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 (ofŽcial) 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 (ofŽcial) 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 shouldn¹t 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 Russell¹s Žrst paradox Russell¹s Žrst 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-deŽned set. Does it contain itself? If we assume that it does, it is not a member of M according to the deŽnition. 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 deŽnition 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: DeŽnition A: ------------- ( http://www.cut-the-knot.org/selfreference/russell.shtml ) Sets are deŽned 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 deŽnition 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 deŽnition 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 Russell¹s Žrst paradox > Russell¹s Žrst 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-deŽned set. Does it contain itself? If we assume that it > does, it is not a member of M according to the deŽnition. > 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 > deŽnition 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: > DeŽnition A: > ------------- > ( http://www.cut-the-knot.org/selfreference/russell.shtml ) > Sets are deŽned 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 deŽnition 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 deŽnition 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 isn¹t 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 deŽned. === Subject: a comment on searching for 3 orthogonal 10x10 squares While trying to Žnd 3 such orthogonal squares, I used over 300 sets of duals, but the best I could do was Žt 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 Žnish 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 I¹ve also heard that a lot of people who take time off don¹t ever go back to school, that it¹s 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 Želd (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 isn¹t worth going to graduate school. Finally, I¹d 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 isn¹t > 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 don¹t say it isn¹t worth going to graduate school; just say it isn¹t worth going to graduate school for me. === Subject: Re: Advice about applying to Math Graduate School > Finally, I¹d 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. That¹s a very good idea; In fact, I¹d like to do that, in the opposite way (I¹m 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 I¹ll lay your soul to waste - The Rolling Stones, Sympathy for the Devil === Subject: Algorithm to Žnd the n-root of a equation. Hi guys. im trying to create a program to Žnd 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 y¹s 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 Žnd the root, but if i choose another number, i Žnd 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 Žnd 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 >I¹m trying to Žnd 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 Žnd W, I can use W=G*inv(A) (1) >or W=pinv(A*pinv(G)) (2). >(1) works generally Žne 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 Žnd a W without inverting A ? Are there any other >possibilites ? Any help will be appreciated. In general, to solve matrix-vector equations you shouldn¹t invert the matrix (but LU factorization can be useful...). Of course, A should be invertible, otherwise solutions either won¹t exist or won¹t 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 Œthinking outside the box.¹ Look at the works of Emmy Noether on symettry in physical systems or Kurt Goedel¹s works. Barwise and Etchemendy describe Goedel as the Žrst hacker because of the foresight his work required to deal with the complexity of Žnding 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 Œnuff. I suppose I should clarify: I¹m 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. I¹ve worked in education (community college + technical school + > for-proŽts, 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, Žrst 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 dreamer¹s shop) because I > felt like a robot at this place. Later in life, I also worked at a > for-proŽt 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 don¹t make enough for the BMW, but as long as you do make enough to feed > yourself. > So I¹m back for the Ph.D., so¹s I can research, which is the Želd I¹ve > dreamed about forever, but for various reasons have never done until now. > Herc, I¹ll 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 I¹m curious if any companies are speciŽcally 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? I¹m 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 I¹m 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. I¹d 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 Žsh at www.monster.com its the biggest employment forum, say you¹re graduating this year! Herc I¹ll add your question here so I didn¹t terminate the thread > go that route, but I¹m curious if any companies are speciŽcally 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 I¹ve 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 Žnd 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, I¹d *think* Žnancial Žrms, insurance companies, > >computer companies would hire mathematicians. I haven¹t 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 you¹re willing to work on this side of the Atlantic. MS > >Research has positions if you¹re in computer-related math (e.g., crypto), > >with locations in the Seattle area, Cambridge, England etc. Of course, > >there¹s the NSA, if you¹re willing. (Sorry, Leigh Ann, I don¹t have speciŽc > >info on Australia.) > > The recruiter at some point will ask something like What can you do for > the company? It¹s getting popular these days to try to model Žnancial > systems with the methods of abstract mechanics from physics. Insurance > companies hire statisticians. I¹ve 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 you¹ve learned? What are other people doing > with the math you¹ve learned? Hi guys, particularly in Žnance, 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 Žnancial services Žrm 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 Žnance - 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, I¹ve 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 Žnance, 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 Žt into the trading environment that the above divisions reside in - Male dominated (?) There have been some academics who¹ve made it into the industry - for instance, my mentor is leaving to work for Susquahanna, a derivatives trading Žrm. But in general, given two strong candidates from math and Žnance, there is a tendency for people to hire in Žnance. Anyway, this is my experience from Sydney, Australia, so it may be different elsewhere! And if so, I may consider moving. Otherwise, it¹s 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! It¹s 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 I¹ve 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 Žnd 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 insigniŽcant... > Off the top of my head, I¹d *think* Žnancial Žrms, insurance companies, > computer companies would hire mathematicians. I haven¹t 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 you¹re willing to work on this side of the Atlantic. MS > Research has positions if you¹re in computer-related math (e.g., crypto), > with locations in the Seattle area, Cambridge, England etc. Of course, > there¹s the NSA, if you¹re willing. (Sorry, Leigh Ann, I don¹t have speciŽc > info on Australia.) > > Are there any good listings of companies that hire math Ph.D.s? I¹ve tried a > few books, but the ones I¹ve looked at tend to list categories (e.g., > actuary, teacher), not companies or speciŽc positions. I can¹t speak for > Leigh (or anyone else), but if the job were tempting enough, I¹d travel > throughout the English-speaking world. if you Žnd 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, > > > > > > I¹m just Žnishing up my PhD in mathematics (focus on stochastic > > > calculus) and I am looking at possible career paths. I¹ve always had > > > some interest in economics, and actually taught a Žrst year Žnancial > > > mathematics course at university. Not knowing too much about math > > > careers in Žnance, does anyone here have any ideas? > > > > > > > Don¹t 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 I¹ve 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 won¹t stop inside their head. I¹ve 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 won¹t stop inside their head. I¹ve > 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? DeŽne music. Axioms are not necessary. Norm === Subject: Re: Are all mathematicians music lovers? > DeŽne music. Axioms are not necessary. The noises recorded on CD¹s 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. I¹ve 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. It¹s to give pleasure to civilized human beings. > Actually, there are people > who go crazy from repetitive tunes that won¹t stop inside their head. As I am writing this, it¹s already around 400 times that I have listened to the Scherzo from Beethoven¹s Sonata #3 for Piano and Cello in A Opus 69, non-stop. The melody¹s 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. > I¹ve > been able to drive otherwise calm-and-collected math/science types berserk > by saying music is stupid. And I¹ve been able to ward off barbarian subtypes by playing Bach Cantatas non-stop very loudly. What¹s 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. It¹s to give pleasure to civilized human > beings. Actually, cavemen made plenty of music with their bone žutes and stone drums. Music does have a tendency to inžate wishful thinking tendencies though. > > Actually, there are people > > who go crazy from repetitive tunes that won¹t stop inside their head. > As I am writing this, it¹s already around 400 times that I have listened > to the Scherzo from Beethoven¹s Sonata #3 for Piano and Cello in A Opus > 69, non-stop. > The melody¹s 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. > > I¹ve > > been able to drive otherwise calm-and-collected math/science types berserk > > by saying music is stupid. > And I¹ve been able to ward off barbarian subtypes by playing Bach > Cantatas non-stop very loudly. What¹s 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 won¹t stop inside their head. I¹ve > 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? > > DeŽne music. Axioms are not necessary. > > The noises recorded on CD¹s 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 Žnd quite excellent! Perfect background for studying functional analysis. though that would probably be the > quickest way to make enemies. I¹ve 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 won¹t stop >> inside their head. I¹ve 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 don¹t 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 don¹t listen to music, so she thought most people in the world don¹t 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 won¹t stop inside > their head. I¹ve 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. (It¹s 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 efŽcient. 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. You¹ll Žnd 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 won¹t stop inside their head. Yes. There are also people who go crazy from working in a post ofŽce 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. > I¹ve 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? I¹ve 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 let¹s 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. That¹s Žne 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 Žnd 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 doesn¹t 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 don¹t happen to enjoy it. Just like saying music is stupid simply because you don¹t enjoy it or like it would not be a courageous statement, just a rather stupid statement on your part. ============================================================= ========= It¹s not denial. I¹m 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? I¹m not a mathematician, but I¹ll 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 don¹t think that has anything to do with arts or music. Correct me if I¹m wrong, but didn¹t 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 don¹t 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 fulŽlling 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 Borel¹s death as an excuse to attach Serre. >Which later Bourbaki volumes do you believe are Serre¹s 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 fulŽlling 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 Žnd Serre¹s 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. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: Re: Armand Borel >Personally, I Žnd Serre¹s 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 don¹t understand why you have it in for Serre. I don¹t believe any reasonably objective person could Žnd his work obscure or pedantic. I imagine Serre must be kicking himself for not proving Fermat¹s Last Theorem, as he must have all the tools at his Žngertips, 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 Žnd 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 Žnd Serre¹s 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 don¹t understand why you have it in for Serre. >I don¹t believe any reasonably objective person >could Žnd 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 won¹t Žnd out who has written what of Bourbaki, but that superŽciality 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 Fermat¹s Last Theorem, >as he must have all the tools at his Žngertips, >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 Žnd 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 Žrst 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 satisŽed. If Andre Weil was not present, and that method was not used, one would expect standards to drop. Also, it would be more difŽcult to select good topics, if the idea is merely to write an encyclopedia, and not speciŽcally inžuence better mathematical logical description. Bourbaki itself stimulated a movement in math towards better logical clarity, making it unnecessary. Hans Aberg * Anti-spam: remove remove. from email address. * 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 don¹t). But even free speech doesn¹t 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, Weil¹s autobiography includes a picture of the 1951 Bourbaki summer meeting, showing Weil, Cartan and Serre (and others), and Pierre Dugac¹s 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 don¹t). But even free speech doesn¹t 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, Weil¹s autobiography includes a picture of the 1951 Bourbaki summer >meeting, showing Weil, Cartan and Serre (and others), and Pierre Dugac¹s >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 Želds, but continue beyond that seems pointless. Hans Aberg * Anti-spam: remove remove. from email address. * 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 Želds, but continue beyond > that seems pointless. Who says that was the object of the game? I¹ve 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 Žt 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 haven¹t 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 Žnd 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 won¹t miss him nor his math: Armand Borel was known as being intimidating and of his own math research he said that the most difŽcult part was to keep track of the arrows (i.e., the implications). So you wouldn¹t go to his math looking for something deep and subtle or any other traits of genius level science. Borel¹s 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 fulŽlling 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 didn¹t 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 scientiŽc value itself. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: Re: Armand Borel Finally dead > Well, I won¹t 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 fulŽlling 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 Borel¹s death as an excuse to attach Serre. Which later Bourbaki volumes do you believe are Serre¹s work, and what exactly is your complaint about them ? Personally, I Žnd Serre¹s 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. It¹s 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 won¹t 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 won¹t 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. >It¹s 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 won¹t 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 (ofŽcial) age? Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: Re: Calculus is irrational? >Hi All, >If calculus assumes inŽnity to come to its answers It doesn¹t. >(for example, the >limit of a function, we sum to inŽnity to Žnd an answer) No, we don¹t. >and because inŽnity is irrational (inŽnity being deŽned by p/0 No, that¹s not the deŽnition of inŽnity. >and >rational number deŽned by p/q where q <> 0) >is it fair to say that any answer given to us by calculus is by >deŽnition 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 inŽnity: Fact: The sum of 1/2^j, j = 1, ... inŽnity, is 1. Now the way that¹s stated it certainly appears to involve something inŽnite - there¹s that inŽnity right there in the statement. But here¹s what the fact means, _by deŽnition_: Fact, translated by inserting deŽnitions: 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 there¹s nothing inŽnite at all in the translation. _All_ the inŽnities in elementary calculus actually vanish this way, if you insert the deŽnitions. >I¹m 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 deŽned 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 inŽnity; the limit > > > of a function, when it exists, is a speciŽc number with a very > > > speciŽc property. > > > > Yes, but, in some functions (not all of course) the limit is found by > > approaching inŽnity. > > But not by ever being at inŽnity, whatever that means. Exactly my point. Lets take the problem of a bouncing ball. This was taken from a friend¹s post in another newsgroup: ==================== Say the Žnite time between each bounce gets progressively shorter by a factor of 10. So the Žrst 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, you¹ll 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 inŽnite 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 inŽnity](1 / (10 ^ i)) now you can work out that that doesn¹t add up to inŽnity: Let¹s assume that x = Sum[i=1 to inŽnity](1 / (10 ^ i)) then, multiplying both sides by 10: 10x = 10*Sum[i=1 to inŽnity](1 / (10 ^ i)) = Sum[i=1 to inŽnity](10 / (10 ^ i)) = Sum[i=1 to inŽnity](1 / (10 ^ i-1)) = Sum[i=0 to inŽnity](1 / (10 ^ i)) = 1 + Sum[i=1 to inŽnity](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 inŽnity](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 Œem 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 inŽnity, 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 inŽnity (even if we don¹t need to literally add to inŽnity 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 doesn¹t mean that the answer is comletely rational. Thats all I¹m trying to point out. Mike Helland === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. [.series problem snipped.] >Or >Sum[i=1 to inŽnity](1 / (10 ^ i)) >now you can work out that that doesn¹t add up to inŽnity: >Let¹s assume that x = Sum[i=1 to inŽnity](1 / (10 ^ i)) >then, multiplying both sides by 10: >10x = 10*Sum[i=1 to inŽnity](1 / (10 ^ i)) > = Sum[i=1 to inŽnity](10 / (10 ^ i)) > = Sum[i=1 to inŽnity](1 / (10 ^ i-1)) > = Sum[i=0 to inŽnity](1 / (10 ^ i)) > = 1 + Sum[i=1 to inŽnity](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 inŽnity](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 Œem 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 inŽnity, No, we do not (or rather, we do not have to do any such thing). The expression Sum{i=1 to inŽnity}( 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 inŽnite sum adds up to 1/9. But we are not at inŽnity 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 inŽnity (even if we don¹t need to literally add to >inŽnity and can use the shortcuts of Calculus). No, we do not. ============================================================= ========= It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ============================================================= ========= Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Calculus is irrational? > InŽnity is NOT A NUMBER. But are you arguing that inŽnity is indeed a rational quantity? Do you have any cites that state that inŽnity is not a quantity? I¹ve found several that deŽne it as a limitless quantity, including the online Mathworld: http://mathworld.wolfram.com/InŽnity.html === Subject: Re: Calculus is irrational? > > ... Are we > on our way (say 100 years from now) to making inŽnity into an entity > treated as any other number? We¹re 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 you¹ve said have themselves been incoherent. (Limits are irrational. Huh? So 0 is irrational, because it¹s 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 . Didn¹t you see that? >Mike Helland ************************ David C. Ullrich === Subject: Re: Calculus is irrational? [...] >So Sum[i=1 to inŽnity](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 Œem 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 inŽnity, >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 inŽnity No we _don¹t_. Why not has been explained. > (even if we don¹t need to literally add to >inŽnity 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 doesn¹t mean that the answer is comletely rational. Thats all I¹m >trying to point out. But this thing you¹re 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. >> InŽnity is NOT A NUMBER. >But are you arguing that inŽnity is indeed a rational quantity? >Do you have any cites that state that inŽnity 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 +inŽnity and -inŽnity are NOT [emphasis in the original] real numbers. The phrase f(x) approaches +inŽnity is akin to saying that f(x) approaches the unapproachable; it is a colloquialism for f(x) increases without bound. [...] Furthermore, since +inŽnity and -inŽnity are not numbers, it is inappropriate to manipulate these symbols using rules of algebra. There are ways of dealing with inŽnity 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 compactiŽcations). But there is no need to do so in order to do calculus. The answer to your question of whether inŽnity is a rational or irrational quantity is simply it is neither, because it is not considered a quantity for the purposes of calculus. > I¹ve >found several that deŽne it as a limitless quantity, including the >online Mathworld: >http://mathworld.wolfram.com/InŽnity.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. ============================================================= ========= It¹s not denial. I¹m 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 inŽnity through inference, or implication, or some other kind of induction? A young person who Žrst 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 deŽnition may be difŽcult, the meaning seems understandable. G C === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Are we left to understand inŽnity through inference, or implication, or some >other kind of induction? The point is that in calculus there is no need to understand inŽnity as anything other than a colloquialism with a very precise meaning which can be expressed entirely in terms of the usual real numbers. For inŽnity in other contexts (topology, set theory, cardinal arithmetic, etc) you are left to understand inŽnity through precise, formal deŽnition that will be given in those contexts, or through agreement as to what they represent as a colloquialism. >A young person who Žrst 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 deŽnition may be difŽcult, the meaning seems understandable. One can deŽne inŽnity 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. ============================================================= ========= It¹s not denial. I¹m 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 inŽnite >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 inŽnite number of bounces. There can¹t be an inŽnite number of bounces, because there is no number called inŽnity. >So the total amount of time the ball spends bouncing is.... >1/10 + 1/100 + 1/1000...... >Or >Sum[i=1 to inŽnity](1 / (10 ^ i)) Sum to inŽnity is just a convenient shorthand for something else, which can be expressed precisely, does not involve the word, or concept, inŽnity. >now you can work out that that doesn¹t add up to inŽnity: You don¹t need to work it out. If you add two real numbers together, you get another real number. You don¹t get inŽnity, because inŽnity isn¹t a real number. If you add a lot of real numbers together, you still don¹t get inŽnity, because the answer is always a real number. You can¹t add an inŽnite number of terms together, because there is no such number. When mathematicians talk about the sum of an inŽnite series, they don¹t mean the result obtained by adding an inŽnite 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 deŽnitions: > > 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 there¹s nothing inŽnite at all in the > translation. _All_ the inŽnities in elementary > calculus actually vanish this way, if you insert > the deŽnitions. But, how is saying for all N different from 1 to inŽnity? I understand the semantics are different, but isn¹t it saying the same thing? Also, you are changing the wording of the answer so that it doesn¹t 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 Žnal leap is where the irrationality now lies. I understand exactly what you¹ve done, you¹ve limited the scope of Calculus so that it is not irrational, however, you¹ve done it in a way that to present a Žnal 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 deŽnitions: >> >> 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 there¹s nothing inŽnite at all in the >> translation. _All_ the inŽnities in elementary >> calculus actually vanish this way, if you insert >> the deŽnitions. >But, how is saying for all N different from 1 to inŽnity? Presumably you mean for all n>N and from N to inŽnity. > I >understand the semantics are different, but isn¹t it saying the same >thing? Yes. But your mistake is thinking that from N to inŽnity means that you must at some point be plugging in something called inŽnity, or that at some point you must consider something called inŽnity. shorthand for all n greater than N. You never consider anything which is not a positive integer. You certainly never consider anything called inŽnity. ============================================================= ========= It¹s not denial. I¹m 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 Žnd that solution. >Mike If the solution can be expressed as a ratio of integers, then the solution is rational. Otherwise it¹s not. dy/dx = 2x is not rational or irrational. At least, not until x is speciŽed. -- 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, inŽnity 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 inŽnity, 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 you¹re the late Armand Borel, you and your friend J-P Serre go quite a lot further from there.) So inŽnity 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 you¹re the late Armand Borel, you and >your friend J-P Serre go quite a lot further from there.) Having now Žnished reading the news that accumulated during the blackout, I want to add that my reference to Borel and Serre (and, implicitly, to their bordiŽcation) have nothing whatever to do with Hans Aberg¹s comments on the two of them in the thread on Borel¹s death. Lee Rudolph === Subject: Re: Calculus is irrational? >David C. Ullrich >> Fact, translated by inserting deŽnitions: >> >> 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 there¹s nothing inŽnite at all in the >> translation. _All_ the inŽnities in elementary >> calculus actually vanish this way, if you insert >> the deŽnitions. >But, how is saying for all N different from 1 to inŽnity? I >understand the semantics are different, but isn¹t it saying the same >thing? Presumably that was a typo for ŒBut, how is saying for all n > N different from 1 to inŽnity?¹ In any case, I don¹t quite follow the question. There is _no_ mention of anything inŽnite above. The above _is_ what the sum of 1/2^j, n = 1..inŽnity = 1 _means_. >Also, you are changing the wording of the answer so that it doesn¹t >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..inŽnity = 1 means. That¹s an example of a bit of calculus that appears to involve something inŽnite but actually doesn¹t, if you insert the deŽnitions. You say I¹m changing the question, but I have no idea what original question you have in mind. What ball? >In this case, that Žnal leap is where the irrationality now lies. I >understand exactly what you¹ve done, you¹ve limited the scope of >Calculus so that it is not irrational, however, you¹ve done it in a >way that to present a Žnal solution irrationality is required. >Am I mistaken? You¹re 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..inŽnity? A: 1. Exactly how was irrationality required to present that solution? What¹s irrational is the way you continue to try to explain what¹s irrational about calculus, in spite of the fact that none of the mathematicians in the crowd think you¹re 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 there¹s nothing irrational involved. (I mean really. It¹s 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 don¹t know what you mean by quantity. Do you? > > Quantity. Number. The choice of name isn¹t that important. It¹s what you > can do with the thing (whether it¹s called a quantity, a number, or > something else) that¹s important. Well, the way I interpret it a number would imply a speciŽc quantity. Reasonable? === Subject: Re: Calculus is irrational? Arturo Magidin > >In order to come to the answer of 1/9, we have to be at inŽnity, > No, we do not (or rather, we do not have to do any such thing). The > expression > Sum{i=1 to inŽnity}( 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 inŽnite sum adds up to > 1/9. But we are not at inŽnity in any sense. I agree. So here¹s 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 inŽnite sums add up to 1/9. So either: calculus isn¹t 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 Žnd that solution. > If the solution can be expressed as a ratio of integers, then the solution > is rational. Otherwise it¹s not. Sorry, you¹re 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 it¹s 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 don¹t know what you mean by quantity. Do you? > > > > Quantity. Number. The choice of name isn¹t that important. It¹s what > > you can do with the thing (whether it¹s called a quantity, a number, or > > something else) that¹s important. > Well, the way I interpret it a number would imply a speciŽc quantity. > Reasonable? Yes. But I don¹t know what your point is. And I¹m curious about your use of the word speciŽc above. Are you thinking that something could be _a_ quantity without being a _speciŽc_ quantity? If so, please give an example. David === Subject: Re: Calculus is irrational? David C. Ullrich > You¹re 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..inŽnity? > A: 1. > Exactly how was irrationality required to present > that solution? In order to sum all from 1 to inŽnity, one would have to reach inŽnity to get the precise answer. I¹m making the assumption that for a human to reach inŽnity is an irrational suggestion because inŽnity 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 inŽnity, 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 Žnd 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 shouldn¹t be too hard too admit that those solutions cannot be regarded as absolutely rational. I realize that Calculus does not literally sum to inŽnity 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 deŽnition, any *real* number that is not > rational. InŽnity, 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. InŽnity lies outside of human rationality. Depending on inŽnity to Žnd 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, I¹m 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. I¹m not saying that these solutions are wrong, invalid, useless, or even something we should avoid. I¹m 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 inŽnity, >> No, we do not (or rather, we do not have to do any such thing). The >> expression >> Sum{i=1 to inŽnity}( 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 inŽnite sum adds up to >> 1/9. But we are not at inŽnity in any sense. >I agree. >So here¹s 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 don¹t ->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 isn¹t 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. >or >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. ============================================================= ========= It¹s not denial. I¹m 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 >> You¹re 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..inŽnity? >> A: 1. >> Exactly how was irrationality required to present >> that solution? >In order to sum all from 1 to inŽnity, one would have to reach >inŽnity to get the precise answer. It¹s 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 Žnd the exact value of sum 1/2^j, j = 1..inŽnity we do _not_ have to reach inŽnity. _By deŽnition_, if we Žnd 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..inŽnity exactly. Without ever reaching inŽnity. Words in math mean what the deŽnitions _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..inŽnity, exactly, without ever reaching inŽnity. > I¹m making the assumption that for >a human to reach inŽnity is an irrational suggestion because inŽnity >lies outside the limits of human rationality. You¹re also making the assumption that reaching inŽnity has something to do with calculus. It does not. You can continue to say it does - it still doesn¹t. You can continue to tell people they¹re missing your point - that doesn¹t change the fact that the things you¹re saying are nonsense, and it doesn¹t change the fact that _you_ are missing important points (the fact that you¹re missing something is shown by your continued belief that Žnding limits has something to do with reaching inŽnity. It doesn¹t.) >If that is the case, even though we can use valid and effective tools >like Calculus to determine what is happening as we approach inŽnity, >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 Žnd 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 shouldn¹t be too hard too admit that those solutions >cannot be regarded as absolutely rational. It would be very easy to admit that. But there¹s no reason to admit it because it¹s simply not so. The fact that _you_ don¹t understand this doesn¹t mean it¹s false. >I realize that Calculus does not literally sum to inŽnity 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 deŽnition, any *real* number that is not >> rational. InŽnity, 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. InŽnity lies outside of human >rationality. Depending on inŽnity to Žnd 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, I¹m 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 - you¹ve given no evidence. Or rather no valid evidence. >I¹m not saying that these solutions are wrong, invalid, >useless, or even something we should avoid. I¹m just saying that it >does step outside of rationality. Its interesting to see so much >resistence to this claim. What you¹ve 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 don¹t get it or you¹re just typing for the fun of seeing people reply. If you think it¹s interesting when your behavior makes people wonder whether you¹re simply stupid, well good for you. >Mike ************************ David C. Ullrich === Subject: Re: Calculus is irrational? > > ... InŽnity lies outside of human > rationality. Why so? InŽnity crops up a lot in mathematics. -- G.C. === Subject: Re: Calculus is irrational? Arturo Magidin > REMARK. It should be emphasized that the simbols +inŽnity and > -inŽnity are NOT [emphasis in the original] real numbers. The > phrase f(x) approaches +inŽnity 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 +inŽnity and >> -inŽnity are NOT [emphasis in the original] real numbers. The >> phrase f(x) approaches +inŽnity 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 deŽned 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. ============================================================= ========= It¹s not denial. I¹m 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 speciŽc quantity. > > Reasonable? > > Yes. But I don¹t know what your point is. I¹m not making a point here but defending myself. It was said that my reasoning does not hold because I said inŽnity is a number, so I clariŽed that I did not state inŽnity was a number. I stated that it is a quantity, even though not a speciŽc quantity. > And I¹m curious about your use > of the word speciŽc above. Are you thinking that something could be > _a_ quantity without being a _speciŽc_ quantity? If so, please give an > example. The only example I can think of at the moment would be inŽntiy. === Subject: Re: Calculus is irrational? Arturo > >> Sum{i=1 to inŽnity}( 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 inŽnite sum adds up to > >> 1/9. But we are not at inŽnity in any sense. > > > >I agree. > > > >So here¹s 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 don¹t ->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 don¹t 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, you¹re 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 speciŽc quantity. >> > Reasonable? >> >> Yes. But I don¹t know what your point is. >I¹m not making a point here but defending myself. It was said that my >reasoning does not hold because I said inŽnity is a number, so I >clariŽed that I did not state inŽnity was a number. I stated that it >is a quantity, even though not a speciŽc quantity. Which does not really make sense. If it is a quantity, then surely it is a ->speciŽc<- one? Look: inŽnity can be made very precise. If you do that, you can work with it and it would constitute, according to your classiŽcation, a rational concept. This is the case when you work on the Riemman Sphere (a one point compactiŽcation of the Complex plane) for example, where it is a very speciŽc point in a manifold, and going to inŽnity also has a very precise, speciŽc, topological meaning, and where it ->does<- make sense to state that you are at inŽnity or evaluating a holomorphic function at inŽnity and so on. But that requires a precise deŽnition. Or you can deŽne inŽnite as a very precise adjective with a very precise meaning in formal set theory: a set is inŽnite if and only if it can be put in one-to-one bijective correspondence with a proper subset of itself. And then you can deŽne an inŽnite collection of different inŽnities, 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<- deŽne inŽnity in a very precise formal way using non-standard analysis, but then, again, you are talking about a very speciŽc, well-deŽned, well-understood object that you can handle. In most cases, however, calculus is presented without having to appeal to a concept of inŽnity except in colloquial expressions or formal deŽnitions, where it appears as part of a term of art. Things like when x tends to inŽnity, or the limit equals inŽnity and so on. But these are terms of art, which are given precise, formal, deŽnitions that do not in any way invoke an undeŽned or deŽned concept of inŽnity. The word is just used as part of shorthands for precise explicit expressions. As for the cases where it is given a precise deŽnition, 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 inŽnity tend to be nonintuitive in their behavior, and you have made the jump into considering them to be Œnon-rational¹ because of that? >> And I¹m curious about your use >> of the word speciŽc above. Are you thinking that something could be >> _a_ quantity without being a _speciŽc_ quantity? If so, please give an >> example. >The only example I can think of at the moment would be inŽntiy. Which is not a quantity in the context of standard calculus. ============================================================= ========= It¹s not denial. I¹m 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 inŽnity to come to its answers (for example, the > limit of a function, we sum to inŽnity to Žnd an answer) > > and because inŽnity is irrational (inŽnity being deŽned by p/0 and > rational number deŽned by p/q where q <> 0) > > is it fair to say that any answer given to us by calculus is by > deŽnition irrational as it assumes irrationality in the solution? > > I¹m 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 deŽned as irrational? > > Mike Helland I see your point. I have been taught that the real numbers form a Želd. But I was totally unsuccessful when I tried grazing sheep on them. Likewise, I Žnd 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 inŽnity to come to its answers (for example, the > > limit of a function, we sum to inŽnity to Žnd an answer) > > > > and because inŽnity is irrational (inŽnity being deŽned by p/0 and > > rational number deŽned by p/q where q <> 0) > > > > is it fair to say that any answer given to us by calculus is by > > deŽnition irrational as it assumes irrationality in the solution? > > > > I¹m 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 deŽned as irrational? > > > > Mike Helland > > > I see your point. I have been taught that the real numbers form a Želd. > But I > was totally unsuccessful when I tried grazing sheep on them. You¹ve never seen the grazing goats problem? === Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Arturo >> >> Sum{i=1 to inŽnity}( 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 inŽnite sum adds up to >> >> 1/9. But we are not at inŽnity in any sense. >> > >> >I agree. >> > >> >So here¹s 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 don¹t ->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 don¹t 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, >you¹re 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. ============================================================= ========= It¹s not denial. I¹m 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 Žrst time. What do you mean by outside the real world? My deŽntion of the real world would be everything within our rationality. Outside the real world would be outside our rationality. It sounds like we¹re 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 Žrst 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 Žnite number of bounces, the ball still continues to bounce. If you can produce me a ball that satisŽes these hypothesis, I will be very impressed. So will the physicists, since you will have overthrown the second law of thermodynamics. >My deŽntion of the real world would be everything within our >rationality. Your deŽnition 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 we¹re in exact agreement, are we not? No. I disagree with your deŽnition of rationality; I disagree with your deŽnition of real world; and I disagree with your application of these labels, even assuming your deŽnitions, in the way you have applied them. It seems like we are hardly in agreement. ============================================================= ========= It¹s not denial. I¹m 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 coefŽcient >Is it true that if the n-th cyclotomic polynomial has exactly one >negative coefŽcient then n is of the form 2^i*3^j ? It¹s true at least for n up to 2000. If C_n is the n¹th 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 don¹t know how to prove that these are the only cases where C_n has one negative coefŽcient, though. Of course since the zeros of C_n are symmetric under t -> 1/t, the coefŽcients satisfy a_j = a_{d-j} where d is the degree, and if you have one negative coefŽcient 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 coefŽcient > >Is it true that if the n-th cyclotomic polynomial has exactly one > >negative coefŽcient then n is of the form 2^i*3^j ? > > It¹s true at least for n up to 2000. > If C_n is the n¹th 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 don¹t know how to prove that these are the only cases where C_n has one > negative coefŽcient, though. > Of course since the zeros of C_n are symmetric under t -> 1/t, > the coefŽcients satisfy a_j = a_{d-j} where d is the degree, > and if you have one negative coefŽcient 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 Žnite set of primes p_1,p_2,...,p_m such that if n-th cyclotomic polynomial has exactly k negative (or positive) coefŽcients then the prime factors of n belong to the set p_1,p_2,...,p_m . Bill === Subject: Re: Dedekind Cuts > > >>I¹m trying to understand what is described by Parzynski and Zipse, in >>Introduction to Mathematical Analysis as Dedekind¹s Theorem - R is >>Complete. They describe real numbers as rays (non-empty proper subsets >>of the rationals having no Žrst element and any y greater than an >>element of the ray is also an element of the same ray). >>The proof starts with deŽning A¹ = 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. Hardy¹s 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 > > >>I¹m trying to understand what is described by Parzynski and Zipse, in >>Introduction to Mathematical Analysis as Dedekind¹s Theorem - R is >>Complete. They describe real numbers as rays (non-empty proper subsets >>of the rationals having no Žrst element and any y greater than an >>element of the ray is also an element of the same ray). >>The proof starts with deŽning A¹ = R - A, where A is any ray in R. >>Also U = {x (element of) Q | x>W for every W (element of) A¹} >>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? > > > It¹s hard to explain the notation without seeing it in context. > But showing that the reals are complete, after deŽning 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 Žll S with rays using some rule that is so esoteric it has yet to be discovered. Wishing to make things difŽcult, 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 deŽning 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 deŽne 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 deŽning complex numbers. But I think rays are complete in the sense that they cannot be aggregated in ways that will deŽne 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 deŽne 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 deŽning complex numbers. > But I think rays are complete in the sense that they cannot be > aggregated in ways that will deŽne 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. I¹m not sure that I follow your argument, but Conway¹s deŽnition 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 >> >> >I¹m trying to understand what is described by Parzynski and Zipse, in >Introduction to Mathematical Analysis as Dedekind¹s Theorem - R is >Complete. They describe real numbers as rays (non-empty proper subsets >of the rationals having no Žrst element and any y greater than an >element of the ray is also an element of the same ray). >The proof starts with deŽning A¹ = R - A, where A is any ray in R. >Also U = {x (element of) Q | x>W for every W (element of) A¹} >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? >> >> >> It¹s hard to explain the notation without seeing it in context. >> But showing that the reals are complete, after deŽning 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 Žll S with rays using some rule that is so >esoteric it has yet to be discovered. Wishing to make things difŽcult, > 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 deŽning 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 deŽne 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 deŽning complex numbers. >But I think rays are complete in the sense that they cannot be >aggregated in ways that will deŽne 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 don¹t know that then the _Žrst_ question you should be asking is what it means, not questions about the meaning of the notation in the proof! There is presumably a deŽnition of R is complete somewhere in the book you¹re looking at. That deŽnition 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. It¹s 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 that¹s what these Dedekind cut things are naturally adapted to do; if you¹d deŽned the real numbers in some other way then (ii) might be more natural (in particular it¹s possible to deŽne 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 deŽnition.) 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 deŽne 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 deŽning complex numbers. >>But I think rays are complete in the sense that they cannot be >>aggregated in ways that will deŽne 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. > > > I¹m not sure that I follow your argument, > but Conway¹s deŽnition of surreal numbers > in effect generalises Dedekind sections, > and gives rise to new numeric types. > If I understand Conway¹s surreal numbers, their deŽntion 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. (I¹m 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, I¹m not trying to defend my deŽntion of completeness, I¹m trying to understand the conventional argument that completeness žows 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 Conway¹s surreal numbers, their deŽntion 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. [...] |(I¹m 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 don¹t 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 isn¹t 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, deŽnes negative ordinals. Keith Ramsay === Subject: Re: Dedekind Cuts > > >>I¹m trying to understand what is described by Parzynski and Zipse, in >>Introduction to Mathematical Analysis as Dedekind¹s Theorem - R is >>Complete. They describe real numbers as rays (non-empty proper subsets >>of the rationals having no Žrst element and any y greater than an >>element of the ray is also an element of the same ray). >>The proof starts with deŽning A¹ = 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. Hardy¹s classic A Course of Pure Mathematics. In Section 17 > (p.28 of the 10th edition) he recapitulates the deŽnition 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 > Dedekind¹s 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. It¹s 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 Zipse¹s book, but Ken¹s mention of Hardy¹s 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 Žrst 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 efŽciently 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 Zipse¹s book, but Ken¹s mention of Hardy¹s 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 Žrst > 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 I¹ve looked it up, the Hadamard matrix is indeed a solution to my problem, but the set S that I deŽned above seems to be much larger, since it doesn¹t 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 coefŽcient >> 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, there¹s Kummer¹s 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 deŽne pi^2 in the following nifty way: pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... Which means you can deŽne it using members of the *Želd* of rationals. However, pi is transcendant and is itself not a rational. I¹m curious about the rule mathematicians use to exclude pi^2/6 from the Želd of rationals, as it itself is the result of an inŽnite sum of members of that Želd. Is that it? Mathematicians simply exclude inŽnite sums from the Želd of rationals? Or do they rely on the deŽnition of a rational as the ratio of a/b, where Œa¹ and Œb¹ 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 (Professor¹s name omitted): Professor ****: You assertion is easily proven false. Please consider the following. inŽnity. 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 inŽnite 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 deŽnition *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 inŽnite 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 coefŽcients. Do you agree with the professor, who I remind is a *leading* mathematician? === Subject: Re: Field of rationals and pi > Euler found out that he could deŽne pi^2 in the following nifty way: > > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > > Which means you can deŽne it using members of the *Želd* of > rationals. However, pi is transcendant and is itself not a rational. > Another way to see that pi is an inŽnite 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 inŽnite 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. > I¹m curious about the rule mathematicians use to exclude pi^2/6 from > the Želd of rationals, as it itself is the result of an inŽnite sum > of members of that Želd. > > Is that it? Mathematicians simply exclude inŽnite sums from the > Želd of rationals? Or do they rely on the deŽnition of a rational > as the ratio of a/b, where Œa¹ and Œb¹ 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 inŽnite sum of rationals. So that property doesn¹t 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 deŽne pi^2 in the following nifty way: > > > > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > > > > Which means you can deŽne it using members of the *Želd* of > > rationals. However, pi is transcendant and is itself not a rational. > > > Another way to see that pi is an inŽnite 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 inŽnite 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. > > I¹m curious about the rule mathematicians use to exclude pi^2/6 from > > the Želd of rationals, as it itself is the result of an inŽnite sum > > of members of that Želd. > > > > Is that it? Mathematicians simply exclude inŽnite sums from the > > Želd of rationals? Or do they rely on the deŽnition of a rational > > as the ratio of a/b, where Œa¹ and Œb¹ 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 inŽnite sum of > rationals. So that property doesn¹t 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 deŽne pi^2 in the following nifty way: >pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... >Which means you can deŽne it using members of the *Želd* of >rationals. However, pi is transcendant and is itself not a rational. >I¹m curious about the rule mathematicians use to exclude pi^2/6 from >the Želd of rationals, as it itself is the result of an inŽnite sum >of members of that Želd. >Is that it? Is what what? >Mathematicians simply exclude inŽnite sums from the >Želd of rationals? InŽnite sums are not excluded from the rationals; every rational number _is_ an inŽnite sum. But there¹s no rule that says an inŽnte sum of rationals has to be rational, which is possibly what you meant. >Or do they rely on the deŽnition of a rational >as the ratio of a/b, where Œa¹ and Œb¹ are integers? Saying they rely on the deŽnition is a strange way to put it. That _is_ the deŽnition, so that _is_ what a rational number _is_. By deŽnition. >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 (Professor¹s name omitted): >Professor ****: > >You assertion is easily proven false. Please consider the following. > >inŽnity. > >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 inŽnite 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 deŽnition *does* exclude it. No. It proves that there are an inŽnite number of units in a certain ring, but that ring is _not_ Z[pi]. Because Z[pi] is what the deŽnition below says it is - those inŽnite 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 _inŽnite sums_ of elements of Z[pi]. That ring is actually R, the real numbers. So (assuming you did the algebra right, I haven¹t checked) you¹ve 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 inŽnite 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 coefŽcients. >Do you agree with the professor, who I remind is a *leading* >mathematician? Well duh, of course we do. Not because he¹s a leading mathematician, but because he¹s telling the truth. Look up the deŽnition of polynomial somewhere. A polynomial _is_ a Žnite polynomial, by deŽnition. > ************************ David C. Ullrich === Subject: Re: Field of rationals and pi > InŽnite sums are not excluded from the rationals; > every rational number _is_ an inŽnite sum. But there¹s > no rule that says an inŽnte sum of rationals has to be > rational, which is possibly what you meant. There¹s a subtle point of confusion I hope you can clarify for me. In the previous post by Arturo Magidin, he said (I think) that inŽnite sums *are* excluded from the rationals. But evidently he was referring to the operation of taking inŽnite sums, rather than the value of the sum itself. Certainly any rational number *can* be expressed as an inŽnite sum, whereas irrational and transcendental numbers may *only* be expressed that way. Could you be a little more speciŽc 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. >> InŽnite sums are not excluded from the rationals; >> every rational number _is_ an inŽnite sum. But there¹s >> no rule that says an inŽnte sum of rationals has to be >> rational, which is possibly what you meant. >There¹s a subtle point of confusion I hope you can clarify for me. In the >previous post by Arturo Magidin, he said (I think) that inŽnite 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, inŽnite sums are excluded; that is, a ring structure on a set does not, in and of itself, give you a notion of inŽnite sums or inŽnite 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 man¹s 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 Žgures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indeŽnite 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 >There¹s a subtle point of confusion I hope you can clarify for me. In the >previous post by Arturo Magidin, he said (I think) that inŽnite sums *are* >excluded from the rationals. But evidently he was referring to the >operation of taking inŽnite sums, rather than the value of the sum itself. >Certainly any rational number *can* be expressed as an inŽnite sum, >whereas irrational and transcendental numbers may *only* be expressed that >way. InŽnite sums are excluded only in the same way that square roots are excluded - that is, the Želd axioms don¹t ensure that that a Želd is closed under that operation. The term inŽnite sum is really a metaphor: shorthand for the limit of the sequence of partial sums. You can¹t therefore assume automatically that anything true about Žnite sums is also true about inŽnite ones. Of course, if the rationals *were* closed under (convergent) inŽnite sums, they would be the reals! -- Richard -- Spam Žlter: to mail me from a .com/.net site, put my surname in the headers. FreeBSD rules! === Subject: Re: Žnite 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 I¹d 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: Žnite 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 I¹d 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 Žxed 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, Žnd automorphism groups, etc. Derek Holt. === Subject: Re: Žnite 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 Žnite (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: Žnite 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 > Žnite (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. Let¹s take the simplest case, where the Želd is K=GF(2), so we have the group G=SL_3(K) of 168 elements. If memory serves me right (it¹s 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 can¹t 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 Žnd elements (in a non-split torus?) that aren¹t conjugate to their inverse transposes. May be the graph automorphisms for groups of other type will work, but it doesn¹t 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 I¹ve been able to Žnd develops it in a rigorous way, and I¹m repulsed by nonrigorous math (plus it doesn¹t 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: You¹re 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 couldn¹t 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 I¹ve been able to Žnd develops it in a rigorous way, and I¹m repulsed by >nonrigorous math (plus it doesn¹t 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: >You¹re 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 couldn¹t 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 it¹s not clear how helpful this is - the rules that are true are more or less obvious, and they¹re 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, >EJ ************************ 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? It¹s not silly. It doesn¹t do what I thought she wanted to do. Turns out that she didn¹t want to do what I thought she wanted to do... ************************ David C. Ullrich === Subject: Re: formal rules for big-oh manipulation |OK, I didn¹t actually say so, but I¹m 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 don¹t remember where I read this) is that in this context = is directional. It¹s 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. That¹s 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 sufŽciently close to the limit, because of division by zero). Typically the limit is as the independent variable goes to inŽnity, 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 deŽned 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. That¹s maybe the most basic thing. Then there are a number of little lemmas such as O(f)O(g)=O(fg). I don¹t 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 I¹ve been able to Žnd develops it in a rigorous > way, and I¹m repulsed by nonrigorous math (plus it doesn¹t 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: > > You¹re 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 couldn¹t 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 sufŽciently nice function Želd F, where limits exist at oo, the Big-Oh notation is essentially of valuation-theoretic origin since functions Žnite 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 sufŽces to consider just function Želds containing (a small subŽeld of) the Hardy exp-log Želds. There are effective algorithms for asymptotic calculation in such Želds 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 Žnd 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 I¹ve been able to Žnd develops it in a rigorous way, and I¹m repulsed by | > nonrigorous math (plus it doesn¹t 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 haven¹t seen one, but if you search around the web for | course notes dealing with big-O, you¹ll Žnd 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 that¹s how big-O is deŽned. | | >(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 haven¹t told us what a distinct graph is (speciŽcally, 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. > > J The vertices are labeled and connected in a closed ring; ie, polygon by Œn¹ 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 speciŽc graph may be 2, 3 or 4 but never >4. To clarify the confusion about different paths; each speciŽc 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 deŽnitions, axioms and rules.) >The formalism is still programing. I¹ve 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. > > I¹ve been wondering whether anyone else would point this out... > That¹s what I¹ve 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 Charlie¹s 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 speciŽes 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 I¹ll lay your soul to waste - The Rolling Stones, Sympathy for the Devil === Subject: Re: Fraud in Computer Science Publishing > programs. How do you deŽne 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 deŽne 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 deŽne 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 isn¹t 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 Žnal word as to the deŽnition that it is conveying? 14. Do you think that a simple wff may require complex programs to implement it (deŽning 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 deŽne 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 that¹s 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 isn¹t programming? Not that the computer can understand. If the computer can understand it, it¹s 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 he¹s doing a proof, he¹ll probably just say is a proper factor. If he¹s working with computer, he¹ll use whatever language he Žnds suitable for the task. Might be Maxima, Lisp, PC, or something else. What¹s 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. It¹s not because some people, however talented they are, have decided to use a speciŽc 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 Žrst. > 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 doesn¹t 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 speciŽcations 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 don¹t 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 Žnal > word as to the deŽnition that it is conveying? I¹m 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 deŽne some actions. Same thing for most languages. > 14. Do you think that a simple wff may require complex programs to > implement it (deŽning 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 Želd > >> > > 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) > >> > > n]d(sigma) > > > >= 0. > > > > > > > >> > >>Hint: Stokes¹s Theorem/Divergence Theorem. > >> > >>Note that the div(curl(v)) = 0. > >> > >>Dale. > >> > > > > I was able to show that div(curl(v)) = 0 but can¹t see the connection > > between div(curl(v)) and double integral [(.84 x v) 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 Želd 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 I¹m sorry but I don¹t 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 mensaje|nTCW_a.5$Op.2@news02.roc.ny: > 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] Shoudn¹t 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? I¹ve 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 you¹re interested in. A bijection is a 1-1 (injective) and onto (surjective) map. There¹s 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. It¹s important to make sure you know which deŽnition of 1-1 is being used by Schaum. In any event, an isomorphism is always deŽned to be a bijection. You didn¹t say what topic you¹re studying, but since you mentioned linear, I¹m guessing it¹s linear algebra, and you¹re 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 there¹s a bijection, there¹s not a linear bijection. === Subject: Re: isomorphic mapping Žshfry, You mentioned: There¹s 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 you¹re interested in. A bijection is a 1-1 (injective) and > onto (surjective) map. There¹s 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. It¹s important to make sure you know which deŽnition of > 1-1 is being used by Schaum. In any event, an isomorphism is always > deŽned to be a bijection. > You didn¹t say what topic you¹re studying, but since you mentioned > linear, I¹m guessing it¹s linear algebra, and you¹re 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 there¹s a bijection, > there¹s 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 doesn¹t 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 speciŽc 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 deŽned more precisely (which can¹t 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 sufŽcient 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 isn¹t sufŽcient 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 doesn¹t 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 speciŽc >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 deŽned more precisely (which can¹t 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 sufŽcient 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 isn¹t sufŽcient 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, Œsurjection¹ and Œonto¹ are not synomous, and Œinjectiion¹ and Œ1-1¹ are not either. Properly speaking, injective and surjective are deŽned in terms cancellation 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 can¹t remember off-hand how much weaker you can get, and am too lazy to Žgure it out now), surjective is equivalent to Œonto a dense subset¹. >No, see above >>Peter >Ulysse Larry (this space unintentially left blank ..... make obvious deletion for email === Subject: Re: isomorphic rings nojb. > > > > I¹m 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 haven¹t 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 Schaum¹s 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.¹s, 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 don¹t 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 Schaum¹s 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 Schaum¹s 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 doesn¹t matter which order you read them in. You don¹t 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 (I¹m 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 there¹s usually lots of prcatical examples and applications. === Subject: Re: math courses >> Most books on abstract algebra are reasonably self contained (I¹m 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. You¹ll 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-deŽned commentary around and have it pass for mathematics. We live in a more reŽned 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 Žne in science Žction 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 Cantor¹s 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, they¹ll 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 režexive axiom (A=A). And, yes, perhaps with the režexive 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 Cantor¹s 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 Cantor¹s transŽnitude 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, Cantor¹s 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 Pythagoras¹ society who Žrst 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 Cantor¹s 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 killŽle didn¹t work properly. So I found your friendly If ever you will post something relevant in our 21-st century, I¹ll Žnd 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 / 5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADv IrOrduRArpW7D Ržb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/ LpBYalbA2qiLZKP16G+KHLowIYojSm UTys+/ LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3X L8v547w/ uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+ Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/u9oA6vhAAAAAElFTkSuQmCC > It is as simple as this: if Cantor¹s 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 Cantor¹s 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. Wasn¹t it meant as irony, then? I thought it was a parody of the recent thread about Cantor¹s inžuence 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-deŽned commentary around and have it pass for > mathematics. We live in a more reŽned 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. It¹s 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, they¹ll accept the Axiom of Choice. If not, then not. I¹m not to happy with feel like, but apart from that, that is the nature of AC. What¹s 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 > režexive 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. I¹m 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 difŽcult to verbalize. > If you send me an email I¹ll 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 Žnite number of non-isomorphic Žnite subgroups . > > > >Let me ask what is the maximal order of a Žnite subgroup of GL(n,Z) > >? > > > > 2^n n! > > Derek Holt. Shouldn¹t 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 Žnite number of non-isomorphic Žnite subgroups . >> > >> >Let me ask what is the maximal order of a Žnite subgroup of GL(n,Z) >> >? >> > >> >> 2^n n! >> >> Derek Holt. >Shouldn¹t the answer for n=2 be 12 ? Yes. Sorry, I missed out `for sufŽciently large n¹ ! 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 Žnds the maximal (order?) subgroups for smaller values of n. The proof uses the classiŽcation of Žnite simple groups. You can Žnd the references in the paper by Kuzmanovich & Pavlichenkov in the American Math. Monthly 109 (2002), 173-186 which was mentioned earlier (wasn¹t 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 Žnite number of non-isomorphic Žnite subgroups . > >> > > >> >Let me ask what is the maximal order of a Žnite subgroup of GL(n,Z) > >> >? > >> > > >> > >> 2^n n! > >> > >> Derek Holt. > > > >Shouldn¹t the answer for n=2 be 12 ? > > Yes. Sorry, I missed out `for sufŽciently large n¹ ! > 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 > Žnds the maximal (order?) subgroups for smaller values of n. The > proof uses the classiŽcation of Žnite simple groups. > > You can Žnd the references in the paper by Kuzmanovich & Pavlichenkov in > the American Math. Monthly 109 (2002), 173-186 which was mentioned earlier > (wasn¹t that by you?) > > Derek Holt. Yes that was me but I don¹t have access to the paper of Feit . Let me just ask if for small values of n the number of non-isomorphic Žnite subgroups of GL(n,Z) is known ? === Subject: Meaning of curly d? I have some review books for differential equations, and they don¹t explain the difference between e.g. dM/dx and (curly)dM/dx. What¹s the difference?? === Subject: Re: Meaning of curly d? on Friday 15 > I have some review books for differential equations, and they don¹t explain > the difference between e.g. dM/dx and (curly)dM/dx. What¹s 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 -- Don¹t be afraid, I¹m 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 don¹t explain >the difference between e.g. dM/dx and (curly)dM/dx. What¹s the >difference?? The second one stands for a partial derivative, while the Žrst 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 couldn¹t solve it (a multidirectional limit, according to Maple¹s 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 couldn¹t solve it (a multidirectional limit, according to |Maple¹s 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 don¹t 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 don¹t know but I would assume there is. If the limit is through complex values it¹s 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 there¹s some way of analyzing how rapidly they approach 0 as they approach the common zero locus from different directions. Note that it¹s 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 mensaje|nbhjla2$c1c$1@news-reader5.wanadoo.fr: > I was trying to compute the (trivial) limit: > limit((x^3+y^3)/(x^2+y^2),{x->0,y->0}), > but Maple couldn¹t solve it (a multidirectional limit, according to > Maple¹s 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) > I¹ve recently demonstrated here that the determination of income > distribution by the theory of supply and demand makes no sense. You¹re 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. I¹d 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 that¹s supposed to be constant and you show that, after you solve your linear equations, it doesn¹t stay constant on both sides of that point. Since it¹s off by 3/11 of a bushel etc, the theory that claims it¹s constant is wrong. But how can real-world economists do much with linear algebra? It¹s mostly a nonlinear world. Linear algebra is a *convenience*, something you use to look at small changes where the nonlinearities won¹t show up too badly. If the economists you want to disprove are *depending* on linearities then you¹re completely right to sneer at them, but surely they aren¹t. Are they? When people bring this point up you point out that economics students are presented regularly with linear models and it¹s wrong for them to be taught that way. I¹d say those models are a convenience and it¹s wrong if they¹re taught to depend on them too much. Likely they are. When the only tool you have is a hammer.... But that doesn¹t handle the bigger problem that you want to disprove economic arguments which should not have been presented in linear terms in the Žrst place. Mostly, I don¹t 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 you¹d get something like Masses attract other masses. The earth has a great big mass so things fall down. It explains but it doesn¹t predict very well. Similarly, you can express Gresham¹s 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 Žxed, 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. There¹s a traditional numerical example. Then there are a lot of studies that show it can be asserted given various simplifying assumptions. Then there¹s the argument on faith that it should work in the real world. And there¹s the argument by real example, that prosperous nations tend to advocate free trade while poor nations often don¹t. Somehow a lot of people who read about economics have solidiŽed this spiderweb of assertions and assumptions into an inžexible belief. But then, that happens in physics too. When the laser was Žrst 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 gresham¹s law, or supply-and-demand. We won¹t get studies showing the compatibility until there¹s a demand for them.... === Subject: Re: My (Un)Originality (was Re: Reminder: Wages, Employment Not Determined By Supply, Demand) > > > I¹ve recently demonstrated here that the determination of income > > distribution by the theory of supply and demand makes no sense. > You¹re 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 Œexceptions¹: 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-Bawerk¹s Œaverage period of production¹) 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. > I¹d 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 what¹s linear in my examples or what slope he¹s 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, Leontief¹s 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 Žt 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 coefŽcients: > How close can f(n) be to an integer, when n is an integer? > How can I Žnd 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 inŽnitely often; moreover, the Euclidean algorithm >gives us an efŽcient way to walk around GL(2,Z) looking >for approximate solutions to b n - m = 0. >For other polynomials f, I suppose it¹s 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 inŽnitely >often, but I don¹t know how to Žnd good values of n . >(In fact, I don¹t even recall off-hand how to do this for >f(n) = a + b n with nonzero real coefŽcients 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 Žrst polynomial. First, Žnd 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 Žnd 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 Žnding this example, exhaustive search is probably more efŽcient. Rich Burge === Subject: Non-differentiability of continuous functions with partial derivatives of Žrst order Is there any practical way to tell if a function f: U->F (U is an open set of a Žnite-dimensional real vector space E, and F a Žnite-dimensional real vector space) which is continuous on U and has Žrst order partial derivatives (not all continuous) is differentiable? I¹m 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 Euclid¹s Žfth 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 Žrst four postulates: You can do _____ > >> > >>The Žfth 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 Žfth postulate is the one that is most obviously > >>violated. > > > > > > And another of Euclid¹s axioms is violated in this case too. In the > > Hyperbolic case _only_ the Žfth 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. I¹d have to look up which one is usually changed on the > sphere. The second one: Any straight line segment can be extended indeŽnitely 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~~~ What¹s happened? Cooling down? -- G.C. === Subject: Re: one to one holomorphic functions >> Let D¹ be the closed unit disk in the complex plane: D¹:= {z in C : >> |z|<=1}. >> Let f be holomorphic on a neighborhood of D¹. Assume that the >> restriction of f to D¹ is one-to-one. Prove that in fact f is >> one-to-one on a neighborhood of D¹. >That looks false to me. If f(z) = (z-1)^2, then f is 1-1 on D¹ but not in >any neighborhood of 1. Hmm. On the one hand that seems right, but otoh it can¹t be false, since it¹s an exercise in _Krantz_... (inside joke, sorry). ************************ David C. Ullrich === Subject: Re: one to one holomorphic functions > > Let D¹ be the closed unit disk in the complex plane: D¹:= {z in C : > > |z|<=1}. > > Let f be holomorphic on a neighborhood of D¹. Assume that the > > restriction of f to D¹ is one-to-one. Prove that in fact f is > > one-to-one on a neighborhood of D¹. > That looks false to me. If f(z) = (z-1)^2, then f is 1-1 on D¹ but not in > any neighborhood of 1. That¹s a nice example, Wade. -- your old student, Bill H. === Subject: Re: one to one holomorphic functions I bet if you add that f¹ never vanishes on D¹ it would be true. > > > > >> Let D¹ be the closed unit disk in the complex plane: D¹:= {z in C : > >> |z|<=1}. > >> Let f be holomorphic on a neighborhood of D¹. Assume that the > >> restriction of f to D¹ is one-to-one. Prove that in fact f is > >> one-to-one on a neighborhood of D¹. > > > >That looks false to me. If f(z) = (z-1)^2, then f is 1-1 on D¹ but not in > >any neighborhood of 1. > > Hmm. On the one hand that seems right, but otoh it can¹t be > false, since it¹s an exercise in _Krantz_... (inside joke, sorry). > > ************************ > > David C. Ullrich === Subject: Re: one to one holomorphic functions > That¹s a nice example, Wade. > -- your old student, Bill H. Hi Bill. Glad to see you¹re still interested in complex analysis. Best, W. === Subject: Re: one to one holomorphic functions > I bet if you add that f¹ never vanishes on D¹ 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 D¹. 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 f¹(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 popular, it is called Jan Ken and in the USA and England they call it Rock-Paper-Scissors. The game is quite simple. Two players sit facing each other and on a sign they Žrst stamp three times with the right Žst in the open left hand and then they simultaneously make a gesture with their right hand. There are three different gestures: Rock - closed Žst; Paper - stretched hand; Scissors - two Žngers 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 determine who has to do a certain job. The Japanese then say Jan Ken 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 Žnd 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 I¹m one of the bidders, its up to $70 and I¹m planning a last minute $100 bid as that¹s 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 won¹t 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 afŽliate programs for webmasters to advertise to receive 2% of the sale price of a HOUSE! 3/ That¹s how Amazon and 1000¹s of others started large scale business by starting afŽliate 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 I¹ve seen was TheOnlineRealty.com and they go for $100s. Imagine you¹re 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 afŽliate 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 we¹ll 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 who¹s 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 Žnd as ordinary jobs, >if anyone knows something about it let me know, I recommend you buy one of those training manuals. You¹ll 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 who¹s 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 Žnd 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 that¹s funny. Steve > -- > Ghost Rider > aa # 2011 > EAC Nonexistent Director of Alcohol, Tobacco and Bad Puns > How can you just obey? > [Greg Lake, InŽnite 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 that¹s funny. > Steve no that¹s what all work from home businesses are! here¹s 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 I¹ve 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. What¹s with the squared quantities on the left-hand side? We¹re only supposed to work in an octant of three-dimensional space? Can¹t 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 Žrst post, it just |sounded too silly. But this time, I¹m serious :) you could also put the string in between two complementarily curved pieces of glass, or something like that, if you¹re 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 deŽne arc length.The metric is ds^2= du^2+ G(u,v) dv^2 . We have geodesic radial lines and parallels. When the question Žrst 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 Želd 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 Žnger 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 > don¹t have that much of a mathematical knowledge, because I¹m a high > school student. > For convenience reasons, the situation is as following: > F=K¹*p1*p2/r^2 > p1*p2=1/K¹ (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, that¹s with constant > acceleration, with this one I can¹t Žgure it out. > Tal P Good luck with that dimensionally incorrect magnetic monopole Želd. 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: Žnite 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 I¹d 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 identiŽed a desired subgroup of index 6 in S_6) What I end up getting doesn¹t 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: Žnite groups) in message <3F3CAAE8.C07FEADF@utu.Ž>: [...] > 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 identiŽed a desired subgroup of index 6 in S_6) > What I end up getting doesn¹t 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 don¹t 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 ClassiŽcation Theorem!). Using only fairly elementary group theory, it¹s straightforward, but rather tedious, to show that there¹s 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: Žnite groups) > > of S_6. It is slightly harder to Žnd such a map which deŽnes 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 I¹ve seen. Even I managed to Žgure 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 Žxed 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 Žxed. 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: Žnite groups) Visiting Assistant Professor at the University of Montana. >> >> of S_6. It is slightly harder to Žnd such a map which deŽnes 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 I¹ve seen. Even I managed to Žgure >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. ============================================================= ========= It¹s not denial. I¹m 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 (Žnally - 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 Žne detail*, the process will just produce an unpredictable sequence of numbers. What does he mean by that ? - what is unpredicatable about numbers in their *Žne detail*. Also, is there some mathematical basis for most numbers are irrational ? Isnt this an issue of an uncountable inŽnity - or can you say that this type of inŽnity 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 (Žnally - 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 Žne 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 > *Žne detail*. Also, is there some mathematical basis for most numbers are > irrational ? Isnt this an issue of an uncountable inŽnity - or can you say > that this type of inŽnity is *bigger* than the set of rational numbers ? The rational numbers form a countable set because one can make an inŽnite list of them, q_1, q_2, q_3, ... which leaves none out and repeats none. It¹s 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. > What¹s 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; it¹s 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 I¹m looking for. http://www.wikipedia.org/wiki/Lambert¹s_W_function Starling Who could do it numerically, but where¹s 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 efŽciently count. Geometric series are particularly useful in Žnancial percent interest calculations. Check an intermediate or college algebra textbook and look at some of the example and word problems to Žnd 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 Žle 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 Žnite 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. I¹d say the odds are good that there is a counterexample <= 10^16 :) Then too, there is a probabilistic argument that integers don¹t exist: Pick a real number at random, and the probability is zero that it¹s 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. = Docs 4 A Collatz conjecture pixel evaluation 5 Œ 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: Set graphics screen mode to 350 X 640 pixels 15 DEFDBL A: Double precision for any variable beginning with A 20 A=1:A3=A:A4=1:Y=1:A5=640:A6=640: PSET(A,Y): A is starting integer (seed) and turns on pixel x(A) = 1 and Y = 1: This line never used again. 30 A1$=STR$(A): 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 30:¹ 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: Œ Handles odd integers of seed and its sequence. Ect. 320 A4=A4+1:A=A4:A3=A4:GOSUB 500:GOTO 30: ŒRetrieves 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: Œ Sets Up A for right row (Y) 510 If Y>350 THEN Y=1:GOTO 540: Œ 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: Œ 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: Œ 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 540:¹ A branch from 500 where Else means Y=1 540 A6=A5:Y=1:RETURN: Œ 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 Žrst solid white band or seed band grows in width. Please note, when Žrst 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 Œregular¹ 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 >Žrst 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 deŽnite resolution to the Twin¹s Paradox. This result No it doesn¹t. The Twin¹s Paradox isn¹t 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, they¹ll 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. That¹s 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 Žnding them online somewhere. But my search was unsuccessful. So I copied a pair into a Maple worksheet from Marshall Hall¹s Combinatorial Theory, page 194. According to this source the following pair is due to Ostrowski and Van Duren. It serves to show that Mann¹s 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 have¹m: 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, I¹m sure someone is going to tell me where I could have obtained them. :-) --Edwin === Subject: Re: Two orthogonal Latin squares of order 10 charset=Windows-1252 > I was trying to save myself the trouble of typing in two orthogonal Latin > squares of order 10 by Žnding them online somewhere. But my search was > unsuccessful. So I copied a pair into a Maple worksheet from Marshall Hall¹s > Combinatorial Theory, page 194. According to this source the following > pair is due to Ostrowski and Van Duren. It serves to show that Mann¹s > 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 have¹m: > 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, I¹m 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 didn¹t 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 Žnd, 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 you¹ve nicely illustrated the problem of eliminating the vowels. Another one: She¹s not meeting me at the meat market, so the attempt at mate is moot, Mort. turns into Sh¹s nt mtng m t th mt mrkt, s th ttmpt t mt s mt, Mrt. > > V. -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: What is an algebraic integer? > >Not factorable over the ring of coefŽcients. > > 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, doesn¹t ring of coefŽcients refer to the > integers? Does it refer to some ring that is deŽned for each polynomial, > based upon its speciŽc coefŽcients? > > What about x+3 and x-5, taken as individual polynomials? Are they > considered irreducible? >>I think the irreducible bit in the deŽnition is not really necessary. >>As you noticed, the fact that x^2 - 2x - 15 has roots -3 and 5 doesn¹t >>make -3 and 5 algebraic integers according to the deŽnition, 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 deŽnitions 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 Žnd a >monic (irreducible) polynomial with integer coefŽcients that has A as >a root, then A is an algebraic integer. This deŽnition, of itself, >isn¹t decidable, since my failure to Žnd a polynomial with A as a root >isn¹t proof that no such polynomial exists. It might only be due to >my lack of extreme cleverness. Whether there¹s an effective procedure depends on how you¹re given the number as input. If you¹re given algebraic _numbers_ x and y, and their minimal poynomials, there is an effective procedure for Žnding the minimial polynomial of x + y, xy and x/y (I believe). Hence if you¹re given an algebraic integer expressed, say, as an algebraic combination of radicals, there is an effective procedure for Žnding its minimial polynomial. And then if you have the minimal polynomial for an algebraic number you decide whether it¹s an algebraic integer by looking to see whether the (primitive) minimial polynomial is monic. ************************ David C. Ullrich === Subject: Re: What is an algebraic integer? > > > I¹ve been seeing this term a lot in Œs proofs. Can someone > > please give me the deŽnition, 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 coefŽcients and a_n = 1. > > For example sqr 3, 1 + sqr 2, (1 + sqr 5)/2. > > Don¹t feed the troll, maybe it¹ll go away. Why does a_n have to equal one? So it can¹t be a solution to a polynomial with rational coefŽcients? 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 a¹s 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? >> >> > I¹ve been seeing this term a lot in Œs proofs. Can someone >> > please give me the deŽnition, 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 coefŽcients and a_n = 1. >> >> For example sqr 3, 1 + sqr 2, (1 + sqr 5)/2. >> >> Don¹t feed the troll, maybe it¹ll go away. >Why does a_n have to equal one? So it can¹t be a solution to a >polynomial with rational coefŽcients? The deŽnition of algebraic integers stipulates that a_n = 1. If you allow arbitrary integers as a_n, then you get all elements of the Želd 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 a¹s being >rational, then those two numbers are not alegbraic integers. That¹s 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 Želd - 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? It¹s 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 coefŽcients is again an algebraic integer. But you cannot, in general, take quotients. ============================================================= ========= It¹s not denial. I¹m 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? > > It¹s 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 coefŽcients 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 coefŽcients zero. Multiply by 2^(n-1). At x = 1/2, the Žrst 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 coefŽcients, 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 coefŽcients, 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 sufŽciently 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 Bach¹s 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 insufŽcient. Out of this insufŽciency, 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 Žrst 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 Želds 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 Žxed 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 Žlled 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 Žrst 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 Želds > 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 Žxed 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 Žlled 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 who¹s 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 it¹s possile to > have a proŽtable home internet business. > A lot of the stuff on e-bay, for example, is there because you just can¹t > buy the stuff locally and many trade suppliers don¹t 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 can¹t buy > easily: > 1) High efŽciency 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 proŽle. You don¹t > want to have your suppliers cut you off and you don¹t 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 Žnd and meet your customers needs the > proŽts 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 Žnd 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 Žnd 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 Žnd 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 deŽne 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 Žnd 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 speciŽc. 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 deŽning the relations between the above concepts, we Žnd that the structure concept has more interesting information than the quantity concept because: 0^0 = inŽ^0 = 1 = content exists and we can¹t 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 deŽning the relations between the above concepts, we Žnd that the structure concept has more interesting information than the quantity concept because: 0^0 = inŽ^0 = 1 = content exists and we can¹t 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.