not suggesting that this tells us which square rootof -1 is i and which is -i. Orientable is not the same as oriented!C is orientable but not oriented unless we make a choice of i or -i,and either one will do as well as the other to give us an orientation!The fact that Riemann surfaces are orientable simply seemed tooirresistibly nice to not share. I hope my post is not too confusing.I agree with your post that says there is no way to tell i from -i.It is true that one can add structure to C to make i and -i distinguishable,but then C embeds into this new structure in two ways.(At least that is === all that I can see.)David C. HarrisSubject: Re: Factorial/Exponential Identity, Infinity> I like to think about infinity, and the mathematical work around> infinity, because I enjoy deriving novel results and especially> extending them to show them non-trivial.> I derived this some time ago:> lim n->oo n! / (2 (n/2)! 2^n) = 1 = (2 (n/2)! 2^n) / n!> By Sterling's approximation, n! has some resemblance to n^n in the large.> Thus by my reckoning lim n->oo n! / (2 (n/2)! 2^n) = oo> Also when n = 2> 4 = (2 (n/2)! 2^n)/n! /= 1> What it is there is an identity relating the factorial to the binary> exponent. What it says there is that> lim n-> oo n! / (2(n/2)!) = 2^n> Mathematical that makes no sense as> lim... is a constant and 2^n a variable.I think the identity is correct, actually. It's obvious that n! > 2^nin the large. That starts at n=4 where 24>16, for n=5 120>32, and forall following integers. What happens though is you divide n! by(n/2)! and 2.About the limit and having the variable on both sides of the equation,the idea there is that each side of that equality could be multipliedor divided by the corresponding sides of some other equality with ndiverging.lim_n->oo n!/(2(n/2)!) = lim_n->oo 2^nThere's only one variable, n. If it was x on the left and y on theright then it wouldn't be true uness x=y.The expresion n! has a value less than n^n for any integer n greaterthan one.I'm interested in this Stirling function(s) and approximation(s), I amnot familiar with them and read about them.Assume the identity is correct. What good is it? I derived it fromassuming that half of the infinite binary strings have equal numbersof zeros and ones.Consider the rational binary numbers of these forms:.0101010101....1010101010...I think those are 1/3 and 2/3 but I don't know. Let's see:1/4 + 1/16 + 1/64 + 1/256 + ...1/2 + 1/8 + 1/32+ 1/128 + ...How do I evaluate those binary rational numbers? I want to knowreduced fractional forms for those and all other rationals betweenzero and one with equal numbers of zeros and ones in their === binaryexpansion.Hi.RossSubject: Re: Factorial/Exponential Identity, Infinity> lim_n->oo n!/(2(n/2)!) = lim_n->oo 2^n> There's only one variable, n. If it was x on the left and y on the> right then it wouldn't be true uness x=y.Wrong! The variable on each side is a bound (or dummy) variable which has n meaning outside of the limit statement, so changing the variable within either limit statement has no effect on the === meaning of that limit statement.Subject: Re: Finite subgroups of GL(n,Z)> Is it true that for every n>=1 the group GL(n,Z) as only a finite number> of non-isomorphic finite subgroups ?> If G is a subgroup of GL(n,Z) then G acts faithfully on Z^n> which is contained in R^n. There is a positive definite quadratic> form Q on R^n (an inner product) which is invariant under G> (take any positive definite form, and sum its images under G).> We can represent G as a subgroup of O(n), the orthogonal group> and then G fixes a lattice L (corresponding to Z^n).> Suppose first that G acts irredcibly on R^n. In that case let> v be a nonzero vector of shortest length in L. The images> of v under G must span R^n as a vector space, as otherwise> their R-span would be a G-invariant subspace of R^n (contrary> to irreducibility of the G-action). This means that R^n is spanned> by the set L_1 of the nonzero vectors of minimal length in L.> The group G permutes L_1 and only the identity acts trivially> on L_1 since L_1 spans R^n. Hence G is isomorphic to a subgroup> of Sym(L_1), i.e., |G| <= |L_1|!> I claim that the size of L_1 is bounded by a function of n.> This follows from the kissing number bounds in the theory of lattices.> If v and w lie in L_1 and |v-w| < |v| = |w| then as v-w in L this> contradicts v being a shortest nonzero vector in L. Hence the> spherical caps on the sphere of radius |v| centred at the points> of L_1 and having radius pi/6 don't meet. These have positive> area on the sphere and so there's an upper bound for |L_1|> (depending on n). We conclude that for each n there> is a bound on the order of finite G acting faithfully on GL(n,Z).> What about non-faithful G? If G acts nonfaithfully, there is> a subgroup A of Z^n that it preserves with A isomorphic to Z^m> 0 < m < n. Then also G preserves A' = QA intersect Z^n. Now> Z^n/A' is isomorphic to Z^{n-m}. G acts on A' (isomorphic to Z^m)> and Z^n/A' (isomorphic to Z^{n-m}). An element of G acting trivially> on both must be the identity. Inductively there is a bound C_m> on the size of the image of G in the action on A' and a bound C_{n-m}> on the size of the image of G in the action on Z^n/A'. Then> |G| <= C_m C_{n-m}. So there is a bound on |G| for all finite subgroups of> GL(n,Z).> The answer is yes.Is the same assertion true for the === groupp GL(n,K) where K is a number field ?Subject: Homo Highhttp://www.msnbc.com/news/945134.asp...State Conservative Party Chairman Mike Long criticized the creation ofthe school. Is there a different way to teach homosexuals? Is there gay math?This is wrong, Long said. There's no reason these children should betreated separately.Yes, I can think of plenty of gay math problems.e.g. Bill the hetero male spreads Human Papilloma Virus (Cervical Cancer) to3 girls. The girls have an average of 2.75 partners. Cervical cancer has a15% fatality rate. Determine how manystraight girls will die of cervical cancer after 10 generations.Q1: In a nightclub full of 10 het men and 2 het women, determine how manypairings are possible.A1: 10 * 9 = 90.Q2: In a nightclub full of 10 gay men and 2 lesbian women, determine howmany pairings are possible.A2: (9 * 7 * 5 * 3 * 1) + (1) = 945 + 1 = 946.Q3: In the het night club, a female has a 40% chance of being attracted toany of the poorly dressed het men. If an unpaired het male has a 2% chanceof going on a shooting rampage, what is the likelihood of a shooting rampageoccurring in the het club?A3: Case 1: Both women find an attractive het. The chances of this are 0.4* 0.4 = 16%. Then the other 8 men * 0.02 = .16Case 2: One of the women finds an attractive het. The chances of this are2(0.4 * 0.6) = 48%. Then the other 9 men * 0.02 = .18Case 3: Neither woman finds an attractive het. The chances of this are 0.6* 0.6 = 36%. Then the 10 men * 0.02 = .2Adding up, (.16)(.16) + (.48)(.18) + (.36)(.2) = .0256 + .0864 + .072 =18.4% of a shooting rampage.Q4: A child of a straight couple has a 3% chance of having a major birthdefect. A person with a major birth defect has a 15% chance of becoming afuture violent criminal or rapist. A straight couple with a monster babyhas a 17% chance of getting a divorce. A child of a single mother has a 47%chance of becoming a future violent criminal or rapist. Determine theoverall odds of a monster baby born to a straight couple to develop into afuture predator.A4: (.17)(.47) + (1 - .17)(.15) = .0799 + .1245 = === about 20%.Subject: Re: Johnny Can't Add But Suresh Venktasubramanian CanA lot of the problem with America's primary and secondary school is thatparents refuse, in the main, to back up teachers. Parents do not want therechildren to get bad grades. So, teachers teach down to their students. Aslippery slope! In addition to that, school psychologists are too concernedfact, American childern suffered from inflated self-esteem!Moreover, American society in general disdains intellectuals. It iscooler, if not outright funny, to be stupid.People coming from other parts of the world value education. They are usedto competing with many more students/people than the average American is.To stand out in some of the Asian countries where there is such aridiculously large over-population, takes a lot of work.Lurch> Johnny Can't Add> But Suresh Venktasubramanian Can> By Fred Reed> The other day I went to the Web site of Bell Labs, one of> the country's premier research outfits. I clicked at> random on a research project, Programmable Networks for> Tomorrow. The scientists working on the project were> Gisli Hjalmstysson, Nikos Anerousis, Pawan Goyal, K. K.> Ramakrishnan, Jennifer Rexford, Kobus Van der Merwe, and> Sneha Kumar Kasera.> Clicking again at random, this time on the Information> Visualization Research Group, the research team turned> out to be John Ellson, Emden Gansner, John Mocenigo,> Stephen North, Jeffery Korn, Eleftherios Koutsofios, Bin> Wei, Shankar Krishnan, and Suresh Venktasubramanian.> Here is a pattern I've noticed in countless organizations> at the high end of the research spectrum. In the> personnel lists, certain groups are phenomenally over-> represented with respect to their appearance in the> general American population: Chinese, Koreans, Indians,> and, though it doesn't show in the above lists, Jews.> What the precise statistical breakdown across the world> of American research might be, I don't know. An awful lot> of personnel lists look like the foregoing.> Think about this: Asians make up a small percent of the> population, yet there are company directories in Silicon> Valley that read like a New Delhi phone book. Many of our> premier universities have become heavily Asian, with many> of these students going into the sciences. If Chinese> citizens and Americans of Chinese descent left tomorrow> for Beijing, American research, and graduate schools in> the sciences and engineering, would be crippled.> Jews are two or three percent of the population. On the> rough-cut assumption that Goldstein is probably Jewish,> and Ferguson probably isn't, it is evident that Jews are> doing lots more than their share of research-and, given> that people named Miller may well be Jewish, the name-> recognition approach probably produces a substantial> undercount. I asked a friend, researching a book on> Harvard, the percentage of Asian and Jewish students.> Answer: Asians close to 20%. Jews close to 25%-> unofficial, because you are allowed to list by gender,> ethnicity, geography, but not religion. Our last taboo.> None of this is original with me. In 1999, the National> Academy of Sciences released a study noting that over> half of U.S. engineering doctorates are awarded to> foreign students. Where are Smith and Jones?> Why are members of these very small groups doing so much> of the important research for the United States? That's> easy. They're smart, they go into the sciences, and they> work hard. Potatoes are more mysterious. It's not> affirmative action. They produce. The qualifications of> these students can easily be checked. They have them. The> question is not whether these groups perform, or why, but> why the rest of us no longer do. What has happened?> It is not an easy question, but a lot of it, I think, is> the deliberate enstupidation of American education.> Again, the idea is not original with me. Said the> American Educational Research Association of the NAS> report, Serious deficiencies in American pre-college> education, along with wavering support for basic> research, were cited by the panel as major contributors> to this problem.> Consider mathematics. In the mid-Sixties I took freshman> chemistry at Hampden-Sydney College, a solid school in> Virginia but not nearly MIT. It was assumed-assumed> without thought-that students knew algebra cold. They had> to. You can't do heavy loads of highly mathematical> homework, or wrestle with ideas like integrating> probability densities over three-space, or do endless> gas-law and reaction-rate calculations, if you aren't> sure how exponents work.> Remedial mathematics at the college level was unheard of.> The assumption was that people who weren't ready for> college work should be somewhere else. No one thought> about it. Today, remedial classes in both reading and> math are common at universities. We seem to be dumbing> ourselves to death.> I recently had children go through the high schools of> Arlington, Va., a suburb of Washington. I watched them> come home with badly misspelled chemistry handouts from> half-educated teachers, watched them do stupid, make-work> science projects that taught them nothing about the> sciences but used lots of pretty paper.> The extent of scholastic decline is sometimes> astonishing. So help me, I once saw, in a middle school> in Arlington, a student's project on a bulletin board> celebrating Enrico Fermi's contributions to Nucler> Physicts (Scripps-Howard National Spelling Bee> 2001, Sean Conley; 2000, George Thampy; 1999, Nupur> Lala).> It appears that a few groups are keeping their standards> up and the rest of us are drowning our children in self-> indulgent social engineering, political correctness, and> feel-good substitutes for learning.> Some of our growing dependency is hidden. We do not> merely rely on small industrious groups in America and on> foreigners working here. Increasingly the United States> contracts out its technical thinking to Asia.> If you read technically aware publications like Wired> magazine (and how many people do?), you find that major> American corporations have more and more of their> computer programming done by people in, for example,> India. In cities like Bombay, large colonies of Indians> work for U.S. companies by Internet. This again means> that counting names at American institutions> underestimates the growth of intellectual dependence.> The Indians, and others, have discovered the suddenly> important principle that intellectual capital is> separable from physical capital. To program for Boeing,> you don't have to be anywhere near Seattle. Nor do you> need an aircraft plant. All you need is a $700 computer,> a book called something like How to Program in C++, and a> fast Internet connection. Crucial work like circuit-> design can now be done abroad by bright people who don't> need chip factories. They need workstations, the> Internet, and engineering degrees.> This too we would be wise to ponder. Americans often> think of India chiefly as a land of ghastly poverty.> Well, yes. It is also a country with about three times> our population and a lot of very bright people who want> to get ahead. They're professionally hungry. We no longer> are.> People speak of globalization. This is it, and it's just> beginning. Where will it take us? How long can we> maintain a technologically dominant economy if we are, as> a country, no longer willing to do our own thinking? If> we rely heavily on less than 10 percent of our own> population while employing more and more foreigners> abroad?> It's not them. It's us. I've heard the phrase, the Asian> challenge to the West. I don't think so. When Sally Chen> gets a doctorate in biochemistry, she's not challenging> America. She's getting a doctorate in biochemistry. Those> who study have no reason to apologize to those who don't.> The Mathematical Association of America runs a contest> for the extremely bright and prepared among high-school> students. It is called the United States of America> Mathematics Olympiad, and it provides a means of> identifying and encouraging the most creative secondary> mathematics students in the country.> An unedited section of a list of those recently chosen:> Sharat Bhat, Tongke Xue, Matthew Peairs, Wen Li, Jongmin> Baek, Aaron Kleinman, David Stolp, Andrew Schwartz, Rishi> Gupta, Jennifer Laaser, Inna Zakharevich, Neil Chua,> Jonathan Lowd, Simon Rubinsteinsalze, Joshua Batson,> Jimmy Jia, Jichao Qian, Dmitry Taubinsky, David Kaplan,> Erica Wilson, Kai Dai, Julian Kolev, Jonathan Xiong,> Stephen Guo.> Q.E.D.> Fred Reed> f.v.reed@att.net> First appeared in The American Conservative.> - - - - - - -> Fred Reed,columnist for The Washington Times, former> Marine, streety police writer, occasional terrified war> correspondent,and afficionado of raffish bars, offers> weekly his unique, often satirical and arguably> opinionated views on ...everything. You'll grind your> teeth. (He denies that he gets a kickback from the dental> lobby, though no one believes him.) But you'll think.> I'm an equal-opportunity irritant, says Fred> democratically. Visit his website here:> http://www.fredoneverything.net/> Source - http://mensnewsdaily.com/archive/r/reed/03/reed072703.htm> Jai Maharaj> http://www.mantra.com/jai> Om Shanti> Shubhanu Nama Samvatsare Dakshinaya Nartana Ritau> Kark Mase Krishna Pakshe Indu Vasara Yuktayam> Pushya Nakshatr Vajr-Siddhi Yog> Chatushpad-Naag Karan Amavasya Yam Tithau> Hindu Holocaust Museum> http://www.mantra.com/holocaust> Hindu life, principles, spirituality and philosophy> http://www.hindu.org> http://www.hindunet.org> The truth about Islam and Muslims> http://www.flex.com/~jai/satyamevajayate> o Not for commercial use. Solely to be fairly used for the> educational purposes of research and open discussion. The contents of> this post may not have been authored by, and do not necessarily represent> the opinion of the poster. The contents are protected by copyright law> and the exemption for fair use of copyrighted works.> o If you send private e-mail to me, it will likely not be read,> considered or answered if it does not contain your full legal name,> current e-mail and postal addresses, and discussion. Views expressed by others> are not necessarily === those of the poster.Subject: Re: lambda calculus> It seems to me that the lambda calculus is the formulation that makes the> Church-Turing thesis most plausible. How come I've never seen a book that uses> it to prove the unsolveability of the halting problem? Does anybody know of a> source like that?How so? There's basically 3 models of computation shown to be equivalent,TMs, recursive functions, lambda abstractions (almost 2GL, 3GL, 4GL).But each methods' approach to the Halting problem only requires that functions areenumerated and compute a parameter, an actual prototype calculation of themethod used is not required of the proof.Hence we write TM(n) = xyz just as easily as Lambda x. utm n.Would be a nice book if it exists, but most lamdba calculus work is on === optimisation,so that one day we might use it!HercSubject: Re: Mapping of integers to reals ~ Cantor's disproof> : > : >> : >> : >> Cute TM, but I think you miss the point of the problem. You need to> : >> find a function (or TM) that can map every irrational to a unique> : >> integer.> : >> : >> That's easy: I can define a function that maps every irrational to the> : >> unique integer 17.> : >> : >> Wow, you did it... I'm in awe of your genius.> : > : >Nothing to it. Mapping every irrational to a *unique* integer is> : >trivial: it's inherent in the idea of a function that every element of> : >the domain is mapped to a unique element of the range. Now, if you'd> : >wanted to map every irrational to a *different* integer, that would be> : >much harder. Impossible, in fact.> > : Huh? I just mapped every irrational to 18.> : That *is* a different integer from 17, isn't it?> What would be impossible is to map every irrational to an integer such> that for any two unequal irrationals, the integers to which they are> mapped in that mapping are also unequal.> Of course, both of you realized that already...> hello John,> This is part of the claim of the thread have you read the whole thread?> The integers they map to are the Turing Machine numbers that generate> them. Two different outputs from 2 machines implies the machines> are different, as TM's are deterministic.> HercWhat about mapping every irrational onto a rational? Can that be done?Or better yet: Mapping every transcendental number onto a, uh, what isthe nickname or nontranscendental? algebraic? Gemoetric? Polynomialic?Whatever.(What is the nickname for nontranscendental anyways?)Well can you map every transcendental to a... uh... that?(...Starblade Riven === Darksquall...)Subject: Re: Mapping of integers to reals ~ Cantor's disproof>I'm using the fact that H(n) is noncomputable ...> That *fact* is wrong. H(n) *is* computable (for every n). H(n) is simply> an integer; in fact, quite restricted: a member of {0, 1}. But H (the> function) is NOT computable.> Daryl said it well: you need a *different* computable function (a constant> function will do) for each n. (In fact, you only need two such computable> functions). Computability of a *function* (as opposed to a value in that> function's range) is a different matter.> All you're saying is that every real number is computable, because we have> ten constant functions into {0,1,2,3,4,5,6,7,8,9} that together permit any> digit of any real number to be computed -- and those ten functions are all> computable! Wowee.> Michel.Actually, by that logic, it can be done just for 0 and 1, since everyreal number can be expressed as an infinitely long set of sums ofbinary fractions. Rational numbers would be repeated, and irrationalnumbers must be as well, since each irrational number lies between tworational numbers. Take two rational numbers A and B, and oneirrational number C, along with an integer n. If you take A Just don't ask me for a 24-prime set, I don't have one yet :).>Nor I, but I've found the following n-prime sets:>n = 17: eight 2's and nine 3's>n = 28: 15 2's and 13 1'sI believe === five 1's and nineteen 2's will work for n=24. -- ErickSubject: Re: Please recommend some entertainment math books> Please recommend some entertainment math books. > My undergraduate major was math, but, no graduate training, I had> graduate training in electrical engineering.> I am looking for some entertainment math books for my background. The> books should cover some new and depth results. Most entertainment math> books are for general public, too simple and too shallow. I am looking> for some easy and fun to read books with depth, and suitable for math> undergraduate major. (Not trivial subject of chaos or fractal, or> things like those).> Ideally the book can cover some recent active research results, but,> in someway, people without graduate training can understand and enjoy.John Conway, The Sensual Quadratic Form. Julian Havil, Gamma. Robin Wilson, Four Colors Suffice. Donald Saari, Chaotic Elections. Ian Stewart, Another Fine Math You've Got Me Into. David Gale, Tracking the Automatic Ant. Colin Adams, The Knot Book. Charles Livingston, Knot Theory. Alf van der Poorten, Notes on Fermat's Last Theorem.-- Gerry Myerson === (gerry@maths.mq.edi.ai) (i -> u for email)Subject: Re: Please recommend some entertainment math books>Please recommend some entertainment math books. >[...]>I am looking for some entertainment math books for my background. The>books should cover some new and depth results. Most entertainment math>books are for general public, too simple and too shallow. I am looking>for some easy and fun to read books with depth, and suitable for math>undergraduate major.How aboutEdwards HM. Riemann's Zeta Function.?The following do not exactly fit the bill as far as the depth you request, but I think you will find them very enertaining nonetheless. I certainly did.Abbot EA. Flatland. (Stewart also has an annotated version, which I have not yet read.)Aczel AD. Fermat's Last TheoremDavis PJ, Hersch R. The Mathematical Experience. Really, absolutely delightful.Not mathematics, but with some mathematical content:Carroll L. The Annotated Alice. (Annotated by M. Gardner. Again, delightful.)If you want to teach yourself some basic mathematics with which you are not yet familiar, and which will allow yourself to understand more advanced mathematics, you might take a look at any of these short textbooks. As someone else has put it, these are not exactly bedtime reading.Bartle RG. The Elements of Integration.Halmos PR. Naive Set Theory.Lindley DV. Making Decisions.Moore TO. Elementary General Topology. (Actually, Munkres JR, Topology is a better, in-depth book, but it is not short.)Ross SM. Applied Probability Models with Optimization Applications. (Again, Ross's Intro. to Probability Models is a more fleshed-out text.)Ross SM. An Introduction to Mathematical Finance.Spivak M. Calculus on Manifolds.Perhaps you will enjoyNasar S. A Beautiful Mind.Struik DJ. A Concise History of Mathematics.-- Stephen J. Herschkorn === herschko@rutcor.rutgers.eduSubject: Re: Please recommend some entertainment math books> Please recommend some entertainment math books.> My undergraduate major was math, but, no graduate training, I had> graduate training in electrical engineering.> I am looking for some entertainment math books for my background. The> books should cover some new and depth results. Most entertainment math> books are for general public, too simple and too shallow. I am looking> for some easy and fun to read books with depth, and suitable for math> undergraduate major. (Not trivial subject of chaos or fractal, or> things like those).> Ideally the book can cover some recent active research results, but,> in someway, people without graduate training can understand and enjoy.David Wells' _Curious and Interesting Geometry_ is lots of fun. Read it atbedtime and you will be getting back up to find a pencil and paper. EricWeisstein's MathWorld is available as a book, published by CRC press. Iwould urge you to buy it directly === from Eric's site.Subject: Re: stuck proving simple(?) tautology> Given a binary relation G(x,y) and T1 defined as:> T1(x,y) = I / G(x,y) / G(x,y) / T1(x,y)> and T2 defined as:> T2(x,y) = I / G(x,y) / T2(x,y) / T2(x,y)> where I is identity relation, prove that T1 = T2.Huh? I opened this message expecting a sentence in FOL. But T1 andT2 are predicates, and you want to show that they're the same... Butthe FOL doesn't have rules of inference for doing this. I suspectyou're using a second-order logic. And I suspect that you mustdisambiguate your sentences, as wedges and vees don't mix well withoutparentheses. And I suspect that what you're trying to prove isn'tcalled a tautology, since it doesn't seem to follow from the meaningsof the truth functional operators in your === ambiguous wffs.Alex SollaJuniorReed CollegeSubject: Re: stuck proving simple(?) tautology> Given a binary relation G(x,y) and T1 defined as:> T1(x,y) = I / G(x,y) / G(x,y) / T1(x,y)> and T2 defined as:> T2(x,y) = I / G(x,y) / T2(x,y) / T2(x,y)> where I is identity relation, prove that T1 = T2.A couple of questions:Then I represents x=y ?What's with the repeated (redundant) G(x,y) in the definition of T1and T2(x,y) in the definition of T2?At this point, I am not so sure that T1 and T2 are uniquely defined(can tell better after the above questions are answered), though Iguess T1 could = T2 if they === are equally ambiguous!Charlie VolkstorfCambridge, MASubject: Re: stuck proving simple(?) tautology > Given a binary relation G(x,y) and T1 defined as:> T1(x,y) = I / G(x,y) / G(x,y) / T1(x,y)Huh? That's not a definition and it's confusing. Do you mean I / G / (G/T1) ?I presume / or, / and.> and T2 defined as:> T2(x,y) = I / G(x,y) / T2(x,y) / T2(x,y)> where I is identity relation, prove that T1 = === T2.Ditto.Subject: Taking Calculus without taking precalcuMy friends that have taken precalculus have said they pretty much kneweverything and that it was a waste and that they had wished they hadgone straight to calculus. I've been thinking about doing the same, Ihave a firm grounding in trigonometry got a A in it. Would yourecommend against me just outright skipping precalculus? The onlycatch is I would have to write a challenge letter requesting I skipthe prerequisite of precalculus. Any suggestions on what to say inthere a math === professor would be looking at the letterSubject: Re: Taking Calculus without taking precalcu> My friends that have taken precalculus have said they pretty much knew> everything and that it was a waste and that they had wished they had> gone straight to calculus. I've been thinking about doing the same, I> have a firm grounding in trigonometry got a A in it. Would you> recommend against me just outright skipping precalculus? The only> catch is I would have to write a challenge letter requesting I skip> the prerequisite of precalculus. Any suggestions on what to say in> there a math professor would be looking at the letterI always thought precalc was for students who had algebra II (quadratic equations, graphing polynomials, etc.) but not trig. I seem to remember that precalc covers the trig functions, the exp and log function, rules for exponents, and the concept of inverse functions (as in exp and log). If you feel good about most of that, you should take calculus === directly.Subject: Re: Taking Calculus without taking precalcu> My friends that have taken precalculus have said they pretty much knew> everything and that it was a waste and that they had wished they had> gone straight to calculus. I've been thinking about doing the same, I> have a firm grounding in trigonometry got a A in it. Would you> recommend against me just outright skipping precalculus? The only> catch is I would have to write a challenge letter requesting I skip> the prerequisite of precalculus. Any suggestions on what to say in> there a math professor would be looking at the letterThere is no subject of precalculus; it is something invented by bureaucrats and is to be avoided if at all possible. Your A in trig, unfortunately, is meaningless without further information. A's are handed out in high school these days to students who can pass the metal detector test. The most important thing is your actual understanding and facility with trig, algebra, and geometry. By all means write the challenge letter if you are sure of your abilities in these === areas.Subject: Re: Taking Calculus without taking precalcu> My friends that have taken precalculus have said they pretty much knew> everything and that it was a waste and that they had wished they had> gone straight to calculus. I've been thinking about doing the same, I> have a firm grounding in trigonometry got a A in it. Would you> recommend against me just outright skipping precalculus? The only> catch is I would have to write a challenge letter requesting I skip> the prerequisite of precalculus. Any suggestions on what to say in> there a math professor would be looking at the letter> There is no subject of precalculus; it is something invented bybureaucrats> and is to be avoided if at all possible. Your A in trig, unfortunately, is> meaningless without further information. A's are handed out in high school> these days to students who can pass the metal detector test. The most> important thing is your actual understanding and facility with trig,> algebra, and geometry. By all means write the challenge letter if you are> sure of your abilities in these areas.With great respect for Wade and his many excellent posts, I must disagree onthis one. Precalculus is basically college algebra + trigonometry +sometimes an introduction to transcendental functions. Show me a studentwho called it a waste, and nine times out of ten, I'll show you a studentwho can't graph a logarithm function or even something like y = sin(x-pi).One of the best students I ever had took every course twice because of herfeelings of insecurity. For whatever reason, it works great.But anyway, here is a simple test. If you can sit down right now, andwithout books or notes, and taking all the time you want, derive a formulafor tan(a+b+c) in terms of tan(a), tan(b), and tan(c), then you don't needprecalculus. Otherwise, I would say === you do.Subject: Re: Taking Calculus without taking precalcu> Would you>recommend against me just outright skipping precalculus? The only>catch is I would have to write a challenge letter requesting I skip>the prerequisite of precalculus. Any suggestions on what to say in>there a math professor would be looking at the letterHow is your retention of Intermediate Algebra? You performed excellently inTrigonometry, but how is your retention of that main topic? If your school hasa qualification examination for skipping PreCalculus, you could take advantageof it to see if you could be recommended to skip to Calculus. If your retention is very good of Intermed Algebra and the Trigonometry, thenPreCalculus may very well be a waste of time; you could later review anysubtopic you need from any old PreCalculus textbook. Calculus will mostly justrequire some basic trigonometry, and all of intermediate algebra. You may findthat the study of Calculus itself will force you THEN to review what you need. Your friends generally have the right idea IF you retain your === IntermediateAlgebra and Trigonometry knowledge.G CSubject: Re: Tell me your paradoxes!>(2) In the town of Placerville (CA) the barber shaves everyone who doesn't>shave himself. Who shaves the barber?Not a paradox. The barber can shave himself without === violating anything you'vesaid.DougSubject: Re: Tell me your paradoxes!>(2) In the town of Placerville (CA) the barber shaves everyone who doesn't>shave himself. Who shaves the barber?>Not a paradox. The barber can shave himself without violating anything>you've>said.Yes. I suppose the barber would have to only shave those who don't shavethemselves. That nobody in === Placerville shaves probably won't help me any.RichSubject: Re: Tell me your paradoxes!>Have you got any paradoxes to share? Write them here!> (1) How many kinds of infinity are there? An infinite number? If so, what> kind of infinity? Is this really a paradox, or just mindbooglingly confusing withoutactually being paradoxical?> (2) In the town of Placerville (CA) the barber shaves everyone who doesn't> shave himself. Who shaves the barber?She shaves herself, obviously.Yours, Mike === all,I have a question about linear transform. What is the definition of theeigenvalue of linear transform? How to show the linear transformation T:f(x) -> integrate[f(t), t from 0 === to x] has no eigenvalues?-WalalaSubject: Re: Why are martingales called martingales?> ...>> Jacqueline Pichoche's _Dictionnaire Etymologique du francais_>> (Robert, 1979) clarifies this, and, I think, settles where>> the betting system got its name. My translation:>>[snip def supporting above]>Maybe it settles it, maybe not. The previously-suggested >bridle idea is supported by Webster's 1913 Dictionary>(according to http://www.hyperdictionary.com anyhow):>3. (Gambling) The act of doubling, at each stake, that which has >been lost on the preceding stake; also, the sum so risked; -- >metaphorically derived from the bifurcation of the martingale >of a harness. [Cant] --Thackeray. That is, hyperdictionary's>entry makes no mention of this usage of martingale deriving from>Martigues, but instead says it derives from martingale as harness.> My 1924 printing of the 1909 edition of the Merriam-Webster> New International Dictionary (commonly called Webster's First> International) doesn't have that citation from Thackeray. I> don't know which Thackeray it was, but assume it to be the> English novelist and literary man of that name. In that case> certainly (and in any case, maybe with less certainty) I would> greatly favor the scholarly opinion of Jacqueline Picoche > (a professional lexicographer writing in 1979) over the> (very likely less scholarly, and in any case many decades> earlier) opinion of Thackeray. The metaphorical derivation> given by Thackeray has all the earmarks of folk-etymology> (if not his own fertile imagination). Also, though Picoche> doesn't give a citation for jouga a la martegalo, she does> date it to the 18th century, whereas the Oxford English > Dictionary's earliest citation for martingale in the> gaming sense is from 1815 (ah, and followed by Thackeray-the-> novelist, in a novel, 1854) as a noun, and from 1823 as> a verb (whose etymology asks the reader to compare to French > martingaler without positively claiming that that's its source;> the OED etymology of the *noun* martingale is even less help).And to think that for all these years I thought the game-theorist David Gale had named it in honor of his brother, Martin....-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)