mm-2789 === Subject: Re: JSH: Factorization P(x) = 2(x(x+1)/2 + 1) >>If 7 and 22 are *not* units, I'm pretty sure they won't have non-unit >>divisors, but I'm not sure how to go about showing that. Maybe when >>I've got some time later I'll look at it. > That's why the usual definition is better: > LEMMA. If R is a ring, and x and y in R are coprime in the sense that > there exist a and b in R such that ax+by = 1, then for any ring that > contains R, x and y are also coprime (in the same sense). > Proof: a and b serve as witnesses in both R and the larger ring. QED > REMARK. If R is a ring, x and y in R are coprime in the sense that > any common divisor in R of x and y is a unit in R, then it is possible > for there to be a larger ring S, containing R, where x and y are no > longer coprime (in that sense). > Example: R= Z[sqrt(-5)]; x = 2, y = (1+sqrt(-5)), S= > Z[sqrt(-5),sqrt(2)]; note that y is a multiple of sqrt(2), since > (1+sqrt(-5)) = sqrt(2)*sqrt(3+sqrt(-5)). > PROP. Let R be a ring. If x and y are coprime in R in the sense that > there exists a and b in R such that ax+by=1, then x and y are coprime > in R in the sense that any common divisor in R is a unit in R. > Proof. Let u be a common divisor of x and y. Then it is a divisor of > ax, and it is a divisor of by, so it is a divisor of ax+by=1. Divisors > of 1 are units. So u is a unit. QED > So in ANY ring that contains the integers, 7 and 22 are coprime (under > either definition). -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Repeat: White Noise Dilemma > Incidentally, I had a thought on the applied side. In physics, for > example, if we apply a strong force over a short period time, like > when two billiard balls collide, we can describe that by a function > f(t) which equals the strength of the force at time t. (Or something > like that. Sorry, I'm no physicist.) I think it's called an > impulse(?). Anyway, it's the integral of f that determines the overall > effect of the impulse. If we want to assume the entire impulse is > applied at a single instant in time, so that f is suuported at the > origin, then f *must* be a delta function, or there will be no effect > at all. If, for example, we wanted to apply the impulse f(t) which > is zero everywhere except f(0)=1, then we're saying that we apply some > finite amount of force over an arbitrarily small period of time, which > in the limit amounts to applying no force at all. Maybe your process > is analogous to this. Maybe a noise which satisfies the properties > you prescribe would amount to no noise at all. I think this is almost right. I have some heuristics and some mathematics to support this. Those processes X_r(t)=[B(t+r)-B(t)]/r are converging to white noise. Formally, white noise is all that abstract business about distributions and what not. But what's really happening as r-->0? Well, X_r becomes wilder and wilder. It's oscillations become more frequent, with roughly the same number of ups and downs. Normally, this would be bad because it would cause things to cancel out. What I mean is that, given some function f(t), if we look at int{X_r(t)f(t)dt}, then the fact that there are as many ups as downs of X_r(t) and the fact that these ups and downs are so close together would cause this integral to tend to zero. In particular, all the Fourier coefficients of X_r(t) would go to zero. What prevents this from happening in the case of white noise is that the magnitudes of the ups and downs grow in their variance as r-->0. So some are very large, some are very small, and they are unable to cancel each other out, leaving something random and nontrivial in the limit. In your case, you would want to look at the processes Y_r(t)=[B(t+r)-B(t)]/sqrt{r}. The difference here is that you've bounded the variance. So as r-->0, the ups and downs will have roughly the same magnitude and all the Fourier coefficients will go to zero. So it's not that your process results in no noise at all; it results in a noise that too regular. It's like looking at sin(nt) as n-->infty. It's not converging to zero, but if you look at int{sin(nt)f(t)dt} for any reasonable function f, this will tend to zero. Here's some sketchy computations. See the Oksendal SPDE reference for more details on the notation and concepts. Let W(x)=int{f(t-x)dB(t)}, where f is a Schwartz function. (If f is sufficiently close to a delta function, then W(x) is close to white noise.) Let g(x) be some reasonable function, say of compact support. Then int{W(x)g(x)dx} = int{int{f(t-x)dB(t)}g(x)dx} = int{-int{B(t)f'(t-x)dt}g(x)dx} = -int{B(t)int{f'(t-x)g(x)dx}dt}. Let F(t)=int{f(t-x)g(x)dx}=int{f(x)g(t-x)dx}, so that int{W(x)g(x)dx} = -int{B(t)F'(t)dt} = int{F(t)dB(t)}. Hence, the variance of int{W(x)g(x)dx} is int{[F(t)]^2 dt}. For the process you're trying to construct, we would want to take a sequence of positive f's that converge to zero everywhere except the origin, where they converge to 1 (still assuming sigma=1). Then, unlike a sequence which coverges to the delta function, int{f(x)dx} would go to zero. Now, note that |F(t)| <= (sup|g(x)|)*int{f(x)dx}. So we see that int{W(x)g(x)dx} would actually converge to zero as the f's converged to the desired function and we have a noise that behaves the way the heuristics expected it would. === Subject: Re: Key core error argument, stepped out good move. now, when are you going to finish the short proof of Fermat's so-called Last Theorem --is it NP-soluble or just Googol undecidable?-- and how long will it have become, then? maybe it's just Shakespearean Sonnet Primate Team Time solvable. I think it's been stated, before, that you can use examples with much smaller numbers, but having the same mathematical concept content, what ever in Hell that's supposed to be. is there really any point in numbering a proof, unless it involves several lemmas? now, let me be less kind to Devlin's sophistry --and I wish I'd gone to B&N, when he was there, to say it-- he's just lying, because the Pythagorean Theorem, and its extension to Pythagorean Triples is anything but trivial, although it is *simple*, or elementary. but, then, you're just using him to beg the question, only in a more direct way then you always do; eh?... or, do you consider yourself a rightwinger or leftwinger in all things, including proofs? > And I'm adding in more changes as the word coprime that I used > before has become a flashpoint, so I'm taking it out. > I also number out the main steps, so if anyone thinks there's a > problem and wishes to reply back, they need to give at least *some* > numbers. > When is a proof? http://www.maa.org/devlin/devlin_06_03.html > While valid in an idealistic sense, the right wing definition of a > proof has the problem that, except for trivial examples, it is not > clear that anyone has ever seen such a thing. > http://mathforprofit.blogspot.com --ils duces d'Enron! http://tarpley.net/bush8.htm http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Re: naive geometry questions >The other surface is the catenoid, formed by rotating a catenary, Q: How do you make a catenoid? A: Pull its tail. Lee Rudolph === Subject: Re: Continuum > Methinks I finally disproved CH by finding something with cardinality of C, > all think by looking in places noone else looks. You would need to find something of cardinality strictly greater than Aleph-0 and strictly less than C to disprove CH. If all you did was find something of card C, that's not very impressive. For example the reals have card C. === Subject: Re: Key core error argument, stepped out >sci.physics snipped >Newsgroups trimmed. Again. [deletia] >>4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I >>have >>P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) >>P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). Note that all you have done is add and subtract 3 to define b_3(x); >that is, you are writing b_3(x) = (a_3(x) - 3) so 5a_3(x) + 7 = 5(a_3(x)-3+3) + 7 > = 5(a_3(x)-3) + 15 + 22 > = 5b_3(x) + 22. The exact same process that you decried when I used it. You claimed >that doing this ma[de] no sense mathematically. Do you still make >that claim? Just curious. >>That is a lie from Arturo Magidin as in fact he just subtracted and >>added 3 on the same line. I'm focusing on constant terms, not trying >>to hide a correct argument with meaningless operations like >>subtracting and adding 3. >No it's not, James. You might want to check your facts before you >accuse others of lying. >>You obviously are the one who didn't check facts as what I said IS correct. > Actually, no. You said three things, and you implied a fourth. Of the > three things you said, one is false. Therefore, what you said is NOT > correct. > The three things you said are: > (a) I lied; I had interpretted the lie to refer to you coming up with the idea first. In either case, you accurately described what he did, and did it first. > (b) I just subtracted and added 3 on the same line; > (c) You are focusing on constant terms. > The thing you implied was that I was trying to hide a correct > argument with meaningless operations. > Statements (b) and (c) are correct. Statement (a) is false. Your > definition of b_3 MEANS that you've added and subtracted 3 to go from > 5a_3(x)+7 to 5b_3(x) + 22. Just because you did not say so explicitly > You said three things. One of them is false. Therefore, you claim that > what you said IS correct is false. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Was Lucy the split image - Jabriol is a creationist. Jabriol is indeed a creationist as are all Jehovah's Witnesses. He's been trying to disprove evolution for the past several years here on Usenet, using what he's learned from the Watchtower publications, with no success. You can't reason with a religious fanatic. Zulo....... Jesus, save me from your followers. > Sorry, I first thought you were a creationist. > nope I am not === Subject: Re: Continuum >> Methinks I finally disproved CH by finding something with cardinality of C, >> all think by looking in places noone else looks. >You would need to find something of cardinality strictly greater than >Aleph-0 and strictly less than C to disprove CH. If all you did was >find something of card C, that's not very impressive. For example the >reals have card C. Yeah, but *everyone* looks *there*. Lee Rudolph === Subject: Re: Key core error argument, stepped out Adjunct Assistant Professor at the University of Montana. >>[deletia] >4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I >have P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). >>Note that all you have done is add and subtract 3 to define b_3(x); >>that is, you are writing >>b_3(x) = (a_3(x) - 3) >>so >>5a_3(x) + 7 = 5(a_3(x)-3+3) + 7 >> = 5(a_3(x)-3) + 15 + 22 >> = 5b_3(x) + 22. >>The exact same process that you decried when I used it. You claimed >>that doing this ma[de] no sense mathematically. Do you still make >>that claim? Just curious. That is a lie from Arturo Magidin as in fact he just subtracted and >added 3 on the same line. I'm focusing on constant terms, not trying >to hide a correct argument with meaningless operations like >subtracting and adding 3. >>No it's not, James. You might want to check your facts before you >>accuse others of lying. >You obviously are the one who didn't check facts as what I said IS correct. >> Actually, no. You said three things, and you implied a fourth. Of the >> three things you said, one is false. Therefore, what you said is NOT >> correct. >> The three things you said are: >> (a) I lied; >I had interpretted the lie to refer to you coming up with the idea >first. I don't remember (and I do not see in the original post) any claim of priority. In fact, nowhere did I say first. All I said is what james had done, in this case, and noted that it was the exact same thing James had attacked recently. In fact, when he attacked it I pointed out that it was the same thing James had done in his Advanced Polynomial Factorization first Lemma (that was done in a different thread), so I certainly did not claim priority in any way! Not that James first came up with the idea of writing a function f as f(x) = g(x) + f(a) for some specific value a; it's just that it's not terribly interesting for the vast majority of functions... Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to this extent, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Riemann's Zeta Function by H. M. Edwards OK, so backtracking a few lines: Take Gamma(1-s)(-2*pi*n)^s-1. Summing > over all integers n other than n=0 and using 3) -Zeta(s)-Sigma(Gamma > 1-s/2*pi*i)Int_|x+-2*pi*n|=Epsilon ((-x)^s)/((e^x)-1) * dx/x=0 then > gives Zeta(s)=Sigma_n=1^infinity(Gamma(1-s)[(-2*pi*n)^s-1 + (2*pi*n)^s-1]. I > don't see how 3) is implemented to give this result. Sorry ahead of > time for any Ughs. Well can I present you with an ugh for each asterisk above :-) Let's translate this. Does > Gamma(1-s)(-2*pi*n)^s-1 > mean > Gamma(1-s)(-2pi n)^s-1 > or > Gamma(1-s)(-2pi n)^{s-1}. > The latter seems to make more sense in this context. > You are correct > I can't make head nor tail of what you write for 3). > Is n the summation variable? what has happened to the first integral. > Is the equals sign before the epsilon really meant to be there? > Yes. I lifted this verbatim from p 13. OK, maybe not quite verbatim! I presume you are using the contour integral argument for continuing > Gamma(s)zeta(s). One introduces a countour C_e in three parts: > imaginary axis from -infinity to -e, circle radius e about origin, > imaginary axis from -e to -infinity and take the integral > of z^{s-1} e^z on C_e (with branch cut on negative real axis). > Thse integral f(s) is independent of e by Cauchy's theorem. > It is also an entire function of s: convegence is nice since e^t -> 0 > rapidly as t -> -infinity. Really dumb question. is e in the contour integral an epsilon, or the basis of the natural logarithm. (I am getting confused by both uses of e.) > The integral of z^{s-1} e^z on the first part of the contour is > integral_e^infinity t^{s-1} exp(-pi i(s-1)) e^{-t} dt > = - integral_e^infinity t^{s-1} exp(-pi is) e^{-t} dt. > Similarly on the third part of the contour it is > integral_e^infinity t^{s-1} exp(pi is) e^{-t} dt. > These add to > 2i integral_e^infinity t^{s-1} sin(pi s) e^{-t} dt. > If Re(s) > 0 the integral over the circle of radius e is > O(e^Re(s)). Letting e -> 0 we get that What is this O Is this Big-O notation? > f(s) = 2i sin(pi s) Gamma(s) > for Re(s) > 0. Now we consider > g(s) = integral_{C_e} z^{s-1} e^z/(1-e^z) dz > where we insist e < 2pi (so that e^z =/= 1 for 0 < |z| < e ). > For Re(s) > 1, as with f(s), we can take e -> 0 so that > g(s) = 2i sin(pi s) integral_0^infinity t^{s-1} e^{-t}/(1-e^{-t}) dt > = 2i sin(pi s) Gamma(s)zeta(s). Got it. Consider > G_N(s) = integral_{C_{(2N+1)pi}} z^{s-1} e^z/(1-e^z) dz. > By Cauchy's theorem the difference G_N(s) - g_e(s) is 2pi i times > the sum of the residues of the poles of the integrand > at +-2pi i, +- 4pi i, ..., +- 2Npi i, that is > 2pi i sum_{n=1}^N [-(2pi ni)^{s-1} - (-2pi ni)^{s-1}] > = 2pi i (2pi)^{1-s} sum_{n=1} (-2) cos(pi(s-1)/2)/n^{1-s} > = something nasty times sum_{n=1}^N 1/n^{1-s} Hmm. Edwards gets sin(s pi/2). However I have seen the functional equation written both with cos and sin, so I am sure you must be right. Now at least I see what [-(2pi ni)^{s-1} - (-2pi ni)^{s-1}] is all about If Re(s) < 0, as N -> infinity, G_N(s) -> 0. This is because > |e^z/(1-e^z)| = 1/(1-e^{-z}) and on the contours C_{(2N+1)pi} > e^{-z} is bounded away from 1. Thus for Re(s) < 0 > Gamma(s)zeta(s) = something nasty times zeta(1-s). OK === Subject: Re: Continuum by C i ment continuum, as in CH. Goedel would be proud. But since I am not a mathematician, I will take the proof with me to my grave. > Methinks I finally disproved CH by finding something with cardinality of C, > all think by looking in places noone else looks. > You would need to find something of cardinality strictly greater than > Aleph-0 and strictly less than C to disprove CH. If all you did was > find something of card C, that's not very impressive. For example the > reals have card C. === Subject: Re: Skeptical Inquirer UFO ah, so; am I to infer that PBK works both sides of the conspiracy?... and here, I thought it was Skull and Bones. do you know of the cache taht Roswell has with WW2?... I know of two things, that Art Bell (et al ad vomitorium) never mention (although others have mentioned one of them, on his show ... I stopped listening to taht **** over 3 years ago .-) as the co-author exposed him during the book tour, Corso's thesis is that human beings cannot create ideas, withe corrolary that they were not made in the image of God. (I didn't know, when he came to borders without Corso, that he was deathly ill; probably made it easier to drop his little clues; are they in the book?... the funny things was, I was the only one in the audience who asked any hard questions; everyone else must have been above top secret !-) > mysterious aerospace reports and events (see www.jamesoberg.com) and share > what I've found, for discussion, with no directives, constraints, or other > Jim Oberg > Phi Beta Kappa, Ohio Wesleyan University, 1966 > on by systematically ridiculing those of us who suspect the truth about this > phenomenon, all the while being privy to above-top-secret knowledge that > would prove us right? One wonders.... > C. SCOTT LITTLETON > President, Phi Beta Kappa Alumni in Southern California --ils duces d'Enron! http://tarpley.net/bush8.htm http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Re: Continuum >> Methinks I finally disproved CH by finding something with cardinality of C, >> all think by looking in places noone else looks. >You would need to find something of cardinality strictly greater than >Aleph-0 and strictly less than C to disprove CH. If all you did was >find something of card C, that's not very impressive. For example the >reals have card C. > Yeah, but *everyone* looks *there*. > Lee Rudolph I looked where everyone already looked. === Subject: Re: [JSH] On the Rewriting of a Polynomial Most readers will be able to see that Dik W. and Ghost ITM are cheating, by using more than a handheld calculator. Well, I've never been wrong, yet! > Actually it is quite simple. Q(a) is one of the many possible polynomials. > And it works the other way around, when you have a1(x), a2(x) and a3(x) > roots of Q(a), then P(x) = (5 a1(x)+7)(5 a2(x)+7)(5 a3(x)+7). > Q(a) gives the values for (a1+a2+a3), (a1 a2+a1 a3+a2 a3) and (a1 a2 a3). > Writing out the factorisation of P(x) as a single expression and filling > in these three values gives the original form of P(x). --ils duces d'Enron! http://tarpley.net/bush8.htm http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Re: Key core error argument, stepped out > [.snip.] >> But for two of the factors of P(x), the constant terms is 7, which is >> coprime to 22. Therefore, *none* of the constant terms can have 7 as >> a factor. >> >> (By saying that 7 is coprime to 22, I'm making a choice as to where >> the proof is going. Since I've been talking about algebraic integers, >> where 7 is coprime to 22, it's natural to go with a choice where 7 is >> coprime to 22.) > This is the heart of it. JSH says >[1] 5*b3 + 22 >has constant term 22, and that constant term is coprime to 7. >Correct. > Here, I believe, is how the thinking goes. If you have a >polynomial with integer coefficients, say, > F(x) = a*x^3 + b*x^2 + c*x + 22, >then that *polynomial* is coprime to 7, because one of the coefficients >itself is coprime to 7. That is true whether the coefficients are >integers or algebraic integers. There is no possible common factor >of F(x) with 7. The constant term, F(0), is 22, and that is all >you need to know to say that the polynomial is coprime to 7. > This statement seems to me to be correct. One says that a polynomial >with integer coefficients is coprime to an integer m if *any one* of >its coefficients is coprime to m. Here the constant coefficient happens >to be coprime to 7, and that is enough. > The statement is true IF by coprime you mean coprime in Z[x], and > you mean have no common divisors other than units. However, it is > not true if you mean coprime in Z^Z; the example of x^2+x and 2 > comes to mind. Right. The definition I intended was, all the coefficients are coprime to 7. I am trying to give JSH as much benefit of the doubt as possible on this. Why? Not sure. Perhaps I shouldn't. It doesn't matter anyway. > Using Dot's proof that the values of a polynomial with algebraic > integer coefficients are always divisible by an integer if and only if > each coefficient is a multiple of that integer (in the algebraic > integers) gives you that the statement is correct for polynomials in > A[x]. I had not noticed Dot's theorem. The x^2 + x example seems to belie it: for any integer x, x^2 + x is divisible by 2, but the coefficients are not multiples of 2, and both of the coefficients are algebraic integers. You must therefore be thinking that the domain is algebraic integers as well. > Of course in [1] above, b3 is not a polynomial. It is a function >of x. It takes values in the algebraic integers. But perhaps the >same reasoning that applies for polynomials applies for b3(x). > But it does not. Yes, of course I know that. Again I am trying to guess at JSH's thinking. One could make up a definition: Definition. Two functions f(x) and g(x) which take on integer values when x is an integer are COPRIME if there exists a number x0 such that f(x0) and g(x0) are coprime in the integers. Whether such a definition is worth considering, I don't know. By this definition, f(x) = 2 and g(x) = x^2 + x are not coprime. > [.snip.] > Figuring out how JSH is thinking at any given point is not >necessarily much of a reason to celebrate. > One possiblity that occurred to me yesterday is that James has not yet > realized that any function f(x) from A to C can be written as > f(x) = g(x) + c > where c is a constant, and g is a function that takes any specified > value at any specified point. That is, if a is in A and b is in C, > then there always exists a function g(x) from A to C and a constant c > in A such that f(x) = g(x) + c, and g(a)=b. > Just take c = f(a)-b, and let g(x) = f(x)-f(a)+b. > So, given ANY f(x) function from A to C, there is a function g(x) such > that g(0)=0 and f(x) = g(x)+c, c a constant. > He has found ONE function that has the right c, namely, > (5a_1(x)+7)/7; so he thinks this is the ONLY function that has the > right c. Likewise, (5a_2(x)+7)/7 works, so it must be the only one > that works; and (5b_3+22)/1 works, so it must be the only one that > does. He does not realize that for ANY complex-valued functions > w_1(x), w_2(x), w_3(x), such that w_1(0)=w_2(0)=7, w_3(0)=1, and > w_1(x)*w_2(x)*w_3(x)=49, one can write (5a_1(x)+7)/w_1(x), > (5a_2(x)+7)/w_2(x), and (5b_3(x)+22)/w_3(x) as a function which is 0 > at 0, plus 1, a function which is 0 at 0, plus 1, and a function > which is 0 at 0, plus 22. He found one possibility, he thinks there > is only one possibility. And that could be the mistake. As noted in my post, I think his problem is at a lower level. He might say that 7*x^3 + 6*x^2 - 7*x + 7 is coprime to 7 because 6 is coprime to 7. I think he is going by the definition I described above: if a polynomial Q(x) has *any coefficient* which is coprime to 7, then that polynomial is coprime to 7. Therefore if the constant term is coprime to 7, the whole thing must be. Then he somehow generalizes this idea to all other functions. It is just *verbal* reasoning, not mathematical reasoning, and as such it misses the point. He doesn't even need that the polynomial is coprime to 7. What he needs is that specific *evaluations* of the polynomial are coprime to 7. He is clearly not understanding this, so he keeps repeating the bit about constant terms like a shaman singing a chant. I liked what you said about it yesterday: he assigns almost magical powers to the value of his functions at x = 0. Nora B. > Why do you take so much trouble to expose such a reasoner as > Mr. Smith? I answer as a deceased friend of mine used to answer > on like occasions - A man's capacity is no measure of his power > to do mischief. Mr. Smith has untiring energy, which does > something; self-evident honesty of conviction, which does more; > and a long purse, which does most of all. He has made at least > ten publications, full of figures few readers can criticize. A great > many people are staggered to this extent, that they imagine there > must be the indefinite something in the mysterious all this. > They are brought to the point of suspicion that the mathematicians > ought not to treat all this with such undisguised contempt, > at least. > -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan > Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: Ring problem > Suppose R is a ring with s = s ^ 2 for each s in R. Why s + s = 0? how about this one? (-a)^2 = (-a) (-a)^2 = (-a)(-a) = a^2 = a Therefore, a = (-a), or a+a = 0 === Subject: Re: Using De L'Hopital for solving equations The question now is : Does exist a particular function g(x) where : lim (x->t) g(x) = 0 and lim (x->t) g ' (x) = 0 ? === Subject: Re: JSH: Advanced Polynomial Factorization the FBI is probably going to pay him through his blog, before they dump him in the river. [sorry, if it's not you folks .-] > Who is the one who calls people liars for disagreeing? Who is the one > who resorts to cursing? Who is being childish? > http://mathforprofit.blogspot.com/ > Have you made any profit yet? --ils duces d'Enron! http://tarpley.net/bush8.htm http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Averaging Errors: Restricted #-of-Divisors Function Although this post is regarding a topic of its own, it is a continuation, in a way, of the thread at: Let d(r,m) = the number of divisors, k, of m, where 1 <= k <= m^(1/r) for each k. (So, d(1,m) = d(m) {sometimes called tau(m)}, the standard number-of-divisors function.) So, let e(r,m) = d(r,m) - d(m)/r. Then, for r = any fixed integer >=2, is: limit{m -> oo} (1/m) sum{k=1 to m} e(r,k) = c *(1 -2/r), where c is Euler's constant (.5772...)?? (Maybe this WOULD have been the limit if a limit existed...) Leroy Quet === Subject: UFO Bogus Physics re: Nick Cook's Hunt for the Zero Point http://www.salon.com/books/review/2002/08/05/zero_gravity/ --- In ItalianPhysicsCenter@yahoogroups.com, Jack Sarfatti How about this from a PhD physicist, in the minority, who has studied the UFO subject: I did some investigating and interviewed retired senior USG executives who laid the Nazi flying saucer (disguised as UFOs) claim (a la Cook's book) to rest - it didn't exist and it didn't fly. Which is pretty much what I have also been saying about Cook's book BTW. I'm interested in any debunkery of Cook's book. What did this Ph.D. fellow discover precisely? You need to ask him. I have bcc'd him on this. He can reveal his ID if he wishes. He has USG Intelligence connections for information. The point is that we at ISSO 1999-2000 at checked out many of the bogus claims on zero point energy by fringe people and found them to be essentially worthless. Creon Levit on leave from NASA Ames did a very thorough job on several of the claims. More information on this is in my two books from 2002 Destiny Matrix and Space-Time and Beyond II and in my semi-popular Society for Literature and Science http://qedcorp.com/APS/StarGate1.mov Note of clarification I do not include the Haisch-Puthoff zero point energy program nor Puthoff's PV gravity program in the same crackpot category as ALL the other work mentioned by Nick Cook in his book. My objections to the HRP program are that they do not ask the right questions, are basically superficial in their formulations, make some errors of interpretation of their formalisms and most importantly have not led to any testable predictions nor any clarifying explanations of significant problems and mysteries, e.g the UFO. They are not bogus physics simply wrong physics in my humble opinion. A few specifics: 1. On the zero point origin of inertia - it is a mistake to look only at the virtual photons. It is the virtual electron-positrons that is most important. 2. There is no vacuum coherence in their idea set. That throws the baby out with the bathwater because guv Einstein's metric field for curved spacetime emerges from that non-perturbative vacuum coherence of the virtual electron-positron pairs primarily. So does the dark energy/matter that is 96% of the universe that does not appear anywhere in their models. They have been working on this for almost 20 years with little to show really. 3. There is no PV (i.e. no quantum polarized fluctuations) in Puthoff's PV math. 4. Puthoff and Ibison mis-interpret the physical meaning of their isotropic radial r coordinate in their toy model K = e^2GM/c^r Puthoff is very interested in UFOs and that is a primary motivation for this zero point gravity work. Nowhere do Puthoff and Haisch et-al squarely face the number String Tension = c^4/8piG = 10^19 Gev per 10^-33 cm which prevents any plausible explanation of UFO metric engineering with their brute force approach. Space-time geometry is simply too stiff to bend with the energy schemes they have in their paradigm. They are missing some very essential new concepts. Cook is a very approachable fellow; have you cross-checked with him also? -Andrew him about the flaky stuff he was pushing so he erased me from his book. That told me he was not intellectually honest, but had some hidden non-scientific agenda. I find this disturbing since he is associated with Jane's Defence Weekly in UK. === Subject: Re: new way of describing ellipse for math it takes five points to determine a conic section, not four ... I'm not sure if that's correct, but an infinity of conics will pass through the vertices of a ractangle, as they're not in general position. so, the use of two rectangles on the same center (?) is overdetermined, in a way, but it seems to work. obviously, any 3 of the corners of a rectangle determine the circle. > |-/----------------/-| > | / / | > |-/----------------/-| > ellipse. I wonder why the 4 corners of a solo rectangle is unable to > make > it unique. I do not see the mechanism for why 1 rectangle was not --ils duces d'Enron! http://tarpley.net/bush8.htm http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >>Let's be serious for once. >>Consider an object being accelerated by a idealistic jet of water or a >>continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >>pattern? >>accelerated beyond the operating speed of the accelerating fields, ie at 'c'. I >>hope it might also produce a relationship that is equivalent to mass >>'appearing' to increase with velocity by gamma.) > What is being ignored is the fact that no matter what >in the same direction will still be going faster by precisely c ! > Why don't you find a useful agenda, you are doing >nothing but costing a lot of people a lot of money on a >worthless and futile effort to show that henry is smarter >than Einstein and all physicists since. What I proposed is a simple and interesting mechanical problem. I gather you cannot solve the rather elementary differential equation. OR - Maybe you fear that I am right and this will give the same relationship that SR give for fictitious 'mass increase' >Joe Fischer Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: I can't stand it anymore >Why are people using the racist IQ tests as God's given ruler to >measure intlligence? Probably for the same reason that people feel the need to generalize about all IQ tests. The only surprising thing about this post was that you didn't also cross-post it to alt.flame, alt.fan.rush-limbaugh, and a dozen other unrelated newsgroups. Doug === Subject: Re: Key core error argument, stepped out > ... >In integers, 2 and 3 are coprime as are 12 and 13, as it simply means >they don't share non-unit factors. > >However, in reals, NO numbers are coprime, as for instance, 2(3/2) = >3, so 2 is a factor of 3. > > But 2 is a unit, so your claim here is false. > > Hmmm...now that's interesting, every real but 0 is a unit, eh? > > Yup. Every element of a field, except 0, is a unit. > > Fascinating perspective. > > Isn't it? > So you have coprime with one clear meaning in a ring like integers, > but things are different in a field. Is it? The definition of coprime in a ring with unit (like the integers and the reals) is equivalent to: two elements a and b are coprime if there are elements c and d in the ring such that a*c + b*d = 1. (Things get a bit more complex if the ring has no unit.) You use (apparently) the definition that a and b are coprime if every number that divides both a and b is a unit. Well, that again is equivalent for both the integers and the reals. So where is the difference in meaning? > Notice then that 3 and 6 are coprime, but 6 still has 3 as a factor, > so in fact, coprime simply loses any relevance. Yup in the reals 3 and 6 are coprime, but 6 has 3 as a factor. 4 also has 3 as a factor. 4/3 is a real. Coprimeness indeed loses any relevance in a field, as does having a factor of. Or would you say that 4 does *not* have 3 as a factor in the reals? If so, how would you define have a factor of? (I assume your meaning is is divisible by.) > I think that's telling. Is it? But that is the way reals work! > Part of my point here is that the mathematics you've taken for > granted, with lots of definitions that seem ok, goes off into some > interesting places. ? > Actually, in the real numbers, ANY TWO NONZERO NUMBERS ARE > COPRIME. That's because, given any two nonzero real numbers x and y, > any common divisor of x and y is a unit. > > Well, then every real number but 0 is a unit follows from that > position. > > As that is the definition of a field, it looks about right. > So mathematicians take these positions, which not only go against > common sense, they don't make sense in general, pushing definitions. The definition of a field is that it is a ring (commutative) where each non-zero element has a multiplicative inverse. As having a multiplicative inverse is the definition of being a unit, I see no problem. It is easily seen that the reals form a field... > So you have this broken word coprime which has no use at all if > you're in the field of real numbers. Right. In a field coprime, divisor of are of no use. Why does that surprise you? > I guess you could say that and it doesn't change things in any > meaningful way. > > But it's *fascinating* that Arturo Magidin pushed that position! > > Are you contradicting it? > Nope. I'm just highlighting how screwed up things get when > mathematicians are left to their own devices. What is screwed up about it? How would *you* define coprimeness for numbers a and b in the reals? > Well consider then, he's saying that 3 is not a factor of 6 in reals > because they're coprime! > Hmmm...that's my statement. It looks off in retrospect, as instead > coprime is broken, so that what it means in integers isn't what it > means in reals. It means exactly the same thing. Namely in the ring you are working in there are numbers c and d such that a*c + b*d = 1. > So you have 3 is coprime to 6, 3 is still a factor of 6, but it's a > unit, or trivial factor, which in one sense is ok, as every real but 0 > is a factor of every other real, but the word coprime is now a > liability. Eh? Chose a = 3, b = 6, c = 1/3 and d = 0 and see that a*c + b*d = 1. How would *you* define coprimeness in a field? > And you know what? I think the way mathematicians usually go, he's > right!!! > > And indeed. But apparently you have no idea about the mathematical > meaning of coprime. > For those who wonder, coprimeness is usually defined to ignore > *trivial* factors, where unit factors are, of course, trivial. Nope. That is *not* the usual definition. As always you do not know the mathematical definitions. > So 2 is coprime to 3 in integers, but they both have 1 as a trivial > factor, of course. > But notice how the word coprime gets broken when you end up where > *every* factor is trivial! > Then you can say 3 is coprime to 6, in reals. > Most of you do a quick switch in your heads when you're operating in > the real world, so that you handle the problem, and operate in a > particular ring based on your particular needs at the time. > Mathematicians, well, they basically do the same thing, but *claim* to > be more precise. > Fun stuff, eh? > > Yeah. > HELL YEAH!!! Mathematicians have broken or screwed up terms all over > the place, but tend to make ad hoc rules to handle them. > For instance, with coprime an ad hoc rule would be NOT to use the > word with a field!!! Why? The definition is in a ring. As a field is also a ring, why should it not be used in a field? > The math world is spectacularly illogical and quirky, and uses lots of > end-runs and ad hoc rules to cover up messes, which is why I say the > situation is like Ptolemy's circles. > The physics community should take note of the reality versus just > ignoring it as we usually do. Tsk. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Geometry And Newtonian Mechanics of Action Device > And by the way, Al, did you ever think that perhaps > Abhi's language might be French, for instance? > Abhi speaks fluent crackpot. Then where should you be placed on a list if you admit to understanding Crackpoteze..... For if you did not understand it you could not realize he was fluent in it..... > -- > Pyriform Paul R. Mays ---------------------------------------------------------------------------- - Some where within the Quantum State Http://Paul.Mays.Com/story.html http://paul.mays.com/mayday.html http://paul.mays.com/rainy.html Science is facts. Just as houses are made of stones, so is science made of facts. But a pile of stones is not a house and a collection of facts is not necessarily science. - Jules Henri Poincare (1854~1912) === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >Explain to me how this model applies to an attractive >electric field. >What are the ping-pong balls and where are they coming >from? Don't worry about it Randy. I doubt if you know what a differential equation is. > - Randy Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >> Explain to me how this model applies to an attractive >> electric field. >> What are the ping-pong balls and where are they coming >> from? >I will seriously sit back and seriously enjoy the reply. >Good one :-)) >Dirk Vdm Same applies to you. Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: Using De L'Hopital for solving equations > The question now is : > Does exist a particular function g(x) where : > lim (x->t) g(x) = 0 and lim (x->t) g ' (x) = 0 ? Certainly, no problem. g(x) = (x-t)^2 comes to mind. Or g(x) = cos(x) - 1. Any function that touches or crosses the but also f' must be zero. In other words, your method won't work unless the function f *and* its derivative have the same root t. And I don't see how you could know that without solving both of them. === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> accelerated beyond the operating speed of the accelerating fields, ie at 'c'. >Explain that phrase, please. >As you know, the accelerating electic field in an accelerator >So what do you mean by: >the operating speed of the accelerationg field? You don't know what happens in close proximity to the accelerating charge. Back radiation? >Paul Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: A Double-Sum Congruence Here is a trite little theorem, yet possessing some beauty. (And so I post...) Let p = prime, and let m be any positive integer <= p^p -1. Then: m --- --- | 1/p | | m | / / |------| is congruent to --- --- |_ j _| k=1 j|k j<= k^(1/p) 1/p [m ] --- --- | m | / / |-----| (mod p) --- --- |_ j _| k=1 j|k Linear-mode: sum{k=1 to m} sum{j|k, j<= k^(1/p)} floor(m^(1/p)/j) is congruent to sum{k=1 to floor(m^(1/p))} sum{j|k} floor(m/j) (mod p) (I think.) Leroy Quet === Subject: Re: Geometry And Newtonian Mechanics of Action Device > OK, let us see Pythagores in Action.. > ______________________________________________________ > Pythagores Theorem: > In a right angled triangle, the square of hypotenuse is equal to the > sum of the squares of the other two sides. > c^2 = a^2 + b^2 > ______________________________________________________ > Now draw X, Y axis on paper. We refer origin as B instead of > conventional O. > Line AC of length c is on X axis and initially point C of line AC > coincides with origin B. Let us move or slide line AC along X axis by > distance d(where d< slides along Y axis in vertical direction. Now, we have a right angle > traingle ABC. > In this triangle AC = c, AB = a = c-d, BC = b > According to Pyathagores theorem, in this right angle triangle > c^2 = a^2 + b^2 > We replace a by c-d as AB = a = c-d > Hence c^2 = (c-d)^2 + b^2 > c^2 = c^2-2cd+d^2 + b^2 > We get, b^2 = 2cd-d^2 > ****************** > b = sqrt[d(2c-d)] > ****************** > This is the basic equation which controls this action device. Please > note that when, initially, we move line AC along X axis by very small > distance d, point C slides along Y axis and BC = b > d > Please tell me whether you have understood this or not. > -Abhi. Understood, yes. What's the point? This tells us nothing about the mysterious Action Device. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >> accelerated beyond the operating speed of the accelerating fields, ie at 'c'. I >> hope it might also produce a relationship that is equivalent to mass >> 'appearing' to increase with velocity by gamma.) >> We have: >> S= -------m/s->Vo--------->M->v >> A stream of 'matter' emerges from source S at velocity Vo and m 'mass units' >> per second. It strikes unconstrained object (mass M), which subsequently >> accelerates away. Both energy and momentum must be conserved during this >> operation. >> The problem is to find the x/t relationship for three different situations. >> 1) nonelastic case where the jet ends up having the same velocity as the object >> after collision. >> 2) where the jet bounces back perfectly elastically. >> 3) (maybe) where the jet is absorbed into the object. >> WRT the source frame, the equation describing the nonelastic jet case is: >> M.dv/dt=m(Vo-v) or: >v = Vo*(1 - exp(-m*t/M)) That's roughly what I got - but I thought there was more to it? Are you sure that's the full story? >> (d2x/dt2)=K(Vo-dx/dt) >> Here, the water ends up traveling at M's velocity. >> In the elastic case, the water (or stream of ping pong balls) ends up moving >> backwards at Vo-2v, so the change in momentum per second is 2m(Vo-v) >v = Vo*(1 - exp(-2*m*t/M)) OK here's a harder problem. When the jet strikes the object, it is collected inside the object so that it increases the weight of the object. What is the x/t equation? Note, this is not the reverse of the 'rocket engine' problem because in that case the jet velocity is constant wrt the ship not the ground. >> Do you all agree? >> I now wish to solve this equation. My maths is a bit rusty but I can probably >> work it out eventually. I get something starting with a factor e^-K (which is >> good). I'm not sure about the rest. Any offers? >> What are mathematicians for anyway? >There is no reason for making this more complicated than it is, Henry. >The mass M will in all cases end up moving with the speed Vo. >Or rather, v will approach Vo asymptotically. >But you knew this, didn't you? >And this have absolutely nothing to do with what happens >Which you also knew, didn't you? Not at all Paul. I have to compare the curve given by the above solution with that predicted by SR for a charge accelerated in a constant field. Can you tell me what that might be? >Paul Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: Skeptickal Inquirer UFO Lt.Col. Corso, may he rest in peace, used his onetime position at the Roswell bomber base to illucidate his silly story, We dumb Americans found this little pile of ****, and I sent it out to the assembled brethren of the MIC, and we reverse-engineered RADAR, fiber optics, transistors ... um, le'see ... balsa wood and mylar and latex balloons, to boot! do you know of the cache that Roswell has with WW2?... really, the whole thing could be denial, denial, denial. Corso's essential thesis was: human being cannot create ideas; it's got to be cargo-cult from upstairs! his coauthor exposed him on the booktour, which Corso had to sit out, as he was dying, but I don't know if it's actually in the book, as well. >history to now, i.e. about 7 billion years ago. This dark energy is the >same as Kip Thorne's exotic matter in his 1986 Star Gate paper and it >is the same stuff needed for Alcubierre's weightless warp drive metric >of the early 1990's and it is the same negative matter needed for >exotic propulsion that British Ministry of Defense Chief Scientist >Herman Bondi told us Cornell students about in ~ 1960 that Stalin's top > physics Spy Master Y. Terletskii was also very keen on. > http://www.rense.com/general44/nmxx.htm > one of the few middle of the road outlets for information of > the strange bizarre and unusual. > such tools of destruction. To bring man to the point > where he was capable of handling the technology without > destroying the earth. > That would be the best case scenario. > I have no reason to doubt the credibility of this man. > Mr. Phillips further states that there are records and filmed > documentation of meetings in California in 1954 between ETs > and leaders of the USA. He lists a few of the technologies > we were able to develop because of the ETs: computer chips, > lasers, night vision, bulletproof vests, and concludes, > Are these ET people hostile? Well, if they were hostile, > with their weaponry they could have destroyed us a long time > ago or could have done some damage. Mr. Phillips now develops > technologies that can help eliminate environmental pollutants > and reduce the need for fossil fuels: energy generation > systems that use natural energies from planet Earth. > And speaking of debunkers, I look at the people on the above > page, not to mention a person like Corso, and I examine > their credibility. Then I look at the Amazing Randi, > and I want to laugh out loud. > We are not alone in the universe, that I am sure of. --ils duces d'Enron! http://tarpley.net/bush8.htm http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >> accelerated beyond the operating speed of the accelerating fields, ie at 'c'. I >> hope it might also produce a relationship that is equivalent to mass >> 'appearing' to increase with velocity by gamma.) >Please define what you mean by the operating speed of an >accelerating field. I've been in physics research for over 20 >years, and I've never heard that term. >Do you mean phase velocity, group velocity, how fast the >operators turn the knobs, what? >*static* fields, while most RF acclerating structures use >standing waves, so your point is dead on arrival. >Why are you blathering on about jets of water and ping pong >balls? They are nothing like the acceleration of an EM >field. Even if they were, what you're describing is a rocket, >and a rocket can easily exceed the velocity of it's propellant. >If you'd ever taken a physics class, you'd know that the >final speed of a rocket starting at rest is (in the >absense of gravity): > v = v_0 * ln(m_initial/m_final) Here the jet speed is constant wrt the rocket not the ground. It is NOT the same problem. >where v_0 is the propellant (or water or ping pong ball) velocity, >so as long as that mass ratio is greater than e, you're point- >even though it was flawed to begin with - is *still* dead on arrival. >Maxwell's Equations and the Lorentz Force Equation have >been around for a very long time, and no one's ever found >a flaw with them. If you're going to talk about accelerating >fields, please do so in the proper language - or is it >that you *can't*? >Just because you think the Earth is flat doesn't stop >other from sailing around it. Find a hobby you can understand. Oh what an obvious genius we have here! If the x/t curve for my 'object' turns out to be about the same as the one for a charge in a constant field, your conceited smirk might fade away. Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: Cantor's diagonal number: why do we just ASSUME such a number exists? > Mike Deeth says... >I have obtained empirical evidence that Cantor's diagonal number >does not exist, through a long thought experiment. > The diagonal number isn't just one number. You failed to see that the original post was an attempt to get into the Guinness book of records: highest crank index of all times. I have not actually counted the score, but it must be very high. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Sum{k|m, k<= sqrt(m)} mu(k) If mu(k) is the Mobius(Moebius) function, then sum{k|m, k<= sqrt(m)} mu(k) gives us the sequence: 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0,... (Sequence A068101 in the EIS: http://www.research.att.com/projects/OEIS?Anum=A068101 ) I may have asked this already a long time ago, but what can be said about this sequence, such as its asymptotics, the expected number of 0's, +-1's, +-2's, etc below any given value, and (perhaps) even what its closed-form might be?? All I can deduce right now is that, unspectacularly, sum{m=1 to n} sum{k|m, k<= sqrt(m)} mu(k) = sum{1<=k<=sqrt(n)} mu(k) (floor(n/k) +1 -k). Leroy Quet === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days > Let's be serious for once. > Consider an object being accelerated by a idealistic jet of water or a > continuous 'stream of elastic ping pong balls'. What is its subsequent velocity > pattern? > accelerated beyond the operating speed of the accelerating fields, ie at 'c'. I > hope it might also produce a relationship that is equivalent to mass > 'appearing' to increase with velocity by gamma.) > We have: > S= -------m/s->Vo--------->M->v > A stream of 'matter' emerges from source S at velocity Vo and m 'mass units' > per second. It strikes unconstrained object (mass M), which subsequently > accelerates away. Both energy and momentum must be conserved during this > operation. > The problem is to find the x/t relationship for three different situations. > 1) nonelastic case where the jet ends up having the same velocity as the object > after collision. > 2) where the jet bounces back perfectly elastically. > 3) (maybe) where the jet is absorbed into the object. > WRT the source frame, the equation describing the nonelastic jet case is: > M.dv/dt=m(Vo-v) or: >>v = Vo*(1 - exp(-m*t/M)) >That's roughly what I got - but I thought there was more to it? >Are you sure that's the full story? > (d2x/dt2)=K(Vo-dx/dt) > Here, the water ends up traveling at M's velocity. > In the elastic case, the water (or stream of ping pong balls) ends up moving > backwards at Vo-2v, so the change in momentum per second is 2m(Vo-v) >>v = Vo*(1 - exp(-2*m*t/M)) >OK here's a harder problem. >When the jet strikes the object, it is collected inside the object so that it >increases the weight of the object. What is the x/t equation? >Note, this is not the reverse of the 'rocket engine' problem because in that >case the jet velocity is constant wrt the ship not the ground. > Do you all agree? > I now wish to solve this equation. My maths is a bit rusty but I can probably > work it out eventually. I get something starting with a factor e^-K (which is > good). I'm not sure about the rest. Any offers? > What are mathematicians for anyway? >>There is no reason for making this more complicated than it is, Henry. >>The mass M will in all cases end up moving with the speed Vo. >>Or rather, v will approach Vo asymptotically. >>But you knew this, didn't you? >>And this have absolutely nothing to do with what happens >>Which you also knew, didn't you? >Not at all Paul. >I have to compare the curve given by the above solution with that predicted by >SR for a charge accelerated in a constant field. >Can you tell me what that might be? >>Paul Incidentally, particularly in the case of the elastic 'ping-pong ball drive', momentum is balanced but does the (kinetic) energy equation match? Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: Two coin flip/ clarification for C Bond >> >>[...] can't fill in > that blank, prior to some kind of inspection, or 'look'. The short answer, which I'm sure you've been given many times, is > simply Sorry, Chum, no, we don't mean the statement with a blank was > made, and then the blank was unprejudicially filled. If we _did_ mean > that, of course your value of 50% is correct. But we hereby clarify > that that is _not_ what we mean. End of discussion. We need to clarify who _we_ is. I hope it's not too presumptuous of me to inform you that by we I > mean, er, everyone involved in the discussion except yourself. >I'll accept that, except that we don't have a concensus. If we >>did, it wouold be even more embarrassing, as, either Moritz is >>absolutely correct, or, he's wrong. [...] > [...] >>No, it's a little probability question, and the majority have it >>wrong. We are answering the question which we think the questioner >>meant. Moritz is by no means alone in his quest. [...] > >>First, it's a matter of right and wrong. Then, Moritz has spent a lot >>of time on this question. He understands it well. Maybe, if we get >>off our high horse, and pick his brain a little, we can get more out >>of it than just this one question. [...] >We know that there had to be a look. Was court card decided prior to >>the look? If so, how were we supposed to know, we weren't told about >>it. This one gets a little ambiguous. Moritz doesn't say that >>ambiguity doesn't exist, he just says that our question isn't >>ambiguous. >>At first gasp Moritz thinks the complement would be, Two decks were >>cut and* at least one is not a court card. [...] >Here, we make false statement. Listen to Moritz, he'll show you what >>can be said prior to the look. [...] >> >> An off-topic comment, but one you should be interested in. As has >> been explained to you before, when you refer to yourself in the >> third person this way it comes off sounding _extremely_ bizarre. >Doctor Ullrich, >I'm writing this real slow. (I know that you think slow). >I'm bizarre, I'm extremely bizarre, that's okay with me. The truth of >my argument is my only concern. You have dodged my argument, all you >do is kibitz about me. > Uh, no. I've explained very clearly what I think of your argument, > many times. The fact that you're willing to repeat the argument > ad infinitum, regardless of what anyone says about it, but I'm > not interested in infinitely many repetitions of what I have to > say about it, does not imply that I've dodged it. > Regarding the kibitzing about you, it's just something I > thought you might want to know. Not that you're going to > convince people you're Right regardless, but I thought > you might want to know that talking about yourself > in the third person like that doesn't help - it makes >The following is my argument: >Assumptions: >Heads and Tails are equally likely. >The problem statement is true. >Question: > Two coins were flipped and at least one is a head. What are the >chances for two heads? >Solution: >I. Heads and tails were equally likely. >II. The problem statement is true. >III. The problem statement is complete. >IV. There must have been an inspection of at least one coin. >V. The statement gives no evidence of inspecting more than one coin. >To flip two coins three ways, both coins must be inspected. >VI. The flip was a first time event. >VII. The statements at least one is a head and at least one is a >tail were equally likely. >VIII. To get an answer other than one half, heads must have been >chosen, prior to inspection.. >IX. To get an answer other than one half, there must be more >information. The answer is 1/2. You dodged it again. Here it is, in numbered sequence. If my argument is wrong, one of those nine is false. Can you find a false sstatement? You can't, because there isn't one. Yet, you say my argument is false. I'm bizarre, so you dodge my argument. You're a Doctor? Methinks there is something fishy. It doesn't matter what you think of me. My argument makes you look foolish. Eldon Moritz >Either my argument is correct, or, one of those nine statements is >wrong. Prove one wrong and I'll send the $1000 to the American Heart >Association in your honor. >We'll let Mitchell Jones be the referee. >Eldon > >> >> >> ************************ >> >> David C. Ullrich > ************************ > David C. Ullrich === Subject: Re: Geometry And Newtonian Mechanics of Action Device >> Abhi speaks fluent crackpot. > Then where should you be placed on a list if > you admit to understanding Crackpoteze..... > For if you did not understand it you could not > realize he was fluent in it..... Not so. I know enough French and German to recognise fluency in others, even though I am not fluent myself. Same with Crackpot. -- Pyriform === Subject: Re: simple variational/control problem? >The task is to choose h(t) that maximizes the integral: >int_a^b e^(-rt) * (p(t) - c(t)) * h(t) dt, Let's call this int_a^b f(t) h(t) dt >subject to: >int_a^b h(t) dt = W (some given constant) >and where p(t)-c(t) is concave down (ie, upside-down-parabola-like). >My book says that it is evident, due to the simple form of the >integrand in the objective, that the required h(t) is given by: >h(t) = W * delta (t - t_m), I hope the book doesn't call this a function... >where t_m that time giving the maximum value of e^(-rt) * (p(t) - >c(t)). >I don't dispute the truth of the claim, I just dont see what's so >evident about it - can anyone out there explain the obviousness of the >Dirac delta as the solution to this problem? Is it required that h >= 0? Otherwise (except in the trivial case where f is constant) the claim is false and there is no such maximum for the integral. Namely, taking points c,d in the interval where f(c) > f(d), you can get an arbitrary value v by taking h(t) = (W + x) delta(t-c) - x delta(t-d) where x = (v - W f(c))/(f(c) - f(d)). Suppose M = max(f(t): a <= t <= b). Then (if h >= 0) int_a^b h(t) f(t) dt <= M int_a^b h(t) dt = M W = int_a^b f(t) delta(t-t_m) dt 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: Magnetism: compare magnetic fields Looking for a direct mathmatical comparison between the magnetic field produced by an electronic current (i) and that of a fixed magnet of a given strength. === Subject: Re: JSH: Factorization P(x) = 2(x(x+1)/2 + 1) > ... > > My point is that you have to focus on the *factorization* and its > > validity in particular rings. > > > > Some factorizations will be valid in one ring, but not another. > > > > Yes, and your factorisation > > P(x)/49 = (5 a1/7 + 1)(5 a2/7 + 1)(5 b3 + 22) > > is in general not valid in the ring of algebraic integers. So what > > are you trying to show? > > > > That's the point. I *prove* that if you have coprimeness between 7 > > and 22 in the ring in which the factorization is valid, where 7 is NOT > > a unit (and neither is 22), then the constant terms of the factors > > that result from dividing P(x) by 49 *MUST* be coprime to 7. > > If you are talking here about the factorisation > P(x)/49 = (5 a1(x)/7 + 1)(5 a2(x)/7 + 1)(5 b3(x) + 22) > being valid in that ring (for all x), that is vacuously true. No > proof needed, 1 is coprime to 7 and 22, and 22 is coprime to 7 in > any ring that contains 1, 7 and 22. Even if in that ring 7 and/or 22 > are units. So we need no proof in that case. > > Oh yeah, so ignore my statement that 7 is NOT a unit Dik Winter as > you're showing your lack of reasonableness. > > You are ignoring my statement that in *every* ring that contains 1, 7 and 22, > they are coprime. Whether 7 is a unit or not. > Are you just totally stupid? That's irrelevant Dik Winter as the > point is that 7 is NOT a factor of 22, which my saying that it's not a > unit points out. Why do you not read what I write, rather than what you think I write? > I'm getting sick of stupid games from stupid people who apparently > have nothing better to do with *their* time. > Now are you or are you not intelligent enough to understand what it > means for 7 NOT to be a factor of 22? Yup. That means that you think your factorisation of P(x)/49 is valid in some ring where 7 is not a unit. I am not convinced. I *know* it is not valid in the algebraic integers, because there is ample proof that for varying x the factors of 49 distribute differently amongst the three factors of the polynomial when you wish to stay in the algebraic integers. Furthermore, I *know* that in the algebraic integers (5 b3(x) + 22) is *not* coprime (either the standard definition or your definition) to 7 for most values of x. So you should construct your ring such that both (5 a1(x) + 7) and (5 a2(x) + 7) are divisible by 7 and that (5 b3(x) + 22) is coprime to 7 *for all x*, and that 7 is not a unit. I am not satisfied that such a ring does exist. Moreover, you wish to divide all three factors by a constant. Why do you assume it must be a constant? Given w1(x), w2(x) and w3(x) such that their product is 49, and w1(0) = 7, w2(0) = 7 and w3(0) = 1. Wouldn't a distribution like: P(x)/49 = (5 a1(x)/w1(x)+7/w1(x))(5 a2(x)/w2(x)+7/w2(x))(5 b3(x)/w3(x)+22/w3(x)) be feasable? If I calculate correctly, the constant terms in this factorisation are now 1, 1 and 22, just as you wish. So why can this not satisfy your requirements? Assuming the w's are constant is just plain stupid if you wish to stay in a particular ring. The functions w depend on divisibility considerations within the ring, and so are not continuous in any sense. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Usenet Posting Guide? > Sometime I miss the days when getting to USENET required configuring > UUCP, compiling, installing and configuring B News or C News, finding > and negotiating with an admin somewhere for a news feed... or, if you > were a student, getting an account from your university news admin. I am still using 'rn'. An improvement over 'readnews', but until now I have seen nothing better. (And, no, no graphical interfaces for me, please. I have problems when I use the mouse too much.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: My victories get lost >There are quite a few posters who reply in my threads. Now I've >noticed that I trounce one poster and then another pops up and starts >yapping. Later some poster who got his ass kicked is back trotting >out the *same* crap. > That's your impression. In _fact_ you have never trounced > anyone here. David Ullrich is a ing piece of dog. I think it's funny that I can call a professor at Oklahoma State University a ing piece of dog knowing that he'll keep replying in my threads. You see, he has to keep replying pushing the same old lies. He's stuck. He's trapped in something that he can't get out of, so it doesn't matter what I call him, or what I say about him, he has to come back. You see I'm the person who has the correct math argument, so posters like David Ullrich or Arturo Magidin are *compelled* to reply out of fear that if they go away, then I'll get some people who'll pay attention to the truth. So David Ullrich, the math professor at Oklahoma State University, is demeaned by me as the piece of ing dog he is, and he *has* to keep coming back. James Harris === Subject: Re: Continuum > by C i ment continuum, as in CH. Goedel would be proud. But since I am not > a mathematician, I will take the proof with me to my grave. You can't know how much I approve of your well-mannered decision. Rick === Subject: Re: Cantor's diagonal number: why do we just ASSUME such a number exists? > Mike Deeth says... > >I have obtained empirical evidence that Cantor's diagonal number >does not exist, through a long thought experiment. The diagonal number isn't just one number. >You failed to see that the original post was an attempt to get into the >Guinness book of records: highest crank index of all times. I have >not actually counted the score, but it must be very high. Someone else said that, but I didn't know that he meant it literally. But yes, he goes right down the list of things not to do in the crackpot index. It's a significant accomplishment to be the world's foremost crackpot. -- === Subject: Re: elliptic integral of first kind > Up to here, you are correct (as far as I can tell). The integral > formula however is only correct if r' must be symmetric in r and r' it is the other way round if r Integrate(0,oo) J(rk) J(r'k) dk = 2/(pi r) K((r'/r)^2) if r' Integrate(0,oo) J(rk) J(r'k) dk = 2/(pi r') K((r/r')^2) if r'>r > Sorry, for beeing so late with this remark, but I only just found > time for it. Also forget about my comment about E(-4 r/...), it > was just a stupid error on my part. Hello Dr. Ulm, assuming that i have botched the integration, and the above integrals are in fact correct....the electric field still comes out as looking like e_r ~ K(a/r) for r>a e_r ~ K(r/a) for ra heading to r=a make sense to me intuitively. but the above also suggests that there is a singularity at on the rim when approaching it from any direction. why would it be the case for r Looking for a direct mathmatical comparison between the magnetic field > produced by an electronic current (i) and that of a fixed magnet of a > given strength. See Biot-Savart Law. Also look up solenoids. A tightly wound solenoid has a magnetic field that is exactly like that of a natural bar magnet. The workings of a magnetic are due to the spin of electrons in the atoms of the magnet and the fact that the spins are aligned over large regions of the magnet. All magnetism, as far as anybody knows, is do to electric charges in motion. No one has ever seen a magnetic monopole. === Subject: Re: Uncle Al is Sadistic . >How many people die in America due to economic pressure on its citizens >for every American that achieves wealth? An interesting question; but does it assume wealth is a zero-sum game? That assumption seems not to be historically supported. But since I am reading this in a math newsgroup, let's do some arithmetic. >Minimum wealth being defined >to sustain a yearly cost of $40KUS/yr for thirty years in ones >retirement. Wow! You certainly have a high threshhold for minimum wealth! I certainly hope you are not retiring in the US Social Security system; maximum payout there is about $15K (US)/yr. Your notion of minimum wealth would put most people at or under the minimum _even while they are at the height of the earning period_! Also note that you need not plan for 30 years of retirement. Save what you will and then, upon retirement, buy a life annuity from a reputable carrier. They will happily fund your 30 or more years in retirement with the money saved from the contracts purchased by all those who only lived for 10 years or less. More precisely, of every 100 people in a US age cohort, around 82 will reach their 65th birthday. About 63 will live another 10 years; about 34 of them will make it another 10 years after that; only 6 will see their 95th birthday and none (well, 0.2 of them) will be around only plan for less than 20 years of retirement. (Less, actually, now that the nominal retirement age is above 65.) >Approximately $2.7MUSis needed as a base. A game in the U.S. >is to achieve capital growth over a lifetime. A $60K salary over 40 >years is $2.6M. I have no idea where you got the figure of $2.7M . Thirty years of $40K/yr payouts costs $1.2M up front -- much less, if you put the money you do have at that point into a savings vehicle which pays any returns along the way. You mentioned taxes and so on too, but if $60K/yr is sufficient before you retire, how can it be insufficient after you retire? Even $60K/yr for 30 years only costs $1.8M. It seems to me that in order to get a $2.7M-up-front figure, you must postulate a combination of conditions something like this: that you need $60K/yr before taxes (or other costs) after retirement; that you earn nothing on your mega-dollar savings; that you continue to have the expense of saving for retirement (?!); and that inflation averages 2.7% per year. Would you describe yourself as a pessimist? BTW, you must be saving in an _extremely_ conservative retirement fund if you manage a net ROI of approximately 0.4% per annum across 4 decades. You called it a game; I think you need a new coach! I must emphasize that you're failing to illustrate the severity of the real situation if you assume a $60K salary is available during your saving-for-retirement phase. That's very much above average for an individual wage-earner in the US; using real data would show that the nationwide situation is indeed more grim than you indicate. > People in the US should be OUTRAGED. The social/economic/political > system is skewed for the wealthy to continue their wealth at the > expense of the majority in this system. Personally I'm inclined to agree, but you haven't substantiated it with this model. With accurate data, it's easy to show that most people will be able to save only very little for retirement. But it's much harder to lay the blame for this at the feet of any particular group; after all, life certainly comes with no guarantee that people should be able to enjoy a long retirement on their savings! dave === Subject: Re: Continuum > by C i ment continuum, as in CH. Goedel would be proud. But since I am not > a mathematician, I will take the proof with me to my grave. > You can't know how much I approve of your well-mannered decision. > Rick Lucky me, living in bliss ... === Subject: Re: [JSH] On the Rewriting of a Polynomial > To reprise James Harris' proof, or at least the first two steps > (I've included '*' so that GP/Pari works): > [begin excerpt] > 1. Let P(x) = 14706125 * x^3 - 900375 * x^2 - 17640 * x + 1078, where x is > in the ring of algebraic integers, notice that P(x) has a constant > term that is 1078. > 2. It can be shown that > P(x)= 7^2*(2401*x^3 - 147*x^2 + 3*x)*(5^3) - 3*(-1 + 49*x)*(5)*(7^2)+7^3 > where the *same* polynomial has been put in a form which allows a > factorization into non-polynomial factors so that I have > P(x) = (5*a_1(x) + 7)(5*a_2(x)+ 7)(5*a_3(x) + 7) > where the a's are roots of > Q(a) = a^3 + 3*(-1 + 49*x)*a^2 - 49*(2401*x^3 - 147*x^2 + 3*x). > [end excerpt] > Now, here's where I have a problem. How did Mr. Harris get to > this point? You can get there quickly by noticing that if you let a = -7/5 then -5^3Q(a=-7/5) = P(x) Similarly in the factored from Q(a)=(a-a_1)(a-a_2)(a-a_3) let a = -7/5 then -5^3Q(a) = (5a_1 + 7)(5a_2 + 7)(5a_3 + 7) = P(x) This confirms that the P(x) is a correct expression for the a_1, a_2 and a_3 and that they are indeed algebraic integers. The problem starts when James divides them by 7 and declares that because a_1/7 and a_2/7 are not necessarily algebraic integers, something is wrong. I can't work out what he means. For example P(x) = x(x + 1) is a multiple of 2 for all x integer therefore P(x)/2 is integer for all x integer P(x)/2 = x/2*(x + 1) (the only way it can go when you set x = 0, or something) But x/2 is not an integer for all x integer. Presumably there is something wrong with integers as well. HTH, Phil. === Subject: Re: Key core error argument, stepped out >> >> For crying out loud. You should stop cutting and pasting incorrect >> stuff. In fact, you should stop cutting and pasting, period. Why not >> post a link to your original post, instead, if all you are going to do >> is repeat it verbatim, ERRORS INCLUDED? >> >> [.snip.] >Why don't you off Arturo Magidin? > Three times I offered to do so if you told me to stop posting > replies. You declined to do so. > Are you doing so now, in your oh-so-courteous way? > Just say so: Do you want me to stop posting with comments about your > statements? Yes or no? You're ing kidding me!!! You mean you're asking *me* whether or not I want you to stop posting in my threads? Quit posting in my threads Arturo Magidin. off. James Harris === Subject: Re: Key core error argument, stepped out through two of the factors, so > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > > 300125 x^3 - 18375 x^2 - 360 x + 22 > from reverse use of the distributive property, which gives a constant > term coprime to 7, as required. Let's take a simpler example and see how your argument works. Let Q(x) = 7(25x^2 + 30x + 2) [1] = 7(x^2 + x)(5^2) + 7(x - 1)(5) + 7^2 and write Q(x) = (5a_1(x) + 7)(5a_2(x) + 7) [2] Now we mirror your construction and observe that we may take a_1(x) and a_2(x) to be roots of r(x) = a^2 - (x - 1)a + 7(x^2 + x) [3] We see that Q(0) = 14 = (7)(2), and r(0) = a^2 + a has roots a_1(0) = 0, a_2(0) = -1. We see that this assignment is consistent with your factorization [2], since Q(0) = (5(0) + 7)(5(-1) + 7) = (7)(2) [4] and we see that we can rewrite [4] as Q(0)/7 = (5(a_1(0)/7) + 1)(5a_2(0) + 7) = (1)(2) or, emphasizing the constant term in [1] we may write, letting b_2(0) = a_2(0) + 1, Q(0)/7 = (5(a_1(0)/7) + 1)(5b_2(0) + 2) = (1)(2) Now, however, you would have us write each factorization over the algebraic integers as Q(x)/7 = (5(a_1(x)/7) + 1)(5b_2(x) + 2) [5] regardless of the (rational integer) value of x. This constraint in [5] does indeed work in some cases. For example, if we take x = -1 we see that Q(-1) = -21 = (7)(-3) r(-1) = a^2 + 2a and we see that we may take a_1(-1) = 0, a_2(-1) = -2 and that doing so gives us the factorization Q(-1) = (5(0) + 7)(5(-2) + 7) and we see that Q(-1)/7 = (5(a_1(-1)/7) + 1)(5a_2(-1) + 7) = (1)(-3) The problem is that your restriction in [5] depends strongly on the polynomial r(x) used to define the terms a_1(x) and a_2(x). For x = 0, -1 the factorization in [5] worked because of the way r(x) splits into particular linear factors, but consider the case when x = 1. Now we have Q(1) = 399 = (7)(57) = (7)(3 * 19) r(1) = a^2 + 14 which gives us a_1(1) = sqrt(-14), a_2(1) = -sqrt(-14) and so we have the factorization Q(1) = (5 sqrt(-14) + 7)(-5 sqrt(-14) + 7) which is indeed equal to 399, as expected. However, it's immediately obvious that sqrt(-14) isn't divisible by 7 in the algebraic integers, so the factorization in [5] does not hold. Instead of distributing the value 7 as 7 in the first factor and 1 in the second, we have the distribution sqrt(7) in both factors, so we have the factorization in algebraic integers Q(1)/7 = [(5 sqrt(-14) + 7)/sqrt(7)] * [(-5 sqrt(-14) + 7)/sqrt(7)] In fact, your constraint on the way 7 (or 49, in the example you have been using) is split among the terms in your nonpolynomial factorization is valid only in a small number of cases for x. Rick === Subject: Re: Magnetism: compare magnetic fields | Looking for a direct mathmatical comparison between the magnetic field | produced by an electronic current (i) and that of a fixed magnet of a | given strength. Keyword: solenoid FrediFizzx === Subject: Derivatives II I'm reposting this because it didn't show up on my newsreader, only > Phil Holman grava .88 la saucisse et au marteau: > f(p) = q > where f(140) = 15000 > and f'(140) = -100 > R = pq > What is dR/dp (p=140) > I get (f'*p) + (f+f')*dp > = (-100*140) + (14900*1) > = -14000 + 14900 > = +900 > R = pf(p) > dR/dp = f(p) + pf'(p) > So dR/dp |p =140 = f(140) + 140*f'(140) > = 15000 - 14000 > = 1000 If we apply this in a practical example where p=price, q=quantity sold, R=revenue. If we increase the price from 140 to 141, we decrease the quantity sold from 15000 to 14900. This will increase the revenue by 900 and not 1000. I guess the problem here is with a changing derivative between a p=140 and p=141. This gave me a real problem and hence my hacked solution. With no definition of f(p) except for a single point the function could be defined in an infinite number of ways. Does anyone have any other comments about this. Phil Holman === Subject: Re: Future of mathematics >which do you claim? That mere humans really ARE that predictable that you can tell what a mehum is going to do or say even a decade in advance. === Subject: c-number Howdy everybody, I'm reading Peskin and Schroeder's Intro to QFT and in this book the author constantly refers to c-numbers... without saying what a c-number is. I couldn't find c-number on mathworld.wolfram.com or physicsworld.wolfram.com. Can someone please tell me what the hell a c-number is? adam === Subject: Re: Math dependency logic permission for an emailed response. That's stupid. Given that 7 is a NUMBER it can't just change, and > given that dividing P(x) by 49 *does* change it to 1, then 1 can't > change as a variable dependent on x. But x is a number too. And it can't change. > Well what is it then? I told you, it's a number. > Obviously x is a symbol that can stand in for LOTS of different > numbers, which is what makes algebra so powerful. x is a symbol that stands for whatever you like. You can't tell without more information; it might be an arbitrary set, a number, an ultrafilter, a topology, or whatnot. x might stand for a number in some context. In the context here, x stands for a number. Which means that x is a number. > On the other hand, 2 is just a number. 2 stands for a number: the number 2. Unlike x, 2 always stands for the same thing. We don't have to have that convention; it's purely a matter of convenience. We might use the symbol 2 for something completely different. Thomas === Subject: Re: Math dependency logic permission for an emailed response. > Ultimately my argument relies on numbers like 7 being NUMBERS, not > variables dependent on x, and on the distributive property. 7 is a number, but so is x. Neither can change (in a given context); each refers to exactly one number at a time. === Subject: Re: new way of describing ellipse for math > (snipped) If you use two rectangles, I believe you can generate a both a unique > ellipse or circle by connecting their verticies, so long as the > rectangles do not have any identical verticies. I suppose if you were > to take a limit case, you might be able to generate some equation that > would give the result you're looking for. Eg: > |-/----------------/-| > | / / | > |-/----------------/-| Connect the eight outer verticies, and you get an ellipse. As the > verticies converge on each other, you maintain the ellipse. If you > took the limit as V1 -> V1', V2 -> V2' etc, you might be able to > generate a single equation for the ellipse in terms of the verticies > of the rectangle(s). Also, I wonder Aaron, if your suggestion of 2 rectangles to make a > unique > ellipse. I wonder why the 4 corners of a solo rectangle is unable to > make > it unique. I do not see the mechanism for why 1 rectangle was not > sufficient. > Because the differences between the first and second rectangle provide the slopes at the points of contact. To put it another way - the relationship between the aspect ratios of the two rectangles determines the aspect ratio of the elipse. > Perhaps the answer is that with 2 rectangles the one rectangle > provides the > major axis and the other is the minor axis. So I really do not get > away from > axis. I suppose there is a Projective Geometry theorem then that would > say something to the effect that 8 points of the 8 corners of the 2 > rectangles > is the equivalency of a major axis and minor axis. Question: since a unique circle is formed from a rectangle that > circumscribes > the rectangle. Then a unique ellipse formed from 2 rectangles has 2 > unique > circles. What relationship do those 2 unique circles have with the > ellipse??? A.P. > My guess would be that the diameters of the circles correspond to the > major/minor axis. I think there will always be a major/minor axis in > some form for your problem, since they are part of what defines the > ellipse. You got it. Did Arch? === Subject: One algebra question Hi I am learning algebra by myself I am trying to do some problems but I can't solve this one could you please do me a favor to give me some help here is the question: Let f be a homomorphism defined on a finite group G and let H subset G a) show that |f(G):f(H)| divides |G:H| b) show that |f(H)| divides |H| === Subject: Re: Cantor's diagonal number: why do we just ASSUME such a number exists? permission for an emailed response. > On my webpage I have posted a proof of the 1-1 mapping, N -> R. However, as > a corrolary to my proof, I demonstrate that N itself is uncountable. It's > there for anyone to view, and unless someone can find a flaw in my proof, we > will have to agree that it is correct. The definition of countable is that a set can be in a 1-1 mapping with N. A set is uncountable if it is infinite and not countable. Since the identity mapping is a 1-1 mapping from N->N, N is countable. Thomas === Subject: Re: Geometry And Newtonian Mechanics of Action Device >OK, let us see Pythagores in Action.. >______________________________________________________ >Pythagores Theorem: >In a right angled triangle, the square of hypotenuse is equal to the >sum of the squares of the other two sides. >c^2 = a^2 + b^2 >______________________________________________________ >Now draw X, Y axis on paper. We refer origin as B instead of >conventional O. >Line AC of length c is on X axis and initially point C of line AC >coincides with origin B. Let us move or slide line AC along X axis by >distance d(where d<slides along Y axis in vertical direction. Now, we have a right angle >traingle ABC. >In this triangle AC = c, AB = a = c-d, BC = b >According to Pyathagores theorem, in this right angle triangle >c^2 = a^2 + b^2 >We replace a by c-d as AB = a = c-d >Hence c^2 = (c-d)^2 + b^2 >c^2 = c^2-2cd+d^2 + b^2 >We get, b^2 = 2cd-d^2 >****************** >b = sqrt[d(2c-d)] >****************** >This is the basic equation which controls this action device. Please >note that when, initially, we move line AC along X axis by very small >distance d, point C slides along Y axis and BC = b > d >Please tell me whether you have understood this or not. Yes I understand Abhi, you are speaking about this. http://dbarkertv.com/UPDATE.htm So why do you think your life is in danger? Why not just come forward. You can talk here. People here are just pretending they don't know what goes on in this world. We are all alright. Just like the song says. Cheaptrick-Surrender.mp3 Although I know at times, it seems like just being alright is not enough. At least that is what Tatu says in this song. Tatu - All The Things She Said.mp3 If you'd like to hear those songs now, you need but visit this site and download Kazaa Lite. Here is the page where you can choose the version of Kazaa Lite you want to download. http://home.hccnet.nl/h.edskes/mirror.htm I use this version http://le4.edskes.com/klitekpp210e.exe found on that page. But then you might want the french version. =*= Rick === Subject: Re: Key core error argument, stepped out As a stopped clock is right twice a day... : > HELL YEAH!!! Mathematicians have broken or screwed up terms all over : > the place, but tend to make ad hoc rules to handle them. That is overstating the case. Math is a very broad thing. There are a great many subfields. It should not surprise anyone that terms and concepts that are central in one subfield (no pun intended) are sufficiently analogous to concepts in some other subfield as to make the same word appropriate, but to leave the applicability less central. : > For instance, with coprime an ad hoc rule would be NOT to use the : > word with a field!!! Well, yes, precisely, THAT rule WOULD be ad hoc -- that's PRECISELY WHY nobody has adopted that rule. : Why? The definition is in a ring. As a field is also a ring, why should : it not be used in a field? This is a stupid question, Dik. As ANYbody who knows what YOU know knows, IN A FIELD, THE CONCEPT IS DEGENERATE. Almost ANY pair of things is coprime. It doesn't SAY ANYTHING ABOUT a pair of elements of a field to say they are coprime. Even if you decide to KEEP using the term, you wind up HAVING no use for it. But even when Dik is wrong, the person he is arguing against is much more wrong. : > The math world is spectacularly illogical and quirky, and uses lots of : > end-runs and ad hoc rules to cover up messes, which is why I say the : > situation is like Ptolemy's circles. No, really, it isn't. By definition, it isn't. There is nothing messy about the fact that the notion of coprimeness, defined for rings in general, becomes degenerate in certain special kinds of rings (e.g. fields). For crying out loud, ADDITION becomes degnerate when one of the addends is 0. That doesn't mean that we start disallowing expressions like 7+0. If 7 was what you wanted to say, then it's true, you wouldn't ever need 7+0 to do THAT, but you MIGHT need 7+0 as an intermediate step in a proof or calculation. Somebody just needs to stick to the math and just ing quit it with all these lame generalizations about the content of mathematicians' character. It is the content of their proofs that matters. -- --- It's difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America === Subject: Re: new way of describing ellipse for math > If you are looking for a unique elipse defined only by an inscribed > rectangle, you are out of luck. Four points arranged as the corners of > a rectangle define a whole family of elipses including one circle. > You have not read yesterday's suggestion of 2 inscribed rectangles. > I am not sure if 2 inscribed rectangles defines a unique ellipse or if it > even creates an ellipse. > Anyway, if it does then 9 points uniquely define a ellipse where 8 are the > corners of 2 rectangles and 1 point is what I call the *center of the ellipse*. > And in keeping with this method then 5 points uniquely define a circle > where 4 corners and the center of the circle which coincides with the center > of the square inscribed. Arch, It only takes two points to define a circle if one is the center, three if they lie on the circle. > An elipse is not a 'squashed` circle, it is a rotated one. > (Conic Sections - remember?) > In some meaningful sense a rectangle is a squashed square. Likewise, > a ellipse is a squashed circle and now I have to engineer and create the > mathematics to conform to that idea. > This is important for physics in that the Unified Force is one force > where all the other forces are broken symmetry (the squashing). In three > dimensions the circle becomes a sphere and the ellipse becomes ellipsoids > or lobes. Thus the hydrogen atom is the nearest to the unified force and > the uranium atom far away in broken symmetry. > Whether you like it or not, I believe that the relationship you are > looking for > does involve the rotation posted earlier. > I have not thrown out the rotation. I simply am saying that I can get at it > in a simplier method than rotation. An equivalent but much more simple > method. > Consider the circle with its inscribed square. (You didn't say you > had trouble with that.) If you rotate this figure about one side of > the square, and project it, you get a rectangle, (the square with one > set of opposite sides forshortened), and a unique circumscribed > elipse, (the circle rotated). > This is Projective Geometry. I think a simplier method, but equivalent > will be inscribed and circumscribed. I studied Projective Geometry for > one year in college and know that points can be equivalent to rotations. Didn't do too well eh? > Hint: if you wanted to graph this elipse, you could multiply one set > of ordinates from the graph of the circle with its center at the > projected intersection of the diagonals of the square by the cosine of > the angle of rotation. > This is High School stuff. If you have trouble with these concepts, > you are wasting your time in attempting to understand, much less > modify, string theory. > your mind is too restless with the creation of new ideas or the assembling of > a new method of looking at things and as a result, your mind bulks and then > tosses out these ad hominems. If you cannot stand new ideas or new approaches > then just do not ever respond to any of my posts and stick to your teaching > in schools, for you do not belong in any of my adventures which invariably seek > new ideas, new methods. > I have made this detour of ellipse,sphere,spheroid, ellipsoid because the original > intent of String theory in the 1960s was that the Euler Gamma Function seemed > to clarify some rules of the StrongNuclear Force, or the nucleus of atoms. And since nuclei are > symmetrical in spherical or ellipsoid directly implies that the mathematics of the geometry of nuclei > involve distorting spheres into ellipsoids. > Why the s orbital makes precisely p orbitals or d orbitals or f orbitals. So if we link up Schrodinger > Equation with Euler Gamma Function that broken symmetry > becomes squashing. > If the StrongNuclear force is really NuclearElectrons and that the StrongNuclear > force is paired to the WeakNuclear Force as a Coulomb Unification then this > ellipse defined by 2 rectangles where one rectangle is the StrongNuclear force and the other rectangle > is the WeakNuclear Force. > When all is said and done, I expect to link and tie Schrodinger Equation with > Euler Gamma Function with circle/sphere/ellipse/ellipsoid for the StrongNuclear > paired to the WeakNuclear Force. > Coulomb Unification of Forces in Physics: > Coulomb force, perfect and spheriod, region of existence is the > nucleus protons to the electrons > / > / > StrongNuclear to WeakNuclear, broken coulomb symmetry, and > region of existence is the nucleus of atoms > / > / > Gravity to Antigravity, again a coulomb force when combined and > region of existence is the electronic-space of atoms > Trouble with Stringtheory is that they never really applied it to the nuclear > region of atoms and to fully garner the understanding of the strongnuclear force, > and instead they tried to overblow stringtheory as some theory of everything. > Archimedes Plutonium, a_plutonium@hotmail.com > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies === Subject: Citation for a theorem in topology? I seem to have shown the following. It seems too fundamental not to be known. Can anyone supply a citation? Theorem: Suppose X is a closed subspace of the normal space Y and X is a subspace of the completely regular space Z. Then the amalgamated sum Y +_X Z is completely regular. Moreover both Y and Z embed as subspaces of it. === Subject: Re: A 'basic' topology question about interiors <3FA725C2.6040506@rutcor.rutgers.edu> NNTP-Posting-User: bbscott >>Of interest perhaps are these formulas: >>When U subset A subspace S >>cl_A (U) = A / cl_S (U) >>int_A (U) = A / int_S (SA / U) >Ben Scott > And thank *you* for using my preferred plural of formula! > -- > Stephen J. Herschkorn herschko@rutcor.rutgers.edu Of course. But y'know, 'you say formulas, I say formulae...' Ben Scott === Subject: Re: Key core error argument, stepped out > Using Dot's proof that the values of a polynomial with algebraic > integer coefficients are always divisible by an integer if and only if > each coefficient is a multiple of that integer (in the algebraic > integers) gives you that the statement is correct for polynomials in > A[x]. > I had not noticed Dot's theorem. The x^2 + x example seems to belie > it: for any integer x, x^2 + x is divisible by 2, but the coefficients > are not multiples of 2, and both of the coefficients are algebraic > integers. You must therefore be thinking that the domain is algebraic > integers as well. I think you are reasoning on cross-purposes. As I remember, Dot's theorem was about polynomials over the algebraic integers. As a polynomial over the integers, x^2 + x is always divisible by 2, but not as a polynomial over the algebraic integers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: The Mystery Of Mass (was: The need of geodesics in physics) >I don't know where you got that notion from. QT says nothing about m >changing discretely. Mass is quantized on a discrete spectrum in QFT, but the origin of the mass matrix and its spectrum is unknown. What *is* known only furthers the mystery: the mass operator doesn't commute with the SU(2) symmetry generators corresponding to the W and anti-W. The mystery only deepens, considering that [anti-]W can only interact with and right helicity states is relative. But interacting vs. not interacting can't be relative too! Therefore, the helicity states Which means, in turn, that their mass is not intrinsic, but dynamic in origin, arising from an interaction with an otherwise unknown or unseen/undiscovered field. The mass matrix, then, just describes Were it not for this extra kink, all the charge states of all the completely indistinguishable from one another. The mass matrix mixes between these groups, and the mass eigenstates for each charge state are actually mixtures across the 3 groups with 3 separate eigenvalues. The mystery deepens yet further when you step back and loop at the charge states. They form a pattern: they reside at the points of a 6 dimensional Cartesian lattice forming within it a 5-dimensional hypercube! Indeed, you can write down an orthonormal basis (X,A,B,C,D,E), and the 32 vertices are precisely those for the 32 combinations of vectors xX + (x-a)A + (x-b)B + (x-c)C + (x-d)D + (x-e)E for which a,b,c,d,e = +/- 1/2 with x undetermined (possibly with one set for each of the 3 groups). And the mystery deepens yet further when you consider what the W and anti-W actually do; and what the 6 (charged) gluons in SU(3) do. The W switches a (+a,-b) to a (-a,+b); anti-W the other way. The gluons switch a (+c,-d) to a (-c,+d); and the other 5 combinations involving c, d and e. Left and right are distinguished solely by one of the a or b, say, by +b vs. -b, with left being (+,-) and (-,+) and right being (+,+) or (-,-); so (+,-) & (+,+) are the left & right another. === Subject: Re: Graduate algebra book > You still haven't said what topics in algebra you consider advanced > enough to belong in a graduate algebra course. I'd be curious to see > the syllabus for an undergraduate introduction to abstract algebra > course which covered: > The thing here is that you seem to be equating undergraduate algebra > with introductory algebra. I would say that for *most* universities, > these two are not the same. The university I went to for undergrad (after > 3 courses in linear algebra) have SIX upper-year undergradute courses in > abstract algebra (granted, two of these are not intended for pure/applied > math majors, but still the other four courses cover all the topics you > listed earlier.) Well, I think the U.S. and other countries are wildly different in their structure of an undergraduate curriculum. I don't have a problem admitting that my university is not so terrific. We have 2 algebra courses which are available to undergraduates at my university, the more advanced of which is almost identical to the introductory course, albeit slightly more fast paced. It so happens that I _am_ an undergraduate and I am trying to get a better education than what my university offers to the typical student. What I have a problem with, is listening to the egos of those people who feel like they are superior to everyone else for no particular reason. All Mr. Chapman had to say was generally those topics are covered at the undergraduate level, instead of the snide remark 'thats graduate algebra?' Or no remark about it at all would have been fine too. I also clearly stated in my original post that I would have been finished with Hungerford by the time I started reading this new post, so another helpful response would have been if you're going to have finished reading hungerford, then perhaps what you want is not a first year graduate book, but something more advanced such as __________________. Actually, upon checking a few of the top universities' websites, I find that while the undergraduate curriculum generally does offer algebra courses which talk about galois theory and the topics I mentioned, they are generally at the near-graduate level. Thus, for any AVERAGE university, they will not be offered until the graduate level. And even at these better universities, not everyone takes every single course, and I imagine not everyone takes 5 or 6 semesters of algebra as an undergraduate. Should they take 5 semesters of analysis and 5 more of topology as well? how is anyone ever supposed to graduate? === Subject: Re: naive geometry questions >The other surface is the catenoid, formed by rotating a catenary, > Q: How do you make a catenoid? > A: Pull its tail. How do you make a hormone? -- === Subject: Re: [JSH] On the Rewriting of a Polynomial > Most readers will be able to see that Dik W. and Ghost ITM are cheating, > by using more than a handheld calculator. Well, I've never been wrong, > yet! Ah, you are also cheating. You are using more than a handheld calculator -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: My victories get lost >There are quite a few posters who reply in my threads. Now I've >noticed that I trounce one poster and then another pops up and starts >yapping. Later some poster who got his ass kicked is back trotting >out the *same* crap. > That's your impression. In _fact_ you have never trounced > anyone here. > David Ullrich is a ing piece of dog. > I think it's funny that I can call a professor at Oklahoma State > University a ing piece of dog knowing that he'll keep replying > in my threads. Mr. Harris, is there no level to which you will stoop to propagate your fierce lies? You are pathetic and you can continue repeating your lies and conspiracy theories, but the truth has and will always be against you. Go see a doctor! === Subject: Re: How did Euler determine Euler's Constant? > I know that : > Euler's constant = Lim (n-> infinity) [ Sum(i/j) from j=1 to n - ln(n)] > but, I can't see how one actually evaluates this relation to get the value > of Euler's constant. > MB Details are given in H. Goldstine, A History of Numerical Analysis from the 16th through the 19th Century, Springer, 1977, page 129. He used the Euler-Maclaurin formula (he had just derived it using a generating function approach) and found the value to 16 places. The 16th one was wrong. Not bad for a blind man. === Subject: Re: JSH: My victories get lost Yes it is true that every one of your victories has gotten lost. It is also true that every one of your victories has not gotten lost. It is also true that every one of your victories is the pope. It is also true that every one of your victories has any predicate you want including four wheel drive, but not as it happens, existence. - William Hughes (with a bit of help from Tom Stoppard) === Subject: Re: Determinant of a special matrix > Adj(Adj(A))=det(A)^(n-2) A >Or references if it is already proven by someone (which will > probably be the case, though I cannot find it). Google gives three hits on Adj(Adj(A))=det(A)^(n-2) A. Judging from your email address ( I assume nl stands for Netherlands?) you should be able to read this one easily :-) [PDF] D:DocsOnderwijsMatrixtheorieTent 02-05tent 02-05.dvi File Format: PDF/Adobe Acrobat - View as HTML ... c) Als n 3 en ?(A ) < n, dan geldt adj(adj(A )) = 0 (d) Als n = 2, dan geldt adj(adj(A )) = A (e) Voor n 3 geldt adj(adj(A )) = (det A ) n?2 A. Normering 1 ... www.win.tue.nl/~wscomalo/math-onderwijs-2F400/ tent-02-05.pdf - Similar pages === Subject: Re: Key Factorization Proof [...] Why did you post this to sci.math.num-analysis with not even a cross-post to sci.math? V. -- email: lastname at cs utk edu homepage: cs utk edu tilde lastname === Subject: Re: Key core error argument, stepped out ... > : Why? The definition is in a ring. As a field is also a ring, why should > : it not be used in a field? > This is a stupid question, Dik. As ANYbody who knows what YOU know knows, > IN A FIELD, THE CONCEPT IS DEGENERATE. Almost ANY pair of things is > coprime. It doesn't SAY ANYTHING ABOUT a pair of elements of a field > to say they are coprime. Even if you decide to KEEP using the term, > you wind up HAVING no use for it. > But even when Dik is wrong, the person he is arguing against is > much more wrong. Exactly what is wrong about what I said? (And pray, keep the caps-lock key in the unpressed position.) Did I say that every use would be degenerate? Is it wrong to use a degenrerate concept? if so, what is wrong about it? > : > The math world is spectacularly illogical and quirky, and uses lots of > : > end-runs and ad hoc rules to cover up messes, which is why I say the > : > situation is like Ptolemy's circles. > No, really, it isn't. By definition, it isn't. > There is nothing messy about the fact that the notion of > coprimeness, defined for rings in general, becomes degenerate > in certain special kinds of rings (e.g. fields). > For crying out loud, ADDITION becomes degnerate when one of the > addends is 0. That doesn't mean that we start disallowing > expressions like 7+0. If 7 was what you wanted to say, then it's > true, you wouldn't ever need 7+0 to do THAT, but you MIGHT need > 7+0 as an intermediate step in a proof or calculation. > Somebody just needs to stick to the math and just ing > quit it with all these lame generalizations about the content > of mathematicians' character. It is the content of their proofs > that matters. > -- > --- > It's difficult ... you need to be united to have any > strength, but internal issues have to be addressed. > --- E. Ray Lewis, on liberalism in America -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: I can't stand it anymore > not all Asians are equal. And the > Japanese thinks (may be not so much anymore) > they are superior to all other Asians. AHahahhaah.....ahahahaha.....easy babe, easy. This is a hilarious chapter in the annals of human behavior..... Serious Japanese insistence about their superiority reaches back into the mist of time.........ever since the two peoples diverged. According to the Japanese there was that Japanese Princess who ed with and got impregnated by a monkey. Their offspring became the m/patriarch of all the Chinese to come.............meanwhile on the Asian mainland the lore of yore insists that in the grey mist of time there existed a horny Chinese Princess that ed with and got impregnated by a monkey. Their off spring became the m/patriarch of all the Japanese to come.............Meanwhile back in those times of bygone wonders the Jewish proboscis superiority was established by their claim to be God's chosen people, to which the Pope's and his troops later on objected to and retorted that God had given this torch to Jesus and his Xian Church was the superior gig now, which left the Yiddles in the dust. So, while in the West the good folks established exclusive superiority over others by telling their God(s) what to do, the much more practically inclined Asians achieved the same goal by the insistence on some gross bestiality, a threat that was suffiecnt for them to get to the top of the pecking order. Ain't it a wonderful world..........AHhahahahaha.....ahahaahnson PS: somebody bring forth such lores about other cultures' orginal attempts and explanations which were needed to give them their raison d'etre at the top. === Subject: Re: Conversion of red phosphorus to white phosphorus Nice, but quasi green bob suffers from Plumbum paranoia of legendary dimensions which forced him to hallucinate that his NG comments will enhance mankind's survival against the doomsday prospects of lead in message which he delivered under great inner torment and psychotic fear, because he says... > I wonder if you are a danger to others at times. His comment here is embedded in a master piece of a post, that teeters on the intellectual niveau of a petrock, the belletristic values of an ameba and his understanding of the field with the depth comparable to that of a flatlander's. His efforts nevertheless elicit a faint chuckle, at best, but, me being magnanimous will bless him with a roaring AHAhahahhaha.......ahahahahanson === Subject: Re: Ring problem Justin Young grava .88 la saucisse et au marteau: > how about this one? > (-a)^2 = (-a) > (-a)^2 = (-a)(-a) = a^2 = a > Therefore, a = (-a), or a+a = 0 Actually, that's false, because you say that (-1)(-1) = 1. It should be -1. -- Nicolas === Subject: Re: Ring problem > Justin Young grava .88 la saucisse et au marteau: > how about this one? > (-a)^2 = (-a) > (-a)^2 = (-a)(-a) = a^2 = a > Therefore, a = (-a), or a+a = 0 > Actually, that's false, because you say that (-1)(-1) = 1. It should be > -1. He didn't say that (-1)(-1) = 1. In fact, 1 may not even be in the ring. (-a)(-a) = a^2 is true in *all* rings, not just the ring mentioned in the original problem. -- Bill Hale === Subject: Re: Ring problem William Hale grava .88 la saucisse et au marteau: > (-a)(-a) = a^2 is true in *all* rings, not just the ring mentioned > in the original problem. -- Nicolas, who didn't use rings nor fields since 3 years. === Subject: Re: Repeat: White Noise Dilemma >>I find it odd (2n+1) that no one has responded to this. I'm guessing >>that either a) everyone thinks it is a homework problem (it is not), >>or b) no one knows the answer. Come on, all you big strong mathematicians - >>certainly you can answer a little ol' question like this, can't you? ... >>Here is a question that has had me in a quandary for several >>years now. >>Let X(t) be a zero-mean IID process with variance of >>sigma^2. Now we know that since this function is >>IID, it has a white PSD and therefore its autocorrelation >>function at lag 0, R_XX(0), should be b*delta(tau), where >>delta(tau) is the usual Dirac delta function and b is some >>constant. > What are you using as a definition of PSD (power spectrum density)? The Wiener-Khinchine theorem states that the PSD of a wide-sense stationary random process, Sxx(w), is the Fourier transform of its autocorrelation function Rxx(tau) = E[X(t)X(t+tau)]. > This would seem to be a problem for a process such as X(t) in which > t--> X(t) is not measurable, or continuous in L^2. What do you mean by t-->X(t)? -- % Randy Yates % ...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall. %%%% % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr === Subject: Re: Magnetism: compare magnetic fields > Looking for a direct mathmatical comparison between the magnetic field > produced by an electronic current (i) and that of a fixed magnet of a > given strength. See: http://scienceworld.wolfram.com/physics/Magnetism.html Note the differences in o diamagnetism o Paramagnetism Ferromagnetism http://scienceworld.wolfram.com/physics/Ferromagnetism.html The development of extremely strong magnetic properties in certain materials which occurs when magnetic domains (regions at most 1 mm in dimension) become aligned in the absence of an applied field, below a temperature known as the Curie temperature. The net magnetization depends on the magnetic history (the hysteresis effect). Above the Curie temperature, these materials become paramagnetic. Iron, nickel, cobalt, and gadolinium are ferromagnetic at room temperature. Ferromagnetism is believed to be caused by magnetic fields generated by the electrons' spins in combination with a mechanism known as exchange coupling, which aligns all the spins in each magnetic domain. === Subject: Re: elliptic integral of first kind > let n(r') = constant =1. and lets evaluate this in the plane i.e. z=0 > V(r,0) = 1/2 Integrate(0,oo) J(rk)*J(r'k) dk Integrate(0,a) r'dr' > the integral over the bessel functions (i think) is > V(r,0) = 1/(pi*r) * Integrate(0,a) E((r'/r)^2) r' dr' > Up to here, you are correct (as far as I can tell). The integral > formula however is only correct if r' must be symmetric in r and r' it is the other way round if r Integrate(0,oo) J(rk) J(r'k) dk = 2/(pi r) K((r'/r)^2) if r' Integrate(0,oo) J(rk) J(r'k) dk = 2/(pi r') K((r/r')^2) if r'>r dr' but i meant Integrate(0,a) K((r'/r)^2) r' dr' I have to slow down and double check everything when i post on these groups...i seem to rush a bit. i appologise. i did actually use K not E in all my calculations from that point on and the rest would seem correct. i have gone over it all again and i do get what i originaly posted as... V(r,0) = 1/(pi*r)* [ r^2 *E((a/r)^2) + (a-r)*(a+r)K((a/r)^2)] electric field is = -1/r*dV/dr e(r) = [ K((a/r)^2) - E((a/r)^2) ] / pi The next thing is to find out what happens to E for the interior of the disc! I dont think its as simple as swapping r' and r. I *think* we have to break it up into two integrals over two domains: i.e. we break up the disc into two domains: 1) points r' that belong to [0,r) 2) points r' that belong to [r,a] if this is the case we have V(r,0) = Integrate(0,r) K((r/r')^2) dr' + 1 Integrate(r,a) K((r'/r)^2) r' dr' ok but the first integral does not converge over the domain of integration...so lets offset it by a small positive constant b to get V(r,0) =Integrate(0,r-b) K((r/r')^2) dr' + Integrate(r+b,a) K((r'/r)^2) r' dr' this avoids the singularity, while still maintaining the applicability of both forms under the integrals for the given domains. at the end we let b->0 Integrating this we have V(r,0) = r^2 * E( (b-r)^2 / r^2 ) + a*E((r/a)^2) - (b+r)*E( r^2/(b+r)^2) + b*(b-2*r)*K((b-r)^2/r^2) letting b->0 im not sure! i dont know what happens in the limit of z*K(z -1) for z->0 i am assuming we cannot discount it...and i am assuming that it probably blows up!. difficult. im not sure this is the right way to get at potential for the domain of the disc. cheers moth === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >> accelerated beyond the operating speed of the accelerating fields, ie at 'c'. I >> hope it might also produce a relationship that is equivalent to mass >> 'appearing' to increase with velocity by gamma.) >Please define what you mean by the operating speed of an >accelerating field. I've been in physics research for over 20 >years, and I've never heard that term. >Do you mean phase velocity, group velocity, how fast the >operators turn the knobs, what? >*static* fields, while most RF acclerating structures use >standing waves, so your point is dead on arrival. >Why are you blathering on about jets of water and ping pong >balls? They are nothing like the acceleration of an EM >field. Even if they were, what you're describing is a rocket, >and a rocket can easily exceed the velocity of it's propellant. >If you'd ever taken a physics class, you'd know that the >final speed of a rocket starting at rest is (in the >absense of gravity): > v = v_0 * ln(m_initial/m_final) > Here the jet speed is constant wrt the rocket not the ground. It is NOT the > same problem. OK, so you have something which is neither a model for a rocket NOR an EM field. Of what use is that supposed to be, exactly? >where v_0 is the propellant (or water or ping pong ball) velocity, >so as long as that mass ratio is greater than e, you're point- >even though it was flawed to begin with - is *still* dead on arrival. >Maxwell's Equations and the Lorentz Force Equation have >been around for a very long time, and no one's ever found >a flaw with them. If you're going to talk about accelerating >fields, please do so in the proper language - or is it >that you *can't*? >Just because you think the Earth is flat doesn't stop >other from sailing around it. Find a hobby you can understand. > Oh what an obvious genius we have here! > If the x/t curve for my 'object' turns out to be about the same as the one for > a charge in a constant field, your conceited smirk might fade away. But of course, since you can't even do *this* simple math, you'll never know, will you? Grown-ups deal with it all the time, and it's nothing involving ping pong balls or water or any such nonsense. If you can't handle math, you can't handle math, but don't pretend there's anything profound about that except your ignorance. -E > Henri Wilson. > See why relativity is wrong: > http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: Derivatives Phil Holman grava .88 la saucisse et au marteau: > f(p) = q > where f(140) = 15000 > and f'(140) = -100 R = pq > If we apply this practically. p=price, q=quantity sold and R=revenue. > So if we increase the price from 140 to 141, we decrease the quantity > sold from 15000 to 14900. This will increase the revenue by 900 and > not 1000. I guess the problem here is with a changing derivative > between a p=140 and p=141. This gave me a real problem and hence my > hacked solution. Do you have any other comments about this. Actually, you're applying a discrete derivative, which is not the same thing. For you, dR/dp = R(p+1) - R(p) where f(p+1) = f(p)+ 1*f'(p). In this case, your formula is right and we have dR/dp = (p+1)(f(p)+f'(p)) - pf(p) = f(p) + (p+1)f'(p) = 15000 + 141*(-100) = 900 But you have to keep in mind that the discrete derivative isn't the same thing as the true derivative. Furthermore, in your case, you're forced to use an approximation of f(p+1) using the first order at p (I don't know if this is the way to say it in English, but I think it is understandable). -- Nicolas === Subject: Re: Unit interval homemorphic to Circle help >If I = [0,1], and I / ~ is the quotient space of I obtained by identifying 0 >and 1, then the circle S^1 is homeomorphic to I / ~. >This is intuitively clear, but how can I prove this? >I think I want to show that p : I ---> S^1 given by p(x) = e^(2ix*pi) is a >quotient map. > That's exactly what you must show. > In this case it is quite easy (fortunately). > The map p is already continuous, as I and S^1 are compact metric the map > p is > automatically closed (i.e., if A is closed then so is p[A]). Why does a continuous map from a compact space to a compact space necessarily have to be a closed map? Is this what you are implying? > This implies that p is a quotient map. > KP > -- > E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI > PHONE: +31-15-2784572 TU Delft > FAX: +31-15-2786178 Postbus 5031 > URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft > the Netherlands === Subject: Re: Ring problem >> how about this one? >> (-a)^2 = (-a) >> (-a)^2 = (-a)(-a) = a^2 = a >> Therefore, a = (-a), or a+a = 0 > Actually, that's false, because you say that (-1)(-1) = 1. > It should be -1. No. (-a)(-b) = ab in any Ring R (with 1) due to the Law of Signs: ----------- a b + a(-b) + (-a)(-b) since over/underlined = 0 ---------------- by the distributive law => a b equals (-a)(-b) -Bill Dubuque === Subject: Re: Key core error argument, stepped out [cut] > : Why? The definition is in a ring. As a field is also a ring, why should > : it not be used in a field? > This is a stupid question, Dik. As ANYbody who knows what YOU know knows, > IN A FIELD, THE CONCEPT IS DEGENERATE. Almost ANY pair of things is > coprime. It doesn't SAY ANYTHING ABOUT a pair of elements of a field > to say they are coprime. Even if you decide to KEEP using the term, > you wind up HAVING no use for it. I don't agree with your position here. (Actually, from your other comments that I cut, you don't seem to be consistant maintaining the above position.) I don't agree with your position (and others maintaining your position also) because there may be some general theorem proved already that has one of the pre-conditions that a certain pair of elements are coprime, which theorem you wish to apply to a certain type of field that also meets the other conditions of the given theorem, so that then you can use the conclusion of that theorem in some proof you are working on for that type of field. I can't think off-hand of such a theorem for coprime, but here follows a similar example. The concept of unit is degenerate in a field, as already mentioned. But, you can use the concept of integral over a ring applied to algebraic over field to obtain the less general result that the sum and product of two algebraic numbers are algebraic numbers. A second example might be a polynomial ring over a UFD is a UFD, but applied to a field. This may not be a good example: first, the result applied to a field might be already proved first; and second, I believe the proof for UFD works in quotient field anyway. -- Bill Hale === Subject: Re: Bible 1, Darwin 0! And we are talking pure archeology? > Forbidden archeology by Cremo, Thompson! Like the name says: Forbidden I urge you not to read it! > Have you been to a museum lately? They have these things call Dinosaurs > that lived millions of years ago. Don't you think if the bible is the > word > of the lord that the good lord would have maybe warned us about gigantic > carnivorous creatures roaming the earth! Afterall, he did mention the > measly little snake. > Lurch > The argument from my Southern Baptist Preacher relatives in > Kentucky ( I have several) say that the world was created complete > with the bones and other found elements of the historical record created > intact to give a old construct when it was created 6000 years ago.... > And I cannot dispute it... Because any old stuff sticking out of the ground > was made by the hand of god sticking out of the ground and any test only > confirms gods ability to set a good stage for mans little drama to play > out.... > We are but god theatrical group and nothing more than entertainment for > his and his kids enjoyment in the evening, them throwing in an occasional > ELE to start the next act....... That's why it's important to note that Genesis contradicts Genesis. If you're looking for facts, the Bible is bull. Mark Folsom === Subject: Ex(~x=x), counterpart theory, and contingent identity In another post I suggested that John Correy's ideas about non-reflexive identity might have a rational formulation with respect to something called counterpart theory. (I think categoreal would be the right term here--in my humble opinion, the apparent self-contradictions associated with John's intuitions arise from vagueness and ambiguity associated with standard presuppositions rather than irrational error on his part. For the record, Langholm's investigation of determinability and indeterminability in first-order contexts includes incoherent formulas. Exclusion negation is not informationally well-behaved.) The link, http://www.sussex.ac.uk//Users/muralir/kct_final.pdf <>((x=x) & ~(x=x)) is discussed as well as what is actually done in counterpart theory to exclude such an incoherent result. :-) mitch === Subject: how to identify data from different distributions? Hi sorry to bother you guys. This problem suddently comes into my mind. Given two data sets from two different distributions, if I mixed them together, i.e., you don't know which point comes from which distribution. Is it possible to use some learning or statistical algorithms to separate the two data set out? For example, X=(x1,x2,...,xn) ~ Gaussia Y=(y1,y2,...,yn) ~ Uniform Given (x1, y1, x2,x3,y2,x4,y3,...), can you separate out X and Y? === Subject: Re: I can't stand it anymore > I have been biting my tongue about the IQ test but I can't any more. > Which IQ test is this? Whichever one people were referrign to in sci.chem, sci.phy, and sci.math. I never checked into the those tetss. All I heard about IQ test is from the media when that book bell curve came out and from people using the term Asian. > Martin Hogbin === Subject: Re: how to identify data from different distributions? roy grava .88 la saucisse et au marteau: > For example, > X=(x1,x2,...,xn) ~ Gaussia > Y=(y1,y2,...,yn) ~ Uniform > Given (x1, y1, x2,x3,y2,x4,y3,...), can you separate out X and Y? Try looking for Kernel PCA on Google. Basically, you learn the eigenfunctions of the covariance matrix with respect to a kernel K. Every eigenfunction will correspond to a distribution (or am I mistaken?) -- Nicolas === Subject: Re: I can't stand it anymore > I have been biting my tongue about the IQ test but I can't any more. How reliable is a test that use the term Asian to represent the most > diverse of ethnic and cultural groups? [off-topic crapsnip] Apparently you've just failed another test -- the one > that measures your ability to choose suitable newgroups > for a given topic, and vice versa. > Or may be math, physics, and chemistry are the subjects Amanda got her C-'s. :-) I was referring to people mentioning IQ tests in discussion about determinging intelligence of different race in all these 3 groups. But, how could you possibly realized that? After all, you got F in all these 3 subjects. === Subject: Re: I can't stand it anymore > amanda replied: > I have been biting my tongue about the IQ test but I can't any more. How reliable is a test that use the term Asian to represent the most > diverse of ethnic and cultural groups? > Are you Asian? Yes. Not oriental though but grew up in that region. > Rich === Subject: Re: JSH: Factorization P(x) = 2(x(x+1)/2 + 1) > So in ANY ring that contains the integers, 7 and 22 are coprime > (under either definition). >> That's true in any ring R since R contains a homomorphic image of Z, >> Z -> Z*1_R, i.e. any ring is a Z-algebra. But any ring homomorphism >> must preserve the relation 22 - 3(7) = 1. > Hmmm... Only if you assume that ring morphisms map 1 to 1, which > is not necessarily a given either. Even assuming rings have a 1, > the zero map is usually considered a valid homomorphism, > and your conclusion would be incorrect there. For Rings (with 1, as I assume above) ring morphisms must preserve 1, so the zero map is not a morphism of Rings with 1. If, as you claim, one considered the zero map as a valid homomorphism of Rings with 1 then basic theorems on rings would fail, e.g. the image of the zero morphism would fail to be a subring (except if the target ring is 0). Therefore my above quoted statement is in fact correct as written. >> By the way, as I mentioned in a prior post [1], coprime is >> a highly overloaded term whose meaning depends upon context. >> JSH is using one of the most common definitions and it is >> incorrect to criticize him for that. > I will quibble that what is most common depends on context as well. > Most ring theorists I know would object to using the definition > depending on common divisors, since to them 'prime' refers to ideals, > almost never to elements; and most number theorists would certainly > disagree that the definition via common divisors is 'the most common' > (for the latter the definition ->is<- invariably related to ideals, > never to elements). This issue is not what _should_ be the proper definition but, rather, what _is_ past and current usage. The fact of the matter is that due to the way this and related ideas evolved, the denotation of coprime was - and still is - highly overloaded. Thus it is best to explicitly specify its meaning in any context where possible ambiguity exists. > To me, and particularly given that JSH's work is taking place in > subrings of the ring of all algebraic integers, it seems that the > most common usage from algebraic number theory should prevail. But often such subrings are of infinite degree over Q and are not Dedekind domains. Therefore they lie outside the scope of classical algebraic number theory. Hence common usage may not even apply. -Bill Dubuque === Subject: any info on a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n Hi anybody could please give me some info on a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n it looks like geometric and/or binomial series but nothing is clear (to me). === Subject: Re: c-number > Howdy everybody, > I'm reading Peskin and Schroeder's Intro to QFT and in this > book the author constantly refers to c-numbers... without > saying what a c-number is. I couldn't find c-number > on mathworld.wolfram.com or physicsworld.wolfram.com. > Can someone please tell me what the hell a c-number is? > adam I haven't read Peskin and Schroeder but if their usage of the term c-number is the same as everyone else's usage then they are referring to commuting variables. For example, Bryce DeWitt introduces the concepts of c-numbers (commuting variables) and a-numbers (anticommuting variables) in his book Supermanifolds. If a,b are c-numbers then a*b = b*a while if x,y are a-numbers we have x*y = -y*x. More generally, denoting the parity of an element x of a Z2-graded algebra by |x| we have x*y = (-1)^{|x|.|y|}*y*x, where |x| = 0 (resp. 1) if x is a c-number (resp. a-number). davidoff === Subject: Re: any info on a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n GR56 > a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n > it looks like geometric and/or binomial series but > nothing is clear (to me). The ratio of each term to the next is a/b, which is independant of n, so it's a geometric series. If the sum is s, we get (b-a)s = b^(n+1) - a ^(n+1) and therefore a formula for s. Equally good is (1 - a/b)s = etc. LH === Subject: Re: Gaussian Continued Fractions > Conjugates of alpha have the same norm; is there always a conjugate of > alpha with purely positive coefficients? If so, then one could take > the algebraic integer nearest zero in the cell alpha lies in. This > guarantees that the coefficients remain positive Oops! The coefficients are all positive on the remainder, but certainly not its inverse. I haven't worked out if there's always some integer that gives all positive coefficients on the inverse of the remainder while keeping the magnitude of the remainder within some bound. Another definition of positive would be to consider those numbers with positive real part; I think this is preserved with the definition of greatest integer above. Mike Stay === Subject: Re: RFI: Anybody know if this is true/known? > yet] If X is odd, X^2 = 1 mod 4; If X is even, X^2 = 0 mod 4 If all 3 of a,b,c are odd then d^2 = 3 mod 4 (impossible) If 2 of a,b,c are odd, then d^2 = 2 mod 4 (impossible) -Tralfaz > There exist no solutions to: > a^2+b^2+c^2=d^2 with > gcd(a,b)=gcd(b,c)=gcd(c,a)=1 > If a, b, c, d are integers and > a^2 + b^2 + c^2 = d^2 > then at least two of a, b, c are even. > -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html > Needless to say, I had the last laugh. > Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: I can't stand it anymore : Nope. You favor the excellent and tolerate the mediocre. Success and : failure are not indistinguishable. Well, it's pretty obvious to us which side of that line *you're* on. For how many years have you been promising us kg-sized diamonds? Considering that you cannot even figure out how to post to usenet, it doesn't come as a big surprise to me that you are a failure in the more important things in life, e.g., being able to tell the difference between the truth and a lie. ----- Richard Schultz schultr@mail.biu.ac.il Department of Chemistry, Bar-Ilan University, Ramat-Gan, Israel Opinions expressed are mine alone, and not those of Bar-Ilan University ----- . . . for while he was not dumber than an ox, he was not any smarter. -- James Thurber, _My Life and Hard Times_ === Subject: A NOTy problem. G'day G'day Folks, I have been puzzling over what ought to be a simple logic problem but am at a loss to come up with a method. A crime is committed by one of the following; Alan, Bob, Chris or Dave. Each makes a statement to the police but three of them lie. Alan says, I didn't do it. Bob says, Alan is lying. Chris says,Bob is lying. Dave says, Bob did it. Usually one can solve such problems with a grid L. N. I. G. A. B. C. D. Where A. B. C. D represent their names and L. = lying, N. = not_lying, I. = innocent, G. = guilty. Somehow that doesn't seem to get anywhere is this case. Maybe it is because N = NOT(lying) and G = NOT(innocent) and one has to use another method say Boolean logic or Karnaugh maps. I am less interested in knowing the answer to the puzzle than understanding why it is different from the routine logic puzzles. Best wishes, -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: UFO Bogus Physics what use do you get from these postings, when you never reply to your critics?... do you think that we're going to drop our gaurd, letting go of Eternal Vigilance? Ode to the Reichsaucer?... or can it all be ascribed to Puthoff's remote viewing research? I cannot imagine the repartee to follow your address to the Austin Powers Society for Science and Lit.! > http://www.salon.com/books/review/2002/08/05/zero_gravity/ > I did some investigating and interviewed retired senior USG > executives > who laid the Nazi flying saucer (disguised as UFOs) claim (a la > Cook's book) to rest - it didn't exist and it didn't fly. > erased me from his book. That told me he was not intellectually honest, > but had some hidden non-scientific agenda. I > find this disturbing since he is associated with Jane's Defence Weekly --ils duces d'Enron! === Subject: Re: Continuum > by C i ment continuum, as in CH. Goedel would be proud. But since I am not > a mathematician, I will take the proof with me to my grave. Or you could write an SMS saying you have found A Marvelous Proof, but that there is not enough characters available in the message to give any details... Then for the next 300 years, mathematicians will try to re-create your proof. Of course, it all depends on your reputation as a mathematician...... -Michael. === Subject: Re: Bible 1, Darwin 0! And we are talking pure archeology? Forbidden archeology by Cremo, Thompson! Like the name says: Forbidden I urge you not to read it! Have you been to a museum lately? They have these things call Dinosaurs > that lived millions of years ago. Don't you think if the bible is the > word > of the lord that the good lord would have maybe warned us about gigantic > carnivorous creatures roaming the earth! Afterall, he did mention the > measly little snake. > Lurch The argument from my Southern Baptist Preacher relatives in > Kentucky ( I have several) say that the world was created complete > with the bones and other found elements of the historical record created > intact to give a old construct when it was created 6000 years ago.... > And I cannot dispute it... Because any old stuff sticking out of the > ground > was made by the hand of god sticking out of the ground and any test only > confirms gods ability to set a good stage for mans little drama to play > out.... > We are but god theatrical group and nothing more than entertainment for > his and his kids enjoyment in the evening, them throwing in an occasional > ELE to start the next act....... > That's why it's important to note that Genesis contradicts Genesis. If > you're looking for facts, the Bible is bull. > Mark Folsom I don't agree it's totally bull.. While I don't in anyway think there is anything like god as constructed in its words it still has some historical value on both a sociological standpoint and in the study of physiology with a bit of geography to boot. Some is good data but must be considered as a man made group of many writings put together by committee ( mainly a king) for a purpose. But within its text are references that can be verified and are worth noting. While the B'Levers will point to the occasional reference of a verifiable event, person or subject as proof that the god construct is valid, they are mistaken. As with any bit of prose, parts may be valid and parts may be complete fabrications and parts may be incorrect interpretations of valid parts. Paul R. Mays ---------------------------------------------------------------------------- - Some where within the Quantum State Http://Paul.Mays.Com/story.html http://paul.mays.com/mayday.html http://paul.mays.com/rainy.html All truth passes through three stages: First, it is ridiculed Second, it is violently opposed Third, it is accepted as self-evident. - Arthur Schopenhauer === Subject: Re: A NOTy problem. > I am less interested in knowing the answer to the puzzle than > understanding why it is different from the routine logic puzzles. I am not familiar with the routine logic puzzles, so I don't know why its solution would be different. However, it seems to be straigtforward to solve. You did not make use of the information that exactly three are lying. I would guess that the routine puzzle would be that exactly three are telling the truth. The same method can be used to solved both types of problems. This method must be a routine method to solve puzzles like this. -- Bill Hale === Subject: Re: Cantor's diagonal number: why do we just ASSUME such a number exists? > I am of the firm belief that you and the author of this thread are one and > the same (and possibility just an alias of JSH, which would explain his > immaturity). Both of you sign your name in the same way and right your age > afterwards. Both of you refer to proofs but do not provide links to them > (Charlie, neither you nor Justin have websites with your name). You both > sign with different names than the email address states. Lastly, both of > you are wrong. You cannot state that the natural numbers are uncountable. > By definition, a countable set is that of the same cardinality as the > naturals. By the way, Nathan how have you been through University at age > 11, and Charlie, how have you finished law school by age 9? > Steven Hey, I just assumed it was a big joke. Did you notice how the OP scored absolutely top marks in the crackpot test, by satisfying *all* criteria? Not a coincidence, me thinks.... -Michael. === Subject: Re: Classic wave equation vs. Klein-Gordon >>The ordinary one-dimensional wave equation may be read to say that the >>tension creates a restorative force proportional to the curvature >>(second derivative), which in turn produces an acceleration. >>The one-dimensional Klein-Gordon equation can be read the same way, >>except that we have added a force term proportional direction to the >>displacement -- the string is joined to its equilibrium position by >>springs. >>Comment. >The way the Klein-Gordon equation is usually derived (in quantum >theory) is by taking the relativistic formula E^2 = (pc)^2 + >(mc^2)^2 and converting it to an operator equation by replacing E with >i hbar d/dt and p with -i hbar d/dx, so you end up with a second order >diff eq in x and t in the form of a classical wave equation with an >additional term in the form of a mass-related constant times the wave >function. Write down the equations of motion for a bunch of point masses joined by springs... then take the continuum limit and you end up with the wave equation (the nearest neighbor terms due to the springs turn into spacial dirivatives) Write down the equations of motion for a bunch of point masses joined by springs and also held in place by zero length springs .. then take the continuum limit and you end up with the K-G equation (the nearest neighbor terms due to the springs turn into spacial dirivaties and the zero length spring terms turn into the mass term.) If you need more info you should find a book on classical field theory... or the first chapter of a book on quantum field theory. adam === Subject: Re: How to find probability mass function of this? >>Let U and V be independent random variables each with Uniform(0,1) >>distribution. Determine the joint density of X = maj(U,V) and Y= U+V. >> >I get this: >P(X= (1/2)(ab)^2 And that answer is wrong. >since: >P(UP(U1? Care to try again before we provide a solution? Do you know about the use of conditional probability? >How do you like this solution? Please feel more than free to point out >any eventuall flaws. I think this part of mathematics is hilariously >interesting. I really enjoy going into it. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Sets before logic > Random thoughts on creating a theory of sets prior to a theory of > propositions and quantifiers: > Let's start with the empty set, 0, and logical identity, =, then we can > define T, for true, by > T =def 0 = 0 need a reduction rule, how do we know {{},{{}}} = {{{}},{}} is T. does (0 = 0) = T, or (0 = 0) -> T ? The functional meaning of equals and equivalent is 'can be substituted with', a kind of transformation check, or a partition or branch with equivalent formula in search space. Herc === Subject: Re: How to define a function to be smooth? >Hey all >When we say a function f(t) is smooth, does this mean that >f has infinite differentials with respect to t? >Or any other formal definition on this? IIRC, in my undergraduate days, one professor (W. Gottschalk) defined smooth to mean having continuous nth derviatives for sufficiently large n, where sufficiently large depended on context. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: c-number : : Howdy everybody, : : I'm reading Peskin and Schroeder's Intro to QFT and in this : book the author constantly refers to c-numbers... without : saying what a c-number is. I couldn't find c-number : on mathworld.wolfram.com or physicsworld.wolfram.com. : : Can someone please tell me what the hell a c-number is? I think it means a classical number, as opposed to a q-number (quantum number). : : : adam : === Subject: Re: Skeptickal Inquirer UFO > I had mentioned yesterday that I believed that since major > contact appeared to take place just after Hiroshima, > that these sorts of things were a beacon, inviting > investigation. > Taking this further, we have every reason to believe that > some sort of galactic government then intervened on earth, > and essentially took control of the earth. Covertly, > and not pushing any overt agenda, other than to > slowly and methodically assist mankind in dealing with > such tools of destruction. To bring man to the point > where he was capable of handling the technology without > destroying the earth. > That would be the best case scenario. > It would appear however, that their mandate was quite > strict in that they could not necessarily interfere in > the affairs of men, otherwise we would certainly not have > seen subsequent wars and our civilization would not be where > it is today. We would expect a more utopian society based > on intelligence, rather than the continued power struggles > we see. > However, people, are just people, and even under threat > of death, or sworn to secrecy cannot keep a secret and things > do leak out. You're making a lot of assumptions here. First, you're assuming that the ET's exist; second you're making the assumption that they would care one way or another about what happens to humans (no reason why they should) third, and much more profoundly, you're making the assumption that they ET's are CAPABLE of doing something about it even if they wanted to. I don't mean in the sense of technology, necessarily, but it is naive to think that a species that evolved on another planet (assuming that evolution is a universal law) would be able to speak, see, listen, think, smell, feel or taste just because we can. Who says the ET's are even in a dimension that we can see (they could be two dimensional, restricted only to moving on a plane), they would be completely incapable of seeing us. They could be only visible at a different frequency... X-ray, infrared, and we would be completely incapable of seeing them. And quite frankly, if the alien gov't was controlling all world gov'ts simultaneously, they've done a pretty wretched job of keeping us from killing ourselves off (we're not dead yet, but we're in a much more likely position to do it after the arms race in the Cold War) You're assuming that SOMEHOW the ET's are able to detect a very small nuclear explosion from a considerable distance, despite the fact the amount of radiation it caused would be negligible at any distance outside the Earth's atmosphere, and then are able to travel here in 2 Earth years or less and convince world leaders to let them rule the planet. The whole thing seems a little far-fetched to me... === Subject: NOVA strings and branes === Subject: NOVA Look at Elegant Universe NOVA tonight. http://www.pbs.org/wgbh/nova/elegant/ Brian Greene telephoned an ET Gray on brane universe next door a millimeter away using presumably Ray Chiao's gravity radio transducer of off-brane world gravity waves to EM waves. ;-) Also long part on Star Gate time and space-travel. Ed Witten & Co http://superstringtheory.com/people/witten.html would be considered real kooks by the SI CSICOPS if their actual words were given to the CSICOPS without identifying who they were in a blind fold Turing Test. The tidy classical universe of James Oberg & Co is not the Universe of Ed Witten & Co. BTW Q to Ed Witten: How can the cosmological constant be so close to zero but not zero? Ed Witten's answer: I really don't know. It's very perplexing that astronomical observations seem to show that there is a cosmological constant. It's definitely the most troublesome, for my interests, definitely the most troublesome, observation in physics in my lifetime. In my career that is. My answer to the same question is at http://qedcorp.com/APS/StarGate1.mov For the record, I do not think that the intelligence agencies of the the major powers, US in particular, have the ET UFO anti-gravity technology on the shelf and are hiding it. In other words I disagree with Nick Cook's book Hunt for the Zero Point and with Stephen Greer's thesis in the Disclosure Project. It is quite obvious that no human physicist today understands how such stuff would work. We are only now beginning to get to that level of understanding of how metric engineering would work by manipulating dark energy. It may be possible that USG et-al has retrieved damaged craft and alien creatures from the Universe Next Door perhaps only a millimeter away across the extra bosonic space dimensions like Robert Bigelow's NIDS people reported on his Utah Ranch, or that may have happened at Roswell in 1947 etc. Even if that were true, and I am not saying it is true, the fact is that none of the intelligence people who might have that stuff have the slightest understanding how that hypothetical and/or alleged alien technology actually works. It's The Sorceror's Apprentice. Hi Jim, OK, I believe you. But I do suspect that you may have been subtly conditioned by your NASA(?) experience to accept uncritically (despite your overt commitment to critical thinking) the Government's long-standing public position re the phenomenon in question, that is, that UFOs (and aliens) are all either misidentifications of natural phenomena, hoaxes, or hallucinations, mass and otherwise. This, I suspect-taking you at your word that you're not cynically mouthing a party line that you know to be false-is reinforced by the logically flawed assumption that they shouldn't exist, therefore they don't exist. Clearly, it's very difficult for most mainstream members of the scientific and engineering establishment-that is, those not privy to (or part of) any Government-sponsored disinformation campaign-to accept even the remote possibility that we human beings are, in effect, Wogs vis-a-vis what I've come to think of as an Alien Raj, an extraterrestrial (or possibly extra-dimensional) colonial establishment that has maintained clandestine hegemony over this planet for at least the last eleven or twelve millennia, if I read the late Upper Paleolithic evidence correctly. No, of course I can't as yet prove any of this definitively. But the vast amount of observational evidence, as gathered by people like Richard Dolan, et al., is extremely persuasive, as is the comparative mythological and folkloric evidence, which you may not be aware of (this is my anthropological specialty), and which, after stripping away the pre-modern glosses and metaphors, closely resembles what has been reported by contemporary abductees and other experiencers. And now that physicists like Jack Sarfatti are beginning to give us a handle on these remarkable craft appear to work, the probability of their presence, both today and in the past, is significantly reinforced. Again, once we get over the ego problem, and can accept-and live with-at least the possibility that we've long since been trumped by others when it comes to the technology of space exploration, we can begin to approach this supremely important phenomenon objectively, and from a variety of perspectives, including that of the cultural anthropologist, free from the hysterical sneering, ridicule, and marginalizing that all too frequently tarnishes the discussion of UFOs, whether by mainstream scientists or by the media. (That much of this sneering and ridicule is the result of a conscious policy on the part of the Government, or some elements thereof, to discredit anyone who takes the subject seriously so as to keep the truth from becoming general knowledge-perhaps due to a legitimate fear of massive and socially disruptive nativistic movements that would put the New Guinean Cargo Cults to shame-is a can of peas which I won't re-open any further in this message.) Scott C. SCOTT LITTLETON President, Phi Beta Kappa Alumni in Southern California Professor of Anthropology, Emeritus Occidental College Los Angeles, CA 90041 TEL (323) 255-5477 FAX (323) 982-0264 http://www.oxy.edu/~yokatta/home.htm Any sufficiently advanced technology is indistinguishable from magic --Sir Arthur C. Clarke I think we're property. . . --Charles Fort -----Original Message----- === Subject: Fw: Skeptical Inquirer UFO fyi ----- Original Message ----- === Subject: Re: Skeptical Inquirer UFO I can solve this one. The answer, for me at least, is 'No'. My agenda is to find out about mysterious aerospace reports and events (see www.jamesoberg.com) and share what I've found, for discussion, with no directives, constraints, or other external controls over my activities whatsover. You may stop wondering, and start thinking, now. Jim Oberg Phi Beta Kappa, Ohio Wesleyan University, 1966 Or do debunkers like Sheaffer, Oberg, Klass, and the rest have another, more devious agenda, that is, to further the Government's long-standing policy of keeping the lid on by systematically ridiculing those of us who suspect the truth about this phenomenon, all the while being privy to above-top-secret knowledge that would prove us right? One wonders.... C. SCOTT LITTLETON President, Phi Beta Kappa Alumni in Southern California === Subject: Re: More symmetry between derivative and antiderivative? > What about functions not defined at zero, such as f(x) = 1/x? > The antiderivatives of this function are too important to ignore, > but setting limits of integration to include zero is a no-no. I consider the physically relevant space either according to the traditional arrow of time (-inf, 0) or matching to R+ (0, inf), i.e. neither zero nor infinity are included. You are correct, 1/x does not exist exactly at x. Its antiderivative ln(x) + C contains anyway an infinite constant C. Only ln(x1)-ln(x0) = ln(x1/x0) makes sense in real-world physics. I presume, there is no reason not to hide C within x2=0. The suggested leaning of integration on the limes x towards zero seems to fit quite naturally to a sperical rather than cubic notion of the world. It resolves several nasty problems. However, one cannot expect it to make the function 1/x more generally adequate to nature. === Subject: Re: Repeat: White Noise Dilemma >>I find it odd (2n+1) that no one has responded to this. I'm guessing >>that either a) everyone thinks it is a homework problem (it is not), >>or b) no one knows the answer. Come on, all you big strong mathematicians - >>certainly you can answer a little ol' question like this, can't you? ... >>Here is a question that has had me in a quandary for several >>years now. >>Let X(t) be a zero-mean IID process with variance of >>sigma^2. Now we know that since this function is >>IID, it has a white PSD and therefore its autocorrelation >>function at lag 0, R_XX(0), should be b*delta(tau), where >>delta(tau) is the usual Dirac delta function and b is some >>constant. > What are you using as a definition of PSD (power spectrum density)? > The Wiener-Khinchine theorem states that the PSD of a wide-sense stationary > random process, Sxx(w), is the Fourier transform of its autocorrelation > function > Rxx(tau) = E[X(t)X(t+tau)]. But what is the *definition* of the term PSD? > This would seem to be a problem for a process such as X(t) in which > t--> X(t) is not measurable, or continuous in L^2. > What do you mean by t-->X(t)? The mapping of R into L^2 that sends time t to r.v. X(t). -- A. === Subject: Re: any info on a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n > Hi anybody could please give me some info on > a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n > it looks like geometric and/or binomial series but > nothing is clear (to me). Multiply it by a-b and see what you get. === Subject: Re: JSH: My victories get lost > David Ullrich is a ing piece of dog. > I think it's funny that I can call a professor at Oklahoma State > University a ing piece of dog knowing that he'll keep replying > in my threads. Yes, it is funny that a ing piece of dog can call a professor at Oklahoma State University a ing piece of dog. I think it is more than five years since I posted that there is a difference of at least 30 in our IQs, and I know which direction, and you know it as well. === Subject: Re: I can't stand it anymore >I have been biting my tongue about the IQ test but I can't any more. >How reliable is a test that use the term Asian to represent the most >diverse of ethnic and cultural groups? >If you want to know more about the diversity, ask me and I will give >you tons of specific examples. Here is one: An example of cultural >differences (that affects their educational goals) of the two groups - >ethnically the same and speak the same EXACT langauge - in Burma who >are the descendants of the people who came to Burma from a town known >as Surat, India during the British colonial days. > Now, do not assume that they are racially the same as the majority of >northern Indian population, i.e Indo-Aryan. What I am aware of is that >they are descendants of those who came from Central Asia (to india) >and may be MIXED with the locals of India (I haven't checked into >that) like the majority of the current day Pakistanis. >See the complexity yet? >Add to that the Afghans, particularly the Pathan, which also form a >comunity in Burma. >And then you have all these different native groups in Burma, namely >Burman, Mon, Shan, Karen, Kachin, Chin, etc. where the Karen with a >good size Christian population now claiming that they are one of the >lost tribes. > The Mon are the first group who adopted Bhuddism which made them the >most literate group *in the old old days* through the monastry >education. Shans are the stauch Bhuddists too. I read somewhere on >the Internet that Mon originated from ancient India. They look like >typical oriental though some have darker skin while others are very >light-skinned) instead of the dravidians of ancient Inida. >Shan is ethincally similar to some groups in Thialand from the area >that borders Burma. Talking about Thailand, not all Thias are >ethincally the same. >Now, the Chinese would claim that the high score of IQ tests is >because of them Chinese, i.e not all Asians are equal. And the >Japanese thinks (may be not so much anymore) they are superior to all >other Asians. >Does that term *Asian* include the Arabs? >What about the people of Egypt (I avoid calling them Egyptians to >differentiate from the ancient Egyptians) some of whom are Arabs, some >of whom are descendants of anciant Egyptians, some are descendnats of >a group called Berber (my spellig may be wrong). >And how many Ethiopians look or act like a typical African. >Why are people using the racist IQ tests as God's given ruler to >measure intlligence? http://metropolis.japantoday.com/tokyo/recent/feature.asp In the off chance you are interested in whether or not the Japanese people came through the star gate or whether they crashed and are decended from the ones who crashed in China, I am inclined to believe that they came through the stargate. But that does not detract from the Chinese who have equal right to be here. We aren't going to change anything at this point. I have seen on the net, some very Japanese looking ET's and one in particular that looked very Japanese. But for the most part, the ones with the large cranium, look more Chinese. They are scientists as far as I know. The ones that were here last year were an advanced race. They beamed my son home from school if you can image that. At lunch time. The really odd thing about it was, that for about 20 minutes after, his eyes were slanted down. Then his appearance returned to normal. His conscious self didn't know anything about it. I think he just assumed he walked home. At the time a bunch of us had been discussing the future as per The 5th Element. And I was arguing for rejuvenation machines. Seems silly I know. But no more silly than this... http://www.rense.com/general41/flying.htm or this... http://dbarkertv.com/UPDATE.htm As I understand it, this is a reptoid ship. Were the ones who arrived first, and notified the glactic government, after Hiroshima, distant relatives of the Japanese people? That is a nice notion, is it not? Ignore the stupid pictures of the alien autopsy. Some humans are very sick mentally and spiritually. Those are the kind of people who make mock ups like that to hide their own evil deeds and to take advantage of their fellow man. Thats the price we pay for freedom. We have to tolerate some people who truly should not be allowed to exist. So what is wrong with the Japanese people? They are still suffering from the old, how could God, allow this to happen? That is a very good question. That will undoubtedly be answered one day. But take it from me, there is nothing wrong with the Japanese people. They are beautiful people. And do have a very valid inherent right to be here. But try to think of people as individuals, not as a race. Do not ask what is wrong with Japanese people, say what is wrong with some human beings. You cannot blame a victim for a crime. And the answer again, is the price of freedom. The freedom to be bad as well as good, makes people free. That does not however mean, there are no consequences for people's actions. Be thankful for your good nature. That is a blessing. Some may see that as a weakness, but then they usually find out the hard way, that there is only one true power in the universe. Trying to compete with him, is a very silly idea that will only lead to misery. What Japan has to offer humanity, is invaluable to the future of mankind. So don't go changing. :-) -*- === Subject: Re: Geometry And Newtonian Mechanics of Action Device OK, let us see Pythagores in Action.. > ______________________________________________________ Pythagores Theorem: > In a right angled triangle, the square of hypotenuse is equal to the > sum of the squares of the other two sides. c^2 = a^2 + b^2 > ______________________________________________________ > You would be more credible if you started by spelling his name > correctly. The Pythagoran theorem is trivial special case of the > general triangle theorem. Look it up to learn why the right angle is > so convenient. If I spell your name as Yncal Al, will it change your persona, actions, understanding of Geometry and Physics... I am trying to explain step by step Geometry, Newtonian Mechanics of this action device. Please, Uncle Al, concentrate on meaning, not on words. I don't have much time left. -Abhi. === Subject: Re: A 'basic' topology question about interiors > Of course. But y'know, 'you say formulas, I say formulae...' formulae formulae formulas formularum formulis formulis ?? Woohoo! > Ben Scott === Subject: DESIGN OF A PRIMITIVE NANOFACTORY Design of a Primitive Nanofactory Chris Phoenix Director of Research, Center for Responsible Nanotechnology http://CRNano.org Abstract: Molecular manufacturing requires more than mechanochemistry. A single nanoscale fabricator cannot build macro-scale products. This paper describes the mechanisms, structures, and processes of a prototypical macro-scale, programmable nanofactory composed of many small fabricators. Power requirements, control of mechanochemistry, reliability in the face of radiation damage, convergent assembly processes and joint mechanisms, and product design are discussed in detail, establishing that the design should be capable of duplicating itself. Nanofactory parameters are derived from plausible fabricator parameters. The pre-design of a nanofactory and many products appears to be within today's capabilities. Bootstrapping issues are discussed briefly, indicating that nanofactory development might occur quite soon after fabricator development. Given an assembler, a nanofactory appears feasible and worthwhile, and should be accounted for in assembler policy discussions. . . . More here: http://www.jetpress.org/volume13/Nanofactory.htm Jai Maharaj http://www.mantra.com/jai Om Shanti Shubhanu Nama Samvatsare Dakshinaya Jeevan Ritau Tula Mase Shukl Pakshe Mangal Vasara Yuktayam Poorvaprostapad-Uttaraprostapad Nakshatr Vyaghat-Harshan Yog Vishti-Bav Karan Ekadashi-Dvadashi Yam Tithau Hindu Holocaust Museum http://www.mantra.com/holocaust Hindu life, principles, spirituality and philosophy http://www.hindu.org http://www.hindunet.org The truth about Islam and Muslims http://www.flex.com/~jai/satyamevajayate o Not for commercial use. Solely to be fairly used for the educational purposes of research and open discussion. The contents of this post may not have been authored by, and do not necessarily represent the opinion of the poster. The contents are protected by copyright law and the exemption for fair use of copyrighted works. o If you send private e-mail to me, it will likely not be read, considered or answered if it does not contain your full legal name, current e-mail and postal addresses, and live-voice telephone number. are not necessarily those of the poster. === Subject: Re: I can't stand it anymore And yes, as far as we know, the same thing happened in India years before. But that is not as bad as mass extinctions in prehistoric times. So why do these things happen? In the course of evolution, it must be necessary at times to test certain limits. Abuse of power is one that humans keep getting wrong. And by power I don't necessarily mean strength. But any abuse of power of power over another is wrong. To properly handle power you need to have a lot of other qualities. Like compassion. And the ability to cook small fish. ;) Look at the legends of Atlantis. Abuse of power. Mu, same. Only totalitarian power. Look at George Orwell. He addresses the abuse of power in 1984 and looks at the crappy part of human nature in Animal Farm. Another part of being free to make mistakes, is to not be sure that there is someone watching you decide. Maybe there is, maybe there isn't. People who act in good conscience, don't concern themselves with it at all. They are more free than one who tries to hide trickery or deceit. I think it was because of the treatment of women. That the people had gone too far beyond normal sensuality into fetishes. They had lost touch with nature. Lost touch with their own humanity and were going too far with process and order and formality, and subservience. People need to be free. Japan is a very free nation today, because of the sacrifices of their parents. Good or bad, right or wrong, the problem appears to be fixed. Japan is far better off today than India, or Pakistan, or any dinosaur, or Neanderthal. In fact it is one of the best countries in the world today. Does it still have social problems? What country doesn't? Would a person from America, rather go to Japan for a visit, or to South Africa? Is South Africa better off now, that the western world has left them to their own devices? Better for them perhaps, but I would not go there or take my family there. I would however, love to visit Japan. -*- === Subject: Re: c-number >: Howdy everybody, >: I'm reading Peskin and Schroeder's Intro to QFT and in this >: book the author constantly refers to c-numbers... without >: saying what a c-number is. I couldn't find c-number >: on mathworld.wolfram.com or physicsworld.wolfram.com. >: Can someone please tell me what the hell a c-number is? >I think it means a classical number, as opposed to a q-number (quantum >number). Okay, then what is a classical number? adam === Subject: Re: A NOTy problem. Hello > I have been puzzling over what ought to be a simple logic problem > but am at a loss to come up with a method. > A crime is committed by one of the following; Alan, Bob, Chris or > Dave. > Each makes a statement to the police but three of them lie. > Alan says, I didn't do it. > Bob says, Alan is lying. > Chris says,Bob is lying. > Dave says, Bob did it. Sounds like Dave plays no role here. Either Bob or Alan is telling the truth (if not, than Bob lies and Alan is not lieing, which is absurd). Either Chris or Bob is telling the truth (same proof) But there is only one truth-teller, hence Bob tells the truth. Alan did commit the crime! -hopefully, he's living in New Zealand so he won't be grilled like chicken-. === Subject: Re: Key core error argument, stepped out >[deletia] >4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I >>have >>P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) >>P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). Note that all you have done is add and subtract 3 to define b_3(x); >that is, you are writing b_3(x) = (a_3(x) - 3) so 5a_3(x) + 7 = 5(a_3(x)-3+3) + 7 > = 5(a_3(x)-3) + 15 + 22 > = 5b_3(x) + 22. The exact same process that you decried when I used it. This is what I interpretted as priority---^^^^^^^^^^^^^^^ You claimed >that doing this ma[de] no sense mathematically. Do you still make >that claim? Just curious. >>That is a lie from Arturo Magidin as in fact he just subtracted and >>added 3 on the same line. I'm focusing on constant terms, not trying >>to hide a correct argument with meaningless operations like >>subtracting and adding 3. No it's not, James. You might want to check your facts before you >accuse others of lying. >>You obviously are the one who didn't check facts as what I said IS correct. >Actually, no. You said three things, and you implied a fourth. Of the >three things you said, one is false. Therefore, what you said is NOT >correct. >The three things you said are: >(a) I lied; >>I had interpretted the lie to refer to you coming up with the idea >>first. > I don't remember (and I do not see in the original post) any claim of > priority. In fact, nowhere did I say first. All I said is what james > had done, in this case, and noted that it was the exact same thing > James had attacked recently. > In fact, when he attacked it I pointed out that it was the same thing > James had done in his Advanced Polynomial Factorization first Lemma > (that was done in a different thread), so I certainly did not claim > priority in any way! Not that James first came up with the idea of > writing a function f as f(x) = g(x) + f(a) for some specific value a; > it's just that it's not terribly interesting for the vast majority of > functions... Ah. Ok. In any case, I do recall him attacking that procedure before he adopted it here. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: More symmetry between derivative and antiderivative? > What about functions not defined at zero, such as f(x) = 1/x? Just an addition: With ln(b)-ln(a) = ln(b/a) the number of parameters reduces from two to one in this case, too. === Subject: Re: More symmetry between derivative and antiderivative? What about functions not defined at zero, such as f(x) = 1/x? The antiderivatives of this function are too important to ignore, > but setting limits of integration to include zero is a no-no. > I consider the physically relevant space either according to the > traditional arrow of time (-inf, 0) or matching to R+ (0, inf), > i.e. neither zero nor infinity are included. Physical relevance is irrelevant to the mathematics. > You are correct, 1/x does not exist exactly at x. > Its antiderivative ln(x) + C contains anyway an infinite constant C. Infinite constants are nonsense, at least in the mathematics of Reimann integration. > Only ln(x1)-ln(x0) = ln(x1/x0) makes sense in real-world physics. > I presume, there is no reason not to hide C within x2=0. You will, no doubt, presume whatever nonsense you wish about real-word physics, but that does not make it true or relevant to mathematics. The realities of mathematics are quite independent of your view of physics. [garbage snipped] === Subject: Re: Your Attention Please, Sci.* NG > Abhi: >This is very serious attempt indeed. My life is at stake. > I hope you have a will. I will show you meaning of Force of Life. Trust me. >I request >you to discuss and understand this very seriously and then execute >this action device to bring before this world. I just want to give justice to this invention. > Then burning at the stake is probably not out of the question. >I will begin my attempt in new post titled, Geometry And Newtonian >Mechanics of Action Device > I understand alt.test has a very large readership. posts, with different words by different people. Your past experience have overpowered your rationality. I would have done the same thing you are doing. Though we people come everyday and post our views in same NG, we live in different parts on this earth, with different countries, langauge, culture, circumstances. We just fail to understand circumstances at other end. But one thing is certain. You people change my emotions from sadness to flames whenever I came here. It really makes me forget problems in my personal life. Sometime bad things turn out to be good. -Abhi. === Subject: Re: Cantor's diagonal number: why do we just ASSUME such a number exists? A8EwTYfhf*u~,Eu,tf6$HN*MY&)u0G =N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@J6m5.EN?>Zh Xh;Y V|',x(js'Jfq02joVpj|#x linux) > the crackpot index IN ORDER. didn't realize it was a parody with method. -- Jesse Hughes I often told you of the dangers of hubris, and most importantly of all, I TOLD you that I wanted to change the institution of mathematics worldwide. -- James Harris, on the evils of pride === Subject: Re: Your Attention Please, Sci.* NG This time, I will try my best to explain Geometry and Newtonian > Mechanics behind the working of Action Device. If we know root of any > problem, we can prepare counter measures to solve that problem. This > is how our nature works. This Action Device is based on understanding source and exact nature > and mechanism of Gravity and counter mechanism is designed so that > things does fall or accelerates towards centre of earth but > accelerate very slowly towards centre of virtual planet, gravity > of which is greater than that of earth. This very slow acceleration > towards centre of another virtual planet gives us illusion of things > hanging in air or space. For example, if some ball is lying on surface > of earth, it gives us illusion of being stationary on surface of earth > but actually it is accelerating very slowly towards centre of earth. > http://b5.sdvc.uwyo.edu/bab5/snds/argcstpd.wav > http://w0rli.home.att.net/youare.swf > http://www.mazepath.com/uncleal/sunshine.jpg Now, please give me URL of somebody who is really interested in understanding Gravity and carry out this invention in right spirit. Do you have any URL stored in your hard disk? -Abhi. === Subject: Re: any info on a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n > Hi anybody could please give me some info on > a^n+a^(n-1)b+a^(n-2)b^2+..+ab^(n-1)+b^n > it looks like geometric and/or binomial series but > nothing is clear (to me). For a = b, it is n*a^n = n*b^n. For a and b distinct, it is [a^(n+1) - b^(n+1)]/[a-b] === Subject: Re: Graduate algebra book > Well, I think the U.S. and other countries are wildly different in > their structure of an undergraduate curriculum. I don't have a > problem admitting that my university is not so terrific. We have 2 > algebra courses which are available to undergraduates at my > university, the more advanced of which is almost identical to the > introductory course, albeit slightly more fast paced. It so happens > that I _am_ an undergraduate and I am trying to get a better education > than what my university offers to the typical student. Good for you. So you really want a decent undergraduate text, rather than a postgrad text, to compensate for the impoverished curriculum at your university. > What I have a > problem with, is listening to the egos of those people who feel like > they are superior to everyone else for no particular reason. Who's that then? > All Mr. > Chapman had to say was generally those topics are covered at the > undergraduate level, instead of the snide remark 'thats graduate > algebra?' Aaaaaaaah! > Actually, upon checking a few of the top universities' websites, I > find that while the undergraduate curriculum generally does offer > algebra courses which talk about galois theory and the topics I > mentioned, they are generally at the near-graduate level. Thus, for > any AVERAGE university, they will not be offered until the graduate > level. What is an AVERAGE university? > And even at these better universities, not everyone takes > every single course, and I imagine not everyone takes 5 or 6 semesters > of algebra as an undergraduate. Should they take 5 semesters of > analysis and 5 more of topology as well? how is anyone ever supposed > to graduate? It's quite possible to study a reasonable amount of algebra, a reasonable amount of analysis and a reasonable amount of topology all in three years. (And also a reasonable amount of applied maths and statistics if that's your thing.) I'm a bit baffled about your reference to semesters of algebra --- do you only study one topic per semester at your university? Anyway, you'd be better off studying some maths rather than flaunting the chip on your shoulder. I'd second the suggestion of Dummit/Foote as a suitable book, and also re-iterate mine of Rotman's recent Advanced Modern Algebra (this does some topics, e.g., the homological theory of local rings, not often found in texts at this level). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: Covering a rectangle > Is it possible to cover a 1 x sqrt(2) rectangle with a finite number of > disjointed squares (with eventually different measures)? No. Covering a rectangle with squares is possible iff the ratio of the sides is rational. This follows from the relation between square coverings and electrical networks: see for instance, Bollobas's _Graph Theory_. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: Riemann's Zeta Function by H. M. Edwards >> I presume you are using the contour integral argument for continuing >> Gamma(s)zeta(s). One introduces a countour C_e in three parts: >> imaginary axis from -infinity to -e, circle radius e about origin, >> imaginary axis from -e to -infinity and take the integral >> of z^{s-1} e^z on C_e (with branch cut on negative real axis). >> Thse integral f(s) is independent of e by Cauchy's theorem. >> It is also an entire function of s: convegence is nice since e^t -> 0 >> rapidly as t -> -infinity. > Really dumb question. is e in the contour integral an epsilon, or the > basis > of the natural logarithm. (I am getting confused by both uses of e.) No Real Mathematician thinks of the logarithm having a basis:-) If you want to use epsilon instead of e you're welcome: using e does save typing. >> The integral of z^{s-1} e^z on the first part of the contour is >> integral_e^infinity t^{s-1} exp(-pi i(s-1)) e^{-t} dt >> = - integral_e^infinity t^{s-1} exp(-pi is) e^{-t} dt. >> Similarly on the third part of the contour it is >> integral_e^infinity t^{s-1} exp(pi is) e^{-t} dt. >> These add to >> 2i integral_e^infinity t^{s-1} sin(pi s) e^{-t} dt. >> If Re(s) > 0 the integral over the circle of radius e is >> O(e^Re(s)). Letting e -> 0 we get that > What is this O Is this Big-O notation? Yes. Very useful! >> >> Consider >> G_N(s) = integral_{C_{(2N+1)pi}} z^{s-1} e^z/(1-e^z) dz. >> By Cauchy's theorem the difference G_N(s) - g_e(s) is 2pi i times >> the sum of the residues of the poles of the integrand >> at +-2pi i, +- 4pi i, ..., +- 2Npi i, that is >> 2pi i sum_{n=1}^N [-(2pi ni)^{s-1} - (-2pi ni)^{s-1}] >> = 2pi i (2pi)^{1-s} sum_{n=1} (-2) cos(pi(s-1)/2)/n^{1-s} >> = something nasty times sum_{n=1}^N 1/n^{1-s} > Hmm. Edwards gets sin(s pi/2). How does sin(s pi/2) relate to cos((s-1)pi/2)? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Key Core Error Argument The following proof steps through a rather basic argument which is key in proving an over one hundred year old error resulting from previously unexpected consequences resulting from the definition of the ring of algebraic integers. Note that ultimately the proof relies on 22 NOT having 7 as a factor, and constant terms like 7 and 22, being constant, and not variables dependent on x, which may seem like odd things to emphasize, but I've faced posters who've gotten away with challenging those truths because people seem unaware that's what they're doing. 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is in the ring of algebraic integers, notice that P(x) has a constant term that is 1078. 2. It can be shown that P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 where you should note that using v = -1 + 49x, gives P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 where the *same* polynomial has been put in a form which allows a factorization into non-polynomial factors so that I have P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) where the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). 3. Now let x=0, so P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) as the cubic defining the a's at x=0 is a^3 - 3a^2, which has roots, 0, 0 and 3, and I've picked a_1(0) and a_2(0) to equal 0, which leaves a_3(0) with a value of 3. 4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I have P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). 5. Now P(x) has a factor of 49 as P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 which means that (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) has a factor of 49. 6. However, the constant term of P(x)/49 is 22, which is verified by again setting x=0, which gives P(0)/49 = 22. But for two of the factors of P(x), the constant terms is 7, which is NOT a factor of 22. Therefore, *none* of the constant terms of P(x)/49 as they multiply to give 22 can have 7 as a factor. (By saying that 7 is NOT a factor of 22, I'm making a choice as to where the proof is going. Since I've been talking about algebraic integers, where 7 is NOT a factor of 22, it's natural to go with a choice where 7 is NOT a factor of 22.) Given that the constant terms are independent of x's value, it must be the case that dividing P(x) by 49 divides the two constant terms equal to 7, by 7. 7. But to divide 7 from those constant terms requires dividing through two of the factors, so (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 from reverse use of the distributive property, which gives constant terms that don't have 7 as a factor, as required. Notice that it's a rather short and direct argument, where if you accept that 22 does not have 7 as a factor, it's obvious enough what the constant terms of the factors must be as you go from 7, 7 and 22, necessarily to 1, 1, and 22, when you divide P(x) by 49. James Harris http://mathforprofit.blogspot.com/ === Subject: Re: Key core error argument, stepped out > Using Dot's proof that the values of a polynomial with algebraic > integer coefficients are always divisible by an integer if and only if > each coefficient is a multiple of that integer (in the algebraic > integers) gives you that the statement is correct for polynomials in > A[x]. I had not noticed Dot's theorem. The x^2 + x example seems to belie > it: for any integer x, x^2 + x is divisible by 2, but the coefficients > are not multiples of 2, and both of the coefficients are algebraic > integers. You must therefore be thinking that the domain is algebraic > integers as well. > I think you are reasoning on cross-purposes. As I remember, Dot's theorem > was about polynomials over the algebraic integers. As a polynomial over > the integers, x^2 + x is always divisible by 2, but not as a polynomial > over the algebraic integers. Oh yeah, I finally realized where you were cheating Dik Winter, as using v=-1+49x my polynomial is simply enough P(x) = (v^3+1)5^3 - 3v(5)7^2 + 7^3 which I'm sure you conveniently forgot when you came up with your examples. Nevertheless my explanations before were correct, but maybe now you can start to see more of the particulars of why your hacks put you in a field. What I hate about posters like you is that you don't seem to care about the truth, but only about convincing sci.math and other newsgroup readers. James Harris === Subject: Re: Math dependency logic > Ultimately my argument relies on numbers like 7 being NUMBERS, not > variables dependent on x, and on the distributive property. > 7 is a number, but so is x. Neither can change (in a given context); > each refers to exactly one number at a time. What does time have to do with it? How do you define time with mathematical argument Thomas Bushnell, BSG? James Harris === Subject: Re: One algebra question > Let f be a homomorphism defined on a finite group G and let > H subset G Do you mean that H is a subgroup of G? > a) show that |f(G):f(H)| divides |G:H| > b) show that |f(H)| divides |H| Let K be the kernel of f. Note that |f(H)| = |H|/|H intersect K| .... -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: JSH: Factorization P(x) = 2(x(x+1)/2 + 1) > ... > > > My point is that you have to focus on the *factorization* and its > > > validity in particular rings. > > > > Some factorizations will be valid in one ring, but not another. > > Yes, and your factorisation > > P(x)/49 = (5 a1/7 + 1)(5 a2/7 + 1)(5 b3 + 22) > > is in general not valid in the ring of algebraic integers. So what > > are you trying to show? > > That's the point. I *prove* that if you have coprimeness between 7 > > and 22 in the ring in which the factorization is valid, where 7 is NOT > > a unit (and neither is 22), then the constant terms of the factors > > that result from dividing P(x) by 49 *MUST* be coprime to 7. > > If you are talking here about the factorisation > > P(x)/49 = (5 a1(x)/7 + 1)(5 a2(x)/7 + 1)(5 b3(x) + 22) > > being valid in that ring (for all x), that is vacuously true. No > > proof needed, 1 is coprime to 7 and 22, and 22 is coprime to 7 in > > any ring that contains 1, 7 and 22. Even if in that ring 7 and/or 22 > > are units. So we need no proof in that case. > > Oh yeah, so ignore my statement that 7 is NOT a unit Dik Winter as > > you're showing your lack of reasonableness. You are ignoring my statement that in *every* ring that contains 1, 7 and 22, > they are coprime. Whether 7 is a unit or not. Are you just totally stupid? That's irrelevant Dik Winter as the > point is that 7 is NOT a factor of 22, which my saying that it's not a > unit points out. > Why do you not read what I write, rather than what you think I write? > I'm getting sick of stupid games from stupid people who apparently > have nothing better to do with *their* time. Now are you or are you not intelligent enough to understand what it > means for 7 NOT to be a factor of 22? > Yup. That means that you think your factorisation of P(x)/49 is valid > in some ring where 7 is not a unit. I am not convinced. I *know* it That's because you're *refusing* to be rational!!! Now I've found another way your hacks go away from my argument as the polynomial I use, with v=-1+49x is simply P(x) = (v^3+1)5^3 - 3v(5)7^2 + 7^3 which shouldn't look totally unfamiliar to you, but you're a hack, apparently dedicated to convincing newsgroup readers of what you believe, without concern about the actual mathematical truth. Just my problem you and your cabal have been successful enough that I have to break you. > is not valid in the algebraic integers, because there is ample proof that > for varying x the factors of 49 distribute differently amongst the three > factors of the polynomial when you wish to stay in the algebraic integers. And my point remains that I've found an error in core so your claim of a proof using core is stupid in context as I've *repeatedly* explained it to you. Why do you refuse to be logical Dik Winter? Why do you refuse to follow a stepped out mathematical proof? What's wrong with you? > Furthermore, I *know* that in the algebraic integers (5 b3(x) + 22) is > *not* coprime (either the standard definition or your definition) to 7 > for most values of x. You're irrational Dik Winter, desperate to hold on to what you think you know, as if mathematics need care about your beliefs. You've been refuted. I've shown yet again how your hack diverges from my work. Tell the truth, or go away. James Harris === Subject: Re: JSH: Attacking a proof is attacking yourselves > Now I've found out that even though I've stepped out my proof and > explained in detail there are posters who just keep trotting out the > same things despite getting refuted. Maybe it's because _you_ keep trotting out the same thing despite being shown the errors in PAINFUL DETAIL! > 7 and 22 are NUMBERS Are they. Blimey! Didn't realise that. I don't think that 22 was covered in my analytic number theory course...7 was, obviously, since it's a prime variable...er....number. > Now I'm sick of it. Good. So am I. I've kept fairly quiet about all this ridiculous goings-on for over 7 years (that's a number, not a variable!) but I really have had enough. Shut up and go and do something useful. I've spent many years studying mathematics, number theory in particular, and have bumped into numerous people like you. None quite as determined though. :-) Most reasonable people will admit that they have made an error and go away and work on it. Please do so. >You people are going to play by some rules. Er...how about the 'usual rules of mathematics'? > ...anti-math behavior on display... Hmm, I wonder where most of this comes from? > After all, you people are attacking or sitting by while people attack > the idea that 7 and 22 are constant!!! Not at all! I am quite happy with your assertion that 7 and 22 are constants. > And how can any of you not understand what it means for 22 not to have > 7 as a factor in a ring where the only integer units are -1 and 1? A very good question. Do _you_ understand what it would mean? > These are *basic* points people... Certainly are! > YOU ATTACK THE VALIDITY OF BASIC THINGS THAT PEOPLE CAN BE CERTAIN > MUST BE TRUE!!! Hmm. 'Goddamn proof'? Best not to bring religion into a discussion about mathematics. Always a bad mixture! > Oh yeah, make no mistake, what I've given is a mathematical proof. Indeed it is, albeit wrong. Don't feel bad though, even the best mathematicians have made some corking errors! > I've proven that *as a group* mathematicians can be so irrational as > to question that 22 and 7 are constant. Er, no. I don't think that even 'modern math' teaching asserts that 7 and 22 are anything other than constant. I could be wrong though. > You are attacking your own interests by attacking mathematical proof. Ooo no, you seem to misunderstand the idea of publishing work on a forum such as this...only by getting others to examine our work do we find errors that, by definition, we cannot see ourselves. In an on-topic aside, look at the progress of Wiles' own proof of FLT; when colleagues spotted error(s) he took a nice walk, went back to his desk, and worked on fixing what was broken. What he did _not_ do is proclaim that his detractors (read: peer reviewers) did not understand what he was doing (doubtless, many of them did not understand _all_ of his work but they examined what they _did_ understand) and that they were all ganging up on him! Having been in a similar situation (albeit on a much smaller problem) and, in fact I'm still working on it after 3 years (I probably will never get there but I'm learning a lot in the process of not getting there!) In short, whilst mathematical proofs are wonderfully powerful, the process of producing them can be very difficult and strewn with errors! > James Harris ttfn (no doubt) JasonG === Subject: Re: More symmetry between derivative and antiderivative? >What about functions not defined at zero, such as f(x) = 1/x? >The antiderivatives of this function are too important to ignore, >but setting limits of integration to include zero is a no-no. >>I consider the physically relevant space either according to the >>traditional arrow of time (-inf, 0) or matching to R+ (0, inf), >>i.e. neither zero nor infinity are included. > Physical relevance is irrelevant to the mathematics. Really? You caused me to crosspost this message. Maybe, mathematicians believe they are independent. However, they are just not aware to what extent their discipline benefited not just from funding but also from people who dealt with practical applications. I will try and give the gist of an utterance by Oliver Heaviside: Mathematicians said, this series does not converge. On that condition it will be of use. He hated unjustified rigor. Ironically, his revenge was made as rigorous as possible. >>You are correct, 1/x does not exist exactly at x. >>Its antiderivative ln(x) + C contains anyway an infinite constant C. > Infinite constants are nonsense, at least in the mathematics of > Reimann integration. Even if I am merely an old engineer with minimal training in mathematics, I am aware of some facts concerning Riemann type integration. Perhaps nobody defined integrals exclusively for open domains while the world excessively integrates from minus infinite to plus infinite. Is this correct? >>Only ln(x1)-ln(x0) = ln(x1/x0) makes sense in real-world physics. >>I presume, there is no reason not to hide C within x2=0. > You will, no doubt, presume whatever nonsense you wish about > real-word physics, but that does not make it true or relevant to > mathematics. The realities of mathematics are quite independent of > your view of physics. I was told, every paying costumer is the king at least in America. Doesn't physics pay for mathematics? Doesn't physics provide tailor-made models of nature? === Subject: teaching descriptive statistics / proposal When teaching descriptive statistics, it is usual to interpret the moments of a distribution in R as follows: moment of order 1 (mean) : location centered moment of order 2 (variance): dispersion reduced centered moment of order 3 (skewness): degree of asymmetry reduced centered moment of order 4 (kurtosis): degree of flatness Now let us look a bit more to the skewness (goes back to 1895). It is known that some non symmetric distributions have a null skewness. Nevertheless, owing to its simplicity, skewness is still used. Here comes my proposal: Replace the skewness by the chiral index Chi. Having a sample and a pocket calculator, the chiral index is simpler to compute than the skewness, and it is such that: Chi = 0 <=> symmetric distribution (i.e. indirect symmetry) The chiral index exists also for multivariate distributions. (http://www.mdpi.net/entropy ; This journal is free for readers) Sections 2.9 and 4.2; chiral index for univariate distributions: equations 2.9.3 to 2.9.5. Michel Petitjean, Email: petitjean@itodys.jussieu.fr ITODYS (CNRS, UMR 7086) ptitjean@ccr.jussieu.fr http://petitjeanmichel.free.fr/itoweb.petitjean.html === Subject: Re: Uncle Al is Sadistic . ------------------------------------------- > The first African, a grad student in Chemistry department whom I met >ten years ago definitely was not dull. He was very intelligent and >good in his study. > 3 Africans I met in Computer science department in the last 4 years >were way above average in thier programming skills in the midst of >Chinese and Indian grad students who are the overwhelming majority in >that department. One was so good that he even got a parttime teachign >position to teach an undergrad programming course, while he's still a >student himself. the way of uni recruitment in Africa - shake a palm tree, those who drop, go to US or Europe, the rest lives as normal. The strogest survive. === Subject: Re: Was Lucy the split image - Jabriol is a creationist. > Jabriol is indeed a creationist as are all Jehovah's Witnesses. He's been > trying to disprove evolution for the past several years here on Usenet, > using what he's learned from the Watchtower publications, with no success. > You can't reason with a religious fanatic. and of course they need you to tell them this? tell them also how you love non jw-pedophiles. and how you agree with sex among children, when you send your own son to wild parties at motels.. oh by the way since you are in an informative mood.. tell them how you have been caught lying so much.. that nobody belives you anymore.. calling me a creationist does not make me one... and in fact you being a JW groupie should know that JW's are not creationist... However we also know, thta you need to lie, because, you cant get your point across by telling the truth.. now go get a new husband and service him like a house wife should, because the first one bashed you like a romper room punching bag, and the second run you out of Nashville, because you send to much time on the net.. === Subject: Asymtotic normal distribution Hello everybody, I'm working on a project and I want to prove the following. Let Ggr ~ BIN( m/2 , pgr) Gro ~ BIN( m/2 , pro) Zgr ~ BIN( n/2 , pgr) Zro ~ BIN( n/2 , pro) I've already proved that S is a unbiased estimator for the ratio n/m is: S = Sqrt{ (Zgr Zro)( Ggr Gro) } What I want to prove now is that LOG(S) ~ Normal distributed. I already have: LOG(S)=1/2LOG(Zgr) +1/2LOG(Zro) - 1/2LOG(Ggr) - 1/2LOG(Gro) = .. Can somebody help me with this prove? === Subject: Re: Key core error argument, stepped out ... > Using Dot's proof that the values of a polynomial with algebraic > integer coefficients are always divisible by an integer if and only if > each coefficient is a multiple of that integer (in the algebraic > integers) gives you that the statement is correct for polynomials in > A[x]. > > I had not noticed Dot's theorem. The x^2 + x example seems to belie > it: for any integer x, x^2 + x is divisible by 2, but the coefficients > are not multiples of 2, and both of the coefficients are algebraic > integers. You must therefore be thinking that the domain is algebraic > integers as well. > > I think you are reasoning on cross-purposes. As I remember, Dot's theorem > was about polynomials over the algebraic integers. As a polynomial over > the integers, x^2 + x is always divisible by 2, but not as a polynomial > over the algebraic integers. > Oh yeah, I finally realized where you were cheating Dik Winter, as > using v=-1+49x my polynomial is simply enough > P(x) = (v^3+1)5^3 - 3v(5)7^2 + 7^3 > which I'm sure you conveniently forgot when you came up with your > examples. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Using De L'Hopital for solving equations > but also f' must be zero. In other words, your method won't > work unless the function f *and* its derivative have the > same root t. And I don't see how you could know that without > solving both of them. ''(x) ) won't work. But if I consider only the first derivate of the functions? So I have the limit: lim (x->t) ( f (x) / g (x) - f '(x) / g '(x) ) = 0 f (x) and g (x) are 0 per hypotesis, and the first derivates of functions are not necessarily to be 0, because their quotient represents the result of the limit not in form 0/0. === Subject: Re: Key Core Error Argument > The following proof steps through a rather basic argument which is key > in proving an over one hundred year old error resulting from > previously unexpected consequences resulting from the definition of > the ring of algebraic integers. > Note that ultimately the proof relies on 22 NOT having 7 as a factor, > and constant terms like 7 and 22, being constant, and not variables > dependent on x, which may seem like odd things to emphasize, but I've > faced posters who've gotten away with challenging those truths because > people seem unaware that's what they're doing. > 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is > in the ring of algebraic integers, notice that P(x) has a constant > term that is 1078. > 2. It can be shown that > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > where you should note that using v = -1 + 49x, gives > P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 > where the *same* polynomial has been put in a form which allows a > factorization into non-polynomial factors so that I have > P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) > where the a's are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > 3. Now let x=0, so > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's at x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3, and I've picked a_1(0) and > a_2(0) to equal 0, which leaves a_3(0) with a value of 3. > 4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I > have > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) > P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). > 5. Now P(x) has a factor of 49 as > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 > which means that > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) > has a factor of 49. > 6. However, the constant term of P(x)/49 is 22, which is verified by > again setting x=0, which gives P(0)/49 = 22. > But for two of the factors of P(x), the constant terms is 7, which is > NOT a factor of 22. Therefore, *none* of the constant terms of > P(x)/49 as they multiply to give 22 can have 7 as a factor. > (By saying that 7 is NOT a factor of 22, I'm making a choice as to > where the proof is going. Since I've been talking about algebraic > integers, where 7 is NOT a factor of 22, it's natural to go with a > choice where 7 is NOT a factor of 22.) > Given that the constant terms are independent of x's value, it must be > the case that dividing P(x) by 49 divides the two constant terms equal > to 7, by 7. > 7. But to divide 7 from those constant terms requires dividing > through two of the factors, so > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > > 300125 x^3 - 18375 x^2 - 360 x + 22 > from reverse use of the distributive property, which gives constant > terms that don't have 7 as a factor, as required. Define three functions w1(x), w2(x), w3(x), such that w1(x).w2(x).w3(x) = 49 for all x, and w1(0) = w2(0) = 7, w3(0) = 1. Now (5 a1(x)/w1(x)+7/w1(x))(5 a2(x)/w2(x)+7/w2(x))(5 b3(x)/w3(x)+22/w3(x)) = 300125 x^3 - 10375 x^2 - 360 x + 22 So why is that *not* a possibility? If I understand your terminology correctly (you still have *not* defined the concept constant term as you use it), the constant terms of the three factors are 1, 1 and 22. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Algebraic Closure > What is the algebraic closure of a finite field? > I'd be happy just to know the algebraic closure of Z/<2>. On the algebraic closure of two, by H.W. Lenstra, Jr. (Nederl. Akad. Wet., Proc., Ser. A 80, 389-396 (1977)). === Subject: Re: Key core error argument, stepped out A8EwTYfhf*u~,Eu,tf6$HN*MY&)u0G =N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@J6m5.EN?>Zh Xh;Y V|',x(js'Jfq02joVpj|#x linux) > Sorry to butt in, but we untold milllions listening in can take this > tripe in silence no longer. The poster James Harris *should* be > apologizing to this newsgroup and others for leading so many of us > astray for YEARS, but instead, he's quibbling as he appears > hell-bent on continuing to obfuscate. James S. Harris has led so many of us astray for YEARS? Who? If you're one of those victims, I wouldn't be so quick to publicly proclaim so. Good thing that you've obfuscated your email address, because I think comments like this will draw Nigerian scams and penis enlarging spam like honey draws flies. Perhaps you mean that JSH has *attempted* to lead so many of us astray for years -- although, I'm not sure that's accurate either. Instead, he's attempted to convince us of his mathematical arguments, which he probably believes are valid but which evidently are not -- not that I care to enter the discussion with him on this point. -- Contrariwise, continued Tweedledee, if it was so, it might be, and if it were so, it would be; but as it isn't, it ain't. That's logic! -- Lewis Carroll === Subject: Re: Naive Q: Set theory, logic - which comes first? > [.snip.] >Natural numbers pose a special problem since they may be viewed in two >different ways. > Actually, they can be defined in many, in many, many different ways, > and viewed in many, many, many different ways. It seems to me that there are essentially two ways of viewing them. >We may view them either as something defined by >induction, or as the smallest algebraic structure which is closed >under the operations 0, 1 , +, and * (theese being subject to the >usual laws). > As far as I can tell, {0,1} satisfies the hypothesis you give, through > suitable definitions. Presumably, you mean the smallest subset of > R/Q/Z which contains 0, 1 and is closed under + and *... Yes, or natural numbers as an algebraic structure characterized elsehow (for example, adding an axiom saying that there are infinitely many elements would do the trick). > But there are plenty of other ways to characterise the natural > numbers; they are the smallest non-finite ordinal; they are the first > singular cardinal; Here you view the natural numbers as an algebraic structure. The number 1 is simultaneously a natural number, an integer, a real number, an ordinal number, a cardinal number, and who knows what else. This is because the natural numbers form a substructure of the various other structures. Suppose now we define natural numbers in the same way that we define finite lists or binary trees. I do not see any way of viewing the latter two structures as algebraic structures, and therefore I think it is fair to say that natural numbers defined this way are not an algebraic structure. Maybe I am wrong, though. Mattias === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >Explain to me how this model applies to an attractive >electric field. >What are the ping-pong balls and where are they coming >from? > Don't worry about it Randy. I doubt if you know what a differential equation > is. Brilliant :-) Dirk Vdm === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) > Let's be serious for once. Consider an object being accelerated by a idealistic jet of water or a > continuous 'stream of elastic ping pong balls'. What is its subsequent velocity > pattern? >>Explain to me how this model applies to an attractive >>electric field. >>What are the ping-pong balls and where are they coming >>from? >> Don't worry about it Randy. I doubt if you know what a differential equation >> is. >Brilliant :-) All right. Now tell me which diff. eq. is on the number 2 ball in the Mass. Millions lottery. /BAH Subtract a hundred and four for e-mail. === Subject: Differential Equation Hi there. Does the differential equation (d^2/dt^2-grad^2+m^2)f=0 for m real have a solution in the space of Schwartz functions? I don't believe so, but would feel much better, if one of you could gie me a definite answer. Benjamin Bahr === Subject: Re: Usenet Posting Guide? > I'm considering writing up a guide to posting on newsgroups. Why? Why would your guide be better than any of the very many guides available... > I wouldn't put in a lot of technical things, nor would I talk about > other news readers or means of posting besides Google Groups ...especially when your guide is obviously going to be so limited? > Or I could talk about the real Nettiquette, and which ones you not only can break but you *must* break and which 'rules' *must* you break? > if you're trying to break through. === Subject: Re: Solving definite integral problem - newbie question > I am studying for an exam and don't have access to my instructor > today. > Can someone help me with the following? I have a graph of the right > half of a bell curve, a portion of which I must solve. > The portion is from 0 to 1 (that is x=0 to x=1) and the equation is > y=e^(-x^2/2). > That is, e raised to (negative x squared, divided by two). > I just need some help in the proper approach. I have been given the > answer, but so far nothing that I have tried has helped me to get the > answer. > Pam Pam === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) > Let's be serious for once. Consider an object being accelerated by a idealistic jet of water or a > continuous 'stream of elastic ping pong balls'. What is its subsequent velocity > pattern? >>Explain to me how this model applies to an attractive >>electric field. >>What are the ping-pong balls and where are they coming >>from? >Don't worry about it Randy. I doubt if you know what a differential equation >is. Well, you could write one and we could see. But you don't write equations, do you? Meanwhile, why don't you explain to me how this model applies to an attractive electric field. What are the ping-pong balls and where are they coming from? Or do you really think nobody noticed that you avoided answering any questions about your model? - Randy === Subject: Re: Bible 1, Darwin 0! And we are talking pure archeology? >> It's an old tradition. On sci.physics, Think for yourself! has always >> meant Learn to think the way I do! It was never an acceptable option >> that thinking for one's self should lead the thinker to conclusions of his >> own. >Nonsense. No, then what to all the independent thinkers who want us to throw off the religious SR thinking and QM brainwashing really want for us to do if we throw off our brainwashing and religious indoctrination? What do they mean by brainwashing and religious? - Randy === Subject: Re: Key core error argument, stepped out A8EwTYfhf*u~,Eu,tf6$HN*MY&)u0G =N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@J6m5.EN?>Zh Xh;Y V|',x(js'Jfq02joVpj|#x linux) >> Fascinating perspective. >> Isn't it? > So you have coprime with one clear meaning in a ring like integers, > but things are different in a field. No, it has the same meaning in the field, but the special properties of a field make that meaning trivial. > Notice then that 3 and 6 are coprime, but 6 still has 3 as a factor, > so in fact, coprime simply loses any relevance. > I think that's telling. > Part of my point here is that the mathematics you've taken for > granted, with lots of definitions that seem ok, goes off into some > interesting places. But everyone else here seems to already *know* that the standard definition of coprime yields a trivial concept for fields. Mathematicians have *not* taken their definitions for granted. They accept that a concept may reduce to triviality in some settings. A function f:X -> Y is 1-1 if, for all x,x' in X, if f(x) = f(x'), then x = x'. If X is a singleton or empty, then every function is 1-1. So, for certain sets X, this definition yields a trivial concept. (Similarly, if Y is empty, then all functions with codomain Y are 1-1.) So what? That's the price of consistency (a price, apparently, that JSH is reluctant to pay). >> Actually, in the real numbers, ANY TWO NONZERO NUMBERS ARE >> COPRIME. That's because, given any two nonzero real numbers x and y, >> any common divisor of x and y is a unit. >> >> Well, then every real number but 0 is a unit follows from that >> position. >> As that is the definition of a field, it looks about right. > So mathematicians take these positions, which not only go against > common sense, they don't make sense in general, pushing definitions. > So you have this broken word coprime which has no use at all if > you're in the field of real numbers. What is your preferred definition of coprime for fields? For what elements x and y of, say, R ought the sentence x is coprime to y be true? For which ought it be false? If you think that coprime *shouldn't* be trivial for fields, suggest how you would define it to avoid triviality. -- Jesse Hughes [I]t's the damndest thing. There's something wrong with every last one of you, and I *never* thought that was a possibility. But now I feel it's the only reasonable conclusion. --JSH sees some sorta light === Subject: Re: Naive Q: Set theory, logic - which comes first? >Two questions whose answers I am seeking are: >1) What exactly might we mean by representation? >2) How do we define a mathematical object if we do not want to >identify it with some representation of it? > This is done quite often. The Peano Postulates CHARACTERIZE > the positive (or non-negative) integers. There are many other > ways to characterize them. So we say that 0 is a natural number, that the successor operation takes us from a natural number to another natural number, and then list properties that natural numbers satisfy. But what we say is consistent with their being sets, functions in lamda calculus, or something else. We need to exclude such interpretations, but how? Mattias === Subject: Re: Cantor's diagonal number: why do we just ASSUME such a number exists? >I am of the firm belief that you and the author of this thread are one and >the same (and possibility just an alias of JSH, which would explain his >immaturity). Both of you sign your name in the same way and right your age >afterwards. Both of you refer to proofs but do not provide links to them >(Charlie, neither you nor Justin have websites with your name). You both >sign with different names than the email address states. Lastly, both of >you are wrong. You cannot state that the natural numbers are uncountable. >By definition, a countable set is that of the same cardinality as the >naturals. By the way, Nathan how have you been through University at age >11, and Charlie, how have you finished law school by age 9? You're thinking linearly. Nathan's age does not progress linearly. He's been 11 many times. Gets up to about 13 or 14 and recycles. - Randy === Subject: Re: Unit interval homemorphic to Circle help >>The map p is already continuous, as I and S^1 are compact metric the map >>p is >>automatically closed (i.e., if A is closed then so is p[A]). >> >Why does a continuous map from a compact space to a compact space >necessarily >have to be a closed map? Is this what you are implying? Yes, as A is closed it is compact, as p is continuous p[A] is compact, as p[A] is compact it is closed in the metric space S^1. KP -- E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI PHONE: +31-15-2784572 TU Delft FAX: +31-15-2786178 Postbus 5031 URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft the Netherlands === Subject: Re: Uncle Al is Sadistic . >>I think part of the problem is that we have such poor measures of what >>it is to be educated. The College Board only claims that the SAT is >>a very good predictor of first-year college grades; the average >>correlation between SAT scores and _freshman_ grades is +0.52; the >>correlation to being educated and smart is almost certainly quite a >>bit smaller. >Aha. The only problem is, we've (currently) no measure which gives a >of deciding (in this field as in any other) that you won't use any >measure until you've one which is perfect. I'm not arguing that we need a perfect measure. My point is that whether the measures used today are good enough is open to debate. We base a very large proportion of our judgments of who is smart and capable of being educated on the results of multiple choice examinations. There is at the very least a significant body of anecdotal evidence that the skills required to do well on such tests are different from those we would otherwise consider desirable in students. My personal experience with standardized testing is somewhat mixed. On the one hand, I did well enough on standardized tests to end up admitted to graduate school in chemistry (and yes, I did graduate). On the other hand, I'm fond of using a personal story as a cautionary tale; my junior high school guidance counselor advised me not to take algebra as a high school freshman because I didn't have the necessary aptitude. (To this day, I'm still not sure how he arrived at that judgment, although I'm pretty sure it had something to do with test scores; I wonder if he knows that I actually enjoyed P Chem). My experience on a university faculty graduate admissions committee also indicated that standardized test scores are a highly imperfect measure. [snip discussion on the Washington DC school district] I don't have actual budget figures. I was not looking to defend the performance of DC schools; my first statement acknowledged that they are a mess. My point was that there are a lot of confounding factors that don't get controlled in the SAT points per dollar metric. Also, DC will always be an outlier in such comparisons because their statistics often end up being the orange in a barrel of apples. Their finances are not exactly comparable to other states because they have all the difficulties of an urban school system without the less-expensive suburban and rural systems that get factored into the performance of every other state. Their finances aren't exactly comparable to other big city systems because the DC school board handles some functions that are paid for at the state level in other jurisdictions. [Mr. Meron continues:] . >For sure. But they are not going to be solved by hiding behind ohh, >it is so complicated, we can't really measure what we're doing, so >lets just keep spending and hope for the best. Anything that is >fundamentally sound can only benefit from some serious questioning. I agree that serious questioning is beneficial. Nevertheless, I'm not hiding behind that particular straw man; I would observe that unintended consequences can be particularly surprising when one uses simple and perhaps inadequate measures to solve a complex problem. Urging caution so that we better understand the nature of the problem before we fix it is not the same as saying keep on spending and hope for the best. George ********************************************************************** Dr. George O. Bizzigotti Telephone: (703) 610-2115 Mitretek Systems, Inc. Fax: (703) 610-1558 3150 Fairview Park Drive South E-Mail: gbizzigo@mitretek.org Falls Church, Virginia, 22042-4519 ********************************************************************** http://www.newsfeeds.com - The #1 Newsgroup Service in the World! -----== Over 100,000 Newsgroups - 19 Different Servers! =----- === Subject: Re: Graduate algebra book Perhaps the use of snotty is different in the UK than it is here in > the US. Chez nous, snotty is often used by the person who feels > abused. It is used to indicate that one's counterpart is elevated by > airs of superiority, unable to maintain a tone of equality. Among us, > it would be called a put-down, rather than abuse. > But an unwarranted use of a perjorative term is surely abuse > in anyone's lexicon? A fair point to raise. I first say that I am not party to the dispute, nor do I intend to be. I graduated in math over 35 years ago and have hardly used it since then. I read this newsgroup only for interest. (And, if I ever get to retire, I may want to study algebra and other topics again.) That said, let me observe that I have seen contentious remarks in this thread. Rather than raise the level of contention, it may be preferable to make remarks that are dismissive put-downs. We cannot all be like Victoria, whose famous dismissive put-down was We are not amused. She had position, whereby her remark carried some weight. Frankly I think the best answer to the original post, wanting an appropriate graduate-level text, might come from faculty rather than from those who have completed a certain level of studies. I do not have time during the week to search but I am certain I have seen, someplace on the Web, a complete book which is intended to be an appropriate first-year graduate text in algebra. Perhaps I may have time on the weekend (the 11th is a holiday and I am taking the 10th off) to research that point. David Ames === Subject: Re: Usenet Posting Guide? > I'm considering writing up a guide to posting on newsgroups. > Why? Why would your guide be better than any of the very many guides > available... Actually there are several already in existence. I will have to check with my friend for the URL, but there is a good one. Unfortunately people take advantage of ignoring it too often. Norma > I wouldn't put in a lot of technical things, nor would I talk about > other news readers or means of posting besides Google Groups > ...especially when your guide is obviously going to be so limited? > Or I could talk about the real Nettiquette, > Just out of interest, which 'rules' can you bend? > and which ones you not only can break but you *must* break > and which 'rules' *must* you break? > if you're trying to break through. === Subject: Re: Jaynes' book on probability - thoughts? As a matter of interest, didn't Jayne's book used to be available on-line? Is it still? Or am I mistaken? -- === Subject: Re: Equation Object converting? > Probably can do so with a quick macro. > To what version of Word are you converting? That is MS Word 2002. === Subject: Re: Product of Reals >> Is it possible to determine what the uncountable product of all real >> numbers in the interval [0.5, 1.5] is? Intuitively it seems like 1, >> but is this concept studied in general anywhere (e.g. the concept of >> uncountably infinite products or sums of real numbers)? >> Lookup the term summable family. This is a concept which allows definition >> of convergence for any numbers a_i with i in I where I is an arbitrary >> index set. By applying exp-log-formalism (transform your product into a >> sum) you get the analogue theory for products. >> A necessary condition for such a family to converge is that its support (the >> largest set J subset I such that a_j <> 0 forall j in J) is countable - >> this means that your product cannot converge. >> Because equally well than your reasoning I could argue that it converges to >> any other real number <> 1. Cf. Riemann's rearrangement theorem. > This is something that I have wondered about. Occasionally in > previous threads the idea of summing an uncountable set of numbers has > come up. It is usually commented this is not possible unless all but > a countable number are zero. However my reaction has always been: is > there any definition for infinite sums of any sort other than the > common series. Suppose {a_i} is a family of real numbers, where the subscripts i range over some (possibly uncountable) index set I. If S is a finite subset of I, we can consider the finite sum A_S = sum_{s in S} a_s. We say the family {a_i} for i in I is summable (and the sum = L) if, for every epsilon > 0, there exists a finite subset S of I (depending on epsilon) such that | A_S' - L | < epsilon for every finite set S' such that S subseteq S' subseteq I. This can be stated more concisely by saying that the net of finite sums converges to L. > I wondered about summing the values of a function f from a set to R or > C. I wondered what sort of structure the set would have to have and > what class of functions could be handled. Even when the set is > countable, the answer is not obvious since the sequence can affect the > result. I mentioned in a recent thread, the distinction between > absolutely and conditionally convergent series. In the definition above there is no structure imposed on the index set I. It does not even have an order. It does turn out that with this definition, a summable family must necessarily be zero except for an countable number of terms. If uncountably many terms are nonzero, then for some n we must have | a_i | > 1/n for uncountably many i, implying that the sum diverges. > This is the first reference that I have noticed to a generalisation of > infinite sums. Unfortunately I could not find much on Summable > families on the net. A search of the groups revealed little but this > thread. A search of the web found more but many seemed to mention it > only in passing or were in formats that I could not read. Mathworld > did not seem to have anything. > Can you point me somewhere or give a bit more detail? > For non-countable sets, could measure theory be considered a > generalisation of these sums? Yes. In fact, counting measure is a particular type of measure, and the net of finite sums approach amounts to integration with respect to counting measure. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Naive Q: Set theory, logic - which comes first? ..................... > The problem is exclusively related to the phrase defined by induction. > just as it has been for the last several thoushand years. > Since you have to know something about geometry rather than > natural numbers to define inductive logic. This is totally false. In fact, one can do geometry without using inductive logic at any point, and a key part of the Peano Postulates is the induction postulate. BTW, inductive logic is usually used for inference from observations, and as such, has nothing to do with mathematical induction. Geometry is unnecessary for this, but properties of the real numbers as such is of great importance. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Naive Q: Set theory, logic - which comes first? ....................... >> Natural numbers pose a special problem since they may be viewed in two >> different ways. We may view them either as something defined by >> induction, or as the smallest algebraic structure which is closed >> under the operations 0, 1 , +, and * (theese being subject to the >> usual laws). >> Isn't the field of two elements a smaller algebraic structure which is >> closed under the operations 0, 1 , +, and * (theese (sic) being subject to >> the usual laws) than the natural numbers. >You are right! Of course, Z_2 is the smallest such structure. One >needs to add some axiom that forces there to be infinitely many >elements. This is not enough. The algebraic closure of Z_p has this property, and it is not a smallest. The closure of Z_p under solutions of quadratics is (I believe) one of the smallest, although this takes proof. I do not know how to define characteristic 0 without first having the integers. Alternatively, one could add < to the list of primitives; this will get the natural numbers, even without *. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: B-Spline Extension Hi All, I am trying to fit a b-spline to a set of 3d points. I have a couple of questions that I can't see the answer to at the moment. 1. How, if at all, can I extend the curve beyond the range of the points I have. I want to fit a curve to my data points, then extrapolate to see where the curve is heading. 2. Does the curve have to go through the start and finish points. Can I make it so it just fits the data points better, rather than going directly through the start and end points I have. Adam === Subject: Re: More symmetry between derivative and antiderivative? > While dx/dt at x has only one parameter x, > the integral from x1 to x2 has two parameters x1 and x2. > Iff x2 was always equal to zero, then derivative and antiderivative > would be more similar to each other. > Mathematicians might feel this idea somewhat cheeky. > However, it has a physical background. >> Actually, the fundamental theorem says that the former is the opposite >> of the latter when the first bound remains constant and is the point >> where the function takes the value 0. >> I hope I've been clear. >I do not even understand what fundamental theorem you are referring to. >Please get more specific. >Antiderivative is the same like integral. Therefore one could expect >some similarity. >I blame Rn Descartes for the missing fix point of our coordinates. Antiderivative is not the same as integral; it gives a means of calculating integrals. Integral is FAR older than derivative. In fact, integrals with respect to discrete measures were evaluated approximately 5000 years ago, and all measures (and most integrals) are limits of these. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Science and Vectors One of my students has given me a chemistry math problem which he found and I can't see how to solve it. It reads like this: Consider a molecule such as cyclohexane in the trans configuration in which as the bonds are the same length |> c | | b d | | <---------------< | | | | >----------------> e | | a | | f < | (Sorry for the bad diagram but the arrows chase each other round a->b->c->d->e->f) and the angle theta between two consecutive bonds is constant throughout the molecule. (Notice also that a and d are parallel). The angle of skew alpha between two bonds are separate by a single intermediate one is defined as the angle which these two bonds appear to form when viewed along the intermedite one. Show that this is given by cos alpha = - cos theta / 2 (cos (theta/2))^2 Does any one know how to solve this? I would really appreciate it! Sarah === Subject: Re: naive geometry questions >The other surface is the catenoid, formed by rotating a catenary, > Q: How do you make a catenoid? > A: Pull its tail. > How do you make a hormone? :-)) Who says Geometry! when he has grown up? Dirk Vdm === Subject: Re: I can't stand it anymore amanda replied: >>amanda replied: >I have been biting my tongue about the IQ test but I can't any more. >How reliable is a test that use the term Asian to represent the most >diverse of ethnic and cultural groups? >>Are you Asian? > Yes. Not oriental though but grew up in that region. So you recognize and understand the word. It has utility and meaning. Why do you get so upset about it? Many hate the word American and raise all manner is idiotic arguments as to why it is wrong (and most of them defend the term native American). Do you have similar issues with the word American, or issues like the ones you have with the word Asian? Rich >>Rich === Subject: Re: Your Attention Please, Sci.* NG This is very serious attempt indeed. My life is at stake. > Some day in history, things are supposed to stop falling. Why Not NOW? > -Abhi. Pull the ripcord, Abhi. Pull it NOW! [Another Darwin Award entry on the way, I think] Tony. === Subject: some coin tossing If I toss a fair coin n times, what is the probability that the outcome of my experiment will contain m consecutive heads (m>The map p is already continuous, as I and S^1 are compact metric the map >>p is >>automatically closed (i.e., if A is closed then so is p[A]). >Why does a continuous map from a compact space to a compact space >necessarily >have to be a closed map? Is this what you are implying? > Yes, as A is closed it is compact, as p is continuous p[A] is compact, > as p[A] is compact it is closed > in the metric space S^1. > KP > -- > E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI > PHONE: +31-15-2784572 TU Delft > FAX: +31-15-2786178 Postbus 5031 > URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft > the Netherlands === Subject: Re: Uncle Al is Sadistic . >> largely *inherited*. This fairly obvious fact is largely denied by >> society and researchers, and if it true, this fact will have serious >> societal effects that are not being dealt with at all. > The genetic inheritance of mental efficiency is NOT obvious. It is very > hard to isolate genetics from upbringing since they are so intertwined. > The only way to do it with any accuracy is with studies on identical > twins who have been brought up seperately and differently. Actually, there are a lot of studies that make the connection pretty obvious. I don't have the book here, though, and it's been years since I read it; I can't remember the details. If you're curious you can either find a copy of _The Bell Curve_ yourself, or I can get back to you on Thursday. -- - Laurel * * * http://amberdine.com === Subject: Re: Key core error argument, stepped out Adjunct Assistant Professor at the University of Montana. For crying out loud. You should stop cutting and pasting incorrect > stuff. In fact, you should stop cutting and pasting, period. Why not > post a link to your original post, instead, if all you are going to do > is repeat it verbatim, ERRORS INCLUDED? [.snip.] >>Why don't you off Arturo Magidin? >> Three times I offered to do so if you told me to stop posting >> replies. You declined to do so. >> Are you doing so now, in your oh-so-courteous way? >> Just say so: Do you want me to stop posting with comments about your >> statements? Yes or no? >You're ing kidding me!!! You mean you're asking *me* whether or >not I want you to stop posting in my threads? I asked you three times before and you decline to ask. I was wondering if you had a martyr complex of some sort. >Quit posting in my threads Arturo Magidin. I will not stop posting in your threads (no such thing, this is a public forum); but I will keep my word from the offer before. I will not follow up on anything else you post, not even by piggybacking. I may reply to others who ask specific mathematical questions. Just remember, next time you try to get on your high horse about being interested in truth, that you are a hypocrite. Next time you claim that the fact that people don't reply means you are correct, remember what you just did. And next time you complain about people being rude, remember your post here, and you oh-so-courteous parting shot: > off. [Gabriele etti] has left a vast body of writings... in which he has attempted to prove the truth of his unorthodox interpre- tation of medieval literature. They present a formidable record of unsystematic research in which we see an enthusiast plunging farther and farther and farther from the logic of facts and good sense until truth is lost in the dreadful nightmare of an idee fixe. There is no real evolution of the Theory although it grows and expands until it embraces ever wider horizons. The numerous inaccuracies of deduction, mis-statements of historical fact, and self-contradictions...have caused critics to turn away from them in disgust... [...] It is impossible to read far... without realizing that we have to deal with a work of faith and imagination rather than of reasoning. There is an appearance of reason, for the author is set on proving by logic the truth of what he already believes by intuition. The truth is plain to him and he cannot comprehend why others do not immediately accept it, but as they desire demonstration he has multiplied his proofs. It is the redundancy and confusion of a prophet expounding by a familiar method the truth revealed to his own simple soul in a flash of inspiration... In such work as this... it is idle to look for the calm reasoning of a scholar; we do not find it, and there is little or no advantage in attacking the obvious inconsistencies and absurdities that abound. -- E.R. Vincent, _Gabriele etti in England_, quoted in _The Shakespearan Ciphers Examined_, by William F. Friedman and Elizebeth S. Friedman Arturo Magidin magidin@math.berkeley.edu === Subject: Re: A NOTy problem. === Subject: A NOTy problem. > I have been puzzling over what ought to be a simple logic problem >A crime is committed by one of the following; >Alan, Bob, Chris or Dave. >Each makes a statement to the police but three of them lie. >Alan says, I didn't do it. >Bob says, Alan is lying. >Chris says,Bob is lying. >Dave says, Bob did it. If Chris told truth: Alan told truth; two soothe sayers. Thus Chris lied; Bob told truth; Alan did it. Alan lied; Dave is wrong ---- === Subject: Re: Key core error argument, stepped out Adjunct Assistant Professor at the University of Montana. [.snip.] >> Using Dot's proof that the values of a polynomial with algebraic >> integer coefficients are always divisible by an integer if and only if >> each coefficient is a multiple of that integer (in the algebraic >> integers) gives you that the statement is correct for polynomials in >> A[x]. > I had not noticed Dot's theorem. The x^2 + x example seems to belie >it: for any integer x, x^2 + x is divisible by 2, but the coefficients >are not multiples of 2, and both of the coefficients are algebraic >integers. But the values of x^2+x are not multiples of 2 for every algebraic integer value of x: if x=i, then you i-1, which is not a multiple of 2. > You must therefore be thinking that the domain is algebraic >integers as well. Indeed. [.snip.] === Subject: Re: JSH: My victories get lost >>There are quite a few posters who reply in my threads. Now I've >>noticed that I trounce one poster and then another pops up and starts >>yapping. Later some poster who got his ass kicked is back trotting >>out the *same* crap. >> That's your impression. In _fact_ you have never trounced >> anyone here. >David Ullrich is a ing piece of dog. You know, the _next_ time you complain to my employer about me being mean to you it's going to be even funnier than the last time... >I think it's funny that I can call a professor at Oklahoma State >University a ing piece of dog knowing that he'll keep replying >in my threads. >You see, he has to keep replying pushing the same old lies. >He's stuck. He's trapped in something that he can't get out of, so it >doesn't matter what I call him, or what I say about him, he has to >come back. >You see I'm the person who has the correct math argument, so posters >like David Ullrich or Arturo Magidin are *compelled* to reply out of >fear that if they go away, then I'll get some people who'll pay >attention to the truth. >So David Ullrich, the math professor at Oklahoma State University, is >demeaned by me as the piece of ing dog he is, and he *has* to >keep coming back. Yeah, you got all that right. Doesn't change the fact that you've never trounced anyone here, except in your imagination. >James Harris ************************ David C. Ullrich === Subject: Re: JSH: Factorization P(x) = 2(x(x+1)/2 + 1) Adjunct Assistant Professor at the University of Montana. >> So in ANY ring that contains the integers, 7 and 22 are coprime >> (under either definition). > That's true in any ring R since R contains a homomorphic image of Z, > Z -> Z*1_R, i.e. any ring is a Z-algebra. But any ring homomorphism > must preserve the relation 22 - 3(7) = 1. >> Hmmm... Only if you assume that ring morphisms map 1 to 1, which >> is not necessarily a given either. Even assuming rings have a 1, >> the zero map is usually considered a valid homomorphism, >> and your conclusion would be incorrect there. >For Rings (with 1, as I assume above) ring morphisms must preserve 1, >so the zero map is not a morphism of Rings with 1. Granted; like I said, if you assume that ring morphisms map 1 to 1. On the other hand, you lose some things by assuming that: you don't get isomorphic copies of the rings in direct products. Some conventions are better than others, depending on the situation. > If, as you claim, >one considered the zero map as a valid homomorphism of Rings with 1 >then basic theorems on rings would fail, e.g. the image of the zero >morphism would fail to be a subring (except if the target ring is 0). >Therefore my above quoted statement is in fact correct as written. If by Ring you meant with 1 and by homomorphism you meant preserve 1, then ->of course<- they were correct. But you must grant that not ->everyone<- means both things, and that there was at least some sense in my stating so explicitly... [.snip.] === Subject: Re: How to define a function to be smooth? >Hey all >When we say a function f(t) is smooth, does this mean that >f has infinite differentials with respect to t? That's _often_ what it means - sometimes it means less than that. >Or any other formal definition on this? >Fred ************************ David C. Ullrich === Subject: Re: How to define a function to be smooth? > Hey all When we say a function f(t) is smooth, does this mean that > f has infinite differentials with respect to t? Or any other formal definition on this? >>The number of continuous derivatives that a function possesses is a >>measure of its smoothness. >Well, yes. But more directly responsively to Fred: yes, in >many areas of mathematics, When we say a function f(t) is >smooth and don't specify anything further, we often mean >that f is infinitely differentiable. This is particularly >common among differential topologists. I have a feeling that >real analysts, for instance, will tend to be more precise here; >but maybe not. We may have a larger zoo of measures of smoothness than you do, but infinitely differentiable is still what I assume smooth means if there's no reason to assume something else is meant. >Lee Rudolph ************************ David C. Ullrich === Subject: Re: Key Core Error Argument [snip umpteenth repeat of previous flawed argument] I thought you were going to just post your proof elsewhere and stop cluttering up this newsgroup. You've given a URL, now go away. -- 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: I can't stand it anymore In sci.chem Doug Norris writted: :>Why are people using the racist IQ tests as God's given ruler to :>measure intlligence? : Probably for the same reason that people feel the need to generalize : about all IQ tests. Doug, Why does everyone in the US insist on generalising about people's need to generalise about IQ tests? Just curious... Gavin === Subject: Re: Skeptical Inquirer UFO I'm sensing that you're offering some interesting insights to this discussion, but frankly I can't figure out what it is you're trying to say. Sorry! > ah, so; am I to infer that PBK works both sides > of the conspiracy?... and here, > I thought it was Skull and Bones. > do you know of the cache taht Roswell has with WW2?... > I know of two things, that Art Bell (et al ad vomitorium) never mention > (although others have mentioned one of them, > on his show ... I stopped listening to taht **** > over 3 years ago .-) > as the co-author exposed him during the book tour, > Corso's thesis is that human beings cannot create ideas, > withe corrolary that they were not made in the image of God. > (I didn't know, when he came to borders without Corso, > that he was deathly ill; probably made it easier > to drop his little clues; are they in the book?... the funny things was, > I was the only one in the audience who asked any hard questions; > everyone else must have been above top secret !-) > mysterious aerospace reports and events (see www.jamesoberg.com) and share > what I've found, for discussion, with no directives, constraints, or other > Jim Oberg > Phi Beta Kappa, Ohio Wesleyan University, 1966 > on by systematically ridiculing those of us who suspect the truth about this > phenomenon, all the while being privy to above-top-secret knowledge that > would prove us right? One wonders.... > C. SCOTT LITTLETON > President, Phi Beta Kappa Alumni in Southern California > --ils duces d'Enron! > http://tarpley.net/bush8.htm > http://www.wlym.com/PDF-SpReps/SPRP13.pdf === Subject: Re: DESIGN OF A PRIMITIVE NANOFACTORY > Design of a Primitive Nanofactory http://www.geocities.com/drjosemariachi/jay_faq.html#bb Troll FAQ for Jai Maharaj (Hindi for cracked athletic cup) -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Key Core Error Argument > Define three functions w1(x), w2(x), w3(x), such that w1(x).w2(x).w3(x) = 49 > for all x, and w1(0) = w2(0) = 7, w3(0) = 1. Now > (5 a1(x)/w1(x)+7/w1(x))(5 a2(x)/w2(x)+7/w2(x))(5 b3(x)/w3(x)+22/w3(x)) = > 300125 x^3 - 10375 x^2 - 360 x + 22 > So why is that *not* a possibility? If I understand your terminology > correctly (you still have *not* defined the concept constant term > as you use it), the constant terms of the three factors are 1, 1 and 22. Let's say you have f(x)/g(x) in the ring of algebraic integers. That is, both f(x), and g(x) give algebraic integers for algebraic integer x. Then necessarily, you have *another* algebraic integer function I'll call h(x). Understand? Now then, if you have h(x), then you can set x=0 to get the constant term. Otherwise you'd have a backdoor to making numbers like 7, that are constants, into variables, which would be mathematically inconsistent. So if you have 7/w_1(x), and it gives algebraic integers for any algebraic integer x, then you have some function h(x) = 7/w_1(x), which STILL has to have a constant term of 1, found by checking at x=0. It's a simple but powerful technique Dik Winter, and no matter how much you try, you can't find a way to invalidate its results or make them inconsistent. So despite all your efforts Dik Winter, you end up with the same result: constant terms for the factors that are 1, 1, and 22. Apparently, you think writing the functions as ratios changes things, but *if* they're in the ring of algebraic integers, you can flatten them out. For other readers who don't understand what my original post is, it's a math proof. Yup, an actual real math proof, which means that it begins with a truth and proceeds by logical steps to a conclusion which MUST be true. Attacking a math proof, as you may know, is silly, as ultimately it relies on rather basic facts, which are simple enough that everyone accepts them, and in my case the proof relies on some basic algebra, the constancy of 7 and 22, and 22 not having 7 as a factor. I mention that because when it comes to knowing that you have found a proof, simplicity rules. Now if any of you have decided that algebra might be bunk, or if you think that 7 and 22 are variables, or wish to claim that 7 is a factor of 22, in the appropriate rings, then that's another matter, of course, as in you're no longer a part of the intellectual community, but are then a crank. James Harris === Subject: Re: JSH: Attacking a proof is attacking yourselves > Now I've found out that even though I've stepped out my proof and > explained in detail there are posters who just keep trotting out the > same things despite getting refuted. > Maybe it's because _you_ keep trotting out the same thing despite being > shown the errors in PAINFUL DETAIL! That's not true. I've stepped out the proof into 7 main steps. If you think there's an error, point it out. Math society needs to either play by rules or accept the consequences of not playing by the rules, which is that the world not believe what you say!!! If mathematicians can't accept a proof as simple as mine, where not accepting it means questioning basic algebra, the fact that 22 and 7 are constants, and that 22 does not have 7 as a factor in the ring of algebraic integers, then why should the world believe you on anything? Maybe you're confident that you can fool the world, like magicians with illusions, so that you can keep going confidently supported by a deluded world. But can you really fool yourselves? Think about the corrosion upon math society as student after student learns that even a math proof can be questioned if enough mathematicians don't care for the result. What good will a math prize be then, eh? What will it matter when everyone knows the game is rigged, so who knows what's really true, and what's just a truth that mathematicians have decided they want? Attack the proof and you attack the foundations that make your work meaningful. James Harris === Subject: continuation of partial measure Suppose we are given a set X, a family S of subsets of X. Also we have a function M_S : S -> [0, oo]. Let F be the sigma algebra generated by S. What are necessary and sufficient conditions on M_S such that there exists a measure M_F : F -> [0, oo] that coincides with M_S on S? -- reverse my forename for mail! === Subject: Re: B-Spline Extension > Hi All, > I am trying to fit a b-spline to a set of 3d points. I have a couple of > questions that I can't see the answer to at the moment. > 1. How, if at all, can I extend the curve beyond the range of the points I > have. I want to fit a curve to my data points, then extrapolate to see > where the curve is heading. > 2. Does the curve have to go through the start and finish points. Can I > make it so it just fits the data points better, rather than going directly > through the start and end points I have. Look up information on the difference between uniform and open-uniform B-splines. Note that you're talking about a bunch of b-splines, not one b-spline, so only the first and last b-spline are relevant to point ) B-spline is an interpolation curve, not an extrapolation (it's not designed to 'fit' points, rather than be an estimate). For a curve that completely fits a given set of data-points, you might be better of with Lagrange curves. Look up the recursive algoritm of Neville for those. -- Quaternion === Subject: Re: Two coin flip/ clarification for C Bond > But then what is the point of conversing with someone who doesn't read > what you write? End of conversation. > Randy, > them, either deliberately or because you ignored what > preceded it AND what I said about it in a followup. > You continue to quote my entire texts, obviously unread as > you make no response to anything there, up to the last couple > of lines, which you again read out of context. > As I said, end of conversation. Bye bye. Randy, I'll write this slowly. (I know, that like Ullrich, you think slowly) I read your post. I responded where I disagreed. Where I didn't respond, I agreed. You're good at catching typo's. Eldon Moritz === Subject: Re: Real Roots of Polynomial >[...] >> For a given polynomial you can build a particular matrix from its >> coefficients, such that you can read off the information about the >> numbers of (real) zeroes from the rank and signature. >> I have forgotten the details, it should be in any book on real algebra. >> Marc > Is this not the Routh-Hurwitz Criterion? I do not know. I have met the criterion under the name (Jacobi, Borchardt, Hermite)-Theorem Like Sturm, it works for a real-closed field, but it is enough to consider R (the reals) here. The method runs like this: One may assume f is of degree n and has leading coefficient 1 Let a1, ..., an be the zeroes of f in C. For any natural number (including 0) define the rth-Newton sum w(r,f) = a1^r + ... + an^r Then each w(r,f) lies in R because it is stable under complex-conjugation. One can compute the w(r,f) recursively without knowledge of the a1,...,an. Build the nxn Hermite-matrix H(f) of f, defined via H(f)[i,j] = w(i+j,f) with indices i,j =0... n-1 Now there is Theorem (Jacobi, Borchardt, Hermite) rank(H) = # of distinct zeroes of f in C signature(H) # of distinct zeroes of f in R Marc === Subject: Re: continuation of partial measure > Suppose we are given a set X, a family S of subsets of X. Also we have a > function M_S : S -> [0, oo]. > Let F be the sigma algebra generated by S. What are necessary and sufficient > conditions on M_S such that there exists a measure M_F : F -> [0, oo] that > coincides with M_S on S? That's not so easy. See the pi-lambda theorem and monotone class theorem for partial answers. === Subject: A 3rd Grade Word Problem---HELP My 3rd grade son brought this word problem home the other day and he was given the answer (127). His job, for extra credit, was to figure out how to get 127. I'm no genius but not a dope either. I couldn't figure out how to get 127. Nobody in the whole neighborhood could figure out how to get 127. Is this something of a trick question or is there something in the wording that I am missing? Any Help???? The Problem: You know a very good story. On Sunday you tell the story to a friend. On Monday you tell it to two new people. (So far, a total of three people have heard the story). Each day after Monday, you double the number of new people you tell the story to. What will be the total number of people that will have heard your story after you tell it on Thursday? === Subject: Re: Sets before logic >Random thoughts on creating a theory of sets prior to a theory of >propositions and quantifiers: >Let's start with the empty set, 0, and logical identity, =, then we can >define T, for true, by Without at least a grammar of propositions, what does this mean? In fact, what is a set? In most set theories, set is a primitive notion, but in any case, we need to already have rules of logic to discuss anything. > T =def 0 = 0 >Let's define ordered pair a la Kuratowski, then we can define >conjunction by How do you define ordered pair without logic? To do so, one needs to prove theorems. The Kuratowski definition is not the only one, and in fact one can do a lot without them. > phi & psi =def = How does this define conjunction? One needs logical methods to prove theorems. >(So formulae are sets--but why not? If our logic is to be infinitary, >with infinite conjunctions say, we'll want formulae to be sets anyway.) >Now if-then is defined by > phi -> psi =def (phi & psi) <-> phi >(<-> is, of course, just another way of writing =; I told you we'd >want formulae to be sets.) I object! <-> is NOT logical identity. >Throwing caution to the wind, we'll allow ourselves to have a universal >set, so that we can define the universal quantifier by > (for all x)phi =def {x | phi} = {x | x = x} > F (for false) =def (for all p)p >(p a formula without free variables.) > ~phi =def phi -> F > phi v psi =def (phi -> psi) -> psi > (exists x)phi =def ~(for all x)~phi. >Now, what theory of sets will yield what logic of propositions and >quantifiers? What logic do we need to be able to carry out he set operations? -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: B-Spline Extension I have quickly looked up what you suggested. However, where I was looking all they were talking about was the interpolating curve that is created. How do I go about explorating this, or what other 3d curve fitting method do you suggest. The need to explorate is crutically important to what I am trying to achieve. Adam > Hi All, > I am trying to fit a b-spline to a set of 3d points. I have a couple of > questions that I can't see the answer to at the moment. > 1. How, if at all, can I extend the curve beyond the range of the points I > have. I want to fit a curve to my data points, then extrapolate to see > where the curve is heading. > 2. Does the curve have to go through the start and finish points. Can I > make it so it just fits the data points better, rather than going directly > through the start and end points I have. > Look up information on the difference between uniform and open-uniform > B-splines. > Note that you're talking about a bunch of b-splines, not one b-spline, so > only the first and last b-spline are relevant to point ) > B-spline is an interpolation curve, not an extrapolation (it's not designed > to 'fit' points, rather than be an estimate). For a curve that completely > fits a given set of data-points, you might be better of with Lagrange > curves. Look up the recursive algoritm of Neville for those. > -- > Quaternion === Subject: Re: Uncle Al is Sadistic . >>How many people die in America due to economic pressure on its citizens >>for every American that achieves wealth? > An interesting question; but does it assume wealth is a zero-sum game? > That assumption seems not to be historically supported. No. Capital wealth is distributed at ones death. Taxes first if not sheltered. > But since I am reading this in a math newsgroup, let's do some arithmetic. >>Minimum wealth being defined >>to sustain a yearly cost of $40KUS/yr for thirty years in ones >>retirement. > Wow! You certainly have a high threshhold for minimum wealth! I certainly > hope you are not retiring in the US Social Security system; maximum > payout there is about $15K (US)/yr. Your notion of minimum wealth would > put most people at or under the minimum _even while they are at the height > of the earning period_! If you read SS program information one can discover the notion that SS is an assist to ones retirement. Certainly below the poverty line if one only lives on what is advertised as an assistance to ones retirement. I would not call that wealth. There is a even a question of solvency of SS especially if starve the beaster philosophy continues. > Also note that you need not plan for 30 years of retirement. Save > what you will and then, upon retirement, buy a life annuity from a > reputable carrier. They will happily fund your 30 or more years in > retirement with the money saved from the contracts purchased by all > those who only lived for 10 years or less. hypothesis; I saved 15,000 dollars over my work life. Insurance carriers are going to fund me over thirty years at 40K/year? I don't think so. > More precisely, of every 100 people in a US age cohort, around 82 > will reach their 65th birthday. About 63 will live another 10 years; > about 34 of them will make it another 10 years after that; only 6 will > see their 95th birthday and none (well, 0.2 of them) will be around > only plan for less than 20 years of retirement. (Less, actually, now > that the nominal retirement age is above 65.) >>Approximately $2.7MUSis needed as a base. A game in the U.S. >>is to achieve capital growth over a lifetime. A $60K salary over 40 >>years is $2.6M. Certainly my math error here! 40K over 40 years is 1.6M not 2.7M I had used the 60k/year salary times 40 years to come up with a 2.7M figure which should be 2.4M > I have no idea where you got the figure of $2.7M . Thirty years of > $40K/yr payouts costs $1.2M up front -- much less, if you put the > money you do have at that point into a savings vehicle which pays any > returns along the way. You mentioned taxes and so on too, but if > $60K/yr is sufficient before you retire, how can it be insufficient > after you retire? Medical Bills. Check out the cost curves and extrapolate Even $60K/yr for 30 years only costs $1.8M. It seems > to me that in order to get a $2.7M-up-front figure, you must postulate > a combination of conditions something like this: that you need $60K/yr > before taxes (or other costs) after retirement; that you earn nothing on > your mega-dollar savings; that you continue to have the expense of saving > for retirement (?!); and that inflation averages 2.7% per year. > Would you describe yourself as a pessimist? Yes. Life expectancy and costs are projected to increase for certain social strata. You must not remember the double digit inflation just a few short years ago. Certainly the average inflation is not 2.7%;over what time? Can you provide a reference for inflation over the past 40 years? > BTW, you must be saving in an _extremely_ conservative retirement fund if > you manage a net ROI of approximately 0.4% per annum across 4 decades. > You called it a game; I think you need a new coach! There are no guarantees that savings will appreciate and these days financial types take into consideration the idea of deflation. > I must emphasize that you're failing to illustrate the severity of the > real situation if you assume a $60K salary is available during your > saving-for-retirement phase. That's very much above average > for an individual wage-earner in the US; using real data would show > that the nationwide situation is indeed more grim than you indicate. I was under the impression that the average is 46K/person not 46K per family. This is a two wage earner nation on average is it not? >>People in the US should be OUTRAGED. The social/economic/political >>system is skewed for the wealthy to continue their wealth at the >>expense of the majority in this system. > Personally I'm inclined to agree, but you haven't substantiated it > with this model. With accurate data, it's easy to show that most people > will be able to save only very little for retirement. But it's much harder > to lay the blame for this at the feet of any particular group; after > all, life certainly comes with no guarantee that people should be able > to enjoy a long retirement on their savings! > dave capital base for retirement is something that most Americans will not achieve in their lifetime under present business and government climates. boer === Subject: Re: Key Core Error Argument > Apparently, you think writing the functions as ratios changes things, > but *if* they're in the ring of algebraic integers, you can flatten > them out. I don't get it. I flunked flattening out. Can you explain? === Subject: Re: Key Core Error Argument Your proof is hard to follow. Is this a good rewrite of your proof? 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is in the ring of algebraic integers. 2. P(0) = 1078 3. P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 4. Let v = -1 + 49x, then P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 5. Let F(a(x)) = a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) 6. Let the roots of F(a(x)) be a_1(x), a_2(x) and a_3(x). 7. P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) 8. P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) = 1078 9. F(a(0)) = a^3 - 3a^2, which has roots 0, 0, and 3. 10. Let a_1(0) = 0, a_2(0) = 0, and a_3(0) = 3. 11. Let a_3(x) = b_3(x) + 3 12. P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) 13. P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). 14. P(x) has a factor of 49 as P(x)/49 = 300125 x^3 - 18375 x^2 - 360x + 22 15. Then (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) has a factor of 49. 16. The constant term of P(x)/49 is P(0)/49 = 22. 17. But for two of the factors of P(x), the constant terms is 7, which is NOT a factor of 22. Therefore, *none* of the constant terms of P(x)/49 as they multiply to give 22 can have 7 as a factor. Note: (By saying that 7 is NOT a factor of 22, I'm making a choice as to where the proof is going. Since I've been talking about algebraic integers, where 7 is NOT a factor of 22, it's natural to go with a choice where 7 is NOT a factor of 22.) 18. Given that the constant terms are independent of x's value, it must be the case that dividing P(x) by 49 divides the two constant terms equal to 7, by 7. 19. But to divide 7 from those constant terms requires dividing through two of the factors, so (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 from reverse use of the distributive property, which gives constant terms that don't have 7 as a factor, as required. 20. Notice that it's a rather short and direct argument, where if you accept that 22 does not have 7 as a factor, it's obvious enough what the constant terms of the factors must be as you go from 7, 7 and 22, necessarily to 1, 1, and 22, when you divide P(x) by 49. > The following proof steps through a rather basic argument which is key > in proving an over one hundred year old error resulting from > previously unexpected consequences resulting from the definition of > the ring of algebraic integers. > Note that ultimately the proof relies on 22 NOT having 7 as a factor, > and constant terms like 7 and 22, being constant, and not variables > dependent on x, which may seem like odd things to emphasize, but I've > faced posters who've gotten away with challenging those truths because > people seem unaware that's what they're doing. > 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is > in the ring of algebraic integers, notice that P(x) has a constant > term that is 1078. > 2. It can be shown that > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > where you should note that using v = -1 + 49x, gives > P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 > where the *same* polynomial has been put in a form which allows a > factorization into non-polynomial factors so that I have > P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) > where the a's are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > 3. Now let x=0, so > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's at x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3, and I've picked a_1(0) and > a_2(0) to equal 0, which leaves a_3(0) with a value of 3. > 4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I > have > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) > P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). > 5. Now P(x) has a factor of 49 as > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 > which means that > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) > has a factor of 49. > 6. However, the constant term of P(x)/49 is 22, which is verified by > again setting x=0, which gives P(0)/49 = 22. > But for two of the factors of P(x), the constant terms is 7, which is > NOT a factor of 22. Therefore, *none* of the constant terms of > P(x)/49 as they multiply to give 22 can have 7 as a factor. > (By saying that 7 is NOT a factor of 22, I'm making a choice as to > where the proof is going. Since I've been talking about algebraic > integers, where 7 is NOT a factor of 22, it's natural to go with a > choice where 7 is NOT a factor of 22.) > Given that the constant terms are independent of x's value, it must be > the case that dividing P(x) by 49 divides the two constant terms equal > to 7, by 7. > 7. But to divide 7 from those constant terms requires dividing > through two of the factors, so > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > > 300125 x^3 - 18375 x^2 - 360 x + 22 > from reverse use of the distributive property, which gives constant > terms that don't have 7 as a factor, as required. > Notice that it's a rather short and direct argument, where if you > accept that 22 does not have 7 as a factor, it's obvious enough what > the constant terms of the factors must be as you go from 7, 7 and 22, > necessarily to 1, 1, and 22, when you divide P(x) by 49. > James Harris > http://mathforprofit.blogspot.com/ === Subject: Re: A 3rd Grade Word Problem---HELP > My 3rd grade son brought this word problem home the other day and he was > given the answer (127). His job, for extra credit, was to figure out how > to get 127. I'm no genius but not a dope either. I couldn't figure out how > to get 127. Nobody in the whole neighborhood could figure out how to get > 127. Is this something of a trick question or is there something in the > wording > that I am missing? Any Help???? > The Problem: > You know a very good story. On Sunday you tell the story to a friend. On > Monday you tell it to two new people. (So far, a total of three people > have > heard the story). Each day after Monday, you double the number of new > people you tell the story to. What will be the total number of people > that will have heard your story after you tell it on Thursday? I don't see 127 either. It should be 31. If you keep telling the story up till and including Saturday, then you would have 127. Have a tolerable existence. Eli === Subject: Re: Why is math so difficult for some people? > Herman Rubin >>Not where I went to school. A perfectly valid prrof would be marked >>down if it was not elegant; quite properly, IMHO. >> I see no reason for this. This is not the way mathematics is >> done; first get the proof, then MAYBE look for elegance. >Taking away the elegance from ANY proof, transforms the proof into a >tedious, boring and mechanical procedure, which should interest no true >mathematician. >> Also, >> it is quite common for elegant proofs to hide the concepts; >I'd rather have an elegant proof which hides some of the concepts yet gives >the overall idea, than a mechanistic and non-interesting proofwhich boggles >the mind with inane details which are of no interest to the reader. Many of the so-called elegant proofs completely hide the concepts. Proving the Central Limit Theorem for sums of independent identical distributions is certainly the most elegant, but it hides virtually everything; it has no probability in it at all. This is also the case when the distributions are not identical. However, the somewhat clumsy Lindeberg proof extends to the martingale case, and gives an idea of what is happening. I can give lots of somewhat unrelated proofs, but does anyone really understand the theorem? Similarly, Cramer's proof of the Levy-Cramer theorem that the sum of two independent random variables is not normal unless they both are is probably the only easy proof, but it likewise obscures the ideas. Any time characteristic functions are used to prove a probability theorem, the concepts are not even present. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: A 3rd Grade Word Problem---HELP > I don't see 127 either. It should be 31. If you keep telling the story up > till and including Saturday, then you would have 127. 30 if you don't count yourself. /David === Subject: Re: I can't stand it anymore >I have been biting my tongue about the IQ test but I can't any more. >How reliable is a test that use the term Asian to represent the most >diverse of ethnic and cultural groups? >If you want to know more about the diversity, ask me and I will give >you tons of specific examples. Here is one: An example of cultural >differences (that affects their educational goals) of the two groups - >ethnically the same and speak the same EXACT langauge - in Burma who >are the descendants of the people who came to Burma from a town known >as Surat, India during the British colonial days. > Now, do not assume that they are racially the same as the majority of >northern Indian population, i.e Indo-Aryan. What I am aware of is that >they are descendants of those who came from Central Asia (to india) >and may be MIXED with the locals of India (I haven't checked into >that) like the majority of the current day Pakistanis. >See the complexity yet? >Add to that the Afghans, particularly the Pathan, which also form a >comunity in Burma. >And then you have all these different native groups in Burma, namely >Burman, Mon, Shan, Karen, Kachin, Chin, etc. where the Karen with a >good size Christian population now claiming that they are one of the >lost tribes. > The Mon are the first group who adopted Bhuddism which made them the >most literate group *in the old old days* through the monastry >education. Shans are the stauch Bhuddists too. I read somewhere on >the Internet that Mon originated from ancient India. They look like >typical oriental though some have darker skin while others are very >light-skinned) instead of the dravidians of ancient Inida. >Shan is ethincally similar to some groups in Thialand from the area >that borders Burma. Talking about Thailand, not all Thias are >ethincally the same. >Now, the Chinese would claim that the high score of IQ tests is >because of them Chinese, i.e not all Asians are equal. And the >Japanese thinks (may be not so much anymore) they are superior to all >other Asians. >Does that term *Asian* include the Arabs? >What about the people of Egypt (I avoid calling them Egyptians to >differentiate from the ancient Egyptians) some of whom are Arabs, some >of whom are descendants of anciant Egyptians, some are descendnats of >a group called Berber (my spellig may be wrong). >And how many Ethiopians look or act like a typical African. >Why are people using the racist IQ tests as God's given ruler to >measure intlligence? > http://metropolis.japantoday.com/tokyo/recent/feature.asp > In the off chance you are interested in whether or not the Japanese > people came through the star gate or whether they crashed and > are decended from the ones who crashed in China, I am inclined > to believe that they came through the stargate. But that does not > detract from the Chinese who have equal right to be here. > We aren't going to change anything at this point. > I have seen on the net, some very Japanese looking ET's > and one in particular that looked very Japanese. But for the > most part, the ones with the large cranium, look more > Chinese. They are scientists as far as I know. The ones > that were here last year were an advanced race. They beamed > my son home from school if you can image that. At lunch time. > The really odd thing about it was, that for about 20 minutes after, > his eyes were slanted down. Then his appearance returned to normal. > His conscious self didn't know anything about it. I think he just > assumed he walked home. > At the time a bunch of us had been discussing the future as per > The 5th Element. And I was arguing for rejuvenation machines. > Seems silly I know. But no more silly than this... > http://www.rense.com/general41/flying.htm > or > this... > http://dbarkertv.com/UPDATE.htm > As I understand it, this is a reptoid ship. > Were the ones who arrived first, and notified the glactic government, > after Hiroshima, distant relatives of the Japanese people? > That is a nice notion, is it not? > Ignore the stupid pictures of the alien autopsy. > Some humans are very sick mentally and spiritually. Those are the > kind of people who make mock ups like that to hide their own evil > deeds and to take advantage of their fellow man. > Thats the price we pay for freedom. We have to tolerate some people > who truly should not be allowed to exist. > So what is wrong with the Japanese people? > They are still suffering from the old, how could God, allow this > to happen? > That is a very good question. > That will undoubtedly be answered one day. > But take it from me, there is nothing wrong with the Japanese people. > They are beautiful people. > And do have a very valid inherent right to be here. > But try to think of people as individuals, not as a race. > Do not ask what is wrong with Japanese people, say what is wrong with > some human beings. You cannot blame a victim for a crime. > And the answer again, is the price of freedom. > The freedom to be bad as well as good, makes people free. > That does not however mean, there are no consequences for > people's actions. > Be thankful for your good nature. > That is a blessing. > Some may see that as a weakness, but then they usually find out the > hard way, that there is only one true power in the universe. > Trying to compete with him, is a very silly idea that will > only lead to misery. > What Japan has to offer humanity, is invaluable to the future > of mankind. Except what they have offered during their fascist days. They never apologized to us victims. My grandfather, a successful business man was murdered by the facist thugs at the prime of his life - he was barely 40 - for having close business associations with the British subjects). Don't tell me that the thugs did not torture him. These beautiful Japanese didn't tell their offsprings the atrocities their government committed in Burma, Korea, and China. > So don't go changing. :-) Nope. We just sit and watch the Chinese takeover. Heck, Burma (Myanmar) should now be referred to as a Chinese colony. Burmese military dictatorhip is made up of those with Chinese blood. Of course, they are fully Burmanized. === Subject: Re: I can't stand it anymore > I was referring to people mentioning IQ tests in discussion about > determinging intelligence of different race in all these 3 groups. > But, how could you possibly realized that? After all, you got F in > all these 3 subjects. Seems to me that *you're* the one that brought it up. So are you a Troll or just clueless? === Subject: Re: Those Tricky Natural Numbers . . . >In Set Theory, is an element of N also a proper subset of N? It is in the von Neumann representation of the ordinals, or in any representation in which an ordinal number is precisely the set of all smaller ordinal numbers. However, one can give other representations. >For example, I want to prove that a cardinal k belonging to N is less >than the cardinal of N itself, aleph null. If an element of the >natural numbers also a proper subset of N, then I can show k < aleph >null by showing that a set A with cardinal k is a proper subset of the >natural numbers. Since a natural number n can't be mapped onto a >proper subset of itself, This takes proof; what is the formulation of natural number? I think I can say that aleph null can't be >mapped onto a proper subset of itself. This is CLEARLY false. The mapping n -> n+1 does this. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Uncle Al is Sadistic . > 3 Africans I met in Computer science department in the last 4 years > were way above average in thier programming skills in the midst of > Chinese and Indian grad students who are the overwhelming majority in > that department. One was so good that he even got a parttime teachign > position to teach an undergrad programming course, while he's still a > student himself. > You will always find bright smart people in just about any naturally > occuring group of humans. Race is nothing. Culture is everything. That's my point. > === Subject: Re: B-Spline Extension > I have quickly looked up what you suggested. However, where I was looking > all they were talking about was the interpolating curve that is created. > How do I go about explorating this, or what other 3d curve fitting method > do you suggest. The need to explorate is crutically important to what I am > trying to achieve. When you explorate a B-Spline curve (which is a collection of B-Spline segments that flow neatly into each other), you can only take the first and last B-Spline of the curve and stretch them. Suppose you work with third degree B-Spline curves, then you have two three-degree polynomials that you want to draw over a bigger domain than just from begin-to end datapoint. That is why I suggested to look up this information since you don't seem to knwo what a B-Spline really is. B-splines also have little or nothing to do with data-points, you build them using *control* points. Data-points are things only used in Lagrange curves. However, there are also n'th order B-Spline curves, for n datapoints. This means that for each control point you add, you get another degree to your polynomial. So you work with *one* n-th degree polynomial. (as opposed to stickign with third degree polynomials and then just adding another B-spline segment to the curve for each new control point) They are called 'Bezier' curves, and you can stretch them too of course, they are polynomial in nature (the functions x(t), y(t) and z(t) are anyway) There is a simple equivalence between Lagrange and Bezier curves using a matrix that is filled with coefficients created by the algoritm of Casteljau. Lagrange curves are, as said, the curves you make to fit data-points. As for NURBS or anything, they're not what you need, they are basically B-splines with the added quality that you can perform non-affine transformations on their control-points and then the shape of the curve will change in the same manner. Well there's a lot of theory you have to go through I think, because stretching a curve, what you want to do, is very starightforward if you'd understand the theory, and there's more than I can type here. I do know something about curves, but perhaps someone with a clearer pen might help you better here. -- Quaternion === Subject: Re: A NOTy problem. > G'day G'day Folks, > I have been puzzling over what ought to be a simple logic problem > but am at a loss to come up with a method. > A crime is committed by one of the following; Alan, Bob, Chris or > Dave. > Each makes a statement to the police but three of them lie. > Alan says, I didn't do it. > Bob says, Alan is lying. > Chris says,Bob is lying. > Dave says, Bob did it. > Usually one can solve such problems with a grid I find such grids helpful, but not perfect. It's hard to track some statements with them, and to keep track of what ifs. > Where A. B. C. D represent their names and > L. = lying, > N. = not_lying, > I. = innocent, > G. = guilty. > Somehow that doesn't seem to get anywhere is this case. > Maybe it is because N = NOT(lying) and G = NOT(innocent) Well because of that, an easier grid to use would be T G A B C D (T for telling truth) Then you just need a method to indicate whether the statement is true or false for each one. For instance, an X means its false and a dot means it's true. Thus, each column will have one dot and three X's when you're done. If you put a dot, you have to put three corresponding X's. > I am less interested in knowing the answer to the puzzle than > understanding why it is different from the routine logic puzzles. It's not. But you do need to be able to backtrack and reset your grid after a false hypothesis. For instance, let's start with the first statement. > Alan says, I didn't do it. Suppose that's true. Then Alan gets an O in column one (he's telling the truth) and an X in column two (he's not guilty). T G A O X B X C X D X Once we put an O, everybody else gets an X. Now Bob says Alan is lying. We assumed Alan is a truth teller, so Bob is lying. We already marked that. Chris says Bob is lying. This is true. But we already marked Chris as a liar. Inconsistency. Alan is not the truth teller. Restart the grid with somebody else as truth teller and X under Alan for column one. Of course, if Alan is lying, then this statement is false: Alan says, I didn't do it. So no matter who the truth teller is, Alan did it. T G A X O B X C X D X Now we just have to see if putting an O under B, C, or D works out consistently. But actually, I see that Bob's statement is now true. T G A X O B O X C X X D X X Is this consistent with the other two statements? Better check (if not, there is no solution). > Chris says,Bob is lying. If Chris is a liar, then this statement is wrong. Which it is. It checks. > Dave says, Bob did it. If Dave is a liar, then Bob didn't do it. And he didn't. The grid system works fine, you just set up an improper grid. - Randy === Subject: Re: One algebra question > Hi I am learning algebra by myself > I am trying to do some problems > but I can't solve this one > could you please do me a favor to give me some help > here is the question: > Let f be a homomorphism defined on a finite group G and let > H subset G > a) show that |f(G):f(H)| divides |G:H| > b) show that |f(H)| divides |H| The claims are false unless H is actually a subgroup. If that is the case, let N be the kernel of f. Then |f(G)| = |G:N| and |f(H)| = |H:Ncap H| so that |f(G):f(H)| = |G:N|/|H:Ncap H| = |G:H|/|N:Ncap H| which the makes a) obvious and b) already is. === Subject: A 3rd Grade Word Problem---HELP > I don't see 127 either. It should be 31. If you keep telling the story up > till and including Saturday, then you would have 127. > 30 if you don't count yourself. > /David Surely 31 is the number not including yourself ? === Subject: Re: Magnetism: compare magnetic fields > Looking for a direct mathmatical comparison between the magnetic field > produced by an electronic current (i) and that of a fixed magnet of a > given strength. > See: http://scienceworld.wolfram.com/physics/Magnetism.html > Note the differences in > o diamagnetism > o Paramagnetism > Ferromagnetism > http://scienceworld.wolfram.com/physics/Ferromagnetism.html > The development of extremely strong magnetic properties in certain > materials which occurs when magnetic domains (regions at most 1 mm in > dimension) become aligned in the absence of an applied field, below a > temperature known as the Curie temperature. The net magnetization > depends on the magnetic history (the hysteresis effect). Above the > Curie temperature, these materials become paramagnetic. Iron, nickel, > cobalt, and gadolinium are ferromagnetic at room temperature. > Ferromagnetism is believed to be caused by magnetic fields generated > by the electrons' spins in combination with a mechanism known as > exchange coupling, which aligns all the spins in each magnetic > domain. === Subject: Re: A 3rd Grade Word Problem---HELP > Surely 31 is the number not including yourself ? Oh, sorry, I overlooked that it starts on sunday with one person. My weeks start on mondays :) /David === Subject: Re: Uncle Al is Sadistic . >> 3 Africans I met in Computer science department in the last 4 years >> were way above average in thier programming skills in the midst of >> Chinese and Indian grad students who are the overwhelming majority in >> that department. One was so good that he even got a parttime teachign >> position to teach an undergrad programming course, while he's still a >> student himself. >You will always find bright smart people in just about any naturally >occuring group of humans. Race is nothing. Culture is everything. > She said Africa. I would have asked what country? Ethiopia > used to send their brightest to the States for grad study. It > would be interesting which countries are starting to get their > act together. > /BAH > Subtract a hundred and four for e-mail. None of them Ethiopians. === Subject: Re: [JSH] On the Rewriting of a Polynomial In sci.math, Phil : >> To reprise James Harris' proof, or at least the first two steps >> (I've included '*' so that GP/Pari works): >> [begin excerpt] >> 1. Let P(x) = 14706125 * x^3 - 900375 * x^2 - 17640 * x + 1078, where x > is >> in the ring of algebraic integers, notice that P(x) has a constant >> term that is 1078. >> 2. It can be shown that >> P(x)= 7^2*(2401*x^3 - 147*x^2 + 3*x)*(5^3) - 3*(-1 + 49*x)*(5)*(7^2)+7^3 >> where the *same* polynomial has been put in a form which allows a >> factorization into non-polynomial factors so that I have >> P(x) = (5*a_1(x) + 7)(5*a_2(x)+ 7)(5*a_3(x) + 7) >> where the a's are roots of >> Q(a) = a^3 + 3*(-1 + 49*x)*a^2 - 49*(2401*x^3 - 147*x^2 + 3*x). >> [end excerpt] >> Now, here's where I have a problem. How did Mr. Harris get to >> this point? > -5^3Q(a=-7/5) = P(x) > Similarly in the factored from Q(a)=(a-a_1)(a-a_2)(a-a_3) > let a = -7/5 then > -5^3Q(a) = (5a_1 + 7)(5a_2 + 7)(5a_3 + 7) = P(x) Hm...yes, that looks fairly simple. :-) I'll admit what confused me was the requirement that Q(a,x) have two independent variables. > This confirms that the P(x) is a correct expression for the a_1, a_2 and a_3 > and that they > are indeed algebraic integers. > The problem starts when James divides them by 7 and declares that because > a_1/7 and a_2/7 are not necessarily algebraic integers, something is wrong. > I can't work out what he means. Join the club. :-/ > For example > P(x) = x(x + 1) is a multiple of 2 for all x integer > therefore P(x)/2 is integer for all x integer > P(x)/2 = x/2*(x + 1) (the only way it can go when you set x = 0, or > something) > But x/2 is not an integer for all x integer. > Presumably there is something wrong with integers as well. Well, James has also tried to argue that Z[1/2] contains all reals. :-) (It doesn't even contain 1/3.) > HTH, Phil. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Riemann's Zeta Function by H. M. Edwards >> I presume you are using the contour integral argument for continuing >> Gamma(s)zeta(s). One introduces a countour C_e in three parts: >> imaginary axis from -infinity to -e, circle radius e about origin, >> imaginary axis from -e to -infinity and take the integral >> of z^{s-1} e^z on C_e (with branch cut on negative real axis). >> Thse integral f(s) is independent of e by Cauchy's theorem. >> It is also an entire function of s: convegence is nice since e^t -> 0 >> rapidly as t -> -infinity. Really dumb question. is e in the contour integral an epsilon, or the > basis > of the natural logarithm. (I am getting confused by both uses of e.) > No Real Mathematician thinks of the logarithm having a basis:-) Oops I meant base :-( > If you want to use epsilon instead of e you're welcome: using e > does save typing. True. :-) >> The integral of z^{s-1} e^z on the first part of the contour is >> integral_e^infinity t^{s-1} exp(-pi i(s-1)) e^{-t} dt >> = - integral_e^infinity t^{s-1} exp(-pi is) e^{-t} dt. >> Similarly on the third part of the contour it is >> integral_e^infinity t^{s-1} exp(pi is) e^{-t} dt. >> These add to >> 2i integral_e^infinity t^{s-1} sin(pi s) e^{-t} dt. >> If Re(s) > 0 the integral over the circle of radius e is >> O(e^Re(s)). Letting e -> 0 we get that What is this O Is this Big-O notation? > Yes. Very useful! >> >> Consider >> G_N(s) = integral_{C_{(2N+1)pi}} z^{s-1} e^z/(1-e^z) dz. >> By Cauchy's theorem the difference G_N(s) - g_e(s) is 2pi i times >> the sum of the residues of the poles of the integrand >> at +-2pi i, +- 4pi i, ..., +- 2Npi i, that is >> 2pi i sum_{n=1}^N [-(2pi ni)^{s-1} - (-2pi ni)^{s-1}] >> = 2pi i (2pi)^{1-s} sum_{n=1} (-2) cos(pi(s-1)/2)/n^{1-s} >> = something nasty times sum_{n=1}^N 1/n^{1-s} Hmm. Edwards gets sin(s pi/2). > How does sin(s pi/2) relate to cos((s-1)pi/2)? Oh yeah. They're equal! I think I am going to look at the Davenport book, Chapter 8. If I can find it for less than $58.00! This Edwards book is over my head. :-o === Subject: Re: I can't stand it anymore >But, how could you possibly realized that? After all, you got F in >all these 3 subjects. Oh, I see. You're trolling - why didn't you just say that? Doug === Subject: Re: The need of geodesics in physics >That's a bit confusing, one can set the Lorentz >>Force to generally vanish ie. >>f_u = q*F_uv *U^v =0 >>and in effect decribe EM geodesics, then put >>q*F_uv in the metric forming a non-symmetrical >>metric. The problem with that is that both sides of that equation are tensors. >If they vanish in one frame, then they must vanish in all frames. >Geodesics are based on affine connections, which are not tensors, so >they have non-homogeneous transformation properties. >>Ah, but the geodesic equation is the absolute derivative >>of the 4 velocity U^u = dx^u/ds, ie >>DU^u/ds = U^u;v * U^v =0. > True, but how does that relate to the LF? >>Well multiply by mass (scalar invariant) then, >>m*DU^u/ds =0 >>but QT requires m to change discretely, and >>since dm >0 requires a continuous change in m, >>dm=0, therefore set P^u = m*U^u (the 4- >>momentum) and find >>DP^u/ds =0 (P^u = m*U^u, dm=0). >I don't know where you got that notion from. QT says nothing about m >changing discretely. In fact, the rules of QT maintain continuous >change of mechanical variables (of course under the restrictions of >the operator algebra). What is discrete in QT are the quantized >values of the so-called stationary states, but these tend to be the >exception rather than the rule. Continuous fluctuation tends to occur >everywhere in the quantum world. We should make sure we're talking about the same thing. It is *invariant* energy which can only vary discretely and NOT continuously. A simple atom emits and absorbs energy by photons. >>Is true generally, and applies to LF, so >>f^u=DP^u/ds =0. >Moreover, >geodesic acceleration should be independent of mass, which applies to >forces such as centrifugal, coriolis, and of course, gravitation. >>True for acceleration of point masses, (but not true >>for relatively large masses, that's why acceleration >>is a better term than force in GR, otherwise you'll >>need to include the term k*t_uv >0 we figuring G_uv). >Acceleration due to Lorentz force will be inversely dependent on mass, >making it a no go. >>motion must respect Quantum Theory, Lorentz force does >>not, so the stated objection evaporates. >But why are you making that assumption? I can make that statement >certainly, but I'm excluding a large variety of systems by doing it. >So I don't think it is very realistic. >>Consider, for example f_0 = q*F_0i*U^i, >>which in vectors is approximately >>f_0 = q*E_i (dot) V_i. >>If q moves in direction of E_i then it's >>potential energy will vary continuously and >>this is prohibited by QT, therefore >>E_i (dot) V_i =0 and thus f_0 =0. >>Simply stated, charge q's V_i is always >>perpendicular to E_i. >But potential still varies continuously in QT and its changes are very >easily tracked in the position representation. We're talking about Lorentz force not potential. >I don't know how you came to that conclusion. Fair enough. Just to make clear definitions, define invariant energy P by P*P = P_u*P^u ok? An invariant force would be f = dP/ds, and dP = f*ds ok? The Lorentz force f_u *dx^u = f*ds, ok? f_u*dx^u = q*F_uv *(dx^v/ds)* dx^u ok? Since dx^u*dx^v is symmetrical and F_uv is antisymmetrical F_uv*dx^u*dx^v =0 when summed ok? Then f*ds = 0 = dP ok? That's it! To summarize: The Lorentz Force Equation predicts dP=0. But we know P is not a constant, therefore P varies discretely, ie by quanta. Ken S. Tucker PS: Snippable comment - That LF predicts and is compatible with QT is excellent. It forms a great bridge between GR and QT embodied in DP^u/ds = 0 (absolute derivative). This equation is derived w.r.t LF, QT (therefore EM) and GR, without any added assumptions, ie straight from working in the system. It can be called a *generalized geodesic*, (I'd hestitate to call it THE *universal geodesic*). One thing that is of interest : since P<>0 the Limited Einstein Field Equation G_uv =0 is not applicable (as it applies to massless points) so the EFE G_uv = k*T_uv would be required. L8R KST === Subject: Re: c-number Originator: grubb@lola >> I'm reading Peskin and Schroeder's Intro to QFT and in this >> book the author constantly refers to c-numbers... without >> saying what a c-number is. I couldn't find c-number >> on mathworld.wolfram.com or physicsworld.wolfram.com. >> Can someone please tell me what the hell a c-number is? >> adam >I haven't read Peskin and Schroeder but if their usage of the term >c-number is the same as everyone else's usage then they are referring >to commuting variables. For example, Bryce DeWitt introduces the >concepts of c-numbers (commuting variables) and a-numbers (anticommuting >variables) in his book Supermanifolds. If a,b are c-numbers then >a*b = b*a This is correct. The problem is that Peskin and Schroeder don't go into 'grassman numbers' in any detail at all. DeWitt's book is a good intro. --Dan Grubb === Subject: Re: how to identify data from different distributions? >Hi sorry to bother you guys. >This problem suddently comes into my mind. >Given two data sets from two different distributions, >if I mixed them together, i.e., you don't know which point comes from >which >distribution. Is it possible to use some learning or statistical >algorithms to separate the two data set out? >For example, >X=(x1,x2,...,xn) ~ Gaussia >Y=(y1,y2,...,yn) ~ Uniform >Given (x1, y1, x2,x3,y2,x4,y3,...), can you separate out X and Y? It can be partially used to separate them. The most that one can get by learning or statistical algorithms is the number from each group, and the precise distributions. If an individual has a prior probability h of coming from the first group, the posterior probability is given by Bayes' Theorem. More than this cannot be done for one individual. If it is known exactly how many are from each group, one can get the posterior probability that a particular partition is the correct partition. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Smallest Eigenvalue in a Potential Problem Given an open subset A of the unit square [0,1]x[0,1] with fixed area a:=area(A), it is known that the lowest eigenvalue of the operator u |-> - u_xx - u_yy + chi_A * u i.e. Laplacian of u + [characteristic function of A] times u on the space of functions u on [0,1] x [0,1] with periodic boundary conditions will be positive. But does there exist a positive lower bound for this eigenvalue which is independent of the choice of open subset A as long as we keep the area a=area(A) fixed? Does anyone have an idea? Thomas === Subject: Re: Bible 1, Darwin 0! And we are talking pure archeology? > Forbidden archeology by Cremo, Thompson! > Like the name says: Forbidden > I urge you not to read it! If it is about Bible and Darwin, it is not physics, not math and not archaeology. You are posting in wrong newsgroups. Mikko === Subject: Re: A 3rd Grade Word Problem---HELP >> I don't see 127 either. It should be 31. If you keep telling the >> story up till and including Saturday, then you would have 127. > 30 if you don't count yourself. Nope. 31 not counting yourself. Remember that the first day you tell it to *one* person. 1(Sun) + 2(Mon) + 4(Tue) + 8(Wed) + 16(Thu) That is 31, and is the number of people you have told the story -- not the number of people that know the story (which would be 32, including yourself) Carlos -- === Subject: Re: Conversion of red phosphorus to white phosphorus hanson spewed: > Nice, but quasi green bob suffers from > Plumbum paranoia of legendary dimensions which forced him > to hallucinate that his NG comments will enhance mankind's > survival I'm betting on marsupials. > against the doomsday prospects of lead in message > which he delivered under great inner torment and psychotic fear, > because he says... > I wonder if you are a danger to others at times. Well, are you? Do you take antipsychotics? If not, has a doctor ever suggested that you take them? > His comment here is embedded in a master piece of a post, > that teeters on the intellectual niveau of a petrock, the belletristic > values of an ameba and his understanding of the field with > the depth comparable to that of a flatlander's. That didn't flow very well Hanson. You get a C-. > His efforts > nevertheless elicit a faint chuckle, at best, but, me being > magnanimous will bless him with a roaring > AHAhahahhaha.......ahahahahanson A twitter when you say it aloud. === Subject: Re: Graduate algebra book > Well, I think the U.S. and other countries are wildly different in > their structure of an undergraduate curriculum. I don't have a > problem admitting that my university is not so terrific. We have 2 > algebra courses which are available to undergraduates at my > university, the more advanced of which is almost identical to the > introductory course, albeit slightly more fast paced. It so happens > that I _am_ an undergraduate and I am trying to get a better education > than what my university offers to the typical student. > Good for you. > So you really want a decent undergraduate text, rather than > a postgrad text, to compensate for the impoverished curriculum > at your university. > What I have a > problem with, is listening to the egos of those people who feel like > they are superior to everyone else for no particular reason. > Who's that then? > All Mr. > Chapman had to say was generally those topics are covered at the > undergraduate level, instead of the snide remark 'thats graduate > algebra?' > Aaaaaaaah! Actually, upon checking a few of the top universities' websites, I > find that while the undergraduate curriculum generally does offer > algebra courses which talk about galois theory and the topics I > mentioned, they are generally at the near-graduate level. Thus, for > any AVERAGE university, they will not be offered until the graduate > level. > What is an AVERAGE university? > And even at these better universities, not everyone takes > every single course, and I imagine not everyone takes 5 or 6 semesters > of algebra as an undergraduate. Should they take 5 semesters of > analysis and 5 more of topology as well? how is anyone ever supposed > to graduate? > It's quite possible to study a reasonable amount of algebra, > a reasonable amount of analysis and a reasonable amount of topology > all in three years. (And also a reasonable amount of applied maths > and statistics if that's your thing.) I'm a bit baffled about your reference > to semesters of algebra --- do you only study one topic per semester > at your university? No, but at my university (and in the United States in general) we don't usually take more than one course of the same discipline per semester. If you go to a university's website and find that they have 5 algebra courses, each of them is a semester long, usually with one being a prerequisite for the next (but not always). But when I say semesters of algebra i mean algebra courses, since each course is one semester long. So, for example, if there are 5 algebra courses, the only people who actually end up taking all of them are those people who have a pretty good idea that they are going to specialize in algebra. Because there simply isn't enough time to take so many courses and they aren't generally required anyway. Remember that in the states we have to take lots of courses which are completely irrelevant to anything we plan to do in the future, such as 2 political science courses, 2 history courses, 3 or 4 english courses, some other courses such as psychology, art history, economics, etc, PLUS get a minor in another subject. I realize this is not the case in most european countries, so there is more time to focus on studying math and it's only natural that your undergraduate education will take you farther. === Subject: Re: Pyrazine amide > A classical route to 1,4 pyrazine-2-amide, > a simple 6 ring* , *=N-CH=CH-N=CH-C(CONH2)=* molecule > starts with o-phenylene diamine + Glyoxal ring condensation to > !,4 Benzo diazine (A). -- (A) being oxidized by fracturing the C ring side > with KMnO4/H2O to pyrazine-1,2 carboxylic acid (B). --- (B) being > decarboxylated in hot conc. H2SO4 to 1,4 pyrazine-2-carboxylic acid, > then esterification and ammination to the final 1,4 pyrazine-2-amide. > Question: Has anybody done or does anybody know which/what > catalysts & conditions are beneficial to get the same product via: > 1) Dicyan (Cyanogen) and Acrylamide per > NC-CN + CH2=CH-CONH2 --> *=N-CH=CH-N=CH-C(CONH2)=* > or > 2) 1,2 Dinitrosoethan and Acrylamide per > O=N-CH2-CH2-N=O + CH2=CH-CONH2 -- *=N-CH=CH-N=CH-C(CONH2)=* + 2 H2O [Mark] > Is this [from] hanson, hahahanson, or hahahahanson? Yes, these are different quantum states of nuclear excitement in/of the same entity, where [h] = Planck's constant and [a] refers to Sommerfeld's Finestructure constant. The first quantum level of ha = 2pi*e^2/c yields an action quantum of atomic emotions. Since symbolism in science seldom produces only one singular interpretation but does ususally cause different strokes for different folks, this particular event is no exception. As can be seen, it does evoke in Tom-Tom Crack Potter subconscious, archetypical fears of the hyenas which have terrified Tom's ancestors on the savannas in Africa, which is why he calls me Harry Hyena Hanson. Where he got the Harry from is still a mystery, but it has probably some thing to do with his personal instigations for/against Bolsheviks. [Mark] > Are we talking organic chemistry here? If so, > you might want to cross-post to sci.physics, > as I have so thoughtfully done :-) [hanson] Your motion is acknowledged as a fait-accompli. It had however some unexpected effects in that some of the 5.5 times smarter web lice and net flees that infest these cyber parties grievously objected to the word Pyrazinamide, Pyr-Azin-Amid, insisting that it referred to you, Mark, as being a Beer-Nazi-amidst, in the code lingo used by Odessa. So, see Mark, one man's thoughtfulness (yours) can easily become other folks' nightmares. [Mark] > Seriously tho...there's a lot being done with > metal-mediated chemistry that is cutting- > edge stuff (some of it is even reproducible). > Checked the patent office? Tetrahedron Lett.? > Overall, you sound like you're in a position > to be on top of it. That said, if I find > something in the trash can, I'll pass it on :-) [hanson] sentence. A high yield solution will most probably come from/via some totally unusual or trashy source. So, yeah yeah, if you find something off beat, while rummaging thru trash cans do let me know. Ansaetze thus far are not too impressive as the polymerization speed of the 2 educts is still much faster then the desired ring closure or condensation rates. The most promising results that came across my desk were reaction initiations induced by laser radiation with distinct wave lengths using fine, reflective metal flakes and colored pigments suspended in the solvent for delivering the best activation energy level and intensity to the compounds in sol. [hanson] > PS: This Pyrazin compound is gaining attention again in > the treatment against new tuberculosis strains, > resistant to the latest antibiotics. > hanson [Mark] > What've you got for terminally stupid people resistant > to common sense (now I didn't say current administration > so don't go imputing motives to me where there are none > that can be proven beyond a reasonable doubt in a non-biased > competent court of law or public opinion, both of which > should be on the endangered species list, eh)? [hanson] Your non- or quasi chemical response in your above para went completely over my head in both modes, FIFO or LAFI. hanson > Mark > (Harassment is out. Groping is out. Oogling > is out. So, let's talk about the weather :-) [hanson] Wait, I'd go for a try with unbridled and unabashed Goosing. ahahahhaha.......ahahahanson === Subject: Re: Graduate algebra book > What's a good book for a first year graduate algebra course? > Something with alot of emphasis on factorization, polynomial rings, > fields, PIDs, Galois Theory, and of course all the more basic topics > of algebra as well like groups, ideals, integral domains, etc. > Something that has lots and lots of _good_ exercises? Hungerford is > good, but I'll have finished most of the exercises by the time I want > to start doin the exercises in this new book. Lang is probably good, > but there don't seem to be very many exercises. A severely critical opinion of Hungerford may be found at http://www.math.hawaii.edu/~lee/algebra together with some professional remarks as to what sort of algebra is commendable at the graduate level. An on-line book of first-year graduate mathematics may be found at http://www.math.uiuc.edu/~r-ash/ and this may be a direct answer to your question. Both these references may be seen at http://hilbert.dartmouth.edu/~leejstem/freebook.html#al David Ames === Subject: Re: Jaynes' book on probability - thoughts? >As a matter of interest, didn't Jayne's book used to be available on-line? >Is it still? Or am I mistaken? http://omega.albany.edu:8008/JaynesBook John Bailey http://home.rochester.rr.com/jbxroads/mailto.html === Subject: Re: B-Spline Extension I understand what you have said, and I did know most of this. But you did not answer my question from my last post. How do I go about explorating Lagrange curves, or what other 3d curve fitting method domyou suggest. My initial research into the Lagrange curve, keeps saying they are for interpolation. If you could suggest a web link to somewhere that would explain them better that would be great. Adam > I have quickly looked up what you suggested. However, where I was looking > all they were talking about was the interpolating curve that is created. > How do I go about explorating this, or what other 3d curve fitting method > do you suggest. The need to explorate is crutically important to what I am > trying to achieve. > When you explorate a B-Spline curve (which is a collection of B-Spline > segments that flow neatly into each other), you can only take the first > and last B-Spline of the curve and stretch them. Suppose you work with > third degree B-Spline curves, then you have two three-degree polynomials > that you want to draw over a bigger domain than just from begin-to end > datapoint. That is why I suggested to look up this information since you > don't seem to knwo what a B-Spline really is. > B-splines also have little or nothing to do with data-points, you build them > using *control* points. > Data-points are things only used in Lagrange curves. > However, there are also n'th order B-Spline curves, for n datapoints. This > means that for each control point you add, you get another degree to your > polynomial. So you work with *one* n-th degree polynomial. (as opposed to > stickign with third degree polynomials and then just adding another > B-spline segment to the curve for each new control point) They are called > 'Bezier' curves, and you can stretch them too of course, they are > polynomial in nature (the functions x(t), y(t) and z(t) are anyway) > There is a simple equivalence between Lagrange and Bezier curves using a > matrix that is filled with coefficients created by the algoritm of > Casteljau. > Lagrange curves are, as said, the curves you make to fit data-points. > As for NURBS or anything, they're not what you need, they are basically > B-splines with the added quality that you can perform non-affine > transformations on their control-points and then the shape of the curve > will change in the same manner. > Well there's a lot of theory you have to go through I think, because > stretching a curve, what you want to do, is very starightforward if you'd > understand the theory, and there's more than I can type here. I do know > something about curves, but perhaps someone with a clearer pen might help > you better here. > -- > Quaternion === Subject: Re: Uncle Al is Sadistic . > 3 Africans I met in Computer science department in the last 4 > years were way above average in thier programming skills in > the midst of Chinese and Indian grad students who are the > overwhelming majority in that department. > You will always find bright smart people in just about any naturally > occuring group of humans. Race is nothing. Culture is everything. > > That's my point. What point? Culture Shmulture....that is simplistic politically correct heurist bull of the first kind. If it were that simple then the problem would have been solved a long time ago. Painlessly. But one can easily see the problem being enhanced and aggravated at every step of its emergence and development because, basically... What was thoughtless came by easy. What was easy became habit. What were habits became tradition. What were traditions became culture. What was culture became religion. What is religion becomes thoughtless. ahahahaha...........ahahahahanson === Subject: Re: Graduate algebra book Remember that in > the states we have to take lots of courses which are completely > irrelevant to anything we plan to do in the future, such as 2 > political science courses, 2 history courses, 3 or 4 english courses, > some other courses such as psychology, art history, economics, etc, > PLUS get a minor in another subject. I realize this is not the case > in most european countries, so there is more time to focus on studying > math and it's only natural that your undergraduate education will take > you farther. In France undergraduate students (who plan to graduate in mathematics or physics) study maths (eheh), physics (electronics, mechanics, thermodynamics), chemy, computer sciences, and (a little) English. We don't have to minor in anything, but we can chose either maths or physics as a prevalent subject for our two first years. That means you'll have more maths, or more physics, depending on what you'll do later. Moreover, we have only two years of undergraduate school (ie your BchS is one year more than ours) so we can specialize earlier (this system is going to change in two years in order to follow other european and american systems). === Subject: Re: Uncle Al is Sadistic . >I think part of the problem is that we have such poor measures of what >it is to be educated. The College Board only claims that the SAT is >a very good predictor of first-year college grades; the average >correlation between SAT scores and _freshman_ grades is +0.52; the >correlation to being educated and smart is almost certainly quite a >bit smaller. >>Aha. The only problem is, we've (currently) no measure which gives a >>of deciding (in this field as in any other) that you won't use any >>measure until you've one which is perfect. >I'm not arguing that we need a perfect measure. My point is that >whether the measures used today are good enough is open to debate. Sure. >We base a very large proportion of our judgments of who is smart and >capable of being educated on the results of multiple choice >examinations. There is at the very least a significant body of >anecdotal evidence that the skills required to do well on such tests >are different from those we would otherwise consider desirable in >students. No doubt. Nevertheless, there is a significant correlation between aptitude exam scores and college performance (don't recall exact numbers but it is in the 0.5-0.6 range). Far from perfect, indeed, and that's why you do get anecdotal evidence for cases where it fails. But far from irrelevant as well. Note further that, as with any positive correlation measure of this sort, the value improves when applied to a larger population. Thus, while a difference of, say, 50 points on SAT score between two individuals is not terribly meaningful, a similar difference between the averages of graduating classes of two different high schools is much more meaningful. >My personal experience with standardized testing is somewhat mixed. On >the one hand, I did well enough on standardized tests to end up >admitted to graduate school in chemistry (and yes, I did graduate). On >the other hand, I'm fond of using a personal story as a cautionary >tale; my junior high school guidance counselor advised me not to take >algebra as a high school freshman because I didn't have the necessary >aptitude. (To this day, I'm still not sure how he arrived at that >judgment, although I'm pretty sure it had something to do with test >scores; I wonder if he knows that I actually enjoyed P Chem). My >experience on a university faculty graduate admissions committee also >indicated that standardized test scores are a highly imperfect >measure. I didn't encounter anybody claiming that they're perfect. But imperfect does not mean meaningless and, as I said, the value of the score grows with the size of the population it is applied to. All tests are fallible, on the individual level. Even performance tests are fallible. Aptitude tests, much more so. Still, they do carry information. >[snip discussion on the Washington DC school district] >I don't have actual budget figures. I was not looking to defend the >performance of DC schools; my first statement acknowledged that they >are a mess. My point was that there are a lot of confounding factors >that don't get controlled in the SAT points per dollar metric. Also, >DC will always be an outlier in such comparisons because their >statistics often end up being the orange in a barrel of apples. Their >finances are not exactly comparable to other states because they have >all the difficulties of an urban school system without the >less-expensive suburban and rural systems that get factored into the >performance of every other state. Their finances aren't exactly >comparable to other big city systems because the DC school board >handles some functions that are paid for at the state level in other >jurisdictions. For sure. That's why a better comparison would be obtained keeping DC out of it and comparing various states. You'll still find little if any correlation between spending levels and performance. Meaningful? I don't know. All states are not the same either. >[Mr. Meron continues:] >>For sure. But they are not going to be solved by hiding behind ohh, >>it is so complicated, we can't really measure what we're doing, so >>lets just keep spending and hope for the best. Anything that is >>fundamentally sound can only benefit from some serious questioning. >I agree that serious questioning is beneficial. Nevertheless, I'm not >hiding behind that particular straw man; I would observe that >unintended consequences can be particularly surprising when one uses >simple and perhaps inadequate measures to solve a complex problem. For sure. >Urging caution so that we better understand the nature of the problem >before we fix it is not the same as saying keep on spending and >hope for the best. That's all fine. But, the ongoing process on these issues seems to be the fact that what we did appears to yield no results, indicates that we didn't try hard enough. That's a questionable attitude. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: downloads about philosophy, religion, natural sciences, mathematics and justice Hello! I'm working on a website right now. You can find free downloads about philosophy, religion, natural sciences, mathematics and justice at my website. The archive is already quiet large. I would be happy about your visit and your comment in my guestbook. The internet address is: www.philosophieren.com At the bottom of this message you can find a list of the downloadable texts available. If you're a webmaster, I would be happy if you put a link to my site on your site. Rainer Ebert --------------- --------------- -> Philosophy and Religion Alt.8agyptische Weisheitslehren Ethical texts from the Bible (M. Luthers) Codex Hammurapi (Babylonian laws from the 18th century before Christ) Recht und Rechtschaffenheit Der hippokratische Eid Parmenides Apology The Republic Das Liniengleichnis. Das H.9ahlengleichnis von der Natur des Menschen. Der Sophist Das Gastmahl (Symposion) Thrasymachos' Rede .9fber Nutzen und Vorteil der Ungerechtigkeit. Tugend ist das Finden des rechten Ma¤es Der Tod ist f.9fr uns ein Nichts Holy Bible (King James) Meditationen .9fber die Grundlagen der Philosophie in denen das Dasein Gottes und die Verschiedenheit der menschlichen Seele vom K.9arper bewiesen werden Was ist Aufkl.8arung? The Antichrist Also sprach Zarathustra. Quotations: Friedrich Nietzsche Quotations: Atheism The world as I see it Lexicon: Sein und Zeit - Martin Heidegger Was ist Philosophie? Intro to Philosophy Part 1: Basics Intro to Philosophy Part 2: Semantics Intro to Philosophy Part 3: Metaphysics Theory of Science Theory of Science Logic Interview with Tom Regan: Tiere geh.9aren nicht zwischen zwei Scheiben Brot. Interview with Tom Regan: Auf das junge Gem.9fse kommt es an. Die Philosophie der Tierrechte Materialismus und Metaphysik Selbstbewusstsein: Ein metaphysisches Relikt? Mythen des Materialismus Plato -> Natural Sciences Cosmology The Big Bang Quantum Mechanics Textbook A Brief History of Time Vibrations and Waves The modern revolution in Physics Quantum Mechanics -> Mathematics Mathematik II Lineare Algebra I Linear Algebra Trigonometry -> Law B.9frgerliches Gesetzbuch (BGB) Strafgesetzbuch (StGB) Strafvollzugsgesetz (StVollzG) Bundesverfassungsgerichtsgesetz (BVerfGG) === Subject: Re: c-number >> Howdy everybody, >> I'm reading Peskin and Schroeder's Intro to QFT and in this >> book the author constantly refers to c-numbers... without >> saying what a c-number is. I couldn't find c-number >> on mathworld.wolfram.com or physicsworld.wolfram.com. >> Can someone please tell me what the hell a c-number is? >> adam >I haven't read Peskin and Schroeder but if their usage of the term >c-number is the same as everyone else's usage then they are referring >to commuting variables. For example, Bryce DeWitt introduces the >concepts of c-numbers (commuting variables) and a-numbers (anticommuting >variables) in his book Supermanifolds. If a,b are c-numbers then Eh? I thought it was just a complex number. -- Let us learn to dream, gentlemen, then perhaps we shall find the truth... But let us beware of publishing our dreams before they have been put to the proof by the waking understanding. -- Friedrich August Kekul.8e === Subject: Re: Unit interval homemorphic to Circle help >If I = [0,1], and I / ~ is the quotient space of I obtained by > identifying 0 >and 1, then the circle S^1 is homeomorphic to I / ~. >This is intuitively clear, but how can I prove this? I think I want to show that p : I ---> S^1 given by p(x) = e^(2ix*pi) is > a >quotient map. > That's exactly what you must show. > In this case it is quite easy (fortunately). > The map p is already continuous, as I and S^1 are compact metric the map > p is > automatically closed (i.e., if A is closed then so is p[A]). > Why does a continuous map from a compact space to a compact space > necessarily > have to be a closed map? Is this what you are implying? If X is a compact space and Y is a Hausdorff space, then any continuous map f: X --> Y is closed. This follows from: (1) If X is compact and A subset X is closed, then A is compact. (2) If Y is Hausdorff and C subset Y is compact, then C is closed since continuous maps preserve compactness. > This implies that p is a quotient map. > KP > -- > E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI > PHONE: +31-15-2784572 TU Delft > FAX: +31-15-2786178 Postbus 5031 > URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft > the Netherlands === Subject: Re: A NOTy problem. This post not CC'd by email >> G'day G'day Folks, >> I have been puzzling over what ought to be a simple logic problem >> but am at a loss to come up with a method. >> A crime is committed by one of the following; Alan, Bob, Chris or >> Dave. >> Each makes a statement to the police but three of them lie. >> Alan says, I didn't do it. >> Bob says, Alan is lying. >> Chris says,Bob is lying. >> Dave says, Bob did it. >> Usually one can solve such problems with a grid >I find such grids helpful, but not perfect. It's hard to >track some statements with them, and to keep track of >what ifs. >> Where A. B. C. D represent their names and >> L. = lying, >> N. = not_lying, >> I. = innocent, >> G. = guilty. >> Somehow that doesn't seem to get anywhere is this case. >> Maybe it is because N = NOT(lying) and G = NOT(innocent) >Well because of that, an easier grid to use would be > T G >(T for telling truth) G'day G'day Randy, That is much simpler and it handles the negation aspect. >Then you just need a method to indicate whether the >statement is true or false for each one. For instance, >an X means its false and a dot means it's true. >Thus, each column will have one dot and three X's >when you're done. If you put a dot, you have to put >three corresponding X's. >> I am less interested in knowing the answer to the puzzle than >> understanding why it is different from the routine logic puzzles. >It's not. But you do need to be able to backtrack and >reset your grid after a false hypothesis. I thought there had to be a way to avoid backtracking ie a method of doing the grid WITHOUT using an eraser. I imagined solving a Boolean must arrive at the answer using reductio absurdum ie it can't be this answer or that one so it must be whatever is left. >For instance, let's start with the first statement. > Alan says, I didn't do it. >Suppose that's true. Then Alan gets an O in column >one (he's telling the truth) and an X in column >two (he's not guilty). > T G >A O X >B X >C X >D X >Once we put an O, everybody else gets an X. >Now Bob says Alan is lying. We assumed Alan is a >truth teller, so Bob is lying. We already marked that. >Chris says Bob is lying. This is true. But we already >marked Chris as a liar. Inconsistency. Alan is not >the truth teller. Restart the grid with somebody >else as truth teller and X under Alan for column one. >Of course, if Alan is lying, then this statement >is false: > Alan says, I didn't do it. >So no matter who the truth teller is, Alan did it. > T G >A X O >B X >C X >D X >Now we just have to see if putting an O under >B, C, or D works out consistently. >But actually, I see that Bob's statement is now >true. > T G >A X O >B O X >C X X >D X X >Is this consistent with the other two statements? Better >check (if not, there is no solution). >> Chris says,Bob is lying. >If Chris is a liar, then this statement is wrong. Which >it is. It checks. >> Dave says, Bob did it. >If Dave is a liar, then Bob didn't do it. And he didn't. >The grid system works fine, you just set up an improper grid. I love it. Personally I find I need a visual grid of some sort to much. > - Randy -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: A NOTy problem. This post not CC'd by email >> I am less interested in knowing the answer to the puzzle than >> understanding why it is different from the routine logic puzzles. >I am not familiar with the routine logic puzzles, >so I don't know why its solution would be different. >However, it seems to be straigtforward to solve. >You did not make use of the information that exactly three are lying. >I would guess that the routine puzzle would be that exactly >three are telling the truth. G'day G'day William, In the routine logic puzzles the person setting the puzzles provides the grid. This was my inexpert attempt at creating a grid and solving a logic puzzle. >The same method can be used to solved both types of problems. >This method must be a routine method to solve puzzles like this. >-- Bill Hale -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: Asymtotic normal distribution >Hello everybody, >I'm working on a project and I want to prove the following. >Let >Ggr ~ BIN( m/2 , pgr) >Gro ~ BIN( m/2 , pro) >Zgr ~ BIN( n/2 , pgr) >Zro ~ BIN( n/2 , pro) >I've already proved that S is a unbiased estimator for the ratio n/m is: > S = Sqrt{ (Zgr Zro)( Ggr Gro) } > What I want to prove now is that LOG(S) ~ Normal distributed. >I already have: > LOG(S)=1/2LOG(Zgr) +1/2LOG(Zro) - 1/2LOG(Ggr) - 1/2LOG(Gro) = .. > Can somebody help me with this prove? I do not see how one can say that S (or LOG(S)) is unbiased, as the numerator or denominator in the square root can be 0. That said, the variable Ggr/(m/2) is asymptotically (as m goes to infinity) normally distributed with mean pgr and variance 2/(m*pgr*(1-pgr)), etc., and as LOG(S) is a differentiable function of its arguments, it is asymptotically (for m and n large) normally distributed with mean LOG(n/m) and variance given by linearizing the logarithms. This is the usual delta-method. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Naive Q: Set theory, logic - which comes first? >>Two questions whose answers I am seeking are: >>1) What exactly might we mean by representation? >>2) How do we define a mathematical object if we do not want to >>identify it with some representation of it? >> This is done quite often. The Peano Postulates CHARACTERIZE >> the positive (or non-negative) integers. There are many other >> ways to characterize them. >So we say that 0 is a natural number, that the successor operation >takes us from a natural number to another natural number, and then >list properties that natural numbers satisfy. But what we say is >consistent with their being sets, functions in lamda calculus, or >something else. We need to exclude such interpretations, but how? Why? We only characterize what it means for a particular object to be considered as a model for the integers. It is a mistake to attempt to say what an integer is. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: A NOTy problem. This post not CC'd by email >Hello >> I have been puzzling over what ought to be a simple logic problem >> but am at a loss to come up with a method. >> A crime is committed by one of the following; Alan, Bob, Chris or >> Dave. >> Each makes a statement to the police but three of them lie. >> Alan says, I didn't do it. >> Bob says, Alan is lying. >> Chris says,Bob is lying. >> Dave says, Bob did it. >Sounds like Dave plays no role here. G'day G'day Julian, I like the approach. Reducing a four possibility problem to a three possibility problem. My question is how did you do that? Are you working that out from what Dave said or because of the statements that follow? >Either Bob or Alan is telling the truth (if not, than Bob lies and Alan is >not lieing, which is absurd). >Either Chris or Bob is telling the truth (same proof) >But there is only one truth-teller, hence Bob tells the truth. That is fantastic. As I see it you aren't forced to do an eventual back track. Your description fits finding the intersection of two sets. The reductio absurdum is confined to establishing the _initial_ is that the backtracks are confined to a single stage and not to multiple stages in the logic. (Hey, it saves on erasers.) >Alan did commit the crime! -hopefully, he's living in New Zealand so he >won't be grilled like chicken-. Indeed. Best wishes, -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: A NOTy problem. This post not CC'd by email === >Subject: A NOTy problem. > I have been puzzling over what ought to be a simple logic problem >A crime is committed by one of the following; >Alan, Bob, Chris or Dave. >Each makes a statement to the police but three of them lie. >Alan says, I didn't do it. >Bob says, Alan is lying. >Chris says,Bob is lying. >Dave says, Bob did it. >If Chris told truth: Alan told truth; two soothe sayers. >Thus Chris lied; Bob told truth; Alan did it. >Alan lied; Dave is wrong G'day G'day Alan, Well that is brilliant but I am not. Somehow you must have eliminated other entry points to the problem. That is quite a skill backtrack because of further reductio absurdum. Best wishes, -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: Why is math so difficult for some people? > Many of the so-called elegant proofs completely hide the > concepts. Proving the Central Limit Theorem for sums of > independent identical distributions is certainly the most > elegant, but it hides virtually everything; it has no > probability in it at all. This is also the case when the > distributions are not identical. But it shows the power of the Fourier transform. > However, the somewhat clumsy Lindeberg proof extends to > the martingale case, and gives an idea of what is happening. > I can give lots of somewhat unrelated proofs, but does anyone > really understand the theorem? > Similarly, Cramer's proof of the Levy-Cramer theorem that > the sum of two independent random variables is not normal > unless they both are is probably the only easy proof, but > it likewise obscures the ideas. Any time characteristic > functions are used to prove a probability theorem, the > concepts are not even present. Would you say that the elementary proofs of say the prime number theorem due to Selberg-Erdos etc, give more insight than methods using complex analysis and Fourier analysis as done by Wiener, Ikehara and Newman? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: Key core error argument, stepped out >>For crying out loud. You should stop cutting and pasting incorrect >>stuff. In fact, you should stop cutting and pasting, period. Why not >>post a link to your original post, instead, if all you are going to do >>is repeat it verbatim, ERRORS INCLUDED? >> [.snip.] >Why don't you off Arturo Magidin? >>Three times I offered to do so if you told me to stop posting >>replies. You declined to do so. >>Are you doing so now, in your oh-so-courteous way? >>Just say so: Do you want me to stop posting with comments about your >>statements? Yes or no? > You're ing kidding me!!! You mean you're asking *me* whether or > not I want you to stop posting in my threads? > Quit posting in my threads Arturo Magidin. > off. I'm amazed you finally said that. He made the same offer about a month ago, and you didn't answer. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Usenet Posting Guide? > I am still using 'rn'. An improvement over 'readnews', but until now I > have seen nothing better. (And, no, no graphical interfaces for me, > please. I have problems when I use the mouse too much.) I prefer 'tin' but remember 'rn' (and 'readnews') fondly and could use it if necessary. Graphical interfaces do have their uses; for instance, the X Window System lets me open multiple xterm windows to run command-line clients like tin (or rn). -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: Usenet Posting Guide? >I prefer 'tin' but remember 'rn' (and 'readnews') fondly and could use it >if necessary. Graphical interfaces do have their uses; for instance, the >X Window System lets me open multiple xterm windows to run command-line >clients like tin (or rn). What benefit do you get by having two newsreader windows open at once? === Subject: Re: A 3rd Grade Word Problem---HELP > The Problem: > You know a very good story. On Sunday you tell the story to a friend. > On Monday you tell it to two new people. (So far, a total of three > people have heard the story). Each day after Monday, you double the > number of new people you tell the story to. What will be the total > number of people that will have heard your story after you tell it on > Thursday? If I have a few beers, I might solve the problem like this (I still don't get 127, but I get closer.) On Sunday I tell one person. On Monday, I want to double the number of people told, so I tell 2 new people, for a total of 3. On Tuesday, having told 3 people, I want to tell double that number of new people, so I tell 6 more. Now 9 people have heard. On Wednesday, 9 people have heard, and I want to tell double that number of new people, so I tell 18 more. On Thursday, 27 people have heard, so, doubling that I tell 54 more. Now 81 people have heard me tell my story. So 81 seems to be the answer. But wait. The problem tells only about new people I tell the story to, but asks, finally only about people who've heard the story. No qualifying new. It turns out that while you were telling your story to new people, a few old people were standing around, caught in the crossfire, and heard the story also. Obviously, from the story, there were 46 of these old people standing around and they must be added in, getting 81+46 = 127. Beer, now THERE'S a temporary solution. --Homer. Bart === Subject: Re: Key core error argument, stepped out >> Three times I offered to do so if you told me to stop posting >> replies. You declined to do so. >> Are you doing so now, in your oh-so-courteous way? >> Just say so: Do you want me to stop posting with comments about your >> statements? Yes or no? > You're ******* kidding me!!! You mean you're asking *me* whether or > not I want you to stop posting in my threads? We've all seen him asking you if you wanted him to stop replying to you. (Actually, I thought it was more than three times, but I haven't really been counting.) I've always assumed that you ignored the offer because the attention you get from him (and others like him) is the only thing that lends you even the *appearance* of any credibility. > Quit posting in my threads Arturo Magidin. So you've decided to accept the offer. Either you've finally learned to read, or else you've finally realized you can't stay backed into this particular corner forever. (I'd bet on the latter.) > **** off. Please try at least to *pretend* that you are fit for decent company. -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: Two coin flip/ clarification for C Bond > As I said, end of conversation. Bye bye. > Randy, > I'll write this slowly. (I know, that like Ullrich, you think slowly) Getting bored? Trolling for 2-week-old posts to try to kick up old dust? Finding nobody will play with you any more? I walked out of this conversation with that post on Oct 23 and have no interest in returning. - Randy === Subject: Re: Key core error argument, stepped out >> Using Dot's proof that the values of a polynomial with algebraic >> integer coefficients are always divisible by an integer if and only if >> each coefficient is a multiple of that integer (in the algebraic >> integers) gives you that the statement is correct for polynomials in >> A[x]. >> >> I had not noticed Dot's theorem. The x^2 + x example seems to belie >> it: for any integer x, x^2 + x is divisible by 2, but the coefficients >> are not multiples of 2, and both of the coefficients are algebraic >> integers. You must therefore be thinking that the domain is algebraic >> integers as well. >>I think you are reasoning on cross-purposes. As I remember, Dot's theorem >>was about polynomials over the algebraic integers. As a polynomial over >>the integers, x^2 + x is always divisible by 2, but not as a polynomial >>over the algebraic integers. > Oh yeah, I finally realized where you were cheating Dik Winter, as > using v=-1+49x my polynomial is simply enough > P(x) = (v^3+1)5^3 - 3v(5)7^2 + 7^3 When you started this thread you said: 1. Let P(x) = 14706125 x3 - 900375 x2 - 17640 x + 1078, where x is in the ring of algebraic integers, notice that P(x) has a constant term that is 1078. Where did the v's come frome? > which I'm sure you conveniently forgot when you came up with your > examples. Why would he think it's there? > Nevertheless my explanations before were correct, but maybe now you > can start to see more of the particulars of why your hacks put you in > a field. Or not. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Graduate algebra book >> It's quite possible to study a reasonable amount of algebra, >> a reasonable amount of analysis and a reasonable amount of topology >> all in three years. (And also a reasonable amount of applied maths >> and statistics if that's your thing.) I'm a bit baffled about your reference >> to semesters of algebra --- do you only study one topic per semester >> at your university? >No, but at my university (and in the United States in general) we >don't usually take more than one course of the same discipline per >semester. Oh dear, oh dear. I do hope by discipline you mean, for instance, subfield of a subject. If you mean to imply that taking more than one course per semester in all of mathematics is unusual (for someone studying mathematics seriously, e.g., majoring--or even minoring--in it), then either you're wildly wrong or I am living in a dreamworld. How could someone graduate from university with a degree in mathematics having taken only eight courses in mathematics? Lee Rudolph === Subject: Re: Your Attention Please, Sci.* NG > Abhi: >This is very serious attempt indeed. My life is at stake. .. > But one thing is certain. You people change my emotions from sadness > to flames whenever I came here. It really makes me forget problems in > my personal life. Sometime bad things turn out to be good. Perhaps you should spend the effort on solving those problems instead. Some ideas work, some don't but whether this one does or not, it won't change whatever has upset you. Your life will be what you make of it, the same it true for all of us. Good luck. George === Subject: Re: Integral of e^x^2 dx > I was evaluating the integral of e^(x^2) from 2 to 3 and got something > around 12, which I know cannot be right (and I verified it on my TI-83). > David Moran Whoa ... that can't possibly be anywhere _close_ to being correct. After all, the integrand is increasing and exp(2^2) = 54.59815003314423907811 so the integral has to be greater than 54. In fact, I think that a reasonably accurate estimate for the value of your integral is 1428.052253 ... === Subject: NOVA on strings and branes & bogus free energy claims you are absolutely right that there is a cult that has grown up around various FE claims. They fit into 3 general classifications: kooks, cons and idiots. The people in the Bearden cult have tried legal threats to shut down email lists that challenge their claims. they won't respond to errors in their instrumentation - like at: www.phact.org/e/z/beardenreview.htm New free energy people spring up all the time. There is a lot of cross pollenation between FE people and UFO nuts - they both expect undeniable vindicating evidence to arrive very soon - they both think that a lot of poor evidence adds up to good evidence, and they will report glowingly on a claimant with out mention things like a prior history of frauds. Eric Krieg www.phact.org/e/dennis4.html Yes, I also know what you say to be true from first-hand experience. This is what makes legitimate work so difficult. The Skeptical Inquirer lobby does not help by garbling important differences. Perhaps 90% of the FE claims are nonsense, perhaps 99% or even more. In fact so far I am confident that 100% of over-unity FE claims are bogus. That The UFO data is another matter although there too the noise swamps the signal and one must be very careful. It is obvious from NOVA's Elegant Universe last night that the Universe is a lot stranger than anyone imagined. http://www.pbs.org/wgbh/nova/elegant/ My main themes in my two books Destiny Matrix and Space-Time and Beyond II published in 2002 were there last night in NOVA i.e. Star Gate space and time travel, contact with UFO alien intelligence in the Universe Next Door. Yes they were done in a humorous way, but the main point of the NOVA was that such science fiction may not be totally fiction. Just in: Sorry Jack it is not nonsense. Hal knows dam well who I am speaking of and it is time he comes clean. Again are you part of the solution or part of the cover up and censorship. Z Energy was involved in this as well. Puthoff has to search his own soul on this one, regain his own integrity. My integrity and the scientist are not in question. I find it interesting how quick you dismiss all of this without any investigation not even asking who the scientist is. All it takes is Puthoff to say this is nonsense and you drop everything. Do you have a mind of your own? Or do you just ask the government boys what is the truth. Don't you find it hard to consider yourself an authority in these matters when all you know is what you are told by those with a track record of not telling the truth. Sorry Jack you and Puthoff seem to be bedfellows and he has already established his character. Sarfatti's reply: Hal and I have our specific disagreements on matters of zero point energy induced gravity metric engineering theory, which is part of normal scientific method. Hal tells me my refutations of his theory is nonsense all the time. I do not believe him. However, I do believe him on this because that is one of his fields of expertise. However, I do not rely only on Hal in these matters. Hal worked with Creon Levit at ISSO on many of these bogus free energy claims in our ISSO laboratory. Creon did his work with complete integrity and an open mind. There is no coverup, there is no censorship. The fact is that ALL the current over-unity free energy claims that are for example described by Nick Cook are bogus or bad physics and engineering, I asked you what your scientific credentials were and you have not answered. The government boys sometimes ask me what is the truth in these matters. You got it wrong way round. :-) I suggest you carefully study http://qedcorp.com/APS/StarGate1.mov What were the specific technical details in the case you described? Whose lab was blown up BTW? In a message dated 11/4/03 8:56:10 PM, sarfatti@pacbell.net quotes a statement by: >Statement: It was so far over unity it scared them. He (Puthoff) refused to come up with > the money they boast to have for anyone who demonstrates such a > device. Several other top scientists were there. This same device was > demonstrated to other top scientists in the field. Shortly afterwards > demands were made for the device, threats to family etc. His lab was > blown up and these idiots keep saying no such device exists. Jack: This sounds like fiction to me. ;-) Puthoff: Definitely fiction! Hal Puthoff This is nonsense. Putoff had a demonstration of a zero point free energy device that put out phenomenal energy. Oh really? What are your credentials in physics? It was so far over unity it scared them. He refused to come up with the money they boast to have for anyone who demonstrates such a device. Several other top scientists were there. This same device was demonstrated to other top scientists in the field. Shortly afterwards demands were made for the device, threats to family etc. His lab was blown up and these idiots keep saying no such device exists. This sounds like fiction to me. ;-) With friends like you, Hal does not need enemies. :-) I have told Hal of my concern that a cult has grown up around these mindless free energy claims using his name in the most outlandish fabrications like yours here. The next thing you will ask is prove it. Show us the device or it does not exist. Been there done that, will not expose the scientist again to the so called free energy zero point community. Most are there to suppress it, some are there to control and capitalize on it, most want to steal it. When the integrity returns, the devices will come. not before. Do you really think the US government the top of which is now governed by the oil companies want such a device to get out? Then why does everyone look to government officials as an authority on whether or not zero point devices exist. It is like asking the wolf to guard your chickens and trusting him to give you an accurate head count. Might want to ask these retired government officials if they are afraid of loosing their pensions, doing a little jail time, maybe loosing their lives or the lives of their families. Might want to ask them if they would stand up for the truth, the right of the American public which they were sworn to serve to know the truth? Ask them if they new any of the people on the shuttles that died needlessly to keep a lie going?Might want to ask them how it feels in their soul to lie? Sending people up on giant firecrackers hoping they can drop like a rock and land is ridiculous with the hidden technologies at area 52 etc. Hey ask them about the other areas from 1 to 52 and beyond. What is being done there? How about hunt for integrity as a new book? ----- Original Message ----- === Subject: Nazi Saucers? re: Nick Cook's Hunt for the Zero Point http://www.salon.com/books/review/2002/08/05/zero_gravity/ --- In ItalianPhysicsCenter@yahoogroups.com, Jack Sarfatti How about this from a PhD physicist, in the minority, who has studied the UFO subject: I did some investigating and interviewed retired senior USG executives who laid the Nazi flying saucer (disguised as UFOs) claim (a la Cook's book) to rest - it didn't exist and it didn't fly. Which is pretty much what I have also been saying about Cook's book BTW. I'm interested in any debunkery of Cook's book. What did this Ph.D. fellow discover precisely? You need to ask him. I have bcc'd him on this. He can reveal his ID if he wishes. He has USG Intelligence connections for information. The point is that we at ISSO 1999-2000 checked out many of the bogus claims on zero point energy by fringe people and found them to be essentially worthless. Creon Levit on leave from NASA Ames did a very thorough job on several of the claims. More information on this is in my two books from 2002 Destiny Matrix and Space-Time and Beyond II and in my semi- popular to the Society for Literature and Science online at http://qedcorp.com/APS/StarGate1.mov Note of clarification I do not include the Haisch-Puthoff zero point energy program nor Puthoff's PV gravity program in the same crackpot category as ALL the other work mentioned by Nick Cook in his book. My objections to the HRP program are that they do not ask the right questions, are basically superficial in their formulations, make some errors of interpretation of their formalisms and most importantly have not led to any testable predictions nor any clarifying explanations of significant problems and mysteries, e.g the UFO. They are not bogus physics simply wrong physics in my humble opinion. A few specifics: 1. On the zero point origin of inertia - it is a mistake to look only at the virtual photons. It is the virtual electron-positrons that is most important. 2. There is no vacuum coherence in their idea set. That throws the baby out with the bathwater because guv Einstein's metric field for curved spacetime emerges from that non-perturbative vacuum coherence of the virtual electron-positron pairs primarily. So does the dark energy/matter that is 96% of the universe that does not appear anywhere in their models. They have been working on this for almost 20 years with little to show really. 3. There is no PV (i.e. no quantum polarized fluctuations) in Puthoff's PV math. 4. Puthoff and Ibison mis-interpret the physical meaning of their isotropic radial r coordinate in their toy model K = e^2GM/c^r Puthoff is very interested in UFOs and that is a primary motivation for this zero point gravity work. Nowhere do Puthoff and Haisch et- al squarely face the number String Tension = c^4/8piG = 10^19 Gev per 10^-33 cm which prevents any plausible explanation of UFO metric engineering with their brute force approach. Space-time geometry is simply too stiff to bend with the energy schemes they have in their paradigm. They are missing some very essential new concepts. Cook is a very approachable fellow; have you cross-checked with him also? -Andrew I told him about the flaky stuff he was pushing so he erased me from his book. That told me he was not intellectually honest, but had some hidden non-scientific agenda. I find this disturbing since he is associated with Jane's Defence Weekly in UK. ------------------------- Caveat from Jack Sarfatti Torsion fields may well exist. Orthodox studies suggest that the coupling of electron spins to hypothetical torsion fields are too weak to be of biological importance. However, not enough work has really been done. It is remotely conceivable that there could be bio-torsion effects. However, the theory of the Russian claims below is not at all clear to me nor how torsion waves would be generated and in what strength. Brief explanation of torsion field. Physics today is based on invariants of groups of symmetry transformations. The invariant is what does not change when the perspective or frame of reference of the detectors changes. Symmetries are first introduced globally i.e. same everywhere-when then they are made local. This localization of symmetry explains ALL the well-established fundamental forces of nature i.e. gravity, electromagnetism, weak radio-active and strong sub-nuclear. The global symmetry group of flat-torsionless 4Dim space-time is the 15 parameter conformal group of massless null geodesics or light rays. The conformal group splits into 4 pieces or subgroups. 1 parameter dilation group. 4 parameter nonlinear special conformal translations 4 parameter linear translations 6 parameter space-time rotations i.e. O(1,3) Lorentz group. Rest masses of electrons etc. require a breaking of symmetry of the dilation group at least. Utiyama showed how to derive Einstein's gravity that is torsion free from locally gauging the 4-parameter linear translation group which, in turn, is locally generated by the special relativity total four-momentum Pu. Einstein's curved space-time gravity metric field guv is the compensating field whose new dynamical degrees of freedom restore the translational invariance. Similarly, the torsion field would be the compensating gauge force field from locally gauging the 6-parameter space-time rotation group. There are then also compensating gauge force fields from the one-parameter dilation group and the 4-parameter special conformal transformations. The compensating field for the 1-parameter dilatation group seems to be the inflation complex scalar field in the large-scale limit. i.e. spin 0 Vacuum Coherence Field = (Higgs Amplitude Field)e^i(Goldstone Phase Field) from locally gauging the dilation subgroup. The compensating field for the 4-parameter nonlinear special conformal group seems to be the exotic vacuum unified dark energy/matter w = -1 field modulations around / = 0 where /zpf > 0 is negative zero point pressure anti-gravity dark energy and /zpf < 0 is gravitating positive zero point pressure dark matter. Lepto-quarks are quantized gauge force flux vibrating strings like vortex strings in Type II superconductors where in the core the Vacuum Coherence Field = 0. The ends of the spin 1/2 lepto-quark vortex core strings and spin 1 strings may be stuck to the 3D branes as shown last night on NOVA with off-brane spin 2 gravitons as closed vibrating strings. However these strings are ~ 1 fermi not 10^-33 cm which is why the hadronic alpha' Regge slope is (1 Gev)^-2 i.e. hadronic gravity G* ~ 10^40G(Newton). Einstein's gravity is from modulating the Goldstone phase of the Vacuum Coherence Field in the local gauging of the linear 4-parameter translation group. When there is local restoration of translational invariance is simply Einstein's field equation for the total local stress-energy density Diff(4) tensor tuv(geometry) + Tuv(matter) = 0 where trivially there is local total current conservation tuv(geometry)^;v + Tuv(matter)^;v = 0 where tuv(geometry) = (c^4/8piG)Guv Guv = Ruv - (1/2)Rguv When there is NO TORSION we also have tuv(geometry)^;v = 0 All of the above is for non-exotic vacuum where /zpf = 0. In the exotic vacuum dark energy/matter case we have local stress-energy density current conservation tuv(exotic vacuum)^;v + tuv(geometry)^;v + Tuv(matter)^;v = 0 where tuv(exotic vacuum) = (c^4/8piG*)/zpfguv /zpf = Lp*&-2[1 - Lp*^3(Higgs Field)^2] In Taylor series to first order Higgs field ~ (Non Exotic Vacuum Higgs Field) + Pudx^u/h(Non Exotic Vacuum Higgs Field) Note that the EM-weak-strong forces are from locally gauging the symmetry groups of the extra bosonic space dimensions of hyperspace not from the torsion fields as G. Shipov claims - at least in mainstream physics. The extra fermionic dimensions of hyperspace give supersymmetry transformations between lepto-quark fermions and gauge force bosons. === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> Consider an object being accelerated by a idealistic jet of water or >> continuous 'stream of elastic ping pong balls'. What is its >subsequent velocity >> pattern? Explain to me how this model applies to an attractive >electric field. What are the ping-pong balls and where are they coming >from? > Don't worry about it Randy. I doubt if you know what a differential >equation > is. >>Brilliant :-) >All right. Now tell me which diff. eq. is on the number 2 ball >in the Mass. Millions lottery. Ask an SRian. He claims to know everything. >/BAH >Subtract a hundred and four for e-mail. Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? >Explain to me how this model applies to an attractive >electric field. >What are the ping-pong balls and where are they coming >from? >>Don't worry about it Randy. I doubt if you know what a differential equation >>is. >Well, you could write one and we could see. But you don't write >equations, do you? >Meanwhile, why don't you explain to me how this model applies to an >attractive electric field. What are the ping-pong balls and where are >they coming from? This is called theoretical research Randy. The aim is to try to match a hypothetical physical process with the prediction of an equally hypothetical maths theory, namely relativity. >Or do you really think nobody noticed that you avoided answering any >questions about your model? Paul Anderson understood perfectly. Is he sarter than you?. > - Randy Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: cardinality of monotone set-functions Content-Length: 399 Originator: rusin@vesuvius I suspect the solution to the following is known: Let X be a finite set. A set-function on X is a function f: 2^X rightarrow 2^X. f is monotone if A subset B implies f(A) subset f(B). What is the cardinality of the set of monotone set-functions on X ? (Actually, I'm looking for the cardinality of the set functions such that f(A)subset A for all A.) Any help is appreciated. Federico === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> same problem. >OK, so you have something which is neither a model for >a rocket NOR an EM field. Of what use is that supposed to >be, exactly? I want to drive a boat across a river by hosing it with water from the bank. I want to know how fast it will travel. I want to know exactly how it reaches the maximum speed. By the way the river water has no viscosity. >>where v_0 is the propellant (or water or ping pong ball) velocity, >>so as long as that mass ratio is greater than e, you're point- >>even though it was flawed to begin with - is *still* dead on arrival. >>Maxwell's Equations and the Lorentz Force Equation have >>been around for a very long time, and no one's ever found >>a flaw with them. If you're going to talk about accelerating >>fields, please do so in the proper language - or is it >>that you *can't*? >>Just because you think the Earth is flat doesn't stop >>other from sailing around it. Find a hobby you can understand. >> Oh what an obvious genius we have here! >> If the x/t curve for my 'object' turns out to be about the same as the one >for >> a charge in a constant field, your conceited smirk might fade away. >But of course, since you can't even do *this* simple math, >you'll never know, will you? >Grown-ups deal with it all the time, and it's nothing involving ping >pong balls or water or any such nonsense. If you can't handle >math, you can't handle math, but don't pretend there's anything >profound about that except your ignorance. I'm sure the x/t equation for moving charges has been calculated many times, including the so-called 'relativistic mass increase'. I just want to compare that curve with a couple af mechanical models. Anything wrong wth that? Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: A 3rd Grade Word Problem---HELP > My 3rd grade son brought this word problem home the other day and he was > given the answer (127). His job, for extra credit, was to figure out how to > get 127. I'm no genius but not a dope either. I couldn't figure out how to > get 127. Nobody in the whole neighborhood could figure out how to get 127. > Is this something of a trick question Yeah, the trick is that 127 isn't the answer (or you've got the problem wrong and it's Saturday, not Thursday): Day new total --------- --- ----- Sunday 1 1 Monday 2 3 Tuesday 4 7 Wednesday 8 15 Thursday 16 31 Friday 32 63 Saturday 64 127 > or is there something in the wording > that I am missing? Any Help???? Very likely that Saturday was the intended day, since that would involve telling it each day for a week. If it was told for over a week, then the days would have to be distinguished. > The Problem: > You know a very good story. On Sunday you tell the story to a friend. On > Monday you tell it to two new people. (So far, a total of three people have > heard the story). Each day after Monday, you double the number of new > people you tell the story to. What will be the total number of people that > will have heard your story after you tell it on Thursday? === Subject: Re: cardinality of monotone set-functions Federico grava .88 la saucisse et au marteau: > I suspect the solution to the following is known: > Let X be a finite set. A set-function on X is a > function f: 2^X rightarrow 2^X. f is monotone > if A subset B implies f(A) subset f(B). > What is the cardinality of the set of monotone > set-functions on X ? > (Actually, I'm looking for the cardinality of > the set functions such that f(A)subset A for all A.) I'd say that you've got C(a,x) sets or card x in A. For each of those sets, your function can take 2^x values. Therefore, you must have Sum(x=0,a) 2^x C(a,x) = 3^a. Am I right ? -- Nicolas === Subject: Re: Key core error argument, stepped out Adjunct Assistant Professor at the University of Montana. > Using Dot's proof that the values of a polynomial with algebraic > integer coefficients are always divisible by an integer if and only if > each coefficient is a multiple of that integer (in the algebraic > integers) gives you that the statement is correct for polynomials in > A[x]. I had not noticed Dot's theorem. The x^2 + x example seems to belie > it: for any integer x, x^2 + x is divisible by 2, but the coefficients > are not multiples of 2, and both of the coefficients are algebraic > integers. You must therefore be thinking that the domain is algebraic > integers as well. >I think you are reasoning on cross-purposes. As I remember, Dot's theorem >was about polynomials over the algebraic integers. As a polynomial over >the integers, x^2 + x is always divisible by 2, I know what you meant, but no, as polynomial over the integers, x^2+x is not always divisible by 2. In fact, it is simply not divisible by 2 It is always divisible by 2 as a FUNCTION from the integers to the integers, meaning that the value of x^2+x is divisibly by 2 for every integer value x. That's part of the entire point and what has long caused James confusion, really: the difference between factoring polynomials and factoring values of polynomials. > but not as a polynomial over the algebraic integers. In case anyone is interested, here's a link to Dot's post: The statement of her result is: Theorem 1. Suppose A is the ring of integers of a number field K. Let f and g be two elements of A[x] that are coprime in K[x]. If g_eval divides f_eval in A^A then g is a constant. Theorem 2. Suppose A is the ring of integers of an algebraic closure K of Q. Let f and g be two elements of A[x] that are coprime in K[x]. If g_eval divides f_eval in A^A then g divides f in A[x]. g_eval means the image of the polynomial in A^A, i.e., the polynomial considered as a function. So, if a polynomial with algebraic integer coefficients f(x) has the property that for every algebraic integer a, f(a) is divisible by a fixed algebraic integer b, then every coefficient of f(a) is a multiple of b. In addition, it is worth noting that James's conclusion would be true ->if<- the factors of P(x) were polynomials. Then it would follow by applying Dedekind's Prague Theorem (aka Kummer's Lemma), as I noted some little time ago, and then the fact that one of the polynomials has constant term coprime to f would imply that the each of the other 2 has all coefficients divisibly by f. James's ->argument<-, however, would still be invalid. === Subject: Re: simple variational/control problem? >My book says that it is evident, due to the simple form of the >integrand in the objective, that the required h(t) is given by: >h(t) = W * delta (t - t_m), > I hope the book doesn't call this a function... lol - no worries - the book explicitly points out that this is cheating, since the variational apparatus developed there was for PWS functions - that's what immediately preceded the intuitive analysis I asked about, in fact. > Is it required that h >= 0? ack! I was writing from memory - that one slipped my mind - my bad - yes, h is nonnegative. ( h is a harvest rate, and the rest of the integrand is present-value-profit.) sorry about the omission, cdj === Subject: diagonalise a matrix how to diagonalise a matrix using elementry operations what is advantage ofusing other methods === Subject: Re: Derivatives > Phil Holman grava .88 la saucisse et au marteau: > f(p) = q > where f(140) = 15000 > and f'(140) = -100 R = pq > If we apply this practically. p=price, q=quantity sold and R=revenue. > So if we increase the price from 140 to 141, we decrease the quantity > sold from 15000 to 14900. This will increase the revenue by 900 and > not 1000. I guess the problem here is with a changing derivative > between a p=140 and p=141. This gave me a real problem and hence my > hacked solution. Do you have any other comments about this. > Actually, you're applying a discrete derivative, which is not the same > thing. For you, dR/dp = R(p+1) - R(p) where f(p+1) = f(p)+ 1*f'(p). > In this case, your formula is right and we have dR/dp = (p+1)(f(p)+f'(p)) - > pf(p) = f(p) + (p+1)f'(p) = 15000 + 141*(-100) = 900 > But you have to keep in mind that the discrete derivative isn't the same > thing as the true derivative. Furthermore, in your case, you're forced > to use an approximation of f(p+1) using the first order at p (I don't > know if this is the way to say it in English, but I think it is > understandable). before. The charge dissipation of a capacitor is seen as a continuous function of time but if you are a physicist, an electrical charge is made up of electrons with a discrete charge. It may be ok to assume a continuous discharge function when the number of electrons is large but when we get down to only a few, it is not the case. Your English is excellent by the way. Phil Holman === Subject: Re: Why is math so difficult for some people? > Similarly, Cramer's proof of the Levy-Cramer theorem that > the sum of two independent random variables is not normal > unless they both are is probably the only easy proof, but > it likewise obscures the ideas. Any time characteristic > functions are used to prove a probability theorem, the > concepts are not even present. Do you happen to know of a probabilistic proof of the Cram.8er-L.8evy theorem? -- A. === Subject: Re: Why is math so difficult for some people? >> Many of the so-called elegant proofs completely hide the >> concepts. Proving the Central Limit Theorem for sums of >> independent identical distributions is certainly the most >> elegant, but it hides virtually everything; it has no >> probability in it at all. This is also the case when the >> distributions are not identical. >But it shows the power of the Fourier transform. I do not disparage the power of the Fourier transform as a method of getting numerical answers, but in most cases in which it is used, it conceals the important ideas. In some cases, such as the Central Limit Problem, the concealment is great, as the representation, which is admittedly not easy to write down without characteristic functions, has an easier probability interpretation than the characteristic function admits. I am responsible for the current procedures based on the characteristic function for the generation of random variables with stable distributions, as the method I saw based on the Ibragimov-Chernin representation of the density of extremal stable random variables was carried over to the Zolotarev extension of their formula. Alas, I do not know of any other situation where the procedure yields anything useful. >> However, the somewhat clumsy Lindeberg proof extends to >> the martingale case, and gives an idea of what is happening. >> I can give lots of somewhat unrelated proofs, but does anyone >> really understand the theorem? >> Similarly, Cramer's proof of the Levy-Cramer theorem that >> the sum of two independent random variables is not normal >> unless they both are is probably the only easy proof, but >> it likewise obscures the ideas. Any time characteristic >> functions are used to prove a probability theorem, the >> concepts are not even present. >Would you say that the elementary proofs of say the prime number >theorem due to Selberg-Erdos etc, give more insight than methods >using complex analysis and Fourier analysis as done by Wiener, >Ikehara and Newman? I will have to pass on these, not having enough familiarity with them. The occurrence of prime numbers is sufficiently erratic that I do not feel that any of the approaches has achieved what Erdos would call appropriate for THE BOOK. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: How do you integrate the function f(x) = x/(tanx) [0,pi/2]? > Georg Goerg >> How do you integrate the function f(x) = x/(tanx) ? > There is a tricky way. Let S be the definite integral. > S=int{0 to pi/2} t cos t / sin t > = int{0 to pi/4} t cos t / sin t + int{pi/4 to pi/2} t cos t / sin t > = int{0 to pi/4} t cos t / sin t + int{0 to pi/4} (pi/2 - t) sin t / cos > t = int{0 to pi/4} t [ cos t / sin t - sin t / cos t ] - pi/2 log cos > pi/4 = int{0 to pi/4} t [ cos 2t / sin 2t ] - pi/2 log 1/sqrt(2) > = S/2 + pi/4 log 2 > and therefore > S = pi/2 log 2. > Another way: > pi cot pi.t = 1/t + 2t/(tt-1) + 2t/(tt-4) + 2t/(tt-9) + ... > Now multiply by t and integrate termwise. It's quite cute. Try it. > Larry I worked on the indefinite integral a bit. We want to find I= int(x/tan(x) dx). Let u= tan x, x= arctan(u), dx= du/(1+u^2). Then we get int( arctan(u)/u(1+u^2) du). Now separate 1/ u(1+u^2) into partial fractions. Then we get I= int(arctan(u)/u) du ) - int( u*arctan(u)/(1+u^2) du). The first integral is a standard nonelementary integral. One can use arguments similar to those in Wiener's posts to prove this. But it can be written as 1/2*I*dilog(1+I*u)-1/2*I*dilog(1-I*u). The second one also seems to be nonelementary, but I haven't a proof as yet. Interestingly enough, I managed to reduce it to integrating (arctan(u))^2 and Maple was unable to do this last integral. Maple expressed our second one as -1/2*ln(2)*arctan(u)+1/4*I*dilog(1/2+1/2*I*u)-1/8*I*ln(1+I*u)^2-1/4*I*dilog( 1/2-1/2*I*u)+1/8*I*ln(1-I*u)^2 a combination of arctan(x), logs and dilogs. It seems we have an interesting result here: arctan(x) has an elementary antiderivative, but (arctan(x))^2 doesn't! What do you all think? Ray Steiner === Subject: Re: A 3rd Grade Word Problem---HELP > My 3rd grade son brought this word problem home the other day and he was > given the answer (127). His job, for extra credit, was to figure out how to > get 127. I'm no genius but not a dope either. I couldn't figure out how to > get 127. Nobody in the whole neighborhood could figure out how to get 127. > Is this something of a trick question or is there something in the wording > that I am missing? Any Help???? > The Problem: > You know a very good story. On Sunday you tell the story to a friend. On > Monday you tell it to two new people. (So far, a total of three people have > heard the story). Each day after Monday, you double the number of new > people you tell the story to. What will be the total number of people that > will have heard your story after you tell it on Thursday? I get 31 reading it in what is to me the obvious way. Looks to me like there's a cheap trick, some way to read the words in a non-obvious way. - Randy === Subject: Re: some coin tossing > If I toss a fair coin n times, what is the probability that the outcome of > my experiment will contain m consecutive heads (m This is NOT a binomial distribution > If the exact probability is difficult to compute, can we have a good upper > bound on the probability? Here's a guy who's looked at probabilities of successive heads and found a connection with Fibonacci numbers: http://www.bjmath.com/bjmath/probable/flips.htm > A trivial upperbound is (n-m+1)(0.5^m) - Randy === Subject: Re: More symmetry between derivative and antiderivative? > What about functions not defined at zero, such as f(x) = 1/x? > Just an addition: With ln(b)-ln(a) = ln(b/a) the number of parameters > reduces from two to one in this case, too. That only considers one out of infinitely many cases of functions with antiderivatives not defined at zero, so it hardley solves the problem. === Subject: Re: Calculate Annual Percentage Rate (APR) of an Adjsutable Rate Mortgage (ARM)? > .... > Does anyone know or know where to find the procedure/formula for > calculating the APR of an ARM? > .... You may get some help from the sci.math FAQ page at http://db.uwaterloo.ca/~alopez-o/math-faq/node76.html#SECTION001140000000000 000000 Ken Pledger. === Subject: Re: Why is math so difficult for some people? > I donÇt get it.IÇm a perfect 0 at math. > Some people have no problems at all with it. > Am I too dumb for math? I've decided that I am. Having decided that, I then had to ask myself whether it mattered. Luckily I decided that it didn't. -- G.C. === Subject: Re: A 3rd Grade Word Problem---HELP > I get 31 reading it in what is to me the obvious > way. Looks to me like there's a cheap trick, some way > to read the words in a non-obvious way. I really hate it when someone makes me waste my time that way!! I remember once this supposedly smart puzzle that went like Three hunters with their three dogs were hunting. They shoot at a bunch of ducks, the three dogs go running to pick up the dead ducks, each one picks one duck, and when they came back there was exactly one duck. How do you explain that?. The stupid solution to the puzzle was that one of the dogs was called each one -- and that dog picked up one duck... Yeah, you look at it and it may seem fun (it doesn't to me -- but I can see how someone might find it humorous), but we were told that puzzle once in the middle of a conversation where we were telling serious logic and mathematics puzzles... Arrrgh, I hated it!!! Carlos -- === Subject: Re: some coin tossing Manan Sanghi grava .88 la saucisse et au marteau: > If I toss a fair coin n times, what is the probability that the outcome of > my experiment will contain m consecutive heads (m This is NOT a binomial distribution > If the exact probability is difficult to compute, can we have a good upper > bound on the probability? I define d as the beginning of the consecutive head throws. Therefore, if d > 1, you must have a tail at instant d-1. d must also be < n-m + 1 So I guess the probability is 1/2^m (at d = 1) + (n-m-1)/2^(m+1) = (n-m-1)/2^(m+1) Isn't that correct ? -- Nicolas === Subject: Re: Sets before logic : >Random thoughts on creating a theory of sets prior to a theory of : >propositions and quantifiers: : : >Let's start with the empty set, 0, and logical identity, =, : Without at least a grammar of propositions, what does : this mean? Here, one simply needs to be charitable. Even if he hasn't fleshed out a full grammar of propositions, it is clear that he gives SOME individual propositions by example. So let's just admit those and see where it takes us. For now, it is clear that x=y is a proposition iff x and y are terms and that x =def y meta-asserts substitutivity of x for y in propositions, while introducing new terms x as = to old terms y. > then we can define T, for true, by > T =def 0 = 0 The relationship between = and =def still needs clarifying. Does x =def y entail x=y? In the above example, does T =def 0=0 entail T=(0=0) ? If it does, then we already have (despite the fact that he hasn't stated a grammar for them) an infinite number of propositions, starting with 0=0, T=(0=0), (0=0)=T, T=T, (0=0)=(0=0), (0=0)=(T=T), (T=T)=(0=0), and (T=T)=(T=T). Even though he hasn't stated it, it is sort of obvious that T is going to have to NOT be equal to 0. He is just sort of dividing an infinitary propositional universe into 2 disjoint equivalence classes, one =0 and another =(0=0). : In fact, what is a set? In most set theories, : set is a primitive notion, Or even an irrelevant one; e.g., in first-order ZF, EVERYthing is a set; the notion is not so much primitive as trivial. is the definition of both set and epsilon; whenever you find a model of those axioms, you find that the class of all sets is precisely that model's domain. : but in any case, we need to : already have rules of logic to discuss anything. We can usually have discussions at the meta-level in a meta-language withOUT having fully formalized rules for its logic. > Let's define ordered pair a la Kuratowski, then : How do you define ordered pair without logic? This question is backwards; the real question is, how do you define logic without presuming ordered pairing? Any grammar of propositions where predicates take arguments is going to have to use ordering to separate the predicate from the arguments, and if there is more than one argument, it is probably going to have to use it again on the arguments. The only way you can escape ordering is to descend from the 1st-order languag to the 0th-order one where the atomic propositions don't take arguments and where the operators on them (like & and V) are commutative and associative. But assuming that we keep on cheating, that we continue to depend on ordered pairing while denying that we have defined it, HR does have a point. In ZF, in order to define Kuratowski ordered pairing, you have to use the Axiom of (unordered) Pairing. In this framework, if we don't have that axiom in advance, then we CAN'T define ordered pairing a la Kuratowski. That would leave GC with no choice but to take ordered pairing as a primitive, in which case he needs neither Kuratowski's definition nor anybody else's. But since most people think ordered pairing is MORE complex than unordered, GC would automatically be coming up with something that was using bigger stronger tools to accomplish something that others had already accomplished with smaller simpler tools. That's usually a BAD thing. : The Kuratowski definition is not the only one, and in fact one : can do a lot without them. : : > phi & psi =def = : : How does this define conjunction? One needs logical : methods to prove theorems. This is wrong; it defines conjunction by defining conjunction. It Is A *DEFINITION*! He is insisting that this IS conjunction and MEANS conjunction. The burden of proof is on you to show how it could be otherwise, not on him to show how it could be. : >(So formulae are sets--but why not? If our logic is to be infinitary, : >with infinite conjunctions say, we'll want formulae to be sets anyway.) : : >Now if-then is defined by : : > phi -> psi =def (phi & psi) <-> phi : : >(<-> is, of course, just another way of writing =; I told you we'd : >want formulae to be sets.) : : I object! <-> is NOT logical identity. You MAY not object! It's HIS definition! You have to object to his theorems, NOT his axioms, and the ONLY way you GET to object is by showing that 1 of his theorems is a contradiction. If your objection is that he's going to make it impossible to distinguish <-> from <=>, well, guess what, you can't do that in FOL, either. Logical identity is modal in FOL; No individual connective of THAT language expresses provability or tautologyhood; you have to DEFINE provability, and if the original axiom-set is not rich enough, then you CAN'T define it in the object-language. All he is saying here is that neither of <-> nor = means logical identity; they just mean identity of truth value. But the distinction is only meaningful in a context where the same statements can have multiple models, and he hasn't shown that to be possible here yet. But I am not really willing to go further with the rest of this because the let's define ordered pairs a la Kuratowski step really is just completely inadmissible. === Subject: Re: Key Core Error Argument > The following proof steps through a rather basic argument which is key > in proving an over one hundred year old error resulting from > previously unexpected consequences resulting from the definition of > the ring of algebraic integers. > Note that ultimately the proof relies on 22 NOT having 7 as a factor, > and constant terms like 7 and 22, being constant, and not variables > dependent on x, which may seem like odd things to emphasize, but I've > faced posters who've gotten away with challenging those truths because > people seem unaware that's what they're doing. > 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is > in the ring of algebraic integers, notice that P(x) has a constant > term that is 1078. > 2. It can be shown that > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > where you should note that using v = -1 + 49x, gives > P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 > where the *same* polynomial has been put in a form which allows a > factorization into non-polynomial factors so that I have > P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) > where the a's are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). I have previously pointed out that, when x = 1, *another* solution for the a's is a1 = 1, a2 = 1562, a3 = 28. This set of numbers satisfies all your requirements for x = 1, because P(1) = (5*1 + 7)*(5*1562 + 7)*(5*28 + 7) = 13789188. Maybe you should explain why you don't want this solution, but instead prefer the solutions which satisfy a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > 3. Now let x=0, so > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's at x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3, and I've picked a_1(0) and > a_2(0) to equal 0, which leaves a_3(0) with a value of 3. > 4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I > have > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) > P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). > 5. Now P(x) has a factor of 49 as > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 > which means that > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) > has a factor of 49. > 6. However, the constant term of P(x)/49 is 22, which is verified by > again setting x=0, which gives P(0)/49 = 22. Yes. This is OK. > But for two of the factors of P(x), the constant terms is 7, which is > NOT a factor of 22. Therefore, *none* of the constant terms of > P(x)/49 as they multiply to give 22 can have 7 as a factor. Also true. > (By saying that 7 is NOT a factor of 22, I'm making a choice as to > where the proof is going. Since I've been talking about algebraic > integers, where 7 is NOT a factor of 22, it's natural to go with a > choice where 7 is NOT a factor of 22.) > Given that the constant terms are independent of x's value, it must be > the case that dividing P(x) by 49 divides the two constant terms equal > to 7, by 7. No, this is your central error. You factor your function P(x) into three parts, g1, g2, and g3. Each part is a function of x: thus g1(x), g2(x), g3(x), where g1(x)*g2(x)*g3(x) = P(x). Let's focus on g3(x). It is g3(x) = 5*b3(x) + 22. The constant term is indeed 22, and 22 is indeed coprime to 7. Now suppose I pick a particular value of x. Say, x = 1. I have g3(1) = 5*b3(1) + 22. You say 22 is coprime to 7. I accept that. We all do. But why would that imply that 5*b3(1) + 22 is also coprime to 7? After all, you can have two numbers, each of which is coprime to 7, but their sum is NOT coprime to 7. For example, 20 + 22. Right? How does 22 being coprime to 7 prove that 5*b3(1) + 22 is also coprime to 7 ? Give me a straightforward, honest answer to this simple question, and I will start thinking you have something. Nora B. > 7. But to divide 7 from those constant terms requires dividing > through two of the factors, so > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > > 300125 x^3 - 18375 x^2 - 360 x + 22 > from reverse use of the distributive property, which gives constant > terms that don't have 7 as a factor, as required. > Notice that it's a rather short and direct argument, where if you > accept that 22 does not have 7 as a factor, it's obvious enough what > the constant terms of the factors must be as you go from 7, 7 and 22, > necessarily to 1, 1, and 22, when you divide P(x) by 49. > James Harris > http://mathforprofit.blogspot.com/ === Subject: Re: Homological algebra by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9VELis06821; >Can someone suggest a good book on Homological algebra?? The standrad modern text for a first course in H.A. is Weibel's An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics, 38) Its not that expensive, and it does things in the correct order. If you want to go furhter, you can try Gelfand & Manin HTH DL === Subject: Re: Was: Convergence on a space with no topology by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9VIOTn25039; >Convergence on a space with no topology >It was mentioned that: >Hey, how about including the context? > (S,d) is a metric space and > for all n in N, A_n subset S >The following does appear to be equivilant... >1. A_n -> A >What's your definition of A_n -> A ? Am I to presume it is #2? >2. A_n subset A forall n and for all p > 0, p in R for all a in A >exists q in N for all n > q exists b in A_n: d(a,b) < p If you believe me, you may assume the definition of your choice, since I believe them to be equivilant (for me, that means I have an outline lazy boy proof of this). The earlier definition, which I presume us both to have been using throughout our previous discussions, was A_n -> A <-> A_n subset A forall n and for all a' in A exists (a'_n) subset X such that a'_n -> a and a'_n in A_n for all n. >A stricter form of convergence: >Ya, uniform convergence. Agreed, uniform convergence is the way I've been refering to it in my notes, too, although I was yet to label it as such. >1. A_n ->* A >2. A_n subset A forall n and for all p > 0, p in R exists q in N >for all a in A for all n > q exists b in A_n: d(a,b) < p >So you work out a definiton of convergence for P(S). >A question then to ask is there a metric or topology for P(S) >based upon d of (S,d) that will give you that convergence. >Do you need to restrict S? >Consider if compact S helps. Ok, but lets not forget my all important B*, either. >Anyway the question has already be answered, the Hausdorff metric. >As it turns out, it's a metric for bounded closed subsets of S. >Thus for that metric, P(S) could be restriced to compact sets. see below >Now the inverse question, as the Hausdorff metric does give >convergence of A_n for compact subsets of P(S), can you convert >Hausdorff metric convergence into something more akin to what >you're wanting? >Whenever you work out a definiton of A_n -> A, will it give >convergence for more that compact subsets? If not, then you've >done nothing more that what Hausdorff did many decades ago. I think we agree here a lot, but, as for me, I remain unconvinced on this point. Take the example given on this page: N_n -> N. We know (no proof as of yet) that from this follows convergence in the Hausdorff metric. Since !(N_n ->* N), uniform convergence and convergence in the Hausdorff metric appear quite different, to me at least. see below >-- dh >Here's the extend of my notes on Hausdorff distance > dh(A,B) = max(sup{ d(a,B) | a in A }, sup{ d(b,A) | b in B }) >metric for nonnul closed bounded subsets of metric space (S,d) > dh(A,A) = 0. If dh(A,B) = 0: for all a in A, d(a,B) = 0 > A subset { x | d(x,B) = 0 } = cl B; similarly B subset cl A > dh(A,C) <= dh(A,B) + dh(B,C). Restricting a,b,c to A,B,C resp. > for all a,b, d(a,C) <= d(a,b) + d(b,C) <= d(a,b) + sup_b d(b,C) > for all a, d(a,C) <= d(a,B) + sup_b d(b,C) > sup_a d(a,C) <= sup_a d(a,B) + sup_b d(b,C) > sup_c d(c,A) <= sup_b d(b,A) + sup_c d(c,B) similarily > conclude with max(u+v, x+y) <= max(u,x) + max(v,y) >It'll show why closed bounded is needed just for the metric to be >defined. I prefer to think compact, for compact sets are indeed >closed bounded and I've hunch that compact's needed for convergence. >-- set theory >As you want convergence without a topology, why base A_n -> A upon >an underlying metric? Here's a simple definition of A_n -> A >For A_n subset S, > limsup A_n = /{ { / Aj | j > k } | k in N } > liminf A_n = /{ { / Aj | j > k } | k in N } >Yes, analogous to limsup, liminf of analysis. >Exercise: liminf A_n subset limsup A_n >A_n converges when limsup A_n = liminf A_n > (A definition of Cauchy sequence?) >and we say A_n -> A when > A = limsup A_n = liminf A_n >I think it a better way to go. It's not limited to limit from above or >from below as is yours nor will it haunt you with metrics or topologies. >In addition as it's an already established definition, we won't have the >problems your definitions are presenting. It's a more fruitful approach. Thoughts such as these have been in the back of my mind for a few days, (which, btw., is just a little bit longer than my thoughts on this subject as a whole have been.). Especially concerning non-uniform convergence -> as apposed to uniform convergence. I wondered if I could find example for A_n -> A with A != {x | x in A_n for infinitely many n}. which believe is equivilant either your limsup or liminf def. (forgive me for not knowing this off the top of my head) Just a thought: If B(0, r) subset X is a closed ball of radius r at 0 in a metric space X, and r_n -> r with r_n != r for all n, then B(0,r_n) -> B(0, r) Proof? (I've got another lazy boy proof of it). Three points to consider: 1. B(0,r) != {x | x in B(0,r_n) for infinitely many n} 2. Since B(0,r_n) -> B(0, r), B(0,r_n) should converge to B(0,r) in the Hausdorff metric. over the reals...) does it also follow that B(0,r_n) ->* B(0, r). For the moment I am entirely concetrating on point 3, i.e. trying understand more about uniform convergence. The way I see it as of now, it is inherently unlike set theoretic convergence described above. And note again, as was mentioned in a previous message, it has pleasant property A_n ->* A, then (A_n) is a *Cauchy-sequence. >Projects: >Revamp the defintion of A_n converges into something more > Cauchy sequence like >Determine when or if a topology can be introduced onto P(S) to make > A_n -> A equivalent topologically and set theoretically. >Does convergence by Hausdorff metric give set theory convergence? >Does your definition of convergence give set theory convergence? > Can you make it equivalent to set theory convergence? > I don't think so as it's based upon a metric while set theory > is more general and flexible that that. >Can set theoretic convergence in P(S) trickle down to > convergence in S? Havent had the time yet to go through your points, above. I will look at them again, tomorrow. (but still see below) >-- A simple example regarding both types of convergence: >Let N be the natural numbers (no zero) and N_n = {1,2, ...,n}. >It is easy to see that N_n -> N. However, !(N_n ->* N). >A_n subset A forall n and for all p > 0, p in R for all a in A >exists q in N for all n > q exists b in A_n: d(a,b) < p >Indeed, trivially > A_n -> /{ A_n | n in N } for any ascending sequence >do you have, using the dual superset definition for A_n -> A > A_n -> /{ A_n | n in N } for any descending sequence? >Both are so for the set theory definition. >Proof: >We have to check: >Every sequence of subsets of N convergent to N, isn't >uniformly convergent to N unless the sequence is eventually N. My notes, exactly. >Every sequence of subsets of R convergent to R, isn't uniformly >convergent to N unless the sequence is eventually dense. Yes, I agree. >You claim A_n ->* A iff >A_n subset A forall n and for all p > 0, p in R exists q in N >for all a in A for all n > q exists b in A_n: d(a,b) < p >For all n, let A_n = D be a dense subset of R. >Let A be any set with D subset A subset R. >Then A_n ->* A for all such A's. >Your uniform convergence, and hence your convergence doesn't have unique >limits. Just want to reiterate a statement from before: unique, i.e., it is unique if the set A' subset A your are refering to above is closed in the metric topology of X; (a proof followed) >Moreover, a sequence can even converge to different sets with >different cardinalities. Will have to look at this, later, I've got to run for now. >for all p > 0, p in R exists q in N for >all a in N for all n > q exists b in N_n: |a-b| < p >This is equivilant to (choose p < 1) >exists q in N for >all a in N for all n > q exists b in N_n: a = b >which, in turn, is equivilant to >exists q in N for all a in N for all n > q: a < n + 1 >This is not valid for a = q + 2 and n = q + 1, q.e.d. >Whatever math I dream up is already old hat. > -- William's Metatheorem >---- === Subject: Re: White Noise Dilemma by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA13UrL31982; >Here is a question that has had me in a quandary for several >years now. >Let X(t) be a zero-mean IID process with variance of >sigma^2. Now we know that since this function is >IID, it has a white PSD and therefore its autocorrelation >function at lag 0, R_XX(0), should be b*delta(tau), where >delta(tau) is the usual Dirac delta function and b is some >constant. >However, we also know that the autocorrelation function R_XX(tau) >is defined to be E[X(t)*X(t+tau)], and therefore that R_XX(0) >is E[X^2] (where we have dropped the dependence on t since the >process is IID). Therefore in this sense R_XX(0) = E[X^2] = sigma^2, >which is not infinite. >How do you resolve this discrepancy? Check again. The autocorrelation function is never infinite. Try normalizing phil === Subject: find (x,y) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA1D8Zo02174; so: x+y=127008 x and y have 45 divisors in common. x,y natural === Subject: hypercube/tesseract help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA1Jo9b26896; I am not at all mathematical, please keep in mind when replying. How could I find hypercube images for hebrew letters? Does anyone know how to create them? Is it even possible? === Subject: Re: someone told me ... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA1Jo7B26872; >That, somewhere in the decimal expansion of pi, >there is ASCII code for the date of my birth, the >date of my death, a thumb-scale sketch of my >life on this earth, and even a few JPEG's of me >at various activies during my life. >This is sci.math, and that statement must be >true, or false, or Godel unknowable. >Which is it? If you don't know the answer and >must speculate, please don't. If you do know >the answer, please post. >Ben Well, if normality is unknown for pi, what are some numbers that are proven normal? Every one of them would satisfy Ben's requirement. phil === Subject: Re: new way of describing ellipse for math by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA248C325624; >Playing around with stringtheory today I was in need of a more convenient method of >describing an ellipse. Instead of the two focal points and the major >and minor axis. I need a method to tie in directly to that of square and rectangle. >That the circle is to the ellipse what a square is to a rectangle and dispense with >the notion of focal points and axes. >I want this because in a Coulomb Unification the perfectness of a Coulomb force in >that it is so perfect it comes as a solo force whereas the StrongNuclear pairs with >the WeakNuclear to restore the nuclear region as a NuclearCoulomb and for the space >of electrons where the Gravity pairs with Antigravity restores a Coulomb force there. >Thus all the forces are Unifyed by being Coulomb forces of a particular region. And >it is stringtheory that can give math data as to the StrongNuclear force from Euler's >Gamma Function. >So I need a better way of going from circle to ellipse without foci and axes. >We can consider an ellipse as a squashed circle. We can consider a rectangle as >a squashed evenly square. >Proposal: If I start with a given rectangle and its 4 corners, I cannot construct a >circle by using those 4 corners. However, I can construct a ellipse using those >4 corners, but can I get a unique ellipse from a given rectangle? I think not. But I >want a unique ellipse given a particular rectangle. So, if I determine the diagonal >of that given rectangle and use it as the diameter of a circle, I wonder if I can >thus >use that circle to determine a unique ellipse that inscribes the rectangle. >What I want when finished is an easier way of describing ellipses by saying there is >a unique rectangle and a unique circle given a particular ellipse. So to speak the >circle that is squashed to get the ellipse. >The old pedantics was to fiddle around with major axis and minor axis and F1 >and F2. Remember the old pedantic of string attached to F1 and F2 and that >a pencil in the string will draw the ellipse. >Well, instead of that what I want in this new pedantic is that given a rectangle, >then what is the unique ellipse associated to that rectangle? And it is got from >the diagonals of the rectangle converted to the diameter of a circle and this circle >is then made to fit the 4 corners of the rectangle forming an ellipse. I get rid of >focal points and string. >The reason I want this for stringtheory is that the photon is perfect since the >Coulomb force is perfect and thus the photon would be a circle as a string. But the >Neutrino would be almost a circle since its restmass is tiny but not zero. So the >neutrino string would be almost a circle and almost a square instead of a >rectangle. But in the Strongnuclear force these string-ellipses would be very much >elongated especially when you get to heavy atomic nuclei such as uranium where the >nuclei are the most elongated. >I suspect that in Euler's Gamma Function that a string of stringtheory would rarely >become square or rectangle. That the photon would be the only circular >string and that most every other case of a string would be elliptical. >Can someone in mathematics tell me if my above method of getting ellipses >is a new method or whether some other mathematician has gone down that >path of describing ellipses? I would be awfully surprized if this is a new >method since so much of mathematics is trampled over and picked clean. >Archimedes Plutonium, a_plutonium@hotmail.com >whole entire Universe is just one big atom where dots >of the electron-dot-cloud are galaxies Haven't you ever used a drawing program? To draw an ellipse, you typically start at one corner of a rectangle and end at the diagonally opposite corner. The ellipse fits inside with its axes parallel to the sides of the rectangle. Old Hat. phil === Subject: Re: Repeat: White Noise Dilemma by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA248Cw25630; >I find it odd (2n+1) that no one has responded to this. I'm guessing >that either a) everyone thinks it is a homework problem (it is not), >or b) no one knows the answer. Come on, all you big strong mathematicians - >certainly you can answer a little ol' question like this, can't you? ... >Here is a question that has had me in a quandary for several >years now. >Let X(t) be a zero-mean IID process with variance of >sigma^2. Now we know that since this function is >IID, it has a white PSD and therefore its autocorrelation >function at lag 0, R_XX(0), should be b*delta(tau), where >delta(tau) is the usual Dirac delta function and b is some >constant. >However, we also know that the autocorrelation function R_XX(tau) >is defined to be E[X(t)*X(t+tau)], and therefore that R_XX(0) >is E[X^2] (where we have dropped the dependence on t since the >process is IID). Therefore in this sense R_XX(0) = E[X^2] = sigma^2, >which is not infinite. >How do you resolve this discrepancy? >-- >% Randy Yates % ...the answer lies within your soul >%% Fuquay-Varina, NC % 'cause no one knows which side >%%% 919-577-9882 % the coin will fall. >%%%% % 'Big Wheels', *Out of the Blue*, ELO >http://home.earthlink.net/~yatesc r You got the wrong delta function. You want the one that is zero everywhere except at the origin, where it is one. No infinities please. The autocorrelation you seek is equal to the variance when the two functions overlap and equal to aero otherwise. phil === Subject: Re: Something about standard deviation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA248CE25643; >Standard deviation is calculated by taking the square root of the >average of the square of the deviations from mean...Isn't it? >Why is it like that...?? >A more meaningful approach is taking the average of the magnitudes of >the deviations from the mean..something which is more obvious ? >Is it for the ease of calculations ?? >Please clarify me... >Jean In addition to moments of inertia, electrical power is also best described by variance in an alternating current situation. So variance is useful in the real world while your function isn't phil === Subject: Re: Greek Alphebet by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA1Jo8V26876; >
  In my studies on triginometry, and calculus, I am comming across 
many
>> symbols which appear to be the Greek Alphebet, as there was a chart of
>> the Alphebet in the front of the book. I am wondering if someone could
>> tell me what they all mean, or point me to a site where I could find
>> out.
>> Tom
>You need this rule: The symbols mean whatever the author says they mean.
>Take each one in the context where it is discussed. Somewhere else, the
>same symbol is very likely to mean something different.
>Even the venerable PI does not necessarily mean the constant 3.1415...,
>although that is probably the meaning in the context of the subjects you
>mentioned.
>Lynn Killingbeck
>
=== Subject: Re: Why is math so difficult for some people? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA2ChuR27568; >> I'm a perfect zero in art. Some people have no trouble >> with it. Here is my vision of hell (something that has >> actually happened at various personal-enrichment seminars): >> somebody sits you down with a pile of magazines and a >> pair of scissors and says make a collage representing >> yourself. After all, anybody can make a collage, >> right? >Well, it could be worse. They could be coming in with a >pile of bricks and a bunch of mortar and then asking you >to make a college representing yourself. Then you'll >have to be good at architecture too. Brick are very lucky. === Subject: find (x,y) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA2Chu427562; so: x+y=127008 x and y have 45 divisors in common. x,y natural 127008=(2^5)(3^4)(7^2) === Subject: Who contributed most to mathematics? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA2ChvA27582; This is somewhat of a political statement but here it is anyway. I am a patriotic American who does not cheat on his taxes,does not drive drunk, does not carry a gun or knife, obeys traffic laws, obeys the rights of others, is courteous to other people, does not litter, does not smoke, tries to be courteous to telemarketers (sometimes you really can't), sees the other guy's point of view, even right wing republican's and who is sick and tired of the French bashing by the republican party. I am tired of freedom fries, freedom cuffs, freedom ja vou, freedom bread, freedom curves, freedom connection, freedom kissing and all other bull the right wing disgustingly bestow on France, our ally! I think France has made more contributions to mathematics than any other nation. There is no Nobel prize for math but the French out rank the U.S there too or are very close. If we list, up to now, all the mathematicians professional and amateur including those who immigrated to a country and gave each points 1-100, I would bet France would top the list in total points. My SWAG of some of the others' rank. Italy Germany Russia India Greece England Japan China United States Egypt Hungary Austria Serbia ..... A good historian could probably do this in his head. Why are we bashing France because they did not want to Invade Iraq on the basis of the lies of the Asses of evil BCR? This nation should wake up and not forget who put them here and keep in damn good mind who is taking them away every day. Again, I love my country, just not how is running it. Cino Hilliard === Subject: Re: Why is math so difficult for some people? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA2ChuW27578; > at 04:56 PM, leah_lidtorf@yahoo.com.au (Leah Lidtorf) said: >>I donÇt get it.IÇm a perfect 0 at math. >It could be you, it could be your textbooks, it could be your >teachers. There is a cultural bias against intellectual subjects, and >there is a cultural bias against females in the sciences, so right off >the bat you're facing a double whammy. The public schools are full of >teachers who hate the subjects that they are teaching, and children >pick up on that. Studies have shown that teachers, even female >teachers, tend to be less encouraging to females than to males. >>Am I too dumb for math? >Second, don't be mislead by the popular prejudices. Women can do >Mathematics, some of them well enough that most of us would give our >right arms to have the same talent. Emmy N.9ather comes to mind, but >she's not the only one. This lady was great http://www.agnesscott.edu/lriddle/women/germain.htm Cino === Subject: can't show that ..... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA3DAJZ21959; Hi All, I dont know where is mistake: a and r are vectors, a=a1*i+a2*j+a3*k, a1,a2,a3-const, r=x*i+y*j+k*z fi=fi(r) rot(fi*a x r))=(grad(fi).r).a-(grad(fi).a).r+2*fi*a I use rot(fi*v)=grad(fi)xv+fi*rot(v) and can't get 2*fi*a, always have 3*fi*a. What am I missing??? Tnx Ihor === Subject: Re: congruence arithmatic/modular arithmatic by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA3DB4Z22197; >
 Are there any books available that deal exclusively or almost 
exclusively 
with
>> congruence/modular/clock arithmatic?  Most number theory books briefly 
discuss
>> this subject, but I want more information and information about
>applications of
>me too!
>any websites while we're at it?
>Jesse Samuels
>jcs.ng@asu.edu
>
=== Subject: Re: find (x,y) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA3DAna22183; >so: >x+y=127008 >x and y have 45 divisors in common. >x,y natural >127008=(2^5)(3^4)(7^2) HINT: If x & y have 45 divisors in common, then 127008 has those same divisors (and maybe more). === Subject: Re: Mathematicians' quotes about !math by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA3DAQl22014; When can a theorem be wrong? === Subject: Re: Union of hyperplanes by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA3H3nN06697; >Does anyone know convenient references to the following results? >(1) Let V be a d-dimensional vector space over a finite field F. If >d>1, then V is a union of |F|+1 hyperplanes but not |F| hyperplanes. >(2) If F is an infinite field, or even a division ring, then V is >not a union of finitely many hyperplanes. >S. Glasby A quick proof for (2): we can assume that V=F^d. A hyperplane H in V is given by a linear polynomial f in d variables. A union of finitely many hyperplanes is given as the set of zeros in V of the product of finitely many linear polynomials. Therefore if V is the union of finitely many hyperplanes, then there exists a non-zero polynomial f(X_1,...,X_d) in the polynomial ring F[X_1,...,X_d] such that f(c_1,...,c_d)=0 for every (c_1,...,c_d) in F^d. However if F is infinite, such a polynomial does not exist as one can prove by induction on d: for d=1 the assertion is clear because a polynomial in one variable of degree n has at most n zeros. For d>1 write f=f_0+f_1X_d+f_2X_d^2+...+f_nX_d^n , where the f_i are polynomials in the variables X_1,...,X_(d-1). For every (c_1,...,c_(d-1)) in F^(d-1) the polynomial f(c_1,...,c_(d-1),X_d) in one variable has infinitely many zeros by assumption. Thus by induction hypothesis it is the zero polynomial. Thus for all i and all (c_1,...,c_(d-1)) the equations f_i(c_1,...,c_(d-1))=0 hold. By induction hypothesis this implies that all polynomials f_i are equal to zero. So f is equal to zero. H === Subject: Re: Proving the Cubic Formula in college Calc. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA448dj21873; >I have been trying to find a way to prove the Cubic Formula for some >extra credit in my class. If someone knew how to do it and showed Now if we gave this to you, would you deserve the extra credit? Look up Cardan's formula and derivation Phil === Subject: Re: minimum foam by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA42HCY14115; >Hi Stephen >>Does anyone disagree that space fills with irregular tetrahedra >>such that each vertex is shared by 20 tetradrahedra? NO? Okay. >>Then.... >> >>Place a vertex at the center of each tetrahedron in that maximum >>foam and connect the dots. Each vertex has 4 edges connected. >>Voila! Space fills with irregular pentagonal dodecahedra. >>Minimum foam. >> >>Yes? No? >>Does anyone know if space fills with irregular pentagonal >>dodecahedra? Why or why not? >> If you go to 4 dimensions, then the tetrahedra can be regular. >If I 3 go with dimensions, can I fill space with irregular tetrahedra >such that all edges share 5 tetrahedra and all vertexes share 20 >tetrahedra? >> If you try to do this with distortion in Euclidean 3-space, you will >> get a distorted version of the same result. It is analogous to >> trying to tile the plane with triangles, while allowing only five >> around each vertex, or tiling with three pentagons around each vertex. >So your saying, yes, irregular tetrahedra and irregular dodecahedra >are both space fillers? >Dick Space can be filled with exactly 600 distorted tetrahedra or with exactly 120 distorted dodecahedra. If the 4-dimensional polytopes are drawn on the hypersurface of a hypersphere, they can be projected from a point on the hypersphere to a 3-dimensional hyperplane tangent to the diametrically opposite point on the hypersphere. The projected images of the polyhedra will have spherically curved faces, and the angles between the faces will be the same as on the hypersphere. Unfortunately, their volumes will not be the same, as required by the minimum foam problem. Further distortion cannot remedy the volume of the one polyhedron which has infinite volume. Sorry about that, Stephen === Subject: Maple question by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA4FWL901507; Does anyone know of a way to extract coefficients of certain expressions, like if I have z = A*f(x) + B*g(x), how can I get A (and B) without copying it from the output? I want something like A = get_coeff(z, f(x)) I guess it could be possible to divide z by f(x) and then expand it in a 0th order taylor series to keep only the constant (A). Is there a better way? Daniel === Subject: Problems on elementary functions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA4NKGn03790; Is the function sin x, if x>=0 f(x)= arc sin x, if 0>x>=-1 an elementary function? Is the function sin x / x, if x>0 or x<0 g(x)= 1, if x=0 an elementary function? How to prove your conclusions? === Subject: Re: JSH: My victories get lost by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA51QQX13640; You should do the same as your victories -- get lost. === Subject: real analysis - absolute continuity & variation of fcns by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA564Fn31804; Using the notation of Royden, 1) w.t.s the total variaton of an absolutely continuous function on [a, b] satisfies: T^a _b (f) = int{over [a,b]} |f'| dx. What am I missing? Isn't this result obvious from the FTOC? 2) Also looking for a function of bounded variation that satisfies T^a _b (f) > int{over [a,b]} |f'| dx. ... obviously this function cannot be absolutely continuous by earlier statement. I'm having difficulties thinking up a function of bounded variation that isn't absolutely continuous. === Subject: Problem with Euler's function by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA5DMI127963; Please can anyone help me? How can i prove that phi(n) is even if and only if n>=3? I am right in saying: 1) suppose n>=3, and want to prove phi(n) is even: If n is a prime, then phi(n)=n-1=even, since prime numbers are odd. If n is not prime, then n is product of primes, n=p[1]^a[1] x .... x p[m]^a[m]. for p[i] not equal p[j] if i not equal j. Then phi(n)= product(from i=1 to m) of (p[i]-1) *p[i]^(a[i]-1). since (p[i]-1) is even, then phi(n) is also even. Is this a reasonable proof? 2) suppose phi(n) is even, we know phi(1)=1 and phi(2)=1, phi(3)=2. So n must be at least 3. Is this enough to be a good proof? === Subject: problem with euler's function phi(n) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA5DMQb27968; How can i find all positive integers n such that phi(n) is not a multiple of 3? I have read from a book that there are infinitely many integers n so that 3 does not divide phi(n), but no formal proof is given, it just said that some of the integers are of the form 5^k, k=1,2,..... But this is not helping me to find all the positive integers n though. Can anyone help me? === Subject: Re: Gaussian Continued Fractions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA5DMYw27987; >> Try this: Instead of rounding down, round to the nearest Gaussian >> integer, regardless of its direction. >This wouldn't be the right thing to do, I think. If we consider a >complex number (that we are trying to find the CF of) which is a >negative real, then we would want to round towards zero. >This is because: >if x = [a(1),a(2),a(3),...], >then -x = [-a(1),-a(2),-a(3),...]. >Leroy Quet Using round to nearest, e = (3, -4, 2, 5, -2, -7, 2, 9, ... ). This can be converted to the usual form as follows: 1. At each sign change, insert a term equal to 1 between them while reducing their absolute values by one each. 2. Replace all terms by their absolute values. This gives: e = (2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ... ). (Of course, this is assuming that the original number is positive.) So there is no problem with using round to nearest. Besides Gaussian integers, there are also Eisenstein integers, (a + b w), where a & b are integers and 1 + w + w^2 = 0, and Hurwitz integers which are quaternions (a + b i + c j + d k), where the a, b, c & d are all integers, or else they are all odd half integers. There are 6 Eisenstein units and 24 Hurwitz units. Continued fraction expansions can be formed from each of these by alternately subtracting out the nearest integer and then taking the reciprocal of the result. Analogously, using geometric inversion instead of the reciprocal, there are lattices of any finite dimensionality that can be treated the same way. This relates to the sphere packing problem because this procedure only fails if there is room to pack more spheres of unit diameter in the holes of the lattice. The traditional continued fraction series can be related to a regular tiling of the hyperbolic plane with triangles which have all three angles equal to zero. These triangles can be paired to produce squares, or a triangle can be grouped with its 3 neighbors to produce a regular hexagon. The groups need to be placed symmetrically as if the joining edge is a mirror to make a square or a hexagonal tiling pattern. The Gaussian version is related to a 3-dimensional hyperbolic tiling with octahedra which have 90 degree angles between faces and vertex angles equal to zero. The Eisenstein CF expansion is related to a tiling with tetrahedra which have 60 degree angles between faces and vertex angles equal to zero. By surrounding one of these tetrahedra with four others, a cube is formed, thus making a cube tiling pattern. === Subject: Re: Colors of Infinity : Arbitary Maths by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA5DLxO27949; 10002021415341453416335333134135354135415341534413579+77974341321.1534324456 46543444145643 5341464443245345343453454145453153456453453453453435465422585888856546546546 456465456746844+45646767435467464655753456745675669+++4567456756453453536456 3 746878756453486756453456467465746545646545646787867875424546+456445 456 54 5 5 53 4 4 (protons structure) done by Dr/phisicist Dan === Subject: Factorizations by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA5H43g11605; can anyone help me === Subject: Re: Key Core Error Argument > [snip umpteenth repeat of previous flawed argument] > I thought you were going to just post your proof elsewhere and stop > cluttering up this newsgroup. You've given a URL, now go away. Such a remarkably silly posting. What's the definition of insanity again? How many YEARS have I posted, and how many posters through the years have ordered me to go away? Yet here you are C. Bond doing the same silly thing, as if it matters. You're funny though in your irrationality. > -- > There are two things you must never attempt to prove: the unprovable -- > and the obvious. > -- > Democracy: The triumph of popularity over principle. But you see there's talk and there's substance. My proof is substance, and in fact, there is no error in it. If you disagree, then point out an error in what follows. The following proof steps through a rather basic argument which is key in proving an over one hundred year old error resulting from previously unexpected consequences resulting from the definition of the ring of algebraic integers. Note that ultimately the proof relies on 22 NOT having 7 as a factor, and constant terms like 7 and 22, being constant, and not variables dependent on x, which may seem like odd things to emphasize, but I've faced posters who've gotten away with challenging those truths because people seem unaware that's what they're doing. 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is in the ring of algebraic integers, notice that P(x) has a constant term that is 1078. 2. It can be shown that P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 where you should note that using v = -1 + 49x, gives P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 where the *same* polynomial has been put in a form which allows a factorization into non-polynomial factors so that I have P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) where the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). 3. Now let x=0, so P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) as the cubic defining the a's at x=0 is a^3 - 3a^2, which has roots, 0, 0 and 3, and I've picked a_1(0) and a_2(0) to equal 0, which leaves a_3(0) with a value of 3. 4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I have P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). 5. Now P(x) has a factor of 49 as P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 which means that (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) has a factor of 49. 6. However, the constant term of P(x)/49 is 22, which is verified by again setting x=0, which gives P(0)/49 = 22. But for two of the factors of P(x), the constant terms is 7, which is NOT a factor of 22. Therefore, *none* of the constant terms of P(x)/49 as they multiply to give 22 can have 7 as a factor. (By saying that 7 is NOT a factor of 22, I'm making a choice as to where the proof is going. Since I've been talking about algebraic integers, where 7 is NOT a factor of 22, it's natural to go with a choice where 7 is NOT a factor of 22.) Given that the constant terms are independent of x's value, it must be the case that dividing P(x) by 49 divides the two constant terms equal to 7, by 7. 7. But to divide 7 from those constant terms requires dividing through two of the factors, so (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 from reverse use of the distributive property, which gives constant terms that don't have 7 as a factor, as required. Notice that it's a rather short and direct argument, where if you accept that 22 does not have 7 as a factor, it's obvious enough what the constant terms of the factors must be as you go from 7, 7 and 22, necessarily to 1, 1, and 22, when you divide P(x) by 49. James Harris http://mathforprofit.blogspot.com/ === Subject: Re: Sets before logic > : >Random thoughts on creating a theory of sets prior to a theory of > : >propositions and quantifiers: > : > : >Let's start with the empty set, 0, and logical identity, =, > : Without at least a grammar of propositions, what does > : this mean? > Here, one simply needs to be charitable. Even if he hasn't > fleshed out a full grammar of propositions, it is clear > that he gives SOME individual propositions by example. > So let's just admit those and see where it takes us. > For now, it is clear that x=y is a proposition iff x and y are terms > and that x =def y meta-asserts substitutivity of x for y > in propositions, while introducing new terms x as = to old terms y. > then we can define T, for true, by > T =def 0 = 0 > The relationship between = and =def still needs clarifying. > Does x =def y entail x=y? In the above example, > does T =def 0=0 entail T=(0=0) ? > If it does, then we already have (despite the fact that he > hasn't stated a grammar for them) an infinite number of propositions, > starting with 0=0, T=(0=0), (0=0)=T, T=T, > (0=0)=(0=0), (0=0)=(T=T), (T=T)=(0=0), and (T=T)=(T=T). > Even though he hasn't stated it, it is sort of obvious that > T is going to have to NOT be equal to 0. He is just sort of > dividing an infinitary propositional universe into 2 disjoint > equivalence classes, one =0 and another =(0=0). > : In fact, what is a set? In most set theories, > : set is a primitive notion, > Or even an irrelevant one; e.g., in first-order ZF, EVERYthing > is a set; the notion is not so much primitive as trivial. > is the definition of both set and epsilon; whenever you find > a model of those axioms, you find that the class of all sets > is precisely that model's domain. > : but in any case, we need to > : already have rules of logic to discuss anything. > We can usually have discussions at the meta-level in > a meta-language withOUT having fully formalized rules > for its logic. > Let's define ordered pair a la Kuratowski, then > : How do you define ordered pair without logic? > This question is backwards; the real question is, how > do you define logic without presuming ordered pairing? > Any grammar of propositions where predicates take arguments > is going to have to use ordering to separate the predicate > from the arguments, and if there is more than one argument, > it is probably going to have to use it again on the arguments. > The only way you can escape ordering is to descend from the > 1st-order languag to the 0th-order one where the atomic > propositions don't take arguments and where the operators > on them (like & and V) are commutative and associative. > But assuming that we keep on cheating, that we continue to > depend on ordered pairing while denying that we have defined it, > HR does have a point. In ZF, in order to define > Kuratowski ordered pairing, you have to use the Axiom of > (unordered) Pairing. In this framework, > if we don't have that axiom in advance, then we CAN'T > define ordered pairing a la Kuratowski. I am presupposing a theory of sets. But I'm interested in the question: given _some_ theory of sets, and given my definitions of the logical constants, what logic results? I.e. classical FOL, Intuitionistic FOL, etc. I don't think that all of Herman's objections are valid--as you point out, I can define as I please. > That would leave GC with no choice but to take ordered > pairing as a primitive, in which case he needs neither > Kuratowski's definition nor anybody else's. But since > most people think ordered pairing is MORE complex than > unordered, GC would automatically be coming up with something > that was using bigger stronger tools to accomplish something > that others had already accomplished with smaller simpler tools. > That's usually a BAD thing. > : The Kuratowski definition is not the only one, and in fact one > : can do a lot without them. > : > : > phi & psi =def = : How does this define conjunction? One needs logical > : methods to prove theorems. > This is wrong; it defines conjunction by defining conjunction. > It Is A *DEFINITION*! He is insisting that this IS conjunction > and MEANS conjunction. The burden of proof is on you to show > how it could be otherwise, not on him to show how it could be. > : >(So formulae are sets--but why not? If our logic is to be infinitary, > : >with infinite conjunctions say, we'll want formulae to be sets anyway.) > : > : >Now if-then is defined by > : > : > phi -> psi =def (phi & psi) <-> phi > : > : >(<-> is, of course, just another way of writing =; I told you we'd > : >want formulae to be sets.) > : > : I object! <-> is NOT logical identity. > You MAY not object! It's HIS definition! > You have to object to his theorems, NOT his axioms, and the ONLY > way you GET to object is by showing that 1 of his theorems is a > contradiction. If your objection is that he's going to > make it impossible to distinguish <-> from <=>, well, guess > what, you can't do that in FOL, either. Logical identity > is modal in FOL; No individual connective of THAT language > expresses provability or tautologyhood; you have to DEFINE > provability, and if the original axiom-set is not rich enough, > then you CAN'T define it in the object-language. All he is saying > here is that neither of <-> nor = means logical identity; they > just mean identity of truth value. But the distinction is only > meaningful in a context where the same statements can have multiple > models, and he hasn't shown that to be possible here yet. > But I am not really willing to go further with the rest of this > because the let's define ordered pairs a la Kuratowski step > really is just completely inadmissible. -- G.C. === Subject: Re: problem with euler's function phi(n) Kathy grava .88 la saucisse et au marteau: > How can i find all positive integers n such that phi(n) is not a > multiple of 3? I have read from a book that there are infinitely many > integers n so that 3 does not divide phi(n), but no formal proof is > given, it just said that some of the integers are of the form 5^k, > k=1,2,..... > But this is not helping me to find all the positive integers n > though. Can anyone help me? If n = p1^a1 + p2^a2 + ... + pk^ak, then: phi(n) = n(1-1/p1)(1-1/p2)...(1-1/pk) phi(n) is a multiple of 3 if n is a multiple of 3 at a power strictly greater than 1 or if one of its prime divisors is of the form 3i+1. Otherwise: - if n is a multiple of 3, then n(1-1/3) = 2n/3 will not be a multiple of 3 - if n is not a multiple of 3 and has no prime divisors of the form 3i+1, none of the (pk-1)/pk will be a multiple of 3. - on the contrary, if one of the pk is of the form 3i+1, then (pk-1)/pk will be a multiple of 3. -- Nicolas === Subject: Re: Was Lucy the split image - Jabriol is a creationist. > tell them also how you love non jw-pedophiles. > and how you agree with sex among children, when you send your own son > to wild parties at motels.. > oh by the way since you are in an informative mood.. tell them how you > have been caught lying so much.. that nobody belives you anymore.. > calling me a creationist does not make me one... and in fact you being > a JW groupie should know that JW's are not creationist... > However we also know, thta you need to lie, because, you cant get your > point across by telling the truth.. > now go get a new husband and service him like a house wife should, > because the first one bashed you like a romper room punching bag, and > the second run you out of Nashville, because you send to much time on > the net.. === Subject: Re: I can't stand it anymore Unc, Is it racist to state that Pygmies are short, and Watusis are tall?. -- 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: Key core error argument, stepped out > 7. But to divide 7 from those constant terms requires dividing > through two of the factors, so (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > > 300125 x^3 - 18375 x^2 - 360 x + 22 from reverse use of the distributive property, which gives a constant > term coprime to 7, as required. Let > Q(x) = 7(25x^2 + 30x + 2) [1] > = 7(x^2 + x)(5^2) + 7(x - 1)(5) + 7^2 That's dumb. The factorization I use isn't just pulled out of the blue. More importantly, examples are a waste of time when the proof is in front of you, and the proof itself is so simple. What I hate about posters who decide to go off on a tangent is their need to waste time. Obviously they have an agenda which is to cast doubt on my work, but rather than face it directly, they start chattering about something else, claiming it's related. But over time it's clear that there's some key difference, or worse, even when their own examples support my case they simply fudge. So their purpose is simply to obfuscate, which is why they hide from the clear and direct argument in front of them to send people on wild goose chases. Rick Decker is a particularly abhorrent type as he pretends at times to be reasonable, but he probably realizes by now that he's screwed as my work is correct, and he's had a hand in hiding that fact. Rather than show some common decency, he simply looks for another route to continue hiding the truth. I point that out because I don't think he should remain in a position based on society's faith in his common decency, when he so clearly lacks any. If he had any, he'd be posting retractions, rather than working still to hide the truth. James Harris === Subject: Re: Maple question > Does anyone know of a way to extract coefficients of certain expressions, like if I have z = A*f(x) + B*g(x), how can I get A (and B) without copying it from the output? I want something like A = get_coeff(z, f(x)) Try the op(.) command === Subject: Re: More symmetry between derivative and antiderivative? >What about functions not defined at zero, such as f(x) = 1/x? The antiderivatives of this function are too important to ignore, >but setting limits of integration to include zero is a no-no. >>I consider the physically relevant space either according to the >>traditional arrow of time (-inf, 0) or matching to R+ (0, inf), >>i.e. neither zero nor infinity are included. > Physical relevance is irrelevant to the mathematics. > Really? Really! > You caused me to crosspost this message. How? Did I twist your arm? > Maybe, mathematicians believe they are independent. Do you claim that physics is the only use for mathematical theories? There are applications of mathematics outside of physics which are not in any way dependent on the theories of physics. In that sense, if no other, mathematics IS independent of physics. > However, they are > just not aware to what extent their discipline benefited not just from > funding but also from people who dealt with practical applications. Actuarial sciences and economics and other non-physics areas fund more math than physics does. Perhaps mathematicians should dump physics and only go where the big money is? > I will try and give the gist of an utterance by Oliver Heaviside: > Mathematicians said, this series does not converge. On that > condition it will be of use. He hated unjustified rigor. > Ironically, his revenge was made as rigorous as possible. The whole mathematical theory of asymptotic series deals usefully and rigorously with series that diverge. So what is your point? > >>You are correct, 1/x does not exist exactly at x. >>Its antiderivative ln(x) + C contains anyway an infinite constant C. Infinite constants are nonsense, at least in the mathematics of > Reimann integration. > Even if I am merely an old engineer with minimal training in > mathematics, I am aware of some facts concerning Riemann type > integration. Perhaps nobody defined integrals exclusively for open > domains while the world excessively integrates from minus infinite to > plus infinite. Is this correct? Were you, as an old engineer or otherwise aware that Reimann integrals are defined only on closed intervals? To integrate over an open domain, including the set of all reals, requires an extension of the Reimann definition though some sort of limiting process. I am not sure what you mean by excessively in this context. > >>Only ln(x1)-ln(x0) = ln(x1/x0) makes sense in real-world physics. >>I presume, there is no reason not to hide C within x2=0. > You will, no doubt, presume whatever nonsense you wish about > real-word physics, but that does not make it true or relevant to > mathematics. The realities of mathematics are quite independent of > your view of physics. > I was told, every paying costumer is the king at least in America. > Doesn't physics pay for mathematics? No! See above! Doesn't physics provide tailor-made > models of nature? Yes, but it is not the exclusive provider of models of interest to mathematicians. Very often it has been mathematics developed without physical models that has been later adapted to the needs of physics. Group theory springs to mind as one example. Thus one can produce a strong argument that mathematics does not need physics as much as physics needs mathematics. You seem to want the tail (physics) to wag the dog (mathematics). === Subject: Re: naive geometry questions > Here are two geometry questions from a curious layperson. If they're unclear, > please tell me. Ditto if they're addressed to the wrong forum. > 1) Are there surfaces that can be bent to each other (the way a flat piece of > paper is bent to a cylinder) but not to anything flat? > I'm not sure I understand very well... it depends on what > kind of bending would be allowed. For instance, would > a cube and a sphere be 'bendable' into each other? > Spheres and eggs? > If you think about it, I guess you will have a hard time > explaining (and even defining) what you mean exactly > with 'bending'. How about 'preserving Gaussian curvature'? The surfaces with Gaussian curvature = 0 are developable. E.g. cones, cylinders, the plane. -- G.C. === Subject: Re: Factorizations > can anyone help me 1st - Algebraic (Cyclotomic and Aurefeuillian) 2nd - Sieve/trial-divide 3rd - Pollard's P-1 or Rho. 4th - Lenstra's ECM. 5th - If small (<100 digits) QS, else more ECM And that's prettu much all you need to know. Phil -- Unpatched IE vulnerability: DNSError folder disclosure Description: Gaining access to local security zones Reference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0306/52.html === Subject: Re: problem with euler's function phi(n) > How can i find all positive integers n such that phi(n) is not a multiple of 3? I have read from a book that there are infinitely many integers n so that 3 does not divide phi(n), but no formal proof is given, it just said that some of the integers are of the form 5^k, k=1,2,..... > But this is not helping me to find all the positive integers n though. > Can anyone help me? Phi is a multiplicative ( phi(p^n.q^m) = phi(p^n) * phi(q^m) ) so only need to consider prime powers. What's Phi(p^n) for a) n=1 b) n>1 ? have 3 as a divisor of phi(n). Phil -- Unpatched IE vulnerability: Web Archive buffer overflow Description: Possible automated code execution. Reference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0303/107.html === Subject: Re: Math dependency logic REVISED <2tncqv4kb8gf7chslv93qjbr34aca7h2qa@4ax.com> <3fa5b2a8_3@newsfeed.slurp.net> <63dgqvcuool9gmchue32p952rpatmsm05l@4ax.com> A8EwTYfhf*u~,Eu,tf6$HN*MY&)u0G =N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@J6m5.EN?>Zh Xh;Y V|',x(js'Jfq02joVpj|#x linux) >> I kinda like this when you get all het up - you make a bigger >> fool of yourself than usual. >> Here for example. All I said was that the disputes were not over >> the fact that if you start with something true and apply correct >> reasoning then the conclusion must be true, the disputes >> were over the correctness of his reasoning. You say that >> of course I'm lying? Then you must think that the people >> who have been saying James is wrong have been disputing >> the statement if you start with something true and apply >> correct reasoning then the conclusion must be true. >> You should point out where anyone has disputed that fact. >> . > You yourself disputed that fact. Starting from C1-C4, > C1 AxAy[x=y -> Az(z in x <-> z in y)] > C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] > C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in > A)Classification > C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak > Extensionality) > by applying correct reasoning, one deduces Ex~(x=x). > Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point out > the error. So C1-C4 |- Ex~(x = x)? I assume that's right -- I won't check. Ullrich said it didn't? Maybe, though if so, it's likely just a minor error (perhaps caused by context?). Where does this entail a denial of if you start with something true and apply correct reasoning then the conclusion must be true? Are you just lying? Or, shall we be charitable and assume that you're stupid? -- Well *supposedly* a correct and profound math paper can get published in a 'reputable journal' which means that the journals I've faced so far may lose a lot of their luster once the full story comes out. --- James Harris, on the quality of math journals rejecting his paper === Subject: Re: JSH: Attacking a proof is attacking yourselves A8EwTYfhf*u~,Eu,tf6$HN*MY&)u0G =N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@J6m5.EN?>Zh Xh;Y V|',x(js'Jfq02joVpj|#x linux) > See what I mean? I called David Ullrich a ing piece of dog, > but here he is, yet again. > The math professor trapped by his need to confuse others. You're not trapped by a need to confuse others, right? So, if we call you a ing piece of dog, you'll go away? Just kidding. I love most of your material (though the ing piece of dog bits really aren't so clever --- think over this latest raging asshole without a clue persona). -- Jesse Hughes By definition m is a variable. By definition all then (sic) numbers represented by letters are variables--that's algebra[,] Magidin. -- James Harris shows deep understanding of algebra === Subject: Re: real analysis - absolute continuity & variation of fcns > Using the notation of Royden, > 1) > w.t.s the total variaton of an absolutely continuous function > on [a, b] satisfies: > T^a _b (f) = int{over [a,b]} |f'| dx. > What am I missing? Isn't this result obvious from the FTOC? No, it's not. The FTOC applies to continuous integrands, not L^1 functions. And even when f is C^1, I don't see the FTOC coming in. I see a proof via the mean value theorem and convergence of Riemann sums. That T(f) <= int |f'| is trivial. To go the other way, you need to prove something like this: If g is in L^1, then sum (i=1,n) | int_[x(i-1), xi] g | -> int_[a,b] |g| as n -> oo. Here a = x_0 < x_1 < ... < x_n = b and the mesh size -> 0. > 2) > Also looking for a function of bounded variation that satisfies > T^a _b (f) > int{over [a,b]} |f'| dx. If you allow noncontinuous functions, this is easy. (Recall that every monotone function is BV.) For a continuous example you could look at the Cantor function. === Subject: Re: Problem with Euler's function > Please can anyone help me? > How can i prove that phi(n) is even if and only if n>=3? > I am right in saying: > 1) suppose n>=3, and want to prove phi(n) is even: > If n is a prime, then phi(n)=n-1=even, since prime numbers are odd. > If n is not prime, then n is product of primes, n=p[1]^a[1] x .... x p[m]^a[m]. for p[i] not equal p[j] if i not equal j. > Then phi(n)= product(from i=1 to m) of (p[i]-1) *p[i]^(a[i]-1). > since (p[i]-1) is even, then phi(n) is also even. > Is this a reasonable proof? Add the case where the only prime factor is 2 (i.e. n=2^x). > 2) suppose phi(n) is even, > we know phi(1)=1 and phi(2)=1, phi(3)=2. So n must be at least 3. > Is this enough to be a good proof? No problemo, Phil -- Unpatched IE vulnerability: DNSError folder disclosure Description: Gaining access to local security zones Reference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0306/52.html === Subject: Re: Math dependency logic permission for an emailed response. Ultimately my argument relies on numbers like 7 being NUMBERS, not > variables dependent on x, and on the distributive property. 7 is a number, but so is x. Neither can change (in a given context); > each refers to exactly one number at a time. > What does time have to do with it? > How do you define time with mathematical argument Thomas Bushnell, BSG? Good grief. Time in that sentence refers to scope, it's a repetition of the in a given context paretheses. This is English usage. I refer you to In any given context, x refers to exactly one number (that is, x refers to the number x), never more than one. It certainly doesn't change. It doesn't vary. For example, in the sentence x = 2, x refers to 2. In the sentence x - 2 = 0, the symbol x again refers to 2. In both sentences, x refers to exactly one number, and doesn't vary or change at all. In the sentence for all x, if x is an integer, then either x is even, or x is odd--in that sentence, x doesn't appear at all! That's just how quantification works. In the phrase x + 2, we have two ways of understanding that phrase. We might mean x + 2 to be a function f, such that f:R->R, with the definition that for all x, f(x) = x+2. In that case, the variable x is bound by the quantifier, and so again it doesn't appear. This can be confusing, because for all x, f(x) = x+2 gets abbreviated as f(x) = x+2, and the function f:R->R such that f(x) = x+2 gets abbreviated as x+2. So when x+2 is taken to refer to a function, then the variable x doesn't occur in it, because x+2 would just be shorthand for the function f such that for all x, f(x) = x+2, and neither f nor x are free in that specification thank to the rules for quantifiers. Or, we might mean by x+2 a number, rather than a function. In that case, it refers to exactly one number, and x also refers to exactly one number (though we don't know which absent further information). Thomas === Subject: Re: DESIGN OF A PRIMITIVE NANOFACTORY > Design Shut the up. === Subject: Re: Sets before logic : I am presupposing a theory of sets. I'm sorry; that is just TOO much to presuppose. Moreover, it's ambiguous: there are DIFFERENT theories of sets and it matters WHICH one you presuppose. To presume that they all have a pairing axiom may itself be presuming too much. : But I'm interested in the question: : given _some_ theory of sets, and given my definitions of the logical : constants, what logic results? I repeat, there is MORE than one theory of sets and you canNOT make this work without stating your own underlying set theory. And that is going to turn out to be a harder problem than this one. : I.e. classical FOL, Intuitionistic FOL, etc. OBVIOUSLY -- JEEZUS! -- THAT is going to depend on whether YOUR UNDERLYING SET THEORY WAS DEFINED in Classical FOL xor in Intutionistic FOL! You simply canNOT say, Presume SOME set theory. HOW DOES ANYONE KNOW whether a theory is or isn't a set theory? You really are going to have to define your own set theory. The mere fact that you have alleged the existence of a universal set AUTOMATICALLY takes you AWAY from most of the world's known set theories! At a bare minimum you owe us a simple small axiom set that is consistent with the existence of a universal set, or, if that's too much, a universal class. === Subject: Re: Algebraic Closure >What is the algebraic closure of a finite field? >I'd be happy just to know the algebraic closure of Z/<2>. > Let K be a finite field and L the algebraic closure of K. Then we know that for every integer n there exists exactly one field M within L that has degree n over K. > The field M looks like K[X]/pK[X], where p is an arbitrary irreducibel > polynomial of degree n. > There are algorithms to determine the irreducibel polynomials over the > field K (for example Berlekamps algorithm). > The field L is the union over all the field M, and these fields form > an ascending chain. > Every algebraic computation within L typically involves only finitely > many elements. So you can perform the calculation in one of the M's, > that are under your control. > So in a sense one can say that the algebraic closure of K is >known<. First off, it is easy to show that if F has p^a elements and E has p^b, then F is (isomorphic to) a subfield of E iff a divides b. Armed with this, you can form the tower of fields GF(p), GF(p^2), GF(p^6), GF(p^24),...,GF(p^n!) and their union is the algebraic closure of GF(p). The point is that any algebraic extension of a finite field is a root extraction and one sees that all roots of the elements of GF(p) appear somewhere in that tower. This is about as explicit as you are going to get. === Subject: Re: some coin tossing >Manan Sanghi grava .88 la saucisse et au marteau: >> If I toss a fair coin n times, what is the probability that the outcome of >> my experiment will contain m consecutive heads (m> This is NOT a binomial distribution >> If the exact probability is difficult to compute, can we have a good upper >> bound on the probability? >I define d as the beginning of the consecutive head throws. Therefore, >if d > 1, you must have a tail at instant d-1. d must also be < n-m + 1 >So I guess the probability is 1/2^m (at d = 1) + (n-m-1)/2^(m+1) >= (n-m-1)/2^(m+1) I gave it some thought and I think this is way too optimist, since you can get a similar result by assuming that within the n throws there are n-m+1 independent sub-sequences whose probabilities of containing the required sequence are the same - in reality none of the sub-sequences are independent for m > n/2 so this isn't a very good estimate for other than m << n. Might one first identify the cases where P ~= 0 and then discard these cases to reduce the problem to somewhat simpler parameters? For example we know that the case m > n/2 for just about any n is infinitesimally unlikely so we can discard those. === Subject: Re: some coin tossing Toni Lassila grava .88 la saucisse et au marteau: > I gave it some thought and I think this is way too optimist, since you > can get a similar result by assuming that within the n throws there > are n-m+1 independent sub-sequences whose probabilities of containing > the required sequence are the same - in reality none of the > sub-sequences are independent for m > n/2 so this isn't a very good > estimate for other than m << n. By assuming the throw before the serie was a bad throw, I thought I had avoided the problem. But I may be wrong. I'll think about it later. -- Nicolas === Subject: Maximizing and minimizing in optimization Is it possible to convert a problem where we are trying to maximize a quantity (profit) into a problem where we are trying to minimize a (different) quantity (loss)? If yes, can we do the transform in linear time? In particular, is it possible to transform a Network flow problem where we try to maximize the flow between the source and sink into a shortest path problem where we are trying to find the shortest between two points (say source and sink)? -- C P Pradip === Subject: Re: I can't stand it anymore > amanda replied: > >>amanda replied: >I have been biting my tongue about the IQ test but I can't any more. How reliable is a test that use the term Asian to represent the most >diverse of ethnic and cultural groups? >>Are you Asian? Yes. Not oriental though but grew up in that region. > So you recognize and understand the word. It has utility and meaning. > Why do you get so upset about it? > Many hate the word American and raise all manner is idiotic arguments > as to why it is wrong (and most of them defend the term native American). > Do you have similar issues with the word American, or issues like the > ones you have with the word Asian? What gave you the idea that I was upset about the term Asian? My only point was that IQ tests to determine people's intelligence are bs. > Rich >>Rich === Subject: Re: A NOTy problem. > G'day G'day Folks, > I have been puzzling over what ought to be a simple logic problem > but am at a loss to come up with a method. > A crime is committed by one of the following; Alan, Bob, Chris or > Dave. > Each makes a statement to the police but three of them lie. > Alan says, I didn't do it. > Bob says, Alan is lying. > Chris says,Bob is lying. > Dave says, Bob did it. > Somehow that doesn't seem to get anywhere is this case. > Maybe it is because N = NOT(lying) and G = NOT(innocent) and one has > to use another method say Boolean logic or Karnaugh maps. > I am less interested in knowing the answer to the puzzle than > understanding why it is different from the routine logic puzzles. It depends on the situation as to what method is most useful. I would attempt a trial&error approach. There is exactly one truth teller. If it's Alan, then Bob is lying which means Chris is telling the truth. Contradiction. If it's Bob, the other three are lying and Alan did it. If it's Chris, Bob is lying so Alan told the truth. Contradiction. If it's Dave, then Alan told the truth. Contradiction. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Key Core Error Argument > The following proof steps through a rather basic argument which is key > in proving an over one hundred year old error resulting from > previously unexpected consequences resulting from the definition of > the ring of algebraic integers. > Note that ultimately the proof relies on 22 NOT having 7 as a factor, > and constant terms like 7 and 22, being constant, and not variables > dependent on x, which may seem like odd things to emphasize, but I've > faced posters who've gotten away with challenging those truths because > people seem unaware that's what they're doing. > 1. Let P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078, where x is > in the ring of algebraic integers, notice that P(x) has a constant > term that is 1078. > 2. It can be shown that > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > where you should note that using v = -1 + 49x, gives > P(x) = (v^3+1)(5^3) - 3v(5)(7^2) + 7^3 > where the *same* polynomial has been put in a form which allows a > factorization into non-polynomial factors so that I have > P(x) = (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 a_3(x) + 7) > where the a's are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > 3. Now let x=0, so > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's at x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3, and I've picked a_1(0) and > a_2(0) to equal 0, which leaves a_3(0) with a value of 3. > 4. Further let a_3(x) = b_3(x) + 3, to keep indices matched. Then I > have > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 5(3) + 7) > P(x) = (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22). > 5. Now P(x) has a factor of 49 as > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 > which means that > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) > has a factor of 49. > 6. However, the constant term of P(x)/49 is 22, which is verified by > again setting x=0, which gives P(0)/49 = 22. > But for two of the factors of P(x), the constant terms is 7, which is > NOT a factor of 22. Therefore, *none* of the constant terms of > P(x)/49 as they multiply to give 22 can have 7 as a factor. > (By saying that 7 is NOT a factor of 22, I'm making a choice as to > where the proof is going. Since I've been talking about algebraic > integers, where 7 is NOT a factor of 22, it's natural to go with a > choice where 7 is NOT a factor of 22.) > Given that the constant terms are independent of x's value, it must be > the case that dividing P(x) by 49 divides the two constant terms equal > to 7, by 7. You are still making the mistake of thinking the constant terms of a factor determine the divisibility of the factor. This has been repeatedly demonstrated to be incorrect. Your conclusion in step 6 is wrong. > 7. But to divide 7 from those constant terms requires dividing > through two of the factors, so > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > > 300125 x^3 - 18375 x^2 - 360 x + 22 > from reverse use of the distributive property, which gives constant > terms that don't have 7 as a factor, as required. > Notice that it's a rather short and direct argument, where if you > accept that 22 does not have 7 as a factor, it's obvious enough what > the constant terms of the factors must be as you go from 7, 7 and 22, > necessarily to 1, 1, and 22, when you divide P(x) by 49. > James Harris > http://mathforprofit.blogspot.com/ -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: I can't stand it anymore amanda replied: >>amanda replied: >>amanda replied: >I have been biting my tongue about the IQ test but I can't any more. How reliable is a test that use the term Asian to represent the most >diverse of ethnic and cultural groups? >>Are you Asian? >Yes. Not oriental though but grew up in that region. >>So you recognize and understand the word. It has utility and meaning. >>Why do you get so upset about it? >>Many hate the word American and raise all manner is idiotic arguments >>as to why it is wrong (and most of them defend the term native American). >>Do you have similar issues with the word American, or issues like the >>ones you have with the word Asian? > What gave you the idea that I was upset about the term Asian? You're entire original post, other than the first sentence. > My only point was that IQ tests to determine people's intelligence are bs. Their inventor, years ago, invented them not to determine intelligence, but as a test for slow learners. He had cautions against using the test as it's used today. Nevertheless, the notion that they are totally worthless is also worth examining. The poster 'Uncle Al' (I think it was him) said that they measure your ability to take IQ tests but that's naught but a tautology. IQ tests don't tale place in a vacuum and students should have learned something from all those years in school. Do you think that all tests are worthless? some are worthless? Somewhere in-between? Rich === Subject: Re: naive geometry questions > Here are two geometry questions from a curious layperson. > I think these are wonderful questions; such a curious layperson is > welcome into my classes any time! Let met take them in opposite order. > Cool, huh? Very :-) I wish having had you as a diffgeo professor, in stead of that brilliant but boring man we had. Dirk Vdm === Subject: Re: A NOTy problem. Adjunct Assistant Professor at the University of Montana. >G'day G'day Folks, > I have been puzzling over what ought to be a simple logic problem >but am at a loss to come up with a method. These sort of things can be solved by brute force, by assigning all possible values of truth or false to each and seeing which yield contradictory information. On the other hand, sometimes there are shortcuts based on noticing certain things. >A crime is committed by one of the following; Alan, Bob, Chris or >Dave. >Each makes a statement to the police but three of them lie. >Alan says, I didn't do it. >Bob says, Alan is lying. >Chris says,Bob is lying. >Dave says, Bob did it. There's two easy observations here. Note that Dave and Allan's statements are compatible. If Dave is telling the truth, then so is Alan. Since only one person is telling the truth, Dave must be lying. Note also that it is impossible for BOTH Bob and Chris to be lying. That means that the person telling the truth is either Bob or Chris; which means that Alan must be one of the liars. Which solves the problem: Alan committed the robbery, and is lying; Bob is telling the truth, and both Chris and Dave are lying. === Subject: Re: A 3rd Grade Word Problem---HELP > My 3rd grade son brought this word problem home the other day and he was > given the answer (127). His job, for extra credit, was to figure out how to > get 127. I'm no genius but not a dope either. I couldn't figure out how to > get 127. Nobody in the whole neighborhood could figure out how to get 127. > Is this something of a trick question or is there something in the wording > that I am missing? Any Help???? > The Problem: > You know a very good story. On Sunday you tell the story to a friend. On > Monday you tell it to two new people. (So far, a total of three people have > heard the story). Each day after Monday, you double the number of new > people you tell the story to. What will be the total number of people that > will have heard your story after you tell it on Thursday? As others have found, the answer you were given is wrong. But it is an interesting problem in that the answer is also the largest positive (unsigned, natural) value an N-bit binary number can take on, where N is the number of days the story is repeated. The general answer is 2^N - 1. -- % Randy Yates % ...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall. %%%% % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr === Subject: Re: A 3rd Grade Word Problem---HELP >I don't see 127 either. It should be 31. If you keep telling the story up >till and including Saturday, then you would have 127. >>30 if you don't count yourself. >>/David > Surely 31 is the number not including yourself ? Yes, and don't call me surely... :) -- % Randy Yates % ...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall. %%%% % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr === Subject: Re: I can't stand it anymore > What gave you the idea that I was upset about the term Asian? > My only point was that IQ tests to determine people's intelligence are bs. Ah, of course. It would be far better indeed to administer intelligence tests rather than IQ tests to determine people's intelligence, right? Really, you sound like someone who is disappointed by their own results on such a test. === Subject: VECTORS AND SCIENCE? One of my students has given me a chemistry math problem which he found and I can't see how to solve it. It reads like this: Consider a molecule such as cyclohexane in the trans configuration in which as the bonds are the same length |> c | | b d | | <---------------< | | | | >----------------> e | | a | | f < | (Sorry for the bad diagram but the arrows chase each other round a->b->c->d->e->f) and the angle theta between two consecutive bonds is constant throughout the molecule. (Notice also that a and d are parallel). The angle of skew alpha between two bonds are separate by a single intermediate one is defined as the angle which these two bonds appear to form when viewed along the intermedite one. Show that this is given by cos alpha = - cos theta / 2 (cos (theta/2))^2 Does any one know how to solve this? I would really appreciate it! Sarah === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) >> Let's be serious for once. >> >> Consider an object being accelerated by a idealistic jet of water or a >> continuous 'stream of elastic ping pong balls'. What is its subsequent velocity >> pattern? Explain to me how this model applies to an attractive >electric field. What are the ping-pong balls and where are they coming >from? >>Don't worry about it Randy. I doubt if you know what a differential equation >>is. >Well, you could write one and we could see. But you don't write >equations, do you? >Meanwhile, why don't you explain to me how this model applies to an >attractive electric field. What are the ping-pong balls and where are >they coming from? > This is called theoretical research Randy. > The aim is to try to match a hypothetical physical process with the prediction > of an equally hypothetical maths theory, namely relativity. It's just a theory? Are we a creationist now? Of course the theory is just a theory. The point is, it makes predictions about such things like the maximum speed You have an alternate theory. are accelerated by receiving momentum from little objects moving at c. Fine. If this is your explanation for the limiting speed, please tell me what corresponds to the little ping-pong balls in the case of an attractive force, and where they are coming from. You have now alluded to theoretical research without a theory, differential equations without an equation, and in general spun a lot of words while refusing to answer the question. Could it be that you know that your model is not a model >Or do you really think nobody noticed that you avoided answering any >questions about your model? > Paul Anderson understood perfectly. Is he sarter than you?. If Paul Anderson has given you an answer about your model which you can crib and offer as your answers, by all means feel free to do so. Do you really think nobody has noticed that you've avoided answering any questions about your model? Explain to me how this model applies to an attractive electric field. What are the ping-pong balls and where are they coming from? - Randy === Subject: Some topological questions I think I have solved a problem in my recent dimnsion theory homework. It relies on these claims: A set that separates a space also separates a subspace of that space. The open halfspace R^n+, where the n:th coordinate is positive (and not merely non-negative), is homeomorphic with R^n. If both of thse are true then my answer should be sufficient. Are they true? -- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland -------- -- http://www.helsinki.fi/~palaste --------------------- rules! --------/ It's time, it's time, it's time to dump the slime! - Dr. Dante === Subject: Re: Key Core Error Argument [snip umpteenth repeat of previous flawed argument] I thought you were going to just post your proof elsewhere and stop > cluttering up this newsgroup. You've given a URL, now go away. > Such a remarkably silly posting. Yes! After all our experience of JSH's behavior, expecting him to do anything rqtions is quite silly. > What's the definition of insanity again? It is synonymous with JSH. You cannot reason with JSH, as he is immune to it. > How many YEARS have I posted, and how many posters through the years > have ordered me to go away? > Yet here you are C. Bond doing the same silly thing, as if it > matters. Certainly what JSH posts here doesn't matter, except as an irritant. This continued irritation by JSH has created some pearls, but only in refutation of his inanity. > But you see there's talk and there's substance. My proof is > substance, and in fact, there is no error in it. The only substance in JSH's proofs is one not referred to in polite society. [snipped the remaining oongah] === Subject: Re: Some topological questions Adjunct Assistant Professor at the University of Montana. >I think I have solved a problem in my recent dimnsion theory homework. >It relies on these claims: >A set that separates a space also separates a subspace of that space. So: the set {(x,0) : x in R} separates R^2. Therefore, it also separates the subspace {(a,b): a,b>0} of R^2... Surely you have more hypothesis? You need something like this, I think: Suppose that X is a space, and A is a subset that separates X, so X-A = Y_1 disjoint union Y_2. Let B be a subspace of X, and assume that B intersect Y_1 and B intersect Y_2 are both nonempty. Then Acap B separates B. Is that what you were thinking? >The open halfspace R^n+, where the n:th coordinate is positive (and not >merely non-negative), is homeomorphic with R^n. Let f:(0,infty) -> R be your favorite homeomorphism. Then map R^n+ to R^n by the identity on each coordinate but the n-th, and by f on the nth coordinate. Is that a homeomorphism? === Subject: Re: cardinality of monotone set-functions >Federico grava .88 la saucisse et au marteau: >> I suspect the solution to the following is known: >> Let X be a finite set. A set-function on X is a >> function f: 2^X rightarrow 2^X. f is monotone >> if A subset B implies f(A) subset f(B). >> What is the cardinality of the set of monotone >> set-functions on X ? If f is monotone, then for each x in X the set G(x) = {A in 2^X: x in f(A)} is monotone, i.e. if A is in G(x) and A subset B then B is in G(x). Conversely, if for each x in X the set G(x) is monotone, then f is monotone, i.e. if A subset B then B is in G(x) for all x in f(A), and so f(A) subset f(B). And the family {G(x): x in X} determines f, since f(A) = {x: A is in G(x)}. So the cardinality of the set of monotone set-functions is |X|^N(|X|) where N(|X|) is the cardinality of the set of monotone subsets G of 2^X, or equivalently monotone Boolean functions. That is sequence A000372 from the On-Line Encyclopedia of Integer Sequences, (look up sequence number A000372). There is no known formula for it. See the many references there. >> (Actually, I'm looking for the cardinality of >> the set functions such that f(A)subset A for all A.) That's a different question. Are you also requiring f to be monotone? >I'd say that you've got C(a,x) sets or card x in A. For each of those >sets, your function can take 2^x values. Therefore, you must have >Sum(x=0,a) 2^x C(a,x) = 3^a. >Am I right ? Yes, for the second question (if f is not required to be monotone). 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: cardinality of monotone set-functions Robert Israel grava .88 la saucisse et au marteau: >I'd say that you've got C(a,x) sets or card x in A. For each of those >sets, your function can take 2^x values. Therefore, you must have >Sum(x=0,a) 2^x C(a,x) = 3^a. >Am I right ? > Yes, for the second question (if f is not required to be monotone). Ok. Actually, I didn't even try to know if it was the same question, I answered to the last I saw. -- Nicolas === Subject: Re: Some topological questions Arturo Magidin scribbled the following: >>I think I have solved a problem in my recent dimnsion theory homework. >>It relies on these claims: >>A set that separates a space also separates a subspace of that space. > So: the set {(x,0) : x in R} separates R^2. Therefore, it also > separates the subspace {(a,b): a,b>0} of R^2... > Surely you have more hypothesis? You need something like this, I > think: > Suppose that X is a space, and A is a subset that separates X, so > X-A = Y_1 disjoint union Y_2. Let B be a subspace of X, and assume > that B intersect Y_1 and B intersect Y_2 are both nonempty. Then > Acap B separates B. > Is that what you were thinking? Yes, more or less. I understand that a set that separates a space does not separate a subspace if the set and the subspace are disjoint. However such is not the case in my homework. In my homework the set and the subspace intersect each other. So therefore the subspace is separated by a non-empty set. >>The open halfspace R^n+, where the n:th coordinate is positive (and not >>merely non-negative), is homeomorphic with R^n. > Let f:(0,infty) -> R be your favorite homeomorphism. > Then map R^n+ to R^n by the identity on each coordinate but the n-th, > and by f on the nth coordinate. Is that a homeomorphism? It should be. I used the exact same reasoning in my homework. As my favourite homeomorphism I used f(x) = ln x. -- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland -------- -- http://www.helsinki.fi/~palaste --------------------- rules! --------/ Ice cream sales somehow cause drownings: both happen in summer. - Antti Voipio & Arto Wikla === Subject: Re: Some topological questions >Arturo Magidin scribbled the following: >I think I have solved a problem in my recent dimnsion theory homework. >It relies on these claims: >A set that separates a space also separates a subspace of that space. >> So: the set {(x,0) : x in R} separates R^2. Therefore, it also >> separates the subspace {(a,b): a,b>0} of R^2... >> Surely you have more hypothesis? You need something like this, I >> think: >> Suppose that X is a space, and A is a subset that separates X, so >> X-A = Y_1 disjoint union Y_2. Let B be a subspace of X, and assume >> that B intersect Y_1 and B intersect Y_2 are both nonempty. Then >> Acap B separates B. >> Is that what you were thinking? >Yes, more or less. I understand that a set that separates a space >does not separate a subspace if the set and the subspace are >disjoint. However such is not the case in my homework. In my >homework the set and the subspace intersect each other. So therefore >the subspace is separated by a non-empty set. Again, not enough: take two circles that are tangent at a point; this space is separated by the point of tangency, but the point of tangency does not separate each of the circles separately, even though the intersection is nonempty (in fact, even though they each CONTAIN the entire separating subset). You really need the subspace not just to intersect the separating set, but to intersect both parts that the big space gets separated into. But if you have that, then your conclusion holds. >The open halfspace R^n+, where the n:th coordinate is positive (and not >merely non-negative), is homeomorphic with R^n. >> Let f:(0,infty) -> R be your favorite homeomorphism. >> Then map R^n+ to R^n by the identity on each coordinate but the n-th, >> and by f on the nth coordinate. Is that a homeomorphism? >It should be. Well, prove it! Show that the inverse image of an open ball is a product of open intervals, and that will do it. === Subject: Re: Problems on elementary functions > Is the function > sin x, if x>=0 > f(x)= > arc sin x, if 0>x>=-1 > an elementary function? > Is the function > sin x / x, if x>0 or x<0 > g(x)= > 1, if x=0 > an elementary function? > How to prove your conclusions? Perhaps you should first give us the precise definition of elementary function which you wish to use. David === Subject: Re: Maximizing and minimizing in optimization >Is it possible to convert a problem where we are trying to maximize a >quantity (profit) into a problem where we are trying to minimize a >(different) quantity (loss)? Maximizing f is the same as minimizing -f. Or perhaps what you have in mind is the equivalence in Linear Programming between solving the primal problem (a max problem) and the dual problem (a min problem). > If yes, can we do the transform in linear >time? In particular, is it possible to transform a Network flow >problem where we try to maximize the flow between the source and sink >into a shortest path problem where we are trying to find the shortest >between two points (say source and sink)? Maximum flow is equivalent to minimal cut. If it's on a planar graph, and the source and sink are on the same face, then this is equivalent to a shortest-path problem in the dual graph. 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: Stone's Representation Theorem(s?): help! I'm not english, so I dearly hope none of my terms turns out to be misleading or unaccurate. I'm a math student, I'm going to have an exam that has a part on Stone's Representation Theorem, and I miss, on my notes, a part of it, I'd wish to know if anyone can either help me to complete it or address me to a place where I can find it. Let me clarify the situation: In my course, Stone's Representation theorem has been divided in two parts, called Stone's First Theorem and Stone's second Theorem. The first one is the one i've seen talked about somewhere on this newsgroup, which says that if we take a Boole algebra B, then build the set of the ultrafilters on this Boole algebra, U(B), and then consider the set of parts of U(B), P(U(B)), there is a function psi: B -> P(U(B)) which allows us to both structure U(B) as a topological space and then consider the Boole algebra of its clopen, B(U(B)), subset of P(U(B)), and, considered only between the two structures B and B(U(B)), that is, narrowing the codomain, psi turns out to be an isomorphism between Boole algebras. This First Theorem I know, I have completely, and it's quite simple to demonstrate. The Second Theorem is, it seems, far more difficult, and I definitely miss a part of it. Beyond this, I've never heard about it on sci.math, and I wonder if it exists under some other name, since when the Theorem of Stone is named everyone immediately talks only about the aforementioned isomorphism. The First Theorem ended with finding an isomorphism between two structures (two Boole algebras); the Second theorem aims to find a _homeomorphism_ between two _topological spaces_, passing through a Boole algebra, exactly as the first theorem arrived from a Boole algebra to another one by passing through a topological space. We start from X, topological space, and build the Boole algebra of its clopens, B(X). Then we build the set of the ultrafilters of this Boole algebra, U(B(X)). Then we define a function h: X -> U(B(X)) defined like this: for every x belonging to X, h(x) is {A in B(X) | x is in A}. It's shown quite easily that this subset of B(X) is indeed an ultrafilter, and therefore is indeed an element of U(B(X)). We structure U(B(X)) as a topological space using the topology induced from the old psi-function, which makes U(B(X)) compact and totally disconnected (it was proved in the First Theorem). We then suppose that X is totally disconnected (this hypothesis is always taken as granted , from now on), and from that we deduce that h is injective. Then, some properties get demonstrated, which use in parallel h and psi, and precisely: 1) For every A in B(X), h[X] / psi(A) = h[A] 2) h is a continuous function 3) For every A in B(X), the closure in U(B(X)) of h[A] is psi (A). 4) in particular, the closure of h[X] in U(B(X)) is U(B(X)) itself, and therefore h[X] is dense in U(B(X)). So we have demonstrated that h is continuous, injective, and also that its image is dense in U(B(X)). Here ends the part of the theorem that I have. I have the feeling that we're very near to demonstrate that h is also surjective (maybe, it could be enough to demonstrate that h[X] is closed in U(B(X)), therefore its closure, U(B(X)), is h[X] itself, but I don't know if it's possible to prove it), and that it has an inverse function that is continuous (maybe it could be shown that h is open, or that it's closed, I don't know...), to prove that h is a homeomorphism and finish the theorem. Can anybody either help me to fill the blanks or give me an advice of the right text to look at in my university's library (we have tons of books and publications in english, here) to complete the proof? A big big thank you to anyone who'll want to help me. Federico === Subject: Re: Math dependency logic REVISED I kinda like this when you get all het up - you make a bigger > fool of yourself than usual. Here for example. All I said was that the disputes were not over > the fact that if you start with something true and apply correct > reasoning then the conclusion must be true, the disputes > were over the correctness of his reasoning. You say that > of course I'm lying? Then you must think that the people > who have been saying James is wrong have been disputing > the statement if you start with something true and apply > correct reasoning then the conclusion must be true. You should point out where anyone has disputed that fact. > . >> You yourself disputed that fact. Starting from C1-C4, >> C1 AxAy[x=y -> Az(z in x <-> z in y)] >> C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] >> C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in >> A)Classification >> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak >> Extensionality) >> by applying correct reasoning, one deduces Ex~(x=x). >> Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point out >> the error. >So C1-C4 |- Ex~(x = x)? I assume that's right -- I won't check. >Ullrich said it didn't? Maybe, though if so, it's likely just a minor >error (perhaps caused by context?). Caused by context indeed. He quotes that thing I said really a lot, but he never quotes the post where I actually _said_ it, it's always a followup to a reply to a response. I'd like to see the original, where I make that statement with no > at the start of the lines... >Where does this entail a denial of if you start with something true >and apply correct reasoning then the conclusion must be true? >Are you just lying? Or, shall we be charitable and assume that you're >stupid? It _is_ quite remarkable how slow he's being about this point, how it is that calling a proof incorrect, (correctly or incorrectly) does not involve denying that if you start with something true and apply correct reasoning the conclusion must be true. ************************ David C. Ullrich === Subject: Re: Math dependency logic REVISED Adjunct Assistant Professor at the University of Montana. [.snip.] >>So C1-C4 |- Ex~(x = x)? I assume that's right -- I won't check. >>Ullrich said it didn't? Maybe, though if so, it's likely just a minor >>error (perhaps caused by context?). >Caused by context indeed. He quotes that thing I said really >a lot, but he never quotes the post where I actually _said_ it, >it's always a followup to a reply to a response. I'd like to see >the original, where I make that statement with no > at the >start of the lines... It is clear the context has been compromised. Here is that part of the exchange: -- Begin Insert -- >>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2 >> And this follows from >> AxAy[Az(z in x <-> z in y) -> x=y]. >Dull David hasn't noticed that C3-C4 prove >~AxAy[Az(z in x <-> z in y) -> x=y]. You said set theory, which I took to mean ZF. In ZF they do not prove this. And in NBG they do not prove that Ex~(x=x). You say that I'm too stupid to realize that they do prove this. Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point out the error. -- End Insert -- Exhibit a proof of Ex~(x=x) from C1-C4 and someone will point out the error is clearly meant to be Exhibit a proof of Ex~(x=x) from C1-C4 [in ZF or in NBG] and someone will point out the error. Prior comments were: -- Begin Insert -- >Leave it to a brain-dead analysis teacher to >say something like this. Not only >haven't you the faintest idea why Ex~(x=x) >follows from C1-C4, but you are far too >lazy to attempt to figure this out. >You may be good at something other than >stiffing JSH, but nothing you have ever >written regarding either logic or set >theory gives any indication of this. When you said set theory I assumed you meant ZF. That's what set theory with no qualification means these days. If you meant NGB set theory then no, C1-C4 are not inconsistent with set theory. It does _not_ follow that C1-C4 give an example of something which is not equal to itself, or an example of something which does not exist. It is correct that I have no idea why Ex~(x=x) follows from C1-C4. This is because (assuming that NBG is consistent) NBG has a model in FOL=. In that model everything is equal to itself. -- End Insert -- once again clearly circumscribing the discussion to specific contexts which John Correy is now misrepresenting. === Subject: Re: Mathematicians' quotes about !math > When can a theorem be wrong? When it's not the theorem they asked for. -- === Subject: Re: Who contributed most to mathematics? > I am a patriotic American who does not cheat on his taxes,does not > drive drunk, does not carry a gun or knife, obeys traffic laws, > obeys the rights of others, is courteous to other people, How about having the courtesy not to post this dreck to sci.math? -- === Subject: Re: How do you integrate the function f(x) = x/(tanx) [0,pi/2]? > It seems we have an interesting result here: > arctan(x) has an elementary antiderivative, but (arctan(x))^2 doesn't! > What do you all think? The same is true for x e^(x^2).