mm-1018 === Subject: Re: Transcendental or algebraic? >If x^x=2 then x is transcendental or algebraic? My best guess is transcendental because the equation directly transforms to : LOG(2) = x, using base of x for the logarithm function. [would this be properly typed in text form as LOG_x (2) = x ?] G C === Subject: Elliptic curves If we set X = {(x,y) in C^2 | y^2 = x^3 + ax + b} and Y = {(x,z) in C^2 | z^2 = bx^4 + ax^3 + x} (z = x^2y) and define X and Y by removing points with x = 0. One can create a bijection X --> Y by (x,y) |-> (1/x,x^2y) and if one glues together X and Y along X and Y one gets something like when one completes C by the Riemann sphere. Question is now, how do I visualize this one? I have no background in topology. === Subject: Re: Transcendental or algebraic? > If x^x=2 then x is transcendental or algebraic? Lets examine three possibilities: x is rational (p/q)^(p/q) = 2 leads to (p^p)/(q^p) = 2^q Using just the fact that 2 is an integer implies that the lowest-terms p/q has q=1. There is no solution this way. (I would go a little further: If x and x^x are both rational, x must be an integer.) x is algebraic and irrational Look up Hilberts Seventh Problem. x^x would end up transcendental rather than 2. x is transcendental This is the only possibility left. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: JSH: Open letter to Jim Ferry > Im intrigued by the questions raised by your recent posts, as for > years you were this guy who came up with rather creative ways to > insult me, and now I find it hard not to figure youre just doing so > again. Is it okay to ask you questions, talk in any familiar way, or in any way act as if I wish you to reply or am addressing you? Those proscriptions apply only to David Ullrich, Virgil, and anyone who posts under a palindromic pseudonym, right? Okay then. You ask a good question. Just what is my intent? Im not sure precisely why Im doing what Im doing, but Ill try to answer you earnestly and respectfully. > Now Ill embarrass you a bit as from what Ive read on the web you > have one of the highest IQs out there, so it seems to me theres > probably some reason to what youre doing, and possibly Im wrong > about what it is. This does embarrass me because Im certainly not one of the big fish on sci.math. You must be basing this statement on the fact that I once joined something called Mega, which purports to be a high-IQ society for those of 1-in-a-million intelligence. I now realize that by joining, I was implicitly making this arrogant claim about myself, but I reject that claim. The fact that I was able to ace the math part of their test just indicates that it wasnt hard enough, because its easy to find people better at math than I. Indeed, you can find lots of them on sci.math. And some of them even take the time to analyze your work, James. > Therefore, Im going to give you the opportunity that I feel *I* dont > get, which is the benefit of the doubt. Youre asking me to clarify me position, which I am about to do. I think its fair to say, however, that others have given you the same opportunity, i.e., that theyve asked you to clarify your position. > Tell me succinctly and in a way that will minimize potential > embarrassment for both of us, what it is that your up to, and no, none > of this wild stuff about how great I supposedly am, or how Ive proven > FLT or any of that, as I just want you to say something that fits into > a worldview that makes sense. > Whats your intent? You think that Im mocking you, but youre not entirely sure. David Ullrich thinks its incredible that its not obvious to you that Im mocking you. Well let me clear things up. Yes, Im mocking you. Ive made a series of posts over the last six months in which Ive appeared to be converted to a religion in which you are the Messiah of Mathematical Truth being crucified my the benighted masses. Most people consider that absurd and therefore conclude that I must have been being sarcastic. You, however, do not consider the idea that you are the Messiah of Mathematical Truth to be absurd. You consider it to be essentially correct -- a little off somehow: a little over the top, or emotionally overblown, but basically the correct attitude to take. Long ago I decided that creative mockery is the best response to your presence on sci.math. So I have tried to find new and better ways to mock you. My latest effort has been the most elaborate, as well as the most successful (in my opinion). But it has forced me to put myself in your mindset -- a form of Method acting. And in this mindset, I perceived how abusively this newsgroup treats you, and it made me angry. The abuse that people heap upon you is sickening and wrong. I know that this makes me sound like a hypocrite, but just because I mock you does not mean that Im against you. You may not be able to fathom this, but I honestly feel that Im on your side. I want you to succeed, and the way you can succeed is this: to join me in mocking you! Learn to laugh at yourself, James! Your attitude is really quite amusing, quite ridiculous! Normal people get continuous feedback about their strengths and weaknesses and adjust their self-image accordingly, but youve decided to suppress any negatives, and exaggerate the positives to such a ridiculous extent that youve turned yourself into a runaway self-esteem engine. But you dont perceive the self-esteem: what it feels like to you is constant self-doubt because the pressure of all the negative feedback you suppress is constantly leaking in. So you pump the bellows faster and faster! Yee-hah! Here comes the Narcissistic Personality Disorder Express! Look at Ôer go! Ridiculous! Laughable! Dont you see how funny it is? In fact, its so humorous, it would be worth you getting treatment for your NPD just so you can see how magnificently ridiculous youve been! Theres no point in trying to hide from your ridiculousness, and theres no need either. There is a certain finite amount of ridiculousness where shame is maximized. Youve gone far, far beyond that -- youve traveled into realms of ridiculousness where shame is no longer a factor, or has achieved some negative value called pride. Be proud of your absurdity! Take a bow, James. Youre spectacular! But Im getting ahead of myself. First you need to stop the absurdity before you can look back and laugh at it. Go to a mental health professional. Tell them you have NPD, and if they dont believe you tell them to give you a test. Then get treatment. And then, accept what youve been up until now: spectacularly ridiculous! fantastically absurd! Laugh at yourself for about six weeks straight. And then start building yourself a new and better life as a normal person whos not constantly obsessed with being better or worse than everyone else. Peace be with you, -Jim Ferry === Subject: Re: need help!! i can not integrate from arcsin[2/(3+cosx)] > if u can ,please say me by hupo19@yahoo.com i know answer by i dont know how it solve > thank you > hupo What answer do you have? > my answer is =x*arcsin[2/(3+cosx)] > Well, whats the derivative of that? It is definitely not arcsin[2/(3cosx)] . I am guessing that Ôhupo has plugged in his expression into integrals.com (or whatever that site it) and it interpreted (cosx) as just a variable name, since it will not parse it into cos(x). If hupo changes his cosx and plugs it back in with cos(x) he will find that even the programs can not come up with an indefinite integral (probably because there is no closed form for one.) J === Subject: Re: The role of infinity in math > What is the role of infinity in math: > For part of your answer, I suggest reading > How is it defined? > It depends on the context. In calculus, it is usually a concept, not a > value. In complex analysis (as modeled by the Riemann sphere), it is a > number. In set theory, there are multiple distinct infinities > corresponding to various cardinalities. > Why is it needed? > Its easier to write x -> oo than as x grows larger than any > arbitrary value epsilon. > At what precition does math work? > That depends on what the acceptable error is in your statistical > analysis/approximation. > On a more serious note, Im not sure what youre trying to ask. My > inclination is to so something like infinite precision or absolute > precision. precision. > These are questions which seem to have to accepted answer, what do you > think? > Huh? This question doesnt make sense. You are right, it doesnt. Ill try again: These are questions which seem not to have an accepted answer, what do you think? > -- > Will Twentyman > email: wtwentyman at copper dot net === Subject: Re: Transcendental or algebraic? >If x^x=2 then x is transcendental or algebraic? > My best guess is transcendental because the equation directly transforms to : > LOG(2) = x, using base of x for the logarithm function. and this tells us...? J > [would this be properly typed in text form as LOG_x (2) = x ?] > G C -- Jim Nastos, B.Math, B.Ed | Office: 117 Athabasca Hall MSc Candidate | Office Phone: (780) 492-5046 University Of Alberta | Edmonton, Alberta Department of Computing Science | T6G 2H1 nastos@cs.ualberta.ca | http://www.cs.ualberta.ca/people/grad/nastos.html === Subject: Re: need help!! > It is definitely not arcsin[2/(3cosx)] . > I am guessing that Ôhupo has plugged in his expression into >integrals.com (or whatever that site it) and it interpreted (cosx) as just >a variable name, since it will not parse it into cos(x). I looked brießy at that site and was puzzled because it said the integral of (1/x)*sin(x) was sin(x). Upon reading a little more about how to enter a function I discovered that built in functions must be capitalized and use square brackets e.g. (1/x)*Sin[x]. A bit unusual and it makes you wonder how many students get and believe ridiculous answers. --Lynn === Subject: Solving a recursion relation getting a reasonable lower limit) The relation is: S(1,m)=m S(n+1,m)=sum (i=1 to m-1) of S(n,i) I am specifically interested in S(n,n+1) and believe that this will be exponential in n but am unable to prove it :( Any suggestions? === Subject: Re: The role of infinity in math > >>What is the role of infinity in math: >> >None at all. >Mathematicians (and teachers) would probably do everyone a great >service if they never made reference to infinity; leave the >mysticism to someone else (Buzz Lightyear, perhaps?) > [...] > Hersh,_The_Mathematical_Experience_: Not necessarily. For example, Im not a constructivist. I use ordinals and cardinals all the time. I like classical math and set theory, and after reading D. Rusins posts I think I am in complete agreement with him. Leonard Blackburn > The constructivists regard as genuine mathematics only what can be > obtained by a finite construction. The set of real numbers, or any > other infinite set, cannot so be obtained. > Loc. cit., > Constructivists are a rare breed, whose status in the mathematical word > sometimes seems to be that of tolerated heretics surrounded by orthodox > members of an established church. > Personally, I am *not* of this rare breed. === Subject: Re: help please >can you help find me all the positive integer solutions (x,y) to the equation >2y^2=x^4+8x^3+8x^2-32x+15 >any help is much appreciated. >choogu > > Notice to All: > A bit too late, but I suggest you submit your answer to Mathematics > Magazine rather than help the OP here. The problem is posed in the most > recent issue thereof. dear choogu it has not answer by formols but you can plot it and calcolate it whit area can you plot it? === Subject: Re: there is no such thing as infinity > The program in FORTRAN is simple: > 00001 n=1 > 00002 1 n=n+1 > 00003 print(3,4)n > 00004 if(n.eq.M) then print(3,4)M > 00005 else go to 1 > 00006 end if > 00007 end > It has currently reached about 2.0 x 10^18. You LIE! No output device in the world could keep up with the pace you imply. If it output a million lines per second, your current value would have taken about 60,000 years. Socks === Subject: Re: need help!! >I looked brießy at that site and was puzzled because it said the >integral of (1/x)*sin(x) was sin(x). Upon reading a little more about >how to enter a function I discovered that built in functions must be >capitalized and use square brackets e.g. (1/x)*Sin[x]. A bit unusual >and it makes you wonder how many students get and believe ridiculous >answers. This is hilarious. But the integrator is right on the point, of course. Reminds me of the time I taught calculus... Thomas P.S. But its kind of strange, that he allows sin as a multiplicative constant or is sin = s*i*n ? BTW: Do christian fundamentalists study trigonometry [SCNR] === Subject: Re: Solving a recursion relation > getting a reasonable lower limit) > The relation is: > S(1,m)=m > S(n+1,m)=sum (i=1 to m-1) of S(n,i) > I am specifically interested in S(n,n+1) and believe that this will be > exponential in n but am unable to prove it :( > Any suggestions? How about binomial coefficient?? S(i,m) = m choose i Then S(n,n+1) = n+1 choose n = n+1, not exponential. === Subject: Re: there is no such thing as infinity Fortran says positive infinity = 2147483647 and negative infinity = -2147483648. Weird thing is 2*(positive infinity) = 0. Where should I publish my findings? === Subject: integration by substitution question Hello. Im a college freshman currently taking calc 3 but this question is really more about stuff I was supposed to know at the beginning of calc 2. The problem is, when my textbook describes substitution it starts with backwards chain rule. In fact, I looked at some other texts and some even say that integration by chain rule and integration by substitution is the same thing. However I dont see why chain rule is even necessary. Say you have integral there are 2 ways of looking at this. One way is to say, this is just chain rule done in reverse, so the answer is U^2/2 + C. However, the way its done in class and in all the later chapters in the book pretty much 100% of the time is through manipulating differentials; in this case, dU/dx*dx = dU, so the original integral is U*dU. But this can be done without any mention of chain rule whatsoever! So what am I missing? Im not even going to get into the meaning of differentials in both definite and indefinite integral notation and all that stuff I completely dont understand, either. Any enlightenment would be greatly appreciated though :-) === Subject: Re: Columbia University uses False Arrest, Document Destruction! Nothing. Columbia University is so Liberal far to the Left that it will admit anything even vaguely human (short of a credentialed White male) and give it a PhD for emulating Bush the Lessers National Guard attendence record. Columbia Emeritus Professor Barton Sholod is my cousin. He routinely taught Spanish to spics who could not learn Spanish, and they graduated on time fully degreed without having suffered the hate burden of acquiring detectable education or of paying tuition out of pocket. They had compassionate scholarships. Anybody who cannot succeed in such a milieu is a phenomenon. Try CUNY instead - the standards are lower. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Need help with proving that Z[i] is Euclidean Hi everyone Im a high school student that has a little problem. In a project Im doing in matheamtics, Im trying to prove that Z[i] is Euclidean (thus, Euclides algorithm can be apllied to Z[i] as well). Im stuck and I need help to complete this proof. It would be most helpful if someone would give me a clue on how to proceed to prove this since the way that I started out on doesnt seem to get me anywhere. I tried to show that everytime the divisions algorithm is used for two Gaussian integer, the rest is also a Gaussian integer. By showing that if the divisions algorithm is apllied on this new Gaussian integer, the rest is once again a Gaussian integer, I thought that you could prove with induction that so is the case for every step unitl you dont get any rest. Am I completely wrong or at least a bit on the way? After proving that Euclides algorithm can be apllied to Gaussian integers, I have to prove that the last rest is the gcd for these two Gaussian integers. How to proceed here? Once again, I do not want a complete proof or anything, just a few guidelines on how I should start off. Yours Pierre === Subject: Re: need help!! I am guessing that Ôhupo has plugged in his expression into >integrals.com (or whatever that site it) and it interpreted (cosx) as just >a variable name, since it will not parse it into cos(x). > I looked brießy at that site and was puzzled because it said the > integral of (1/x)*sin(x) was sin(x). Upon reading a little more about > how to enter a function I discovered that built in functions must be > capitalized and use square brackets e.g. (1/x)*Sin[x]. A bit unusual > and it makes you wonder how many students get and believe ridiculous > answers. Ha, I didnt look that far... so I just tried plugging in the integral of: ArcSin[ 2/(3*Cos[x]) ] dx and it returns: Integral ( ArcSin[2*Sec[x]/3] ) dx ... at least we know it is correct interpreting the cosine function. J === Subject: what is the significance of left relative residual and right relative residual matrix? Suppose a matrix A has an appoximation B. Define left relative residual to be L=(A-B)*A^(-1), right relative residual to be R=A^(-1)*(A-B). What are the significance and relationship of their norms? Would one norm always larger than another? -Joenyim === Subject: Re: JSH: how you can prove to the mathematicians you are right >> A successful verification of any of Jamess most dubious claims would >> require a careful explanation of how the proof-checker failed. > Thats a safe assumption. The other side is the following: if > it rejects a JSH proof, what is HE going to say about it? > I have now proven not only that there is an error in Core > Mathematics, but also there is a Core Error in the theory of > proof-checkers!!! > No. In my experience, if ones trying to formalize a proof, but the > result keeps getting spit out by the checker, one always figures that > his formalization is off, not that his basic argument is wrong. No??? I mean, yes, most people would conclude they had screwed up the translation and the formalism, but JSH is not most people. Or do you disagree on that also? > This is one reason that suggestions James formalizes his argument > arent so helpful. When youre in the thick of things, its hard to > tell the difference between a bad argument and problems turning a good > argument into a formal proof. Look: he is not going to put in the kind of effort it would take to do this. Why? (1) Because he resists learning from anyone else; (2) I think he knows he is wrong. Nobody can come as close as he has to understanding our arguments without actually understanding them. Hence his refusal to EVER even quote Deckers main point. If he really believed he is right, the motivation to have his proof validated by machine would be enormous. It is quite possible that he would become famous [Letterman, Leno, Larry King, etc.], even rich. He knows this. Thus if he really believed his own claims, he would be breaking his butt to learn how to use a proof-checker. Clearly he isnt. This is not the first time this has come up. But I think, beneath his patina of arrogant confidence, he knows he is wrong and wasting time with a proof-checker would be an exercise in futility. He would rather prolong the inevitable by interminable squabbling with people here. > Elsewhere in this thread he launches a pre-emptive strike which > amounts to saying that at least one of his concepts cannot be > expressed in the current language of mathematics: too advanced even > for a proof checker. The response there should be, If your proof > cannot be stated in the accepted language of mathematical logic, > then you dont have a proof by anyones standards. > Of course thats correct, but James has no idea what formal languages > are. >> >> Of course, I agree that its unlikely James will do anything that >> requires actual effort. > He does make efforts and actually thinks hard about some things. > But he rarely makes a serious attempt to learn from math books, > courses, or people. If it isnt summarized succinctly on some web > page he ignores it. He *has* learned a few things: (1) Fermats little > theorem; (2) the Barlow-Abel relations; (3) what algebraic integers > and algebraic numbers are; (4) the theorem that roots of non-monic > primitive irreducible polynomials with integer coefficients cannot > be algebraic integers. I dont think he could reliably give the > definition of a ring, certainly not an ideal, probably not a group. > Not much to show for 8 years of effort. Learning how to use a > proof-checker, even the most user-friendly one around, is not going > to be trivial. > Well, youre more impressed by his efforts than I am. Just look at a typical JSH non-rant posting: lots of algebra [often carelessly done to be sure], some ideas [usually unsound], some evidence of thinking and calculating and head-scratching: evidence of effort, I would say, but not evidence of a worthwhile result, and never evidence of an effort to understand much beyond high-school algebra. He agonizes over things, thinks about them, works at them. But when it comes to learning any of the existing body of mathematical theory, he is extremely lazy. No, I am not impressed. Nora B. === Subject: Re: Study groups in science : What positive precautions are you taking to prevent the idiots morons and : kooks from taking it over, as has happened in sci.physics? Perhaps the textbook and the equations will keep people like you away? -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: irrationality of sqrt(2): easy question How to prove a^2 even => a even without using irrationality of sqrt(2)? Doesnt it follow easily from unique factorization into primes? === Subject: Re: Need help with proving that Z[i] is Euclidean Adjunct Assistant Professor at the University of Montana. >Hi everyone >Im a high school student that has a little problem. >In a project Im doing in matheamtics, Im trying to prove that Z[i] >is Euclidean (thus, Euclides algorithm can be apllied to Z[i] as >well). To be Euclidean, there must be some way of measuring the size of the elements. What you want to show is that given two elements a and b of Z[i], with b different from 0, then it is possible to divide a by b, which means finding a ->unique<- quotient q and a ->unique<- remainder r subject to the following two conditions: (1) a = b*q + r; and (2) either r=0, or the size of r is strictly smaller than the size of b. In the integers, the size is measured by the absolute value. What are you using to measure size in Z[i]? The usual thing to use is the norm, which takes values N(a+bi) = a^2 + b^2. >Im stuck and I need help to complete this proof. It would be >most helpful if someone would give me a clue on how to proceed to >prove this since the way that I started out on doesnt seem to get me >anywhere. I tried to show that everytime the divisions algorithm is >used for two Gaussian integer, the rest is also a Gaussian integer. The remainder/residue? Well, thats the definition of division algorithm, so you must actually be trying to prove that you can ->define<- a division algorithm for the Gaussian integers; that is, that you can find q and r as described above. > By >showing that if the divisions algorithm is apllied on this new >Gaussian integer, the rest is once again a Gaussian integer, I thought >that you could prove with induction that so is the case for every step >unitl you dont get any rest. >Am I completely wrong or at least a bit on the way? What you are describing is how you would use the division algorithm to implement Euclids algorithm to find a greatest common divisor between two given algebraic integers. For the induction to work, you need to show that each time you divide, the remainders get smaller, and that they cannot get infinitely smaller. In the integers, you do that because at each step you get a positive integer strictly smaller than the previous remainder, or you get 0, so you must eventually get to zero. But first you must show that it is possible to pick q and r as above, and that they are unique (or at least, unique enough; maybe sometimes you could pick a+bi, or a-bi, or b+ai, or b-ai, or -a+bi, etc, so you have to specify how you are going to pick them...) >After proving that Euclides algorithm can be apllied to Gaussian >integers, I have to prove that the last rest is the gcd for these two >Gaussian integers. How to proceed here? You can do this the same way as you do it for the integers. Say you are given a and b; and a=b*q + r1, as above. You want to show that gcd(a,b) = gcd(b,r1). Then, when you divide b by r1, you will get gcd(r1,b) = gcd(r1,r2), where s is the remainder you get from that division. Continuing that way until you get to the last nonzero remainder, call it d, you will have gcd(a,b) = gcd(b,r1) = gcd(r1,r2) = gcd(r2,r3) = ... = gcd(rn,d) = d, and that will give you what you want. So, first: let d be the gcd of a and b. That means that (i) d divides a and divides b; and (ii) if e divides a and also divides b, then e divides d. We want to show that d is the gcd of b and r1. That is, we need to show that: (i) d divides b and divides r1; and (ii) if e divides b and also divides r1, then e divides d. Since a - b*q = r1, and d divides both a and b, it follows that d divides r1. So d divides b and r1. Now assume that e divides both b and r1. Then it divides b*q + r1, so it must divide a. So e divides b and a, and by (ii), it must divides d. Thus, d satisfies (i) and (ii), so it has to be the gcd of b and r1. Therefore, gcd(a,b) = gcd(b,r1). -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: matrix problem: what can you say about the inf-norm(E*A^(-1)) and inf-norm(A^(-1)*E)? Here inf-norm is the maximal absolute row sum. A is a square matrix, E is a Ôsmall matrix of the same order. I wonder if there exists some theorem/conclusion/hypothesis on the relationship between INF-NORM(E*A^(-1)) and INF-NORM(A^(-1)*E) ? Can anybody give me some pointers? -Jeonyim === Subject: Re: need help!! > It is definitely not arcsin[2/(3cosx)] . > I am guessing that Ôhupo has plugged in his expression into >integrals.com (or whatever that site it) and it interpreted (cosx) as just >a variable name, since it will not parse it into cos(x). > I looked brießy at that site and was puzzled because it said the > integral of (1/x)*sin(x) was sin(x). Upon reading a little more about > how to enter a function I discovered that built in functions must be > capitalized and use square brackets e.g. (1/x)*Sin[x]. A bit unusual > and it makes you wonder how many students get and believe ridiculous > answers. > --Lynn I beleive that those notational requirements are standard for Mathematica. === Subject: Re: integration by substitution question > Hello. Im a college freshman currently taking calc 3 but this > question is really more about stuff I was supposed to know at the > beginning of calc 2. The problem is, when my textbook describes > substitution it starts with backwards chain rule. In fact, I looked > at some other texts and some even say that integration by chain rule > and integration by substitution is the same thing. However I dont > see why chain rule is even necessary. Say you have integral > there are 2 ways of looking at this. One way is to say, this is just > chain rule done in reverse, so the answer is U^2/2 + C. However, the > way its done in class and in all the later chapters in the book > pretty much 100% of the time is through manipulating differentials; in > this case, dU/dx*dx = dU, so the original integral is U*dU. But this > can be done without any mention of chain rule whatsoever! Right, and we can just give you the Fundamental Theorem of Calculus without any justification. Whats behind manipulating differentials? Why does it work? What are differentials anyway? At the beginning calculus level, theyre pretty much just a notational device - a trick really (although a useful one). The chain rule justifies this device. Without understanding this, youre just pushing buttons and barking to your teachers commands. (By the way,its good that youre asking this question!) === Subject: Re: Involutionary Calculus > We call it taking logs. > I give a damn to what you call it or in what manner can the result be > established.If you cannot see the inherent aesthetic appeal behind the > results you better keep your comments with you. Oh no. What will happen if I dont keep my comments with me? === Subject: Re: Oh, how I wonder... >Where is the irrasionality of irrasional numbers? >They seem to be fully rational. Theyre not ratios (of whole numbers). Hence ir-ratio-nal. >Where is the inductive logic in mathematical induction? >It seems to be deduction. >Whats primary with prime numbers? >1 seems to be more primary. >Whats imaginary with imaginary numbers? >-1 seems to be the only imaginary number. >What is platonic with platonic figures? >They seem to be of this world. >Whats defferentiated during differensiation? >No difference is attempted found. It is a mistake to think that the common definitions of the words used in mathematics will tell us something about the structure of the mathematical objects. They have a structure, independent of our language(s). We make clear what structure we wish to study by giving a formal definition. Now, we have the OPTION of using a long name for these things; for example we can refer to convex,face-congruent,face-regular polyhedron if we wish. (That would be the chemists approach.) But we dont like that mouthful, and we have a record that Plato mentioned them, so we OPT to call them Platonic solids. That doesnt imbue them with any additional property beyond what is already a consequence of their definition. Sometimes the choices we have collectively made turn out to be good ones, sometimes not so good. Imaginary number was probably a poor choice, in retrospect. Likewise transcendental. Regular topological space was uninspired. Perfect number was great PR for a pretty unimportant idea; I might put chaos in this category too. Spectral sequence would play well in the popular press if they could only get a sense of what to do with it. But theyre all just names. The trick is to learn the actual definition and see the consequences of that definition. A simple group by any other name would be as cool. dave === Subject: Re: JSH: how you can prove to the mathematicians you are right <87znbpo5p5.fsf@phiwumbda.org> <87smhgokhq.fsf@phiwumbda.org> Discussion, linux) >> No. In my experience, if ones trying to formalize a proof, but the >> result keeps getting spit out by the checker, one always figures that >> his formalization is off, not that his basic argument is wrong. > No??? I mean, yes, most people would conclude they had screwed > up the translation and the formalism, but JSH is not most people. > Or do you disagree on that also? Oh, no, no, no, I do *not* disagree that JSH isnt most people. I dont quote most people, but I quote him regularly. But nothing about JSH indicates that hes more likely to decide that his argument is wrong rather than his attempt to formalize it. On the contrary, hes already shown a certain stubbornness regarding his argument (maybe youve even noticed this). Why would he decide that his formalization is correct and his argument wrong? >> This is one reason that suggestions James formalizes his argument >> arent so helpful. When youre in the thick of things, its hard to >> tell the difference between a bad argument and problems turning a good >> argument into a formal proof. > Look: he is not going to put in the kind of effort it would take > to do this. Why? (1) Because he resists learning from anyone else; > (2) I think he knows he is wrong. I agree on (1) and (2) is not implausible. > Nobody can come as close as he has to understanding our arguments > without actually understanding them. Hence his refusal to EVER even > quote Deckers main point. If he really believed he is right, the > motivation to have his proof validated by machine would be enormous. > It is quite possible that he would become famous [Letterman, Leno, > Larry King, etc.], even rich. And thats just the letter L! When he gets to O, theres Oprah, and at R he gets Rose (or does he get Charlie Rose at C?). >> Well, youre more impressed by his efforts than I am. > Just look at a typical JSH non-rant posting: lots of algebra [often > carelessly done to be sure], some ideas [usually unsound], some evidence > of thinking and calculating and head-scratching: evidence of > effort, I would say, but not evidence of a worthwhile result, and > never evidence of an effort to understand much beyond high-school > algebra. He agonizes over things, thinks about them, works at > them. But when it comes to learning any of the existing body of > mathematical theory, he is extremely lazy. No, I am not impressed. Its the theory that would keep him from learning to formalize his argument. Sure, he diddles about with computations and twisted arguments, but he rarely learns much or puts in any real effort to further his understanding. -- Now Im informing all of you that the people arguing against me are EVIL, yes they are real, live EVIL people as mathematics is that important, so its important enough for Evil itself to send minions like them. -- James Harris on Evils interest in Algebraic Number Theory === Subject: Re: Solving a recursion relation > .... > S(1,m)=m > S(n+1,m)=sum (i=1 to m-1) of S(n,i) > .... > How about binomial coefficient?? S(i,m) = m choose i > .... Perhaps Chris could try proving by induction on i that this is in fact the unique solution. Ken Pledger. === Subject: open set containing all rational points of known measure Is anyone aware of a construction (or a reference to a construction) of a (nontrivial) open set containing all rational points on an interval with a proof of its exact measure. For example, a construction of an open set on [0,1] with measure exactly 1/2, containing every rational point in the interval. (Of course, its an easy exercise to construct a set with these properties with a measure within an arbitrarily good approximation of a desired value.) === Subject: optimal solution Group, I hope someone will give me advice, or point me in the right direction so that I can solve the following type of problem. Given: N THINGS that produce GOOD stuff, but at a COST as follows; COST(i) = exp(slope(i)*(GOOD(i)-offset(i))); i=1,N which implies GOOD(i) = log(COST(i))/slope(i) + offset(i); i=1,N I have a target amount of GOOD I need to get out of my THINGS, but I want to minimize the COST so SUM(GOOD(i)) = Target SUM(COST(i)) is minimum It seems to me that the minimum is reached when the slopes of the GOOD vs COST curves of each THING are equal, or 1/(COST(i) * slope(i)) = SOME_CONST. Whats the best way to go about solving this sort of problem? Chris === Subject: Re: Dense Linear Ordering Im pretty confident that the rational and reals ARE isomorphic. >> > Im curious, I know how to show dense linear orderings without > endpoints are isomorphic. However, how do you show that dense linear > orderings without endpoints admits elimination of quantifiers?? > >> >> Are you aware that the rationals and the reals have dense linear >> orderings without endpoints but are not isomorphic? >Countable dense orderings w/o endpoints are all isomorphic. >Are all such orderings of a given cardinality isomorphic? > Surely not... > For example R and R with an interval removed and the rationals > in that interval stuck back in. > ************************ > David C. Ullrich === Subject: Re: integration by substitution question > .... Say you have integral > there are 2 ways of looking at this. One way is to say, this is just > chain rule done in reverse, so the answer is U^2/2 + C. However, the > way its done in class and in all the later chapters in the book > pretty much 100% of the time is through manipulating differentials .... > So what am I missing?.... Leibniz tried and discarded several ideas for good calculus notation, before settling on dy/dx etc. His notation has some fine mnemonics built into it. For example, the chain rule is written dy dy du -- = -- -- dx du dx fractions. Of course its really more subtle than that, but the simple appearance of that equation makes it easy to remember and use. His notation is helpful also in integration by substitution, where differentials. Thats not a thorough explanation of whats going on, but it makes life easier for all of us. So when you casually use differential notation to skim over the depths of integration by substitution, spare a thought for Leibniz who took the trouble to make it look so easy. Ken Pledger. === Subject: Re: Silly question on limits, tensor products > In a nutshell, I see this line in my text : Let u(t) be a smooth curve in > the vector space U and v(t) be a smooth curve in the vector space V. > lim [ u(t+h) tensor [v(t+h) - v(t)]/h ] > h--> 0 > u(t) tensor dv/dt > Fine and dandy. But I am stepping back a bit and asking myself the > following: There was one step that was skipped. That is : > lim [ u(t+h) tensor [v(t+h) - v(t)]/h ] = > h--> 0 > lim u(t+h) tensor lim ( [v(t+h) - v(t)]/h ) > h ----> 0 h --> 0 > Why is that true? (I know, it is a stupid stupid question) The way I > justify this is by basically hand-waving and saying it makes sense, but I > am not convinced that you can do this type of thing for every occasion in > which a limit breaks up into two limits. When can you do this sort of > thing with limits where one breaks up into two and when cant you? I know > Im being a bit vague, but if someone has a quick counterexample or can > understand what Im trying to say then Id really appreciate the insight. If I have my definition right, a 2-tensor in this situation is a bilinear map T from U x V into R. But then there are numbers a(i,j) such that T((x1,...,xn),(y1,...,yn)) = sum a(i,j)*xi*yj, where we sum over all (i,j). Therefore T is continuous on U x V, which easily implies your result. As for a general discussion fof this sort of thing, I feel its better to deal with these situations as they arise. In time youll get a feel for it and the proofs should take a matter of seconds. === Subject: Re: Need help with proving that Z[i] is Euclidean ... [For a ring] >To be Euclidean, there must be some way of measuring the size of the >elements. What you want to show is that given two elements a and b of >Z[i], with b different from 0, then it is possible to divide a by b, >which means finding a ->unique<- quotient q and a ->unique<- remainder >r subject to the following two conditions: > (1) a = b*q + r; and > (2) either r=0, or the size of r is strictly smaller than the > size of b. ... >The remainder/residue? Well, thats the definition of division >algorithm, so you must actually be trying to prove that you can >->define<- a division algorithm for the Gaussian integers; that is, >that you can find q and r as described above. Well, youre the trained algebraist, and Im not, but I dont see why youre so insistent about the uniqueness of q and r, so long as (1) and (2) are satisfied (and I sort of think I remember P. M. Cohn explicitly not requiring uniqueness, though of course in his case hes throwing away a lot of the other properties most people want in their rings, as well). Even in the integers, e.g., when using continued fractions (or equivalently--and the reason I was once sort of up on this kind of stuff--when finding a reduction algorithm for SL(2,Z) modular symbols), its sometimes handy to go for the smallest positive remainder and sometimes handy to go for the remainder of smallest absolute value. Sure, in the latter case, you could say theres just one remainder, and its that one, but you lose nothing that I can see by saying there are various pairs (q,r), for each of which r is of smaller size than b; and I can imagine cases in which you might want actually to take a continued fraction expansion with some remainders positive (though not necessarily of smallest absolute value) and others negative (ditto). Certainly for the application to g.c.d.s, uniqueness isnt used, is it? All thats used is that the measure of size doesnt admit any infinite descending chains. Lee Rudolph === Subject: Rational sines and cosines Does there exist a rational number r such that 2r is not an integer and both the sine and cosine of pi r are rational? Put another way, does there exist a right triangle with integer sides and all angles of rational degree measure? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: there is no such thing as infinity DaveR wibbled: > You think you can reach the largest number with a FORTAN program? How > crude! Obviously, you need to write this program in C++. Anything worth doing can be done in Perl, in a fraction of the memory space something like C++ would use. -- Next on list: 26 === Subject: Re: Dogs, ßeas, and hairy balls thusly: > I have made an interesting (to me) observation about my dog (a bitch, by > the way). The third element of the subject line is therefore a bit puzzling. Now Ive got ... ah, I see what you mean. -- Paul Townsend I put it down there, and when I went back to it, there it was GONE! Interchange the alphabetic elements to reply === Subject: Re: Need help with proving that Z[i] is Euclidean Adjunct Assistant Professor at the University of Montana. >... >[For a ring] >>To be Euclidean, there must be some way of measuring the size of the >>elements. What you want to show is that given two elements a and b of >>Z[i], with b different from 0, then it is possible to divide a by b, >>which means finding a ->unique<- quotient q and a ->unique<- remainder >>r subject to the following two conditions: >> (1) a = b*q + r; and >> (2) either r=0, or the size of r is strictly smaller than the >> size of b. >... >>The remainder/residue? Well, thats the definition of division >>algorithm, so you must actually be trying to prove that you can >>->define<- a division algorithm for the Gaussian integers; that is, >>that you can find q and r as described above. >Well, youre the trained algebraist, and Im not, but I dont >see why youre so insistent about the uniqueness of q and r, >so long as (1) and (2) are satisfied (and I sort of think I >remember P. M. Cohn explicitly not requiring uniqueness, >though of course in his case hes throwing away a lot of >the other properties most people want in their rings, as well). Fair enough. It does not really matter so long as the remainder drops in size, for the purposes of the Euclidean algorithm. -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Topological property that represents Ôunboundedness When traversing the surface of a sphere, one can go in any direction forever, even a straight line, and never encounter an Ôedge of the surface. What is the topological invariant that represents this property? l8r, Mike N. Christoff === Subject: Re: Topological property that represents Ôunboundedness > When traversing the surface of a sphere, one can go in any direction > forever, even a straight line, and never encounter an Ôedge of the surface. > What is the topological invariant that represents this property? Let me also qualify that Im interested in objects with finite surfaces. l8r, Mike N. Christoff === Subject: Re: Any properties for such matrix > Hi here we define M as a mxm complex matrix > and M^n = I > is there any properties related with this kind of matrix? Its diagonalizable, for a start. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: irrationality of sqrt(2): easy question >> How to prove a^2 even => a even without using irrationality of >> sqrt(2)? > Doesnt it follow easily from unique factorization into primes? Sledgehammer -> nut. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: rank of a random {0,1} matrix Given a nxk random matrix R, k>n, each entry of the matrix is either 1 or 0 with equal probability, i.e., Pr{r_ij=1}=0.5, Pr{r_ij=0}=0.5, what is the probability that the rank of the matrix is n ? === Subject: Re: irrationality of sqrt(2): easy question Adjunct Assistant Professor at the University of Montana. >> Doesnt it follow easily from unique factorization into primes? >Sledgehammer -> nut. In Mexico, the expression is swatting ßies by firing cannons at them... -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Topological property that represents Ôunboundedness >> When traversing the surface of a sphere, one can go in any direction >> forever, even a straight line, and never encounter an Ôedge of the > surface. >> What is the topological invariant that represents this property? > Let me also qualify that Im interested in objects with finite surfaces. Hmmm. Lets compare the real plane R^2 with the unit open disc. In the first you can potter along, and never reach an edge. In the latter one falls off the edge pretty rapidly. So what is the topological distinction between these two surfaces? What in the topology makes one finite and not the other? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: What is the Origin of Space and Time? > Uh, I think it is you who needs to do the homeowork. Everyone knows there > are exactly FIVE dimensions. Every hear about a little something called > ÔTwilight Zone (aka the fifth dimension) !?!?! I bet you feel pretty dumb > right about now. You are only partially correct, sir. Our time space fabric is a combination of 5-dimensional Twilight Zone and 3-dimensional Cartesian Space, which gives EXACTLY 8 dimensions. Buckaroo Banzai was right. Whos feeling pretty dumb right now? === Subject: Re: Any properties for such matrix Hi Helen, Well, if M is not (trivially) the identity matrix, then any eigenvalues of M will be the complex n-th roots of unity. --Eric > Hi here we define M as a mxm complex matrix > and M^n = I > is there any properties related with this kind of matrix? === Subject: Re: there is no such thing as infinity >> You think you can reach the largest number with a FORTAN program? How >> crude! Obviously, you need to write this program in C++. > Anything worth doing can be done in Perl, in a fraction of the memory > space something like C++ would use. And can be written in exponentially more obfuscated a manner than even C or assembler... :) -- Darryl L. Pierce Visit the Infobahn Offramp - What do you care what other people think, Mr. Feynman? === Subject: Re: Aspiring mathematicians, send me your proofs! First off the mark, from Australia, Mark Hurd! See the DC Proof Users Gallery at http://www.dcproof.com/Gallery.htm Keep those submissions coming, folks! Dan Visit DC Proof Online at http://www.dcproof.com > Calling all aspiring mathematicians! > Have your proofs published at my website. Use my DC Proof software (FREE) to > generate your proofs in HTML format (see File / Make HTML File option). > Dont worry about making mistakes. They are impossible in DC Proof! > Send your proofs as HTML attachments to me at: dc@dcproof.com > Be sure to include a caption , an introduction and lots of comments in your > proofs. (See Documenting and Viewing your Proof in the User Reference > Guide.) > Suggested topics: Introductory theorems in logic, set theory, number theory > and group theory. Or introduce any axioms as you see fit -- its YOUR proof! > Suggested limit: 100 lines (ßexible). > Optional: Include your name. If a student, your college, university or > school, and year. > Download my free DC Proof software at > http://www.dcproof.com > Includes self-study tutorial (see excerpts at my homepage). > Dan Christensen > Toronto, Canada === Subject: re:What is the Origin of Space and Time? Wow, is this forum a serious one or not? o_o We cant understand whats not in 3D, so we cannot know. We see 3D, but we cant see it all at the same time, like a person drawn on a piece of paper cant see all the piece of paper. He can see all of a 1D piece of paper at the same time, though. We are in 3D, we can see all of a 2D sheet of paper at the same time. Were immersed in 4D but we cant see even a small part of it at a time. Only a 4D person can, or a 5D person but he can see the whole thing at the same time as well ... :) http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: re:R-Integration A bounded function on [a, b] is Riemann integrable iff the set of points of discontinuity has Lebesgue measure 0. Now for f in your example, clearly its R-integrable with value 0. For g, its not bounded when x -> 0. But g is R-integrable on [x, 1] for any 0 < x < 1 (with integration 0) So if talking about improper Riemann integration, i.e. int_0^1 g = lim_{x->0^+} int_x^1 g. Then g is still integrable. therfore g is bounded and integrable on interval [a, 1] for any a with 0 < a < 1. for the given g , it is not R-integrable is my view correct? http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: hyperbolas I would like to know how to find the gradient of the tangent at the turning point of a hyperbola that doesnt have asymptotes going straight up and straight across. Finding the turning point of a nice simple hyperbola is easy since the gradient is 1 or -1, thats obvious, but Im not quite sure how to find them for other http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: Re: there is no such thing as infinity > Fortran says positive infinity = 2147483647 and negative infinity = > -2147483648. Weird thing is 2*(positive infinity) = 0. > Where should I publish my findings? 2*(positive infinity) = -2 2*(negative infinity) = 0 $ create test.for integer *4 i, j i = 2147483647 j = 2 * i type *, j i = 2147483648 j = 2 * i type *, j end $ fort test $ link test $ r test -2 0 Weird or not, thats twos complement without overßow detection for you. John Briggs === Subject: Re: Columbia University uses False Arrest, Document Destruction! > Can you formulate in a few sentences what exactly is the problem? > That would be helpful for those who dont take the effort of reading whole > sites. By snipping all headers and context, you have lost the bulk of your possible readers. Franz === Subject: Re: Fourier analysis of a radarsignal >>A Fourier transform swaps large-scale and small-scale structure. >>A periodic input produces a sampled output. >>A sampled input produces a periodic output. > Undiluted crap. > Hes basically right though. Please point to the line which is basically right, and explain why it is basically right. Franz === Subject: Re: A finite set that actually has more elements than an infinite one. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1CLmLB16112; >> Horizons: Exploring the Universe by Seeds gives lower-dimensional >> projections as examples of different types of constructions of the >> unverse (closed, ßat and open.) >OK, git, how do you reconcile WMAPS data plus Sloan Digital Sky Survey >data plus Jeffryy Weeks connected dodecahderal universe fit (and its >impled intrinsic chirality) with a 2-D projection? >Google >Jeffrey Weeks dodecahedron 153 hits >Jeffrey Weeks dodecahedral 82 hits >Why dont you tell us how to ßatten a Seifert-Weber dodecahedral >space? >> A plane is given as a ßat universe, the surface of a sphere as a >> closed universe, and the surface of a saddle as an open universe. >You dont know shit about the subject. There are lots of compact >minimal surfaces with that are Euclidean without being ßat planes. >There are fully *eight* simply-connected geometric 3-manifolds with >compact quotients, Why doesn`t Uncle Al just admit right here and now that its not himself he`s refering to, but to his own uncle, Al? Then, we might at least find some sympathy (or at least) some explanations. === Subject: Re: open set containing all rational points of known measure > Is anyone aware of a construction (or a reference to a construction) > of a (nontrivial) open set containing all rational points on an > interval with a proof of its exact measure. For example, a > construction of an open set on [0,1] with measure exactly 1/2, > containing every rational point in the interval. (Of course, its an > easy exercise to construct a set with these properties with a measure > within an arbitrarily good approximation of a desired value.) Suppose {r_1, r_2, ...} are the rationals in (0,1). For each r in (0,1), let d(r) = min(r,1-r). For x >= 1, let f(x) denote the measure of the union of [0,1/x), (1-1/x, 1] and the open intervals centered at r_n of radius d(r_n)/x^n. Then f : [1,oo) -> (0,1], f is continuous and decreasing, f(1) = 1, and f(x) -> 0 as x -> oo. By the intermediate value theorem, f takes on all values in (0,1]. So for each m in (0,1] there is an open set in [0,1] of measure m containing all the rationals in [0,1]. === Subject: Re: prime relative to involution In the following I write pn to mean p_n (p sub n) to save space/time. > Define a natural number n to be hypercomposite if there exist two natural numbers > a and b such that a^b=n (b is not 1),and hyperprime if no such numbers exist. If n has the prime factorization n = p1^a1 p2^a2 p3^a3 ... pn^am, then n is hyperprime iff d = gcd(a1, a2, a3, ... am) = 1. If we let c = p1^(a1/d) p2^(a2/d) ... pn^(am/d) then n = c^d with c hyperprime. > Then, > Every natural number is either a hyperprime or can be represented uniquely > as the result of a left to right involution with a hyperprime base and normal > prime exponents;e.g:81=(3^2)^2,256=((2^2)^2)^2. Trivial -- using the above defn of c and d, factor d into q1 q2 q3 ... qm (with the possibility that qi = qj) and n = ((((c ^ q1) ^ q2) ^ q3) ... ^ qm). Of course this is unique only up to the order of exponents. There doesnt seem to be any reason to do such a factorization, though. > Every natural number is either a hyperprime or can be represented uniquely > as the result of a right to left involution with hyperprime exponents;e.g: > 81=3^(2^2),256=2^(2^3). This is clearly not true; i.e. 4^3 = 8^2, and both 3 and 2 are hyperprime. If you require the base to be hyperprime as well its true; you can just write n = c^d as above, then let d = (c)^(d), d = c ^ d etc. until d(n) is hyperprime and then n = c ^ (c ^ (c ^ c ...)) ^ d(n). > Does any one find this representation theorem important? Probably not. First of all, since hyperprimes can be constructed quite easily out of primes, theyre probably not a much richer concept. Further, these properties are rather trivial, so if someone needs to use them they would quickly develop them on their own. Looking that over, it sounds kinda harsh. I dont mean to burst your bubble here; when I first heard about the Laplace transformation in high school I remember I tried to develop something similar using limits. After working on it on and off for a week I or so eventually found out that it reduced to something silly like T(f(x)) = f(x)/f(x) or something like that. === Subject: Re: [arccos((sqrt(5)-1)/2 )] / pi is irrational > I will use this to show that > if cos x = (1 + sqrt(5))/2 then x is transcendental. (Hence, it is > irrational.) > > But that wasnt the original question. The original question > was whether x / pi is irrational. A whole nother kettle of fish. > To make this question actually look interesting, note that if > cos(x)=( (1+sqrt(5)) / 4 ), then x/pi is equal to 1/5. > (Also, the original question has sqrt(5)-1 instead of sqrt(5)+1 . > Again, cos(x) = ( (sqrt(5)-1)/4 ) implies x/pi = 2/5 . The question is > to show that this 2/5 value turns transcendental when that Ô4 on the > denominator changes to a Ô2.) So sorry for the bleeps in my non-solution. Yes, the original question did involve (sqrt(5)-1)/(2 *p)i not sqrt(5)+1 and Lindemanns lemma and my proof are only true if x ne 0. Of course e^0 = 1. I will go back to the drawing board and see if my original proof can be fixed. Sorry for the whole nother kettle of fish! Ray Steiner === Subject: Re: need help!! > > > > i can not integrate from arcsin[2/(3+cosx)] > if u can ,please say me by hupo19@yahoo.com > > i know answer by i dont know how it solve > thank you > hupo > > What answer do you have? > J > dear jim > my answer is =x*arcsin[2/(3+cosx)] This answer is incorrect, and looks like it came from a computer. The computer misunderstood cosx to be a constant or something, not the cosine of x. A general indefinite solution for this integral is not known; you can, however, use a computer to estimate the definite integral between two points. === Subject: JSH: Math is hard I did a neat trick yesterday by finally realizing that I could directly challenge posters making various claims about factors in the ring of algebraic integers by using my own quadratic: y^2 - by - 7 = 0, which has as one of its roots (b + sqrt(b^2 + 28))/2 With what Ive been calling the Decker quadratic various posters have asserted that you can have some algebraic integer functions w_1(x) and w_2(x), where w_1(x) w_2(x) = 7 which vary as x varies, so I simply used a method to directly check by using a quadratic with a variable b, so that I can let b be *any* algebraic integer. So, in fact, b = w_1(x) - w_2(x), is a possibility, which directly challenges those people. What I end up with, at x=2, from the Decker quadratic is 7 z^4 + b z^3 + (6b^2 + 83) z^2 - 6bz + 252 = 0 which is important because its non-monic, but *all* of its roots are actually checks against various possibilities for w_1(x) and w_2(x). Its easy to see that it takes away an integer possibility for b, but the method reveals that no algebraic integer can possibly work, at all. Why does it have to work? Oddly enough, you can figure that out from what happens at b=0, as then you have y^2 - 7 = 0. ANY polynomial that you end up with must always allow for that possibility if you just let b=0. Ive already noticed attempts by Dik Winter and Rick Decker to dodge the result, which doesnt surprise me, though its ironic that I could use a non-monic primitive with integer coefficients to prove my case; however, its not hard to see why their position cant be true. You see, no matter how high you go (like Winter posted a now clearly false polynomial from Keith Ramsay of degree 22) your polynomial would *have* to allow for b=0, which means that it is non-monic with a leading coefficient with a factor that is 7. You may see posters using various techniques, and making claims or otherwise running from the issue as a lot of people have invested time and energy on a false position: David Ullrich, Arturo Magidin, Dik Winter, C. Bond, Nora Baron, Keith Ramsay, Rick Decker, among others Even just a couple of people would be enough for arguments to go on and on as people with egos refused to deal with theirs being bruised. However, the fact remains that what Ive shown is a clear counterexample to their claims, and you know, and I know that math is hard. These people may not be capable of accepting the mathematical truth. Maybe youre not either, but I hope that some of you will come over to the side of mathematics, to believing things that are actually mathematically correct, no matter how much it hurts, and no matter how many other people refuse to do so. James Harris === Subject: Re: Study groups in science > : What positive precautions are you taking to prevent the idiots morons and > : kooks from taking it over, as has happened in sci.physics? > Perhaps the textbook and the equations will keep people like you away? You evidently did not realise that the morons and kooks I referred to included you. Franz === Subject: Re: 3-D analogue of pythagorean theorem Continued... The question raised by Ausurosh is a very important one for geometry and applied sciences. Perhaps ought to be brought to the notice of all young senior school/college students. I myself had (and still have with respect to visualization) such questions in my mind for forty years plus and without fully satisfying answers. The lengths,areas, volumes and hypervolumes in N dimensional space are componented along mutually perpendicular / orthogonal directions, the sum of the squares conserving Pythogorean Rule is valid. Let [u,v,w, ..] be a short notation/operator for u^2+v^2+w^2+ .. Length, 2Dimns l=[lx,ly] Theorem of Pythogorus, e.g.,forces/ vectors Area , 3Dimns A=[Ax,Ay,Az] Elasticity stress componeting, e.g., vectors/tensors for stress/moment of inertia/curvature Volume, 4Dimns V=[Vx,Vy,Vz,V4] What is this? HyperVolume, 5Dimns H=[Hx,Hy,Hz,H4,H5] and this? Geometrical imagination of length and area in 2D and 3D as right triangle/tritetrahedron are known. We take sections parallel to the planes (x=constant etc.) to come to lower order space. But I cant visualize the sections of HyperVolume into Volume by any means of representation. Hoping someone helps towards the literature. === Subject: Re: The role of infinity in math > The classic philosophy (Platonism) says that all of this stuff > (infinity) is in fact real. And just because we cannot perceive it > does not mean that it does not exist. This is the majority opinion, > not necessarily because the majority believe it, but more because it > is the most practical way of thinking of mathematics What?!!!! (read that as Im stunned in disagreement and intrigue) > would be much more difficult to write and describe. The derivative > of x^2 would not be 2x but would be 2x+1/M, where M is the largest > number. In practice, this number 1/M would be so small that it > wouldnt even matter, so why write it down on paper? Limits and derivatives and in general differential calculus makes sense and is 100% tangible without having to resort to the existence of such largest number, which can so easily be shown does not exist. The limit of x^2 as x approaches 0 is *exactly* 0 -- its not 0 + 1/M. Its not as close to 0 as we want -- no, it is 0, because the definition directly implies so (a definition that is tangible and expressed in tangible terms, unambiguous, and very indisputable, I think). > Pythagorean theorem would not hold, and it would be much more messy to > describe this relationship. But this way of thinking still avoids > paradoxes. How does it avoid the paradox that there is the same number of real numbers in the interval (0,1) as in the interval (0,2) ?? If you state that there are M real numbers in the interval (0,1), where M is the largest number that exist, how do you explain that there is exactly M numbers in the interval (0,2)? (but there is also exactly M + M numbers... What? M + M is M?? I dont think thats an easy thing to explain, or avoid paradoxes) Carlos -- === Subject: Re: optimal solution > Group, > I hope someone will give me advice, or point me in the right direction > so that I can solve the following type of problem. > Given: > N THINGS that produce GOOD stuff, but at a COST as follows; > COST(i) = exp(slope(i)*(GOOD(i)-offset(i))); i=1,N > which implies GOOD(i) = log(COST(i))/slope(i) + offset(i); i=1,N > I have a target amount of GOOD I need to get out of my THINGS, but I want to > minimize the COST > so > SUM(GOOD(i)) = Target > SUM(COST(i)) is minimum > It seems to me that the minimum is reached when the slopes of the GOOD vs > COST curves of each THING are equal, or 1/(COST(i) * slope(i)) = SOME_CONST. > Whats the best way to go about solving this sort of problem? > Chris The method of Lagrange multipliers yields your result as the unique critical point. You can find the value of SOME_CONST using the SUM(GOOD(i)) = Target constraint. Rob Pratt === Subject: A tricky integral by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1CMBLk18462; Does anyone know if there is a function for the following integration with respect to X please : SQRT[1 + A*(SIN(X))^2 + B*(SIN(X))^4] === Subject: Re: Rational sines and cosines > Does there exist a rational number r such that 2r is not an integer > and both the sine and cosine of pi r are rational? The only rational r between 0 and 1/2 such that sin pi r is rational are r = 0, 1/6, and 1/2. The only rational r between 0 and 1/2 such that cos pi r is rational are r = 0, 1/3, and 1/2. By the various symmetries inherent in the trig functions you can now work out all the rational r, etc. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: The Clearest by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1CMOQa19499; clarification of your point. Then, I will tell you a counterargument to your argument based on the hypothesis of the lemma. > The proposed proof of Goldbachs conjecture does not > work. > The heart of the proposed proof is the assertion > (Lemma 2.5, second part) > that for each large odd q, > there is no setting of residues r_i corresponding to > the primes p_i < q/2, This should be p_i < q, not q/2. > satisfying the following two conditions: > (1) for each i, there are at least two odd integers > less than q in the > residue class (x = r_i mod p_i); > (2) each odd prime p p_zeta) is in some > residue class (r_i mod p_i). > The following counting argument shows that this > assertion is false. > Letting pi(x) denote the number of primes less than > x, we know > pi(x)=(x/log x)(1+o(1)). > This implies that for all q sufficiently large we > have > pi(q/4 - 2) > 13 + 0.245 * pi(q). Could you tell me why q/4 - 2 appeared? > Consider the first six odd primes: p_1=3, p_2=5, > p_3=7, p_4=11, p_5=13, > p_6=17, which I will call the small primes. > Each odd prime p_i between 18 and q is not divisible > by any of these six > small pimes; these p_i will be the large primes. > The primes p_i between 18 and (q/4 -2) will be > called medium primes; > some primes are both medium and large. > Consider a choice of residue classes r_i unequal to > 0 modulo p_i for > these six small primes. > The number of choices of such residue classes is > (3-1)*(5-1)*(7-1)*(11-1)*(13-1)*(17-1)=92160. > For each such choice of (r_1,...,r_6), some fraction > of the large primes > p_i fall outside all six residue classes. Here, does fraction of large primes mean some large prime? > The average value of these fractions (averaged over > the 92160 choices) is > (3-2)*(5-2)*(7-2)*(11-2)*(13-2)*(17-2)/92160 = > 22275/92160 < 0.2417. It seems to me that this decimal is related to the first eqaution in your argument, but it is hard for me to see how they are related. > So we can select one setting of (r_1,...,r_6) for > which the fraction of > large primes p_i falling outside all six residue > classes is less than > 0.2417. > (In fact for q sufficiently large, each choice of > (r_1,...,r_6) gives a > fraction sufficiently close to 22275/92160.) > So these six odd primes, and their associated > residue classes r_i mod p_i, > hit a fraction at least 1-0.2417=0.7583 of the large > primes p_i. > For each of the remaining large primes p_k (fewer > than 0.245(q/log q) of > them), and the six small primes, > assign one of the medium primes p_i, and select r_i > so that p_k = r_i mod > p_i. > Since p_i p_i) hits at least > two odd numbers in the range (3,q). > The number of medium primes is pi(q/4)-7 6+0.245*pi(q), so there are > enough medium primes to assign to the > leftover large primes and the six small primes. > The upshot is that we have chosen residue classes > (r_i mod p_i) for the > small and medium primes, such that each (small or > large) > prime falls into one of the residue classes; this > is the F whose > existence Lemma 2.5 (part 2) denies. > Don Coppersmith Now, the reason I could reply to your argument without complete understanding of your argument is the last paragraph: > The upshot is that we have chosen residue classes > (r_i mod p_i) for the > small and medium primes, such that each (small or > large) > prime falls into one of the residue classes; this > is the F whose > existence Lemma 2.5 (part 2) denies. Recall the hypothesis of the lemma that the sequence of primes for moduli of F are defined as all odd primes less than or equal to p_x_, where p_x_ + 1 =< q. Now, in paricular, is your p_i_ for a modulo one of small primes? More precisely, are your p_i_s related to induction hypothesis? It might be meaningful to point out that there does exist a layering outside G_E_. More directly, how is your argument related to induction step? Hisanobu Shinya === Subject: Re: Please help prove! ~~~~~~~~~~>_<~~~~~~~~~ > 2. A system of linear equations over a field F has 5 solutions => F is > isomorphic to Z_5. > Fill in the blank: > The set of solutions of a system of linear equations is a ...... . > The number of elements of a ..... is a power of the characteristic. > 5 is prime. But keep in mind when you fill in the blanks that the system was not said to be homogeneous. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Involutionary Calculus >> >> Consider the following definitions: >> >> 1)The hyperderivative of a function f at a point x is defined as the >> >> limit as h approaches 0,of (f(x+h)/f(x))^(1/h),provided it exists. >> >> 2)The hyperintegral(or more properly the productal) of a function f over an >> interval [a,b] over which it is positive is defined as the >> >> limit as n approaches infinity,of the continued product as k varies from 0 >> to >> n-1,of [f(a+(b-a)k/n)^1/n],provided it exists. >> >> It turns out to be that the two operations are related through following >> theorem: >> Suppose that F has f as its hyperderivative over some interval (a,b) and >> continuous over [a,b] then >> then the hyperintegral of f over [a,b] is F(b)/F(a). >> >> The calculus thus cerated has a structure analogous to the known calculus >> and many theorems have their analogues. >> It remains true,however,that the hyperderivative and the hyperintegral can >> be expressed in terms of the known derivatives and known integrals. >> If this idea is so much beautiful why is not it popular in mathematical >> circles? >> >> We call it taking logs. > I give a damn to what you call it or in what manner can the result be > established.If you cannot see the inherent aesthetic appeal behind the > results you better keep your comments with you. > Uh, right. All hail Ashurosh, the discoverer of the amazing > hypercalculus. > The point is that your assumption that hypercalculus is not > popular is simply incorrect. Its just calculus plus logarithms, > and logarithms are _very_ popular... > ************************ > David C. Ullrich The theory exists independent of logarithms.Only the connection between them becomes obvious through the use of logarithms. === Subject: Re: The role of infinity in math > The classic philosophy (Platonism) says that all of this stuff > (infinity) is in fact real. And just because we cannot perceive it > does not mean that it does not exist. This is the majority opinion, > not necessarily because the majority believe it, but more because it > is the most practical way of thinking of mathematics > What?!!!! (read that as Im stunned in disagreement and intrigue) > would be much more difficult to write and describe. The derivative > of x^2 would not be 2x but would be 2x+1/M, where M is the largest > number. In practice, this number 1/M would be so small that it > wouldnt even matter, so why write it down on paper? > Limits and derivatives and in general differential calculus makes > sense and is 100% tangible without having to resort to the > existence of such largest number, which can so easily be shown > does not exist. I think youve missed the point. http://en.wikipedia.org/wiki/Mathematical_constructivism l8r, Mike N. Christoff === Subject: Re: The Nature of Space-Time The nature of space-time? The nature of cubic feet-seconds? The nature of cubic meter-days? space-time Still funny, still stupid, Still worshipped like the false god it is. Space is not part of time, time is not part of space. They are 2 different measurement factors. They should remain seperated, not joined as if they are one. Got a cubic inch-hour? How many cubic mm-seconds are in it? === Subject: Unusual Numbering Systems Unusual Numbering Systems Most people are familiar with fixed radix numbering systems like base ten and base two. There are also product based numbering systems. A product base uses two series: a base series, B, and the product of the base series, P. All fixed radix numbering systems are also product based numbering system. For base 10: S=(1,10,10,10,...) P=(1,10,100,1000,...) The allowable coefficients for position i are 0 through S_(i+1)-1. The factorial base is an example of a product base. S=(1,2,3,4,...) P=(1,2,6,24,...) The allowable coefficients for the lowest order position are 0 or 1. The coefficient for the second position are 0 through (3-1) or 0,1, and 2. 321 (base !) = 3*6 + 2*2 + 1*1 = 23 (base 10) Another intersting product base is the root of prime powers: S=(1,2,3,2,5,7,2,3,...) P=(1,2,6,12,60,...) 4121 (base RPP) = 4*12 + 1*6 + 2*2 + 1*1 = 59 (base 10) We can combine two fixed radix bases into one product base: S=(1,2,3,2,3,...) P=(1,2,6,12,36,...) 2121 (base 2&3) = 2*12 + 1*6 + 2*2 + 1*1 = 35 (base 10) Recently, I have become interested in what I call series bases. Let f() be a series such that the limit of partial sums of f() = 1 and every real number in the range (0,1) is the sum of a subset of f(). For example: f() = (1/2, 1/6, 1/6, 1/12, 1/36, 1/36, ...) Every distinct value, x, in f() represents a position and the allowable coefficients for the PREVIOUS position are 0 through the number of times x appears in the series. To represent integers we take the inverse of each term in f(). (We need to add 1 as the first term in the inverse series.) Inverse of f() = (1, 2, 6, 6, 12, 36, 36, ...) We see that this f() represents the product base 2&3 given above. Are there series bases that are not product bases? Combine the series for base 2 and base 3, (1/2,1/4,1/8,...) and (1/3, 1/3, 1/9, 1/9, ...), to get f() = (1/2, 1/3, 1/3, 1/4, 1/8, 1/9, 1/9, ...) The limit of this f() is 2. Divide each term by 2. f() = (1/4, 1/6, 1/6, 1/8, 1/18, 1/18, ...) Inverse f() = (1, 4, 6, 6, 8, 18, 18, ...) 0 = 0 1 = 1*1 = 1 (base 10) 10 = 1*4 + 0*1 = 4 11 = 1*4 + 1*1 = 5 20 = 2*4 + 0*1 = 8 21 = 2*4 + 1*1 = 9 100 = 1*6 + 0*4 + 0*1 = 6 101 = 1*6 + 0*4 + 1*1 = 7 110 = 1*6 + 1*4 + 0*1 = 10 111 = 1*6 + 1*4 + 1*1 = 11 120 = 1*6 + 2*4 + 0*1 = 14 121 = 1*6 + 2*4 + 1*1 = 15 1000 = 1*8 + 0*6 + 0*4 + 0*1 = 8 1001 = 1*8 + 0*6 + 0*4 + 1*1 = 9 1010 = 1*8 + 0*6 + 1*4 + 0*1 = 12 1011 = 1*8 + 0*6 + 1*4 + 1*1 = 13 1020 = 1*8 + 0*6 + 2*4 + 0*1 = 16 Every product base has a unique representation for each integer. This series base does not have that property. 8 and 9 have more than one representation. This series base has no representation for 2 or 3. We can get around this problem by having a special rule for single digit representations. Is there a series base that is not a product base and every integer greater than some n has an unique representation? Is there a series base that has a finite representation for every real algebraic number? Russell - 2 many 2 count === Subject: Re: Involutionary Calculus >Consider the following definitions: > > 1)The hyperderivative of a function f at a point x is defined as the > > limit as h approaches 0,of (f(x+h)/f(x))^(1/h),provided it exists. > 2)The hyperintegral(or more properly the productal) of a function f over an > interval [a,b] over which it is positive is defined as the > limit as n approaches infinity,of the continued product as k varies from 0 to > n-1,of [f(a+(b-a)k/n)^1/n],provided it exists. > It turns out to be that the two operations are related through following > theorem: > Suppose that F has f as its hyperderivative over some interval (a,b) and > continuous over [a,b] then > then the hyperintegral of f over [a,b] is F(b)/F(a). > The calculus thus cerated has a structure analogous to the known calculus > and many theorems have their analogues. > It remains true,however,that the hyperderivative and the hyperintegral can > be expressed in terms of the known derivatives and known integrals. > If this idea is so much beautiful why is not it popular in mathematical > circles? > Does it have any use? What good is it? What is the use of knowing the Fermats Last theorem,of Eulers formula for the sum of the reciprocals of the squares of all natural numbers and similar results. The fact that a mathematical theory has no obvious use at some moment of time doesnt mean that its useless. What problems can it solve that are not otherwise solveable, if any? In what ways may it offer simpler explanations of phenomena explained by more convoluted expression? > Solve a problem with it, and challenge others to solve the problem any other way. Rise above the childish problem solving mentality. === Subject: Re: Study groups in science : You evidently did not realise that the morons and kooks I referred to : included you. Of course I understood your pretext. You post it all over the group. I was merely pointing out that your obsession with these categorisations without recourse to understanding or, indeed, textual analysis only serves to decrease the quality of posts in this newsgroup. It is antirationalistic and the antithesis of science. So, these posts of yours are amusingly self-referential of you. -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: Fourier analysis of a radarsignal >>A Fourier transform swaps large-scale and small-scale structure. >>A periodic input produces a sampled output. >>A sampled input produces a periodic output. >Undiluted crap. >>Hes basically right though. > Please point to the line which is basically right, and explain why it is > basically right. The word Ôsample is used a broader. A dirac in time or frequency transforms to unity. A wide function in time has a narrow spectrum and a narrow function of time has a wide spectrum. The eigenfunction(s) transforming to themselves are the gaussian, BTW. A periodic signal has a set of lines in the spectrum. A periodic dirac in time has a periodic dirac in the spectrum. Rene -- Ing.Buero R.Tschaggelar - http://www.ibrtses.com & commercial newsgroups - http://www.talkto.net === Subject: Re: Unusual Numbering Systems > Is there a series base that is not a product base > and every integer greater than some n has an unique > representation? > Is there a series base that has a finite representation > for every real algebraic number? Um, Im not sure if this satisfies your criteria, and you might already be familiar with them, but if you are not you would be interested in Fibonacci bases. J === Subject: Re: A tricky integral > Does anyone know if there is a function for the following integration > with respect to X please : SQRT[1 + A*(SIN(X))^2 + B*(SIN(X))^4] > where A and B are positive constants and the limits are from zero to Y > (pi/2 < Y < pi). Mathematica seems to give an answer. It is very lengthy, quite horrific, in terms of incomplete elliptic integrals of all three kinds. BTW, Id almost be willing to bet that its incorrect for at least some values of A, B, or Y. David === Subject: Re: Dense Linear Ordering >Im pretty confident that the rational and reals ARE isomorphic. Good for you. Got any opinions on whether 2 + 2 = 4? > >> Im curious, I know how to show dense linear orderings without >> endpoints are isomorphic. However, how do you show that dense linear >> orderings without endpoints admits elimination of quantifiers?? >> > > Are you aware that the rationals and the reals have dense linear > orderings without endpoints but are not isomorphic? >>Countable dense orderings w/o endpoints are all isomorphic. >>Are all such orderings of a given cardinality isomorphic? >> Surely not... >> For example R and R with an interval removed and the rationals >> in that interval stuck back in. >> ************************ >> David C. Ullrich ************************ David C. Ullrich === Subject: urgent analysis question Im having difficulty with a problem and was hoping someone might be able to help. The problem is the following: Let X be a Lebesgue null subset of R (the reals) and let f: R -> R be a continuously differentiable function. Prove f(X) is a null set. === Subject: Re: [Help] Borel Cantelli >>I have no idea of showing next problem; >>Let A_1, A_2, ... be events in probability space. Define X_n=A_1+A_2+...+A_n >>and s_n = E(X_n). Suppose s_n ->inf and ||X_n/s_n||_2 ->1. >> ??? You say X_n is the sum of some _events_, ie _sets_, and then >> X_n appears to be a random variable, ie a _function_. This >> makes no sense. Was X_n actually supposed to be the sum of >> the indicator functions of those sets? >Yes, X_n is sum of indicator function of those sets. >>Show that >> {X_n=0} <= (k - X_n)(k+1 - X_n)/k(k+1) >>for each positive integer k. >> Again, you ask us to show that a set is <= a function; I >> dont know what this means (do you really want that >> the indicator function of that set is <= the right side >> or what?) >This is the same, it is the indicator function. Ok. Then you need to show that (k - X_n)(k+1 - X_n)/k(k+1) >= 1 where X_n = 0 and (k - X_n)(k+1 - X_n)/k(k+1) >= 0 elsewhere. Both are very easy and have nothing to do with probability. (The second looked wrong at first, but its true because X_n is an integer, so k - X_n < 0 < k+1 - X_n is impossible.) >> Also I wonder if the above is _really_ what you want >> to prove. The reason I wonder is that it has nothing >> whatever to do with the hypotheses s_n ->inf >> and ||X_n/s_n||_2 ->1... >I would also like to show following; >2)By appropriate choice of k, deduce that Sum_0^inf A_i >= 1 a.s. >(again it is indicator function) This just says that the union of the sets A_i has measure 1. Which is clear from the hypotheses, without the inequality you asked about: If m(X) < 1 then there exists c < 1 such that ||f||_1 <= c||f||_2 for all f supported on X, by... >3)Prove that Sum_m^inf A_i >=1 a.s. for fixed m. (again indicator >function) >4)Deduce that P{omega in A_i i.o.} = 1 >To summarize I used the same symbol for a set and its indicator >function. Why? ************************ David C. Ullrich === Subject: Re: Involutionary Calculus > > Consider the following definitions: > > 1)The hyperderivative of a function f at a point x is defined as the > > limit as h approaches 0,of (f(x+h)/f(x))^(1/h),provided it exists. > > 2)The hyperintegral(or more properly the productal) of a function f over an > interval [a,b] over which it is positive is defined as the > > limit as n approaches infinity,of the continued product as k varies from 0 > to > n-1,of [f(a+(b-a)k/n)^1/n],provided it exists. > > It turns out to be that the two operations are related through following > theorem: > Suppose that F has f as its hyperderivative over some interval (a,b) and > continuous over [a,b] then > then the hyperintegral of f over [a,b] is F(b)/F(a). > > The calculus thus cerated has a structure analogous to the known calculus > and many theorems have their analogues. > It remains true,however,that the hyperderivative and the hyperintegral can > be expressed in terms of the known derivatives and known integrals. > If this idea is so much beautiful why is not it popular in mathematical > circles? > > We call it taking logs. >> I give a damn to what you call it or in what manner can the result be >> established.If you cannot see the inherent aesthetic appeal behind the >> results you better keep your comments with you. >> Uh, right. All hail Ashurosh, the discoverer of the amazing >> hypercalculus. >> The point is that your assumption that hypercalculus is not >> popular is simply incorrect. Its just calculus plus logarithms, >> and logarithms are _very_ popular... >> ************************ >> David C. Ullrich > The theory exists independent of logarithms.Only the connection between them > becomes obvious through the use of logarithms. Thats not the only thing that becomes obvious through the use of logarithms - one other thing is that the theory is really no big deal. ************************ David C. Ullrich === Subject: Re: The role of infinity in math >How many integers are there? How would you possibly answer this question? A. Theres an infinity of them. I would never say that. B. Theres an infinite number of them. Dont say that; it makes people think there are infinite (natural) numbers. C. There are infinitely many of them, or Theres an infinite set of them. Thats OK. I dont see a bit of difference between saying that and saying theres not a finite number of them (But there is at least one!) D. There are aleph-1 of them or the set of them is countably infinite. Look up the definition of countably infinite and tell me this isnt nearly a tautology! So whats your point? I thought I made it clear that mathematics encounters infinite sets and limits and all sorts of situations where we might bandy about the term infinity, but its just a shorthand for something more precise, or else just a synonym for not finite. Or are you saying infinity is the set of integers? I think thats awfully limiting... >> Mathematicians (and teachers) would probably do everyone a great >> service if they never made reference to infinity; leave the >> mysticism to someone else (Buzz Lightyear, perhaps?) >Modern set theory deals a great deal with infinity as a core concept. I think you mean it deals primarily with infinite sets and compares and contrasts them. OK. But I think the emphasis is on other structures (such as the existence of functions of various types between sets, or possible well-orderings on sets). The fact that most of the sets are infinite is a given, and not itself the subject of much attention. (Disclaimer: I am not a set theorist!) Some people do wrestle with the precise question of how we define the predicate S is infinite (usual choice: existence of a bijection S->T where T is a proper subset of S) but I dont think thats considered a core issue of modern set theory! I believe youre reading much too much into the OPs question. This is a secondary-school student who links mathematics with measurement and precision. Perhaps someone has whispered about a thing called infinity and he wants to know what it is and why we care. Not infinite sets; not ordinals and cardinals, just a thing called infinity, which is seen as some kind of number (I guess). I dont believe thats a useful concept in itself. Ive given a talk like My infinity can beat your infinity! to high school students. First thing I do is to ask them to stop thinking about any such things, and instead to think about infinite _sets_ instead. Much more mathematically sound and it still seems to intrigue them. >> When you have a set thats not finite, and people ask you how >> many things youve got, you can say infinitely many, but theyre >> still just ordinary things (points, numbers, whatever); theres >> just a lot of them -- not a finite number of them at all, so, well, >> an in-finite number of them. >You are focusing on the individual elements, but when people talk about >how many elements there are in a set, the desire is to get a meaningful >value. In fact, simply saying that it is infinite is not always enough >detail. Thus the cardinals to measure levels of infinity. You mean, Thus the cardinals to represent different answers to Ôhow many. There are finite cardinals, too, and I think youre reinforcing my point: infinity, per se, is pretty useless: if you want to answer how many, you point to a cardinal. If that cardinal is finite, great. If not, you can say not finite. If you want to give more information, you specify the appropriate cardinal, thus making it clear that the answer infinity is at best a shorthand for not finite. Its not really a useful concept in and of itself. >The Riemann sphere has a point on it that *is* infinity. It is a value, >not a concept, in that model of the complex numbers. Um, right. I said it was a point _called_ infinity, but if now you want to tell me that this is, in fact, the definition of infinity, then I dont object, exactly; but if infinity is _really_ a part of the Riemann sphere, then I wonder what the heck you meant by trotting out infinite sets before? Again I think youre sort of making my point for me: as much as we use the term infinity in mathematics, there really is no single thing that encompasses a synonym for not finite the whole panoply of infinite sets a point on the Riemann sphere etc. not to mention the pop-science connections to Big Bangs or whatever. Its a subtle thing, but careful attention to subtleties can help newcomers. Use infinite, the definable adjective, whenever possible instead of an undefined noun infinity. dave === Subject: Analysis I saw this one before but I forgot the proof. Any hints will help. Prove that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. Steven === Subject: Re: JSH: Math is hard >[...] >You may see posters using various techniques, and making claims or >otherwise running from the issue as a lot of people have invested time >and energy on a false position: >David Ullrich, Arturo Magidin, Dik Winter, C. Bond, Nora Baron, >Keith Ramsay, Rick Decker, among others >Even just a couple of people would be enough for arguments to go on >and on as people with egos refused to deal with theirs being bruised. >However, the fact remains that what Ive shown is a clear >counterexample to their claims, and you know, and I know that math is >hard. >These people may not be capable of accepting the mathematical truth. >Maybe youre not either, but I hope that some of you will come over to >the side of mathematics, to believing things that are actually >mathematically correct, no matter how much it hurts, and no matter how >many other people refuse to do so. Right. Even if _everyone_ else refuses to do so. Everyone on sci.math, every mathematician we harass via email (even the ones who seemed at first to be nicer than the people here), every journal editor on the planet... Guffaw. >James Harris ************************ David C. Ullrich === Subject: Re: What is the Origin of Space and Time? > And here I was, pondering the collapse of a 10- or 26-dimensional universe > in a false vacuum state into 4-dimensional, more stable universe with the > remaining dimensions curled upon themselves... Silly me. ;) Yo, dude, those extra spaces do curl upon themselves ... just like my kitty ... === Subject: Re: (Reference) Book for learning mathematics from the ground up James R Newmans World of Mathematics should be available in good libraries. > mathematics, but since I have completed high-school and > wont be going into college for another few years, I think > the best way to learn would be to do it by myself. I dont > like pressure either, and Im not hoping to major in > mathematics in the future. I started thinking about getting > myself a book, but since Ive completed the most basic > high-school maths, it would be unnecessary to buy a whole > series of books starting with explaining addition > completely. A reference book or some sort of dictionary > with examples and complete definitions would be the ideal. > Is mathematics a far too great field to summarize into one > reference book? It would be great to have a book that I can > look things up in when I get to an advanced level. Of > course, the best way to learn math is to solve problems and > exercises, but that might be out of scope for a reference > book. I might add that I wish to learn mathematics up to, > lets say, a university level. > So my question is: given my need, could any of you please > recommend some good reading for me? My wallet-size isnt > infinite, so one or two major works is about what I can === Subject: Re: Sequence of Diophantine Equations > Hi! > Is there some (huge) positive integer M with the following > property: for any z>M, there exist positive integers > x,y_1,y_2,...,y_z such that > x^x=(y_1)^(y_1)+(y_2)^(y_2)+...+(y_z)^(y_z) ? > (Please remark that the ys are >=1 and need not to be > necessarily distinct). Im afraid youll have to put more conditions on it than that. Otherwise, just let z=x^x and y_1 = y_2 = y_3 = y_4 = ... = 1. Then you get x^x = 1 + 1 + 1 + 1 + ... 1. Therefore, the answer is yes, M = 0. You could put more conditions on it, like requiring y_n > 1 (not y_n >= 1) or requiring that y_n are distinct. Did this arise as part of a solution to a bigger problem, or were you just looking at this for fun? > Ignorantly Yours, > Ady. === Subject: Re: Analysis I think you mean X>-1 John -- John T Lowry 5217 Old Spicewood Springs Rd, #312 Austin, Texas 78731 (512) 231-9391 jlowry100@earthlink.net > I saw this one before but I forgot the proof. Any hints will help. Prove > that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Steven === Subject: Got a speeding ticket and need to fight back Hello everyone, the cops got me speeding 10-15 mph above the speed-limit. My arguement is as follows. Given that the radar has a +/-5 mph standard deviation error, I claim that the cops did not get me beyound resonable doubt. Z = -10/5 = -2 P( Z < -2) = 0.03 3% chance that i didnt speed and the radar was at fault. My arguement to the judge is that 3 in 100 cars will recieve a false speeding ticket because the radar isnt accurate. And i am one of those 3. Even still, the chances that i went just anywhere from below Ô5 mph above the speed limit Z = -5/5 = -1 P(Z < -1) = 0.15% There is a 15% chance(huge chance) that the radar gave a wrong reading. Do u think the judge will buy the arguement? Btw The best way to embarrass them is to teach High school students statistics and statiscal reasoning and embarrass them, by making them appear smarter than the cops -suresh === Subject: Re: Dense Linear Ordering > Im pretty confident that the rational and reals ARE isomorphic. You are overconfident. Is the Stainless Steel Rat rusting? Not only is there no order preserving bijection, f: Q -> R, from the rationals, Q, to the reals, R, which is required for such an isomorphism to exist, to there is no such bijection at all of any sort whatsoever from Q to R. Georg Cantor proved this several different ways. Others have come up with several other proofs since Cantors. Ao the issue is settled beyond all reasonable doubt. Against any such isomorphism existing. === Subject: Re: JSH: Math is hard If its so hard, why not just go shopping? Does this mean theyre coming out with a chubby Doofus Barbie now? -- Wayne Brown (HPCC #1104) | When your tails in a crack, you improvise fwbrown@bellsouth.net | if youre good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: JSH: Math is hard [snip handwaving claptrap intended to resuscitate a dead argument] Why dont you post an argument which begins by stating what you are attempting to prove (instead of a tirade against your critics) and follow it with a step by step proof (instead of clanking your sword)? You constantly post these things which have some vague purpose, prompted by some vague statement, made in some unidentified post, and then proceed to ßing expressions and equations with poorly motivated substitutions, bad math and worse logic, and finally reach some kind of unsupported conclusion which has no identifiable bearing on any previous problem. What was the purpose of this post? You did not refute anything. What do you think you proved, or disproved? Your original line of reasoning has been so conclusively refuted so many times that it really isnt worth anyones time to step through your new arguments. Theyre already known to be patently false. If you want to get serious attention to any new post, state the purpose clearly and show a logical connection between your starting position and your conclusion. You have failed to do that -- again. It appears that you are somehow incapable of constructing a valid proof. -- 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: Analysis > I saw this one before but I forgot the proof. Any hints will help. Prove > that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. Two questions: (1) Whats the linear approximation to (1+x)^n when x>-1? (2) Whats the concavity of the function (1+x)^2? Doug === Subject: Re: optimal solution Fenchels Theorem. That, or Lagrange multipliers. Doug === Subject: Re: pi + e is irrational? > Ok, complete disregard my last two posts. Here is what I meant to say: > Are you sure that it is unknown if pi + e is unknown? Quite apart from all the valid objections everyone else is making, I will note this: > let pi + e = q [...] > e = q A proof that pi = 0 is not likely to be taken seriously by anyone. > and since e is a strictly increasing fuction we have How can a function be strictly increasing over the complex numbers? Theres no ordering relation among complex numbers. Besides, e^2*pi*i = e^0. eyelessgame === Subject: Re: Is it known whether pi + e is irrational? >>I saw this problem on an unsolved problem archive. Why isnt this >>problem solved? I know it must be deeper than I think, so could >>someone explain why this simple proof doesnt work: >>Assume pi + e is irrational >>Then let pi + e = q, where q is a rational number. Let i = sqrt(-1). >>Then >>i*(pi + e) = i*q >>then >>e^(i*(pi + e)) = e^(i*q) >>then >>e^(i*pi)*e^(i*e) = e^(i*q) >>then >>-e^(i*e) = e^(i*q) >>then squaring both sides: >>e^(2*(i*e)) = e^(2*(i*q)) >>and since e is a strictly increasing fuction we have >>(2*i)*e = (2*i)*q >>meaning >>e = q. > If this result would be right, it actually proved something more > surprising: > pi+e = q (from the assumption) > e = q (from the argumentation) > coming to... > pi = 0 (as a new ISO-norm for the exhausted student.... ) > from which we can easily deduce that _all_ numbers are zero. > Therefore I suggest that, instead of calling it a new ISO-norm, > we should call it the new is0-norm. ;-) > David Ahh oh thank you. yea ehh its like one of those 1=0 proofs except instead of dividing by 0 you ignore the fact that i*pi is imaginary, heheh. Speaking of which, there is another proof like that and I was wondering if anyone would be interested in addressing it (for amusement, if anything) i = e^(pi*i/2) = e^(pi*(e^(pi*i/2))/2) = .. = e^(pi*(e^(pi*e^(pi*(...)/2)/2))/2) If we index the first expression 1, the second expression 2, and the nth expression n, then the expression you get as (lim n -> infinite) is composed of nothing but e raised to real powers (since only the highest exponent is imaginary, but there is no highest exponent at infinite). But wouldnt this imply that i is real? Also, if you look at it a different way, it is imaginary: i = e^(pi*i/2) = e^(pi*(e^(pi*i/2))/2) = .. = ....^(pi*e^(pi*(e^(pi*e^(pi*i/2)/2))/2)) If you look at it this way, there is a highest exponent, and it contains i. It would seem that these are two irreconcilably different ways of looking at the same expression? Or are both of these expressions poorly constructed and impossible to define? Please explain in epsilon-delta terms if necessary. Thought this was interesting. Joey === Subject: Re: Analysis > I saw this one before but I forgot the proof. Any hints will help. Prove > that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Two questions: > (1) Whats the linear approximation to (1+x)^n when x>-1? > (2) Whats the concavity of the function (1+x)^2? One question: whats wrong with induction on n? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Sequence of Diophantine Equations > Is there some (huge) positive integer M with the following > property: for any z>M, there exist positive integers > x,y_1,y_2,...,y_z such that > > x^x=(y_1)^(y_1)+(y_2)^(y_2)+...+(y_z)^(y_z) ? > > (Please remark that the ys are >=1 and need not to be > necessarily distinct). > Im afraid youll have to put more conditions on it than that. > Otherwise, just let z=x^x and y_1 = y_2 = y_3 = y_4 = ... = 1. Then > you get x^x = 1 + 1 + 1 + 1 + ... 1. Therefore, the answer is yes, M = > 0. That doesnt answer the question, not the way I read it. Take your M = 0. Take z = 2, thus satisfying the hypothesis z > M. Now try to find x, y_1, and y_2 such that x^x = (y_1)^(y_1) + (y_2)^(y_2). Give up? OP wants M such that it works for *all* z > M. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > Hello everyone, > the cops got me speeding 10-15 mph above the speed-limit. My arguement is > as follows. > Given that the radar has a +/-5 mph standard deviation error, I claim that > the cops did not get me beyound resonable doubt. > Z = -10/5 = -2 > P( Z < -2) = 0.03 > 3% chance that i didnt speed and the radar was at fault. My arguement to the > judge is that 3 in 100 cars will recieve a false speeding ticket because > the radar isnt accurate. And i am one of those 3. > Even still, the chances that i went just anywhere from below Ô5 mph above > the speed limit > Z = -5/5 = -1 > P(Z < -1) = 0.15% > There is a 15% chance(huge chance) that the radar gave a wrong reading. > Do u think the judge will buy the arguement? = Refusing to use proper spelling for English words identifies you as a smartass, and you definitely dont want any judge to think that you are a smartass. I think a German judge would ask you if you are serious, and if you say yes, then he would get an expert witness to examine the camera very carefully. If it turns out that the camera couldnt have been so wrong so that you definitely have been speeding (even one mile above the limit), then you will be convicted and the cost for the expert witness will be added to your bill. And it wont be cheap. That is just common sense, as it prevents people who have no respect for other peoples lives to use lame excuses like yours. And what makes you think that the error of a radar camera would have normal distribution anyway? === Subject: Re: Study groups in science > if you are interested in scientific online workshops, please visit > site > http://de.geocities.com/scienceworkshops/ > Its goal is to organize study groups on scientific topics like quantum > field theory, > probabilistic inference or neural nets, to name a few topics I am > personally interested in > (of course, arbitray topics may be suggested). > The idea is to study some text (which is freely available over the > internet, by setting up > a study plan for 12 weeks and open a discussion groups where > participants can post questions, > or solutions to exercises. > The text must be of academic level, ranging from introductory texts to > post-graduate level. > Participation is free in these groups. > I would like to start the first workshop in fall, so if you are > interested you might either > subscribe to our mailing list or propose your own workshop. > Jochen Gruber The site has great links to useful online books. I dont think you should limit yourself to online books, but should also include online papers such as those of Einstein and classics such as The Principle Of Relativity from Dover books and The Principle Of Quantum Mechanics by Dirac which people are prepared to buy. Its also important that there are a few academics that are prepared to kindly give their time for free. Itll end up the blind leading the blind otherwise. Are you an academic? Funky === Subject: Re: Limit Of # Of Coprime Integers > Leroy, > For the original n=2 case, the multiplier out front should be > 1/m^3 rather than 1/m^2, of course. > But also the numerical value seems wrong; > for the limit I get > product ( (1-1/p) + (1/p)*(1-1/p)^2 ) = 0.428... > (the product being taken over all primes) > while for the expression with zeta(3) I get > 12/pi^2 - 1/zeta(3) = 0.383... > The factor in the infinite product simplifies to > 1 - 2/p^2 + 1/p^3 > which is unfortunate. If it could be expressed as products of > factors of the form > 1 - p^k > then we could write the infinite product in terms of > 1/zeta(k) > but it cant be so expressed, so Im at a loss. > For general n, the infinite product would be instead > product ( (1-1/p) + (1/p)*(1-1/p)^n ) > Don Coppersmith You are right, I was *WRONG*!... I discovered I was wrong this morning when I noted that n*6/pi^2 - sum{k=3 to n+1} 1/zeta(j) is negative for all high enough ns, ...obviously wrong! And I did not even notice myself that I had an incorrect 1/m^2, where I should have had a 1/m^3. But, in any case, I did finally get the n=2 case: limit{m-> oo} (1/m^3) *sum{k=1 to m} sum{j=1 to m} H(m;k,j) = (6/pi^2) *product{p=primes} (1 -1/(p(p+1))), which is what you got. (I seem to remember seeing somewhere that the product in my representation is equal to some famous constant,...or perhaps that was with (p(p-1)) in the denominator of the fraction instead...) As for n > 2, I have not figured that out, nor will I investigate this any more for now. Leroy Quet >Let H(m;k,j) = the number of positive integers <= m which are coprime >to *both* k and j. >(So, for instance, H(m;k,j) = H(m;kj,1).) >(And H(kj;j,k) = phi(jk), the Euler phi function.) > >I believe that >limit{m-> oo} > m m > --- --- > 1 --- > > H(m;k,j) >m^2 / / > --- --- > k=1 j=1 > 12 1 >= ---- - ------- , > pi^2 zeta(3) >which is, in linear-mode, >limit{m-> oo} >(1/m^2) *sum{k=1 to m} sum{j=1 to m} H(m;k,j) >= 12/pi^2 - 1/zeta(3). >(zeta(r) is, generally, sum{j=1 to oo} 1/k^r.) >And, more generally, >limit{m-> oo} > m m m > --- --- --- > 1 ------- > > ... > H(m;k_1,k_2,..,k_n) >m^(n+1) / / / > --- --- --- > k_1=1 k_2=1 k_n=1 > n+1 > --- > 6*n 1 >= ---- - / ------- , > pi^2 --- zeta(j) > j=3 >which is, in linear-mode: >limit{m-> oo} >(1/m^(1+n))* > sum{k_1=1 to m} sum{k_2=1 to m}...sum{k_n=1 to m} H(m;k_1,k_2,..,k_n) >= 6*n/pi^2 - sum{j=3 to n+1} 1/zeta(j) . >And H(m;k_1,k_2,..,k_n) = >the number of integers, j, coprime with (k_1)*(k_2)*(k_3)*...*(k_n) >and such that 1 <= j <= m. >Am I right? At least, is my n=2 case right? >Leroy Quet === Subject: Re: Got a speeding ticket and need to fight back Assuming a US judge, he will say shut up, guilty, pay at the window on your way out. Slainte, Fletch The Lord of Chaos (Suresh Devanathan) > Hello everyone, > the cops got me speeding 10-15 mph above the speed-limit. My arguement is > as follows. > Given that the radar has a +/-5 mph standard deviation error, I claim that > the cops did not get me beyound resonable doubt. > Z = -10/5 = -2 > P( Z < -2) = 0.03 > 3% chance that i didnt speed and the radar was at fault. My arguement to the > judge is that 3 in 100 cars will recieve a false speeding ticket because > the radar isnt accurate. And i am one of those 3. > Even still, the chances that i went just anywhere from below Ô5 mph above > the speed limit > Z = -5/5 = -1 > P(Z < -1) = 0.15% > There is a 15% chance(huge chance) that the radar gave a wrong reading. > Do u think the judge will buy the arguement? > Btw > The best way to embarrass them is to teach High school students statistics > and statiscal reasoning and embarrass them, by making them appear smarter > than the cops > -suresh === Subject: the meaning of Jacobian? Can someone please explain to me what a Jacobian is about? I know how to use one, but I dont know its significance. Someone may answer: A zero jacobian makes a singularity in a manifold because some fractions have a zero denominator. Thats not my point. I want to know what the jacobian is in the first place. For example, the schwarzian derivative measures the difference, if you will, between a given function and a mobius transformation. Its that kind of info that Im looking for. Please respond to the newsgroup and not to my email. Ted Shoemaker === Subject: Re: Resolution to Decker Quadratic Issue > Turns out theres another approach to prove a problem with the old > concepts about the ring of algebraic integers. > > Decker put forward the quadratic > > (5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) > > where his as are roots of > > a^2 - (x - 1)a + 7(x^2 + x). > > Letting x=2, you have > > a_1(2)^2 - a_1(2) + 42 = 0, which gives > > a_1(2) = (1 + sqrt(-167))/2 as one of two solutions. > > Now consider the quadratic > > y^2 - by - 7 = 0, which has as one of its roots > > (b + sqrt(b^2 + 28))/2. > > Note that root is an algebraic integer factor of 7 for all algebraic > integers y, and b. > > Now consider > > (1 + sqrt(-167))/2 = (b + sqrt(b^2 + 28))z/2 > > which is > > (1 + sqrt(-167)) = (b + sqrt(b^2 + 28))z > > and I can divide both sides by 4 to finally get > > 49 z^4 + 7b z^3 + (42b^2 - 581) z^2 - 125bz + 1764 = 0. > Not quite. Not that it affects your argument, but you actually get > 7z^4 + bz^3 +(6b^2 + 83)z^2 - 6bz + 252 = 0 Yeah Jones already noted that mistake in his reply in this thread. > Importantly, for any integer b, such that it is irreducible over Q, > *none* of the solutions for z can be an algebraic integer! > > Okay. So for the bs that make the polynomial above irreducible, > youve shown that > (b + sqrt(b^2 + 28))/2 > doesnt divide > (1 + sqrt(-167))/2 > in the algebraic integers. Not much of a surprise, in spite of > the fact that the former divides 7 and the latter divides 42. It is a surprise as I can let b be any algebraic integer, and thus check for every possible algebraic integer factor of 7. Doing so reveals that for algebraic integer b (b + sqrt(b^2 + 28))/2 is never a factor of (1+sqrt(-167))/2 in the ring of algebraic integers. Its a fascinatingly simple way to end the arguing. > Now then, imagine that there exists some algebraic integer b for which > it is reducible over Q, then the root will be a fraction with a 7 or > 49 in the denominator. > > So consider the root c/7, which gives > > (1 + sqrt(-167)) = (b + sqrt(b^2 + 28))c/7, > > which would force (1 + sqrt(-167)) to be coprime to 7. > Why would that be? The problem for your position is that 7 is prime, so if the polynomial with integer coefficients defining z had a rational solution i.e. was reducible over Q, itd have to have at least one fraction as a root with a denominator of 7. However (b + sqrt(b^2 + 28))/2 is itself a factor of 7, as (b + sqrt(b^2 + 28))/2[(b - sqrt(b^2 + 28))/2] = -7, so for (b + sqrt(b^2 + 28))c/7 would have to be coprime to 7, which would force (1 + sqrt(-167)) to be coprime to 7. So no matter how you look at it, (1 + sqrt(-167))/2 cant have a non-unit factor in common with 7 in the ring of algebraic integers. > Therefore, (1 + sqrt(-167)) has no factors in common with 7 in the > ring of algebraic integers!!! > > But thats simply not true, as Dik reminded us in an earlier response. Such a simple denial is not the mark of a highly intelligent person. You need to face the mathematics, as Ive done when Ive been wrong. After all, isnt it the truth which is important? > > And what makes you > think that the error of a radar camera would have normal distribution > anyway? Its upto the court to prove its not. Normal Distribution wins by popularity vote. SAT scores are normal, error in experiments are normal, grades in a class are normal,etc. If the cops really want to prove its not normal, they have go get the guy who invented the radar or the radar gun, or release the schematic of the radar or obscure patented techology. I am going to get a lot of high school students to use this argument. Its so simple that even a ninth grader can understand and argue against the prosecutor. And its so embarrassing for the police if they do not understand what ninth graders understand. -suresh === Subject: Re: JSH: Open letter to Jim Ferry > Im intrigued by the questions raised by your recent posts, as for > years you were this guy who came up with rather creative ways to > insult me, and now I find it hard not to figure youre just doing so > again. > Is it okay to ask you questions, talk in any familiar way, or in any way > act as if I wish you to reply or am addressing you? > Those proscriptions apply only to David Ullrich, Virgil, and anyone who > posts under a palindromic pseudonym, right? > Okay then. You ask a good question. Just what is my intent? Im not > sure precisely why Im doing what Im doing, but Ill try to answer you > earnestly and respectfully. > Now Ill embarrass you a bit as from what Ive read on the web you > have one of the highest IQs out there, so it seems to me theres > probably some reason to what youre doing, and possibly Im wrong > about what it is. > This does embarrass me because Im certainly not one of the big fish on > sci.math. You must be basing this statement on the fact that I once > joined something called Mega, which purports to be a high-IQ society for > those of 1-in-a-million intelligence. I now realize that by joining, I > was implicitly making this arrogant claim about myself, but I reject that > claim. The fact that I was able to ace the math part of their test just > indicates that it wasnt hard enough, because its easy to find people > better at math than I. Indeed, you can find lots of them on sci.math. > And some of them even take the time to analyze your work, James. Youre not one of the big fish on sci.math, and Im less curious about your past experiences with high IQ societies than you probably think. However, I had a theory, and testing it involved mentioning that facet of your public persona. > Therefore, Im going to give you the opportunity that I feel *I* dont > get, which is the benefit of the doubt. > Youre asking me to clarify me position, which I am about to do. I think > its fair to say, however, that others have given you the same opportunity, > i.e., that theyve asked you to clarify your position. > Tell me succinctly and in a way that will minimize potential > embarrassment for both of us, what it is that your up to, and no, none > of this wild stuff about how great I supposedly am, or how Ive proven > FLT or any of that, as I just want you to say something that fits into > a worldview that makes sense. > > Whats your intent? > You think that Im mocking you, but youre not entirely sure. David > Ullrich thinks its incredible that its not obvious to you that Im > mocking you. > Well let me clear things up. Yes, Im mocking you. Ive made a series > of posts over the last six months in which Ive appeared to be converted > to a religion in which you are the Messiah of Mathematical Truth being > crucified my the benighted masses. Most people consider that absurd and > therefore conclude that I must have been being sarcastic. You, however, > do not consider the idea that you are the Messiah of Mathematical Truth > to be absurd. You consider it to be essentially correct -- a little off > somehow: a little over the top, or emotionally overblown, but basically > the correct attitude to take. So you were lying. I just needed to make sure. The problem is that Id concluded that very intelligent people see a much higher value in telling the truth than others. That lead me to consider the possibility that you were in fact sincere, but deluded and confused, possibly dealing with a lot of emotional pain from a difficult position--considering that I might be right--against tremendous social forces. But you werent being brave. You were just being a smart-ass. James Harris === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > 3% chance that i didnt speed and the radar was at fault. My arguement to the > judge is that 3 in 100 cars will recieve a false speeding ticket because > the radar isnt accurate. And i am one of those 3. That doesnt prove that you were one of those three. Doug === Subject: Re: Mathematics, in general, equals a Maze As for representing a series of connected math-results as a maze, I would assume a more apt analogy would be connected math results as a more general graph, not necessarily as a maze (a maze with only one solution, in any case). Anyway, I wonder if there is anything interesting known about the metamathematical topology of connected math-results. (aside from Godels theorem, of course.) (I mean, there HAS to be many interesting results regarding results in-general.) :) Leroy Quet > In >I know mathematics is a maze. >But it is a maze without end. >It is a maze where goals shift >And directions change. >It is almost impossible, >If it is not impossible, >To climb upon its walls >And view everything whole. > I would add that mathematical proof is > A maze of infinite complexity, > of infinite exits and entrances > (each valid), > and of infinite dead-ends > (each invalid), > a maze of opaqueness and transparency, > of obviousness and mystery, > of unappreciated beauty > only seen as it finally becomes known > to esoteric eyes... > But seriously,... > When I sometimes try to prove a math result, I will often get > sidetracked, following other paths, paths not originally intended, to > some other related results instead. (I might as well prove > *something*!...) > Does anyone else have an excellent example of this? > (I know the maze-metaphor applies to Wiles {et al} proof Fermats Last > theorem, since it turned out that a once-abandoned path was actually > the way to Wiles {et al} ultimate goal.) > And of course, in mathematics, if mathematicians fail to solve > something as originally planned, still their work need not be > completely for nothing, since other related results can still come > from their partial successes. > And more ambitiously, as far as the maze metaphor is concerned, can > viewing mathematics and proofs as a maze or as a graph (one result > leads to another leads to another), lead to any new ways to solve > math-problems in general? > (I bet the Ôright-hand-rule algorithm for solving mazes has no > obvious analog that can be applied to proving math results, as an > example, but you never know...) > ;) > Leroy Quet > Leroy Quet === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > Given that the radar has a +/-5 mph standard deviation error, I claim that > the cops did not get me beyound resonable doubt. In addition, whered you get this +/-5mph standard deviation figure? It appears highly arbitrary. Doug === Subject: abstract algebra question, permutation groups I need to find three elements o in S9 with the property that o^3=(157)(283)(469). The answer is (124586739), (142568793), (214856379) I have been trying to figure out how to work this out for a couple hours but havent gotten anywhere. Can someone help me????? I must not be understanding some part of the theory because I dont know a trick that will work for this one. === Subject: Re: the meaning of Jacobian? > Can someone please explain to me what a Jacobian is about? I know how > to use one, but I dont know its significance. > Someone may answer: A zero jacobian makes a singularity in a manifold > because some fractions have a zero denominator. Thats not my point. > I want to know what the jacobian is in the first place. > For example, the schwarzian derivative measures the difference, if > you will, between a given function and a mobius transformation. Its > that kind of info that Im looking for. > Please respond to the newsgroup and not to my email. > Ted Shoemaker Take a tiny box B around a point (u,v,w) in the domain; it will transform into some set F(B); the absolute value of the Jacobian is the limit of the ratio volume(F(B)) / volume(B) as the box shrinks to zero diameter. (Isnt this obvious from the change-of-variable formula for multiple integrals?) The sign of the Jacobian indicates is the transformation was right-handed (+) or left-handed (-). Now you can see that if the Jacobian is zero then F(B) collapses infinitely faster than B (volumewise), indicating the presence of a singularity. === Subject: Re: Analysis > Whats the linear approximation to (1+x)^n when x>-1? 0. === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > Hello everyone, > the cops got me speeding 10-15 mph above the speed-limit. My > arguement is > as follows. > Given that the radar has a +/-5 mph standard deviation error, I claim > that > the cops did not get me beyound resonable doubt. If I were the judge in this case, I would remind you that perjury, that is, lying under oath is illegal. Then I would ask you where you got this +/- 5mph fact > Z = -10/5 = -2 > P( Z < -2) = 0.03 > 3% chance that i didnt speed and the radar was at fault. My arguement > to the judge is that 3 in 100 cars will recieve a false speeding > ticket because the radar isnt accurate. And i am one of those 3. Assuming the judge understands the math, which is highly unlikely since you say no cops would and it justs embarrasses them. In fact, maybe you will embarrass the judge by showing you are smarter than he is. Im sure that will sit well with him. Besides, if I were the judge, I would say, Fine, then you only have to 97% of the ticket, plus court costs. > Even still, the chances that i went just anywhere from below Ô5 mph > above the speed limit > Z = -5/5 = -1 > P(Z < -1) = 0.15% > There is a 15% chance(huge chance) that the radar gave a wrong > reading. If I go up to the average person on the street and say there is an 85% chance of rain tomorrow at 5:00pm, do you think he would take an umbrella to work with him? If I said you have an 85% chance of winning the lottery, would you buy a ticket? If you tell me you have an 85% chance of having been actually speeding, even just just one mile over, I would say guilty then add a stupidity fine for trying to embarrass me and to reimburse the city for the time the officer has wasted being in court to challenge you. > Do u think the judge will buy the arguement? > Btw > The best way to embarrass them is to teach High school students > statistics and statiscal reasoning and embarrass them, by making them > appear smarter than the cops > -suresh I would think the best way to embarrass them would to be find some pictures of the male officers in womens underwear all huddled around livestock of some sort. But then, maybe Im just sick. Besides, whether you know it or not, most police officers went to high school. So, in say 10 years, all of the officers would know statistics (since you are requiring it be taught to high schoolers) and you plan would fire. In fact, police officers would then know enough statistics to pull you over going 5mph *under* the speed limit and say there is an 85% chance you were speeding. - Tim I hate when people lie with math Timothy M. Brauch Graduate Student Department of Mathematics Wake Forest University === Subject: Re: abstract algebra question, permutation groups : I need to find three elements o in S9 with the property that : o^3=(157)(283)(469). : : The answer is (124586739), (142568793), (214856379) : : I have been trying to figure out how to work this out for a couple hours but : havent gotten anywhere. Can someone help me????? I must not be understanding : some part of the theory because I dont know a trick that will work for this : one. Since the 3 of o^3 divides the 9 of S9, the o will be a permutation of length 9 to get the three length 3 permutations you are looking for. The 1st will map to the 4th, the 4th to the 7th, and the 7th back to the 1st, along with the other cycles starting at 2 and 3 (add the rigor). Now, since you know the answers, look at their (1,4,7) (2,5,8) (3,6,9) members. Now show any others must be identical under cyclic permutation. -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: Unusual Numbering Systems > Is there a series base that is not a product base > and every integer greater than some n has an unique > representation? > Is there a series base that has a finite representation > for every real algebraic number? > Um, Im not sure if this satisfies your criteria, and > you might already be familiar with them, but if you are > not you would be interested in Fibonacci bases. No, I had never seen the Fibonacci bases. Too bad the inverse Fibonacci series doesnt converge. I could make an interesting series base if it did. Russell - 2 many 2 count === Subject: Yamanouchi Symbols, Matrix Representation of Irreps of Sn I was wondering if there is anyone who has read Hamermeshs book Group Theory and its Applications to Physical Problems and has a good understanding of and is able to explain in explicit detail how Hammermesh derived the matrix representations of the irreps of Sn displayed from pages 224 to 230, and their relations to the Yamanouchi symbols displayed beside them. I dont understand what exactly the indices s, r, lamda_s, and, lamda_r refer to, and the formulas that Hamermesh labels (7-102), (7-104), (7-105), and (7-111) do not appear to have any relation to the matrix entries, or at least I cant figure out what that relation is. Any information would be greatly appreciated. Rick === Subject: Re: Got a speeding ticket and need to fight back Approximately f lamda = c f lamda = 186 000 mps f -> frequency in hertz h = 60min = 60 * 60 sec f lamda = 186 000 = 186 000 mps f -> 1000,000,000 s^-1 (gigahertz) lamda = .000186 miles Thats the radars wavelength approximately. Take the radars time window, it internally uses to compute the speed: 1/10 second ( the human eye can see about 30 frames / sec ). Assuming Reaction time of a driver = 1/10 sec approx 0.000186/ .1s = .00186 mps = 6.6 mph > The Lord of Chaos (Suresh Devanathan) > Given that the radar has a +/-5 mph standard deviation error, I claim > that > the cops did not get me beyound resonable doubt. > In addition, whered you get this +/-5mph standard deviation figure? It > appears highly arbitrary. > Doug === Subject: Re: The role of infinity in math >These are questions which seem to have to accepted answer, what do you >think? >>Huh? This question doesnt make sense. > You are right, it doesnt. Ill try again: > These are questions which seem not to have an accepted answer, what do you > think? As youve probably seen, there tends to be a majority opinion, with a few dissenters. This is a lot like most other fields. We also get the occasional crackpot. Nobody tends to get kicked out unless they are doing something that is logically inconsistent. Personally, I would have issues with some of the Ultraintuitionist stuff just because I like mathematical induction. As long as they can make it self-consistent, they can do what they want, though. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: JSH: Math is hard > [snip handwaving claptrap intended to resuscitate a dead argument] > Why dont you post an argument which begins by stating what you are > attempting to prove (instead of a tirade against your critics) and follow > it with a step by step proof (instead of clanking your sword)? You > constantly post these things which have some vague purpose, prompted by > some vague statement, made in some unidentified post, and then proceed to > ßing expressions and equations with poorly motivated substitutions, bad > math and worse logic, and finally reach some kind of unsupported > conclusion which has no identifiable bearing on any previous problem. What > was the purpose of this post? You did not refute anything. What do you > think you proved, or disproved? > Your original line of reasoning has been so conclusively refuted so many > times that it really isnt worth anyones time to step through your new > arguments. Theyre already known to be patently false. If you want to get > serious attention to any new post, state the purpose clearly and show a > logical connection between your starting position and your conclusion. You > have failed to do that -- again. It appears that you are somehow incapable > of constructing a valid proof. I dont think youre being fair. As the subject of his post states, JSH finds math hard. His post proves it. Gib === Subject: Re: Unusual Numbering Systems > Is there a series base that is not a product base > and every integer greater than some n has an unique > representation? Is there a series base that has a finite representation > for every real algebraic number? > Um, Im not sure if this satisfies your criteria, and > you might already be familiar with them, but if you are > not you would be interested in Fibonacci bases. ... > Too bad the inverse Fibonacci series doesnt converge. > I could make an interesting series base if it did. My own, very brief, spiel about Fibo base: http://www3.telus.net/ldh/math/fibo.html LH === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > Hello everyone, > the cops got me speeding 10-15 mph above the speed-limit. My arguement is > as follows. If you really want to beat the ticket do what a lawyer friend of mine did. As part of your discovery demand that the police provide all of the maintenance history of the radar gun. Radar guns have to be regularly calibrated. The judge will (sometimes!) dismiss the case if the police dont provide you with the maintenance records, and the police are usually too lazy to do so. Russell - 2 many 2 count === Subject: Re: JSH: Open letter to Jim Ferry |The problem is that Id concluded that very intelligent people see a |much higher value in telling the truth than others. do you consider yourself to be very intelligent? and do you consider yourself to be very honest? in particular, do you consider yourself to have a very good record of telling the truth in your posts to sci.math? -- [e-mail address jdolan@math.ucr.edu] === Subject: Re: Analysis > I saw this one before but I forgot the proof. Any hints will help. Prove > that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Two questions: > (1) Whats the linear approximation to (1+x)^n when x>-1? Yes I realize that the linear approximation to (1+x)^n when x>-1 IS 1 + nx. Im just lost after this. > (2) Whats the concavity of the function (1+x)^2? Concave upward > Doug === Subject: Re: abstract algebra question, permutation groups > I need to find three elements o in S9 with the property that > o^3=(157)(283)(469). > The answer is (124586739), (142568793), (214856379) Well, thats *one* answer. Another answer would involve (189534726), (193542768), and (126589734). And there are many more. Do you understand why (124586739)^3 = (157)(283)(469)? If you do, you should be well on your way to understanding where (124586739) came from (and if you dont, you dont stand a chance). -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: reformulated problem: how to solve for integer matrix equations? Hi ! Using Maple V R4 I find : {a1 = 4*a3, u4 = -4/a4^2, a4 = a4, u2 = -4/a3/a4, a2 = -a4, u3 = 4/a3/a4, u1 = 1/a3^2, a3 = a3} is a solution. Chris > Here is a simplified reformulation of my previous problem: > How to solve this integer matrix equation? > [ a1^2*u1, a1*a2*u2, a1*a2*u3, a2^2*u4] > [ a1*a3*u1, a1*a4*u2, a3*a2*u3, a2*a4*u4] > [ a1*a3*u1, a3*a2*u2, a1*a4*u3, a2*a4*u4] > [ a3^2*u1, a3*a4*u2, a3*a4*u3, a4^2*u4] > [ 16 16 -16 -4] > [ 4 -16 -4 4] > [ 4 4 16 4] > [ 1 -4 4 -4] > a1, a2, a3, a4, u1, u2, u3, u4 should be 2s positive integer power, such > as -2, +1, etc. > I considered to factor the right hand side(the constant matrix) first: > a1^2*u1 =16=4*4*1 > a1*a2*u2=16=4*4*1 > a1*a2*u3=-16=-4*4*1 > a2^2*u4=-4=-2*2*1 > The try to allocate/assign these 2s factors to the variables a1, a2, a3, > a4, u1, u2, u3, u4... > Since there are more equations than unknowns, sometimes there are > conßicting assignment to the variables, so only approximated assignment can > be made to minimize the mean square error between the left hand side and > right hand side in the above equation... > It is doable by exhaustive search, for example, since there are 16 equations > and 8 unknowns, I can do 8-level iteration like the following: > for a1=[-4, -2, -1, 1, 2, 4] do > for a2=[-4, -2, -1, 1, 2, 4] do > for a3=[-4, -2, -1, 1, 2, 4] do > for a4=[-4, -2, -1, 1, 2, 4] do > for u1=[-4, -2, -1, 1, 2, 4] do > for u2=[-4, -2, -1, 1, 2, 4] do > for u3=[-4, -2, -1, 1, 2, 4] do > for u4=[-4, -2, -1, 1, 2, 4] do > compare MSE between LHSvsRHS for all cases > keep the best one > end > end > end > end > end > end > end > end > The only problem is that one the matrix gets larger, with hundreds of > unknowns, the above method cannot be used any more... > Can anybody give me more ideas and pointers? > -Walala === Subject: Re: Dense Linear Ordering Hmm, What I meant to say is that any countable dense linear orderings without endpoints are isomorphic >> > Im curious, I know how to show dense linear orderings without > endpoints are isomorphic. However, how do you show that dense linear > orderings without endpoints admits elimination of quantifiers?? > >> >> Are you aware that the rationals and the reals have dense linear >> orderings without endpoints but are not isomorphic? >Countable dense orderings w/o endpoints are all isomorphic. >Are all such orderings of a given cardinality isomorphic? > Surely not... > For example R and R with an interval removed and the rationals > in that interval stuck back in. > ************************ > David C. Ullrich === Subject: Re: The role of infinity in math > Some people do wrestle with the precise question of how > we define the predicate S is infinite (usual choice: existence of a > bijection S->T where T is a proper subset of S) but I dont think thats > considered a core issue of modern set theory! I came up with a simple definition of an infinite set in another thread. Let S be a set of natural numbers and let x and y be members of S. Consider these two statements: 1) ExAy(x>=y) 2) AxEy(x 0.000186/ .1s = .00186 mps = 6.6 mph Thats a lot of assumptions. You could have saved the time, and just slowed the hell down. Doug === Subject: Re: Got a speeding ticket and need to fight back Proofs are based on assumptions, unfortunately. Thats how things are proved, based on axioms or things taken for granted. If you did not understand that, its not my fault. Assumptions does not imply invalidity of reasoning. They imply that there are based on solid posulates. -suresh === Subject: Re: Sequence of Diophantine Equations > Hi! > Is there some (huge) positive integer M with the following > property: for any z>M, there exist positive integers > x,y_1,y_2,...,y_z such that > x^x=(y_1)^(y_1)+(y_2)^(y_2)+...+(y_z)^(y_z) ? > (Please remark that the ys are >=1 and need not to be > necessarily distinct). > Ignorantly Yours, > Ady. A partial result??:... x^x/(x-1)^(x-1) ~ x*e. So, (for x not y; z >1) z has to be *at least* in the _vicinity_ of x*e, if all ys = (x-1). (I hope this clue {or clue} does not lead you astray.) Leroy Quet === Subject: Re: Oh, how I wonder... thats the worst mnemonic for pi ive ever seen. -- [e-mail address jdolan@math.ucr.edu] === Subject: Re: Got a speeding ticket and need to fight back Are you a professional lawyer? because you seem to taking dumb. -suresh The Lord of Chaos (Suresh Devanathan) > Proofs are based on assumptions, unfortunately. Thats how things are > proved, based on axioms or things taken for granted. If you did not > understand that, its not my fault. Assumptions does not imply invalidity of > reasoning. They imply that there are based on solid posulates. > -suresh === Subject: Re: easy...analysis problem.... > hello.......genius teacher........ > let function f :E -> R is differentiable on open set E in R > 1. let any a in E , f(a) exist. > show that f(a) = lim {f(a+h) - 2f(a) + f(a-h) / (h^2)} , h->0 > 2. let any a in E , n-derivative f^(n) (a) exist. > express f^(n) (a) by limit of f(x) > ------------------------- > um.....i solved first problem. > lim {f(a+h) - 2f(a) + f(a-h) / (h^2)} > h->0 > = lim [{f(a+h)-f(a)} - {f(a-h+h)-f(a-h)} / (h^2)] > = lim {f(a) - f(a-h)} / h , let h=(-t) > = lim {f(a) - f(a+t)} / (-t) > t->0 > = lim {f(a+t) - f(a)} / t > = f(a) > its right?? > but i cant solve second problem.... You havent solved the FIRST problem, either. Your passage > = lim [{f(a+h)-f(a)} - {f(a-h+h)-f(a-h)} / (h^2)] > = lim {f(a) - f(a-h)} / h , let h=(-t) is ßawed. Hint: apply LHopitals Rule to the original quotient, once. Then use the definition of the second derivative. --Ron Bruck === Subject: Re: Got a speeding ticket and need to fight back high school students who take statistics are smarter than you. Besides, everything i am talking is at the high school level. Any high school can understand what i am saying. You have got to be stupid. You dont understand Ôposulates! you think they are signs of weakness. The Radar works based on posulates. One of them is that speed of light is constant and there is such a thing as wavelength, such a thing as velocity, etc. -suresh The Lord of Chaos (Suresh Devanathan) > Are you a professional lawyer? because you seem to taking dumb. > -suresh > The Lord of Chaos (Suresh Devanathan) message > Proofs are based on assumptions, unfortunately. Thats how things are > proved, based on axioms or things taken for granted. If you did not > understand that, its not my fault. Assumptions does not imply invalidity > of > reasoning. They imply that there are based on solid posulates. > -suresh === Subject: Re: rank of a random {0,1} matrix > Given a nxk random matrix R, k>n, each entry of the matrix is either 1 > or 0 with equal probability, i.e., Pr{r_ij=1}=0.5, Pr{r_ij=0}=0.5, > what is the probability that the rank of the matrix is n ? I cant tell you what the exact value is, but I can tell you that it is greater than the probability that you will get others to do your homework for you. Carlos -- PS: Interesting problem, though! === Subject: Re: urgent analysis question > Im having difficulty with a problem and was hoping someone might be > able to help. The problem is the following: > Let X be a Lebesgue null subset of R (the reals) and let f: R -> R be > a continuously differentiable function. Prove f(X) is a null set. Hint: Cover X by countably many intervals whose lengths add to something small. Now, how large can the length of f(I) be if f is bounded on the interval I? This should lead to a proof in the case X is bounded, and if you can handle that case, well ... === Subject: Re: Unusual Numbering Systems Is there a series base that is not a product base > and every integer greater than some n has an unique > representation? Is there a series base that has a finite representation > for every real algebraic number? Um, Im not sure if this satisfies your criteria, and > you might already be familiar with them, but if you are > not you would be interested in Fibonacci bases. > ... > Too bad the inverse Fibonacci series doesnt converge. > I could make an interesting series base if it did. > My own, very brief, spiel about Fibo base: > http://www3.telus.net/ldh/math/fibo.html Is there a series that sums to the golden ratio? Russell - 2 many 2 count === Subject: Re: Resolution to Decker Quadratic Issue > It is a surprise as I can let b be any algebraic integer, and thus > check for every possible algebraic integer factor of 7. No, you cant find them all that way. If you form a quadratic equation, such as x^2 + bx +7 = 0 and let Ôb change, you will *not* find all possible algebraic integer factors of 7. You still have to deal with cubic equations, such as x^3 + px^2 + qx + 7 = 0 which will reveal other algebraic integer factors of 7 which did not appear in the quadratic. [snip typical triumphant leap to false conclusion by JSH] -- 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: Analysis > I saw this one before but I forgot the proof. Any hints will help. Prove > that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Steven Here is what I have (and dont have). I know that (1+X)^n and (1+nX) intersect at (0,1). Since [ (1+X)^n ] Ô > n for all X >0 and since (1 + nX) Ô = n for all X we know that (1+X)^n > (1+nX) ( but we already knew this from the binomial theorem). The real question is whether or not (1 + X) ^n and (1 + nX ) intersect below X =0 ? === Subject: Re: urgent analysis question > Im having difficulty with a problem and was hoping someone might be > able to help. The problem is the following: > Let X be a Lebesgue null subset of R (the reals) and let f: R -> R be > a continuously differentiable function. Prove f(X) is a null set. Look up Sards Lemma for the general case. For this easy case, note that its enough to prove for BOUNDED Lebesgue null subsets of R. (Write X as the countable union of [-n,n] cap X.) So WLOG we may assume X is contained in an interval [a,b]. f is continuous, therefore bounded on [a,b], say |f(x)| <= M for all x, where M > 0. Given e > 0 we can cover X by countably many open sets (ai,bi) (all subsets of [a,b] WLOG) with sum_i (bi - ai) < e/M. The image f((ai,bi)) is an interval (not necessarily open) whose length is <= M*(bi-ai); therefore the union of the f(ai,bi) is Lebesgue measurable with Lebesgue measure <= sum_i M(bi-ai) < e. Since f(X) is a subset of this union, therefore the Lebesgue outer measure of f(X) is < e. Therefore the Lebesgue outer measure of f(X) is 0, which means its Lebesgue measurable and has Lebesgue measure zero. --Ron Bruck === Subject: Re: JSH: Open letter to Jim Ferry > The problem is that Id concluded that very intelligent people see a > much higher value in telling the truth than others. That places *your* intelligence below zero. > James Harris JSH Motto: No matter how much I succeed in lowering the integrity bar, I can still slither under it. -- 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: plotting n-dimensional data points on a 2d screen Ok.. this question may have been asked hundreds of times, but so... Im looking for the canonical formulas for plotting n-dimensional points. I know how to plot a 3d graph: screenx = d * x / z screeny = d * y / z (where d is the distance of the observer from the 3d object) I assume that to plot 4d, 5d, 6d, nd would just be an extension of these equations somehow. I assume I could play around and figure it out, but I figure I might just as well ask. I assume that others might find it helpful too for this answer to So.. is there any canonical way to plot n-dimensional points? (Note: I know there are some elaborate multidimensional graphing techniques for data-mining and such... thats not what Im looking for. Im just looking for the simple and easy way at this point-- it doesnt have to even be that *usable* beyond 4d or so, but you should at least be able to see something.) === Subject: Re: Any properties for such matrix > Hi here we define M as a mxm complex matrix > and M^n = I > is there any properties related with this kind of matrix? Yeah, there are a lot of them. Among the obvious are that it has inverse M^(n-1), that all the eigenvalues are complex roots of unity, etc. === Subject: Re: Oh, how I wonder... >Where is the irrasionality of irrasional numbers? >They seem to be fully rational. > Theyre not ratios (of whole numbers). Hence ir-ratio-nal. >Where is the inductive logic in mathematical induction? >It seems to be deduction. >Whats primary with prime numbers? >1 seems to be more primary. >Whats imaginary with imaginary numbers? >-1 seems to be the only imaginary number. >What is platonic with platonic figures? >They seem to be of this world. >Whats defferentiated during differensiation? >No difference is attempted found. > It is a mistake to think that the common definitions of the > words used in mathematics will tell us something about the > structure of the mathematical objects. They have a structure, > independent of our language(s). We make clear what structure > we wish to study by giving a formal definition. Now, we have > the OPTION of using a long name for these things; for example > we can refer to convex,face-congruent,face-regular polyhedron > if we wish. (That would be the chemists approach.) But we > dont like that mouthful, and we have a record that Plato > mentioned them, so we OPT to call them Platonic solids. > That doesnt imbue them with any additional property beyond > what is already a consequence of their definition. > Sometimes the choices we have collectively made turn out to > be good ones, sometimes not so good. Imaginary number was > probably a poor choice, in retrospect. Likewise transcendental. > Regular topological space was uninspired. Perfect number was > great PR for a pretty unimportant idea; I might put chaos in > this category too. Spectral sequence would play well in the > popular press if they could only get a sense of what to do with it. > But theyre all just names. The trick is to learn the actual > definition and see the consequences of that definition. > A simple group by any other name would be as cool. > dave Dude, I think he was kidding... === Subject: Re: Got a speeding ticket and need to fight back > The cops got me speeding 10-15 mph above the speed-limit. > My argument is as follows. > Given that the radar has a +/-5 mph standard deviation error, > I claim that the cops did not get me beyond reasonable doubt. > They dont need to. > Driving is a privilege and not a right. > All the cops need to show is that it was more likely, > given the evidence, that you were speeding anyone can he likes. Why should there be laws against racism? Hey, why even have a court? ÔDriving is a privilege. The cops can make any rule they want. Why even goto court to fight a ticket? Wait a second, by your reasoning ÔNational security is a privilege. Or is it a right? spelling has nothing to do with math. If you didnt learn that in high school, you are never going to learn it, in the rest of your life! Embrassing!!! completely! high school students are a lot smarter than you! its a fact. Did u pass math/physics by a C+ ? -suresh === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > They dont need to. > Driving is a privilege and not a right. > All the cops need to show is that it was more likely, > given the evidence, that you were speeding anyone can he likes. Why should there be laws against racism? Hey, why even have a court? ÔDriving is a privilege. The cops can make any rule they want. Why even goto court to fight a ticket? Wait a second, by your reasoning ÔNational security is a privilege. Or is it a right? spelling has nothing to do with math. If you didnt learn that in high school, you are never going to learn it, in the rest of your life! Embrassing!!! completely! high school students are a lot smarter than you! its a fact. Did u pass math/physics by a C+ ? > -suresh === Subject: please help I need to prove that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. Here is what I have (and dont have). I know that (1+X)^n and (1+nX) intersect at (0,1). Since [ (1+X)^n ] Ô > n for all X >0 and since (1 + nX) Ô = n for all X we know that (1+X)^n > (1+nX) ( but we already knew this from the binomial theorem). The real question is whether or not (1 + X) ^n and (1 + nX ) intersect below X =0 ? === Subject: help with proof of distribution like function: analysis i will first state the problem, and then show you what i have thus far. I am looking at the kernel of the fundamental solution of the heat diffusion equation i.e. K(x,t) = Exp(-x/(4*t))/sqrt(4*pi*t) t>0 The problem is this: does Limit (x+-->0) -2*Integral(0,t) dK(x,t-s)/dx * (g(s) - g(t)) ds = 0 ? [1] I think it does - at least i remember that this result is true from when i first studied the heat equation - but i dont remember the details, which is what i am trying to reconstruct here. note: -integral(0,t) dK(x,t-s)/dx ds = 1/sqrt(pi)*integral(x/(2sqrt(t)),oo) exp(-p^2) dp =1 for x>0 for x<0 -integral(0,t) dK(x,t-s)/dx ds = -1 [2] thus noting this property what we are really trying to show is Limit (x+-->0) -2*Integral(0,t) dK(x,t-s)/dx * g(s) ds = g(t) [3] we have written this in the form of [1] because i think thats more appropriate for a proof. Now i am particularly bad at proofs (my brain just isnt wired that way i suppose), however this is what i have so far. assume g(t) is continuous at t. Since g(t) is continuous at t, for each e>0 there exists a d>0 such that |g(t) - g(s)| < e/2 for all |t-s|<=d ok. consider the integral I = -2*Integral(0,t) dK(x,t-s)/dx * (g(s) - g(t)) ds which i choose to write as I = -2*Integral(0,t-d) dK(x,t-s)/dx * (g(s) - g(t)) ds -2*Integral(t-d,t) dK(x,t-s)/dx * (g(s) - g(t)) ds :=I1 + I2 look at I2 first. from the assumption |g(t) - g(s)| < e/2 for all |t-s|<=d i think we can write |I2| <= -e*integral(t-d,d)dK/dx ds < e/2 using the above property of [2]. So thats good (or is it?). im quite sure that i am missing many important points that would make all this rigorous - but i am more interested in the end result at the moment. Next we have to show that |I1| >How many integers are there? > How would you possibly answer this question? > A. Theres an infinity of them. I would never say that. > B. Theres an infinite number of them. Dont say that; it makes people > think there are infinite (natural) numbers. > C. There are infinitely many of them, or Theres an infinite set of them. > Thats OK. I dont see a bit of difference between saying that and > saying theres not a finite number of them (But there is at least one!) > D. There are aleph-1 of them or the set of them is countably infinite. > Look up the definition of countably infinite and tell me this isnt > nearly a tautology! Id say C, or Theres aleph-0 of them Probably C. > So whats your point? I thought I made it clear that mathematics encounters > infinite sets and limits and all sorts of situations where we might > bandy about the term infinity, but its just a shorthand for something > more precise, or else just a synonym for not finite. Or are you saying > infinity is the set of integers? I think thats awfully limiting... Probably none. At this point we seem to be splitting hairs of one of the angels dancing on the head of a pin. >Mathematicians (and teachers) would probably do everyone a great >service if they never made reference to infinity; leave the >mysticism to someone else (Buzz Lightyear, perhaps?) >>Modern set theory deals a great deal with infinity as a core concept. > I think you mean it deals primarily with infinite sets and compares and > contrasts them. OK. But I think the emphasis is on other structures (such > as the existence of functions of various types between sets, or possible > well-orderings on sets). The fact that most of the sets are infinite is a > given, and not itself the subject of much attention. (Disclaimer: I am not > a set theorist!) Some people do wrestle with the precise question of how > we define the predicate S is infinite (usual choice: existence of a > bijection S->T where T is a proper subset of S) but I dont think thats > considered a core issue of modern set theory! > I believe youre reading much too much into the OPs question. See my reply to the OP. I just found your response odd. Perhaps its just me. > This is > a secondary-school student who links mathematics with measurement and > precision. Perhaps someone has whispered about a thing called infinity > and he wants to know what it is and why we care. Not infinite sets; > not ordinals and cardinals, just a thing called infinity, which is > seen as some kind of number (I guess). I dont believe thats a useful > concept in itself. > Ive given a talk like My infinity can beat your infinity! to high > school students. First thing I do is to ask them to stop thinking about > any such things, and instead to think about infinite _sets_ instead. > Much more mathematically sound and it still seems to intrigue them. >When you have a set thats not finite, and people ask you how >many things youve got, you can say infinitely many, but theyre >still just ordinary things (points, numbers, whatever); theres >just a lot of them -- not a finite number of them at all, so, well, >an in-finite number of them. >>You are focusing on the individual elements, but when people talk about >>how many elements there are in a set, the desire is to get a meaningful >>value. In fact, simply saying that it is infinite is not always enough >>detail. Thus the cardinals to measure levels of infinity. > You mean, Thus the cardinals to represent different answers to Ôhow many. > There are finite cardinals, too, and I think youre reinforcing my point: > infinity, per se, is pretty useless: if you want to answer how many, > you point to a cardinal. If that cardinal is finite, great. If not, you > can say not finite. If you want to give more information, you specify > the appropriate cardinal, thus making it clear that the answer infinity > is at best a shorthand for not finite. Its not really a useful concept > in and of itself. >>The Riemann sphere has a point on it that *is* infinity. It is a value, >>not a concept, in that model of the complex numbers. > Um, right. I said it was a point _called_ infinity, but if now you want > to tell me that this is, in fact, the definition of infinity, then I > dont object, exactly; but if infinity is _really_ a part of the > Riemann sphere, then I wonder what the heck you meant by trotting out > infinite sets before? Again I think youre sort of making my point for me: > as much as we use the term infinity in mathematics, there really is > no single thing that encompasses > a synonym for not finite > the whole panoply of infinite sets > a point on the Riemann sphere > etc. > not to mention the pop-science connections to Big Bangs or whatever. > Its a subtle thing, but careful attention to subtleties can help > newcomers. Use infinite, the definable adjective, whenever possible > instead of an undefined noun infinity. It is, IMHO, a concept best used carefully and not introduced too soon. Unfortunately, it gets tossed around a lot. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: urgent analysis question > Im having difficulty with a problem and was hoping someone might be > able to help. The problem is the following: > Let X be a Lebesgue null subset of R (the reals) and let f: R -> R be > a continuously differentiable function. Prove f(X) is a null set. If (a,b) is an interval, how large can f((a,b)) be? (f is bounded in [-N,N] so work with the intersection of the image in [-N,N] to start with; use the MVT to get f((a,b)) bounded by some multiple of |b-a| ...) === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > spelling has nothing to do with math. If you didnt learn that in high > school, you are never going to learn it, in the rest of your life! > Embrassing!!! completely! high school students are a lot smarter than > you! its a fact. > Did u pass math/physics by a C+ ? > -suresh Spelling has a lot to do with math. For example, I spell the word postulate as p-o-s-T-u-l-a-t-e, not p-o-s-u-l-a-t-e, as you did in a previous post. Also, perhaps it is one of the failings of math student or perhaps math teachers should stress it more, but math is so much more than 2 + 2 = 4 and other symbols. Try to prove something using symbols only. Hell, even your Ôproof about the standard deviation of the radar gun used words galore. Before you go making statements such as the standard deviation of the radar gun is +/- 5mph, you should actually learn what the standard deviation is. I mean, you said yourself that Any high school student can understand what I am saying. If I recall correctly from my last three years of teaching statistics, the standard deviation cannot be negative. Maybe you should actually pay attention in class. Oh, and by the way, the speed of light is not constant [http://www.telegraph.co.uk/news/main.jhtml? xml=/news/2002/02/17/waa117.xml&sSheet=/portal/2002/02/17/por _right.html ] (http://tinyurl.com/2nrdb) or [http://www.news.harvard.edu/gazette/2001/01.24/01- stoplight.html] (http://tinyurl.com/3cnzk) - Tim Timothy M. Brauch Graduate Student Department of Mathematics Wake Forest University === Subject: Re: Oh, how I wonder... > To me, integration makes perfectly sense but, > Where is the irrasionality of irrasional numbers? > They seem to be fully rational. > Where is the inductive logic in mathematical induction? > It seems to be deduction. > Whats primary with prime numbers? > 1 seems to be more primary. Well, if you define primary as 1 a : first in order then 1 seems more primary. But primary also can be defined as 2 b : not derivable from other colors, odors or tastes So if one adds numbers to that list, primes are not derivable from other numbers. > Whats imaginary with imaginary numbers? > -1 seems to be the only imaginary number. Its a joke. Imaginary means not real, where this use of real refers to existence. The mathematical use of real doesnt refer to existence just as the mathematical use of primary doesnt refer to order. -1 happens to be a real number mathematically. But the square root of -1 is not a real number in the mathematical sense. So its called imaginary even though it does exist. Try explaining that to a bunch of drunks at a party, its a barrel of laughs. They could have called the numbers that are not prime the Secondary Numbers, but instead the name composite was chosen. > What is platonic with platonic figures? > They seem to be of this world. > Whats defferentiated during differensiation? > No difference is attempted found. === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > spelling has nothing to do with math. If you didnt learn that in high > school, you are never going to learn it, in the rest of your life! > Embrassing!!! completely! high school students are a lot smarter than you! > its a fact. Ah, youre trolling! Why didnt you just say so in the first place? Doug === Subject: Re: JSH: Math is hard Apparently to hard for JSH. === Subject: Re: JSH: how you can prove to the mathematicians you are right >Based on previous posts, I think what you want to prove is: > The expression > > (f_1(x) + 7)(f_2(x) + 1) = 7P(x) > > where f_1(x), and f_2(x) are algebraic integer functions, meaning that > for an algebraic integer x they give an algebraic integer, where > > f_1(0) = f_2(0) = 0, > > and P(x) is a polynomial that is also an algebraic integer function, > and P(0)=1 > > can only be true in the ring of algebraic integers if f_1(x)/7 > is an algebraic integer for all x. >Unfortunately, a proof of this will be hard, because it is false. >(it is true if we add the condition that f_1 and f_2 are polynomials [1]) > - William Hughes >[1] Actually, though I have made this claim before, I recently > tried to prove it and failed. The claim is true for the ordinary > integers, but there the proof involves prime factorization of > the polynomial coefficients (this is not going to work in the > algebraic integers). I thought I had a way of substituting > GCDs, but this fell apart when I tried to fill in the details. > I guess the statement it is true if we add the condition that > f_1 and f_2 are polynomials should be classed as a conjecture > (HnFnLC [2]). Can anyone provide a proof? >[2] Hughes neither First nor Last Conjecture > Yes. It follows from Dedekinds Prague Theorem, though I am sure Bill > Dubuque will shortly post explaining that I am doing things the hard > way (as usual): > DEDEKINDS PRAGUE THEOREM (Lemma 2 in Chapter 2, Section 5, of David > Hilberts Zahlbericht) If the coefficients a1,a2,....,b1,b2,... of two > polynomials in one variable > F(x) = a1*x^r + a2*x^{r-1} + ... > G(x) = b1*x^s + b2*x^{s-2} + ... > are algebraic integers, and the coefficients of the product > F(x)*G(x) = c1*x^{r+s} + c2*x^{r+s-1} + ... > are all divisible by the rational integer w, then each of the numbers > a1*b1, a1*b2, a1*b3,...,a2*b1, a2*b2,... is also divisible by w. > (Also proven Kronocker, Mertens, and Hurwitz). > So assume we have > (f_1(x) + 7)(f_2(x) + 1) = 7P(x) > and f_1, f_2, and P are polynomials with algebraic integer > coefficients. Since 7 divides the coefficients of the product, and > f_2(x)=0, then for each coefficient of f_1(x), 1 times that > coefficient must be a multiple of 7. my approach to proving things in the integer case was essentially to prove Dedekinds Prauge Theorem for the case F(x) and G(x) in Z[x]. (I note that if p|ab then p|a or p|b then do a lot of messy bookkeeping). On the other hand, I do have a tendency to do things the hard way so I would not be at all suprised to find there is an easier way. I was interested to note that the statment of the theorem requires w to be a rational integer. (are there any known counterexamples if w is an algebraic integer but not at rational integer?) James examples have always used a rational integer (indeed a prime), however some of his formulations allow for the use of an arbitrary algebraic integer. -William Hughes === Subject: Re: Fourier analysis of a radarsignal Clarification and erratum: A sampled input produces a periodic output. To clarify, the ratios between time coordinates of the samples need to be rational. (This is usually the case in engineering.) A non-periodic input produces a continuous-spectrum output. This statement is ßawed because the input could also be composed of discrete spectral components that are not harmonically related (having an irrational frequency ratio) and therefore not periodic. A signal composed of a number of discrete frequency components that are not harmonically related will be quasi-periodic, meaning that there will be regular recurrences of a waveform that is similar to any given fragment. The higher the degree of accuracy of match, the longer the period between such recurrences. === Subject: Re: Rational sines and cosines >>Does there exist a rational number r such that 2r is not an integer >>and both the sine and cosine of pi r are rational? >> >The only rational r between 0 and 1/2 such that sin pi r is rational >are r = 0, 1/6, and 1/2. >The only rational r between 0 and 1/2 such that cos pi r is rational >are r = 0, 1/3, and 1/2. >By the various symmetries inherent in the trig functions you can now >work out all the rational r, etc. So the answer to my question is no. How do we know these facts? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: The role of infinity in math > Some people do wrestle with the precise question of how > we define the predicate S is infinite (usual choice: existence of a > bijection S->T where T is a proper subset of S) but I dont think thats > considered a core issue of modern set theory! > I came up with a simple definition of an infinite set in another thread. > Let S be a set of natural numbers and let x and y be members of S. > Consider these two statements: > 1) ExAy(x>=y) > 2) AxEy(x If statement (1) is true then S is finite. > If statement (2) is true then S is infinite. But (1) only works for well-ordered sets, and not all sets are well-ordered, e.g., the set of negative integers satisfies (1) so must be finite by your reckoning. How do you tell if an ordered-but-not-well-ordered set is finite? In fact sets need not be ordered at all. How do you tell whether an unordered set is finite? Your definitions are simple enough, but wrong. === Subject: Re: Unusual Numbering Systems > Is there a series that sums to the golden ratio? There are series that sum to any given real number, namely the first differences of any sequence converging to that number form such a series, and there are infinitely many such sequences. === Subject: Re: Dense Linear Ordering > Hmm, What I meant to say is that any countable dense linear orderings > without endpoints are isomorphic Which specifically excludes the reals. === Subject: universal set with 3 valued logic by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1D3GA411021; this is an attempt at axiomatizing a universal set using three valued logic to avoid russells paradox. i have changed the subsets axiom so that if binary logic is applied, then it remains the same axiom. i have changed the foundation axiom so that U can be an element of U yet if U did not exist, the change to the foundation axiom would not exist (it is formulated as IF there is a universal set then it MAY contain itself). some results: cantors diagonal argument fails to prove that U does not map onto P(U). U=P(U), where = means equals, not in bijection with. if U<=x for any x, where this means domination, then U~x where ~ means is bijection with. thus U is the largest set. it immediately follows that the set of all functions from U to U is in bijection with U and so OMEGA to the OMEGA is OMEGA. this also shows that the OMEGA product of OMEGA is OMEGA. i then dabble into an investigation of U as a boolean ring and discuss some more elementarily derivable properties. i discuss the concept of a universal limit and convergence without reference to a topology or metric. this is still a rough draft and all feedback is appreciated so that i can publish it one day. thank you brian === Subject: Re: plotting n-dimensional data points on a 2d screen by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1D3GAD11010; For 4D, I have always plotted the 3D function and then looked at a moving picture for the 4D object and then just imagine that all the pictures I am seeing happend all at once. === Subject: Re: The role of infinity in math > Some people do wrestle with the precise question of how > we define the predicate S is infinite (usual choice: existence of a > bijection S->T where T is a proper subset of S) but I dont think thats > considered a core issue of modern set theory! > I came up with a simple definition of an infinite set in another thread. > Let S be a set of natural numbers and let x and y be members of S. > Consider these two statements: > 1) ExAy(x>=y) > 2) AxEy(x If statement (1) is true then S is finite. > If statement (2) is true then S is infinite. > But (1) only works for well-ordered sets, and not all sets are > well-ordered, e.g., the set of negative integers satisfies (1) so must > be finite by your reckoning. I said S is a set of natural numbers. That wouldnt include negative integers. > How do you tell if an ordered-but-not-well-ordered set is finite? That would be more complicated. Any set of natural numbers can be well ordered. > In fact sets need not be ordered at all. How do you tell whether an > unordered set is finite? My method wont tell you. Cant all finite sets be ordered? > Your definitions are simple enough, but wrong. My definition works for sets of natural numbers. That is all I claimed. Russell - 2 many 2 count === Subject: Re: please help > I need to prove that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Here is what I have (and dont have). I know that (1+X)^n and (1+nX) > intersect at (0,1). Since [ (1+X)^n ] Ô > n for all X >0 and since (1 + nX) > Ô = n for all X we know that (1+X)^n > (1+nX) ( but we already knew this > from the binomial theorem). > The real question is whether or not (1 + X) ^n and (1 + nX ) intersect below > X =0 ? 3 ways to see it: 1. (1+X)^n is strictly concave up on (-1,oo) and so stays above all of its tangent lines, including y = 1 + nX, on that interval. 2. Let f(x) = (1+X)^n - (1+nX). Then f > 0 to the right of 0 and f < 0 to the left of 0. By the mean value theorem, f(x) > 0 for nonzero x. 3. As Gerry Myerson suggested, prove the blasted thing by induction. === Subject: Re: Got a speeding ticket and need to fight back > The Lord of Chaos (Suresh Devanathan) > Hello everyone, > the cops got me speeding 10-15 mph above the speed-limit. My arguement > is > as follows. > If you really want to beat the ticket do what a lawyer friend of mine did. > As part of your discovery demand that the police provide all of the > maintenance history of the radar gun. Radar guns have to be > regularly calibrated. The judge will (sometimes!) dismiss the case if > the police dont provide you with the maintenance records, > and the police are usually too lazy to do so. > Russell > - 2 many 2 count In Colorado, radar guns must be calibrated with a tuning fork that has been validated by a particular state laboratory within 36 months of the date of the ticket, and many police organizations on Colorado have not bothered to get the validation. === Subject: Re: What is the Origin of Space and Time? >> And here I was, pondering the collapse of a 10- or 26-dimensional >> universe in a false vacuum state into 4-dimensional, more stable universe >> with the remaining dimensions curled upon themselves... Silly me. ;) > Yo, dude, those extra spaces do curl upon themselves ... just like my > kitty ... And, yknow, those extra dimensions have a natural harmonic that makes them purr just like a kitten as well...could that be the answer to Schroedingers cat mystery? -- Darryl L. Pierce Visit the Infobahn Offramp - What do you care what other people think, Mr. Feynman? === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > The Lord of Chaos (Suresh Devanathan) message > > They dont need to. > > Driving is a privilege and not a right. > > All the cops need to show is that it was more likely, > > given the evidence, that you were speeding > > Well, it is a privilege is some circumstances. For example, you need a permit if you want to work in this country if you are not a citizen. Owning a business is a privilege. I could make a lot of money selling opium or child pornography, but I cant get a license to do so. Anytime you need a license to do something, that something is a priviliege. Otherwise, you wouldnt need a license to do it. Russell - 2 many 2 count === Subject: Re: Graph Theory: Number of maximal Matchings >thank you for your hints. Unfortunately my bipartite graphs are not >regular. Well, the same upper bound will work in the case where the maximum degree is k (since removing edges can only decrease the number of possible matchings). 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: Unusual Numbering Systems > Is there a series that sums to the golden ratio? > There are series that sum to any given real number, namely the first > differences of any sequence converging to that number form such a > series, and there are infinitely many such sequences. Do you happen to know one? I am too lazy to compute one by hand. I did a search and found numerous methods using continued fractions, but no simple series. Russell - 2 many 2 count === Subject: A question about Kripke semantics and physics (Was: Re: Study groups in science) > if you are interested in scientific online workshops, please visit > site > http://de.geocities.com/scienceworkshops/ > Its goal is to organize study groups on scientific topics like quantum > field theory, > probabilistic inference or neural nets, to name a few topics I am > personally interested in > (of course, arbitray topics may be suggested). > What positive precautions are you taking to prevent the idiots morons and > kooks from taking it over, as has happened in sci.physics? Hi Franz! Lets just defuse the reference to morons and kooks by deferring to Shannons discussion concerning the statistical character of language. I want to ask you a question about the theories physicists are promulgating these days. So, once again, let me start with a quick Google search to establish context. If I use the search string Ô26 dimensions string physics I get about 959 hits. For my part, I could care less about the details. It is the mathematics through which your colleagues express their explanations that is important to me. The 26 dimensions are particularly interesting here because their symmetries and invariants involved offer me an opportunity to ask you how to think about truth and falsity in physical theory. Now, Galathaea has been wanting to talk about Heyting algebras and quantum logic. Section 6.4 of the paper http://www.illc.uva.nl/Publications/ResearchReports/MoL-2001- 09.text.pdf is entitled Finite projective formulas in two variables Strangely enough, there are precisely 26 Heyting algebras associated with 2-universal models discussed here. Moreover, there is not a single mention of quantum logic. Now, isnt this just an amazing coincidence? Logicians tell mathematicians about ÔT and ÔF, the physicists are talking about 26 dimensions, and mitch knows just where to find a paper specifying the 26 2-generated Heyting algebras. Actually, I do not think it is coincidental. Unfortunately, whereas I would love to offer an explanation, you would simply engage in more vulgarity. But, here is a little something to consider. If I am interpreting Kip Thornes account correctly, Einstein applied a Kantian notion of space and time to his formulation of special relativity. By formulating an absolute metric, Minkowski dragged your precious theories of everything into the foundational arguments of mathematics. Einstein thought that was a terrible thing--until he recognized that he could enhance his personal reputation further with general relativity. How convenient! The key combinatorial concept that ties it all together is that of a nerve defined by Eduard Cech. Apparently, these associate simplicial complexes to open coverings. You do understand the significance of open coverings in calculus and intuitionistic semantics, dont you? But, Cech was not a physicist. Rather, he was just someone else whose ideas your colleagues could use without proper citation because, after all, it was mathematics. Here is a little something about the man in his professional capacity: Whenever he was doing something in mathematics, he always strove to achieve a thorough understanding of the subject. The result was that even outside his fields of research he had an extensive knowledge and deep insight into many other areas of mathematics. This feature of his personality also had some other consequences. While he was not conceited and talked easily to people with little formal education, he expected in his fellow professors the same qualities that he himself possessed. This did not contribute to smooth relations with some people as he was not diplomatic but, on the contrary, quite forthright in expressing his opinions. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/ Cech.html Im going to guess that he had few problems with statistical entropy and sentence construction. :-) Unlike you, I am fairly certain that I am looking at a situation that derives from the fundamental theorem of algebra. But, physicists are in the business of theories of everything and philosophical logicians are in the business of telling mathematicians about truth. My only concern, as always, is the foundation of mathematics. If you could offer some insight--being a physicist whose education has served him well for a long career--it would be greatly appreciated. I had been under the impression that physical theories corresponded with physical fact. But, then, facticity does not seem to be very important to distinguished professors who spend more time at faculty tea than the library. By the way, Franz. The vocabulary word for the day is impudence. If anyone else has any explanations to offer or papers that might clarify this apparent correspondence, I would appreciate that as well. :-) mitch === Subject: Re: universal set with 3 valued logic > this is an attempt at axiomatizing a universal set using three valued > logic to avoid russells paradox. I noticed that you mention that MP is not a tautology. I am curious what you think about formulas that are not false, but still arent true. Russell - 2 many 2 count ps: I was able to read your page earlier, but now I am having trouble opening the link. === Subject: Re: JSH: Open letter to Jim Ferry > The problem is that Id concluded that very intelligent people see a > much higher value in telling the truth than others. Another problem is that you seem to place such a low value on telling the truth. What does your own conclusion imply about your own intelligence? === Subject: Re: Got a speeding ticket and need to fight back hey buddy, heres some advice: avoid my theard. I am looking for intelligent people to talk to and debate with. Not some loser who has a lot of time, to waste mine. Heres my problem with you. A) you are dumb B) you do not understand mathematics So, go away. -suresh > The Lord of Chaos (Suresh Devanathan) > spelling has nothing to do with math. If you didnt learn that in high > school, you are never going to learn it, in the rest of your life! > Embrassing!!! completely! high school students are a lot smarter than > you! > its a fact. > Ah, youre trolling! Why didnt you just say so in the first place? > Doug === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > hey buddy, heres some advice: avoid my theard. Hell, no. I wont avoid your thread, either. Anyone whose best argument in a mathematical discussion is you dont understand mathematics and you are dumb deserves what he gets. I hope they get you for contempt of court. Doug === Subject: Re: Math Too Advanced For Mainstream Economists RV> I have written up a demonstration that wages and employment RV> need not be determined by the intersection of well-behaved RV> supply and demand curves for labor: RV> MW> So, what serious model does this work improve upon, MW> and what does it tell us about the observed world that MW> we did not already understand? MW> Responses come, yet no answer to the original question, so again I MW> ask: MW> Does this model show something of interest to any MW> serious researcher? What do we observe that it MW> explains better than competing approaches? MW> The latest response contains the following: MW> I remain of the opinion that discarding theories about the MW> world that are self-contradictory improves ones understanding MW> of the world. Poor Mark Witte continues to insult the intelligence of his readers. He presents that quote as if that is a new answer on my part. He refuses to acknowledge that he has been ignoring an answer to his question for some time. Personally, Im of the opinion that ones ability to understand the world is improved by throwing out internally inconsistent theories. Personally, Im of the opinion that ones ability to understand the world is improved by throwing out internally inconsistent theories. I remain of the opinion that discarding theories about the world that are self-contradictory improves ones understanding of the world. I remain of the opinion that discarding theories about the world that are self-contradictory improves ones understanding of the world. MW> ...the problem is that no post on this thread indicates what MW> self-contradictory theories we should be discarding. Is it MW> supply and demand models for labor? I have explained in this MW> thread how the linked post is not related to simple supply and MW> demand curve analysis of labor markets and only a misunderstanding MW> of the supply and demand approach would make anyone think so, such MW> as with the misuse in the context of tools like IRR. That this MW> confusion exists on the part of the original author is made clear MW> by this quote from the linked webpage: So much for the theory that wages and employment are determined by the intersection of well-behaved supply and demand curves in the labor market. Poor Mark Witte continues to insult your intelligence, dear reader. Note that poor Mark Witte makes no attempt whatsoever to demonstrate any misuse of tools like IRR. He thinks his mere statement will distract you from his previous error in asserting that the Internal Rate of Return will not generally be different at different levels of costs, when revenues are unchanged. And poor Mark Witte seems to think readers, if any, are not aware that, in mainstream economics, labor demand curves are supposed to be derived from optimizing behavior of the firm. I provide a correct analysis of such optimizing behavior. I do not end up with a labor demand curve. In fact, I do not even end up with a firm that will necessarily adopt a more labor-intensive techninue or hire more workers at a lower wage. Somehow Mark Witte thinks you will be confused enough to believe his echo of my point, that such analysis does not yield well-behaved labor demand curves, into thinking that I have not shown the foundation for well-behaved labor demand curves is lacking. If poor Mark Witte thought otherwise, he could always try to present an argument. For example, he might construct an equilibrium of the firm at each possible level of the wage in my example. And then he might show how to draw a labor demand function. Or he could show how to find the equilibrium of the firm under some other description of technology. I do not here address poor Mark Wittes pretence that empirical evidence relevant to my example has not been cited on this thread. > Question: Does the originator of this thread believe that a > well-behaved supply and demand model is never appropriate for > explaining how labor markets function? No such claim has been advanced here. Perhaps poor Mark Witte could outline under what special case conditions one can derive a labor demand function. By the way, I also havent suggested firms will always want to adopt less labor-intensive techniques at lower wages. But if one thought optimizing competitive firms always adopt more labor intensive techniques at lower wages, one might outline some special case assumptions that yield this conclusion. But the empirical evidence suggests that poor Mark Witte will always beg all arguments rather than put forth a statement that is not fallacious. It is better to be left with an empty mind than one filled with nonsense - with deductive inconsistencies and fanciful empirical hypotheses. -- Paul Samuelson since both groups of versions of marginalist equilibrium theory - the long-period versions and the neo-Walrasian versions - encounter what appear to be radical and insurmountable difficulties, one must conclude that at present there is no defensible neoclassical theory (in the sense of explanation) of prices and distribution. The onus is on the neoclassicals to show that this is not so. Unless and until they succeed, it seems reasonable to turn to different, non-neoclassical approaches to value and distribution (and employment and growth). -- Fabio Petri, Professor Hahn on the Ôneo-Ricardian Criticism of Neoclassical Economics, in _Value, Distribution, and Capital: Essays in Honour of Pierangelo Garegnani. Routledge, 1999. -- Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/ Bukharin.html To solve Linear Programs: .../LPSolver.html r c A game: .../Keynes.html v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Problem of the Week Trig Question I saw an interesting problem, Im sure some of you are familiar with it, its the Purdue (Im a Boilermaker fan, btw :) Problem of the Week #5 for the Spring Series [ http://www.math.purdue.edu/academics/pow/ ]. And the only reason I thought I could tackle a problem like this is because weve done things similar in our Trig class before, that is, longitude/latitude type problems and I thought with some reading and investigating, I could solve this. I asked my math teacher, he said I probably would not need spherical trig but mentioned the azimuth (distance to the northern point from the horizon). So, my question is, given that problem, what aspects should I read up on in Trig to help me solve it? Im not asking for a solution, just some opinions and what in John K. === Subject: Re: Got a speeding ticket and need to fight back You are spewing this thread, into nonsense. Look, heres my solid agruement, if you missed my post. If you cannot talk the law of physics, or law of averages or statistics, i am not here to argue with you. Either you have no focus,or just plain dumb. I will post arguement again,if you missed it. If you cannot do math or statistics or physics, dont post, at all. Your posts are pointless, completely to me and just to everybody else. If you are one of those mensa folks, you are an embarrassment to their community. I was here to ask help from a statistical and scientific viewpoint. If you cannot answer that, you stand testimonial to , question of Ôprobability of finding an idiot in mensa?. I already have talked to 3 idiots, out of 5 or 6, ones, i have met there. So, there is about 1/2 probabilty that a member of mensa is a complete idiot. -suresh > The Lord of Chaos (Suresh Devanathan) message > The Lord of Chaos (Suresh Devanathan) > message > They dont need to. > > Driving is a privilege and not a right. > > All the cops need to show is that it was more likely, > > given the evidence, that you were speeding > Well, it is a privilege is some circumstances. > For example, you need a permit if you want > to work in this country if you are not a citizen. > Owning a business is a privilege. > I could make a lot of money selling opium or > child pornography, but I cant get a license to > do so. > Anytime you need a license to do something, > that something is a priviliege. Otherwise, > you wouldnt need a license to do it. > Russell > - 2 many 2 count === Subject: linear operator Hey everyone, The definition of a linear operator is th following: T(x+y) = T(x) + T(y) and T(cx) = cT(x). Now in most cases the first rule is logically independent from the second. Yet if T:V->W and both v and w are vector spaces over the rationals, the second rule follows from the first. Can anyone please show me why? I can t seem to find the relation. -Greg Ryslik === Subject: Re: Rational sines and cosines > >>Does there exist a rational number r such that 2r is not an integer >>and both the sine and cosine of pi r are rational? >> >The only rational r between 0 and 1/2 such that sin pi r is rational >are r = 0, 1/6, and 1/2. >The only rational r between 0 and 1/2 such that cos pi r is rational >are r = 0, 1/3, and 1/2. >By the various symmetries inherent in the trig functions you can now >work out all the rational r, etc. > So the answer to my question is no. How do we know these facts? 2 cos pi r = exp(ir) + exp(-ir) is an algebraic integer (because exp(ir) and exp(-ir) are algebraic integers) and a rational number (because, by hypothesis, cos pi r is rational) so its an ordinary integer (because the only rationals that are algebraic integers are the ordinary integers) so its -2, -1, 0, 1, or 2 (since absolute value of cos pi r is at most 1) so we know what r can be. Similarly for sin pi r. There may be a way to do it without invoking algebraic number theory, but then again the algebraic number theory Ive invoked is first-week-of-class stuff. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > Approximately > f lamda = c > f lamda = 186 000 mps > f -> frequency in hertz > h = 60min > = 60 * 60 sec > f lamda = 186 000 > = 186 000 mps > f -> 1000,000,000 s^-1 (gigahertz) > lamda = .000186 miles > Thats the radars wavelength approximately. Take the radars time > window, it internally uses to compute the speed: 1/10 second ( the > human eye can see about 30 frames / sec ). Assuming Reaction time of a > driver = 1/10 sec approx > 0.000186/ .1s = .00186 mps = 6.6 mph [http://www.bushnell.com/productinfo/speedgun/speedster.html] The Speedster uses digital technology and DSP (Digital Signal Processing) to provide accurate real-time measurements to +/- 1.0 MPH. Commercial radar guns are accurate to about +/- 1mph. I would assume police officers use something just as good, if not better. But that is just an assumption. - Tim Timothy M. Brauch Graduate Student Department of Mathematics Wake Forest University === Subject: Re: Unusual Numbering Systems Russell Easterly > Is there a series that sums to the golden ratio? > There are series that sum to any given real number, namely the first > differences of any sequence converging to that number form such a > series, and there are infinitely many such sequences. > Do you happen to know one? > I am too lazy to compute one by hand. 2 - 1/2 + 1/6 - 1/15 + 1/40 - 1/104 + etc = phi. The denominators are the products of consecutive Fibonacci numbers. I got this just by writing lim (f_{n+1} / f_n} = phi and converting the sequence to a series, using f_n squared = f_{n+1} f _{n-1} +- 1 LH === Subject: Re: Analysis > I think you mean X>-1 > John > -- > John T Lowry > 5217 Old Spicewood Springs Rd, #312 > Austin, Texas 78731 > (512) 231-9391 > jlowry100@earthlink.net > I saw this one before but I forgot the proof. Any hints will help. Prove > that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Steven With x > -1; Induction works: The case for n=1 clearly holds, and (1+x)^{n+1} = (1+x)^n(1+x) >= (1+nx)(1+x) = 1+nx+x+nx^2 = 1+(n+1)x+nx^2 >= 1+(n+1)x. === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > You are spewing this thread, into nonsense. Look, heres my solid > agruement, if you missed my post. If you cannot talk the law of > physics, or law of averages or statistics, i am not here to argue > with you. Either you have no focus,or just plain dumb. I will post > arguement again,if you missed it. If you cannot do math or statistics > or physics, dont post, at all. Your posts are pointless, completely to > me and just to everybody else. If you are one of those mensa folks, > you are an embarrassment to their community. I was here to ask help > from a statistical and scientific viewpoint. If you cannot answer > that, you stand testimonial to , question of Ôprobability of finding > an idiot in mensa?. I already have talked to 3 idiots, out of 5 or 6, > ones, i have met there. So, there is about 1/2 probabilty that a > member of mensa is a complete idiot. > -suresh By that reasoning, I ßipped a coin once and got heads all three times. I guess the probability of getting heads must be 100%. - Tim Timothy M. Brauch Graduate Student Department of Mathematics Wake Forest University === Subject: Re: help with proof of distribution like function: analysis please note amendments: > i will first state the problem, and then show you what i have thus far. > I am looking at the kernel of the fundamental solution of the heat diffusion > equation i.e. > K(x,t) = Exp(-x/(4*t))/sqrt(4*pi*t) t>0 > The problem is this: does > Limit (x+-->0) -2*Integral(0,t) dK(x,t-s)/dx * (g(s) - g(t)) ds = 0 ? > [1] > I think it does - at least i remember that this result is true from when i > first studied the heat equation - but i dont remember the details, which is > what i am trying to reconstruct here. > note: > -integral(0,t) dK(x,t-s)/dx ds = 1/sqrt(pi)*integral(x/(2sqrt(t)),oo) > exp(-p^2) dp =1 for x>0 as x-->0. > for x<0 -integral(0,t) dK(x,t-s)/dx ds = -1 [2] as x-->0. > thus noting this property what we are really trying to show is > Limit (x+-->0) -2*Integral(0,t) dK(x,t-s)/dx * g(s) ds = g(t) [3] > we have written this in the form of [1] because i think thats more > appropriate for a proof. Now i am particularly bad at proofs (my brain > just isnt wired that way i suppose), however this is what i have so far. > assume g(t) is continuous at t. Since g(t) is continuous at t, for each > e>0 there exists a d>0 such that |g(t) - g(s)| < e/2 for all |t-s|<=d > ok. consider the integral > I = -2*Integral(0,t) dK(x,t-s)/dx * (g(s) - g(t)) ds > which i choose to write as > I = -2*Integral(0,t-d) dK(x,t-s)/dx * (g(s) - g(t)) ds -2*Integral(t-d,t) > dK(x,t-s)/dx * (g(s) - g(t)) ds :=I1 + I2 > look at I2 first. from the assumption |g(t) - g(s)| < e/2 for all |t-s|<=d > i think we can write > |I2| <= -e*integral(t-d,d)dK/dx ds < e/2 as x-->0. > using the above property of [2]. So thats good (or is it?). im quite sure > that i am missing many important points that would make all this rigorous - > but i am more interested in the end result at the moment. Next we have to > show that |I1| |I1| = |-2*Integral(0,t-d) dK(x,t-s)/dx * (g(s) - g(t)) ds | = > x/(2*sqrt(pi))|Integral(0,t-d) exp(-x^2/(4*(t-s))/(t-s)^(3/2) (g(s) - g(t)) > ds | > this i am not sure about obtaining. i *think* i need to use an integral > inequality say, > |integral f*g| <= Sqrt((integral f^2 )*(integral g^2)) > but i cant see how to get e/2. the idea being that if we can show |I2| then |I| showing classic signs of there being some sort of delta function involved > perhaps there is another way though all this mess. any help appreciated. > cheers > moth === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > Either you have no focus,or just plain dumb. Yeah, hes dumb, and Im dumb, and everyone else is just plain dumb. Look, I can see how embarrassing this must be for you - you came in here to have a nice troll, and youre getting whacked around more than Paul Reubens jewels at an all-night theater. Really, though, you brought it upon yourself. Doug === Subject: Re: Got a speeding ticket and need to fight back > The Lord of Chaos (Suresh Devanathan) The point of the math was to estimate the order of the error, ie 6.6 mph, because for one, i am not aware of the internals of the radar device. I used the analysis to give a good reason, as to where the +/-5 mph came from. And as you can see, from your own findings of +/- 1 mph , the math is self-consistant. Unless the company that makes the radar, makes their internal patents public, i have a solid case. Btw, i am an electrical engineer. I can definitely look into the radar and i can see for myself, if it has no obvious ßaws. I am sure, it has tons of ßaws in the design. > [http://www.bushnell.com/productinfo/speedgun/speedster.html] > The Speedster uses digital technology and DSP (Digital Signal > Processing) to provide accurate real-time measurements to +/- 1.0 MPH. > Commercial radar guns are accurate to about +/- 1mph. I would assume > police officers use something just as good, if not better. But that is > just an assumption. > - Tim > Timothy M. Brauch > Graduate Student > Department of Mathematics > Wake Forest University === Subject: Re: Got a speeding ticket and need to fight back I told you to Ôgo away. Do u speak english? Can you read or write in english? -suresh > The Lord of Chaos (Suresh Devanathan) > Either you have no focus,or just plain dumb. > Yeah, hes dumb, and Im dumb, and everyone else is just plain dumb. > Look, I can see how embarrassing this must be for you - you came in here > to have a nice troll, and youre getting whacked around more than Paul > Reubens jewels at an all-night theater. Really, though, you brought it > upon yourself. > Doug === Subject: Re: Got a speeding ticket and need to fight back > The Lord of Chaos (Suresh Devanathan) >> Either you have no focus,or just plain dumb. > Yeah, hes dumb, and Im dumb, and everyone else > is just plain dumb. > Look, I can see how embarrassing this must be for > you - you came in here to have a nice troll, and > youre getting whacked around more than Paul Reubens > jewels at an all-night theater. Really, though, you > brought it upon yourself. sounds like damage control to me. and what do you know about the experince of paul reubens jewels at an all-night theater? never mind... forget i asked. :) rgrds, === Subject: Re: Got a speeding ticket and need to fight back If you dont understand Ôgo away, heres a link to dictionary.com http://www.dictionary.com Look up go and away. And put it together and you will understand what i m saying. In case, you do not understand, what together means, you can still look it up at http://www.dictionary.com -suresh The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? Can you read or write in > english? > -suresh > The Lord of Chaos (Suresh Devanathan) > Either you have no focus,or just plain dumb. > Yeah, hes dumb, and Im dumb, and everyone else is just plain dumb. > Look, I can see how embarrassing this must be for you - you came in here > to have a nice troll, and youre getting whacked around more than Paul > Reubens jewels at an all-night theater. Really, though, you brought it > upon yourself. > Doug === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? Can you read or write in > english? > -suresh Hmm, I think the real question is, can *you* write English? - Tim Timothy M. Brauch Graduate Student Department of Mathematics Wake Forest University === Subject: Re: linear operator sure, the first rule imply that cT(x)=T(cx) for c any integer. let a, b be integers then (a/b)*bT((1/b)x)= T((a/b)x) but (a/b)*bT(x/b)= (a/b)*T(x) here you go > Hey everyone, > The definition of a linear operator is th following: T(x+y) = T(x) + > T(y) and T(cx) = cT(x). Now in most cases the first rule is logically > independent from the second. Yet if T:V->W and both v and w are vector > spaces over the rationals, the second rule follows from the first. Can > anyone please show me why? I can t seem to find the relation. > -Greg Ryslik === Subject: Re: rank of a random {0,1} matrix > Given a nxk random matrix R, k>n, each entry of the matrix is either 1 > or 0 with equal probability, i.e., Pr{r_ij=1}=0.5, Pr{r_ij=0}=0.5, > what is the probability that the rank of the matrix is n ? I guess I found the answer. Cooper explained this in (Theorem 1 in [1]) that: If $V$ is a $n times k$ binary random matrix with entries independently and identically distributed as $Pr{v_{ij} = 1} =0.5 $ and $Pr{v_{ij}=0}=0.5$, then [ lim_{k to infty} Pr{ rank(V) = n } = prod_{j=k-n+1}^{infty} left( 1- 0.5^{j} right ) ] [1] C. Cooper. textit{On The Distribution of Rank of a Random Matrix Over a Finite Field}, Random Structures and Algorithms 17(3:4), 2000 === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? > Can you read or write in english? never mind the ticket thing, how did you come by the lord of chaos thing? rgrds, === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? Can you read or write in > english? Do u speak English? What - are you too much in a hurry to actually write out the word you? Doug === Subject: Re: Got a speeding ticket and need to fight back > and what do you know about the experince of > paul reubens jewels at an all-night theater? I can read a newspaper. How about yourself? Doug === Subject: Re: plotting n-dimensional data points on a 2d screen > For 4D, I have always plotted the 3D function and then > looked at a moving picture for the 4D object and then > just imagine that all the pictures I am seeing happend all > at once. Unfortunately, thats not an option here... the plot will actually be tracking a realtime process! I have seen pictures of n-dimensional plots like Im talking about before, but never with a description of the algorithm. === Subject: Re: need help!! > > > > i can not integrate from arcsin[2/(3+cosx)] > if u can ,please say me by hupo19@yahoo.com > > i know answer by i dont know how it solve > thank you > hupo > > What answer do you have? > J > > dear jim > my answer is =x*arcsin[2/(3+cosx)] > This answer is incorrect, and looks like it came from a computer. The > computer misunderstood cosx to be a constant or something, not the > cosine of x. A general indefinite solution for this integral is not > known; you can, however, use a computer to estimate the definite > integral between two points. hi 1.if you say that my answer is wrong then what is it? 2.if my qustion has not answer from anyway theni think that if we CAN plot it , and we CAN calculate area [for example from -1 to +1] we recived answer thank you for yuor help hupo === Subject: Re: Got a speeding ticket and need to fight back may be, you missed the title of my theard. Read it: Got a speeding ticket and neet to fight back Ofcourse, i am in a hurry to find an excellent arguement. -suresh > The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? Can you read or write in > english? > Do u speak English? What - are you too much in a hurry to actually > write out the word you? > Doug === Subject: Re: Got a speeding ticket and need to fight back > The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? > Can you read or write in english? > never mind the ticket thing, > how did you come by > the lord of chaos thing? Thats translation of my name. Suresh = lord of the rain, lord of chaos,... Kumar = Prince, handsome,.... Devanathan = King of the Gods > rgrds, === Subject: Re: Got a speeding ticket and need to fight back >> and what do you know about the experince of >> paul reubens jewels at an all-night theater? > I can read a newspaper. How about yourself? like everybody else who reads newspapers, i filter for interest. :) rgrds, === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) >> The Lord of Chaos (Suresh Devanathan) >> I told you to Ôgo away. Do u speak english? >> Can you read or write in english? >> never mind the ticket thing, >> how did you come by >> the lord of chaos thing? > Thats translation of my name. > Suresh = lord of the rain, lord of chaos,... > Kumar = Prince, handsome,.... > Devanathan = King of the Gods your given name, or an assumed name? rgrds, === Subject: Re: The role of infinity in math >>How many integers are there? > How would you possibly answer this question? > A. Theres an infinity of them. I would never say that. > B. Theres an infinite number of them. Dont say that; it makes people > think there are infinite (natural) numbers. > C. There are infinitely many of them, or Theres an infinite set of them. > Thats OK. I dont see a bit of difference between saying that and > saying theres not a finite number of them (But there is at least one!) > D. There are aleph-1 of them or the set of them is countably infinite. > Look up the definition of countably infinite and tell me this isnt > nearly a tautology! > Id say C, or Theres aleph-0 of them > Probably C. > So whats your point? I thought I made it clear that mathematics encounters > infinite sets and limits and all sorts of situations where we might > bandy about the term infinity, but its just a shorthand for something > more precise, or else just a synonym for not finite. Or are you saying > infinity is the set of integers? I think thats awfully limiting... > Probably none. At this point we seem to be splitting hairs of one of > the angels dancing on the head of a pin. >Mathematicians (and teachers) would probably do everyone a great >service if they never made reference to infinity; leave the >mysticism to someone else (Buzz Lightyear, perhaps?) >>Modern set theory deals a great deal with infinity as a core concept. > I think you mean it deals primarily with infinite sets and compares and > contrasts them. OK. But I think the emphasis is on other structures (such > as the existence of functions of various types between sets, or possible > well-orderings on sets). The fact that most of the sets are infinite is a > given, and not itself the subject of much attention. (Disclaimer: I am not > a set theorist!) Some people do wrestle with the precise question of how > we define the predicate S is infinite (usual choice: existence of a > bijection S->T where T is a proper subset of S) but I dont think thats > considered a core issue of modern set theory! > I believe youre reading much too much into the OPs question. > See my reply to the OP. I just found your response odd. Perhaps its > just me. For what its worth, I tend to agree with you. Im not a mathematician but Rusins comments dont seem to jibe with the way I have learned the concept of infinity. Especially comments that seem to reduce the concept of Ôinfinite set to merely Ônon-finite set. But perhaps it is appropriate for the OPs level of familiarity with math in general. l8r, Mike N. Christoff === Subject: problem Does there exist a real function which has a graph which is everywhere dense overe its plane in the sense that for every circle in the plane of its graph there exists points on the graph that belong to the interior of the circle. An explicit example will be highly appreciated. === Subject: Re: Got a speeding ticket and need to fight back >> and what do you know about the experince of >> paul reubens jewels at an all-night theater? > I can read a newspaper. How about yourself? > like everybody else who reads newspapers, > i filter for interest. Good one, chief. I hear the WB network is hiring comedy writers. Doug === Subject: Re: The role of infinity in math > Some people do wrestle with the precise question of how > we define the predicate S is infinite (usual choice: existence of a > bijection S->T where T is a proper subset of S) but I dont think > thats > considered a core issue of modern set theory! I came up with a simple definition of an infinite set in another thread. > Let S be a set of natural numbers and let x and y be members of S. > Consider these two statements: 1) ExAy(x>=y) > 2) AxEy(x If statement (2) is true then S is infinite. > But (1) only works for well-ordered sets, and not all sets are > well-ordered, e.g., the set of negative integers satisfies (1) so must > be finite by your reckoning. > I said S is a set of natural numbers. > That wouldnt include negative integers. It wouldnt work for negatives either, which is what I said. > How do you tell if an ordered-but-not-well-ordered set is finite? > That would be more complicated. > Any set of natural numbers can be well ordered. > In fact sets need not be ordered at all. How do you tell whether an > unordered set is finite? > My method wont tell you. > Cant all finite sets be ordered? > Your definitions are simple enough, but wrong. > My definition works for sets of natural numbers. > That is all I claimed. By implication you claimed more. You said that you had simple definition in another thread, then stated this one, without making clear that it was not your other definition. And I complained that this one did not work for arbitrary sets. === Subject: Re: Got a speeding ticket and need to fight back > and what do you know about the experince of > paul reubens jewels at an all-night theater? >> I can read a newspaper. How about yourself? >> like everybody else who reads newspapers, >> i filter for interest. > Good one, chief. I hear the WB network is > hiring comedy writers. send them your r.8esum.8e, you might serve as inspiration. :) rgrds, === Subject: Re: Got a speeding ticket and need to fight back @nwrdny02.gnilink.net: >> Good one, chief. I hear the WB network is >> hiring comedy writers. > send them your r.8esum.8e, you > might serve as inspiration. > rgrds, This thread has been more entertaining than anything Ive seen on the WB. - Tim Timothy M. Brauch Graduate Student Wake Forest University === Subject: Re: Unusual Numbering Systems > Is there a series that sums to the golden ratio? > There are series that sum to any given real number, namely the first > differences of any sequence converging to that number form such a > series, and there are infinitely many such sequences. > Do you happen to know one? > I am too lazy to compute one by hand. > I did a search and found numerous methods using continued fractions, > but no simple series. > Russell > - 2 many 2 count Work out the decimal representation for the golden ratio, then the series that adds on one more digit of that expansion with each term will do the trick nicely. === Subject: Re: rank of a random {0,1} matrix > Given a nxk random matrix R, k>n, each entry of the matrix is either 1 > or 0 with equal probability, i.e., Pr{r_ij=1}=0.5, Pr{r_ij=0}=0.5, > what is the probability that the rank of the matrix is n ? > > I guess I found the answer. > Cooper explained this in (Theorem 1 in [1]) that: > If $V$ is a $n times k$ binary random matrix with entries independently > and identically distributed as $Pr{v_{ij} = 1} =0.5 $ and > $Pr{v_{ij}=0}=0.5$, > then > [ lim_{k to infty} Pr{ rank(V) = n } = prod_{j=k-n+1}^{infty} > left( 1- 0.5^{j} right ) ] > [1] C. Cooper. textit{On The Distribution of Rank of a Random Matrix Over > a Finite Field}, Random Structures and Algorithms 17(3:4), 2000 If Cooper is really talking about matrices over a finite field, then his answer is not to your question. E.g., the matrix 1 1 0 0 1 1 1 0 1 has rank 3 as a real matrix but rank 2 over the field of two elements. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: please help > I need to prove that (1+X)^n >= (1+nX), n is a natural #, X is a real number >= -1. > Here is what I have (and dont have). I know that (1+X)^n and (1+nX) > intersect at (0,1). Since [ (1+X)^n ] Ô > n for all X >0 and since (1 + nX) > Ô = n for all X we know that (1+X)^n > (1+nX) ( but we already knew this > from the binomial theorem). > The real question is whether or not (1 + X) ^n and (1 + nX ) intersect below > X =0 ? For n = 0 or 1, (1+x)^n = (1+n*x). For integers n > 1, let f(x) = (1+x)^n - (1+n*x), then show f(0) = 0, and f(x) = 0 at x = 0 (and only at x = 0), and f(x) > 0 at x = 0. === Subject: Re: there is no such thing as infinity > Ive thought really hard about this one and came to the conclusion > that there is no scientific evidence of infinity existing. The highest > number that anyone has ever measured to according to Isaac Asimov in > his book Science and Human Thought is only about 5.0 x 10^48. No one > has ever gotten past that number. Doesnt this sound weird? > Whats to say that eventually there is a number where it is impossible > to count higher than? If someone were to find this number and prove > that it is in fact the highest number, then that person would > undoubtably be rich and famous. > I am currently running a computer program that will eventually find > this magic number (I hope and pray) that I call M for short. It > counts and counts and counts and my theory is that it will eventually > stop at M. I am looking for collaborators in this experiment so that I > can use their computer time. The program in FORTRAN is simple: > 00001 n=1 > 00002 1 n=n+1 > 00003 print(3,4)n > 00004 if(n.eq.M) then print(3,4)M > 00005 else go to 1 > 00006 end if > 00007 end > It has currently reached about 2.0 x 10^18. Just as Einstein proved > that there is no aether, I am convinced that I will prove that there > is no infinity and then write a book or two. > Dr. Ben Zona First of all infinity is not a number or any such thing. On the contrary to the claim, infinity alone exists. As such, its appearance, the universe is only as observed. And so, while infinity exists, the observers universe is simply observed. Now, this observers universe forms the foundation for the observer entangled quantum theory and observer related relativity theory, the modern theories of physics. S S Shastry === Subject: Re: JSH: Open letter to Jim Ferry >> Im intrigued by the questions raised by your recent posts, >>as for >> years you were this guy who came up with rather creative >>ways to >> insult me, and now I find it hard not to figure youre just >>doing so >> again. >> Is it okay to ask you questions, talk in any familiar way, or >>in any way >> act as if I wish you to reply or am addressing you? >> Those proscriptions apply only to David Ullrich, Virgil, and >>anyone who >> posts under a palindromic pseudonym, right? >> Okay then. You ask a good question. Just what is my intent? >> Im not >> sure precisely why Im doing what Im doing, but Ill try to >>answer you >> earnestly and respectfully. >> Now Ill embarrass you a bit as from what Ive read on the >>web you >> have one of the highest IQs out there, so it seems to me >>theres >> probably some reason to what youre doing, and possibly Im >>wrong >> about what it is. >> This does embarrass me because Im certainly not one of the >>big fish on >> sci.math. You must be basing this statement on the fact that >>I once >> joined something called Mega, which purports to be a high-IQ >>society for >> those of 1-in-a-million intelligence. I now realize that by >>joining, I >> was implicitly making this arrogant claim about myself, but I >>reject that >> claim. The fact that I was able to ace the math part of >>their test just >> indicates that it wasnt hard enough, because its easy to >>find people >> better at math than I. Indeed, you can find lots of them on >>sci.math. >> And some of them even take the time to analyze your work, >>James. >Youre not one of the big fish on sci.math, and Im less >curious about >your past experiences with high IQ societies than you probably >think. JSH in his best Rosanne Rosannadanna voice: Nevermind. >However, I had a theory, and testing it involved mentioning >that facet >of your public persona. JSHs attempt at regaining control. >> Therefore, Im going to give you the opportunity that I >>feel *I* dont >> get, which is the benefit of the doubt. >> Youre asking me to clarify me position, which I am about to >>do. I think >> its fair to say, however, that others have given you the >>same opportunity, >> i.e., that theyve asked you to clarify your position. >> Tell me succinctly and in a way that will minimize potential >> embarrassment for both of us, what it is that your up to, >>and no, none >> of this wild stuff about how great I supposedly am, or how >>Ive proven >> FLT or any of that, as I just want you to say something >>that fits into >> a worldview that makes sense. >> >> Whats your intent? >> You think that Im mocking you, but youre not entirely sure. >> David >> Ullrich thinks its incredible that its not obvious to you >>that Im >> mocking you. >> Well let me clear things up. Yes, Im mocking you. Ive >>made a series >> of posts over the last six months in which Ive appeared to >>be converted >> to a religion in which you are the Messiah of Mathematical >>Truth being >> crucified my the benighted masses. Most people consider that >>absurd and >> therefore conclude that I must have been being sarcastic. >>You, however, >> do not consider the idea that you are the Messiah of >>Mathematical Truth >> to be absurd. You consider it to be essentially correct -- a >>little off >> somehow: a little over the top, or emotionally overblown, but >>basically >> the correct attitude to take. >So you were lying. I just needed to make sure. >The problem is that Id concluded that very intelligent people >see a >much higher value in telling the truth than others. JSH shrinking his world a little bit more. >That lead me to consider the possibility that you were in fact >sincere, but deluded and confused, possibly dealing with a lot >emotional pain from a difficult position--considering that I >might be >right--against tremendous social forces. >But you werent being brave. You were just being a smart-ass. JSH summing up JSH. >James Harris Milo === Subject: Re: Unusual Numbering Systems Too bad the inverse Fibonacci series doesnt converge. > I could make an interesting series base if it did. What do you mean by inverse fibonacci series ? If you mean the sum of 1/f(n) then, yes, it does converge and thats a simple consequence of the fact that the fibonacci sequence grows exponentially. (Converges to about 3.35988566) J === Subject: Re: Got a speeding ticket and need to fight back The Lord of Chaos (Suresh Devanathan) > may be, you missed the title of my theard. Read it: Got a speeding ticket > and neet to fight back Ofcourse, i am in a hurry to find an excellent > arguement. Why? You know you were speeding. The honest thing is to admit it, pay the fine and learn to drive within the limit. > -suresh > The Lord of Chaos (Suresh Devanathan) > I told you to Ôgo away. Do u speak english? Can you read or write in > english? > Do u speak English? What - are you too much in a hurry to actually > write out the word you? > Doug === Subject: Re: problem > Does there exist a real function which has a graph which is everywhere dense > overe its plane in > the sense that for every circle in the plane of its graph there exists points > on the graph that belong to the interior of the circle. > An explicit example will be highly appreciated. Let E = {sqrt(2)/m : m = 1,2,3 ...}. Let r_1, r_2, ... be an enumeration of the rationals. Then the sets E_n = r_n + E are pairwise disjoint, and each is countably infinite. So for each n there exists a function f_n from E_n onto the rationals. Now define f : R -> R by setting f = f_n on each E_n, and anything you like on the rest of R. The graph of this f will be dense in the plane. === Subject: Re: Resolution to Decker Quadratic Issue ... stuff deleted ... > So no matter how you look at it, (1 + sqrt(-167))/2 cant have a > non-unit factor in common with 7 in the ring of algebraic integers. >Therefore, (1 + sqrt(-167)) has no factors in common with 7 in the >ring of algebraic integers!!! >> But thats simply not true, as Dik reminded us in an earlier response. > Such a simple denial is not the mark of a highly intelligent person. > You need to face the mathematics, as Ive done when Ive been wrong. > After all, isnt it the truth which is important? >>Rick > You need to look carefully at the argument Rick Decker, and think > carefully before you post again, as not only are you on the line here, > but so is Hamilton College. > You misstep here, and you may let down an awful lot of people for no > good reason. > Fighting mathematics just isnt the way to go. Its just not logical. > James Harris Still with the implied threats? Why not grow a pair, and get on with making good on the threats you are continually implying? Are you *that* much of a girly-boy? I thought as much. You have charged Rick Decker with fraud. You have suggested that you may instigate civil proceedings. You have stated that there will be unpleasant consequences from the actions that people on this newsgroup are taking, specifically in their denial of your ascendancy to the Throne of Mathematics. Yet you do nothing other than to prance around wearing that silly outfit. What outfit, you ask? Why, its the poor me, no one treats me fairly, Im the misunderstood genius, but only dont ever ask me to prove anything because I cant be bothered, plus I couldnt write a proof if my life depended on it satin baby-doll PJs, complete with pink fuzzy bunny-rabbit slippers. You may as well wear a rhinestone tiara and ballerina slippers. Problem is, you dont think anyone else can see through the bullshit. Your outfit is as plain as the nose on your face, to coin a phrase, and everyone who knows a lick of mathematics can see it a mile away. Make threats, or dont. I dont care. Do you know what they call a person who makes threats and doesnt follow through? A coward. A bully. A sissy. A man doesnt threaten. Thats the job of a person who doesnt have the courage to be a man. Heres the fact: youre wrong. Youve always been wrong, and until you actually *learn* some mathematics, youll *continue* to be wrong. The arguments that prove you wrong are beyond your grasp, for the simple reason that the *ONLY* mode of argument you comprehend is the elementary algebraic manipulation. You are outgunned and outclassed in every meaningful sense of the word, and your continuing attempts at using high-school algebra to confound the Evil Mathematics Cabal are as pathetic as they are amusing. If you spent half the time and effort as youre currently doing, but applied it towards *learning* some algebra, you would stand a much better chance at doing something worthwhile. Instead, your romantic notion of being the tortured genius is only serving to make you the complete model of the Usenet crackpot. Perhaps you should take Jim Ferrys recent note to heart. Dale.