mm-2449 === Subject: [OT] de mortuis nil nisi bene (was: Armand Borel Finally dead) > Does that apply to Idi Amin as well, who also died at > the same (official) age? The wonderful thing about math is: you may compare everything to everything and ponder about the more or less interesting interrelationships. You may make as many stupid mistakes as you like when dealing with funny theories - nobody is really harmed by that. But outside math you are responsible for what you do and say. Besides true and false you are confronted with right and wrong. I wanted to express my belief that talking badly about people, who cannot defend themselves, is wrong. I reject your polemics and wish you all the best. Rainer Rosenthal r.rosenthal@web.de === Subject: Re: [OT] de mortuis nil nisi bene (was: Armand Borel Finally dead) > > Does that apply to Idi Amin as well, who also died at > > the same (official) age? > The wonderful thing about math is: you may compare > everything to everything and ponder about the more > or less interesting interrelationships. > You may make as many stupid mistakes as you like when > dealing with funny theories - nobody is really harmed > by that. > But outside math you are responsible for what you do and > say. Besides true and false you are confronted with > right and wrong. I wanted to express my belief that talking > badly about people, who cannot defend themselves, is wrong. That one shouldn't speak ill of the dead is a strange belief--it surely does less harm than speaking ill of the living. (Well, not surely but most often at least.) > I reject your polemics and wish you all the best. > Rainer Rosenthal > r.rosenthal@web.de -- G.C. === Subject: About Russell's first paradox Russell's first paradox ----------------------- ( http://www.wikipedia.org/wiki/Russell%27s_paradox ) Consider the set M to be The set of all sets that do not contain themselves as members. Formally: A is an element of M if and only if A is not an element of A. In the sense of Cantor, M is a well-defined set. Does it contain itself? If we assume that it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, than it has to be a member of M, again according to the very definition of M. Therefore, the statements M is a member of M and M is not a member of M both lead to a contradiction. So this must be a contradiction in the underlying theory. Some example: if we had an entry on list of all lists which do not contain themselves, then that list must be either incomplete (if it does not list itself) or incorrect (if it does). ----------------------------------------------------------------- A structural point of view: Definition A: ------------- ( http://www.cut-the-knot.org/selfreference/russell.shtml ) Sets are defined by the unique properties of their elements. One may not mention sets and elements simultaneously, but one notion has no meaning without other. Let us take as an exapmle, the set (W=N+0) of all natural numbers: {0,1,2,3,...}. By using the empty set, we can show that W has the structure of a set that contain itself as a member of itself: 0 = { } 1 = {{ }} = {0} 0. | | 2 = {{ },{{ }}} = {0,1} 0. . | | 1|____| | | 3 = {{ },{{ }},{{ },{{ }}}} = {0,1,2} 0. . . . | | | | 1|____| |____| | | 2|__________| | | 4 = {{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}} = {0,1,2,3} 0. . . . . . . . | | | | | | | | 1|____| |____| |____| |____| | | | | 2|__________| |__________| | | 3|______________________| | | 0 1 2 3 4 {0,1,2,3,...}={{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}},... ____^ ----^------ ----------^------------ ---^ _________| | | ______________________| | ___________________________| By definition A, the set of all sets that contain themselves as members, must have some kind of the above self structural similarity over scales, by a recursive process. Also by definition A, the set of all sets that do not contain themselves as members, must not have this property, therefore the set of all sets that do not contain themselves as members, must not contain itself as a member of itself. Through this structural point of view, there is no paradox. What do you think? === Subject: Re: About Russell's first paradox > Russell's first paradox > ----------------------- > ( http://www.wikipedia.org/wiki/Russell%27s_paradox ) > Consider the set M to be The set of all sets that do not contain > themselves as members. Formally: A is an element of M if and only > if A is not an element of A. In the sense of Cantor, M is a > well-defined set. Does it contain itself? If we assume that it > does, it is not a member of M according to the definition. > On the other hand, if we assume that M does not contain itself, > than it has to be a member of M, again according to the very > definition of M. > Therefore, the statements M is a member of M and M is not a > member of M both lead to a contradiction. > So this must be a contradiction in the underlying theory. > Some example: > if we had an entry on list of all lists which do not contain > themselves, then that list must be either incomplete > (if it does not list itself) or incorrect (if it does). > ----------------------------------------------------------------- > A structural point of view: > Definition A: > ------------- > ( http://www.cut-the-knot.org/selfreference/russell.shtml ) > Sets are defined by the unique properties of their elements. > One may not mention sets and elements simultaneously, > but one notion has no meaning without other. > Let us take as an exapmle, the set (W=N+0) of all natural numbers: > {0,1,2,3,...}. > By using the empty set, we can show that W has the structure of > a set that contain itself as a member of itself: > 0 = { } > 1 = {{ }} = {0} > 0. > | > | > 2 = {{ },{{ }}} = {0,1} > 0. . > | | > 1|____| > | > | > 3 = {{ },{{ }},{{ },{{ }}}} = {0,1,2} > 0. . . . > | | | | > 1|____| |____| > | | > 2|__________| > | > | > 4 = {{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}} = {0,1,2,3} > 0. . . . . . . . > | | | | | | | | > 1|____| |____| |____| |____| > | | | | > 2|__________| |__________| > | | > 3|______________________| > | > | > 0 1 2 3 4 > {0,1,2,3,...}={{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}},... > ____^ ----^------ ----------^------------ ---^ > _________| | | > ______________________| | > ___________________________| > By definition A, the set of all sets that contain themselves as > members, must have some kind of the above self structural > similarity over scales, by a recursive process. > Also by definition A, the set of all sets that do not contain > themselves as members, must not have this property, > therefore the set of all sets that do not contain themselves > as members, must not contain itself as a member of itself. > Through this structural point of view, there is no paradox. > What do you think? The Russell set antinomy isn't a problem for modern set theory. For example, in the case of ZF the Russell set and other antinomies have been eliminated (as far as can be told) by restricting the way in which sets can be defined. === Subject: a comment on searching for 3 orthogonal 10x10 squares While trying to find 3 such orthogonal squares, I used over 300 sets of duals, but the best I could do was fit 26 squares of the 3rd layer into a tri-orthogonal arrangment before the problem exhausted the possibilities. Most of the duals were found on the net, as James Buddenhagen pointed out, but transforming the row/columns/diagonals easily led to other pairs. === Subject: Advice about applying to Math Graduate School I am very interested in applying for a PhD in Math, and am going to finish my Masters in Math in a year. However, I am thinking I might want to take a year off, because I am a little burnt out with school and want to get myself motivated for the PhD track. I love math, but I think a year off might be a good idea. Is this ill advised? If I take a year off, I may forget a lot of math that I learned in college, and I've also heard that a lot of people who take time off don't ever go back to school, that it's a lot easier to pursue Graduate studies right out of Masters or Undergrad. Another issue I have is this : If I do apply for graduate school, I would only apply to say that top 3 or 4 schools in my area of interest, which I will decide soon enough. I am of the opinion that it is only worth it to go to graduate school if you are at a top school so you can bump heads and study with the top professors in your field (and of course, better school ==> better chances of teaching at a better school). Is this a bad way of thinking? I think if you want to do anything decent in Math, you need to go to the best school(s) and study under the best professor(s). Otherwise, it isn't worth going to graduate school. Finally, I'd be interested in looking outside of the United States to possibly pursue a PhD, possibly France or England. However, I have no idea what the good Math programs are in international countries. Is there a good reference for this? Tim === Subject: Re: Advice about applying to Math Graduate School > I think if you want to do anything decent in Math, you need to go to the > best school(s) and study under the best professor(s). Otherwise, it isn't > worth going to graduate school. True, if you intend a career as a research mathematician at a top school. However, there are other things for Ph.D. mathematicians to do. For some of them, studying at a top school is not necessary. So don't say it isn't worth going to graduate school; just say it isn't worth going to graduate school for me. === Subject: Re: Advice about applying to Math Graduate School > Finally, I'd be interested in looking outside of the United States to > possibly pursue a PhD, possibly France or England. However, I have no idea > what the good Math programs are in international countries. That's a very good idea; In fact, I'd like to do that, in the opposite way (I'm from France). I can give you a few pointers to the very best institutions in France. You might want to check out the website of the Ecole Polytechnique http://www.polytechnique.edu (may be down at the moment; there was a failure recently) and of the ENS Ulm (Paris): http://www.ens.fr/international/ I am not sure about international students, but for french ones, it is often frowned upon to take one year off. You may want to check that too. Sam -- So if you meet me, have some courtesy, have some sympathy, and some taste Use all your well-learned politesse, or I'll lay your soul to waste - The Rolling Stones, Sympathy for the Devil === Subject: Algorithm to find the n-root of a equation. Hi guys. im trying to create a program to find the roots of a equation on n-degrees. One method that i know its this (dont know the name of the method, sorry) lets say that i have the following equation: y^4-3y^3-11y^2+27y+18=0 i take out the exponents and what multipies the y's in the following form: exponents -> 4 3 2 1 0 multiplicators-> 1 -3 -11 27 18 since 1 multiples y^4, -3 multiplies y^3, -11 multiplies y^2, 27 multiplies y^0 and 18 multiplies y^0. then we choose a number, lets say 2 and do the following with the multiplicators: 1 -3 -11 27 18 2 -2 -26 2 number=2 -------------------- 1 -1 -13 1 20 so i take the n exponent (4th exponent in this case, which its 1), then multiply it by the number (2), the result, add it to the n-1 (-3), the result, multiply it by the number (2), the result, add it to the n-2 exponent (-11), the result multiply it by the n-3 exponent, and so on.. in the last example, i cant find the root, but if i choose another number, i find it, lets say -3 1 -3 -11 27 18 3 0 -33 -18 number=3 -------------------- 1 0 -11 -6 0 <- so the 4th root its 3 Questions: this its the best method to find roots (it envolves guessing what number to choose)? what its the name of this method ? there its any rule about choosing the number to try ? And sorry for my bad english. === Subject: Re: A matrix inversion problem >I'm trying to find a matrix W so W*A=G where >G is [1 0 0 0; 0 0 0 0; 0 0 1 0; 0 0 0 0]; >To find W, I can use W=G*inv(A) (1) >or W=pinv(A*pinv(G)) (2). >(1) works generally fine but my problem is that I do not >want to invert matrix A. >Therefore, I came up with (2), however the min-norm solution >W then does not always end up satisfying W*A=G. >How can I find a W without inverting A ? Are there any other >possibilites ? Any help will be appreciated. In general, to solve matrix-vector equations you shouldn't invert the matrix (but LU factorization can be useful...). Of course, A should be invertible, otherwise solutions either won't exist or won't be unique. In this case you can say (row i of W)*A = (row i of G). Rows 2 and 4 of G are 0, so those rows of W are 0. So just solve A^T x = [1 0 0 0 ]^T and A^T x = [0 0 1 0] and the transposes of the vectors obtained are rows 1 and 3 of G respectively. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Any idea for a math career? > As far as mathematicians having a narrow focus, I have always known > mathematicians to be broadly learned and read. Most mathematicians know a > little physics, chemistry, computer science, as well as, literature. > Personally, I have B.A. degrees in both Math and English literature. > Lurch I think that since mathematicians (and logicians) deal with abstract objects (which can represent pretty much anything, so long as an axiomatic description can be formulated), they should be the most adept at 'thinking outside the box.' Look at the works of Emmy Noether on symettry in physical systems or Kurt Goedel's works. Barwise and Etchemendy describe Goedel as the first hacker because of the foresight his work required to deal with the complexity of finding the Goedel number. If thinking outside the box is equivalent to uncritical and unrigorous, a plain-jane english lit major is the ideal candidate. === Subject: Re: Any idea for a math career? > Fair 'nuff. I suppose I should clarify: I'm in my mid-30s, and yes, I have > been in both the real world (as a software developer in Seattle) and in > academia (Seattle / Tacoma areas + St. Louis), so I know all the pros and > cons of money vs. no money, corporate vs. academia from personal experience. > My conclusion (feel free to disagree; most do) is that money only goes so > far. I've worked in education (community college + technical school + > for-profits, see below) and also as a software developer.When I was in > software development, I made decent pay (not great, not all Seattle > programmers work at MS) as a video games scripter, first at a small (but > recognized) company where the pay was OK, but not great, but no one cares > because this career is their life dream. Later, I worked at a huge, > well-known corporate company where everyone was in it primarily for the > money. I made some great $, but left (back to the dreamer's shop) because I > felt like a robot at this place. Later in life, I also worked at a > for-profit technical school (never again!), where the students counted > zilch, except insofar as they were worth cold, hard cash to the school. My > conclusion from comparing the various places: working *only* for the money > is degrading and dehumanizing, working for your dream is amazing, even if > you don't make enough for the BMW, but as long as you do make enough to feed > yourself. > So I'm back for the Ph.D., so's I can research, which is the field I've > dreamed about forever, but for various reasons have never done until now. > Herc, I'll try like hell to be in that top 20%. :) > A more precise way of wording my original question: are there places that > hire mathematicians to do mathematical research (as opposed to, say, > programming, or actuary)? Obviously, universities do, and most likely I will > go that route, but I'm curious if any companies are specifically interested > in high-level mathematics. > I can give a partial answer to my own question: > Wolfram > IBM Yorktown > MS Research (crypto, wavelet compression) > Any other companies to add to the list? I'm willing to relocate, when the > time comes (about three years from now). man the closest I got to publishing a game was uploading www.c-h-e-s-s.com I'm contributing to this thread from experience as an employment councellor and high school teaching, though my job hunting knowledge is way under your level of industry. Assume you only make the top 30%, you have to be smart then its not open invitation. I'd pin it down to companies with 1000 or more employees having a research division, usually engineers. A lot of companies with research will already have an association with universities. But in your case you can fall back on programming augmented by the mathematics. You can fish at www.monster.com its the biggest employment forum, say you're graduating this year! Herc I'll add your question here so I didn't terminate the thread > go that route, but I'm curious if any companies are specifically interested > in high-level mathematics. > I can give a partial answer to my own question: > Wolfram > IBM Yorktown > MS Research (crypto, wavelet compression) > Any other companies to add to the list? === Subject: Re: Any idea for a math career? > >I will be in a similar position in a few years since I've just started a > >Ph.D. program. Are you saying that math Ph.D.s are locked into academia? I > >enjoy academia and might very well take this path, but I find it disturbing > >that there might be few (no?) other paths to take. After all, teaching at a > >university is often a means to end (e.g., tuition) rather than a declaration > >of career choice. > >Off the top of my head, I'd *think* financial firms, insurance companies, > >computer companies would hire mathematicians. I haven't done a whole lot of > >research yet, but I have found that IBM, for instance, has a research center > >in Yorktown, NY, where mathematicians are free to research math, not just > >computer apps, if you're willing to work on this side of the Atlantic. MS > >Research has positions if you're in computer-related math (e.g., crypto), > >with locations in the Seattle area, Cambridge, England etc. Of course, > >there's the NSA, if you're willing. (Sorry, Leigh Ann, I don't have specific > >info on Australia.) > The recruiter at some point will ask something like What can you do for > the company? It's getting popular these days to try to model financial > systems with the methods of abstract mechanics from physics. Insurance > companies hire statisticians. I've heard of a guy that got his Ph.D. in > physics doing general relativity, and he got a job at Firestone because he > was good with the mathematics of curved surfaces, and tires have curved > surfaces. > What can you do with the math you've learned? What are other people doing > with the math you've learned? Hi guys, particularly in finance, and the areas which I thought were applicable are (starting from the most technical to the least): - Interest rate and credit derivatives - This is perhaps the most quantitative division in a financial services firm or investment bank, and involves pricing derivative products on bonds or other interest securities - Equity and index derivatives - Similar to above, but pricing derivatives on shares and indices - Structured products (sometimes known as risk management) - Helping companies hedge their assets by purchasing options, futures, etc - Structured and project finance - Helping companies borrow money or raise debt cheaply whilst minizing default risk. People working here also need to have a strong knowedge of legislation and tax This is not conclusive, so please feel free to add where necessary - and let me know if I have missed any! In Australia, the market for these positions is extremely small, and for the companies that are hiring, I've noticed they prefer hiring candidates with a Masters of Finance in options and derivative securities, rather than a mathematician. The reason being is that these courses are more focussed towards finance, and graduates come away not only with most of the quantitative skills, but also knowing the trading strategies and how the industry works. As I understand it, the perception is that math grads have these undesirable qualities: - Not commercial - Too theoretical - If you are from academia, you probably will not fit into the trading environment that the above divisions reside in - Male dominated (?) There have been some academics who've made it into the industry - for instance, my mentor is leaving to work for Susquahanna, a derivatives trading firm. But in general, given two strong candidates from math and finance, there is a tendency for people to hire in finance. Anyway, this is my experience from Sydney, Australia, so it may be different elsewhere! And if so, I may consider moving. Otherwise, it's back to academia for me I guess - not that I have not enjoyed it (I really have), but would like to try something different the PhD! It's gonna be weird when someone called me Dr ... === Subject: Re: Any idea for a math career? Even if you retain a very small portion of the Mathematics that you studied, what you may use in the REAL world of industry is often much more than what most people have been able to retain, or study. Many people employed industrially can not formulate a simple linear algebraic equation with a simple ratio and then rearrange this equation using inverse operations to obtain the expression for a variable they want. Basically, simple aspects of elementary algebra. It smoothed some problem solutions several times for me. G C === Subject: Re: Any idea for a math career? > I will be in a similar position in a few years since I've just started a > Ph.D. program. Are you saying that math Ph.D.s are locked into academia? I > enjoy academia and might very well take this path, but I find it disturbing > that there might be few (no?) other paths to take. After all, teaching at a > university is often a means to end (e.g., tuition) rather than a declaration > of career choice. is your phd from a top ranked school? if not, good luck. every phd mathematician from non-ranked schools i know, is one of the following 1. teaching and sometimes researching in a mediocre academic institution where academic standards are unimportant. 2. a bureaucrat for the us government. 3. at the brink of poverty working as an adjunct teacher. exceptions exists, although they are insignificant... > Off the top of my head, I'd *think* financial firms, insurance companies, > computer companies would hire mathematicians. I haven't done a whole lot of > research yet, but I have found that IBM, for instance, has a research center > in Yorktown, NY, where mathematicians are free to research math, not just > computer apps, if you're willing to work on this side of the Atlantic. MS > Research has positions if you're in computer-related math (e.g., crypto), > with locations in the Seattle area, Cambridge, England etc. Of course, > there's the NSA, if you're willing. (Sorry, Leigh Ann, I don't have specific > info on Australia.) > Are there any good listings of companies that hire math Ph.D.s? I've tried a > few books, but the ones I've looked at tend to list categories (e.g., > actuary, teacher), not companies or specific positions. I can't speak for > Leigh (or anyone else), but if the job were tempting enough, I'd travel > throughout the English-speaking world. if you find one or two, feel welcome to post it. > Mark > If I have not seen as far as others, it is because giants were standing on > my shoulders. -- Hal Abelson. > > > Hi all, > > > > I'm just finishing up my PhD in mathematics (focus on stochastic > > > calculus) and I am looking at possible career paths. I've always had > > > some interest in economics, and actually taught a first year financial > > > mathematics course at university. Not knowing too much about math > > > careers in finance, does anyone here have any ideas? > > > Don't go out into the real world, with your teaching experience become a > lecturer. > > Herc === Subject: Re: Any idea for a math career? > > I will be in a similar position in a few years since I've just started a > > Ph.D. program. Are you saying that math Ph.D.s are locked into academia? In the US, the largest non-academic employer of mathematicians is the National Security Agency. === Subject: Are all mathematicians music lovers? It seems every math/science type I meet is a Bach lover, having fallen for the propaganda that music expands the mind. Actually, there are people who go crazy from repetitive tunes that won't stop inside their head. I've been able to drive otherwise calm-and-collected math/science types berserk by saying music is stupid. Are there any mathematicians today who have the courage to say music is stupid? === Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover, having fallen for > the propaganda that music expands the mind. Actually, there are people > who go crazy from repetitive tunes that won't stop inside their head. I've > been able to drive otherwise calm-and-collected math/science types berserk > by saying music is stupid. Are there any mathematicians today who have the > courage to say music is stupid? Define music. Axioms are not necessary. Norm === Subject: Re: Are all mathematicians music lovers? > Define music. Axioms are not necessary. The noises recorded on CD's or other media that people listen to. The sounds are produced by instruments or singing, with a melody and rhythm. I suppose I could make a music recording of truck engines, lawn mowers, and car alarms, then play it out loud, though that would probably be the quickest way to make enemies. I've thought about doing that in the grand canyon, to see how tourists react. === Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover, having fallen for > the propaganda that music expands the mind. Huh? What propaganda? Music is sheer pleasure and its prime objective is not to expand the mind. It's to give pleasure to civilized human beings. > Actually, there are people > who go crazy from repetitive tunes that won't stop inside their head. As I am writing this, it's already around 400 times that I have listened to the Scherzo from Beethoven's Sonata #3 for Piano and Cello in A Opus 69, non-stop. The melody's meme yesterday was making very powerful attempts to occupy my entire mind, by displacing all else, including daily activities. It wants me to continuously hum it, play it on the piano, and reherse it to the point of exhaustion. > I've > been able to drive otherwise calm-and-collected math/science types berserk > by saying music is stupid. And I've been able to ward off barbarian subtypes by playing Bach Cantatas non-stop very loudly. What's your point? > Are there any mathematicians today who have the > courage to say music is stupid? That kind of animal would probably come from your side of the Atlantic only, which is famous for things like that and crap music in general. -- Ioannis http://users.forthnet.gr/ath/jgal/ ___________________________________________ Eventually, _everything_ is understandable. === Subject: Re: Are all mathematicians music lovers? > > It seems every math/science type I meet is a Bach lover, having fallen for > > the propaganda that music expands the mind. > Huh? What propaganda? Music is sheer pleasure and its prime objective > is not to expand the mind. It's to give pleasure to civilized human > beings. Actually, cavemen made plenty of music with their bone flutes and stone drums. Music does have a tendency to inflate wishful thinking tendencies though. > > Actually, there are people > > who go crazy from repetitive tunes that won't stop inside their head. > As I am writing this, it's already around 400 times that I have listened > to the Scherzo from Beethoven's Sonata #3 for Piano and Cello in A Opus > 69, non-stop. > The melody's meme yesterday was making very powerful attempts to > occupy my entire mind, by displacing all else, including daily > activities. It wants me to continuously hum it, play it on the piano, > and reherse it to the point of exhaustion. Sounds like quite the prison. > > I've > > been able to drive otherwise calm-and-collected math/science types berserk > > by saying music is stupid. > And I've been able to ward off barbarian subtypes by playing Bach > Cantatas non-stop very loudly. What's your point? Proves my point. So you warded off barbarian subtypes by becoming a barbarian yourself. > > Are there any mathematicians today who have the > > courage to say music is stupid? > That kind of animal would probably come from your side of the Atlantic > only, which is famous for things like that and crap music in general. Actually, your side of the Atlantic has a longer history of killing people for listening to music. === Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover, having fallen for > the propaganda that music expands the mind. Actually, there are people > who go crazy from repetitive tunes that won't stop inside their head. I've > been able to drive otherwise calm-and-collected math/science types berserk > by saying music is stupid. Are there any mathematicians today who have the > courage to say music is stupid? Forget math/science types, what about the whole population of humans. I have never met a single person who does not derive a lot of pleasure from at least some kind of music. -- Stephen Montgomery-Smith stephen@math.missouri.edu http://www.math.missouri.edu/~stephen === Subject: Re: Are all mathematicians music lovers? > > Define music. Axioms are not necessary. > The noises recorded on CD's or other media that people listen to. The > sounds are produced by instruments or singing, with a melody and rhythm. > I suppose I could make a music recording of truck engines, lawn mowers, > and car alarms, then play it out loud, Throw in some fellows screaming in German, and you have Einsturzende Neubauten -- whose music I find quite excellent! Perfect background for studying functional analysis. though that would probably be the > quickest way to make enemies. I've thought about doing that in the grand > canyon, to see how tourists react. === Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover Although trite, it appears to me that the majority of mathematicians and people interested in mathematics enjoy music. I would not agree, however, that they enjoy because it offers some vague promise to expand the mind, but that being a connoisseur of any aesthetic pursuit leads one to value or adopt other like interests. Just as an elegant proof is interesting to a certain type of person, so is an elegant fugue. === Subject: Re: Are all mathematicians music lovers? >> It seems every math/science type I meet is a Bach lover, having >> fallen for the propaganda that music expands the mind. Actually, >> there are people who go crazy from repetitive tunes that won't stop >> inside their head. I've been able to drive otherwise >> calm-and-collected math/science types berserk by saying music is >> stupid. Are there any mathematicians today who have the courage to >> say music is stupid? > Forget math/science types, what about the whole population of humans. > I have never met a single person who does not derive a lot of pleasure > from at least some kind of music. Well, as a counter-example, I offer the fans of the Dixie Chicks. Bart === Subject: Re: Are all mathematicians music lovers? > Forget math/science types, what about the whole population of humans. I have > never met a single person who does not derive a lot of pleasure from at least > some kind of music. Outside of the Western world, there are countries where people have more freedom to say they don't like music, without risk of social ostracism. In addition to East Asian countries, Prophet Mohammad was thought to have hated the arts and music; he destroyed the Vedic temples that once existed at Mecca. Conservative Muslims today refrain from listening to music. There is also a lesser known subculture of people in the Western world who communicate with grammar rules completely different from Western languages, who have a unique parallel form in which they can talk about 3 objects at once. I met a lady who belonged to that subculture once, whose friends don't listen to music, so she thought most people in the world don't listen to music. === Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover, having fallen > for the propaganda that music expands the mind. Actually, there > are people who go crazy from repetitive tunes that won't stop inside > their head. I've been able to drive otherwise calm-and-collected > math/science types berserk by saying music is stupid. Are there any > mathematicians today who have the courage to say music is stupid? You have already answered your own question in that paragraph; the answer is yes, because you have this courage. (It's not really courageous.) Also, you might as well describe other things as equally stupid. I suggest you try a few of these: Living is stupid, because everybody dies anyway. Sports are stupid, because they are wastes of time. Relaxation is stupid, because you could be working. Coffee is stupid, because caffeine pills are more efficient. I can list many others, too. For instance, did you know that Tetris and literature are also stupid? Try looking at society once in a while and take a few notes. You'll find that almost everybody spends much of their life doing stupid things. I suggest that you commence a good bitching at the entire population about how their lives are stupid, at this instant. As it happens, a lack of music is equally stupid as its presence. === Subject: Re: Are all mathematicians music lovers? Visiting Assistant Professor at the University of Montana. >It seems every math/science type I meet is a Bach lover, having fallen for >the propaganda that music expands the mind. Well, I like Bach (not my favorite composer by any means), but the reason I like it is not that I believe music expands the mind. Rather, I enjoy listening to it. > Actually, there are people >who go crazy from repetitive tunes that won't stop inside their head. Yes. There are also people who go crazy from working in a post office day in and day out; and people who go crazy by blows to the head; and people who go crazy after reading and re-reading certain books; etc. > I've been able to drive otherwise calm-and-collected math/science >types berserk by saying music is stupid. Well, you might get similar expletives from me if you were to say that music is stupid, or that reading is stupid, or any number of other statements that I think are over simplistic and, well, stupid. So what? I've seen otherwise calm and collected people go berserk simply because someone told them their political ideas were stupid, or their statement was stupid, or their ideals are stupid. And let's not forget the granddaddy of them all: that their religion is stupid. > Are there any mathematicians today who have the >courage to say music is stupid? Why does it take courage? I know there are people who get no enjoyment from music; whether it is because they are tone-deaf or simply because they do not enjoy music. That's fine with me. For them, listening to music would ->be<- stupid, as would be dedicating lots of time to trying to create music. Just like, for me, it would be stupid to go to a modern art exhibit: because I find that I do not like modern art, and therefore attending such an exhibit would be a monumental waste of time for me, from which I would not derive any enjoyment. I say it freely: I dislike practically all art made after the impressionists (with Escher a notable exception). That might label me a philistine among some circles, but it doesn't take courage for me to say it. On the other hand, it would take something other than courage for me to say that art is stupid just because I don't happen to enjoy it. Just like saying music is stupid simply because you don't enjoy it or like it would not be a courageous statement, just a rather stupid statement on your part. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Are all mathematicians music lovers? ... > Are there any mathematicians today who have the > courage to say music is stupid? I'm not a mathematician, but I'll go out on a limb and courageously suggest that trolling is stupid and responding to trolls is even stupider. === Subject: Re: Are all mathematicians music lovers? on Saturday 16 > Prophet Mohammad was thought to have hated > the arts and music; he destroyed the Vedic temples that once existed at > Mecca. I don't think that has anything to do with arts or music. Correct me if I'm wrong, but didn't these temples contain representations of God? If I am not mistaken, representation of God in a pictural form is not allowed by Islam. > Conservative Muslims today refrain from listening to music. I don't know where you hold this fact from. In fact, Muslim prayers are quite melodic. It almost seems they are sung. Sam -- [...] but the delight and pride of Aule is in the deed of making, and in the thing made, and neither in possession nor in his own mastery; wherefore he gives and hoards not, and is free from care, passing ever on to some new work. - J.R.R. Tolkien, Ainulindale (Silmarillion) === Subject: Re: Armand Borel >... >> Serre with Jean Leray, Andre Weil, Henri Cartan in the Bourbaki group: >> Clearly, Serre must have joined the Bourbaki group after the others had >> left :-) and the original Bourbaki idea had ceased fulfilling its purpose >> (namely, to clear up the logical mess in the early 20th century math). >> Judging from the later Bourbaki volumes, clearly Serre is not up the >> standards of the original Bourbaki group. >It seems very odd to me to take Borel's death as an excuse to attach Serre. >Which later Bourbaki volumes do you believe are Serre's work, Probably all after the original guys had stopped. The book on homological algebra for example. >and what exactly is your complaint about them ? I mentioned that: >the original Bourbaki idea had ceased fulfilling its purpose >> (namely, to clear up the logical mess in the early 20th century math). The later Bourbaki seems to have misunderstood that idea, instead making some kind of very limited encycolopedia, not really useful for anything serious. >Personally, I find Serre's work at least as lucid as >any of the others you mentioned. Well, that is your opinion. But that is not so if one knows something about why and how the original Bourbaki weas created, and the talents of those people involved. * Email: Hans Aberg * Home Page: === Subject: Re: Armand Borel >Personally, I find Serre's work at least as lucid as >>any of the others you mentioned. > Well, that is your opinion. > But that is not so if one knows something about why and how the original > Bourbaki weas created, and the talents of those people involved. I really don't understand why you have it in for Serre. I don't believe any reasonably objective person could find his work obscure or pedantic. I imagine Serre must be kicking himself for not proving Fermat's Last Theorem, as he must have all the tools at his fingertips, and the mixture of abstract and down-to-earth computation involved seems very much his metier. If Serre was really responsible for all the recent works of Bourbaki, as you claim, my opinion of him climbs even higher. I find the recent volumes have at least the same clarity as the early ones. (I was just looking last week at the proof that an integral domain in which each non-zero ideal is expressible as a product of prime ideals is necessarily a Dedekind domain -- it arose from something someone said in this newsgroup -- and I found the account in Commutative Algebra chapter 7 both clear and interesting. I guess I should thank Serre, according to you ?) -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Armand Borel >>Personally, I find Serre's work at least as lucid as any of the others you mentioned. >> Well, that is your opinion. >> But that is not so if one knows something about why and how the original >> Bourbaki weas created, and the talents of those people involved. >I really don't understand why you have it in for Serre. >I don't believe any reasonably objective person >could find his work obscure or pedantic. It is not the clarity, but the selection of topics: Original Bourbaki selected topics that were unclear in the written literature, and aimed at clearing up that logical mess. Having success with such a thing clearly requires dealing with the topics at some depth, not merely moving shallowly at the surface. They succeeded most notably in fundamental algebra, but failed for example in areas like statistics (measure theory) where Bourbaki did not catch on. If one takes the homological algebra volume, it is a subselection of facts much better described elsewhere in the literature. So Bourbaki does not have anything here to contribute, and the reason is that it breaks off from the original design objective. You won't find out who has written what of Bourbaki, but that superficiality is clearly a trademark of Serre: He is not kind of guy that works up new logical foundations, but rather zooms in onto some facts already worked up by others, and then picks the logical pieces together without inventing anything new. Very smart and elegant, but not deep and subtle or creative. >I imagine Serre must be kicking himself >for not proving Fermat's Last Theorem, >as he must have all the tools at his fingertips, >and the mixture of abstract and down-to-earth computation involved >seems very much his metier. Serre is not the kind of person who ever could have sat down and proved deep and subtle things like FLT: He is the predator who quickly moves in and combines certain things without going into depth or adding new creative insights. His stuff is socially advanced, not creatively advanced. >If Serre was really responsible for all the recent works of Bourbaki, >as you claim, my opinion of him climbs even higher. >I find the recent volumes have at least the same clarity as the early ones. I only noticed that the homological algebra volume, which came in the seventies or eighties I think, was substandard relative to the older >(I was just looking last week at the proof that an integral domain >in which each non-zero ideal is expressible as a product of prime ideals >is necessarily a Dedekind domain -- >it arose from something someone said in this newsgroup -- >and I found the account in Commutative Algebra chapter 7 >both clear and interesting. >I guess I should thank Serre, according to you ?) I found the Algebra and Commutative Algebra books useful, but I do not During the days of Andre Weil, one of the original Bourbaki members said that they had meetings deciding what to write about. Then a member would be selected for a first write. Then at the next meeting one discussed the writeout, and if that was not satisfactory, another member would be selected for a rewrite, which could be even a complete rewrite. And so on, until one was satisfied. If Andre Weil was not present, and that method was not used, one would expect standards to drop. Also, it would be more difficult to select good topics, if the idea is merely to write an encyclopedia, and not specifically influence better mathematical logical description. Bourbaki itself stimulated a movement in math towards better logical clarity, making it unnecessary. * Email: Hans Aberg * Home Page: === Subject: Re: Armand Borel dead > Serre with Jean Leray, Andre Weil, Henri Cartan in the Bourbaki group: > Clearly, Serre must have joined the Bourbaki group after the others had > left Obviously, you consider free speech to be more important than elementary decency (I don't). But even free speech doesn't give you the right to screw up the facts, buddy. The retiring age at Bourbaki is 50, so that would yield 1954 for Cartan and 1956 for Weil. OTOH, Serre joined Bourbaki in 1949. Besides, Weil's autobiography includes a picture of the 1951 Bourbaki summer meeting, showing Weil, Cartan and Serre (and others), and Pierre Dugac's biography of Dieudonn.8e includes one of the 1954 meeting, showing Weil and Serre (and others). So next time, do your homework before posting. Hugo === Subject: Re: Armand Borel dead >> Serre with Jean Leray, Andre Weil, Henri Cartan in the Bourbaki group: >> Clearly, Serre must have joined the Bourbaki group after the others had >> left >Obviously, you consider free speech to be more important than elementary >decency (I don't). But even free speech doesn't give you the right to screw >up the facts, buddy. I thought the idea was that free speech admits one to speak out about the things that one considers indecent or wrong. If, in a subculture, speaking out about the things one considers indecent or wrong is in itself is considered indecent, then that subculture does not have much of free speech, and will not evolve properly with respect to more transparent and accountable parts of society. >The retiring age at Bourbaki is 50, so that would yield 1954 for Cartan and >1956 for Weil. OTOH, Serre joined Bourbaki in 1949. >Besides, Weil's autobiography includes a picture of the 1951 Bourbaki summer >meeting, showing Weil, Cartan and Serre (and others), and Pierre Dugac's >biography of Dieudonn.8e includes one of the 1954 meeting, showing Weil and >Serre (and others). It is always good with people keeping track of the facts. :-) So there were a few years of overlap. Interesting, but it does not really prove anything as for the quality of the Bourbaki volumes: I know that some of the original Bourbaki members said that it was really Andre Weil that was the driving force. It would have been so as long as he decided to be actively involved. I found the algebra and commutative algebra volumes useful, but the later homological algebra I found pointless. Also, the original idea of Bourbaki, which I take it was to bring logical accuracy to some selected messy early twentieth century mathematical fields, but continue beyond that seems pointless. * Email: Hans Aberg * Home Page: === Subject: Re: Armand Borel dead > I found the algebra and commutative algebra volumes useful, but the later > homological algebra I found pointless. Also, the original idea of > Bourbaki, which I take it was to bring logical accuracy to some selected > messy early twentieth century mathematical fields, but continue beyond > that seems pointless. Who says that was the object of the game? I've always assumed the aim was to produce a complete logical account of the central themes of pure mathematics. The only part of Bourbaki that seems to fit your description is the part on Set Theory, which has always struck me as the weakest part of all. Surely there were no logical inaccuracies in algebra (or commutative algebra) by the time Bourbaki got to work ? In any case, you haven't explained why you attack Serre for his supposed contribution to Bourbaki, when he has produced a vast number of works which would seem a more logical target for your criticism -- criticism which I for one find completely incomprehensible. What exactly do you have against Serre? Your animosity appears to me to be totally irrational. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Armand Borel Finally dead >The following sad news will be of interest to many on >sci.math.research: Well, I won't miss him nor his math: Armand Borel was known as being intimidating and of his own math research he said that the most difficult part was to keep track of the arrows (i.e., the implications). So you wouldn't go to his math looking for something deep and subtle or any other traits of genius level science. Borel's papers gives no clue as to what good they are for humanity or science. Serre with Jean Leray, Andre Weil, Henri Cartan in the Bourbaki group: Clearly, Serre must have joined the Bourbaki group after the others had left :-) and the original Bourbaki idea had ceased fulfilling its purpose (namely, to clear up the logical mess in the early 20th century math). Judging from the later Bourbaki volumes, clearly Serre is not up the standards of the original Bourbaki group. It also very strangely describes Borel as a part of a group at the Institute of Advanced Study (IAS), Princeton, NJ, USA, with Andre Weil, Robert Langlands and Pierre Deligne: Borel had a job at IAS, and that was pretty much it; it was not a group in any other sense. In fact, Borel tried to take up a job at other places, such as ETH, Zurich, but he did not get along well with people, and so ended up being stuck at the IAS, and the IAS ended up being stuck with him. It appeared to me that Borel and Serre together and in competition with each other viewed themselves as being in the lineage some truly great mathematicians, but without having the capacity themselves. Thus, their carriers are full of some very strange things, such as their collected works appearing before they were dead; clearly that didn't happen because the science community felt a great need for it: They must have pushed very hard for it themselves in order to make their careers appear to be something they never could become by force of scientific value itself. * Email: Hans Aberg * Home Page: === Subject: Re: Armand Borel Finally dead > Well, I won't miss him nor his math: ... > Serre with Jean Leray, Andre Weil, Henri Cartan in the Bourbaki group: > Clearly, Serre must have joined the Bourbaki group after the others had > left :-) and the original Bourbaki idea had ceased fulfilling its purpose > (namely, to clear up the logical mess in the early 20th century math). > Judging from the later Bourbaki volumes, clearly Serre is not up the > standards of the original Bourbaki group. It seems very odd to me to take Borel's death as an excuse to attach Serre. Which later Bourbaki volumes do you believe are Serre's work, and what exactly is your complaint about them ? Personally, I find Serre's work at least as lucid as any of the others you mentioned. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Armand Borel Finally dead Hans Aberg Well, I surely will. It's a shame how you write about someone else who cannot reply, who died only days ago. I want to tell you, while you are alive. Afterwards I won't speak badly about you. I think I better keep quiet then. *bah* Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Armand Borel Finally dead > >The following sad news will be of interest to many on > >sci.math.research: > Well, I won't miss him nor his math: > Armand Borel was known as being intimidating and of his own math research Excuse my ignorance but how (if at all) does the recently deceased Borel relate to _the_ Borel--F.8elix .83douard Justin .83mile? (Forgive me if the accents b*gg*r things up.) -- G.C. === Subject: Re: Armand Borel Finally dead >Well, I surely will. >It's a shame how you write about someone else >who cannot reply, who died only days ago. >I want to tell you, while you are alive. Afterwards >I won't speak badly about you. I think I better >keep quiet then. Does that apply to Idi Amin as well, who also died at the same (official) age? * Email: Hans Aberg * Home Page: === Subject: Re: Calculus is irrational? >Hi All, >If calculus assumes infinity to come to its answers It doesn't. >(for example, the >limit of a function, we sum to infinity to find an answer) No, we don't. >and because infinity is irrational (infinity being defined by p/0 No, that's not the definition of infinity. >and >rational number defined by p/q where q <> 0) >is it fair to say that any answer given to us by calculus is by >definition irrational as it assumes irrationality in the solution? Evidently not. As has already been pointed out, your understanding of how calculus works is a few hundred years out of date. Not your fault, in a typical calculus course things things are often very fuzzy. Let me give an example of how calculus does _not_ assume infinity: Fact: The sum of 1/2^j, j = 1, ... infinity, is 1. Now the way that's stated it certainly appears to involve something infinite - there's that infinity right there in the statement. But here's what the fact means, _by definition_: Fact, translated by inserting definitions: If eps is any positive number then there exists a number N (the value of N depending on the value of eps) such that |1 - sum 1/2^j, j = 1...n| < eps for all n > N. Note that there's nothing infinite at all in the translation. _All_ the infinities in elementary calculus actually vanish this way, if you insert the definitions. >I'm not saying that irrational is equal to bad or wrong or >useless because it is obviouslly none of those things, but am I >right in thinking that all solutions that require calculus are >mathematically defined as irrational? No, not even close. >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? Randy Poe > > > Ehr, no. The limit of a function is not a sum to infinity; the limit > > > of a function, when it exists, is a specific number with a very > > > specific property. > > > > Yes, but, in some functions (not all of course) the limit is found by > > approaching infinity. > But not by ever being at infinity, whatever that means. Exactly my point. Lets take the problem of a bouncing ball. This was taken from a friend's post in another newsgroup: Say the finite time between each bounce gets progressively shorter by a factor of 10. So the first bounce takes 1/10 second, the second bounce takes 1/100 of a second, the third one 1/1000 of a second, right? No matter how many bounces down the line you look, you'll always get a positive amount of time: bounce number i will take 1 / (10 ^ i) seconds, so each bounce takes longer than 0 seconds. It is also clear that there is an infinite number of bounces, as after all no matter how small the amount of time gets, you can always divide it by 10 again.... still with me? So the total amount of time the ball spends bouncing is.... 1/10 + 1/100 + 1/1000...... Or Sum[i=1 to infinity](1 / (10 ^ i)) now you can work out that that doesn't add up to infinity: Let's assume that x = Sum[i=1 to infinity](1 / (10 ^ i)) then, multiplying both sides by 10: 10x = 10*Sum[i=1 to infinity](1 / (10 ^ i)) = Sum[i=1 to infinity](10 / (10 ^ i)) = Sum[i=1 to infinity](1 / (10 ^ i-1)) = Sum[i=0 to infinity](1 / (10 ^ i)) = 1 + Sum[i=1 to infinity](1 / (10 ^ i)) but... hang on... that was our original x, so: 10x = 1 + x ! therefore 9x = 1 so x = 1/9 So Sum[i=1 to infinity](1 / (10 ^ i)) = 1/9 That this is true is easy to see, after all, 1/10 is 0.1, 1/100 = 0.01, 1/1000 is 0.001 and so on, so if you add 'em all up, you will get 0.1111111111111111111111111111111111111111111111111...... and as we all know, that is precisely what 1/9 is. In order to come to the answer of 1/9, we have to be at infinity, otherwise, our answewr will always be just short of 1/9. If we use calculus to answer the question with 1/9, we have to assume addition all the way to infinity (even if we don't need to literally add to infinity and can use the shortcuts of Calculus). Because of that assumption, the answer has an element of irrationality. Please do not get me wrong. It is still a GOOD and USEFUL answer. But that doesn't mean that the answer is comletely rational. Thats all I'm trying to point out. Mike Helland === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. [.series problem snipped.] >Sum[i=1 to infinity](1 / (10 ^ i)) >now you can work out that that doesn't add up to infinity: >Let's assume that x = Sum[i=1 to infinity](1 / (10 ^ i)) >then, multiplying both sides by 10: >10x = 10*Sum[i=1 to infinity](1 / (10 ^ i)) > = Sum[i=1 to infinity](10 / (10 ^ i)) > = Sum[i=1 to infinity](1 / (10 ^ i-1)) > = Sum[i=0 to infinity](1 / (10 ^ i)) > = 1 + Sum[i=1 to infinity](1 / (10 ^ i)) but... hang on... that >was >our original x, so: >10x = 1 + x ! >therefore >9x = 1 >x = 1/9 >So Sum[i=1 to infinity](1 / (10 ^ i)) = 1/9 >That this is true is easy to see, after all, 1/10 is 0.1, 1/100 = >0.01, >1/1000 is 0.001 and so on, so if you add 'em all up, you will get >0.1111111111111111111111111111111111111111111111111...... >and as we all know, that is precisely what 1/9 is. >In order to come to the answer of 1/9, we have to be at infinity, No, we do not (or rather, we do not have to do any such thing). The expression Sum{i=1 to infinity}( 1/(10^i) ) = 1/9 means nothing more and nothing less than: For every epsilon>0, there exists a positive integer N>0 such that, for all n>=N, |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon. We express this by ->saying<- that the infinite sum adds up to 1/9. But we are not at infinity in any sense. >otherwise, our answewr will always be just short of 1/9. If we use >calculus to answer the question with 1/9, we have to assume addition >all the way to infinity (even if we don't need to literally add to >infinity and can use the shortcuts of Calculus). No, we do not. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? > Infinity is NOT A NUMBER. But are you arguing that infinity is indeed a rational quantity? Do you have any cites that state that infinity is not a quantity? I've found several that define it as a limitless quantity, including the online Mathworld: http://mathworld.wolfram.com/Infinity.html === Subject: Re: Calculus is irrational? > ... Are we > on our way (say 100 years from now) to making infinity into an entity > treated as any other number? We're already there. -- G.C. === Subject: Re: Calculus is irrational? >> The notion of a limit, which you are savaging is perfectly coherent. >Have I said it is not coherent? No. But the things that you've said have themselves been incoherent. (Limits are irrational. Huh? So 0 is irrational, because it's the limit of x as x -> 0?) >Have I said that it is useless? >No. >I went out of my way to emphasize the point that I think Calculus is >useful and is by no means wrong . Didn't you see that? >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? [...] >So Sum[i=1 to infinity](1 / (10 ^ i)) = 1/9 >That this is true is easy to see, after all, 1/10 is 0.1, 1/100 = >0.01, >1/1000 is 0.001 and so on, so if you add 'em all up, you will get >0.1111111111111111111111111111111111111111111111111...... >and as we all know, that is precisely what 1/9 is. >In order to come to the answer of 1/9, we have to be at infinity, >otherwise, our answewr will always be just short of 1/9. If we use >calculus to answer the question with 1/9, we have to assume addition >all the way to infinity No we _don't_. Why not has been explained. > (even if we don't need to literally add to >infinity and can use the shortcuts of Calculus). Because of that >assumption, the answer has an element of irrationality. >Please do not get me wrong. It is still a GOOD and USEFUL answer. But >that doesn't mean that the answer is comletely rational. Thats all I'm >trying to point out. But this thing you're attempting to point out simply makes no sense. That limit above is equal to 1. Exactly what does it _mean_ to say that 1 is not completely rational? >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >> Infinity is NOT A NUMBER. >But are you arguing that infinity is indeed a rational quantity? >Do you have any cites that state that infinity is not a quantity? Calculus, Early Transcendentals, Brief Edition, 7th Edition, by Howard Anton, Irl Bivens, and Stephen Davis. pp. 115, REMARK. It should be emphasized that the simbols +infinity and -infinity are NOT [emphasis in the original] real numbers. The phrase f(x) approaches +infinity is akin to saying that f(x) approaches the unapproachable; it is a colloquialism for f(x) increases without bound. [...] Furthermore, since +infinity and -infinity are not numbers, it is inappropriate to manipulate these symbols using rules of algebra. There are ways of dealing with infinity that make it a quantity of sorts; for example, in set theory. There are ways of dealing with it that make it a point on a manifold (e.g., one point compactifications). But there is no need to do so in order to do calculus. The answer to your question of whether infinity is a rational or irrational quantity is simply it is neither, because it is not considered a quantity for the purposes of calculus. > I've >found several that define it as a limitless quantity, including the >online Mathworld: >http://mathworld.wolfram.com/Infinity.html Mathworld is not always a good source of information. What you need to understand is that he is giving a colloquial meaning of the symbol, not a formal one. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? Are we left to understand infinity through inference, or implication, or some other kind of induction? A young person who first learns to count, may understand very well that there is a number beyond 9,999,999; and a number beyond 9,999,999,999; and whatever number is thought, 1 may be ADDED to it, and 1 may be added to this result, and the addition of 1 can be continued without end... Although a definition may be difficult, the meaning seems understandable. G C === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Are we left to understand infinity through inference, or implication, or some >other kind of induction? The point is that in calculus there is no need to understand infinity as anything other than a colloquialism with a very precise meaning which can be expressed entirely in terms of the usual real numbers. For infinity in other contexts (topology, set theory, cardinal arithmetic, etc) you are left to understand infinity through precise, formal definition that will be given in those contexts, or through agreement as to what they represent as a colloquialism. >A young person who first learns to count, may understand very well that there >is a number beyond 9,999,999; >and a number beyond 9,999,999,999; >and whatever number is thought, 1 may be ADDED to it, and 1 may be added to >this result, and the addition of 1 can be continued without end... >Although a definition may be difficult, the meaning seems understandable. One can define infinity in a meaningful way that is useful to calculus. However, it is somewhat complicated, and is not necessary in order to do calculus, or understand limits, series, sequences, or any other part of calculus. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? > It is also clear that there is an infinite >number of >bounces, as after all no matter how small the amount of time gets, you >can >always divide it by 10 again.... still with me? I understand what you are saying, but you are mistaken. There is not an infinite number of bounces. There can't be an infinite number of bounces, because there is no number called infinity. >So the total amount of time the ball spends bouncing is.... >1/10 + 1/100 + 1/1000...... >Sum[i=1 to infinity](1 / (10 ^ i)) Sum to infinity is just a convenient shorthand for something else, which can be expressed precisely, does not involve the word, or concept, infinity. >now you can work out that that doesn't add up to infinity: You don't need to work it out. If you add two real numbers together, you get another real number. You don't get infinity, because infinity isn't a real number. If you add a lot of real numbers together, you still don't get infinity, because the answer is always a real number. You can't add an infinite number of terms together, because there is no such number. When mathematicians talk about the sum of an infinite series, they don't mean the result obtained by adding an infinite number of terms, they mean the limit of the series of partial sums. This is set out pretty clearly in most calculus / analysis textbooks. Gareth === Subject: Re: Calculus is irrational? David C. Ullrich > Fact, translated by inserting definitions: > If eps is any positive number then there exists a > number N (the value of N depending on the value > of eps) such that > |1 - sum 1/2^j, j = 1...n| < eps for all n > N. > Note that there's nothing infinite at all in the > translation. _All_ the infinities in elementary > calculus actually vanish this way, if you insert > the definitions. But, how is saying for all N different from 1 to infinity? I understand the semantics are different, but isn't it saying the same thing? Also, you are changing the wording of the answer so that it doesn't actually answer the question that was asked. If you are required to present a solution to something like when does the ball stop you need to take your answer of there are no numbers that will make the value less than N and reword to say N if you wish to actually answer the question. In this case, that final leap is where the irrationality now lies. I understand exactly what you've done, you've limited the scope of Calculus so that it is not irrational, however, you've done it in a way that to present a final solution irrationality is required. Mike Helland === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >David C. Ullrich >> Fact, translated by inserting definitions: >> If eps is any positive number then there exists a >> number N (the value of N depending on the value >> of eps) such that >> |1 - sum 1/2^j, j = 1...n| < eps for all n > N. >> Note that there's nothing infinite at all in the >> translation. _All_ the infinities in elementary >> calculus actually vanish this way, if you insert >> the definitions. >But, how is saying for all N different from 1 to infinity? Presumably you mean for all n>N and from N to infinity. >understand the semantics are different, but isn't it saying the same >thing? Yes. But your mistake is thinking that from N to infinity means that you must at some point be plugging in something called infinity, or that at some point you must consider something called infinity. shorthand for all n greater than N. You never consider anything which is not a positive integer. You certainly never consider anything called infinity. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? >Gregory L. Hansen >> Many of the standard exercises in calculus, like the slope of a parabola, >> had been found earlier by other methods. If you get the same answer >> either way, is the answer sometimes rational and sometimes irrational? >Yes, thats exactly my point. A solution can only be as rational as the >premises assumed to find that solution. >Mike If the solution can be expressed as a ratio of integers, then the solution is rational. Otherwise it's not. dy/dx = 2x is not rational or irrational. At least, not until x is specified. -- A good plan executed right now is far better than a perfect plan executed next week. -Gen. George S. Patton === Subject: Re: Calculus is irrational? ... >In this framework, infinity is not a real number. >As such it would neither be >described as a rational number, nor as an irrational number. Now you could >think of it as a number in some sense You could (and people do) also think of it as rational, though not (quite) a number, in at least one standard context. I am fond of recalling to sci.math readers a geometric construction used in (one approach to) the classical theory of modular forms in general, and cusp forms in particular: you start with the upper half plane (all complex numbers x + iy with y > 0) and adjoin to it the so-called rational cusps, namely, the points x with x rational AND the point at infinity, all of them on the extended real line which is the boundary of the upper half plane in the extended complex plane. (Then you blow up the cusps and go from there. If you're the late Armand Borel, you and your friend J-P Serre go quite a lot further from there.) So infinity is a rational cusp. In that senses. Lee Rudolph === Subject: Re: Calculus is irrational? ... >(Then you blow up the cusps >and go from there. If you're the late Armand Borel, you and >your friend J-P Serre go quite a lot further from there.) Having now finished reading the news that accumulated during the blackout, I want to add that my reference to Borel and Serre (and, implicitly, to their bordification) have nothing whatever to do with Hans Aberg's comments on the two of them in the thread on Borel's death. Lee Rudolph === Subject: Re: Calculus is irrational? >David C. Ullrich >> Fact, translated by inserting definitions: >> If eps is any positive number then there exists a >> number N (the value of N depending on the value >> of eps) such that >> |1 - sum 1/2^j, j = 1...n| < eps for all n > N. >> Note that there's nothing infinite at all in the >> translation. _All_ the infinities in elementary >> calculus actually vanish this way, if you insert >> the definitions. >But, how is saying for all N different from 1 to infinity? I >understand the semantics are different, but isn't it saying the same >thing? Presumably that was a typo for 'But, how is saying for all n > N different from 1 to infinity?' In any case, I don't quite follow the question. There is _no_ mention of anything infinite above. The above _is_ what the sum of 1/2^j, n = 1..infinity = 1 _means_. >Also, you are changing the wording of the answer so that it doesn't >actually answer the question that was asked. If you are required to >present a solution to something like when does the ball stop you >need to take your answer of there are no numbers that will make the >value less than N and reword to say N if you wish to actually >answer the question. Huh? What question was I changing? There were no questions about balls stopping in any part of the thread lying above the post I was replying to. I was discussing what the statement the sum of 1/2^j, n = 1..infinity = 1 means. That's an example of a bit of calculus that appears to involve something infinite but actually doesn't, if you insert the definitions. You say I'm changing the question, but I have no idea what original question you have in mind. What ball? >In this case, that final leap is where the irrationality now lies. I >understand exactly what you've done, you've limited the scope of >Calculus so that it is not irrational, however, you've done it in a >way that to present a final solution irrationality is required. >Am I mistaken? You're not so much mistaken as making no sense at all. You keep talking about this irrationality, without ever explaining exactly what you mean. Q: What is the sum of 1/2^j, j = 1..infinity? A: 1. Exactly how was irrationality required to present that solution? What's irrational is the way you continue to try to explain what's irrational about calculus, in spite of the fact that none of the mathematicians in the crowd think you're making any sense. It would be more rational to assume that possibly they understand the story better than you do and to try to understand why there's nothing irrational involved. (I mean really. It's easy to see how a person might develop the sort of views you developed, because the way calculus is presented is often not very rigorous. But the ideas you have about this mysterious irrationality are simply _wrong_. Honest.) >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? David W. Cantrell > > I don't know what you mean by quantity. Do you? > Quantity. Number. The choice of name isn't that important. It's what you > can do with the thing (whether it's called a quantity, a number, or > something else) that's important. Well, the way I interpret it a number would imply a specific quantity. Reasonable? === Subject: Re: Calculus is irrational? Arturo Magidin > >In order to come to the answer of 1/9, we have to be at infinity, > No, we do not (or rather, we do not have to do any such thing). The > expression > Sum{i=1 to infinity}( 1/(10^i) ) = 1/9 > means nothing more and nothing less than: > For every epsilon>0, there exists a positive integer N>0 such that, > for all n>=N, > |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon. > We express this by ->saying<- that the infinite sum adds up to > 1/9. But we are not at infinity in any sense. I agree. So here's my issue. If you are answering the question of the bouncing ball with the terminology there exists no postive integer... ect. then you are not really answering the question that was asked. You are stating something that may be true, but it does not provide you a result for the problem... until you actually make the leap to irrationality by saying that the infinite sums add up to 1/9. So either: calculus isn't allowed to answer the question directly, or it answers it irrationally (but still in a useful and valid way). Is my logic on that mistaken? Mike Helland === Subject: Re: Calculus is irrational? >[...]Is my logic on that mistaken? Why do you keep asking this question? People have explained over and over and over that yes, your logic is mistaken. >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? Gregory L. Hansen > >Yes, thats exactly my point. A solution can only be as rational as the > >premises assumed to find that solution. > If the solution can be expressed as a ratio of integers, then the solution > is rational. Otherwise it's not. Sorry, you're still missing my point. What you said should have been: If the _result of a_ solution can be expressed as a ratio of integers, then the _result of a_ solution is rational. Otherwise it's not. However that says nothing about the solution itself or its rationality. For example, we both know of solutions to problems that use division by zero that could lead to a result that can be expressed rationally. However, despite the result being a rational number, we both agree that the solution itself is not rational as it requires irrationality to come to its result. Agreed? === Subject: Re: Calculus is irrational? > David W. Cantrell > > > I don't know what you mean by quantity. Do you? > > Quantity. Number. The choice of name isn't that important. It's what > > you can do with the thing (whether it's called a quantity, a number, or > > something else) that's important. > Well, the way I interpret it a number would imply a specific quantity. > Reasonable? Yes. But I don't know what your point is. And I'm curious about your use of the word specific above. Are you thinking that something could be _a_ quantity without being a _specific_ quantity? If so, please give an example. David === Subject: Re: Calculus is irrational? David C. Ullrich > You're not so much mistaken as making no sense at all. > You keep talking about this irrationality, without ever > explaining exactly what you mean. > Q: What is the sum of 1/2^j, j = 1..infinity? > A: 1. > Exactly how was irrationality required to present > that solution? In order to sum all from 1 to infinity, one would have to reach infinity to get the precise answer. I'm making the assumption that for a human to reach infinity is an irrational suggestion because infinity lies outside the limits of human rationality. If that is the case, even though we can use valid and effective tools like Calculus to determine what is happening as we approach infinity, despite its effectiveness, it is still adding an element of irrationality into the solution. By irratoinal I am not implying that Calculus is wrong or useless. Thats not my intention or opinion at all. But to find its answers, it does rely on stepping outside the realm of human rationality. Of course, this gives us a whole new outlook on what problems we can solve and allows us to apply those solutions effectively in the real world, but it shouldn't be too hard too admit that those solutions cannot be regarded as absolutely rational. I realize that Calculus does not literally sum to infinity with algebra, but the idea is essentially the same. Calculus is an effective short cut for this process. Mike Helland === Subject: Re: Calculus is irrational? The World Wide Wade > An irrational number is, by definition, any *real* number that is not > rational. Infinity, whatever it is, is not a real number. So your use of > irrational here is invalid. However, I can make my point without relying on rational or irrational numbers at all, and stick strictly to the argument of what is and what is not within human rationality. Infinity lies outside of human rationality. Depending on infinity to find a precise answer (whether algebraically or using the short-cuts of calculus) depends on something outside of human rationality. I only decided to add (ir)rational numbers into the mix because I was getting the argument that results such as 1 or 1/9 are rational numbers. While thats true, I'm more focused on the solutions that provide these results, not the results themselves. I assert that the solutions require one to wander outside of human rationality. I'm not saying that these solutions are wrong, invalid, useless, or even something we should avoid. I'm just saying that it does step outside of rationality. Its interesting to see so much resistence to this claim. Mike === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Arturo Magidin >> >In order to come to the answer of 1/9, we have to be at infinity, >> No, we do not (or rather, we do not have to do any such thing). The >> expression >> Sum{i=1 to infinity}( 1/(10^i) ) = 1/9 >> means nothing more and nothing less than: >> For every epsilon>0, there exists a positive integer N>0 such that, >> for all n>=N, >> |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon. >> We express this by ->saying<- that the infinite sum adds up to >> 1/9. But we are not at infinity in any sense. >I agree. >So here's my issue. If you are answering the question of the bouncing >ball with the terminology there exists no postive integer... ect. >then you are not really answering the question that was asked. Yes, you are. The answer may not be a number, but you are giving a correct mathematical answer. Just because you don't ->like<- the answer, or you would like the answer to be something else, does not mean you are not answering the question that was asked. Note that the original question makes assumptions that make it into a question about a mathematical model of something, not about a real world actual event. There is no ball that can meet the requirements of the problem. So a mathematical answer that may not have any real world counterpart is hardly to be objected to. >So either: calculus isn't allowed to answer the question directly, Calculus can answer the question directly. The phrasing of the question places it squarely within the reach of calculus, and outside the real world. >it answers it irrationally (but still in a useful and valid way). Is >my logic on that mistaken? Yes, it is. As has been pointed out many, many, many times already. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? >David C. Ullrich >> You're not so much mistaken as making no sense at all. >> You keep talking about this irrationality, without ever >> explaining exactly what you mean. >> Q: What is the sum of 1/2^j, j = 1..infinity? >> A: 1. >> Exactly how was irrationality required to present >> that solution? >In order to sum all from 1 to infinity, one would have to reach >infinity to get the precise answer. It's truly amazing how you keep repeating the same errors. People point out why what you say is not so, you _agree_, and then you say it again. Write this down somewhere: No, in order to find the exact value of sum 1/2^j, j = 1..infinity we do _not_ have to reach infinity. _By definition_, if we find a number s with the property that for every eps > 0 there exists N such that |s - sum 1/2^j, j = 1..n| < eps for all n > N then we have found sum 1/2^j, j = 1..infinity exactly. Without ever reaching infinity. Words in math mean what the definitions _say_ they mean, not what it seems like they _should_ mean on the basis of the way the same words are used in non-mathematical English. I have found sum 1/2^j, j = 1..infinity, exactly, without ever reaching infinity. > I'm making the assumption that for >a human to reach infinity is an irrational suggestion because infinity >lies outside the limits of human rationality. You're also making the assumption that reaching infinity has something to do with calculus. It does not. You can continue to say it does - it still doesn't. You can continue to tell people they're missing your point - that doesn't change the fact that the things you're saying are nonsense, and it doesn't change the fact that _you_ are missing important points (the fact that you're missing something is shown by your continued belief that finding limits has something to do with reaching infinity. It doesn't.) >If that is the case, even though we can use valid and effective tools >like Calculus to determine what is happening as we approach infinity, >despite its effectiveness, it is still adding an element of >irrationality into the solution. >By irratoinal I am not implying that Calculus is wrong or useless. >Thats not my intention or opinion at all. But to find its answers, it >does rely on stepping outside the realm of human rationality. Of >course, this gives us a whole new outlook on what problems we can >solve and allows us to apply those solutions effectively in the real >world, but it shouldn't be too hard too admit that those solutions >cannot be regarded as absolutely rational. It would be very easy to admit that. But there's no reason to admit it because it's simply not so. The fact that _you_ don't understand this doesn't mean it's false. >I realize that Calculus does not literally sum to infinity with >algebra, but the idea is essentially the same. Calculus is an >effective short cut for this process. >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? >The World Wide Wade >> An irrational number is, by definition, any *real* number that is not >> rational. Infinity, whatever it is, is not a real number. So your use of >> irrational here is invalid. >However, I can make my point without relying on rational or irrational >numbers at all, and stick strictly to the argument of what is and what >is not within human rationality. Infinity lies outside of human >rationality. Depending on infinity to find a precise answer (whether >algebraically or using the short-cuts of calculus) depends on >something outside of human rationality. >I only decided to add (ir)rational numbers into the mix because I was >getting the argument that results such as 1 or 1/9 are rational >numbers. While thats true, I'm more focused on the solutions that >provide these results, not the results themselves. >I assert that the solutions require one to wander outside of human >rationality. The key word being assert - you've given no evidence. Or rather no valid evidence. >I'm not saying that these solutions are wrong, invalid, >useless, or even something we should avoid. I'm just saying that it >does step outside of rationality. Its interesting to see so much >resistence to this claim. What you've been saying is simply nonsense. People have been trying to explain why - there have been enough perfectly clear explanations that people are beginning to wonder whether you really don't get it or you're just typing for the fun of seeing people reply. If you think it's interesting when your behavior makes people wonder whether you're simply stupid, well good for you. >Mike ************************ David C. Ullrich === Subject: Re: Calculus is irrational? > ... Infinity lies outside of human > rationality. Why so? Infinity crops up a lot in mathematics. -- G.C. === Subject: Re: Calculus is irrational? Arturo Magidin > REMARK. It should be emphasized that the simbols +infinity and > -infinity are NOT [emphasis in the original] real numbers. The > phrase f(x) approaches +infinity is akin to saying that f(x) > approaches the unapproachable; it is a colloquialism for f(x) > increases without bound. Is the phrase increases without bound itself a colloquialism? If not, it would imply that something would have to be increasing. Even if increasing beyond our rational limits, its still must be a quantity of some type to be increased. > Mathworld is not always a good source of information. What you need to > understand is that he is giving a colloquial meaning of the symbol, > not a formal one Mike === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Arturo Magidin >> REMARK. It should be emphasized that the simbols +infinity and >> -infinity are NOT [emphasis in the original] real numbers. The >> phrase f(x) approaches +infinity is akin to saying that f(x) >> approaches the unapproachable; it is a colloquialism for f(x) >> increases without bound. >Is the phrase increases without bound itself a colloquialism? f(x) increases without bound is a term of art, a technical phrase with a precisely defined mathematical meaning. In this context, it means that for any M>0, there exists a delta>0 such that for all x such that 0<|x-x_0|M. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? David W. Cantrell > > Well, the way I interpret it a number would imply a specific quantity. > > Reasonable? > Yes. But I don't know what your point is. I'm not making a point here but defending myself. It was said that my reasoning does not hold because I said infinity is a number, so I clarified that I did not state infinity was a number. I stated that it is a quantity, even though not a specific quantity. > And I'm curious about your use > of the word specific above. Are you thinking that something could be > _a_ quantity without being a _specific_ quantity? If so, please give an > example. The only example I can think of at the moment would be infintiy. === Subject: Re: Calculus is irrational? Arturo > >> Sum{i=1 to infinity}( 1/(10^i) ) = 1/9 > >> means nothing more and nothing less than: > >> For every epsilon>0, there exists a positive integer N>0 such that, > >> for all n>=N, > >> |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon. > >> We express this by ->saying<- that the infinite sum adds up to > >> 1/9. But we are not at infinity in any sense. > >I agree. > >So here's my issue. If you are answering the question of the bouncing > >ball with the terminology there exists no postive integer... ect. > >then you are not really answering the question that was asked. > Yes, you are. The answer may not be a number, but you are giving a > correct mathematical answer. Just because you don't ->like<- the > answer, or you would like the answer to be something else, does not > mean you are not answering the question that was asked. Q: When does the ball stop? My A: 1/9 Your A: For every epsilon>0, there exists a positive integer N>0 such that, for all n>=N: |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon You think your answer adequately answers the question asked? If so, then that is where our disagreement lies. And you don't think it is at all shady to change the answer from 1/9 to for evenr eps>0 ect.ect. for the sake of maintaining rationality? I realize your answer is a true mathematical statement, but for it to have any practical meaning in the context of the question asked, you're going to need to make the leap to 1/9. Personally, I would rather assume the irrationality to come up with the answer that is actually useful. This is why I have been trying to make it clear that irrationality <> bad. Mike Helland === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >David W. Cantrell >> > Well, the way I interpret it a number would imply a specific quantity. >> > Reasonable? >> Yes. But I don't know what your point is. >I'm not making a point here but defending myself. It was said that my >reasoning does not hold because I said infinity is a number, so I >clarified that I did not state infinity was a number. I stated that it >is a quantity, even though not a specific quantity. Which does not really make sense. If it is a quantity, then surely it is a ->specific<- one? Look: infinity can be made very precise. If you do that, you can work with it and it would constitute, according to your classification, a rational concept. This is the case when you work on the Riemman Sphere (a one point compactification of the Complex plane) for example, where it is a very specific point in a manifold, and going to infinity also has a very precise, specific, topological meaning, and where it ->does<- make sense to state that you are at infinity or evaluating a holomorphic function at infinity and so on. But that requires a precise definition. Or you can define infinite as a very precise adjective with a very precise meaning in formal set theory: a set is infinite if and only if it can be put in one-to-one bijective correspondence with a proper subset of itself. And then you can define an infinite collection of different infinities, which again have a precise, formal, meaning (and so constitute a rational concept in your terminology). However, in ->calculus<- there is simply NO NEED to do so. One ->can<- define infinity in a very precise formal way using non-standard analysis, but then, again, you are talking about a very specific, well-defined, well-understood object that you can handle. In most cases, however, calculus is presented without having to appeal to a concept of infinity except in colloquial expressions or formal definitions, where it appears as part of a term of art. Things like when x tends to infinity, or the limit equals infinity and so on. But these are terms of art, which are given precise, formal, definitions that do not in any way invoke an undefined or defined concept of infinity. The word is just used as part of shorthands for precise explicit expressions. As for the cases where it is given a precise definition, experience tells us that the concept tends to behave in ways that seem ->non-intuitive<-, but that does not make them irrational (in your nomenclature). Perhaps that is where all of this is coming from, in that the ->formal<- concepts of infinity tend to be nonintuitive in their behavior, and you have made the jump into considering them to be 'non-rational' because of that? >> And I'm curious about your use >> of the word specific above. Are you thinking that something could be >> _a_ quantity without being a _specific_ quantity? If so, please give an >> example. >The only example I can think of at the moment would be infintiy. Which is not a quantity in the context of standard calculus. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? > Hi All, > If calculus assumes infinity to come to its answers (for example, the > limit of a function, we sum to infinity to find an answer) > and because infinity is irrational (infinity being defined by p/0 and > rational number defined by p/q where q <> 0) > is it fair to say that any answer given to us by calculus is by > definition irrational as it assumes irrationality in the solution? > I'm not saying that irrational is equal to bad or wrong or > useless because it is obviouslly none of those things, but am I > right in thinking that all solutions that require calculus are > mathematically defined as irrational? > Mike Helland I see your point. I have been taught that the real numbers form a field. But I was totally unsuccessful when I tried grazing sheep on them. Likewise, I find that normal subgroups are very strange, and complex numbers are really rather simple. -- Stephen Montgomery-Smith stephen@math.missouri.edu http://www.math.missouri.edu/~stephen === Subject: Re: Calculus is irrational? > > Hi All, > > > > If calculus assumes infinity to come to its answers (for example, the > > limit of a function, we sum to infinity to find an answer) > > > > and because infinity is irrational (infinity being defined by p/0 and > > rational number defined by p/q where q <> 0) > > > > is it fair to say that any answer given to us by calculus is by > > definition irrational as it assumes irrationality in the solution? > > > > I'm not saying that irrational is equal to bad or wrong or > > useless because it is obviouslly none of those things, but am I > > right in thinking that all solutions that require calculus are > > mathematically defined as irrational? > > > > Mike Helland > I see your point. I have been taught that the real numbers form a field. > But I > was totally unsuccessful when I tried grazing sheep on them. You've never seen the grazing goats problem? === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Arturo >> >> Sum{i=1 to infinity}( 1/(10^i) ) = 1/9 >> >> means nothing more and nothing less than: >> >> For every epsilon>0, there exists a positive integer N>0 such that, >> >> for all n>=N, >> >> |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon. >> >> We express this by ->saying<- that the infinite sum adds up to >> >> 1/9. But we are not at infinity in any sense. >> > >I agree. >> > >So here's my issue. If you are answering the question of the bouncing >> >ball with the terminology there exists no postive integer... ect. >> >then you are not really answering the question that was asked. >> Yes, you are. The answer may not be a number, but you are giving a >> correct mathematical answer. Just because you don't ->like<- the >> answer, or you would like the answer to be something else, does not >> mean you are not answering the question that was asked. >Q: When does the ball stop? >My A: 1/9 >Your A: For every epsilon>0, there exists a positive integer N>0 such >that, for all n>=N: |Sum{i=1 to n} (1/(10^i)) - 1/9| < epsilon >You think your answer adequately answers the question asked? Yes. My answer is really that at any position short of 1/9, the ball continues to bounce, while at any position strictly larger than 1/9, the ball has already stopped. Of course, there is no real ball that is bouncing, because no ball can satisfy the hypothesis of your question. You are presenting a MATHEMATICAL ABSTRACTION, and you are receiving a mathematical abstraction as the answer. Just because you can phrase the question in terms that ->sound<- concrete does not make them any less an abstraction. >If so, then that is where our disagreement lies. And you don't think >it is at all shady to change the answer from 1/9 to for evenr >eps>0 ect.ect. for the sake of maintaining rationality? Not at all. I do think it is a bit unrealistic (perhaps even shady) of you to treat the question as if it were about an actual ball when it is not, and then complain if the answer cannot be realized in actuality. >I realize your answer is a true mathematical statement, but for it to >have any practical meaning in the context of the question asked, >you're going to need to make the leap to 1/9. There is no practical meaning because the question is not practical to begin with. You are phrasing it as a question about a ball, but it can have no referent in the real world. Like I said, your question is not a ->practical<- question, it is a question about a mathematical ->abstraction<-. The leap is not in the answer, the leap you are making is in taking a question about a mathematical abstraction and thinking that it is identical to a real world situation. >Personally, I would rather assume the irrationality to come up with >the answer that is actually useful. The point is that there is no need to assume any irrationality in the sense you are using the word. Just because YOU seem to be having trouble wrapping your mind about a concept does not place it beyond the reach of others. In this case, you have taken a real world situation (a ball bouncing). Then YOU made the leap to thinking that you can accurately and perfectly describe the real world behavior of the bouncing ball through a particular mathematical model that does not in reality describe the real world situation, but only an approximation of an idealized version thereof. Then you solve the problem posed by that mathematical model in mathematical terms. Then YOU made the leap to thinking that this answer must be identical to something which accurately describes a real world situation. There are indeed leaps, but those leaps are occurring behind the scenes and not in the question or answer at all. If you understand going in that your question is merely an approximation of a real world answer, then you will understand as well that the answer you get need not be an answer that can be simply interpreted as a statement about the real world. The question you ask is about a mathematical MODEL, and the answer you get is an answer about MODEL, not about the real world situation that the model approximates. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? > Personally, I would rather assume the > irrationality to come up with the answer > that is actually useful. This is why I have > been trying to make it clear that > irrationality <> bad. This is not philosophy. Someone, as ignorant about mathematics as you are, does not get to assume and/or choose what one will or will not accept. First, one must learn a little about mathematics terminology and about mathematics itself. ;-) === Subject: Re: Calculus is irrational? Arturo Magidin > Calculus can answer the question directly. The phrasing of the > question places it squarely within the reach of calculus, and outside > the real world. I missed this the first time. What do you mean by outside the real world? My defintion of the real world would be everything within our rationality. Outside the real world would be outside our rationality. It sounds like we're in exact agreement, are we not? Mike Helland === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. Cc: >Arturo Magidin >> Calculus can answer the question directly. The phrasing of the >> question places it squarely within the reach of calculus, and outside >> the real world. >I missed this the first time. What do you mean by outside the real >world? The question assumes that we have a ball that bounces without end; that is, after any finite number of bounces, the ball still continues to bounce. If you can produce me a ball that satisfies these hypothesis, I will be very impressed. So will the physicists, since you will have overthrown the second law of thermodynamics. >My defintion of the real world would be everything within our >rationality. Your definition of the real world includes a lot of things that most people would exclude. For example, unicorns. > Outside the real world would be outside our rationality. >It sounds like we're in exact agreement, are we not? No. I disagree with your definition of rationality; I disagree with your definition of real world; and I disagree with your application of these labels, even assuming your definitions, in the way you have applied them. It seems like we are hardly in agreement. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Chaos, quadratic and logistic maps I have posted some novel results about solutions to the quadratic and logistic maps. Not completely general, and as Eric Weisstein pointed out, known to Mathematica, but interesting nonetheless: http://homepage.ntlworld.com/little_mm More mathematical miscellany probably to come, as I get more time! Max Little === Subject: Re: Cyclotomic polynomials with one negative coefficient >Is it true that if the n-th cyclotomic polynomial has exactly one >negative coefficient then n is of the form 2^i*3^j ? It's true at least for n up to 2000. If C_n is the n'th cyclotomic polynomial, it seems that when n=2^i*3^j for i,j>0, C_n(t) = t^(n/3) - t^(n/6) + 1. Well, that should be provable by induction, together with C_n(t) = t^(n/2)+1 if n is a positive power of 2 and C_n(t) = t^(2n/3) + t^(n/3) + 1 if n is a positive power of 3. I don't know how to prove that these are the only cases where C_n has one negative coefficient, though. Of course since the zeros of C_n are symmetric under t -> 1/t, the coefficients satisfy a_j = a_{d-j} where d is the degree, and if you have one negative coefficient it can only be a_{d/2}. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Cyclotomic polynomials with one negative coefficient > >Is it true that if the n-th cyclotomic polynomial has exactly one > >negative coefficient then n is of the form 2^i*3^j ? > It's true at least for n up to 2000. > If C_n is the n'th cyclotomic polynomial, it seems that > when n=2^i*3^j for i,j>0, C_n(t) = t^(n/3) - t^(n/6) + 1. Well, > that should be provable by induction, together with > C_n(t) = t^(n/2)+1 if n is a positive power of 2 and > C_n(t) = t^(2n/3) + t^(n/3) + 1 if n is a positive power of 3. > I don't know how to prove that these are the only cases where C_n has one > negative coefficient, though. > Of course since the zeros of C_n are symmetric under t -> 1/t, > the coefficients satisfy a_j = a_{d-j} where d is the degree, > and if you have one negative coefficient it can only be a_{d/2}. > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 More generally it seems that given integer k>=1 , there is a finite set of primes p_1,p_2,...,p_m such that if n-th cyclotomic polynomial has exactly k negative (or positive) coefficients then the prime factors of n belong to the set p_1,p_2,...,p_m . Bill === Subject: Re: Dedekind Cuts >>I'm trying to understand what is described by Parzynski and Zipse, in >>Introduction to Mathematical Analysis as Dedekind's Theorem - R is >>Complete. They describe real numbers as rays (non-empty proper subsets >>of the rationals having no first element and any y greater than an >>element of the ray is also an element of the same ray). >>The proof starts with defining A' = R - A, where A is any ray in R.... >>What is this?.... > I learned these things from the discursive but beautiful explanation > in G.H. Hardy's classic A Course of Pure Mathematics. I bought the book today. I see what you mean. I never before thought of a mathematical explanation as beautiful. Tom Adams === Subject: Re: Dedekind Cuts >>I'm trying to understand what is described by Parzynski and Zipse, in >>Introduction to Mathematical Analysis as Dedekind's Theorem - R is >>Complete. They describe real numbers as rays (non-empty proper subsets >>of the rationals having no first element and any y greater than an >>element of the ray is also an element of the same ray). >>The proof starts with defining A' = R - A, where A is any ray in R. >>Also U = {x (element of) Q | x>W for every W (element of) A'} >>What is this? If R is all the rays then - A removes only one ray >>leaving every ray greater or less than A. This obviously is not the intent. >>The authors then speak of V (element of) A; r (element of) U; and V >>(subset of) U. It seems V can sometimes be an element of a ray; >>sometimes a ray. The lower case r is probably a simple rational, i.e. >>an element of a ray. >>Can someone help me with this notation? > It's hard to explain the notation without seeing it in context. > But showing that the reals are complete, after defining a > real to be a ray as above, is exquisitely simple: > Suppose that S is a nonempty subset of R and S has an > upper bound. We need to show that S has a least upper bound. What you say next is an exceptionally clear explanation of why S has a least upper bound. But let me see if I understand its relevance to completeness. I may develop the power to fill S with rays using some rule that is so esoteric it has yet to be discovered. Wishing to make things difficult, I will not let S have a last element (I can do that?). Still, using the union of rays you just described, I can always associate S with a unique ray: a real number that is the least upper bound of S. This is kind of like defining completeness in your metric space by requiring the limit of every cauchy sequence to be an element of the space. I think we then conclude that one cannot use rays to define a new entity (a new type of number) that is different than a ray. Of course, new types of numbers are possible once we start defining complex numbers. But I think rays are complete in the sense that they cannot be aggregated in ways that will define new numeric types -- not because we currently lack the imagination to build the new numeric types, but because building new numeric types by aggregation is impossible. Is that completeness? Tom Adams === Subject: Re: Dedekind Cuts > I think we then conclude that one cannot use rays to define a new entity > (a new type of number) that is different than a ray. Of course, new > types of numbers are possible once we start defining complex numbers. > But I think rays are complete in the sense that they cannot be > aggregated in ways that will define new numeric types -- not because we > currently lack the imagination to build the new numeric types, but > because building new numeric types by aggregation is impossible. I'm not sure that I follow your argument, but Conway's definition of surreal numbers in effect generalises Dedekind sections, and gives rise to new numeric types. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Dedekind Cuts I'm trying to understand what is described by Parzynski and Zipse, in Introduction to Mathematical Analysis as Dedekind's Theorem - R is Complete. They describe real numbers as rays (non-empty proper subsets of the rationals having no first element and any y greater than an element of the ray is also an element of the same ray). The proof starts with defining A' = R - A, where A is any ray in R. Also U = {x (element of) Q | x>W for every W (element of) A'} What is this? If R is all the rays then - A removes only one ray leaving every ray greater or less than A. This obviously is not the intent. The authors then speak of V (element of) A; r (element of) U; and V (subset of) U. It seems V can sometimes be an element of a ray; sometimes a ray. The lower case r is probably a simple rational, i.e. an element of a ray. Can someone help me with this notation? >> It's hard to explain the notation without seeing it in context. >> But showing that the reals are complete, after defining a >> real to be a ray as above, is exquisitely simple: >> Suppose that S is a nonempty subset of R and S has an >> upper bound. We need to show that S has a least upper bound. >What you say next is an exceptionally clear explanation of why S has a >least upper bound. But let me see if I understand its relevance to >completeness. >I may develop the power to fill S with rays using some rule that is so >esoteric it has yet to be discovered. Wishing to make things difficult, > I will not let S have a last element (I can do that?). Still, using >the union of rays you just described, I can always associate S with a >unique ray: a real number that is the least upper bound of S. This is >kind of like defining completeness in your metric space by requiring the >limit of every cauchy sequence to be an element of the space. >I think we then conclude that one cannot use rays to define a new entity >(a new type of number) that is different than a ray. Of course, new >types of numbers are possible once we start defining complex numbers. >But I think rays are complete in the sense that they cannot be >aggregated in ways that will define new numeric types -- not because we >currently lack the imagination to build the new numeric types, but >because building new numeric types by aggregation is impossible. >Is that completeness? ??? When you ask a question about a proof that the reals are complete one naturally assumes that you already know what it _means_ to say the reals are complete - if you don't know that then the _first_ question you should be asking is what it means, not questions about the meaning of the notation in the proof! There is presumably a definition of R is complete somewhere in the book you're looking at. That definition may look like one of the following: (i) Any (nonempty) subset of R which is bounded above has a least upper bound. (ii) Every Cauchy sequence in R converges to some element of R. It's not hard to show that those two statements are equivalent; either one is what is meant when one says that R is complete. The version I gave a proof of was (i), because that's what these Dedekind cut things are naturally adapted to do; if you'd defined the real numbers in some other way then (ii) might be more natural (in particular it's possible to define the reals as equivalence classes of Cauchy sequences of rationals, wrt a certain equivalence relation; when you do that then (ii) is the notion of completeness that falls out of the definition.) Hints for showing that (i) and (ii) are equivalent: (i) -> (ii): Assume (i), and say (x_n) is a Cauchy sequence of reals. Show that (x_n) has a monotone subsequence, (might be weakly increasing or weakly decreasing) - this is probably the trickiest step. Say (x_n) has an increasing subseqence. Show the sequence is bounded above. Let x be the least upper bound for the set of elements of that subsequence, and show that x_n -> x. (Similarly if (x_n) has a decreasing subsequence). (ii) -> (i): Suppose (ii), and suppose that S is bounded above. Since any set of integers which is bounded below contains a smallest element, you can let x_1 be the smallest integer which is >= every element of S. Let x_2 be the smallest number of the form k/2 which is <= every element of S. _Note_ that the minimiality of x_1 shows that x_1 - 1 < x_2 <= x_1, which implies that |x_2 - x_1| < 1. Etc: Let x_n be the smallest number of the form k/2^n which is >= every element of S, and show that |x_{n+1} - x_n| < 2^(n-1). It follows that (x_n) is Cauchy; say x_n -> x and show that x is a least upper bound for S. >Tom Adams ************************ David C. Ullrich === Subject: Re: Dedekind Cuts >>I think we then conclude that one cannot use rays to define a new entity >>(a new type of number) that is different than a ray. Of course, new >>types of numbers are possible once we start defining complex numbers. >>But I think rays are complete in the sense that they cannot be >>aggregated in ways that will define new numeric types -- not because we >>currently lack the imagination to build the new numeric types, but >>because building new numeric types by aggregation is impossible. > I'm not sure that I follow your argument, > but Conway's definition of surreal numbers > in effect generalises Dedekind sections, > and gives rise to new numeric types. If I understand Conway's surreal numbers, their defintion requires two disjoint sets of real numbers, not necessarily contiguous. If my argument has some validity, I would say surreal numbers are not formed by aggregation of reals into a single set; instead, two independent sets are needed. In this way surreals seem to share a two dimensionality with complex numbers. (I'm curious: If surreals are yet another number type, do they have a different ordinal number and can they be mapped one-one to the real number line? ) But please understand, I'm not trying to defend my defintion of completeness, I'm trying to understand the conventional argument that completeness flows from a proof that an arbitrary set of reals with an upper bound has a l.u.b. Tom Adams === Subject: Re: Dedekind Cuts |If I understand Conway's surreal numbers, their defintion requires two |disjoint sets of real numbers, not necessarily contiguous. No, they are formed from a left set and a right set, whose elements are surreals, not necessarily reals. [...] |(I'm curious: If surreals are yet another number |type, do they have a different ordinal number and can they be mapped |one-one to the real number line? ) I don't know what you mean by the ordinal number of the reals. The surreals are not a set. They are too big to be a set, just like all sets or all ordinals. There are sets of surreals of arbitrarily large cardinality. In particular, there isn't a one-to-one correspondence between them and the real line. The ordinals embed in the surreals by letting each ordinal b correspond to the surreal with an empty right set, and a left set containing the surreals corresponding to all the ordinals < b. The opposite construction, putting elements on the right, defines negative ordinals. Keith Ramsay === Subject: Re: Dedekind Cuts >>I'm trying to understand what is described by Parzynski and Zipse, in >>Introduction to Mathematical Analysis as Dedekind's Theorem - R is >>Complete. They describe real numbers as rays (non-empty proper subsets >>of the rationals having no first element and any y greater than an >>element of the ray is also an element of the same ray). >>The proof starts with defining A' = R - A, where A is any ray in R.... >>What is this?.... > I learned these things from the discursive but beautiful explanation > in G.H. Hardy's classic A Course of Pure Mathematics. In Section 17 > (p.28 of the 10th edition) he recapitulates the definition of Dedekind > cuts of the rationals, then points out how the *reals* can be cut in a > similar way. This gives what are sometimes called Dedekind sections of > the reals, to distinguish them from Dedekind cuts of the rationals. > (Ultimately they are pairs of sets of sets of rationals.) This leads to > Dedekind's Theorem (p.30) which says that any such section always > determines a real number such that everything less than it is in the > left-hand class and everything greater than it is in the right-hand class. > The idea is that cuts of the rationals give you something new (all > the reals), but repeating the construction by cutting the reals gives > nothing new. It's one way of seeing the order-completeness of R. > HTH > Ken Pledger. I may be the only one interested in my misinterpretation of Parzynski and Zipse's book, but Ken's mention of Hardy's description of Dedekind sections of the reals (not the rationals) was the clue I needed. My authors were indeed discussing Dedekind sections of the reals. They proved the set A (the section of the reals) was after all, a cut in the rationals (a ray). They then went on to show R - A had a first element, the completeness missing in the cuts in the rationals. V e r y, v e r y cool! They followed up with a proof of the equivalent, least upper bound property, but not as efficiently as David Ullrich. They left me the exercise of showing the two properties are equivalent. Wish me luck. Tom Adams === Subject: Re: Dedekind Cuts > I may be the only one interested in my misinterpretation of Parzynski > and Zipse's book, but Ken's mention of Hardy's description of Dedekind > sections of the reals (not the rationals) was the clue I needed. My > authors were indeed discussing Dedekind sections of the reals. They > proved the set A (the section of the reals) was after all, a cut in the > rationals (a ray). They then went on to show R - A had a first > element, the completeness missing in the cuts in the rationals. Oops! R - A had the greatest element. === Subject: Re: Equidistantly distributed lattice points > I think the keyphrase the original poster is looking for is > Hadamard matrix. As I've looked it up, the Hadamard matrix is indeed a solution to my problem, but the set S that I defined above seems to be much larger, since it doesn't require the components of the matrix to be +/-1. (In fact, S contains odd squares > 1, while there are no Hadamard matrices of those dimensions.) Anyway, thank you again. Tad === Subject: Re: factors of a choice coefficient >> Is there some known easy formula for ord_p( p^n choose a), where >> 00? a choose b is the familiar a!/(b!(a-b)!) > Well, there's Kummer's result that ord_p(C(n+k,k)) = the > number of carries generated when you add the base-p > representations of n, k. You might be able to exploit > the simplicity of the representation of p^n in base-p > to get something usable ... Sure. Adding p^n-a and a in base p generates n-ord_p(a) carries if 1 <= a <= p^n. So this says ord_p (p^n choose a) = n - ord_p(a). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Field of rationals and pi Euler found out that he could define pi^2 in the following nifty way: pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... Which means you can define it using members of the *field* of rationals. However, pi is transcendant and is itself not a rational. I'm curious about the rule mathematicians use to exclude pi^2/6 from the field of rationals, as it itself is the result of an infinite sum of members of that field. Is that it? Mathematicians simply exclude infinite sums from the field of rationals? Or do they rely on the definition of a rational as the ratio of a/b, where 'a' and 'b' are integers? Continuing in that direction, recently a leading mathematician at a major university in the United States of America (a top 20 university) sent me an email stating that my rule of no other integers being units except -1 and 1 did not exclude pi if you used Z[pi]. I said it did in the following reply (Professor's name omitted): Professor ****: You assertion is easily proven false. Please consider the following. infinity. But then you have pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + 1/9+...)+...1/k^2(1+k^2), which is pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, multiplying out and collecting to the left except for 1, 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - 6(24)...(6k^2) - ... -6(24)(54)...] = 1, which proves that you have an infinite number of units, some of which are 6, 24, and 54, which is the result if you include pi in a ring with integers, so my definition *does* exclude it. Well he replied: Actually, Z[pi] has no units save 1, -1, as a consequence of the fact that pi is transcenddental (not algebraic). Z[pi] does not contain the numbers you are considering above, which (as I read it) are obtained by summing infinite series. The elements of Z[pi] are just those real numbers that can be expressed as f(pi) where f(x) is a FINITE polynomial with integer coefficients. Do you agree with the professor, who I remind is a *leading* mathematician? James Harris === Subject: Re: Field of rationals and pi > Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > Which means you can define it using members of the *field* of > rationals. However, pi is transcendant and is itself not a rational. Another way to see that pi is an infinite sum of rational numbers is to write it like this: pi = 3 + 1/10 + 4/100 + 1/1000 + 5/10000 + . . . In this way you can see that EVERY real number is an infinite sum of rational numbers. For example 1/3 = 3/10 + 3/100 + 3/1000 + . . . However note that even though (as you point out) pi is transcendental and not rational, those two terms are not mutually exclusive. For example sqrt(2) is neither rational nor transcencental. > I'm curious about the rule mathematicians use to exclude pi^2/6 from > the field of rationals, as it itself is the result of an infinite sum > of members of that field. > Is that it? Mathematicians simply exclude infinite sums from the > field of rationals? Or do they rely on the definition of a rational > as the ratio of a/b, where 'a' and 'b' are integers? A rational number is a real number that can be expressed as the ratio of two integers. It can be proved that pi can not be so expressed. As I just noted, every real can be expressed as an infinite sum of rationals. So that property doesn't distinguish the rationals in any way from the rest of the reals. === Subject: Re: Field of rationals and pi > > Euler found out that he could define pi^2 in the following nifty way: > > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > > Which means you can define it using members of the *field* of > > rationals. However, pi is transcendant and is itself not a rational. > Another way to see that pi is an infinite sum of rational numbers is to > write it like this: > pi = 3 + 1/10 + 4/100 + 1/1000 + 5/10000 + . . . > In this way you can see that EVERY real number is an infinite sum of > rational numbers. For example 1/3 = 3/10 + 3/100 + 3/1000 + . . . > However note that even though (as you point out) pi is transcendental > and not rational, those two terms are not mutually exclusive. For > example sqrt(2) is neither rational nor transcencental. That means that rational and transcendental are not collectively exhaustive. They are still mutually exclusive. > > I'm curious about the rule mathematicians use to exclude pi^2/6 from > > the field of rationals, as it itself is the result of an infinite sum > > of members of that field. > > Is that it? Mathematicians simply exclude infinite sums from the > > field of rationals? Or do they rely on the definition of a rational > > as the ratio of a/b, where 'a' and 'b' are integers? > A rational number is a real number that can be expressed as the ratio of > two integers. It can be proved that pi can not be so expressed. > As I just noted, every real can be expressed as an infinite sum of > rationals. So that property doesn't distinguish the rationals in any > way from the rest of the reals. > -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Field of rationals and pi >Euler found out that he could define pi^2 in the following nifty way: >pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... >Which means you can define it using members of the *field* of >rationals. However, pi is transcendant and is itself not a rational. >I'm curious about the rule mathematicians use to exclude pi^2/6 from >the field of rationals, as it itself is the result of an infinite sum >of members of that field. >Is that it? Is what what? >Mathematicians simply exclude infinite sums from the >field of rationals? Infinite sums are not excluded from the rationals; every rational number _is_ an infinite sum. But there's no rule that says an infinte sum of rationals has to be rational, which is possibly what you meant. >Or do they rely on the definition of a rational >as the ratio of a/b, where 'a' and 'b' are integers? Saying they rely on the definition is a strange way to put it. That _is_ the definition, so that _is_ what a rational number _is_. By definition. >Continuing in that direction, recently a leading mathematician at a >major university in the United States of America (a top 20 >university) sent me an email stating that my rule of no other integers >being units except -1 and 1 did not exclude pi if you used Z[pi]. >I said it did in the following reply (Professor's name omitted): >Professor ****: >You assertion is easily proven false. Please consider the following. >infinity. >But then you have > pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + >1/9+...)+...1/k^2(1+k^2), which is > pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, >multiplying out and collecting to the left except for 1, > 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - >6(24)...(6k^2) - ... -6(24)(54)...] = 1, >which proves that you have an infinite number of units, some of which >are 6, 24, and 54, which is the result if you include pi in a ring >with integers, so my definition *does* exclude it. No. It proves that there are an infinite number of units in a certain ring, but that ring is _not_ Z[pi]. Because Z[pi] is what the definition below says it is - those infinite sums are not elements of Z[pi]. What ring does it show has lots of units? It seems to be the ring you get if you take all _infinite sums_ of elements of Z[pi]. That ring is actually R, the real numbers. So (assuming you did the algebra right, I haven't checked) you've shown that 6, 24, etc are units in the real numbers. In other words, that 1/6, 1/24 are real numbers. Congratulations. (There are easier ways, btw.) >Well he replied: >Actually, Z[pi] has no units save 1, -1, as a consequence of >the fact that pi is transcenddental (not algebraic). Z[pi] does not >contain the numbers you are considering above, which (as I read it) >are obtained by summing infinite series. The elements of >Z[pi] are just those real numbers that can be expressed as >f(pi) where f(x) is a FINITE polynomial with integer coefficients. >Do you agree with the professor, who I remind is a *leading* >mathematician? Well duh, of course we do. Not because he's a leading mathematician, but because he's telling the truth. Look up the definition of polynomial somewhere. A polynomial _is_ a finite polynomial, by definition. >James Harris ************************ David C. Ullrich === Subject: Re: Field of rationals and pi > Infinite sums are not excluded from the rationals; > every rational number _is_ an infinite sum. But there's > no rule that says an infinte sum of rationals has to be > rational, which is possibly what you meant. There's a subtle point of confusion I hope you can clarify for me. In the previous post by Arturo Magidin, he said (I think) that infinite sums *are* excluded from the rationals. But evidently he was referring to the operation of taking infinite sums, rather than the value of the sum itself. Certainly any rational number *can* be expressed as an infinite sum, whereas irrational and transcendental numbers may *only* be expressed that way. Could you be a little more specific about what constitutes a valid -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Field of rationals and pi Visiting Assistant Professor at the University of Montana. >> Infinite sums are not excluded from the rationals; >> every rational number _is_ an infinite sum. But there's >> no rule that says an infinte sum of rationals has to be >> rational, which is possibly what you meant. >There's a subtle point of confusion I hope you can clarify for me. In the >previous post by Arturo Magidin, he said (I think) that infinite sums *are* >excluded from the rationals. If I said that, then I was certainly being imprecise. I meant to say that, from the point of view of the ring structure of the rationals, infinite sums are excluded; that is, a ring structure on a set does not, in and of itself, give you a notion of infinite sums or infinite products. Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Field of rationals and pi > Yes, he is correct. And it does not matter if he is a *leading* > mathematician or not. Better to be a correct mathematician than to be a leading mathematician ;-) Gib === Subject: Re: Field of rationals and pi >There's a subtle point of confusion I hope you can clarify for me. In the >previous post by Arturo Magidin, he said (I think) that infinite sums *are* >excluded from the rationals. But evidently he was referring to the >operation of taking infinite sums, rather than the value of the sum itself. >Certainly any rational number *can* be expressed as an infinite sum, >whereas irrational and transcendental numbers may *only* be expressed that >way. Infinite sums are excluded only in the same way that square roots are excluded - that is, the field axioms don't ensure that that a field is closed under that operation. The term infinite sum is really a metaphor: shorthand for the limit of the sequence of partial sums. You can't therefore assume automatically that anything true about finite sums is also true about infinite ones. Of course, if the rationals *were* closed under (convergent) infinite sums, they would be the reals! -- Richard -- Spam filter: to mail me from a .com/.net site, put my surname in the headers. FreeBSD rules! === Subject: Re: finite groups BTW, do you have an easy way to check if PGL(2,9) or M_{10} has such an outer automorphism? It occurred to me that whichever one of them has its order-20 subgroups Frobenius might be a good candidate, but I'd have to tackle them by hand, starting with constructing them as subgroups of Aut(A_6), which seemed a little daunting. -- Jim Heckman === Subject: Re: finite groups >BTW, do you have an easy way to check if PGL(2,9) or M_{10} has >such an outer automorphism? It occurred to me that whichever one >of them has its order-20 subgroups Frobenius might be a good >candidate, but I'd have to tackle them by hand, starting with >constructing them as subgroups of Aut(A_6), which seemed a >little daunting. No, neither of these two groups work. PGL(2,9) has classes of elements of orders 5, 8 and 10 which sre not fixed by the outer automorphism. M_11 just has two classes of elements of order 8 which are interchanged by the outer automorphism. By the way, the 5-generator presentation of the group of order 32 that I gave is the one stored by computers in the tables of small groups. These are not very beautiful presentations, but they are the ones most suitable for general purpose computations with groups. Generally, for a solvable group of order p_1^{a_1} ... p_r^{a_r}, the group is stored as a presentation on n := a_1 + ... + a_r generators, together with n(n+1)/2 relations which specify the powers and conjugates under each other of these generators. (Each generator corresponds to a cyclic composition factor of the group.) Starting from such a presentation, it is routine (for a computer) to carry out structural computations within the group, find automorphism groups, etc. Derek Holt. === Subject: Re: finite groups This is probably nonsense, but when I read the question originally I vaguely wondered if one could get a solution by considering a simple Lie algebra whose Coxeter-Dynkin diagram has an obvious symmetry, eg _|_ , and considering the corresponding finite (Chevalley) groups. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: finite groups > This is probably nonsense, > but when I read the question originally > I vaguely wondered if one could get a solution > by considering a simple Lie algebra > whose Coxeter-Dynkin diagram has an obvious symmetry, > eg _|_ , and considering the corresponding > finite (Chevalley) groups. > -- > Timothy Murphy A very interesting idea. I think that there might be a problem with ALL the elements being mapped to a conjugate, though. Consider the simplest case of A_2, where we have a non-trivial graph automorphism. Let's take the simplest case, where the field is K=GF(2), so we have the group G=SL_3(K) of 168 elements. If memory serves me right (it's been 10 years since I had anything to do with algebraic groups), this graph automorphism is simply the mapping A -> A^{-1}^T (= the transpose of the inverse). (Probably you need compose this with a suitable inner automorphisms to get the right permutation of the root subgroups). This will certainly map a root subgroup onto another, and these are all conjugate (looks very promising at this stage!). However, I think that there might be a problem with the elements of order 7. Let g be a generator of the multiplicative group of GF(8), and let A be an element of G with eigenvalues g, g^2 and g^4. Its image under the above automorphism will have eigenvalues g^3, g^5, g^6, so it can't be conjugate to A. I suspect that this kind of reasoning will eliminate all the graph automorphisms for groups of type A_n for the purposes of the problem at hand (perhaps with the exception of some trivial cases). You can always find elements (in a non-split torus?) that aren't conjugate to their inverse transposes. May be the graph automorphisms for groups of other type will work, but it doesn't look very promising (I would be happy to be proven wrong about this). Jyrki Lahtonen, Turku, Finland === Subject: formal rules for big-oh manipulation For practical purposes I need to know about the big-oh stuff, but none of the books I've been able to find develops it in a rigorous way, and I'm repulsed by nonrigorous math (plus it doesn't stay in my head). I would like to solve this problem by turning the basics of big-oh theory it into a completely formalized body of rules for manipulating expressions, along these lines: You're given an expression E with one free variable type x. You also have a body of rule schemas. If A->B is an instance of one of these schemas, you can replace A with B anywhere in E and the resulting expression will be big-oh of E. (There ought to be a rule with A=5x^3+7x+5 and B=x^3, for example.) Is there any reason why you couldn't formalize a good big useful chunk of big-oh theory along these lines? Has somebody already done so, and put it in writing? Any helps or hints would be mucho appreciado. Peace, EJ === Subject: Re: formal rules for big-oh manipulation >For practical purposes I need to know about the big-oh stuff, but none of the >books I've been able to find develops it in a rigorous way, and I'm repulsed by >nonrigorous math (plus it doesn't stay in my head). I would like to solve this >problem by turning the basics of big-oh theory it into a completely formalized >body of rules for manipulating expressions, along these lines: >You're given an expression E with one free variable type x. You also have a body >of rule schemas. If A->B is an instance of one of these schemas, you can replace >A with B anywhere in E and the resulting expression will be big-oh of E. (There >ought to be a rule with A=5x^3+7x+5 and B=x^3, for example.) >Is there any reason why you couldn't formalize a good big useful chunk of big-oh >theory along these lines? Has somebody already done so, and put it in writing? Well, there are things a person could say, for example (i) f = O(F), g = O(F) => f + g = O(F) (ii) f = o(g) => f + g = O(f) but it's not clear how helpful this is - the rules that are true are more or less obvious, and they're not always going to give the result you want anyway: For example say f = x^2, g = 2x^2. Then f and g are both O(x^2), so it follows from (i) that f + g is O(x^2). Otoh say f = x^2 and g = - x^2. Then f + g is O(x^2) for the same reason as before, but here saying that f + g is O(x^2) is a little silly, since in fact f + g = O(0). >Any helps or hints would be mucho appreciado. >Peace, ************************ David C. Ullrich === Subject: Re: formal rules for big-oh manipulation >> Then f + g is O(x^2) for the same reason as before, >> but here saying that f + g is O(x^2) is a little silly, since in >> fact f + g = O(0). >Why is that silly? Is the Schwarz inequality silly because there exist >orthogonal vectors? It's not silly. It doesn't do what I thought she wanted to do. Turns out that she didn't want to do what I thought she wanted to do... ************************ David C. Ullrich === Subject: Re: formal rules for big-oh manipulation |OK, I didn't actually say so, but I'm assuming that, if A=B is true when you |regard A and B as meaningful terms (with the big-oh symbol denoting an |arbitrary |function), then A->B and B->A are rules that you can use in your |manipulations. One useful observation (and I don't remember where I read this) is that in this context = is directional. It's possible for f+O(g)=h+O(k) to be true but h+O(k)=f+O(g) not be true. In fact = is being used essentially to mean is a subset of. The expressions involving o, O and Theta denote sets of functions. A=B means that the set of functions denoted by A is a subset of the set of functions denoted by B. And of course if there are no occurrences of o, O or Theta, the set is the singleton set of the given function. That's why it becomes a symmetric relation when no o, O, or Theta is there: {f} is a subset of {g} means the same thing as {g} is a subset of {f}. So for example O(x) = O(x^2) is valid but O(x^2)=O(x) is not. Note also that the meanings of o, O, and Theta are relative to a limit. The meaning of f=o(g) is the same as g/|f|->0 in whatever limit is concerned, for instance, (except for the case where f=0 for all points sufficiently close to the limit, because of division by zero). Typically the limit is as the independent variable goes to infinity, but it can just as well be as it goes to zero or another value. If % is some operation on functions, then A%C is defined as the set of results of applying % to a function belonging to A and a function belonging to C. So if A=B and C=D then A%C=B%D. That's maybe the most basic thing. Then there are a number of little lemmas such as O(f)O(g)=O(fg). I don't know anywhere where they go through them systematically. Keith Ramsay === Subject: Re: formal rules for big-oh manipulation > For practical purposes I need to know about the big-oh stuff, but > none of the books I've been able to find develops it in a rigorous > way, and I'm repulsed by nonrigorous math (plus it doesn't stay in > my head). I would like to solve this problem by turning the basics > of big-oh theory it into a completely formalized body of rules for > manipulating expressions, along these lines: > You're given an expression E with one free variable type x. You also > have a body of rule schemas. If A->B is an instance of one of these > schemas, you can replace A with B anywhere in E and the resulting > expression will be big-oh of E. (There ought to be a rule with > A = 5x^3+7x+5 and B = x^3, for example.) > Is there any reason why you couldn't formalize a good big useful > chunk of big-oh theory along these lines? Has somebody already done > so, and put it in writing? > Any helps or hints would be mucho appreciado. For a sufficiently nice function field F, where limits exist at oo, the Big-Oh notation is essentially of valuation-theoretic origin since functions finite at oo form a (valuation) subring O of F. In fact if we abuse notation by writing f/g in O as f/g = O then one can obtain Big-Oh notation by clearing denominators e.g. f/g = O and g/h = O => f/h = O via (f/g)(g/h) = OO = O -> f = O(g) and g = O(h) => f = O(h) in Big-Oh notation For applications to complexity analysis of algorithms it suffices to consider just function fields containing (a small subfield of) the Hardy exp-log fields. There are effective algorithms for asymptotic calculation in such fields due to Rosenlicht, myself, and Shackell, e.g. see some of my earlier posts: Salvy et.al. have implemented some algorithms for asymptotic calculus in Maple which you may find of interest. -Bill Dubuque === Subject: Re: formal rules for big-oh manipulation | > For practical purposes I need to know about the big-oh stuff, but none of the | > books I've been able to find develops it in a rigorous way, and I'm repulsed by | > nonrigorous math (plus it doesn't stay in my head). I would like to solve this | > problem by turning the basics of big-oh theory it into a completely formalized | > body of rules for manipulating expressions, along these lines: | | I haven't seen one, but if you search around the web for | course notes dealing with big-O, you'll find rules of | thumb. There are a few here for instance: | | www.math.uvic.ca/faculty/gmacgill/guide/big-O.pdf | | You probably know that the formal theory involves limits of | ratios of functions, since that's how big-O is defined. | | >(There | > ought to be a rule with A=5x^3+7x+5 and B=x^3, for example.) | | term which means that it is also big-O of its leading term. | | - Randy === Subject: Re: Four Color Graph. > Bill, you are saying that every distinct graph is equally likely then, > but you haven't told us what a distinct graph is (specifically, are you > talking about labeled or unlabeled graphs? connected or > not-necessarily-connected? simple graphs or not? etc...) > For instance, the paths on 3 vertices like 1--2--3 and 1--3--2 are > distinct labeled graphs, but when talking about unlabeled graphs, these > are the same. The vertices are labeled and connected in a closed ring; ie, polygon by 'n' edges. This polygon is a subgraph in every graph. Distinct graphs are created by unique combinations of diagonals. Each graph will have exactly 3n-6 edges. Only simple graphs are considered. The chromatic number for any specific graph may be 2, 3 or 4 but never To clarify the confusion about different paths; each specific diagonal combination represents one and only one distinct graph. Bill J. === Subject: Re: Fraud in Computer Science Publishing >> In my case, after the programming language is axiomatized, nobody >> system (using the definitions, axioms and rules.) >The formalism is still programing. I've been wondering whether anyone else would point this out... >Just because you disguise it in >axiomatic form does not change the underlying realities. In the last >step all that you write will be translated into machine language, if and >when it actually runs. >You either are kidding us, or you believe in symbolic juju and magic >Bob Kolker Sam ************************ David C. Ullrich === Subject: Re: Fraud in Computer Science Publishing >>The formalism is still programing. > I've been wondering whether anyone else would point this out... That's what I've been trying to explain from the beginning. PC is a programming language, though it does not have conditional branching and other common features. I have also noted that, according to Charlie's paper, it is translated to APC (which is similar to pseudo-code in its syntax). Therefore, the system is actually a translator from a high-level non-procedural language to a procedural one. That does not contradict the fact that it is a program generator (Charlie does not seem to accept that idea, though), provided it actually works (since Charlie has not accpted to make his implementation available, even though he supposedly has used it in replying to one of my post) And again, almost any syntax that specifies a task to be performed is a programming language. In fact, English can be one (check out the Shakespeare programming language) Sam -- So if you meet me, have some courtesy, have some sympathy, and some taste Use all your well-learned politesse, or I'll lay your soul to waste - The Rolling Stones, Sympathy for the Devil === Subject: Re: Fraud in Computer Science Publishing > programs. How do you define programming? Writing: FAC(x,I)^LT(x,I)^~(exists A)FAC(A,I)^LT(A,I)^LT(x,A) Is programming, no matter what system translates that input into a program. Sam -- Giving the Linus Torvalds Award to the Free Software Foundation is a bit like giving the Han Solo Award to the Rebel Alliance - Richard Stallman, August 1999 === Subject: Re: Fraud in Computer Science Publishing > > programs. How do you define programming? > > > Writing: > FAC(x,I)^LT(x,I)^~(exists A)FAC(A,I)^LT(A,I)^LT(x,A) > Is programming, no matter what system translates that input into a program. > Sam 1. Do you consider Predicate Calculus wffs to be programs? 2. How do you define a program? 3. Do you know of a better way to sepcify the largest proper factor of a given number? 4. Do you know of a way to specify it that isn't programming? 5. Do you know of a simpler way to specify it? 6. How do you think that a Mathematician would specify it? 7. Did you know that the state-of-the-art in Program Synthesis is to specify the program requirement as a Predicate Calculus wff? 8. Do you see a qualitative difference between a Predicate Calculus wff and a computer program? 9. How about the fact that a Predicate Calculus wff has no assignment, conditional execution, loops or the possibility of not terminating? 10. Do you think that Predicate Calculus wffs and computer programs are at the same level of abstraction? 11. Do you think that Predicate Calculus wffs and computer programs are in a one-to-many relationship in that one wff can be implemented by multiple programs based on different algorithms, but for one program there is essentially only one wff that represents the functionality that it provides (ignoring permutations of the conjuncts/disjuncts and other logical redundancy)? 12. Do you think that computer programs have to be analyzed to determine what function they compute, and in general you cannot do that? 13. Do you think that there is a corresponding process of analysis to detemine what a Predicate Calculus wff is doing, or is it the final word as to the definition that it is conveying? 14. Do you think that a simple wff may require complex programs to implement it (defining simple and complex informally or intuitively)? 15. Do you see value in being able to determine the wff that a particular program computes? 16. Do you see value in being able to determine computer programs that implement a given predicate calculus wff? 17. Does my system determmine programs that compute a given wff? 18. Do you know of any system other than mine that determines programs that compute a given wff? Charlie Volkstorf Cambridge, MA === Subject: Re: Fraud in Computer Science Publishing > 1. Do you consider Predicate Calculus wffs to be programs? Technically, it is only source code. But by extension, we may call that a program. > 2. How do you define a program? A list of instructions that a computer can execute. > 3. Do you know of a better way to sepcify the largest proper factor > of a given number? A better way, maybe not; a way that's as good, yes. And it happens that this way allow to specify things that PC does not. > 4. Do you know of a way to specify it that isn't programming? Not that the computer can understand. If the computer can understand it, it's programming. > 5. Do you know of a simpler way to specify it? Depends on the task to be accomplished. > 6. How do you think that a Mathematician would specify it? It depends. If he's doing a proof, he'll probably just say is a proper factor. If he's working with computer, he'll use whatever language he finds suitable for the task. Might be Maxima, Lisp, PC, or something else. What's your point? > 7. Did you know that the state-of-the-art in Program Synthesis is to > specify the program requirement as a Predicate Calculus wff? This is meaningless. It's not because some people, however talented they are, have decided to use a specific language, that all other languages should not be considered. Furthermore, state-of-the art is a subjective notion. > 8. Do you see a qualitative difference between a Predicate Calculus > wff and a computer program? No. That question is equivalent to the first. > 9. How about the fact that a Predicate Calculus wff has no assignment, > conditional execution, loops or the possibility of not terminating? Lisp programs can have the above characteristics. the difference is, it doesn't force you to stay within these bounds. > 10. Do you think that Predicate Calculus wffs and computer programs > are at the same level of abstraction? Depends on the language used. Check out the Shakespeare programming language. Highly abstract :-) > 11. Do you think that Predicate Calculus wffs and computer programs > are in a one-to-many relationship in that one wff can be implemented > by multiple programs based on different algorithms, but for one > program there is essentially only one wff that represents the > functionality that it provides (ignoring permutations of the > conjuncts/disjuncts and other logical redundancy)? several algorithms. On the other hand, some specifications cannot be expressed with PC. > 12. Do you think that computer programs have to be analyzed to > determine what function they compute, and in general you cannot do > that? I don't understand the question. > 13. Do you think that there is a corresponding process of analysis to > detemine what a Predicate Calculus wff is doing, or is it the final > word as to the definition that it is conveying? I'm not sure I understand the question. The human reader reading PC code will have to analyze it in order to understand the action it performs. However, PC is enough to properly define some actions. Same thing for most languages. > 14. Do you think that a simple wff may require complex programs to > implement it (defining simple and complex informally or > intuitively)? Indeed, just as a simple class, with a few members and a couple methods, is much more complex to implement in assembly language than in Lisp or C++. > 15. Do you see value in being able to determine the wff that a > particular program computes? Yes, because PC is closer to mathematical formalism than other languages. > 16. Do you see value in being able to determine computer programs that > implement a given predicate calculus wff? Yes; compilers for all sorts of languages are always useful :-) > 17. Does my system determmine programs that compute a given wff? I have been refused the opportunity to try it, so I cannot tell. > 18. Do you know of any system other than mine that determines programs > that compute a given wff? The question is biased, since it assumes that your system does in fact perfrom its job. Since I have not been able to verify that assumption, I cannot answer that very question. What I can say, though, is that I know no system that determines programs that compute a given wff. Sam -- The pain of war cannot exceed the woe of aftermath The drums will shake the castle wall, the Ringwraiths ride in black, ride on! Sing as you raise your bow, shoot straighter than before. - Led Zeppelin, The Battle of Evermore === Subject: Re: Hints on multivariable calculus problem > > > Suppose S is a smooth, closed surface. Let v(x,y,z) be a vector field > > > with > continuous second partials on an open, convex set which contains S. Let > > > n be > the outer unit normal of S. Show that double integral [(.84 x v) .87 > > > n]d(sigma) > = 0. > >> >> >> > >>Hint: Stokes's Theorem/Divergence Theorem. > > >>Note that the div(curl(v)) = 0. > > >>Dale. > > > I was able to show that div(curl(v)) = 0 but can't see the connection > > between div(curl(v)) and double integral [(.84 x v) .87 n]d(sigma), where .84 > > means gradient. > > Steven > Not even given the Divergence theorem that states: > int_V (div W) dV = int_S (W . n) dS > where > W is a vector field on the solid V, > S is the boundary of W > n is the unit outward normal on S > dV is volume measure on V > dS is area measure on S > ??? > Looks like a direct plug-in from where I sit. > Where are you getting hung up? > Dale I'm sorry but I don't see this one at all. I looked at this for hours, especially after you said it is just a direct plug-in type proof. Can you please assist a little further. === Subject: Re: Integral posed in problem set: Elementary or not? Steven escribi.97 en el I found the following problem listed on an old problem page. Question: Is the following integral elementary? INT[ sqrt(1 + ln(x)/x) dx ]? Ray Steiner >> If you mean INT[ sqrt(1+ln(x))/x dx], use u=ln(x) for your >> substitution. >> -- >> Will Twentyman >> email: wtwentyman at copper dot net > If you mean INT[ sqrt(1+ln(x))/x dx], you can let u= ln(x) but it > would be a little easier to try u = 1 = ln(x). > Steven You meant u = 1 + ln(x), I suppose ... -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: interval logic on ordinals might write > A[a, b] A[c, d] ( [a, b] nat-include [c, d] <-> (Ax(x pp a -> x ppc) / > Ax(x pp d -> x pp b))) > so that monotonicity > (([a, b] nat-include [c, d]) / ([a, b] < [x, y])) -> [c, d] < [x, y] > ((([x, y] < [a, b]) / ([a, b] nat-include [c, d]) -> [x, y] < [c, d] Shoudn't last expression be rather ?: ((([x, y] < [a, b]) / ([c, d] nat-include [a, b]) -> [x, y] < [c, d] Milan === Subject: inverse of a complex toeplitz matrix Hi all I have seen some fast algos to solve for the inverse of a real Toeplitz matrix. Anyone has any idea how to solve for complex Toeplitz matrix in a fast way too? === Subject: Inverse of a complex toeplitz matrix with least number of divisions Hi all I wonder anyone have any idea how to inverse a complex Toeplitz matrix with least number of divisions, assuming my leading diagonal are ones? I've read the Matrix computation book and it seems to cater to real numbers; the Levinson algo seems to have a couple of divisions. === Subject: Re: Irrational numbers. So, what do you want to know about the sqrt(5), sqrt(405), sqrt(245)? Lurch 2,236067977... 15,65247584... 20,1246118.. === Subject: Irrational numbers. Mail-To-News-Contact: postmaster@nym.alias.net Of Lie Algebra root 2 (5,4),(6,5)=1,414213562 and (4,5),(5,6)=1,414213562 === Subject: isomorphic mapping Hi guys, Schaum says, the way I read it, that an isomorphic mapping is a linear 1-1 mapping. Is that true? If it is not, does it mean then that isomorphic is just a fancy name for an injective mapping? Peter === Subject: Re: isomorphic mapping > Hi guys, > Schaum says, the way I read it, that an isomorphic mapping is a linear 1-1 > mapping. > Is that true? > If it is not, does it mean then that isomorphic is just a fancy name for an > injective mapping? > Peter An isomorphism is a bijection that preserves whatever algebraic structure you're interested in. A bijection is a 1-1 (injective) and onto (surjective) map. There's semantic confusion in the use of 1-1, since a 1-1 correspondence is typically used to mean a bijection; whereas a 1-1 map typically means injective but not necessarily surjective. It's important to make sure you know which definition of 1-1 is being used by Schaum. In any event, an isomorphism is always defined to be a bijection. You didn't say what topic you're studying, but since you mentioned linear, I'm guessing it's linear algebra, and you're talking about isomorphisms between vector spaces. An isomorphism is more than just a bijective mapping. An isomorphism is a bijection that preserves the vector space structure. An example is to consider the real numbers, R, and the x-y plane, R^2. We know from set theory that there is a bijection between R and R^2. There was a recent lengthy thread about this on sci.math. However, R and R^2 are not isomorphic. Even though there's a bijection, there's not a linear bijection. === Subject: Re: isomorphic mapping fishfry, You mentioned: There's semantic confusion in the use of 1-1, since a 1-1 correspondence is typically used to mean a bijection Your correspondence, is that the same as the statement that two sets A and B can be isomorphic meaning that there exist an isomorphism of A ONTO B. In which case there is a bijective mapping, the same as your correspondence. Peter > > Hi guys, > > Schaum says, the way I read it, that an isomorphic mapping is a linear 1-1 > > mapping. > > Is that true? > > If it is not, does it mean then that isomorphic is just a fancy name for an > > injective mapping? > > Peter > An isomorphism is a bijection that preserves whatever algebraic > structure you're interested in. A bijection is a 1-1 (injective) and > onto (surjective) map. There's semantic confusion in the use of 1-1, > since a 1-1 correspondence is typically used to mean a bijection; > whereas a 1-1 map typically means injective but not necessarily > surjective. It's important to make sure you know which definition of > 1-1 is being used by Schaum. In any event, an isomorphism is always > defined to be a bijection. > You didn't say what topic you're studying, but since you mentioned > linear, I'm guessing it's linear algebra, and you're talking about > isomorphisms between vector spaces. > An isomorphism is more than just a bijective mapping. An isomorphism is > a bijection that preserves the vector space structure. > An example is to consider the real numbers, R, and the x-y plane, R^2. > We know from set theory that there is a bijection between R and R^2. > There was a recent lengthy thread about this on sci.math. > However, R and R^2 are not isomorphic. Even though there's a bijection, > there's not a linear bijection. === Subject: Re: isomorphic mapping >Hi guys, >Schaum says, the way I read it, that an isomorphic mapping is a linear 1-1 >mapping. >Is that true? Not quite. Because there are contexts where it makes sense to talk about an isomorphic mapping (also called more simply an isomorphism when it is clear that one speaks about a mapping) but where linear doesn't make sense. For instance for a mapping between general groups ( i.e. not necessarily commutative ones). Isomorphism is a very general concept with a context-dependent meaning, the dependance being on the math. structures considered. Even used for non-algebraic structures like ordered sets or topological spaces (for the latter and some others there are specific names one generally uses instead of isomorphic/-ism). In any case one means by an isomorphism: a bijective mapping preserving the structure - this must be defined more precisely (which can't be done with general structures without going to a higher level of abstraction) but may-be it gives you the intuition ... The answer is *yes*, if you are speaking of a mapping between linear spaces (or more generally modules over a ring). But even then, being a linear injective mapping is not sufficient for an isomorphism if one wants to be precise: the mappings must be *bi*-jective ! And one might also consider structures that include linear spaces or modules but with more in them, e.g. so-called algebras, where the linear structure is completed with a bilinear internal multiplication. Then it isn't sufficient to have a linear bijection, because this extra structure must also be taken in account ... >If it is not, (I suppose you mean: if it is - otherwise it gets weird) >does it mean then that isomorphic is just a fancy name for an >injective mapping? No, see above >Peter Ulysse === Subject: Re: isomorphic mapping >>Hi guys, >>Schaum says, the way I read it, that an isomorphic mapping is a linear 1-1 >>mapping. >>Is that true? >Not quite. Because there are contexts where it makes sense to >talk about an isomorphic mapping (also called more simply >an isomorphism when it is clear that one speaks about a mapping) >but where linear doesn't make sense. For instance for a mapping >between general groups ( i.e. not necessarily commutative ones). >Isomorphism is a very general concept with a context-dependent >meaning, the dependance being on the math. structures considered. >Even used for non-algebraic structures like ordered sets or >topological spaces (for the latter and some others there are specific >names one generally uses instead of isomorphic/-ism). In any case >one means by an isomorphism: a bijective mapping preserving the >structure - this must be defined more precisely (which can't be done >with general structures without going to a higher level of >abstraction) but may-be it gives you the intuition ... >The answer is *yes*, if you are speaking of a mapping between >linear spaces (or more generally modules over a ring). But even then, >being a linear injective mapping is not sufficient for an isomorphism >if one wants to be precise: the mappings must be *bi*-jective ! And >one might also consider structures that include linear spaces or >modules but with more in them, e.g. so-called algebras, where the >linear structure is completed with a bilinear internal >multiplication. Then it isn't sufficient to have a linear bijection, >because this extra structure must also be taken in account ... >>If it is not, >(I suppose you mean: if it is - otherwise it gets weird) >>does it mean then that isomorphic is just a fancy name for an >>injective mapping? And, in speaking of more general structures, it is important to keep in mind that, for most people, 'surjection' and 'onto' are not synomous, and 'injectiion' and '1-1' are not either. Properly properties of compostition (OK, it is even more general, since the morphisms in a category are not necessarily functions). For example, in the category of (completely-regular, I can't remember off-hand how much weaker you can get, and am too lazy to figure it out now), surjective is equivalent to 'onto a dense subset'. >No, see above >>Peter >Ulysse Larry (this space unintentially left blank ..... make obvious deletion for email === Subject: Re: isomorphic rings nojb. > > > > I'm studying for an exam and doing problems from the book > > Algebra: A Module-Based Approach. > > > > In one exercise it asks to show that the rings > > > > Z[X,Y]/(XY-1) and Z[Z] are isomorphic, and that > > > > Z[X]/(X^n-1) and Z[Z_n] are isomorhic > > > > where Z[G] is the group ring for any group G. > > > > I haven't been able to come up with a suitable map... Any ideas? > Hint: Z[X]/(X^n-1) = Z[] = Z[Z_n] > Z[X,Y]/(XY-1) = Z[] = Z[Z] > Do you see the general principle? > -Bill Dubuque === Subject: math courses What are the typical courses a jr and sr takes in math at university? Also, I want to do some self study, and I bought 2 Schaum's outline texts, one on abstract algebra, and one on advanced calculus. Any order of reading recommended? === Subject: Re: math courses At the Jr. and Sr. level, one ususally takes courses such as abstract algebra, analysis, p.d.e.'s, maybe linear algebra, discrete math, complex variables, etc... It really depends on how strong of a program your school has. If it is a teaching college, then you will probably top your senior year with abstract algebra. If your school has a strong program, then you might be taking a.a. and introductory analysis in your Jr. year and advanced analysis in your senior year. As far as the order of your reading goes, I don't think it matters; but, a.a. might be easier to begin with. The proofs in a.a. are not always as tricky as they can be in analysis. Lurch > What are the typical courses a jr and sr takes in math at university? Also, > I want to do some self study, and I bought 2 Schaum's outline texts, one on > abstract algebra, and one on advanced calculus. Any order of reading > recommended? === Subject: Re: math courses Are you kidding/trolling? > What are the typical courses a jr and sr takes in math at university? I bet that almost every University in the world publishes its course details on the internet. > Also, > I want to do some self study, and I bought 2 Schaum's outline texts, one on > abstract algebra, and one on advanced calculus. Any order of reading > recommended? I just bought a book on skydiving and another on wok cookery - which order should I read them in? These are almost completely independent subjects, so it doesn't matter which order you read them in. You don't need any calculus to study abstract algebra, and you can go a long way with calculus without any knowledge of abstract algebra. Of course, you better have a good understanding of basic calculus before reading one on Advanced Calculus. Most books on abstract algebra are reasonably self contained (I'm not familiar with Schaum), so there is not usually a lot of pre-requisite knowledge other than basic number theory and algebra. On the other hand, abstract algebra is rather ... abstract ... and hence if you do this as self study you might understand the proofs but not what it really all means. At least with Calculus there's usually lots of prcatical examples and applications. === Subject: Re: math courses >> Most books on abstract algebra are reasonably self contained (I'm not >> familiar with Schaum), so there is not usually a lot of pre-requisite >> knowledge other than basic number theory and algebra. On the other hand, >> abstract algebra is rather ... abstract .. >Not so. If you read the book on skydiving, then go skydiving and your >chute does not open, you will not get to read the book on cookery. >If you read the book on cookery and try out the recipes, most likely you >will live to read the book on skydiving. >You see order -does- count and the world is not commutative. >Bob Kolker Skydiving is transportation. You'll be the center of attention as others commute with you. After you land, you may need to wok the last few steps. -- Wanted: Experts at choosing the best of 100+ applicants for a position. Register as a California voter by September 22, and vote on October 7. Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California Microsoft Research and CWI === Subject: Mathematicians Living in the Past I have come to the conclusion that many of the regular posters here at sci.math are dwelling in a nostalgic past where people could just magically throw half-defined commentary around and have it pass for mathematics. We live in a more refined world and these days mathematics must be rigorous. Those who say otherwise are a blight upon the mathematical community and must be shunned and ostracized. A good example of this sort of nonsense is the collective works of Cantor. The man came up with a whimsical notion of mathematics which would do fine in science fiction but not in any respected textbook. In order to disguise this as rigorous, mathematicians of a later generation twisted and manipulated the basic axiomatic system upon which mathematics is built until it forced Cantor's faery tales to be rigorous. Well, I will concede. Cantor is 100% true and rigorous, *if* you live in the magical neverland where you can just twist and manipulate axioms however you please. Some posters to this newsgroup seem to live in such a world. If they feel like it, they'll accept the Axiom of Choice. If not, then not. It seems they have augmented their education with a far too liberal dosage of liberal arts. One day, AC is true... the next, poof, it is not. These mathematicians (and I use the word in the loosest possible sense) can prove anything whatsoever by simply rejecting the reflexive axiom (A=A). And, yes, perhaps with the reflexive axiom done away with, some interesting things become true, that does not mean that those things then are proven in any serious system. It is as simple as this: if Cantor's theorems are really the absolute truths some people hold them to be, then you will provide a proof of them using nothing but the postulates of Euclid. But I defy you to do this. I say again, these posters live in the past. In this modern 21st century if these posters made some of the claims they make so boldy here in any mathematics classroom they would be a laughingstock. The propositions of Cantor have been so utterly defeated by so many reputable mathematicians, whose disproofs have been so elementary that even elementary school math teachers can comprehend them in all their rigour, that anyone who still holds these raving hallucinations to be even feasible, ought to be locked away. The folks who discuss Cantor's transfinitude with straight faces are the same who propose new classes of imaginary numbers j such that j*0=1. Even speaking non rigorously, on a level that you liberal arts minors who hold Cantor so dear will understand, Cantor's character is one which does in no way invite us to believe anything he says. It is well known that Cantor plagiarized the few publications he made which actually stand up to rigour, and also that he abused small children sexually. There is an anecdote that the member of Pythagoras' society who first discovered the irrationality of sqrt(2) was put to death by the society because of their reluctance to accept this. History, it seems, repeats itself. The people who still stubbornly hold onto Cantor's lies, refusing to hear the disproofs which are so common and irrefutable, are the reincarnations of those Pythagoreans who refused to believe, nay, who FORBID, the proof that sqrt(2) was irrational. Take my words to heart, and turn from your path. There is still time yet for you to be acknowledged as a true champion of what is true and irrefutable, if you turn from the path of nonsensical conjuring which is Cantor. I speak to you as a friend, concerned for your better good. Nathan the Great Age 11 === Subject: Re: Mathematicians Living in the Past > I speak to you as a friend, concerned for your better > good. > Nathan the Great > Age 11 Due to some trouble with my newsreader my killfile didn't work properly. So I found your friendly If ever you will post something relevant in our 21-st century, I'll find out, because there will be some resonance by posters, whom I respect and enjoy reading. *PLONK* again - may be til you are 7 or so :-) Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Mathematicians Living in the Past 7e3Z2dkQEBC5ubm60r3HAAACeUlEQVR4nIXTQW+bMBQAYChLfcWzUq5pipUr7lPptUNGXBMC8zUZ qnt1aNz392c7pGXSpvkSyV/ee37POFr/Y0X/g56rxd+gb2IQC8X79brdLtovaBgACIsRMVTr4hN4 Ah54xy0rtdbjFRCAupizIhUDB69X2IHAGKCrxyrxcJyA76CQKwZn2ckEXK6h/YwoOld/zJUMEYds ghMIYhj0kjehxqGYFacgalk1PpM+jBOkvg0othFR4bjDVHyjQx+FZ+b29fMEe114SIC6UB/RXoCX enAhGQORAdxp/TaN5N63lNqGJZACO2n9/QL81qfVQjEXAkBKPyoPm7BfPqFLJ0qBt1fwmfTAFiYW MbAC1WmCnyEitcYImtI07fY3FwglBuwMUiqoiLv98gL+4PrNSmXdWGJK7Q4D8JBpWeXGxi6CpqOJ QoOrkOksKzMaRinN7LsI8OChsGOl+ogiE8SeaIAP9/9BRIs1xzgzESWteQ7gDgWQpmRdmcgtNFYJ /5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADvIrOrduRArpW7D Rflb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/LpBYalbA2qiLZKP16G+KHLowIYojSm UTys+/LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3XL8v547w/ uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/u9oA6vhAAAAAElFTkSuQmCC > It is as simple as this: if Cantor's theorems are really the absolute > truths some people hold them to be, then you will provide a proof of > them using nothing but the postulates of Euclid. But I defy you to do > this. > I say again, these posters live in the past. You have undoubtedly proved to me that some posters live in the past. -- Jesse Hughes Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem. -- Alan MacKay === Subject: Re: Mathematicians Living in the Past > > It is as simple as this: if Cantor's theorems are really the absolute > > truths some people hold them to be, then you will provide a proof of > > them using nothing but the postulates of Euclid. But I defy you to do > > this. > > I say again, these posters live in the past. > You have undoubtedly proved to me that some posters live in the past. Wasn't it meant as irony, then? I thought it was a parody of the recent thread about Cantor's influence on math. Herman Jurjus === Subject: Re: Mathematicians Living in the Past > I have come to the conclusion that many of the regular posters here at > sci.math are dwelling in a nostalgic past where people could just > magically throw half-defined commentary around and have it pass for > mathematics. We live in a more refined world and these days > mathematics must be rigorous. Those who say otherwise are a blight > upon the mathematical community and must be shunned and ostracized. > A good example of this sort of nonsense is the collective works of > Cantor. It's not unusual for a mathematician of an earlier age to come up with a good idea that mathematicians of a latter age make rigorous. It happens a lot actually. > If they > feel like it, they'll accept the Axiom of Choice. If not, then not. I'm not to happy with feel like, but apart from that, that is the nature of AC. What's more Godel and Cohen showed that that attitude is safe (if anything is safe). > It seems they have augmented their education with a far too liberal > dosage of liberal arts. One day, AC is true... the next, poof, it is > not. These mathematicians (and I use the word in the loosest > possible sense) can prove anything whatsoever by simply rejecting the > reflexive axiom (A=A). That one can prove anything by rejecting it is a reason for not rejecting it. -- G.C. === Subject: Re: Matrix Traversing Question -- I could really use a moment of help. I'm stuck. > I have a task where I have to traverse a matrix in a way I cannot > formulate into an equation easily. I suspect there is a concept to > address this method already but I do not know how to begin looking for > it. > Could someone offer a bit of personal help? The concept is easy to > demonstrait, but difficult to verbalize. If you send me an email I'll try to see if I can suggest something. Only no attachements, please. All the best, Felix. === Subject: Re: Maximal subgroup of GL(n,Z) > >About 2 weeks ago Robin Chapman proved that the group GL(n,Z) has only > >a finite number of non-isomorphic finite subgroups . > >Let me ask what is the maximal order of a finite subgroup of GL(n,Z) > >? > 2^n n! > Derek Holt. Shouldn't the answer for n=2 be 12 ? Dan === Subject: Re: Maximal subgroup of GL(n,Z) >> >About 2 weeks ago Robin Chapman proved that the group GL(n,Z) has only >> >a finite number of non-isomorphic finite subgroups . >> > >Let me ask what is the maximal order of a finite subgroup of GL(n,Z) >> >? >> > 2^n n! >> Derek Holt. >Shouldn't the answer for n=2 be 12 ? Yes. Sorry, I missed out `for sufficiently large n' ! Apparently, there is an unpublished preprint by Walter Feit, which proves that the group of monomial matrices (which has order 2^n n!) is the unique (up to conjugacy) subgroup of GL(n,Z) of maximal order for n > 10. He also finds the maximal (order?) subgroups for smaller values of n. The proof uses the classification of finite simple groups. You can find the references in the paper by Kuzmanovich & Pavlichenkov in the American Math. Monthly 109 (2002), 173-186 which was mentioned earlier (wasn't that by you?) Derek Holt. === Subject: Re: Maximal subgroup of GL(n,Z) > >> >About 2 weeks ago Robin Chapman proved that the group GL(n,Z) has only > >> >a finite number of non-isomorphic finite subgroups . > >> >> >Let me ask what is the maximal order of a finite subgroup of GL(n,Z) > >> >? > >> >> > >> 2^n n! > >> > >> Derek Holt. > >Shouldn't the answer for n=2 be 12 ? > Yes. Sorry, I missed out `for sufficiently large n' ! > Apparently, there is an unpublished preprint by Walter Feit, which proves > that the group of monomial matrices (which has order 2^n n!) is the unique > (up to conjugacy) subgroup of GL(n,Z) of maximal order for n > 10. He also > finds the maximal (order?) subgroups for smaller values of n. The > proof uses the classification of finite simple groups. > You can find the references in the paper by Kuzmanovich & Pavlichenkov in > the American Math. Monthly 109 (2002), 173-186 which was mentioned earlier > (wasn't that by you?) > Derek Holt. Yes that was me but I don't have access to the paper of Feit . Let me just ask if for small values of n the number of non-isomorphic finite subgroups of GL(n,Z) is known ? === Subject: Meaning of curly d? I have some review books for differential equations, and they don't explain the difference between e.g. dM/dx and (curly)dM/dx. What's the difference?? === Subject: Re: Meaning of curly d? on Friday 15 > I have some review books for differential equations, and they don't explain > the difference between e.g. dM/dx and (curly)dM/dx. What's the > difference?? (curly)d denotes partial differentation. If you see dM/dx, you can assume M only depends on x, and you have the derivative; if you see (curly)dM/(curly)dx, you can assume M depends also on other variables than x, and (curly)dM/(curly)dx is the partial derivative of M wrt x, that is, the derivative considering all variables M depends on except x as constants. Sam -- Don't be afraid, I'm gonna give you the choice I never had... - Lestat in Interview with the Vampire (Ann Rice, 1976) === Subject: Re: Meaning of curly d? >I have some review books for differential equations, and they don't explain >the difference between e.g. dM/dx and (curly)dM/dx. What's the >difference?? The second one stands for a partial derivative, while the first one stands for (ordinary) differentiation. Doug === Subject: Re: Meaning of curly d? === Subject: Multidirectional limits ? I was trying to compute the (trivial) limit: limit((x^3+y^3)/(x^2+y^2),{x->0,y->0}), but Maple couldn't solve it (a multidirectional limit, according to Maple's terminology). I was wondering if there is an algorithm to determine wether Q(a1,...,an)/P(a1,...,an) (where P and Q are n-variable polynomials) has a limit or not . -- Julien Santini === Subject: Re: Multidirectional limits ? |I was trying to compute the (trivial) limit: |limit((x^3+y^3)/(x^2+y^2),{x->0,y->0}), |but Maple couldn't solve it (a multidirectional limit, according to |Maple's terminology). You are presumably assuming that x and y are approaching 0 only through real values. If x and y are allowed to be complex there is no limit. I don't know whether Maple assumes by default that its variables are real or complex. |I was wondering if there is an algorithm to determine wether |Q(a1,...,an)/P(a1,...,an) (where P and Q are n-variable polynomials) has a |limit or not . I don't know but I would assume there is. If the limit is through complex values it's simpler. At points where P<>0 there is of course a limit. At points where P=0 but Q<>0 there is of course no limit. (These are because P and Q are continuous.) If P and Q are polynomials of complex variables, without a common polynomial factor and P=Q=0 at the given point, there are points arbitrarily close where P=0 but Q<>0, and Q/P is arbitrarily large arbitrarily close to the given point. If the variables assume only real values, the interesting case is still the one where the zero locus of P and the zero locus of Q are the same within a neighborhood of the point. Then I suppose there's some way of analyzing how rapidly they approach 0 as they approach the common zero locus from different directions. Note that it's possible for Q/P not to have a limit, but for Q/P to have a limit of 0 on each straight line approaching the point. That kind of thing complicates matters. Keith Ramsay === Subject: Re: Multidirectional limits ? Utilisateur1 escribi.97 en el > I was trying to compute the (trivial) limit: > limit((x^3+y^3)/(x^2+y^2),{x->0,y->0}), > but Maple couldn't solve it (a multidirectional limit, according to > Maple's terminology). > I was wondering if there is an algorithm to determine wether > Q(a1,...,an)/P(a1,...,an) (where P and Q are n-variable polynomials) > has a limit or not . The change to polar coordinates generally allows to distinguish the existence or not of the limit. In this case, easily: L = Lim((x^3 + y^3)/(x^2 + y^2), (x, y), (0, 0)) = Lim(r(cos^3(t) + sin^3(t)), r, 0) = 0 because r -->0 and (cos^3(t) + sin^3(t)) is bounded. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: My (Un)Originality (was Re: Reminder: Wages, Employment Not Determined By Supply, Demand) > I've recently demonstrated here that the determination of income > distribution by the theory of supply and demand makes no sense. You're probably right. Probably supply-and-demand would have to be twisted inside out and sideways to explain a real-world income distribution. > Rather than point out any errors in my exposition, certain > irrationalists objected, incorrectly, that I was not echoing the > literature. These people claimed that I was being original. I'd tend to make a different complaint. It looks to me like you use the same trick every time. You use linear math, and you grant yourself a point where the slope changes. Then you take some relation that's supposed to be constant and you show that, after you solve your linear equations, it doesn't stay constant on both sides of that point. Since it's off by 3/11 of a bushel etc, the theory that claims it's constant is wrong. But how can real-world economists do much with linear algebra? It's mostly a nonlinear world. Linear algebra is a *convenience*, something you use to look at small changes where the nonlinearities won't show up too badly. If the economists you want to disprove are *depending* on linearities then you're completely right to sneer at them, but surely they aren't. Are they? When people bring this point up you point out that economics students are presented regularly with linear models and it's wrong for them to be taught that way. I'd say those models are a convenience and it's wrong if they're taught to depend on them too much. Likely they are. When the only tool you have is a hammer.... But that doesn't handle the bigger problem that you want to disprove economic arguments which should not have been presented in linear terms in the first place. Mostly, I don't yet see that economic ideas are stable enough or simple enough to be fruitfully expressed mathematically. Probably we do better to express them in poetry. Compare to physics. Newton expressed gravitation as a mathematical relationship, and his few parameters could be measured closely enough to predict the motion of the planets. If you tried to say it in english you'd get something like Masses attract other masses. The earth has a great big mass so things fall down. It explains but it doesn't predict very well. Similarly, you can express Gresham's Law in poetic english, Bad money drives out good money. But try to do it like Newton! If the money is done by silver coins and gold coins, with the values fixed, but the value of the gold in gold coins is 20% more than the value of silver in silver coins, and the total number of coins is 100,000 while the total value of transactions in a year is 10,000,000 coins exchanged, how many of the coins must be silver to get 90% of the transactions done with silver coins? It was even worse when I took a brief look at comparative advantage. There's a traditional numerical example. Then there are a lot of studies that show it can be asserted given various simplifying assumptions. Then there's the argument on faith that it should work in the real world. And there's the argument by real example, that prosperous nations tend to advocate free trade while poor nations often don't. Somehow a lot of people who read about economics have solidified this spiderweb of assertions and assumptions into an inflexible belief. But then, that happens in physics too. When the laser was first invented a lot of physicists who believed in quantum mechanics thought it was impossible. It turned out that they had an incorrect intuition about what quantum mechanics said, and it was compatible with quantum mechanics after all. Various unintuitive results may turn out to be compatible with comparative advantage, or gresham's law, or supply-and-demand. We won't get studies showing the compatibility until there's a demand for them.... === Subject: Re: My (Un)Originality (was Re: Reminder: Wages, Employment Not Determined By Supply, Demand) > > I've recently demonstrated here that the determination of income > > distribution by the theory of supply and demand makes no sense. > You're probably right. Probably supply-and-demand would have to be > twisted inside out and sideways to explain a real-world income > distribution. Totally off point. My argument, and that of the literature upon which I draw, is one of internal (in)consistency. Particular theoretical examples have forced the admission, in recent economic literature, that the switch of systems might operate in a direction contrary to the one traditionally assumed. The tendency however has been to label those cases as 'exceptions': as if the principle about capital-intensity had resulted from observed regularities, always liable to exception, and was not a pure deduction from postulates (like Bohm-Bawerk's 'average period of production') now generally admitted to be invalid. Instead, it must be recognized that the traditional principle, drawn from incorrect premises, is itself incorrect. -- P. Garegnani > > Rather than point out any errors in my exposition, certain > > irrationalists objected, incorrectly, that I was not echoing the > > literature. These people claimed that I was being original. > I'd tend to make a different complaint. It looks to me like you use the > same trick every time. You use linear math, and you grant yourself a > point where the slope changes. The above makes no sense to me. Inasmuch as I assume, say, Constant Returns to Scale in my examples, I make the same assumption as the theory I criticize. Terms like linear and slope have meanings. If Jonah Thomas is going to use such terms, he could try to point out what's linear in my examples or what slope he's talking about. Consider Appendix B in (I am no longer committed to calling any curve there a demand function.) I think the bifurcations of equilibria shown point to the possibility of complex (non-linear) dynamics. If you want to see what economists have done with my math when looking at the real world, Leontief's work is interesting. If you want to see economists using non-linear math, I recommend the work of Richard Goodwin and of Barkley Rosser, Jr. Once again, what slope? -- Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html To solve Linear Programs: .../LPSolver.html r c A game: .../Keynes.html v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Re: Near-integer values of polynomials on integers >If f is a polynomial with real coefficients: > How close can f(n) be to an integer, when n is an integer? > How can I find the best values of n ? >If f(n) = b n for some real number b, then the theory of >continued fractions implies that f(n) is within O(1/n) of an >integer infinitely often; moreover, the Euclidean algorithm >gives us an efficient way to walk around GL(2,Z) looking >for approximate solutions to b n - m = 0. >For other polynomials f, I suppose it's true that the fractional >parts of f(n) are uniformly distributed in [-1/2, 1/2), so that >again f(n) ought to be within O(1/n) of an integer infinitely >often, but I don't know how to find good values of n . >(In fact, I don't even recall off-hand how to do this for >f(n) = a + b n with nonzero real coefficients a,b.) >I tested some small values of f(n) = a + b n and f(n) = a + b n + c n^2 >where a=pi, b=e, c=sqrt(2). It certainly does appear from a plot that >the points of the form ( log(n), n | { f(n) } | ) are uniformly >distributed in the region (0, infty) x (0, 1) (where {x} = x - round(x) >is the distance to the nearest integer). >Here are the values of n < 3.10^6 where f(n) is within 1/n of an >integer: > {1, 2, 3, 4, 8, 11, 29, 36, 75, 107, 178, 501, 572, 1037, 2038, 3039, > 4040, 11583, 20127, 29672, 47761, 65850, 83939, 256285, 446720, > 655244, 1054203, 1453162, 1852121, 2251080} >for the linear function and > {1, 2, 3, 5, 24, 25, 28, 42, 79, 139, 354, 394, 467, 1357, 1933, 2173, > 3905, 4097, 10218, 12310, 23629, 34644, 42266, 50277, 222996, 250375, > 262688, 272302, 343133, 1226556, 1781633, 2107651} >for the quadratic. >My question is, how might I have found (some of) these values of n >apart from an exhaustive search? Here is something that may work for the first polynomial. First, find a set of pairs of integers C={(m_i,n_i)} so that each m_i*pi+n_i*e is very nearly integer. Now look for m_i0 and m_i1 s.t. gcd(m_i0,m_i1)=1. Then its possible to find integer p,q s.t. p*(m_i0*pi+n_i0*e)+q*(m_i1*pi+n_i1*e)=pi+k*e. With a bit of luck this k will meet your requirements. For example, suppose C={(19,6),(-73,13)}. Then -23*(19*pi+6*e)-6*(-73*pi+13*e)=pi-216*e=-584.00728 which is within 1/216 of being integer. Given I spent several hours finding this example, exhaustive search is probably more efficient. Rich Burge === Subject: Non-differentiability of continuous functions with partial derivatives of first order Is there any practical way to tell if a function f: U->F (U is an open set of a finite-dimensional real vector space E, and F a finite-dimensional real vector space) which is continuous on U and has first order partial derivatives (not all continuous) is differentiable? I'm particularly interested in the case U=R^n and F=R^p. -- Julien Santini === Subject: Re: non-Euclidean geometry > > I am not a mathematician but I love mathematics. I love the warmth > with which this group responds to my questions. Here is one more: > >> Every web-page I read on non-Euclidean geometry refered to problem > with Euclid's fifth postulate. > >> Why does none talk about geometries with any other postulate > re-examined? > >> Is it not possible or is it not interesting? > >> -Bhu > > >>The postualtes are listed at > >>http://mathworld.wolfram.com/EuclidsPostulates.html > > >>Notice the nature of the first four postulates: You can do _____ > > >>The fifth one, when phrased differently, is the only one that asserts > >>the *unique* *existence* of an object. As a result, it is easier to ask > >>what if when viewing this than the others. > > >>Also, if you look at the surface of a sphere and think of a line as a > >>great circle, the fifth postulate is the one that is most obviously > >>violated. > > And another of Euclid's axioms is violated in this case too. In the > > Hyperbolic case _only_ the fifth postulate is violated. > >> This observation leads naturally to investigating the nature > >>of parallel lines. > > True. Once you change a postulate, the next thing that must be checked > is consistency. I'd have to look up which one is usually changed on the > sphere. The second one: Any straight line segment can be extended indefinitely in a straight line. In more rigorous formulations axioms of betweenness have to replaced with axioms of separation as well. -- G.C. === Subject: Re: oh....my god~~~~ > oh......my..my...my god~~~ What's happened? Cooling down? -- G.C. === Subject: Re: one to one holomorphic functions >> Let D' be the closed unit disk in the complex plane: D':= {z in C : >> |z|<=1}. >> Let f be holomorphic on a neighborhood of D'. Assume that the >> restriction of f to D' is one-to-one. Prove that in fact f is >> one-to-one on a neighborhood of D'. >That looks false to me. If f(z) = (z-1)^2, then f is 1-1 on D' but not in >any neighborhood of 1. Hmm. On the one hand that seems right, but otoh it can't be false, since it's an exercise in _Krantz_... (inside joke, sorry). ************************ David C. Ullrich === Subject: Re: one to one holomorphic functions > > Let D' be the closed unit disk in the complex plane: D':= {z in C : > > |z|<=1}. > > Let f be holomorphic on a neighborhood of D'. Assume that the > > restriction of f to D' is one-to-one. Prove that in fact f is > > one-to-one on a neighborhood of D'. > That looks false to me. If f(z) = (z-1)^2, then f is 1-1 on D' but not in > any neighborhood of 1. That's a nice example, Wade. -- your old student, Bill H. === Subject: Re: one to one holomorphic functions I bet if you add that f' never vanishes on D' it would be true. > >> Let D' be the closed unit disk in the complex plane: D':= {z in C : > >> |z|<=1}. > >> Let f be holomorphic on a neighborhood of D'. Assume that the > >> restriction of f to D' is one-to-one. Prove that in fact f is > >> one-to-one on a neighborhood of D'. > >That looks false to me. If f(z) = (z-1)^2, then f is 1-1 on D' but not in > >any neighborhood of 1. > Hmm. On the one hand that seems right, but otoh it can't be > false, since it's an exercise in _Krantz_... (inside joke, sorry). > ************************ > David C. Ullrich === Subject: Re: one to one holomorphic functions > That's a nice example, Wade. > -- your old student, Bill H. Hi Bill. Glad to see you're still interested in complex analysis. Best, W. === Subject: Re: one to one holomorphic functions > I bet if you add that f' never vanishes on D' it would be true. Right. Proof: Suppose to the contrary that f is not 1-1 on each {|z| < 1 + 1/n}, n > N. Then for each n > N, there exist distinct zn, wn in {|z| < 1 + 1/n} with f(zn) = f(wn). Passing to a subsequence, we may assume zn -> z and wn -> w, where z, w are in D'. By continuity, f(z) = f(w). If z and w are distinct, we contradict the given hypothesis. If z = w, then we have f not 1-1 in each neighborhood of z, contradicting the assumption that f'(z) is not zero. It follows that f is 1-1 in some {|z| < 1 + 1/n}. -- CALL FOR PARTICIPATION --- As part of the 8th Computer Olympiad, taking place in Graz, Austria, on organized. The competition will run on a computer located at IKAT in Maastricht, the Netherlands. Participation is free of charge. We especially encourage students to participate in the competition. What is RoShamBo? ----------------- RoShamBo is a game played all over the world. In Japan, where it is very Rock-Paper-Scissors. The game is quite simple. Two players sit facing each other and on a sign they first stamp three times with the right fist in the open left hand and then they simultaneously make a gesture with their right hand. There are three different gestures: Rock - closed fist; Paper - stretched hand; Scissors - two fingers spread. Now it is time to determine who wins. The following rule applies: The scissors are stronger than paper because they can cut paper. The rock beats the scissors because it blunts the scissors. Paper beats the rock because paper can cover the rock. If both players make the same gesture, nobody wins Of course, the game is performed a number of times, because the thrill of the game is to predict what the opponent will do the next time. In Japan the game is often played at the dinner table or, as a gimmick, to Pon! to start the game. Computer-RoShamBo! ------------------ The purpose of the competition is to write a Java program that can play RoShamBo. Your program will play against all other participating programs and the program winning the most games is the champion. We have chosen the programming language Java because many students learn Java in school and it is independent from the operating system (Windows,Linux, Apple, etc.). Furthermore, it is possible to put your contribution as a Java-applet on a website for everybody who wants to play against it. a mathematical point of view, the best approach is to play randomly (with equal probability on the three gestures). In the long term, nobody can beat you then. However, this does not help you in a competition like this, because we do not play long enough. Therefore, you should try to predict the moves of your opponent and react on that, or you should try to be smarter than your opponent. How do you participate? ----------------------- Surf to http://www.cs.unimaas.nl/~donkers/games/roshambo03 and read the instructions. On this site you will also find links to background information on computer-RoShamBo. roshambo@icga.org that you are going to participate in the competition (subject: participation roshambo). will be held on November 21 and the champions are honoured on November 23 at the Computer Olympiad in Graz. Who is organizing this? ---------------------- for Knowledge and Agent Techology IKAT of the Universiteit Maastricht and the International Computer Games Association ICGA. For questions and remarks mail to: Jeroen Donkers (roshambo@icga.org). === Subject: ot: HouseSale.NET I'm one of the bidders, its up to $70 and I'm planning a last minute $100 bid as that's all I have on credit card. There are companies on the net that specialise in selling only real estate names, and this is one of the best. The name itself won't sell for more than a few hundred, but developed it could *easily* become a multi million dollar domain. 1/ People sign up their houses for free 2/ Being a NET rather than COM it still is a major industry name because houses are sold through networking, and it would be one of the most attractive affiliate programs for webmasters to advertise to receive 2% of the sale price of a HOUSE! 3/ That's how Amazon and 1000's of others started large scale business by starting affiliate programs and using ordinary web sites to get customers. A LOT of requests for domain names are for real estate domain names. About the best of 100s I've seen was TheOnlineRealty.com and they go for $100s. Imagine you're selling your house, can you think of a better name on a banner that you would be interested in than HouseSale.net? Can you think of a better incentive to be an affiliate than 2% or 3% commision on the sale of a house just from some visitors to your website? email chess3@ozemail.com.au if you have more than I do on a credit card you can lend the number to and we'll go 50/50 in the business and I can program the website. Herc -- === Subject: Re: ot: working on the internet >I have an email pen pal who's in a wheelchair asking me about >working from home. >I checked out numerous of these offers a few months ago for myself >and they are all asking for a payment for the training manual. >I assume you can do it but the jobs are as hard to find as ordinary jobs, >if anyone knows something about it let me know, I recommend you buy one of those training manuals. You'll learn a lot. --- John Hattan Grand High UberPope - First Church of Shatnerology john@thecodezone.com http://www.shatnerology.com === Subject: Re: ot: working on the internet > > I have an email pen pal who's in a wheelchair asking me about > > working from home. > > I checked out numerous of these offers a few months ago for myself > > and they are all asking for a payment for the training manual. > > I assume you can do it but the jobs are as hard to find as ordinary jobs, > > if anyone knows something about it let me know, > > Herc > Tell him to start an internet business selling training manuals on how to > start an internet business. Now that's funny. Steve > -- > Ghost Rider > aa # 2011 > EAC Nonexistent Director of Alcohol, Tobacco and Bad Puns > How can you just obey? > [Greg Lake, Infinite Space - Emerson, Lake & Palmer] === Subject: Re: ot: working on the internet > > Tell him to start an internet business selling training manuals on how to > > start an internet business. > Now that's funny. > Steve no that's what all work from home businesses are! here's the lady 15 years ago in the paper, this is a couple years after she was shot, so if anyone does know of any work from home opportunities be great. Herc === Subject: Paraboloidal Coordinates I've been looking at some coordinate systems on Mathworld, and for some of them we get x, y, z in terms like x^2 = (a^2-lambda)(a^2-mu)(a^2-nu)/(b^2-a^2) etc. What's with the squared quantities on the left-hand side? We're only supposed to work in an octant of three-dimensional space? Can't distinguish x=10 from x=-10? -- A good plan executed right now is far better than a perfect plan executed next week. -Gen. George S. Patton === Subject: Re: Physical repn in noneuclidean spaces? |> That makes good sense, in terms of the *mental* exercise |> involved. I was hoping for a discussion of the *hands-on* |> aspect. (1) How, in a concrete sense, do kinesthetic learners |> get involved with noneuclidean geometry? | |When you have a physical, curved object you want to do a construction |on, how about using a string made sticky with some kind of glue? |Assuming, in your words, the student has a steady hand. | |Funny--I abandoned that as a joking reply to your first post, it just |sounded too silly. But this time, I'm serious :) you could also put the string in between two complementarily curved pieces of glass, or something like that, if you're interested in some 2d space that embeds isometrically into 3d euclidean space. -- [e-mail address jdolan@math.ucr.edu] === Subject: Re: Physical repn in noneuclidean spaces? > What then do we use when we move to more complicated surfaces? Use must be made of geodesic polar coordinates to define arc length.The metric is ds^2= du^2+ G(u,v) dv^2 . We have geodesic radial lines and parallels. When the question first occured to me, I imagined in the following way. If from any point you travelled a distance R along a geodesic, and did the same in all directions, and joined the end points,you get a warped or crumpled circle, called the J-circle, locus of equi-distant points from the center.This is a geodesic parallel If the length of J circle perimeter = 2 Pi R you are on Euclidean space. If the length of J circle perimeter < 2 Pi R you are on Elliptic space. If the length of J circle perimeter > 2 Pi R you are on Hyperbolic space. The is true when the curved area covered by J circle is equal , less, or greater than Pi R^2. Each perimeter is a geodesic parallel. > Neither the straightedge nor the taut string is guaranteed to work well on a surface with negative curvature. As a matter of fact, neither the straightedge nor the taut string is guaranteed to work well on a surface even with a positive curvature.The straight edge has to be bent in the osculating plane. Now practically what measuring tool/instrument is to be employed? It should not be taut, but still be straight. We should use the same tool to trace geodesics on a curved surface. There are two ways to trace geodesics, one by using a cellulose tape, and other by using a common steel measuring tape ( width << length) used by engineers in field measurements and surveying.. To trace geodesics, choose the initial direction and start releasing from spool and pasting the cellotape along the chosen direction. Paste only a few millimeters each time by pressing with the index finger on the tape centerline.Every time ensure that you do not turn the tape right or left,but only straight ahead on a bumpy course.This gives zero geodesic curvature. When the steel tape is used, it is more easy as unlike a plastic tape, steel tape cannot be bent in the tangential/rectifying plane. The pasted tape executes all the convex, concave areas ( positive Gauss curvature ) as well as saddle points (negative Gauss curvature ). Physically, we can imagine a motorcyclist riding on a mountainous terrain, with peaks , valleys AND saddle cols. Hope this answers your question. === Subject: Re: Physics problem > Hello! > I wanted to make a simulation of an object being pulled through a > magnet and back, making a sine like function over time, however, I > don't have that much of a mathematical knowledge, because I'm a high > school student. > For convenience reasons, the situation is as following: > F=K'*p1*p2/r^2 > p1*p2=1/K' (without the units) > so F = 1/r^2 [N] > the mass of the object pulled is 1Kg, so the acceleration is: > a = 1/r^2 [m/s^2] > What I need is a function of the position and velocity of the ball > over time, or something like that.. > With gravity on earth, I understand the process, you Integrate the > velocity over dt, and get the distance, that's with constant > acceleration, with this one I can't figure it out. > Tal P Good luck with that dimensionally incorrect magnetic monopole field. TalP understands much less than he would have others believe. Bad start. Ignorance can be cured. Honesty counts up front. [Old Man] === Subject: Q: automorphisms of S_6 (was: finite groups) > BTW, do you have an easy way to check if PGL(2,9) or M_{10} has > such an outer automorphism? It occurred to me that whichever one > of them has its order-20 subgroups Frobenius might be a good > candidate, but I'd have to tackle them by hand, starting with > constructing them as subgroups of Aut(A_6), which seemed a > little daunting. > -- > Jim Heckman While we are at it: I have some trouble visualizing the outer automorphisms of S_6 (or A_6). I can construct these things by starting with the action of S_5 on its 6 Sylow-5 subgroups and extending that action to S_6 (surely straightforward as we have just identified a desired subgroup of index 6 in S_6) What I end up getting doesn't look very nice in the sense that I need to work relatively hard to compute the images of the permutations (at least the images of the permutations not in S_5). My questions are therefore: Q1. Is there a really simple way of seeing that the outer automorphisms of S_6 must exist and what they look like? E.g. a combinatorial or a graph theoretic (or even a geometric) way of constructing these beasts? Q2. Is there something that you would call THE outer automorphism of S_6, a unique one of minimal order, or some other property that would single out an individual outer automorphism (or a single conjugacy class of outer automorphisms)? Am I making sense?? Jyrki Lahtonen, Turku, Finland === Subject: Re: Q: automorphisms of S_6 (was: finite groups) in message <3F3CAAE8.C07FEADF@utu.fi>: [...] > While we are at it: I have some trouble visualizing the outer > automorphisms of S_6 (or A_6). I can construct these things by > starting with the action of S_5 on its 6 Sylow-5 subgroups and > extending that action to S_6 (surely straightforward as we > have just identified a desired subgroup of index 6 in S_6) > What I end up getting doesn't look very nice in the sense that > I need to work relatively hard to compute the images of > the permutations (at least the images of the permutations not > in S_5). > My questions are therefore: > Q1. Is there a really simple way of seeing that the outer > automorphisms of S_6 must exist and what they look like? > E.g. a combinatorial or a graph theoretic (or even a geometric) > way of constructing these beasts? > Q2. Is there something that you would call THE outer > automorphism of S_6, a unique one of minimal order, or > some other property that would single out an individual > outer automorphism (or a single conjugacy class of outer > automorphisms)? > Am I making sense?? To add to what Derek said: I don't know about how easy it is, but in my case I discovered the outer automorphisms of S_6 by myself when searching for relatively small non-abelian simple groups, for fun (and of course not invoking the Classification Theorem!). Using only fairly elementary group theory, it's straightforward, but rather tedious, to show that there's only one such group of order 360, and in the process of constructing a presentation for it the outer automorphisms practically jump out at you. In particular, the symmetry in A_6 of the two conjugacy classes of A_5-isomorphic subgroups makes the outer automorphisms of S_6 even more obvious, to me at least, than S_6 itself (as a subgroup of Aut(A_6)). -- Jim Heckman === Subject: Re: Q: automorphisms of S_6 (was: finite groups) > of S_6. It is slightly harder to find such a map which defines an > outer automorphism of order 2. But one example is: > (1,2) -> (1,2)(3,4)(5,6) > (2,3) -> (1,5)(2,4)(3,6) > (3,4) -> (1,2)(3,5)(4,6) > (4,5) -> (1,3)(2,4)(5,6) > (5,6) -> (1,2)(3,6)(4,5). the simplest description I've seen. Even I managed to figure out the fact that an automorphism of S_n (any n) mapping a 2-cycle to a 2-cycle must be inner. So it just is a remarkable coincidence that there are two conjugacy classes of this same size, when n=6. If my calculations are correct, then it seems to me that another pair conjugacy classes of S_6 of the same size, namely the classes of (1234)(56) and (1234) remain fixed under the outer automorphisms. No reason why they should change, but the classes of (123) and (123)(456) do get swapped, so I was mildly surprised to see this pair stay fixed. Nothing deep in their probably. > There is a unique conjugacy class of outer automorphisms of order 2, > so this example is unique up to renumbering the points. There are other > classes of outer automorphisms which do not have order 2 > (one of order 4, two of order 8, and one of order 10). > Derek Holt. Jyrki === Subject: Re: Q: automorphisms of S_6 (was: finite groups) Visiting Assistant Professor at the University of Montana. >> of S_6. It is slightly harder to find such a map which defines an >> outer automorphism of order 2. But one example is: >> (1,2) -> (1,2)(3,4)(5,6) >> (2,3) -> (1,5)(2,4)(3,6) >> (3,4) -> (1,2)(3,5)(4,6) >> (4,5) -> (1,3)(2,4)(5,6) >> (5,6) -> (1,2)(3,6)(4,5). >the simplest description I've seen. Even I managed to figure >out the fact that an automorphism of S_n (any n) mapping a >2-cycle to a 2-cycle must be inner. I highly recommend the following paper: Lam, T. Y. and Leep, David B. Combinatorial structure on the automorphism group of $Ssb 6$. Exposition. Math. 11 (1993), no. 4, 289--308. MR 94i:20006 >So it just is a remarkable coincidence that there are two >conjugacy classes of this same size, when n=6. Yes. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Question from Chaos I was recently reading through Chaos by James Gleick (finally - after all these years !), and in a paragraph on the dichotomy between mathematicians and physicists, he says Smale could be happy with an example like this : take a number, a fraction between zero and one, and double it. The drop the integer part, the part to the left of the decimal point. Then repeat the process. *Since most numbers are irrational and unpredictable in their fine detail*, the process will just produce an unpredictable sequence of numbers. What does he mean by that ? - what is unpredicatable about numbers in their *fine detail*. Also, is there some mathematical basis for most numbers are irrational ? Isnt this an issue of an uncountable infinity - or can you say that this type of infinity is *bigger* than the set of rational numbers ? -confused and bemuzed -simplicio === Subject: Re: Question from Chaos > I was recently reading through Chaos by James Gleick (finally - after all > these years !), and in a paragraph on the dichotomy between mathematicians > and physicists, he says > Smale could be happy with an example like this : take a number, a fraction > between zero and one, and double it. The drop the integer part, the part to > the left of the decimal point. Then repeat the process. *Since most numbers > are irrational and unpredictable in their fine detail*, the process will > just produce an unpredictable sequence of numbers. If by fraction one means a ratio of two integers m and n, i.e. the number m/n, then repeated doubling and taking the integer part will not give an unpredictable sequence. In base 2, the number 0. b1 b2 b3 b4 b5 ... where b1 , b2 , b3 are either 0 or 1 will give the sequence: 0.b2 b3 b4 ... , 0.b3 b4 b5... , .... except for numbers such as 0.011111111 ... (base 2) = 1/2. > What does he mean by that ? - what is unpredicatable about numbers in their > *fine detail*. Also, is there some mathematical basis for most numbers are > irrational ? Isnt this an issue of an uncountable infinity - or can you say > that this type of infinity is *bigger* than the set of rational numbers ? The rational numbers form a countable set because one can make an infinite list of them, q_1, q_2, q_3, ... which leaves none out and repeats none. It's impossible to make a list of all irrationals . David Bernier === Subject: Re: Question from Chaos This means most in the sense of measure. The set of numbers such that the binary expansion is not random is a set of measure zero. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ Distribution: world === Subject: Re: Solving Sums of Exponential and Linear Terms? > Assuming that the unknown is n: if a > 0 then an obvious real > solution is n=0. If a>1, this solution is unique (something is > strictly increasing). The case 0 double, or no, real solution. Make a sketch. Yeah... a sketch would have helped... my bad. > What's being kind? Guessing what you meant by something like I was worried people would point out obvious solutions using quaternions, set theory and the surface geometry of complex knot > Extend the problems to read > a^n + n = b > and then specialize for b=1. The answer uses a non-elementary > function called LambertW, namely: > n = (-lambertW(log(a)*exp(b*log(a)))+b*log(a))/log(a) > Conduct your own search about lambertW; it's fun. left me smacking my forehead. Finding a symbolic solution for my question has been a stopping block ever since I started studying the function of the position of a falling object with air resistence. The Lambert W function does look interesting; a new type of curve might be just what I'm looking for. http://www.wikipedia.org/wiki/Lambert's_W_function Starling Who could do it numerically, but where's the fun in that? :p === Subject: Summations, Geometric Series, Arithmetic Series Can someone please explain what the uses of G.P and A.P are and how and why we sometimes use summation (sigma) to bound these series? How can we -John === Subject: Re: Summations, Geometric Series, Arithmetic Series >Can someone please explain what the uses of G.P and A.P are and how and >why we sometimes use summation (sigma) to bound these series? How can we >tell an equation is G.P by looking at it? They are kinds of patterns. They can allow you to efficiently count. Geometric series are particularly useful in financial percent interest calculations. Check an intermediate or college algebra textbook and look at some of the example and word problems to find a better idea of how they are useful. You could very well understand geometric and arithmetic sequences and their sums with just elementary algebra knowledge. G C === Subject: Re: Summations, Geometric Series, Arithmetic Series > Can someone please explain what the uses of G.P and A.P are and how and > why we sometimes use summation (sigma) to bound these series? How can we Ap is a + a+d + a+2d + a+3d + ... notice the difference of any two terms is always the same, namely d. Gp is a + ar + ar^2 + ar^3 + ... notice the ratios of any two terms is always the same, namely r. http://www.newsfeeds.com - The #1 Newsgroup Service in the World! -----== Over 100,000 Newsgroups - 19 Different Servers! =----- === Subject: re: symbolic engine May I suggest you try SymbMath.com? 17.5. Interface with Other Software You can run SymbMath from another software as a engine. Another software sends a text file to SymbMath, then run SymbMath in background, get result back from SymbMath. Please read its document for details. www.SymbMath.com === Subject: The Fundamental Theorem of Fundamental Theorems Friends, I have recently stumbled upon a very interesting theorem which I have named The Fundamental Theorem of Fundamental Theorems. I have sent it to several journals for publication and as soon as it is published there I will present it in full detail for you to examine with all your scrutiny and skepticism. In the meantime I will give you an idea of what it does. It is a theorem which takes a natural number as an input and outputs a special case of itself, which tends to be a fundamental theorem by itself. Let me clarify this with an example since it may seem strange and new. Suppose we take the F.T.o.F.T. and set N=18. Why 18? Well, I chose 18 among a host of other arbitrary numbers completely at random, but I use it here because it just happens that with N=18 the F.T.o.F.T. degenerates into a special case often known as The fundamental theorem of calculus. Setting N=86 results in The fundamental theorem of algebra. Setting N=201 results in The fundamental theorem of Galois theory. Setting N=7099 results in The fundamental theorem of finite abelian groups. I have noticed one discrepancy as far as names go: if we set N=4427 in my F.T.o.F.T. it results in the Taniyama-Shimura Conjecture, which is arguably certainly a fundamental theorem, just not named as such. In light of this, perhaps we ought to change its name :-) Unfortunately my theorem is not proven for the case when N < 0. But by experimentation I discovered that setting N=-118 in my F.T.o.F.T., the special case which results is none other than the famous Continuum Hypothesis. For this reason I conclude that it is impossible to prove or disprove that my Fundamental Theorem of Fundamental Theorems holds for negative numbers (or at least for -118). Very interestingly, if we set N=0 the F.T.o.F.T. returns itself. I have experimented with plugging in quite a few numbers for N and every time without failure the result is always a fundamental theorem of some branch of mathematics. Most of these have not even been discovered independantly to my knowledge, and the majority are as useful if not more useful than the fundamental theorems of calculus or algebra. :-) I look forward to sharing more details with you as soon as the publishers reply to my papers I have submitted. Your friend, Nathan the Great Age 11 === Subject: Re: Thoughts on the Collatz conjecture Christian Bau > > Observing the Collatz conjecture you would logically have to believe > > it is true. > > The reason I make this statement is if a counter example was ever > > found and the integer path terminated in an endless loop OTHER THAN > > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > > other integers in this new path(s) and (tree) would also be involved. > > So if this conjecture has been tested for all start values < = 1.2 * > > 10^12 then all the integers involved in this new terminating path(s) > > and tree would have to have integer values > 1.2 * 10^12. > > So I believe a counter example is highly unlikely and thus a very > > strong conjecture. > Another conjecture is that for every prime number p, there is a prime > number p < q <= p+1000. This conjecture has been successfully tested for > all primes <= 10^15. So what do you think of this conjecture? Nice one. I'd say the odds are good that there is a counterexample <= 10^16 :) Then too, there is a probabilistic argument that integers don't exist: Pick a real number at random, and the probability is zero that it's an integer. QED :) Larry === Subject: Re: Thoughts on the Collatz conjecture I am still convinced that the conjecture is true, because after hastily throwing together a computer program that I believe makes the case. My argument here is, as each level of the Collatz tree grows new branches are born derived from a smaller integer start number and so does the density bands (see below) and their symmetry. This causes a symmetrical squeeze play so to speak for other possible counter examples start numbers and trees. I did a computer program where integer start numbers are entered in order where start number n = 1,2,3,4,5,6..n with there associated sequences. Each start # turns on a corresponding numbered pixel and all the path members turn on their respected pixels. These pixels stay on. Naturally some of these pixels are on when they get hit again for an on because of the branching tree affect and also returning back to 4,2,1 and terminating on 1. What happens as each new starting (n) the (5) density bands out ahead of the all white band become more apparent after about 32 y rows of all on (white) pixels. This would be an integer start number the size of n =32*640 = 20480. I set this thing up for reading across --- 640 pixels where x ( n) (start #)= 1 to 640 and y=1 Then where x = 1 to 640 and n=641 to 1280 and y=2 etc. Y is set for a max. of 350 pixels. It bypasses any on pixel that is not viewable on screen where y >350. It creates an interesting effect with 6 distinct bands with each of the 5 bands having distinct symmetrical density patterns out ahead of (n) the seed that is the all white band behind the seed. This density band effect could be do to certain delta factors out ahead of the seed or something to do with 2^n? The short basic program is listed below with plenty of documentation so someone can translate to Java, c++ or any other language. ë = Docs 4 ë A Collatz conjecture pixel evaluation 5 ' This program turns on the appropriate pixel for each starting integer and all its sequence members. Pixels once on, stay on. 10 CLS 12 Screen 9: ë Set graphics screen mode to 350 X 640 pixels 15 DEFDBL A: ëDouble precision for any variable beginning with A 20 A=1:A3=A:A4=1:Y=1:A5=640:A6=640: PSET(A,Y): ë A is starting integer (seed) and turns on pixel x(A) = 1 and Y = 1: This line never used again. 30 A1$=STR$(A): ë Line 30-54 checks to see if integer is odd or even. 40 A2=LEN(A1$) 50 J$=MID$(A1$,A2,1): IF J$= 1 THEN GOTO 200 51 IF J$= 3 THEN GOTO 200 52 IF J$= 5 THEN GOTO 200 53 IF J$= 7 THEN GOTO 200 54 IF J$= 9 THEN GOTO 200 60 A3=A3/2:A=A3:GOSUB 500: IF A =< 1 THEN 320 ELSE 30:' This line handles even integers and goes to subroutine that evaluates the correct x and y pixel to turn on. 200 A3=(A*3) +1:A=A3:GOSUB 500:GOTO 60: ' Handles odd integers of seed and its sequence. Ect. 320 A4=A4+1:A=A4:A3=A4:GOSUB 500:GOTO 30: 'Retrieves the next seed and repeats the whole process creating a new sequence from that seed. 500 IF A>A6 THEN A6=A6+A5:Y=Y+1:ELSE 530: ' Sets Up A for right row (Y) 510 If Y>350 THEN Y=1:GOTO 540: ' If integer value in any sequence is > (350*640) then this line bypasses the pixel command (PSET) because pixel will not be in a viewable area of the screen. 515 IF A>A6 THEN 500: ' Go back to line 500 and add another 640 to variable A6 520 IF A=< A6 AND Y>1 THEN Y=Y-1:A7=Y*A5:A8=A-A7:Y=Y+1:PSET(A8,Y):Y=1:GOTO 540: ' Sets up x(A8) value when y>1 and thus the correct x,y coordinates for any applicable integer with a value > 640. 530 If Y=1 THEN PSET(A,Y): GOTO 540:' A branch from 500 where Else means Y=1 540 A6=A5:Y=1:RETURN: ' Resets variables and returns for next integer. 600 END Please excuse the hastily thrown together code. Should have done a renumber also! You have to think of each row of 640 pixels as rows cut off at that point and then stacked on each other where you can easily view how these density patterns out ahead of the seed number are formed. This would probably go unnoticed if the line stayed continuous as in the number line. This creates 5 distinct and fascinating density band patterns that grow in width as the first solid white band or seed band grows in width. Please note, when first starting out the bottom (last) density pattern starts to show a checkerboard pattern on an angle. If nothing more, its interesting! As always, any evaluations or comments are welcome. Dan === Subject: Re: Triangle law in relativity === >Subject: Triangle law in relativity. >For 'regular' geometry, if we construct one straight line, and then >two other straight lines, forming a triangle, with the length of sides >A, B and C, we get A < B + C. >For relativity, if we have an observer going at a constant speed in an >inertial reference frame who experiences time L, and if we have >another observer starting from the same point in timespace as the >first observer and then moves away and experiences time M, and then >turns around and comes back and experiences time N, then L > M + N. >This result will be independant on any of the reference frames. This >is, of course, assuming SR. >At any rate, does anybody here see the analogy? >This gives a definite resolution to the Twin's Paradox. This result No it doesn't. The Twin's Paradox isn't about whether or not one of them is older when again they meet... We accept SR and the fact that one WILL be older... the paradox is WHICH ONE. How can you say which of the twins moved away from the other and which stayed in the same place. >Asking why one twin ages slower than the other is like asking why if a >person walks in a straight line, they'll get there before a person >moving at the same speed who walks two different straight lines to get >there. Which of the twins walked the straight line, and which walked two different straight lines. That's the paradox. adam === Subject: Two orthogonal Latin squares of order 10 I was trying to save myself the trouble of typing in two orthogonal Latin squares of order 10 by finding them online somewhere. But my search was unsuccessful. So I copied a pair into a Maple worksheet from Marshall Hall's Combinatorial Theory, page 194. According to this source the following pair is due to Ostrowski and Van Duren. It serves to show that Mann's inequality (on the same page) is best possible. So to allow anyone else searching for an online pair of orthogonal Latin squares of order 10 to get a pair by copying and pasting here they are: 0 1 2 3 4 5 6 7 8 9 3 4 0 1 2 7 9 8 6 5 4 3 1 2 0 9 7 6 5 8 1 2 4 0 7 8 5 3 9 6 2 0 3 7 5 6 8 9 4 1 5 7 6 9 8 3 4 1 2 0 8 9 7 5 6 1 2 0 3 4 6 5 9 8 1 4 3 2 0 7 9 8 5 6 3 0 1 4 7 2 7 6 8 4 9 2 0 5 1 3 0 1 9 2 3 8 4 6 5 7 6 7 8 9 5 2 3 1 0 4 9 3 7 4 6 5 8 2 1 0 3 8 2 5 4 7 9 0 6 1 1 4 5 0 7 3 6 9 8 2 2 5 6 1 9 4 0 8 7 3 4 0 1 3 8 6 2 7 9 5 5 6 4 8 0 1 7 3 2 9 8 2 0 7 1 9 5 4 3 6 7 9 3 6 2 0 1 5 4 8 If lists of lists are easier to deal with, here you have'm: A := [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [3, 4, 0, 1, 2, 7, 9, 8, 6, 5], [4, 3, 1, 2, 0, 9, 7, 6, 5, 8], [1, 2, 4, 0, 7, 8, 5, 3, 9, 6], [2, 0, 3, 7, 5, 6, 8, 9, 4, 1], [5, 7, 6, 9, 8, 3, 4, 1, 2, 0], [8, 9, 7, 5, 6, 1, 2, 0, 3, 4], [6, 5, 9, 8, 1, 4, 3, 2, 0, 7], [9, 8, 5, 6, 3, 0, 1, 4, 7, 2], [7, 6, 8, 4, 9, 2, 0, 5, 1, 3]] B := [[0, 1, 9, 2, 3, 8, 4, 6, 5, 7], [6, 7, 8, 9, 5, 2, 3, 1, 0, 4], [9, 3, 7, 4, 6, 5, 8, 2, 1, 0], [3, 8, 2, 5, 4, 7, 9, 0, 6, 1], [1, 4, 5, 0, 7, 3, 6, 9, 8, 2], [2, 5, 6, 1, 9, 4, 0, 8, 7, 3], [4, 0, 1, 3, 8, 6, 2, 7, 9, 5], [5, 6, 4, 8, 0, 1, 7, 3, 2, 9], [8, 2, 0, 7, 1, 9, 5, 4, 3, 6], [7, 9, 3, 6, 2, 0, 1, 5, 4, 8]] Now, I'm sure someone is going to tell me where I could have obtained them. :-) --Edwin === Subject: Re: Two orthogonal Latin squares of order 10 > I was trying to save myself the trouble of typing in two orthogonal Latin > squares of order 10 by finding them online somewhere. But my search was > unsuccessful. So I copied a pair into a Maple worksheet from Marshall Hall's > Combinatorial Theory, page 194. According to this source the following > pair is due to Ostrowski and Van Duren. It serves to show that Mann's > inequality (on the same page) is best possible. > So to allow anyone else searching for an online pair of orthogonal Latin > squares of order 10 to get a pair by copying and pasting here they are: > 0 1 2 3 4 5 6 7 8 9 > 3 4 0 1 2 7 9 8 6 5 > 4 3 1 2 0 9 7 6 5 8 > 1 2 4 0 7 8 5 3 9 6 > 2 0 3 7 5 6 8 9 4 1 > 5 7 6 9 8 3 4 1 2 0 > 8 9 7 5 6 1 2 0 3 4 > 6 5 9 8 1 4 3 2 0 7 > 9 8 5 6 3 0 1 4 7 2 > 7 6 8 4 9 2 0 5 1 3 > 0 1 9 2 3 8 4 6 5 7 > 6 7 8 9 5 2 3 1 0 4 > 9 3 7 4 6 5 8 2 1 0 > 3 8 2 5 4 7 9 0 6 1 > 1 4 5 0 7 3 6 9 8 2 > 2 5 6 1 9 4 0 8 7 3 > 4 0 1 3 8 6 2 7 9 5 > 5 6 4 8 0 1 7 3 2 9 > 8 2 0 7 1 9 5 4 3 6 > 7 9 3 6 2 0 1 5 4 8 > If lists of lists are easier to deal with, here you have'm: > A := [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], > [3, 4, 0, 1, 2, 7, 9, 8, 6, 5], > [4, 3, 1, 2, 0, 9, 7, 6, 5, 8], > [1, 2, 4, 0, 7, 8, 5, 3, 9, 6], > [2, 0, 3, 7, 5, 6, 8, 9, 4, 1], > [5, 7, 6, 9, 8, 3, 4, 1, 2, 0], > [8, 9, 7, 5, 6, 1, 2, 0, 3, 4], > [6, 5, 9, 8, 1, 4, 3, 2, 0, 7], > [9, 8, 5, 6, 3, 0, 1, 4, 7, 2], > [7, 6, 8, 4, 9, 2, 0, 5, 1, 3]] > B := [[0, 1, 9, 2, 3, 8, 4, 6, 5, 7], > [6, 7, 8, 9, 5, 2, 3, 1, 0, 4], > [9, 3, 7, 4, 6, 5, 8, 2, 1, 0], > [3, 8, 2, 5, 4, 7, 9, 0, 6, 1], > [1, 4, 5, 0, 7, 3, 6, 9, 8, 2], > [2, 5, 6, 1, 9, 4, 0, 8, 7, 3], > [4, 0, 1, 3, 8, 6, 2, 7, 9, 5], > [5, 6, 4, 8, 0, 1, 7, 3, 2, 9], > [8, 2, 0, 7, 1, 9, 5, 4, 3, 6], > [7, 9, 3, 6, 2, 0, 1, 5, 4, 8]] > Now, I'm sure someone is going to tell me where I could have obtained them. > :-) > --Edwin Yours is the best I found for cut and paste to maple! I poked around a bit looking for other examples. I found these on the net: Text: http://academic.uofs.edu/faculty/doughertys1/square.htm Pictures: http://buzzard.ups.edu/squares.html http://www-math.cudenver.edu/~wcherowi/grid10.gif http://math.dartmouth.edu/sphere/ http://www.brocku.ca/mathematics/courses/math3p81.phtml PDF: on page 9 of 10 of this document (The document is chapter 9 of an on-line text, and is p 160 of that text): http://www.maths.qmw.ac.uk/~rab/DOEbook/doeweb9.pdf Some may duplicate others, I didn't check. --Jim Buddenhagen === Subject: Re: Two orthogonal Latin squares of order 10 > I poked around a bit looking for other examples. > I found these on the net: > Text: > http://academic.uofs.edu/faculty/doughertys1/square.htm This one can be cut and pasted from the source. With more work one can convert the pair to the following list of lists form, in case someone wants to play with them. And find, for example, a third Latin square of order 10 orthogonal to these two. :-) A := [[0, 7, 8, 6, 9, 3, 5, 4, 1, 2], [6, 1, 7, 8, 0, 9, 4, 5, 2, 3], [5, 0, 2, 7, 8, 1, 9, 6, 3, 4], [9, 6, 1, 3, 7, 8, 2, 0, 4, 5], [3, 9, 0, 2, 4, 7, 8, 1, 5, 6], [8, 4, 9, 1, 3, 5, 7, 2, 6, 0], [7, 8, 5, 9, 2, 4, 6, 3, 0, 1], [4, 5, 6, 0, 1, 2, 3, 7, 8, 9], [1, 2, 3, 4, 5, 6, 0, 9, 7, 8], [2, 3, 4, 5, 6, 0, 1, 8, 9, 7]] B := [[1, 2, 3, 4, 5, 6, 0, 7, 8, 9], [0, 3, 9, 5, 4, 7, 6, 2, 1, 8], [6, 0, 5, 8, 7, 4, 2, 9, 3, 1], [9, 6, 0, 7, 1, 2, 4, 8, 5, 3], [4, 8, 6, 0, 2, 3, 9, 1, 7, 5], [8, 4, 1, 6, 0, 9, 5, 3, 2, 7], [7, 1, 4, 3, 6, 0, 8, 5, 9, 2], [3, 5, 7, 2, 9, 8, 1, 4, 0, 6], [5, 7, 2, 9, 8, 1, 3, 0, 6, 4], [2, 9, 8, 1, 3, 5, 7, 6, 4, 0]] === Subject: uniformly continuous?? hello.sir~ show that f(x) = e^x is not uniformly conti on R --------------------- not uniformly conti <=> given e>0, for each d>0, there is x,y in R |x-y| |f(x)-f(y)| >= e ----------------------- in the solution paper, it have put the x=1/d y=(1/d) + (d/2) i think that this is wrong choice it can apply to f(x)=x^2 can it apply to f(x)=e^x ?? if so, how put the x,y,e?? help me, please. genius doctor === Subject: Vowel dropouts (was Re: Why...) In sci.math, Victor Eijkhout <1fzqoqh.sitdjs138jugwN%see.signature@for.address>: >> >>Like that referee report I got the other day. It complimented me on >> >>keeping the paper short. Then he wanted more material, and the paper >> >>shortened. Right. >> > >> > Lv t th vwls. Nd th vrbs. Rdndnt! >> nd nrdbl. :-) > Nada nerdbolo? Nod innerdouble? Heh...It was supposed to be And unreadable. :-) But you've nicely illustrated the problem of eliminating the vowels. Another one: She's not meeting me at the meat market, so the attempt at mate is moot, Mort. turns into Sh's nt mtng m t th mt mrkt, s th ttmpt t mt s mt, Mrt. > V. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: What is an algebraic integer? >Not factorable over the ring of coefficients. So, a polynomial such as x^2 - 2x -15, which factors to (x+3)(x-5) would not be called an irreducible polynomial (and, presumably, would be called a reducible polynomial)? Or, doesn't ring of coefficients refer to the integers? Does it refer to some ring that is defined for each polynomial, based upon its specific coefficients? What about x+3 and x-5, taken as individual polynomials? Are they considered irreducible? >>I think the irreducible bit in the definition is not really necessary. >>As you noticed, the fact that x^2 - 2x - 15 has roots -3 and 5 doesn't >>make -3 and 5 algebraic integers according to the definition, but >>because the polynomial is reducible you then get the irreducible >>polynomials x+3 and x-5, so -3 and 5 are algebraic integers after all. >Any guesses as to *why* some definitions do include irreducible then? >And, a bigger question. Is there an effective procedure for determining >(without the assumption of closure) if any given number is an algebraic >integer or not? Right now, all that I can see is if you can find a >monic (irreducible) polynomial with integer coefficients that has A as >a root, then A is an algebraic integer. This definition, of itself, >isn't decidable, since my failure to find a polynomial with A as a root >isn't proof that no such polynomial exists. It might only be due to >my lack of extreme cleverness. Whether there's an effective procedure depends on how you're given the number as input. If you're given algebraic _numbers_ x and y, and their minimal poynomials, there is an effective procedure for finding the minimial polynomial of x + y, xy and x/y (I believe). Hence if you're given an algebraic integer expressed, say, as an algebraic combination of radicals, there is an effective procedure for finding its minimial polynomial. And then if you have the minimal polynomial for an algebraic number you decide whether it's an algebraic integer by looking to see whether the (primitive) minimial polynomial is monic. ************************ David C. Ullrich === Subject: Re: What is an algebraic integer? > > I've been seeing this term a lot in James Harris's proofs. Can someone > > please give me the definition, so that I can check those proofs for > > myself? > Algebraic integers are solutions to polynomials > x^n + a_(n-1) x^(n-1) +..+ a_1 x + a_0 = 0 > with integer coefficients and a_n = 1. > For example sqr 3, 1 + sqr 2, (1 + sqr 5)/2. > Don't feed the troll, maybe it'll go away. Why does a_n have to equal one? So it can't be a solution to a polynomial with rational coefficients? Take P = 3x^2 + 2x - 7. The solutions are (-1 + sqrt(22))/3 and (-1 - sqrt(22))/3. Since the only polynomial to which they can be solutions must also be a multiple of 3x^2 + 2x - 7, and since all such multiples must have either a_n equal to a multiple of 3 or the other a's being rational, then those two numbers are not alegbraic integers. Yet, somewhere I heard that any number that can be composed of any operations of addition, subtraction, multiplication, division, and raising to a rational exponent, are all algebraic integers. How can this be so? (...Starblade Riven Darksquall...) === Subject: Re: What is an algebraic integer? >> > I've been seeing this term a lot in James Harris's proofs. Can someone >> > please give me the definition, so that I can check those proofs for >> > myself? >> > Algebraic integers are solutions to polynomials >> x^n + a_(n-1) x^(n-1) +..+ a_1 x + a_0 = 0 >> with integer coefficients and a_n = 1. >> For example sqr 3, 1 + sqr 2, (1 + sqr 5)/2. >> Don't feed the troll, maybe it'll go away. >Why does a_n have to equal one? So it can't be a solution to a >polynomial with rational coefficients? The definition of algebraic integers stipulates that a_n = 1. If you allow arbitrary integers as a_n, then you get all elements of the field of algebraic numbers. Note that a rational number is an algebraic integers iff it is an integer. >Take P = 3x^2 + 2x - 7. The solutions are (-1 + sqrt(22))/3 and (-1 - >sqrt(22))/3. Since the only polynomial to which they can be solutions >must also be a multiple of 3x^2 + 2x - 7, and since all such multiples >must have either a_n equal to a multiple of 3 or the other a's being >rational, then those two numbers are not alegbraic integers. That's right. >Yet, somewhere I heard that any number that can be composed of any >operations of addition, subtraction, multiplication, division, and >raising to a rational exponent, are all algebraic integers. How can >this be so? It is not so. Just like the normal integers, the algebraic integers form a ring but not a field - so they are closed under addition, subtraction, and multiplication, but not under division. It is true (but not obvious) that the algebraic integers are closed under raising to a rational exponent. Derek Holt. because if a is an algebraic integer, then a === Subject: Re: What is an algebraic integer? Visiting Assistant Professor at the University of Montana. [.snip.] >Yet, somewhere I heard that any number that can be composed of any >operations of addition, subtraction, multiplication, division, and >raising to a rational exponent, are all algebraic integers. How can >this be so? It's not. But the sum, multiplication, and rational powers of any algebraic integer is an algebraic integer. In fact, the roots of any MONIC polynomial with algebraic integer coefficients is again an algebraic integer. But you cannot, in general, take quotients. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: What is an algebraic integer? > [.snip.] > >Yet, somewhere I heard that any number that can be composed of any > >operations of addition, subtraction, multiplication, division, and > >raising to a rational exponent, are all algebraic integers. How can > >this be so? > It's not. But the sum, multiplication, and rational powers of any > algebraic integer is an algebraic integer. In fact, the roots of any > MONIC polynomial with algebraic integer coefficients is again an > algebraic integer. But you cannot, in general, take quotients. The simplest example would be 1/2 which is _not_ an algebraic integer. Assume 1/2 is the root of a monic polynomial of degree n, that is a polynomial starting with x^n + ... and with all coefficients zero. Multiply by 2^(n-1). At x = 1/2, the first term is then (1/2)^n * 2^(n-1) = 1/2. All the other terms are integers. 1/2 plus any number of integers cannot add up to zero. === Subject: Which are the fastest algorithms to solve large polynomial systems? Hello. Which are the momentary fastest (serial) algorithms for the solution of large polynomial systems with several variables (rational coefficients, real solutions, up to 50 variables, polynomial degrees up to 50, arbitrarily many polynomials)? === Subject: Re: Which are the fastest algorithms to solve large polynomial systems? > Hello. > Which are the momentary fastest (serial) algorithms for the solution of > large polynomial systems with several variables (rational coefficients, real > solutions, up to 50 variables, polynomial degrees up to 50, arbitrarily many > polynomials)? Surely if we limit ourselves to 50 variables, it cannot be sensible to allow arbitrarily many polynomials in the same problem? Let us assume an equal number of variables and (algebraically independent) constraints. If by momentary fastest you mean a local rate of convergence, then of course a Newton method would be one candidate, since (with sufficiently close initial guess to an solution and no multiple roots involved) the convergence would be quadratic. However the Newton algorithm would require computation of the matrix of gradients at each step and the solution of a system of equations. As a practical matter one usually settles for a quasi-Newton method, in which the gradients are updated or simply frozen for a sequence of steps. === Subject: Re: Who here believes maths is all there is? > I mean, from the Calculus used by physics or perhaps the cellular automata > for others, through to information theory and Turing Machines, its all just > numbers. > Perhaps God said: > Let there be the empty set and the set inclusion operator. God actually said: Let there be Johann Sebastian Bach's Music... and this was good, since it is still the only perfect thing in our existence. Math is far from perfect, however you interpret the word perfect. Then Math followed, trying to explain the music, but it was found insufficient. Out of this insufficiency, math created all else, out of sheer disgust and jealousy for the music. The only thing Math is subservient to, is the Music of JSB. All else is subservient to Math. -- Ioannis http://users.forthnet.gr/ath/jgal/ ___________________________________________ Eventually, _everything_ is understandable. === Subject: Re: Who here believes maths is all there is? > I mean, from the Calculus used by physics or perhaps the cellular automata > for others, through to information theory and Turing Machines, its all just > numbers. > Perhaps God said: > Let there be the empty set and the set inclusion operator. > The rest just sort of follows ... As a teenager, I think that I really used to think this way. I thought that the fundamental building blocks of the universe would be some mathematical construction. Then at the age of 23, I became a Christian, and a lot of my world views changed a bunch. Now I believe that the fundamental building block of the universe is that most seemingly anthropomorphic substance, love. That is, relationships are the most important things, and the physical universe simply hangs off that. I also pondered the question - which is more primary - God or mathematics? That is, is God contrained by the laws of mathematics and logic, or did God exist before mathematics and logic? I guess these are kind of unanswerable questions, but there is a passage in the Bible that perhaps suggest the latter. This is in Proverbs Chapter 8, and is part of a long discourse about wisdom. 22 The LORD brought me [wisdom] forth as the first of his works, before his deeds of old; 23 I was appointed from eternity, from the beginning, before the world began. 24 When there were no oceans, I was given birth, when there were no springs abounding with water; 25 before the mountains were settled in place, before the hills, I was given birth, 26 before he made the earth or its fields or any of the dust of the world. 27 I was there when he set the heavens in place, when he marked out the horizon on the face of the deep, 28 when he established the clouds above and fixed securely the fountains of the deep, 29 when he gave the sea its boundary so the waters would not overstep his command, and when he marked out the foundations of the earth. 30 Then I was the craftsman at his side. I was filled with delight day after day, rejoicing always in his presence, 31 rejoicing in his whole world and delighting in mankind. -- Stephen Montgomery-Smith stephen@math.missouri.edu http://www.math.missouri.edu/~stephen === Subject: Re: Who here believes maths is all there is? Where did Noah put the dinosaurs? > > I mean, from the Calculus used by physics or perhaps the cellular automata > > for others, through to information theory and Turing Machines, its all just > > numbers. > > Perhaps God said: > > Let there be the empty set and the set inclusion operator. > > The rest just sort of follows ... > As a teenager, I think that I really used to think this way. I thought that the > fundamental building blocks of the universe would be some mathematical > construction. Then at the age of 23, I became a Christian, and a lot of my > world views changed a bunch. Now I believe that the fundamental building block > of the universe is that most seemingly anthropomorphic substance, love. That > is, relationships are the most important things, and the physical universe > simply hangs off that. > I also pondered the question - which is more primary - God or mathematics? That > is, is God contrained by the laws of mathematics and logic, or did God exist > before mathematics and logic? I guess these are kind of unanswerable questions, > but there is a passage in the Bible that perhaps suggest the latter. This is in > Proverbs Chapter 8, and is part of a long discourse about wisdom. > 22 The LORD brought me [wisdom] forth as the first of his works, > before his deeds of old; > 23 I was appointed from eternity, > from the beginning, before the world began. > 24 When there were no oceans, I was given birth, > when there were no springs abounding with water; > 25 before the mountains were settled in place, > before the hills, I was given birth, > 26 before he made the earth or its fields > or any of the dust of the world. > 27 I was there when he set the heavens in place, > when he marked out the horizon on the face of the deep, > 28 when he established the clouds above > and fixed securely the fountains of the deep, > 29 when he gave the sea its boundary > so the waters would not overstep his command, > and when he marked out the foundations of the earth. > 30 Then I was the craftsman at his side. > I was filled with delight day after day, > rejoicing always in his presence, > 31 rejoicing in his whole world > and delighting in mankind. > -- > Stephen Montgomery-Smith > stephen@math.missouri.edu > http://www.math.missouri.edu/~stephen === Subject: Re: Who here believes maths is all there is? cj-bubba@mindspring.com asks: >Where did Noah put the dinosaurs? Good question, and well worth treating in a logical way; That story has its value, but that value is not entirely historical fact, but more preferably a story to illustrate ethics and conduct in a literary artistic way. The authors(?) of the story did not know about dinosaurs. Math occurs plainly in a few ways in that great story: pi, and the cubit. Maybe other ways, too. G C === Subject: Re: Who here believes maths is all there is? > Perhaps God said: > Let there be the empty set and the set inclusion operator. Then God created Godel, and that just screwed everything up. === Subject: Re: Why... > > >>Like that referee report I got the other day. It complimented me on > > >>keeping the paper short. Then he wanted more material, and the paper > > >>shortened. Right. That was like a report I got back. The paper itself was very simple, but the editor-in-chief wanted it more mathematical. I added about 2.5 pages of mathematics and a lot more of a complex construction. The editor-in-chief LOVED it. The referees HATED it. Why? Too complicated and too mathematical. === Subject: Re: working on the internet > > I have an email pen pal who's in a wheelchair asking me about > > working from home. > > I checked out numerous of these offers a few months ago for myself > > and they are all asking for a payment for the training manual. > It would lake study, some investment in inventory, etc. but it's possile to > have a profitable home internet business. > A lot of the stuff on e-bay, for example, is there because you just can't > buy the stuff locally and many trade suppliers don't want to piss off > their commercial customers (repairmen and installers) by selling to > civilians. > I give you two examples of stuff that consumers might want but can't buy > easily: > 1) High efficiency variable speed motor controllers > 2) The tiny bits of plastic hardware used on doors, windows, etc. > The trick is to get the word out while keeping a low profile. You don't > want to have your suppliers cut you off and you don't want to be stuck with > lots of inventory if the local Wally World or Home Depot starts to carry > your line. > If you make your website easy to find and meet your customers needs the > profits will come. I had CDBar.com on hold last month but another snap company obtained it, would have been ideal having someone in US packaging the CDs. CDWow.com and CDNow and a few others are raking in sales. When one of my other domains kick off I might have some part time work for someone, easy to start a business with money! Herc === Subject: |Q|<|R| Hello dear people, > In the attached address you can find an overlay > of a new theory of numbers: > http://www.geocities.com/complementarytheory/CATpage.html Yicks, hard to read as words are boken at end of line > I shall appreciate very much your remarks and insights. Downdoad pdf? Oh posh, why not TeX or ascii version? http://www.newsfeeds.com - The #1 Newsgroup Service in the World! -----== Over 100,000 Newsgroups - 19 Different Servers! =----- === Subject: Re: |Q|<|R| Hello dear people, > In the attached address you can find an overlay > of a new theory of numbers: > http://www.geocities.com/complementarytheory/CATpage.html > I shall appreciate very much your remarks and insights. > Yours, > Doron Shadmi === Subject: Re: |Q|<|R| > Hello dear people, > > In the attached address you can find an overlay > > of a new theory of numbers: > > http://www.geocities.com/complementarytheory/CATpage.html > Yicks, hard to read as words are boken at end of line > > I shall appreciate very much your remarks and insights. > Downdoad pdf? Oh posh, why not TeX or ascii version? > http://www.newsfeeds.com - The #1 Newsgroup Service in the World! > -----== Over 100,000 Newsgroups - 19 Different Servers! =----- Hi William Elliot, You can download an Acrobat(pdf) reader for free through my website. A lot of papers are now in this format. Yours, Doron === Subject: Re: |Q|<|R| Your one and only axiom is that In the middle of a quantum leap there are > zero points. In an axiom system, you have to define all terms, including > zero, points and middle, not to mention quantum leap. > Although I did like the pdf layout. > > Hello dear people, > > In the attached address you can find an overlay > > of a new theory of numbers: > > http://www.geocities.com/complementarytheory/CATpage.html > > I shall appreciate very much your remarks and insights. > > Yours, > > Doron Shadmi Hi Peter Webb This is an overlay on the theory and not a technical paper of it. So, please try to follow the ideas, and please be more specific. Doron === Subject: Re: |Q|<|R| Content exists = ({__}~={...}) = 1 and level A is phase transition between 0(=on content) to 1(=content). Complementary Level B: {__} <--> {...} By defining the relations between the above concepts, we find that the structure concept has more interesting information than the quantity concept because: 0^0 = infi^0 = 1 = content exists and we can't distinguish between the contents by the quantity concept. But it can be done by the structure concept because: {__}~={...} and we can learn that the structure concept has more information than the quantity concept. === Subject: Re: |Q|<|R| Content exists = ({__}~={...}) = 1 and level A is phase transition between 0(=on content) to 1(=content). Complementary Level B: {__} <--> {...} By defining the relations between the above concepts, we find that the structure concept has more interesting information than the quantity concept because: 0^0 = infi^0 = 1 = content exists and we can't distinguish between the contents by the quantity concept. But it can be done by the structure concept because: {__}~={...} and we can learn that the structure concept has more information than the quantity concept.