mm-2269 Subject: Re: Problem with Algebraic Integers: Detailed Exposition >James, >You sometimes get into an argument as to whether something is a >function of something else. It might help if you made it clear at the >beginning that P is a function of f, m, u and x. And that the b_i and >w_i are functions of f and m. You might then be able to show, >subsequently, that something that looked at though it depended on >something else, in fact does not. >For instance: >Let f, m, u and x be algebraic integers. >Consider the function: >P(f,m,u,x)=f^2((m^3 f^4 - 3m^2 f^2 + 3m)x^3 - 3(-1+mf^2)x u^2 + u^3 f) >Since the set of algebraic integers is an integral domain, P(f,m,u,x) >is necessarily an algebraic integer. >Aside: Is P(f,m,u,x)/f^2 necessarily an algebraic integer? >It is possible to find functions (of f and m) b_1, b_2, b_3, w_1, w_2, >and w_3 such that >P(f,m,u,x)/f^2 = >(b_1(f,m) x + u w_1(f,m))* >(b_2(f,m) x + u w_2(f,m))* >(b_3(f,m) x + u w_3(f,m)) >Clearly these functions are not uniquely determined. >Aside: Can the functions b_1, b_2, b_3, w_1, w_2, and w_3 all be >chosen such that, if f and m are algebraic integers, b_1(f,m) b_2(f,m) >b_3(f,m) w_1(f,m) w_2(f,m) and w_3(f,m) are all algebraic integers? >If m=0, we have >P(f,0,u,x)/f^2 = >(b_1(f,0) x + u w_1(f,0))* >(b_2(f,0) x + u w_2(f,0))* >(b_3(f,0) x + u w_3(f,0)) >etc >Math Fan >>This does two things which James does not seem to like. >>1) It makes the formulas messier looking. >>2) It forces him to clearly define what he is talking about. >>I made similar suggestions to him a month or two ago and was informed >>that he had no interest in issues of style. After several posts back >>and forth it became clear that he doesn't understand the difference >>between style and clarity, nor does he seem to appreciate that clarity >>often comes at the expense of having simple looking expressions. > Maybe if people responding to him just adopted the P(f,m,u,x) b_i(f,m) > w_i(f,m) terminology that would help. It would make the whole > discussion a little more understandable, to me anyway. It has been done off and on for a while, probably longer than I've been following these threads. Most of the time I work to clearly indicate how variables are parameters of the function. James has yet to include parameters other than the ones he wants us to focus on. >>The fact that this may be part of why his papers don't get accepted >>doesn't seem to be something he understands. > Yes, to get published he would probably have to adopt the P(f,m,u,x) > format eventually. > Math Fan Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Problem with Algebraic Integers: Detailed Exposition >> >>This is paraphrased from the post >>Take the polynomial 65x^3 - 12x + 1, and factor it as >>65x^3 - 12x + 1 = (a1*x + 1)(a2*x+1)(a3*x+1) > Ok. >>with a1, a2, a3 (necessarily) algebraic integers (in fact, minus the >>roots of x^3 - 12x^2 + 65). Let z be any root of that polynomial. That >>is, z will be either -a1, -a2, or -a3. It is trivial that any common >>factor between z and 5 will be a common factor of the corresponding ai. > Hmmm...so you call the following trivial. Interesting. In any event, > I'll destroy your claim, and let's see if you admit the truth. No, he called that statement trivial. z=-a1, or z=-a2, or z=-a3. In each case, a common factor of z and 5 is a common factor of ai and 5. >>Take the following three polynomials >> q(x) = 8 x^2 - 76 x - 185 >> r(x) = 8 x^2 - 4 x - 45 >> s(x) = 4 x^2 - 37 x - 104 >>Then any product of q(z), r(z), s(z) and integers will be an algebraic >>integer, necessarily. >>We have that >> q(z)*r(z) = 64 z^4 - 640 z^3 - 1536 z^2 + 4160 z + 8325 >>Note that >> (64 x + 128)*(x^3 - 12 x^2 + 65) >> = 64 x^4 - 640 x^3 - 1536 x^2 + 4160 x + 8320 >>so (65z + 128)(z^3 - 12z^2 + 65) = 64z^4 - 650z^3 - 1536z^2 + >>4160z+8320. >>But z^3-12z^2+65 = 0. So 64z^4 - 650z^3 - 1536z^2 +4160z+8320 = 0. >>Therefore, >>q(z)*r(z) = 64 z^4 - 640 z^3 - 1536 z^2 + 4160 z + 8325 >> = (64 z^4 - 640 z^3 - 1536 z^2 + 4160 z + 8320) + 5 >> = 5. > I'm not interested in checking the math preferring to assume it's > correct. > At this point then the poster is saying that > q(z) r(z) = 5. So, for example, q(-a1) r(-a1) = 5. >>Likewise, take >> r(z)*s(z) = 32 z^4 - 312 z^3 - 864 z^2 + 2081 z + 4680 >>This time, note that >> (32 x + 72)*(x^3 - 12 x^2 + 65) >> = 32 x^4 - 312 x^3 - 864 x^2 + 2080 x + 4680 >>So (32z+72)*(z^3-12z^2+65) >> = 32z^4 - 312z^3 - 864z^2 + 2080z + 4680. >>But since z^3-12z^2+65 is equal to 0, it follows that >>32z^4 - 312z^3 - 864z^2 + 2080z + 4680=0. Therefore, >>r(z)*s(z) = 32z^4 - 312z^3 - 864z^2 + 2081z + 4680 >> = (32z^4 - 312z^3 - 864z^2 + 2080z + 4680) + z >> = z. > As before, I'm not interested in checking, preferring to assume the > operations were correct, and I note that at this point the poster is > saying that > r(z) s(z) = z. Which means r(-a1) s(-a1) = -a1. >>Therefore, r(z), which is an algebraic integer, is a common factor of >>5 and of z. It only remains to show that it is not a unit. > It's worth considering at this time what the intermediate results are, > assuming as before: > q(z) r(z) = 5 > r(z) s(z) = z. You missed it: q(-ai) r(-ai) = 5 r(-ai) s(-ai) = -ai. r(-ai) is a factor of BOTH 5 and ai and (with the observation below) not a unit. 5 and ai have a common factor that is an algebraic integer for all i in {1,2,3}. 5 is not coprime (by your definition) to a1 or a2 or a3 in the algebraic integers. > Switching the latter around I have > q(z) r(z) = 5 > z = r(z) s(z). > Multiplying them together gives > q(z) z = 5 s(z). > Ok, now let's go back to the posters argument. This completely hides what has been accomplished. >>The claim is that r(z) is a root of f(x) = x^3 - 969 x^2 + 315 x +5. >>If this is so, then r(z) is not a unit in the ring of all algebraic >>integers, since this is a monic irreducible polynomial with integer >>coefficients whose constant term is neither 1 nor -1. > That is true. However, Arturo Magidin is relying on the *definition* > of unit as being an factor of 1 or -1 in the ring of algebraic > integers. Are you agreeing that r(-a1), r(-a2), r(-a3) are NOT units? > As I've proven, algebraic integers are flawed in that you can have > abc = 5, > where neither 'a', 'b', nor 'c' has a non-unit factor in common with > 5, *in the ring of algebraic integers* as you're pushed out of the > ring, when you consider factors they share with 5. Two of them do > share non-unit factors with 5 in a higher ring, where one is a unit in > that ring, which doesn't have the problem that the ring of algebraic > integers does. What does this have to do with the above? What has been shown is that your pet example simply doesn't work. The property you claimed is not so. This suggests the flow is in your work. > The wacky thing about the ring of algebraic integer is that in that > case 'a', 'b' and 'c' are off in some kind of weird zone where they > can't be called factors of 5, in the ring of algebraic integers, and > neither can any of them be called units. > So in citing that result, Arturo Magidin is using the very error that > I've pointed out in a central point in his argument. > Sneaky, eh? Ummm... no. a1, a2, a3 are not coprime to 5. You said they are. They have factors in common with 5. You said they don't. Your discussion here is unrelated. >>As Dale noted, we have >>f(r(z)) = (r(z))^3 - 969 (r(z))^2 + 315 (r(z)) + 5 >> = (8 z^2 - 4 z - 45)^3 - 969 (8 z^2 - 4 z - 45)^2 >> + 315 (8 z^2 - 4 z - 45) + 5 >> = 512 z^6 - 768 z^5 - 70272 z^4 + 70592 z^3 >> + 731136 z^2 - 374400 z - 2067520. >>Letting w(z) = 512 z^3 + 5376 z^2 - 5760 z - 31808 >>we have that >> p(z)*w(z) = 512 z^6 - 768 z^5 - 70272 z^4 + 70592 z^3 >> + 731136 z^2 - 374400 z - 2067520 >>and therefore, f(r(z)) = p(z)*w(z). But we know that p(z)=0, so >>f(r(z))=0. This proves that r(z) is not a unit, and yet is a common >>factor of z and 5. > Well you've proven that r(z) is not a unit, but that doesn't prove the > positive, as the ring of algebraic integers is screwed up, as I've > shown. Look up. I highlighted where it was shown that it is a common factor. Do you need the values explicitly calculated for you? [claims deleted as they do not address the math] >>Letting z be -a1, -a2, and -a3, in turn, you obtain common factors of >>EACH of a1, a2, a3 with 5 in the ring of all algebraic integers, which >>are not units. > Sure, if you assume that the problem I've outlined doesn't exist. Do you need them explicitly calculated? Will honest to goodness numbers convince you? Will Twentyman email: wtwentyman at copper dot net === Subject: Proposed digit symbols for base sixteen A bit ago there was a discussion on this topic. I hope I'm not being a > For base 16, schemes are possible whereby the digit for n+8 is the same as > that for n but rotated 90 degrees. This seems like a clever idea. In fact, why not take it one step further and let n+4 be the same digit as n but rotated 90 degrees? Then we only have to make up 4 symbols. Of course, now we have the restriction that these 4 symbols must not have any 180 degree rotational symmetries. This rules out the symbols '0' and '1'. However, if we modify '0' into a tear-drop shape, then it becomes suitable as a digit, and perhaps a good choice, since it can be drawn in a single stroke and (as far as I know) isn't easily mistaken for any established mathematical symbol. So we'd use this tear-drop symbol for the 0, 4, 8, and 12. Now, we can also modify the digit '1' to be suitable, by adjoining a little mark on the top left side (like the way that '1' is commonly rendered in print, except without the bottom horizontal line). And, it seems reasonable to make use of the reflection of this '1' symbol, i.e., with the mark on the right side. The original we'd use to represent 3, 7, 11, and 15, while the reflection would represent 1, 5, 9, and 13. Then, all that's left is to invent a symbol for 2, 6, 10, and 14. A natural choice seems to the T symbol rotated 45 degrees. This does give a very beautiful system indeed. As the digits increase, the heads (i.e., the fat end of the teardrop, and the head of the 1's and T's) move clockwise around a circle. The odd numbers are rougher in appearance than the evens, and the multiples of four have the most round appearance (the teardrop shape). And, best of all, we can take the 16's complement of a digit (i.e., 16 minus the digit) simply by performing a vertical reflection. In case it's not clear, here's an ASCII diagram of the digits as I'm imagining them: 0 1 2 3 4 5 6 7 /- | / | / | / --- < | --- / | v | / / / / |/ 8 9 10 11 12 13 14 15 ^ | / / / / /| / | / --- | > --- / | -/ | / | This idea could probably be modified to work for other bases as well (also, if we liked, we could switch to counter-clockwise instead of clockwise, and let 0 be horizontally oriented, to match with the standard orientation of the complex plane) - Brent === Subject: Re: Proposed digit symbols for base sixteen > A bit ago there was a discussion on this topic. I hope I'm not being a > For base 16, schemes are possible whereby the digit for n+8 is the same as > that for n but rotated 90 degrees. > This seems like a clever idea. Really? And how shall I type them? G.C. === Subject: Re: Proposed digit symbols for base sixteen 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21: > For base 16, schemes are possible whereby the digit for n+8 is the same as > that for n but rotated 90 degrees. > This seems like a clever idea. In fact, why not take it one step further and > let n+4 be the same digit as n but rotated 90 degrees? I'd find that really confusing. If you're going to attempt to make up digit symbols for base 16 (seems kind of pointless when there's already a well established system: 0123456789abcdef), I'd think it would make more sense to use a binary notation for each symbol: choose four strokes, one to represent each bit, and form a digit by combining a subset of the strokes. Also use a fifth stroke to indicate that the symbol is a digit (and prevent zero from being blank). E.g. let the fifth stroke be a small centered circle, let the ones bit be indicated by a vertical stroke through the center of the circle, let the twos bit be indicated by a pair of vertical strokes tangent to the circle (extending past the height of the circle on both sides for easy visibility), let the fours bit be indicated by a horizontal stroke through the center of the circle, and let the eights bit be indicated by a pair of horizontal strokes tangent to the circle. David Eppstein http://www.ics.uci.edu/~eppstein/ Univ. of California, Irvine, School of Information & Computer Science === Subject: Re: Proposed digit symbols for base sixteen >A bit ago there was a discussion on this topic. I hope I'm not being a If such a transition were made, it would have to be for very powerful reasons, not simply a closer alignment with the language of our masters (the computers) If people would just learn to count in binary, one hand could account for numbers up to 32 and both hands would accomodate counting up to 1024! Even that convenience might not be enough to impel a change. Using American Sign Language to illustrate the concept, the sign for 1 is ASL B, the sign for 3 is ASL F, the BIRD is 27, and so on. Unfortunately binary finger code is richer than American Sign Language. The challenge is to find symbols which have mnemonic resonance with the finger positions and their numeric equivalents. John Bailey http://home.rochester.rr.com/jbxroads/mailto.html === Subject: Re: Proposed digit symbols for base sixteen > For base 16, schemes are possible whereby the digit for n+8 is the same as > that for n but rotated 90 degrees. > This seems like a clever idea. > Really? And how shall I type them? Hey, you might not be able to type them, but how is that relevant to whether it's a clever idea or not? I never claimed it was a useful or practical idea, only a clever one :-) I have no desire to change the established number system. I merely think that it's a bit fun to answer the question: If we, on a whim, felt like replacing the established decimal number system, what would we replace it with?. Granted, toying with such questions may not be as fun as doing real math, but can it really hurt? In the long term, if the number system does ever change, the issue of how to type the digits is not really a big deal; it's not really that hard to tweak fonts and keyboard mappings. - Brent === Subject: Re: Proposed digit symbols for base sixteen > This seems like a clever idea. In fact, why not take it one step further and > let n+4 be the same digit as n but rotated 90 degrees? > I'd find that really confusing. If you're going to attempt to make up > digit symbols for base 16 (seems kind of pointless when there's already > a well established system: 0123456789abcdef) Of course it's pointless. The main reason to change the system is simply because 0-9a-f system is ugly. And there's no point to changing something that's ugly, as long as it serves its purpose. But anyhow, there is actually a practical reason to change the system: as already mentioned, using a-f as digits interferes with their use as variables. So, the current hexadecimal system is a pain to use in programming languages and algebra because prefixes or suffixes are usually needed to identify numbers (i.e., 0x or h). Now, this is probably not enough justification to introduce a new number system, because doing so creates some big practical problems that are much larger than the one we eliminate. > I'd think it would make > more sense to use a binary notation for each symbol: choose four > strokes, one to represent each bit, and form a digit by combining a > subset of the strokes. Also use a fifth stroke to indicate that the > symbol is a digit (and prevent zero from being blank). > E.g. let the fifth stroke be a small centered circle, > let the ones bit be indicated by a vertical stroke through the center of > the circle, let the twos bit be indicated by a pair of vertical strokes > tangent to the circle (extending past the height of the circle on both > sides for easy visibility), let the fours bit be indicated by a > horizontal stroke through the center of the circle, and let the eights > bit be indicated by a pair of horizontal strokes tangent to the circle. That's a clever idea, but to me it seems very cumbersome; to write a 15 with this method requires seven strokes. Most digits in my system require only a single stroke, the worst case being two strokes (the 'T' digits: 2, 6, 10, 14), and, after a little thought, we can modify this worst case so that these also take only a single stroke: instead of using a T, use a line a little loop through the end, resembling something like this: / // / / This is also better because it has a more distinctive shape than T, compared to the '1'-like digits. You said you found my system really confusing, but how so? What exactly is it that you find confusing? - Brent === Subject: Re: Proposed digit symbols for base sixteen >> For base 16, schemes are possible whereby the digit for n+8 is the same as >> that for n but rotated 90 degrees. >> This seems like a clever idea. In fact, why not take it one step further and >> let n+4 be the same digit as n but rotated 90 degrees? >I'd find that really confusing. If you're going to attempt to make up >digit symbols for base 16 (seems kind of pointless when there's already >a well established system: 0123456789abcdef), I'd think it would make >more sense to use a binary notation for each symbol: choose four >strokes, one to represent each bit, and form a digit by combining a >subset of the strokes. Also use a fifth stroke to indicate that the >symbol is a digit (and prevent zero from being blank). There are several points to consider. For one, the use of abcdef is very poor; one has to add 9 to the usual number of the letter to get the hex version; there are better. For another, the use of letters for some is poor. Another point is that the characters should be easily made and easily distinguished when made by hand. Natural of coding, there should be at least two distinguishing marks between different characters, and this not obtained by transposition. Even rotation by 90 degrees can be poor here; sometimes the characters are somewhat rotated. It is not easy to come up with good character sets. 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: p vs. np problem Chip, I updated my paper with your suggestions. Any more feedback is welcome. Craig > Hi Chip, > clear in the paper that the 3rd category that I talk about is really > the category of all algorithms which do not fit into the 1st and 2nd > categories. It was unnecessary and confusing to say what I said > regarding comparisons of strings of bits. > I am prepared to rewrite the paper clearer according to the > suggestions of readers, including yourself. I want to get more > suggestions before I change it, so I don't have to keep adding to the > arxiv site, as that looks bad (they don't erase the old versions). > Again, thank you. > Craig > > It seems to me the difficulty of P vs. NP lies precisely in trying to > corral the notion of what all possible computations might be (for > solving, in your case, the SUBSET-SUM problem). As you know, > it is not enough to prove that various methods of solution require > a more than polynomial number of operations (in the size of the > input); one must somehow demonstrate this for _all_ possible > methods of solution. > > Your paper seems to be quite terse in its treatment of why methods > of solution must fall into one of the three categories described, > and these descriptions themselves seem not to have the clarity of > definition that would be required to convince me of a proof. I'd be > happy to read a revised version with more attention to that aspect. > === Subject: Re: question about dual space >Is it possible to find form of space dual to C_0 >where C_0 contains all of continous f:[0,inf)->R such that >lim f(x) = 0 when x->inf and C_0 is endowned with sup norm ? > Yes. Note that [0,infinity) is homeomorphic to [0,1). The members > of C_0 correspond to continuous functions on [0,1] with f(1) = 0. > Use the Riesz-Markov theorem. I guess it gives the set of all measures on Borel sets of [0,inf) with finite values. >Is C_0 reflexive? > No. Note e.g. that its unit ball has no extreme points, and therefore > (by Krein-Milman) can't be weakly compact. So what is its second dual? Am I right thinking of the set of all Borel mesurable bounded functions on [0,inf} ?? olej === Subject: Re: question about dual space >>Is it possible to find form of space dual to C_0 >>where C_0 contains all of continous f:[0,inf)->R such that >>lim f(x) = 0 when x->inf and C_0 is endowned with sup norm ? >> Yes. Note that [0,infinity) is homeomorphic to [0,1). The members >> of C_0 correspond to continuous functions on [0,1] with f(1) = 0. >> Use the Riesz-Markov theorem. >I guess it gives the set of all measures on Borel sets of [0,inf) with >finite values. Or more precisely, it's M = {(regular) real Borel measures on [0, inf)}. >>Is C_0 reflexive? >> No. Note e.g. that its unit ball has no extreme points, and therefore >> (by Krein-Milman) can't be weakly compact. >So what is its second dual? Am I right thinking of the set of all Borel >mesurable bounded >functions on [0,inf} ?? No. Let X be the subspace of M consisting of measures with finite support (I mean finite, not bounded). Let mu be Lebesgue measure restricted to [0,1] (ie mu(E) = |E intersect [0,1]|.) Now mu is not in the norm closure of X, so the Hahn-Banach theorem shows that there exists L in M* such that L vanishes on X while L mu <> 0. Suppose that L were given by a bounded Borel function f. Then for every x, f(x) = L(delta_x), if delta_x is a point mass at x; now delta_x is in X, so f(x) = 0 for all x. Hence int f d(mu) = 0 <> L(mu). >olej ************************ David C. Ullrich === Subject: Question about Zeta Function Does anybody know of an online proof (or proof sketch) for the functional equation of the Zeta function: Zeta(1-s) = 2 (2pi)^{-s} cos(1/2 pi s) Gamma(s) Zeta(s) Was this equation known to Riemann? -- Daryl McCullough Ithaca, NY === Subject: Re: Question about Zeta Function > Does anybody know of an online proof (or proof sketch) for the > functional equation of the Zeta function: > Zeta(1-s) = 2 (2pi)^{-s} cos(1/2 pi s) Gamma(s) Zeta(s) > Was this equation known to Riemann? Certainly it was known to Riemann. In fact, it was known to Euler. links which will prove the functional equation. If you want a hint to help you find a proof for yourself, one way to do it is to start with the formula pi/sinh(pi x) = sum {n = -oo .. oo} (-1)^n/(x - i*n) (at least that's close to being right) === Subject: Re: Question about Zeta Function > Does anybody know of an online proof (or proof sketch) for the > functional equation of the Zeta function: > Zeta(1-s) = 2 (2pi)^{-s} cos(1/2 pi s) Gamma(s) Zeta(s) > Was this equation known to Riemann? It was known to Euler, 100 years before Riemann, though only for real s and without anything we'd accept as a proof today. It was known to & proved by Riemann. I don't know whether there's a proof online. Have you tried a Google search for, say, functional equation AND zeta? Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Question about Zeta Function >> Does anybody know of an online proof (or proof sketch) for the >> functional equation of the Zeta function: >> Zeta(1-s) = 2 (2pi)^{-s} cos(1/2 pi s) Gamma(s) Zeta(s) >> Was this equation known to Riemann? >It was known to Euler, 100 years before Riemann, though only for >real s and without anything we'd accept as a proof today. It was >known to & proved by Riemann. I definitely question whether Euler could have had what we would accept as a proof. However, if it is proved for real s, it automatically holds for all complex s, as the Zeta function is analytic except for one pole, and the other functions are analytic. 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: Questions on the properties of inter-mingled samples > All messages from thread > Message 1 in thread === > Subject: Questions on the properties of inter-mingled samples > > suppose that you have 2 different sets, S1 and S2. there are many, > many sample points in each one. S1 has a mean of x1 and a standard > deviation, sd, of sd1. moreover, S2 has a mean and sd of x2 and sd2. > moreover, S1 and S2 are slightly correllated with each other with R = > -0.4. > suppose that i created a new set called S* with 45% of its sample > points from S1 and 55% from S2, what would the new standard deviation > and mean be? is there an algorithm/calculation for this? To be concrete, in order to show how silly the problem is, I restate that: You have a vector of (X1, X2) for weight and height; which are correlated with some r. You want to know, if you create an X3 where 45% of the numbers are weights, and 55% are heights, then WHAT is the new mean and SD? Obviously, from the re-statement, the correlation is a red-herring -- unless there is some secret connection not explained. So, Yes. You can describe the two means and two SDs; and have different samples; and have an overall mean and overall SD. These 'details' are the statistics (say) of an ANOVA; here are the within-group means and variances. To get the totals: You can see that the mean is the simply- weighted composite of the two. For the SD: I would probably open a basic statistics book to get the ANOVA formulas, to make sure that I did not mess up the weighting for figuring the sum-of-squares of the group means around the overall mean. [snip, extended version of Q, with nothing new that I noted.] Rich Ulrich, wpilib@pitt.edu http://www.pitt.edu/~wpilib/index.html Taxes are the price we pay for civilization. Justice Holmes. === Subject: Re: Questions on the properties of inter-mingled samples Rich Ulrich, i had a feeling that the mean was going to be easy to calculate. however, the standard deviation of this new, 'composite' sample wasn't going to be easy given their inter-correlations. i have tried to find the answer myself in the libraries, bookstores, and internet, but to no avail. i'm sure that people have done research on variance pooling, because i see some tantalizing sites on the net about this topic. finally, i know that the answer to this problem is not going to be that apparant or commonsensical like you would think. again (for those who don't know my original question): given 3 sets: S1, S2, and S3 with means and standard deviations (SD) of m1, m2, m3, sd1, sd2, and sd3, what is the mean and sd of a new sample composed of a% S1, b% S2, and c% S3, given that the 3 samples are correlated with one another? > suppose that you have 2 different sets, S1 and S2. there are many, > many sample points in each one. S1 has a mean of x1 and a standard > deviation, sd, of sd1. moreover, S2 has a mean and sd of x2 and sd2. > moreover, S1 and S2 are slightly correllated with each other with R = > -0.4. > > suppose that i created a new set called S* with 45% of its sample > points from S1 and 55% from S2, what would the new standard deviation > and mean be? is there an algorithm/calculation for this? > To be concrete, in order to show how silly the problem is, > I restate that: > You have a vector of (X1, X2) for weight and height; which > are correlated with some r. > You want to know, if you create an X3 > where 45% of the numbers are weights, and 55% are heights, > then WHAT is the new mean and SD? > Obviously, from the re-statement, the correlation is > a red-herring -- unless there is some secret connection > not explained. > So, Yes. You can describe the two means and two SDs; > and have different samples; and have an overall mean > and overall SD. These 'details' are the statistics (say) > of an ANOVA; here are the within-group means and variances. > To get the totals: You can see that the mean is the simply- > weighted composite of the two. For the SD: I would > probably open a basic statistics book to get the ANOVA > formulas, to make sure that I did not mess up the > weighting for figuring the sum-of-squares of the group > means around the overall mean. > [snip, extended version of Q, with nothing new that I noted.] === Subject: Re: Questions on the properties of inter-mingled samples > Rich Ulrich, > i had a feeling that the mean was going to be easy to calculate. > however, the standard deviation of this new, 'composite' sample wasn't > going to be easy given their inter-correlations. - You have not shown any way in which the correlation can have *any* effect. I suspect that you do not understand what correlation denotes, because you persist in saying that you have sets instead of saying that you have pairs of numbers -- > i have tried to find the answer myself in the libraries, bookstores, > and internet, but to no avail. i'm sure that people have done > research on variance pooling, because i see some tantalizing sites > on the net about this topic. finally, i know that the answer to this > problem is not going to be that apparant or commonsensical like you > would think. > again (for those who don't know my original question): given 3 sets: > S1, S2, and S3 with means and standard deviations (SD) of m1, m2, m3, > sd1, sd2, and sd3, what is the mean and sd of a new sample composed of > a% S1, b% S2, and c% S3, given that the 3 samples are correlated with > one another? [ snip, rest] Trying my concrete example some more: There are N pairs of (X,Y) where X is weight, Y is height; they are correlated. Means and SDs are known. What is the mean and SD for Z, if a vector Z of length N is constructed that consists of random selections from (X,Y), so that some percentage of the time Z is set to X, and the rest, Y? - the mean and variance of these numbers, Z, are readily determined as I described, if there isn't anything more to the problem than this. (For instance, autocorrelation? Non-random selection? I'm imagining that the seemingly-silly model was inspired by some reality....) Now -- perhaps this is where Harry's intuition is being stimulated -- there could be narrower limits on the *variance* of the variance, or the variance of the *mean*. For instance, if X and Y were identical in the first place, -- r= 1.0, same means and SDs -- then Z would be identical to them, too, and the mean and SD of Z would have no sampling variability relative to the population of the fixed X,Y. If this is intended to represent something real and interesting, it would possibly be helpful to say WHAT. Rich Ulrich, wpilib@pitt.edu http://www.pitt.edu/~wpilib/index.html Taxes are the price we pay for civilization. Justice Holmes. === Subject: Re: Questions on the properties of inter-mingled samples boundary=------------040704090807000209020700 --------------------------------------------------------------------- why do you want moments of the composite distribution? Don't you really want to decompose the distribution into its constituent distributions? >>Rich Ulrich, >>i had a feeling that the mean was going to be easy to calculate. >>however, the standard deviation of this new, 'composite' sample wasn't >>going to be easy given their inter-correlations. >> > - You have not shown any way in which the correlation >can have *any* effect. I suspect that you do not understand >what correlation denotes, because you persist in saying >that you have sets instead of saying that you have >pairs of numbers -- >>i have tried to find the answer myself in the libraries, bookstores, >>and internet, but to no avail. i'm sure that people have done >>research on variance pooling, because i see some tantalizing sites >>on the net about this topic. finally, i know that the answer to this >>problem is not going to be that apparant or commonsensical like you >>would think. >>again (for those who don't know my original question): given 3 sets: >>S1, S2, and S3 with means and standard deviations (SD) of m1, m2, m3, >>sd1, sd2, and sd3, what is the mean and sd of a new sample composed of >>a% S1, b% S2, and c% S3, given that the 3 samples are correlated with >>one another? >> > [ snip, rest] >Trying my concrete example some more: >There are N pairs of (X,Y) where >X is weight, Y is height; they are correlated. >Means and SDs are known. >What is the mean and SD for Z, >if a vector Z of length N is constructed that consists >of random selections from (X,Y), so that some percentage >of the time Z is set to X, and the rest, Y? > - the mean and variance of these numbers, Z, >are readily determined as I described, if there isn't >anything more to the problem than this. (For instance, >autocorrelation? Non-random selection? I'm >imagining that the seemingly-silly model was >inspired by some reality....) >Now -- perhaps this is where >Harry's intuition is being stimulated -- >there could be narrower limits on the >*variance* of the variance, or the >variance of the *mean*. For instance, if >X and Y were identical in the first place, > -- r= 1.0, same means and SDs -- >then Z would be identical to them, too, >and the mean and SD of Z would have >no sampling variability relative to >the population of the fixed X,Y. >If this is intended to represent something real and >interesting, it would possibly be helpful to say WHAT. === Subject: Rational point puzzle Let C_r be the circle of radius r centred at (0,0) in the (x,y)-plane. C_r = {(x,y): x, y are real; x^2 + y^2 = r^2} Now, if C_r contains a rational point (i.e. a point (x,y) for which x and y are both rational) then clearly r^2 is rational. Therefore, if r^2 is irrational, the circle C_r must contain no rational points. A trickier question is the following: If r^2 is rational, must C_r contain a rational point? HJ === Subject: Re: Rational point puzzle > If r^2 is rational, then so is r That will certainly be news to Pythagoras. === Subject: Re: Rational point puzzle Harry Jameson escribi.97 en el > Let C_r be the circle of radius r centred at (0,0) in the (x,y)-plane. > C_r = {(x,y): x, y are real; x^2 + y^2 = r^2} > Now, if C_r contains a rational point (i.e. a point (x,y) for which x > and y are both rational) then clearly r^2 is rational. Therefore, if > r^2 is irrational, the circle C_r must contain no rational points. > A trickier question is the following: If r^2 is rational, must C_r > contain a rational point? > HJ No. If r^2 = N is an integer with a prime factor equal to 3 (mod 4) raised to an odd power, no. Then if x^2 + y^2 = N has a rational solution (a/b)^2 + (c/d)^2 = N Let e = lcm(b, d), and multiply by e^2 (ae/b)^2 + (ce/d)^2 = Ne^2 ae/b and ce/d are integers, but Ne^2 is a integer with a prime factor equal to 3 (mod 4) raised to an odd power and cannot be expressed as a sum of two integer squares. Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Rational point puzzle >> If r^2 is rational, then so is r > That will certainly be news to Pythagoras. Sorry, I cancelled that shortly after sending it. Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Registration fee ?? >> >I am anxious to university registration fee for a term at your nation. >[...] >>Germany, public universities, aprox. 40 EUR per term. >>There are no further tuiton fees. >>Some universities have contracts with regional transportation services, >>resulting in a higher fee (approx. 120 EUR) including free transport >>You can find some additional information at >> or via Google. > Which indicates (but doesn't actually *say*) that, if you are accepted, > the price is the same for foreign students. It's almost worth it to do > your undergrad work in Germany just to save the tuition. Well, you still need to support yourself somehow. The job opportunities for foreign students are not very good and there is no system of student loans comparable to the US. Moreover, at least in the traditional programs the prospective student will face some differences that might look quite odd at first sight: (a) the courses usually are in german (b) it is assumed, that the 'broad general education' part has already done in high school. So you start right away with a major and courses, which US-undergrads would enter in their 3rd year. (c) the 'Vordiplom', which is roughly at the level of the US B.Sc. is not visible outside the university. The first graduation is actually the 'Diplom', which is at the level of the US M.Sc.. Consequently, you may need 5-7 years before you can leave gracefully. Beside these traditional programs, there are also more and more attempts to implement the B.Sc/M.Sc combination. These are not too popular, because they are seen as a political scheme to implement tuiton fees later. But still such a B.Sc programs will be cheaper than their US counterparts. > Hey! Maybe that's why the International Baccalaureate program is > gaining in popularity. > It is interesting that the semester abroad in Europe (Germany > included) costs about the same as US tuition. Is there a double > standard for pricing? The courses seem comparable. Well, they're real > courses, but those are available in the US, too. I do not know. Perhaps US universities need to milk their students more because of less public funding?. Marc === Subject: Regression Problem (To give a simple and fictional example of an engineering problem) A race has seven checkpoints along it. As each runner passes a checkpoint, they get their card timestamped. If the checkpoint is busy, the runner just ignores it and keeps going. The finish line is an ordinary checkpoint. After the race, all the runners put their cards in a box. All of the cards transpire to have at least one timestamp (FWIW, checkpoint failure is roughly 10%). I need a formula which, given any card, will work out a 95% confidence interval for the finish time. Obviously, if the finish time is marked in on the card then the problem is trivial. Presumably, the more data is missing, the wider the confidence interval. I assume that missing timestamps towards the end of the race are more important than missing timestamps towards the start. Greg === Subject: Re: Regression Problem > (To give a simple and fictional example of an engineering problem) > A race has seven checkpoints along it. As each runner passes a > checkpoint, they get their card timestamped. If the checkpoint is > busy, the runner just ignores it and keeps going. The finish line is > an ordinary checkpoint. After the race, all the runners put their > cards in a box. All of the cards transpire to have at least one > timestamp (FWIW, checkpoint failure is roughly 10%). > I need a formula which, given any card, will work out a 95% confidence > interval for the finish time. > Obviously, if the finish time is marked in on the card then the > problem is trivial. Presumably, the more data is missing, the wider > the confidence interval. I assume that missing timestamps towards the > end of the race are more important than missing timestamps towards the > start. You'll have to build a model. Based on all the completely filled cards (a little less than half of all the cars, assuming that the missing values are incurred at random, independently from one station to the next) as well as your experience with this type of race, you should postulate the type of relationship between the times (for instance, speed is more or less constant along the entire course or, speed increases on the last 200 metres or whatever). Once you have that model you can try to estimate the speed for all entries based on the time stamps you got. If the speed is linear, then it seems that the quality of the imputation at the end is a function of the spread between the earliest and latest available time stamps (you need at least two for any kind of estimation) and the distance of the latest time stamp from the end. Intermediate time stamps will only confuse the issue. For nonlinear models you need more work. === Subject: Re: Regression Problem > (To give a simple and fictional example of an engineering problem) > A race has seven checkpoints along it. As each runner passes a > checkpoint, they get their card timestamped. If the checkpoint is > busy, the runner just ignores it and keeps going. The finish line is > an ordinary checkpoint. After the race, all the runners put their > cards in a box. All of the cards transpire to have at least one > timestamp (FWIW, checkpoint failure is roughly 10%). > I need a formula which, given any card, will work out a 95% confidence > interval for the finish time. > Obviously, if the finish time is marked in on the card then the > problem is trivial. Presumably, the more data is missing, the wider > the confidence interval. I assume that missing timestamps towards the > end of the race are more important than missing timestamps towards the > start. If I understand your question, you don't have a priori knowledge of the runner's velocity, but perhaps have certain assumptions about how constant it is. What you have is sparse measurements of known position and time. It seems to me this is a state estimation problem and that I've seen Kalman filters used for this. The error bar grows with time since last estimation. I don't remember the details of either Kalman filters or how they were applied to this particular problem, but the state vector which you are trying to maintain an estimate on is (s(t), v(t)), position and velocity in however many dimensions as a function of time. - Randy === Subject: Re: Regression Problem Hi Randy and Gus I had been intending to use Gus's approach, and hadn't entirely realised that I was assuming linearity. I forgot to mention that everybody starts at t=0, so there is always a known first point. On reflection, I think that velocity is probably approximately normally distributed. The runner analogy is pretty close to what is actually happening. If so, am I right in saying that assuming linearity will give me a reasonable expected value ? Also, I think that I should approximate the std dev of the expected value by applying the population std deviation to the last measured value and taking a linear projection. So if it takes the average runner 90% of total time to reach the penultimate checkpoint, and his/her std dev is 2%, then if a runner has taken 10 minutes to reach that mark and has no finish time, his/her 95% confidence interval for expected total time for the race is Min = 10 mins / 0.94 Max = 10 mins / 0.86 Hopefully that doesn't sound too crazy ? === Subject: Re: Relationship between Undecidability, Incompleteness, and NP Completeness?? > I am a student who is about to start studying Undecidability and > Incompleteness. However, before I began studying it, it struck my > curiosity as to how it related to something else I studied, NP > Completeness. I am very curious to know if these are related and if my > understanding of NP Completeness will help me in this new area. different meaning, NP completeness is about practical completeness, exponential algorithms can still be computed but fall outside the scope for determining NP completeness. don't fall for anyone who tells you things are true without proof, especially if they proved it. Herc === Subject: Re: Relationship between Undecidability, Incompleteness, and NP Completeness?? : There is an *analogy* between NP sets and recursively enumerable : (r.e.) sets. But the analogy is imperfect. In particular, : an easy diagonalization shows that r.e. is different from : co-r.e. But there is a direction to this difference, right? I mean, co-r.e. is harder, right? --- 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: Rigorous definitions of derivative without previous limits > I have not seen how Thomas does it, but I am know of two > good rigorous definitions of derivative which do not > presuppose limit. Bernstein also defines derivatives before limits in `Calculus for mathematicians' (http://cr.yp.to/mathmisc.html#calculus). This seemed completely reasonable to me when I read it, though I'm no expert. -- [mdw] === Subject: Re: School hours > Daylight savings time is for saving energy. Look at this site: > http://www.boulder.nist.gov/timefreq/general/daylightsaving.html Those of us who have to get up early for work end up in consuming the saved energy of the previous evening during the following morning, when we rise in cold darkness. Marc === Subject: Re: School hours (was: anxiety of high school??) > I ask, why not push school openings later into the day? >Schedule the high school 9-4 rather than 8-3, or 9:30-3:30 >rather than 8-2. I've read sleep researchers recommend this. At that age people tend to find it difficult to get up early, they say. For part of the time I was working on my PhD thesis I had to live with construction noise waking me up at 6:30 in the morning daily. I think it's inappropriate to assume that we're all getting up that early, and the noise control ordinance should be less generous to the sources of noise this way. Keith Ramsay === Subject: Re: School hours (was: anxiety of high school??) > I ask, why not push school openings later into the day? >Schedule the high school 9-4 rather than 8-3, or 9:30-3:30 >rather than 8-2. > I've read sleep researchers recommend this. At that age > people tend to find it difficult to get up early, they say. > For part of the time I was working on my PhD thesis I had > to live with construction noise waking me up at 6:30 in > the morning daily. I think it's inappropriate to assume > that we're all getting up that early, and the noise > control ordinance should be less generous to the > sources of noise this way. It's not appropriate. One 1/2 get up 7am. The other half that work get up at 7pm. Which is why we always told PhDs, that you should park your Universities up on a hill in a meadow somewhere. So that it doesn't interfere with progress. === Subject: Re: School hours (was: anxiety of high school??) >>To what country are you refering? USA? >>High school in the USA is generally between the hours of 8:00am - 2 or >>3:00pm. >>Lurch > This is not really a sci.math topic, but I'm unsure where to >post it. > During my college days, 8:00 am classes were unpopular. >Now that I'm in the workplace, I often sleep until 8 am, working >into the evening. I see more colleagues in the office at 8 pm >than when I arrive 8 am. [But the cafeteris is open at 8 am and >close at 8 pm.] I went the opposite way. The best working shift in the world is 4:00-12:00 or 3:00-12:00. The first half has none of the manangement natterings everybody has to endure. The second half (my least productive) can be used to attend to natterings. > When I attend local transportation meetings, I repeatedly hear that >21% of morning commute-hour traffic is school-related. That's because of the traffic backups caused by stopped buses. > I ask, why not push school openings later into the day? >Schedule the high school 9-4 rather than 8-3, or 9:30-3:30 >rather than 8-2. Are there historical reasons, such as >a need to tend the farm in the afternoon, for the early school starts? Oh, no! Don't blame for this one either. The farmer got blamed for the stupid daylight shifting. Every farmer I knew expressed an opinion about city slickers and their notion that cows can tell time. > The only good reason I've heard for moving the clocks backward >in the Fall (i.e., ending daylight savings time) is to have >more sunlight as youngsters go to school -- Which is moot anyway because buses have to pick them up very early to get them to school by nine (around here in US Northeast). >. . delaying school >openings would solve this problem a different way. When I was young our high school started at 8:15 and we got out at 15:35; this covered six periods plus lunch. /BAH Subtract a hundred and four for e-mail. === Subject: Re: School hours (was: anxiety of high school??) Hi Peter, This is Bob Silverman. One reason high schools get out early is because of other than academic activities. e.g. Many students have after school jobs. === Subject: Re: School hours (was: anxiety of high school??) Daylight savings time is for saving energy. Look at this site: http://www.boulder.nist.gov/timefreq/general/daylightsaving.html >To what country are you refering? USA? >High school in the USA is generally between the hours of 8:00am - 2 or >3:00pm. >Lurch > This is not really a sci.math topic, but I'm unsure where to > post it. > During my college days, 8:00 am classes were unpopular. > Now that I'm in the workplace, I often sleep until 8 am, working > into the evening. I see more colleagues in the office at 8 pm > than when I arrive 8 am. [But the cafeteris is open at 8 am and > close at 8 pm.] > When I attend local transportation meetings, I repeatedly hear that > 21% of morning commute-hour traffic is school-related. > I ask, why not push school openings later into the day? > Schedule the high school 9-4 rather than 8-3, or 9:30-3:30 > rather than 8-2. Are there historical reasons, such as > a need to tend the farm in the afternoon, for the early school starts? > The only good reason I've heard for moving the clocks backward > in the Fall (i.e., ending daylight savings time) is to have > more sunlight as youngsters go to school -- delaying school > openings would solve this problem a different way. > -- > Wanted: Experts at choosing the best of 100+ applicants for a position. > Register as a California voter by September 22, and vote on October 7. > Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California > Microsoft Research and CWI === Subject: Re: School hours (was: anxiety of high school??) >To what country are you refering? USA? >High school in the USA is generally between the hours of 8:00am - 2 or >3:00pm. >Lurch > This is not really a sci.math topic, but I'm unsure where to > post it. > During my college days, 8:00 am classes were unpopular. > Now that I'm in the workplace, I often sleep until 8 am, working > into the evening. I see more colleagues in the office at 8 pm > than when I arrive 8 am. [But the cafeteris is open at 8 am and > close at 8 pm.] > When I attend local transportation meetings, I repeatedly hear that > 21% of morning commute-hour traffic is school-related. > I ask, why not push school openings later into the day? > Schedule the high school 9-4 rather than 8-3, or 9:30-3:30 > rather than 8-2. Are there historical reasons, such as > a need to tend the farm in the afternoon, for the early school starts? The justification I heard in our school district had to do with the planning of bus routes: The early timing of the high schools allowed the system to reuse the same buses for middle-school and elementary-school after they'd finished their high-school routes. The school officials said it would be prohibitively expensive to start the high-school day later and give up this bus recycling. - Randy === Subject: Simple 1:1 mapping I am trying to find a function y = f(x, k) so that for a given k, each integer x in the range 0 to 255 yields a unique integer y in the range 0 to 255. I don't mind what the function is provided it is conceptually simple, this is just for an example. The mapping should be pseudo-random in nature, although it doesn't have to satisfy any particular measure of randomness. k is a key, an integer which lets you select different mappings. For instance y = (x*(2*k+1)) mod 256 would almost be suitable, with k taking a value of 0 to 127. I want something which looks a bit more random and maybe has a larger range of k values. Any ideas? === Subject: Re: Simple 1:1 mapping > I am trying to find a function > y = f(x, k) > so that for a given k, each integer x in the range 0 to 255 yields a > unique integer y in the range 0 to 255. > I don't mind what the function is provided it is conceptually simple, > this is just for an example. The mapping should be pseudo-random in > nature, although it doesn't have to satisfy any particular measure of > randomness. > k is a key, an integer which lets you select different mappings. > For instance > y = (x*(2*k+1)) mod 256 > would almost be suitable, with k taking a value of 0 to 127. I want > something which looks a bit more random and maybe has a larger range > of k values. > Any ideas? f(x , k)= x works fine. === Subject: Re: Simple 1:1 mapping === >Subject: Simple 1:1 mapping >I am trying to find a function > y = f(x, k) >so that for a given k, each integer x in the range 0 to 255 yields a >unique integer y in the range 0 to 255. >I don't mind what the function is provided it is conceptually simple Well here's a simple one. f(x,k) = x For any given k, each integer x from 0 to 255 yields a unique integer f(x,k) from 0 t0 255. adam === Subject: Re: Simple 1:1 mapping === >Subject: Simple 1:1 mapping >I am trying to find a function > y = f(x, k) >so that for a given k, each integer x in the range 0 to 255 yields a >unique integer y in the range 0 to 255. >I don't mind what the function is provided it is conceptually simple > Well here's a simple one. > f(x,k) = x > For any given k, each integer x from 0 to 255 > yields a unique integer f(x,k) from 0 t0 255. > adam Adam, Is timing important??--Check the next post Steven === Subject: Re: Simple 1:1 mapping === >Subject: Re: Simple 1:1 mapping >Adam, >Is timing important??--Check the next post >Steven Hell yeah, timing is important. I got my flippant reply out almost 15 minutes faster than you did. adam === Subject: Re: Simple 1:1 mapping >I am trying to find a function > y = f(x, k) >so that for a given k, each integer x in the range 0 to 255 yields a >unique integer y in the range 0 to 255. >I don't mind what the function is provided it is conceptually simple, >this is just for an example. The mapping should be pseudo-random in >nature, although it doesn't have to satisfy any particular measure of >randomness. f(x,k) = (a(k)*x + b(k)) mod 256 might be suitable, where b is any integer-valued function, and a is any odd-integer-valued function. Or if you want to spice it up a bit, f(x,k) = (a(k)*x^2 + b(k)*x + c(k)) mod 256 where a(k) is always even and b(k) is always odd. 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: Simple 1:1 mapping === >Subject: Re: Simple 1:1 mapping >Adam, >Is timing important??--Check the next post >Steven > Hell yeah, timing is important. I got my flippant > reply out almost 15 minutes faster than you did. > adam Adam, Nice job. Steven === Subject: Re: Simple 1:1 mapping > f(x,k) = (a(k)*x + b(k)) mod 256 might be suitable, where b is any > integer-valued function, and a is any odd-integer-valued function. > Or if you want to spice it up a bit, > f(x,k) = (a(k)*x^2 + b(k)*x + c(k)) mod 256 > where a(k) is always even and b(k) is always odd. It is important that the result of the function is unique for every unique input value (in the range 0 to 255). I need the function to fill a lookup table which must be reversible, and of course this cannot be the case if 2 different entries in the table contain the same number. I can see how the first function is reversible, but I can't viualize why they second function is (I am not doubting you, I just can't see it). Is there a simple and convincing explanation for this? Also, what range of values of a(k) will yield different results? (eg b(k)=1 and b(k)=257 yield indistinguishable results, what is the equivalent case for a(k)) I think it is what I am looking for, sufficiently random but very simple. === Subject: Re: sir~differ topology problem > let X = {a,b,c,d,e} > For topology T of X that generated by A={{a},{a,b,c},{c,d}} > find local basis Bc for point c in X As {a,b,c} / {c,d} = {c} a local base for c is { {c} } > i think that local basis seems to come out variously. The local base for a is { {a} } for b is { {a,b,c} } for d is { {c,d} } When the topology is finite, then the intersection of all open sets containing a point p, is the smallest open set containing p. Call that set U_p. Then a local base for p is { U_p } and it's the smallest local base for p and any other local base for p will have to have to contain U_p. The biggest tho least useful, possible choice for a local base of p is the collection of all open sets containing p. In the example above, another local base for c could be { {c}, {c,d} }. === Subject: Re: Solving Sums of Exponential and Linear Terms? >>Does anybody know how one would approach solving something like this >>for n? >>a^n + n = 1 >>Starling >>Who has only had up to Linear Algebra, so be kind. :/ > There is a unique real solution if a >= 1. >Indeed, it is a rational solution. Nothing could be a better clue than that. There are, however, other solutions if 0 < a < 1, and they must be obtained either numerically or with LambertW. 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: Solving Sums of Exponential and Linear Terms? >Does anybody know how one would approach solving something like this >for n? >a^n + n = 1 >Starling >Who has only had up to Linear Algebra, so be kind. :/ There is a unique real solution if a >= 1. Indeed, it is a rational solution. Wanted: Experts at choosing the best of 100+ applicants for a position. Register as a California voter by September 22, and vote on October 7. Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California Microsoft Research and CWI === Subject: Re: Standard Topology >... >>Oh, the triangle inequality. >>I suppose that makes perfect sense since a metric space is flat in a small >>enough region. >That's a really ridiculous statement, which I assume you have >made because you have conflated the notion of a Riemannian metric >on a manifold with the abstract notion of a metric space. >Though if (for instance) John Mitchell were to weigh in with >some argument that a metric space is flat in a small enough >region, I would listen respecfully and no doubt learn something, >in the absence of some such informed comment I will throw out >my opinion that it's very rare indeed for a metric space-- >even a metric space on R^2 which induces the standard topology, >such as you seem to be trying to study--to be flat in a small >enough region for any reasonable sense of flat. I don't >think that the various non-Euclidean Minkowski p-norms on >R^2 are flat, for example. >In particular, I don't see any reasonable sense in which >the triangle inequality is closely related to flatness. >If you care to suggest such a sense, go for it. >Lee Rudolph I've probably confused some material from elsewhere with Metric Spaces on page 10 of Bishop and Goldberg. But for flatness and the triangle inequality, I think there must be some ambiguity in defining a triangle on a curved space, and at any rate you can break the inequality over a large enough region of a curved space. On a sphere, take sections of great circles to be our straight lines and define a triangle: across north pole ^ | |x / y / |z | v across south pole The distances d(x,y)+d(y,z)>... >Oh, the triangle inequality. >I suppose that makes perfect sense since a metric space is flat in a small >enough region. >>That's a really ridiculous statement, which I assume you have >>made because you have conflated the notion of a Riemannian metric >>on a manifold with the abstract notion of a metric space. >>Though if (for instance) John Mitchell were to weigh in with >>some argument that a metric space is flat in a small enough >>region, I would listen respecfully and no doubt learn something, >>in the absence of some such informed comment I will throw out >>my opinion that it's very rare indeed for a metric space-- >>even a metric space on R^2 which induces the standard topology, >>such as you seem to be trying to study--to be flat in a small >>enough region for any reasonable sense of flat. I don't >>think that the various non-Euclidean Minkowski p-norms on >>R^2 are flat, for example. >>In particular, I don't see any reasonable sense in which >>the triangle inequality is closely related to flatness. >>If you care to suggest such a sense, go for it. >>Lee Rudolph >I've probably confused some material from elsewhere with Metric Spaces >on page 10 of Bishop and Goldberg. But for flatness and the triangle >inequality, I think there must be some ambiguity in defining a triangle The triangle inequality in a metric space has nothing to do with triangles, just as it has nothing to do with flatness. It's just the inequality d(x,y) <= d(x,z) + d(z,y). The reason for the _name_ triangle inequality has something to do with triangles... >on >a curved space, and at any rate you can break the inequality over a large >enough region of a curved space. On a sphere, take sections of great >circles to be our straight lines and define a triangle: > across north pole > ^ > | > |x > > > / y > / > |z > | > v > across south pole > >The distances d(x,y)+d(y,z)... >>Oh, the triangle inequality. >> >>I suppose that makes perfect sense since a metric space is flat in a small >>enough region. >That's a really ridiculous statement, which I assume you have >made because you have conflated the notion of a Riemannian metric >on a manifold with the abstract notion of a metric space. >Though if (for instance) John Mitchell were to weigh in with >some argument that a metric space is flat in a small enough >region, I would listen respecfully and no doubt learn something, >in the absence of some such informed comment I will throw out >my opinion that it's very rare indeed for a metric space-- >even a metric space on R^2 which induces the standard topology, >such as you seem to be trying to study--to be flat in a small >enough region for any reasonable sense of flat. I don't >think that the various non-Euclidean Minkowski p-norms on >R^2 are flat, for example. >In particular, I don't see any reasonable sense in which >the triangle inequality is closely related to flatness. >If you care to suggest such a sense, go for it. >Lee Rudolph >>I've probably confused some material from elsewhere with Metric Spaces >>on page 10 of Bishop and Goldberg. But for flatness and the triangle >>inequality, I think there must be some ambiguity in defining a triangle >The triangle inequality in a metric space has nothing to do with >triangles, just as it has nothing to do with flatness. It's just the >inequality d(x,y) <= d(x,z) + d(z,y). The reason for the _name_ >triangle inequality has something to do with triangles... >>on >>a curved space, and at any rate you can break the inequality over a large >>enough region of a curved space. On a sphere, take sections of great >>circles to be our straight lines and define a triangle: >> across north pole >> ^ >> | >> |x >> >> >> / y >> / >> |z >> | >> v >> across south pole >> >>The distances d(x,y)+d(y,z)Really? Exactly what do you mean by d here? Length of line, distance between points. A good plan executed right now is far better than a perfect plan executed next week. -Gen. George S. Patton === Subject: Re: Standard Topology >> >>... >Oh, the triangle inequality. > >I suppose that makes perfect sense since a metric space is flat in a small >enough region. >> >>That's a really ridiculous statement, which I assume you have >>made because you have conflated the notion of a Riemannian metric >>on a manifold with the abstract notion of a metric space. >> >>Though if (for instance) John Mitchell were to weigh in with >>some argument that a metric space is flat in a small enough >>region, I would listen respecfully and no doubt learn something, >>in the absence of some such informed comment I will throw out >>my opinion that it's very rare indeed for a metric space-- >>even a metric space on R^2 which induces the standard topology, >>such as you seem to be trying to study--to be flat in a small >>enough region for any reasonable sense of flat. I don't >>think that the various non-Euclidean Minkowski p-norms on >>R^2 are flat, for example. >> >>In particular, I don't see any reasonable sense in which >>the triangle inequality is closely related to flatness. >>If you care to suggest such a sense, go for it. >> >>Lee Rudolph >I've probably confused some material from elsewhere with Metric Spaces >on page 10 of Bishop and Goldberg. But for flatness and the triangle >inequality, I think there must be some ambiguity in defining a triangle >>The triangle inequality in a metric space has nothing to do with >>triangles, just as it has nothing to do with flatness. It's just the >>inequality d(x,y) <= d(x,z) + d(z,y). The reason for the _name_ >>triangle inequality has something to do with triangles... >on >a curved space, and at any rate you can break the inequality over a large >enough region of a curved space. On a sphere, take sections of great >circles to be our straight lines and define a triangle: > across north pole > ^ > | > |x > > > / y > / > |z > | > v > across south pole > >The distances d(x,y)+d(y,z)>Really? Exactly what do you mean by d here? >Length of line, distance between points. Ok - I had a question about that, but I see above you said take sections of great circles to be our straight lines. So I take it that d(x,y) is the length of the shorter of the two great-circle arcs joining x and y. Now how did you deduce that d(x,y)+d(y,z)let X = {a,b,c,d,e} >find topology T of X that generated by A={{a},{a,b,c},{c,d}} >------------------------------------ >i think that A is not subbasis. >because >A subbasis S for a topology on X is a collection of subsets of X >whose union equals X--topolgy-James R.Munkres book >so, i think that this problem is not possible. You are correct in saying that A is not a subbasis. But it does not follow that the problem is not possible! Same question as yesterday: What is the _definition_ of the topology generated by A? >i ask your advice to my thinking. sir~ ************************ David C. Ullrich === Subject: Re: subbasis?? >>let X = {a,b,c,d,e} >>find topology T of X that generated by A={{a},{a,b,c},{c,d}} >>------------------------------------ >>i think that A is not subbasis. >>because >>A subbasis S for a topology on X is a collection of subsets of X >>whose union equals X--topolgy-James R.Munkres book >>so, i think that this problem is not possible. > You are correct in saying that A is not a subbasis. > But it does not follow that the problem is not possible! Evidently not everyone has the same definition here. Willard, for example, says that a subbase for a topology is a collection of sets having the property that all finite intersections of elements from the collection forms a base for the topology. He follows this with: 5.6 Theorem. Any collection of subsets of a set X is a subbase for some topology on X. I don't have Munkres handy, but I have never heard that version of the definition before. A Google search on subbase topology turned up some definitions equivalent to Willard's. > Same question as yesterday: What is the _definition_ > of the topology generated by A? >>i ask your advice to my thinking. sir~ > ************************ > David C. Ullrich Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: subbasis?? > let X = {a,b,c,d,e} > find topology T of X that generated by A={{a},{a,b,c},{c,d}} That topology is the smallest topology which contains A. A subbase for that topology is A with {a,b,c,d,e} added. > i think that A is not subbasis. > because > A subbasis S for a topology on X is a collection of subsets of X > whose union equals X--topolgy-James R.Munkres book A subbase is a collection of subsets for which the unions of every finite collection of subbase sets, form a base. Now the collection of unions of finite intersections of sets from A, cannot be a base because the space itself would have to be a union of base sets, but none of those base sets contain e. Thus A isn't a subbase. Thus James is basically correct in noting a collection of subsets is a subbase when the great union of the collection is the space itself. Do note three different but intertwined concepts, generated by, subbase, base. -- Technical Quibble The intersection of no sets of A can be considered to be X for obscure technical reasons. So with that iota, A could be consider a subbase. For similar reasons, the nulset doesn't have to be a base set as the union of no bases sets is the empty nulset. This should be easier to see than the intersection of no sets is as big as possible. === Subject: Re: subbasis?? >let X = {a,b,c,d,e} >find topology T of X that generated by A={{a},{a,b,c},{c,d}} >------------------------------------ >i think that A is not subbasis. >because >A subbasis S for a topology on X is a collection of subsets of X >whose union equals X--topolgy-James R.Munkres book >so, i think that this problem is not possible. >> You are correct in saying that A is not a subbasis. >> But it does not follow that the problem is not possible! >Evidently not everyone has the same definition here. Willard, for >example, says that a subbase for a topology is a collection of sets >having the property that all finite intersections of elements from the >collection forms a base for the topology. Well the definition looked a little curious to me as well, but since she seemed to be quoting it verbatim I assumed it was an accurate quotation. >He follows this with: > 5.6 Theorem. Any collection of subsets of a set X is a > subbase for some topology on X. >I don't have Munkres handy, but I have never heard that version of the >definition before. A Google search on subbase topology turned up some >definitions equivalent to Willard's. >> Same question as yesterday: What is the _definition_ >> of the topology generated by A? >i ask your advice to my thinking. sir~ >> ************************ >> David C. Ullrich ************************ David C. Ullrich === Subject: Re: subbasis?? >A subbasis S for a topology on X is a collection of subsets of X >whose union equals X--topolgy-James R.Munkres book >Evidently not everyone has the same definition here. Willard, for >example, says that a subbase for a topology is a collection of sets >having the property that all finite intersections of elements from the >collection forms a base for the topology. > Well the definition looked a little curious to me as well, but > since she seemed to be quoting it verbatim I assumed it > was an accurate quotation. >He follows this with: > 5.6 Theorem. Any collection of subsets of a set X is a > subbase for some topology on X. Reference, 'Counterexamples in Topology' by Steen page 213, note 1 referring page 3. Definitions do vary depending upon 'boundary conditions' for openness of emptyset and whole space set. I give one such version. A subbase for X is a collection of subsets of X for which all finite intersections of subbase sets form a base for X. This requires the abtruse notion that the intersection of the empty set, as viewed as a collection of subsets of X, is X itself. For example X = {0,1}, collection = { {0} }. Then to get {0,1} as subbase set, intersection emptyset = {0,1} Now you have base of { {0}, {0,1} } which gives topology { nulset, {0}, {0,1} } since union emptyset = empty set. The definition that any collection of subsets of X with union collection = X is a subbase, avoids the techical nusiance intersection emptyset = X. Yet still, with that subbase definition, if no finite intersection of subsets gives the empty set (cf example above), the base so derived will require the less abstruse notion that union emptyset = emptyset to included emptyset into the topology. Others such as Steen require all finite intersections of subbase sets along with emptyset and X to be a base. Yet in footnote 1, he points out the redundancy of such additions. === Subject: Re: SymbMath.com: web-based computer algebra system > SymbMath For Java is web-based symbolic math and computer algebra > system, > which runs in any computer with Java. You can play it online. www.SymbMath.com === Subject: Re: Synergetics coordinates Fyi: The nice property that was mentioned in the given link, is based on the Viviani's theorem. http://mathworld.wolfram.com/VivianisTheorem.html Ternary phase diagrams have been used during the last century - perhaps longer. See for example: http://www.sv.vt.edu/classes/MSE2094_NoteBook/96ClassProj/experimental/terna ry2.html Btw: MS-Excel has radar plot property and, on the other hand, xyz-cartesian plot, which can be viewed from any angle so that you can result in synergetics view. (45-45 degrees rotation of the point of view) Imho: I didn't know a certain angle of view in the xyz cartesian coordination system is called synergetics coordinates. ;-) Tapio > What Eric W. Weisstein calls the x,y,z axes of the triangle are rotated > one position to the right from mine in his description, but, it looks > like at least some version of Synergetics coordinates might become > respectable in academia now that they are mentioned at: > http://mathworld.wolfram.com/SynergeticsCoordinates.html > Cliff Nelson === Subject: Re: The Definition of a Manifold <3f314280$7$fuzhry+tra$mr2ice@news.patriot.net> <3f367e78$15$fuzhry+tra$mr2ice@news.patriot.net> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS >After rereading some of the posts and considering the above example I >guess i'll interpret cargo cult mathematics to be that intuitive >application of an mathematical idea without setting up (or checking >for) its formal foundations. Dirac's delta function, heaviside's >function, the imaginary unit, analysis (for a few centuries following >newton) and such. Is this what you mean? Basically. Keep in mind that while the examples I gave could be made rigorous and lead to correct predictions, most examples of Cargo Cult Mathematics are simply wrong. Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: the maximum points which make no angle larger than 90 X-NOTrace: GMJKPKHLOKGCJBDPFBDMHLLKFKDPDCKBBB My question is: to find the largest set of points in space (R^3) that if we choose three of them they don't make angles larger than OR equal to 90. I know the answer is 5 but I don't have the idea to prove .PLEASE help.... thank you. http://newsone.net/ -- Free reading and anonymous posting to 60,000+ groups NewsOne.Net prohibits users from posting spam. If this or other posts made through NewsOne.Net violate posting guidelines, email abuse@newsone.net === Subject: Re: the maximum points which make no angle larger than 90 >My question is: >to find the largest set of points in space (R^3) that if we choose three of >them they don't make angles larger than OR equal to 90. >I know the answer is 5 but I don't have the idea to prove .PLEASE help.... >thank you. Acute problem... One way to get 5 points is to take (0,0,1), (0,0,-1) and the vertices of an equilateral triangle centred at the origin in the xy plane (I'll leave it to you to decide the size of this triangle). I don't know how to prove you can't get 6, though. 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: the maximum points which make no angle larger than 90 > My question is: > to find the largest set of points in space (R^3) that if we choose three of > them they don't make angles larger than OR equal to 90. > I know the answer is 5 but I don't have the idea to prove .PLEASE help.... > thank you. Hmm... If the angles were allowed to be 90 degrees, you could take every 0/1-polytope in R^n, so for n=3 the cube has 2^3=8 vertices. But this is rather trivial, I can't help you with your problem. === Subject: Re: The paternity of Christ >... > >>So are you saying that the Y-chromosome that God made for Jesus would >>somehow be different than the one he made for Adam? > Why would you ask such a silly question? There isn't nearly enough > information in scripture to draw that sort of conclusion. Odds favor > the conjecture that they were different, of course, unless you propose > that Jesus and Adam were identical twins. Never mind all that. Do you propose that Adam existed? > Well, assuming it would make the thread even funnier. ;+) > Some claim that Jesus = God. Which means Mother of Jesus = Mother of God. > So, Jesus' Y chomosome would be God's. However, AFAICT, Nobody says Adam > had God's Chromosomes. God likely made Adam's genes like he made > everything else's -- complete with a billion-year history. > Why do you assume that Jesus and Adam had different Y-chromosomes? > After all, if Jesus was made directly by God and Adam was made > directly by God shouldn't they both have the same Y-chromosome? ;) The question becomes trivial when we note that there are no such things as chromosomes. Ask the Roman Catholic church - they don't believe in such nonsense. You should too. All these icky theories about nonsense like the earth being round (and not even at the centre of the universe) is poppycock. BAH! Chromosomes have no place in religion. Flagellation and cold showers before whipping off to kill the Arabs/Jews/Protestants is all that a good Christian needs. === Subject: Re: The two envelope paradox > A correct calculation of expected gain must include an estimate of the > probabilities of the bounding cases and a weighted assessment of the overall > expectation over those and the non-bounding cases, and the naive calculation > is simply wrong. To me this apparent paradox has only to do with the calculation of the expected gain from switching envelopes, which is dependant on the unspecified underlying distribution. It took me a while to figure out the real expectation, but I think I have it now. Let P(n) = probability that the minimal envelope contains n dollars (and then the other one contains 2*n dollars.) If e1 and e2 are the amounts in the chosen and unchosen envelopes respectively, then I calculate the expected value of the unchosen envelope given that the chosen contains n dollars as E(e2 | e1 = n) = n * [ 2 P(n) + 0.5 P(n/2) ] / [ P(n) + P(n/2) ] Since we don't know the original distribution, we can't say what the expected value is, but we can say If P(n) = P(n/2), then E(e2 | e1 = n) = 1.25 * n If P(n) = 0.5 P(n/2), then E(e2 | e1 = n) = n If P(n) > 0.5 P(n/2), then E(e2 | e1 = n) > n If P(n) < 0.5 P(n/2), then E(e2 | e1 = n) < n David Breese === Subject: Re: The two envelope paradox : OK, I have trouble with this problem, when I do simulations on computer you : get the same average return switching or not. You cannot simulate this problem on a computer. In order to attempt that, you would have to arbitrarily impose some sort of likelihood-distribution on the values in the envelopes. The whole paradox depends on the fact that no such distribution is specified. === Subject: Re: The two envelope paradox > : OK, I have trouble with this problem, when I do simulations on computer you > : get the same average return switching or not. >You cannot simulate this problem on a computer. >In order to attempt that, you would have to arbitrarily impose >some sort of likelihood-distribution on the values in the envelopes. >The whole paradox depends on the fact that no such distribution is >specified. Not really - it has already been pointed out that there are genuine distributions, which can be specified, for which you always have a positive expectation from switching. Derek Holt. === Subject: Re: The two envelope paradox > : OK, I have trouble with this problem, when I do simulations on computer you > : get the same average return switching or not. >You cannot simulate this problem on a computer. >In order to attempt that, you would have to arbitrarily impose >some sort of likelihood-distribution on the values in the envelopes. >The whole paradox depends on the fact that no such distribution is >specified. > Not really - it has already been pointed out that there are genuine > distributions, which can be specified, for which you always have a > positive expectation from switching. can you clarify not really so as to keep the thread on topic, not really that you cannot simulate, or not really it imposes a distribution, or not really the paradox depends on no such distribution? I can imagine a distribution like a bell curve where the values all fall around a central point but are still free to move around to many times the other value, without giving any indication of extremities. Guess it'd be a bell curve over a logarithmic scale. My question is, from the game hosts point of view for simplicity, does the game host lose more money over time by giving the option to swap? Herc === Subject: Re: The two envelope paradox > I can imagine a distribution like a bell curve where the values all fall around > a central point but are still free to move around to many times the other value, > without giving *any* indication of extremities. Literally it would give a trivial indication since it has an above or below average marker. But knowing that 50.1% of the time the other envelope is smaller may as well be 50% considering the 2 to 1 odds you get on half your money. puzzle perplexes me, please pass poor paragraph Herc === Subject: Re: The two envelope paradox > puzzle perplexes me, please pass poor paragraph Perplexing puzzle, please pass poor paragraph pertaining paradox. Herc :-) === Subject: Re: The two envelope paradox >> : OK, I have trouble with this problem, when I do simulations on computer you >> : get the same average return switching or not. >> >>You cannot simulate this problem on a computer. >>In order to attempt that, you would have to arbitrarily impose >>some sort of likelihood-distribution on the values in the envelopes. >>The whole paradox depends on the fact that no such distribution is >>specified. >> Not really - it has already been pointed out that there are genuine >> distributions, which can be specified, for which you always have a >> positive expectation from switching. >can you clarify not really so as to keep the thread on topic, not really that >you cannot simulate, or not really it imposes a distribution, or not really the >paradox depends on no such distribution? I find that sentence difficult to parse! But, I posted the distribution earlier. For n>=1, the first person puts 3^{n-1} and 3^n dollars in the two envelopes with probability 2^{-n}. (This departs slightly from the original problem in that one envelope contains 3 times as much as the other rather than twice as much, but that is just a detail.) The second person knows that this distribution was used. If he opens an envelope and sees $x with x > 1, then the expected contents of the other envelope is 11x/9. One suggested resolution of the paradox (which not everybody agrees with) rests on the fact that the expected value of x is infinite. But computer simulations would not help much, given that all expected values are infinite. Derek Holt. >a central point but are still free to move around to many times the other value, >without giving any indication of extremities. >Guess it'd be a bell curve over a logarithmic scale. >My question is, from the game hosts point of view for simplicity, >does the game host lose more money over time by giving the option to swap? >Herc === Subject: Re: The two envelope paradox >> : OK, I have trouble with this problem, when I do simulations on computer you >> : get the same average return switching or not. >> >>You cannot simulate this problem on a computer. >>In order to attempt that, you would have to arbitrarily impose >>some sort of likelihood-distribution on the values in the envelopes. >>The whole paradox depends on the fact that no such distribution is >>specified. >> >> Not really - it has already been pointed out that there are genuine >> distributions, which can be specified, for which you always have a >> positive expectation from switching. >can you clarify not really so as to keep the thread on topic, not really that >you cannot simulate, or not really it imposes a distribution, or not really the >paradox depends on no such distribution? > I find that sentence difficult to parse! I think its me not you, you seemed to negate Georges 3rd sentence, he made 3. >>You cannot simulate this problem on a computer. >>In order to attempt that, you would have to arbitrarily impose >>some sort of likelihood-distribution on the values in the envelopes. >>The whole paradox depends on the fact that no such distribution is >>specified. I was interested the first. > But, I posted the distribution earlier. > For n>=1, the first person puts 3^{n-1} and 3^n dollars in the two envelopes > with probability 2^{-n}. > (This departs slightly from the original problem in that one envelope > contains 3 times as much as the other rather than twice as much, but that > is just a detail.) > The second person knows that this distribution was used. If he opens an > envelope and sees $x with x > 1, then the expected contents of the other > envelope is 11x/9. > One suggested resolution of the paradox (which not everybody agrees with) > rests on the fact that the expected value of x is infinite. > But computer simulations would not help much, given that all expected > values are infinite. Why does that matter? We can assume a distribution as long as the properties of the problem hold. e.g. the max amount is $10,000,000,000. Assuming the mininum payout is $1, this gives us 2^32, 32 possible money _values_ up to 10 billion, so being at the extreme of the distribution only happens 3% of the time which is insignificant compared to the 11/9 or 1.25 advantage. Herc > Derek Holt. >a central point but are still free to move around to many times the other value, >without giving any indication of extremities. >Guess it'd be a bell curve over a logarithmic scale. >My question is, from the game hosts point of view for simplicity, >does the game host lose more money over time by giving the option to swap? >Herc === Subject: Re: The two envelope paradox > : OK, I have trouble with this problem, when I do simulations on computer you > : get the same average return switching or not. > >You cannot simulate this problem on a computer. >In order to attempt that, you would have to arbitrarily impose >some sort of likelihood-distribution on the values in the envelopes. >The whole paradox depends on the fact that no such distribution is >specified. Not really - it has already been pointed out that there are genuine > distributions, which can be specified, for which you always have a > positive expectation from switching. >> >>can you clarify not really so as to keep the thread on topic, not really that >>you cannot simulate, or not really it imposes a distribution, or not really the >>paradox depends on no such distribution? >> I find that sentence difficult to parse! >I think its me not you, you seemed to negate Georges 3rd sentence, he made 3. Oh right, sorry! My `not really' referred to the final statement: >The whole paradox depends on the fact that no such distribution is >specified. >You cannot simulate this problem on a computer. >In order to attempt that, you would have to arbitrarily impose >some sort of likelihood-distribution on the values in the envelopes. >The whole paradox depends on the fact that no such distribution is >specified. >I was interested the first. >> But, I posted the distribution earlier. >> For n>=1, the first person puts 3^{n-1} and 3^n dollars in the two envelopes >> with probability 2^{-n}. >> (This departs slightly from the original problem in that one envelope >> contains 3 times as much as the other rather than twice as much, but that >> is just a detail.) >> The second person knows that this distribution was used. If he opens an >> envelope and sees $x with x > 1, then the expected contents of the other >> envelope is 11x/9. >> One suggested resolution of the paradox (which not everybody agrees with) >> rests on the fact that the expected value of x is infinite. >> But computer simulations would not help much, given that all expected >> values are infinite. You could certainly attempt to simulate this distribution on a computer. But, since you expected gain is infinite, and the actual gain would inevitably be finite, the simulation would have very little chance of agreeing with the theoretical expectation! Derek Holt. >Why does that matter? >We can assume a distribution as long as the properties of the problem hold. >e.g. the max amount is $10,000,000,000. >Assuming the mininum payout is $1, this gives us 2^32, >32 possible money _values_ up to 10 billion, so being at the >extreme of the distribution only happens 3% of the time which >is insignificant compared to the 11/9 or 1.25 advantage. >Herc >> Derek Holt. >>a central point but are still free to move around to many times the other value, >>without giving any indication of extremities. >> >>Guess it'd be a bell curve over a logarithmic scale. >> >>My question is, from the game hosts point of view for simplicity, >>does the game host lose more money over time by giving the option to swap? >> >>Herc >> >> >> === Subject: Re: The two envelope paradox Here is an excellent analysis: http://www.u.arizona.edu/~chalmers/papers/envelope.html HJ > very grateful. > The two-envelope paradox', Analysis, 55 (1995), pp. 6-11. > Emanuel Rutten === Subject: Re: This Week's Finds in Mathematical Physics (Week 197) > ... > But it's better, actually, to glom all these different theories > into one big universal theory. The most obvious way to attempt > this is to take the moduli space of elliptic curves and cook up > a formal group law over the algebra of functions on this space > by stitching together all the formal group laws for each specific > elliptic curve. Sounds really cool, how do you do that? > The new version, namely tmf, is a bit sneakier. I think > it's the limit - in the sense of category theory - of the > elliptic cohomology theories for all specific elliptic curves. > The reason this is different than Ell is that some elliptic > curves have nontrivial symmetries! If this is the reason, can't we get it in the old way by using the moduli stack instead of the usual downgraded moduli space? Btw, I'm likely to come up with a lot more question about this cool week's finds when I understand what it says a bit better (now it's the weekend and my brain is kinda on low gear)! Squark ------------------------------------------------------------------ Write to me using the following e-mail: Skvark_Nuclearsto@excite.exe extension in the obvious way) === Subject: Re: This Week's Finds in Mathematical Physics (Week 197) Originator: baez@math-cl-n01.math.ucr.edu (John Baez) >> But it's better, actually, to glom all these different theories >> into one big universal theory. The most obvious way to attempt >> this is to take the moduli space of elliptic curves and cook up >> a formal group law over the algebra of functions on this space >> by stitching together all the formal group laws for each specific >> elliptic curve. >Sounds really cool, how do you do that? It's pretty easy, assuming you know what the heck I'm talking about here. Any specific elliptic curve consists of the solutions (x,y) of a cubic equation like this: y^2 = 4x^3 - g_2 x - g_3 where x and y are complex variables and g_2,g_3 are complex parameters specifying the elliptic curve in question. This may not be obvious starting from the modern definition of an elliptic curve as a torus-like thingie, but it's not hard to see, and I explained how in week13. The guy who figured this out was my thesis advisor's thesis advisor's thesis advisor's thesis advisor's thesis advisor's thesis advisor, Weierstrass. So, everything about an elliptic curve is a function of these numbers g_1 and g_2. In particular, its formal group law involves g_1 and g_2. But, if we take the same formulas and think of g_1 and g_2 as *formal* variables, we are working not over the complex numbers but over the field of rational functions in g_1 and g_2. And, we get a formal group law over this big field. We get all the *specific* formal group laws just by setting the variables g_1 and g_2 to equal specific values. That's almost how we stitch together all the formal group laws for specific elliptic curves into one big fat formal group law. But instead of working with the field of rational functions in g_1 and g_2, we should really work with the field of rational functions on the moduli space of elliptic curves. The point is that different g_1 and g_2 can give isomorphic elliptic curves. I just brought in g_1 and g_2 to make things seem more concrete. :-) >> The new version, namely tmf, is a bit sneakier. I think >> it's the limit - in the sense of category theory - of the >> elliptic cohomology theories for all specific elliptic curves. >> The reason this is different than Ell is that some elliptic >> curves have nontrivial symmetries! >If this is the reason, can't we get it in the old way by using >the moduli stack instead of the usual downgraded moduli space? We should, but I don't know if anyone has mastered the technology to do it this way! There's a technology for getting complex oriented cobodism theories from formal group laws over a field, but when the field is actually the rational functions on a *stack*, it's not really a field anymore and I don't know if people have generalized all the necessary theorems to this case. In fact, Stephan Stolz told me that a lot of the important ideas in this subject have not yet been published. === Subject: Thoughts on the Collatz conjecture Observing the Collatz conjecture you would logically have to believe it is true. The reason I make this statement is if a counter example was ever found and the integer path terminated in an endless loop OTHER THAN 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of other integers in this new path(s) and (tree) would also be involved. So if this conjecture has been tested for all start values < = 1.2 * 10^12 then all the integers involved in this new terminating path(s) and tree would have to have integer values > 1.2 * 10^12. So I believe a counter example is highly unlikely and thus a very strong conjecture. Also if a counter example were found, you may ask, why would another tree start to show up at an arbitrary large integer point? This new tree would have to share areas but not integers with the original tree. With the Collatz tree structure the branches keep spreading and growing as the integers increase in size thus making it more and more unlikely for a counter example showing up @ an arbitray large integer point. Again, if a counter example is found, how can I say another tree will be involved and not just one or more terminating separate sequences that are not tree connected! I believe the original Collatz tree is a good indicator that if there ever was a counter example it would have to involve another Collatz like tree where its trunk would be an endless loop of terminating integers whose values > 1.2 * 10^12 which is highly unlikely. Just thoughts from a laymen. Any comments welcome! Dan === Subject: Re: Thoughts on the Collatz conjecture > Observing the Collatz conjecture you would logically have to believe > it is true. logically? > The reason I make this statement is if a counter example was ever > found and the integer path terminated in an endless loop OTHER THAN > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > other integers in this new path(s) and (tree) would also be involved. arbitrarily large? The cycle would be finite. Of course, a counterexample might be an unbounded sequence. > So if this conjecture has been tested for all start values < = 1.2 * > 10^12 then all the integers involved in this new terminating path(s) > and tree would have to have integer values > 1.2 * 10^12. > So I believe a counter example is highly unlikely and thus a very > strong conjecture. But 1.2 x 10^12 is a pretty small number. Why does highly unlikely become logically true? Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html His mind has been corrupted by colours, sounds and shapes. The League of Gentlemen === Subject: Re: Thoughts on the Collatz conjecture > Observing the Collatz conjecture you would logically have to believe > it is true. > The reason I make this statement is if a counter example was ever > found and the integer path terminated in an endless loop OTHER THAN > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > other integers in this new path(s) and (tree) would also be involved. > So if this conjecture has been tested for all start values < = 1.2 * > 10^12 then all the integers involved in this new terminating path(s) > and tree would have to have integer values > 1.2 * 10^12. > So I believe a counter example is highly unlikely and thus a very > strong conjecture. Another conjecture is that for every prime number p, there is a prime number p < q <= p+1000. This conjecture has been successfully tested for all primes <= 10^15. So what do you think of this conjecture? What do you think of my modified Collatz sequence: c(n+1) = c(n) + 1 if c = 10^1000 c(n+1) = c(n) - 1 if c = 10^1000 + 1 c(n+1) = 3c(n) + 1 if c is odd, c is not 10^1000 + 1 c(n+1) = c(n) / 2 if c is even, c is not 10^1000. This is also tested up to 1.2 * 10^12, yet it fails at 10^1000. === Subject: Re: Thoughts on the Collatz conjecture Dan > Observing the Collatz conjecture you would logically have to believe > it is true. > The reason I make this statement is if a counter example was ever > found and the integer path terminated in an endless loop OTHER THAN > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > other integers in this new path(s) and (tree) would also be involved. ... > With the Collatz tree structure the branches keep spreading and > growing as the integers increase in size thus making it more and more > unlikely for a counter example showing up @ an arbitray large integer > point. ... Some probabilistic arguments have been put forward about the Collatz deal. The existence of a divergent trajectory looks especially unlikely. Here's one bit: www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html Larry === Subject: Re: Thoughts on the Collatz conjecture > [...] So if the Collatz conjecture has been tested for all start > values below 10^12 then all the integers involved in this new > terminating path(s) and tree would have to have integer values > below 10^12. So I believe a counter example is highly unlikely > and thus a very strong conjecture. [...] Why would you think that? Almost all naturals are larger than 10^12 so there is plenty of room for counterexamples. Apparently many believe that because the problem is small then any counterexample should also be small. But this intuition is misguided. Very large numbers can be specified by extremely short programs. For example, a very tiny 6-state Busy Beaver Turing machine can compute a number larger than 10^865, requiring more than 3*10^1730 steps to do so. Contrast those numbers 10^80, the current estimate of the number of atoms in the universe. If you ran this machine for your whole life it would never halt, so you might conjecture that it wouldn't. But it is known to halt. In fact this example is not too far removed from the Collatz conjecture because open problems analogous to the Collatz congruential iteration have surprisingly been discovered by those working on the halting problem and Busy Beaver machines. In general, John Conway has shown that it is undecidable to tell whether or not such iterations will terminate. For much further information along these lines see my old posts on these matters: -Bill Dubuque === Subject: Re: Thoughts on the Collatz conjecture > Observing the Collatz conjecture you would logically have to believe > it is true. > The reason I make this statement is if a counter example was ever > found and the integer path terminated in an endless loop OTHER THAN > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > other integers in this new path(s) and (tree) would also be involved. > So if this conjecture has been tested for all start values < = 1.2 * > 10^12 then all the integers involved in this new terminating path(s) > and tree would have to have integer values > 1.2 * 10^12. > So I believe a counter example is highly unlikely and thus a very > strong conjecture. > Also if a counter example were found, you may ask, why would another > tree start to show up at an arbitrary large integer point? > This new tree would have to share areas but not integers with the > original tree. > With the Collatz tree structure the branches keep spreading and > growing as the integers increase in size thus making it more and more > unlikely for a counter example showing up @ an arbitray large integer > point. > Again, if a counter example is found, how can I say another tree will > be involved and not just one or more terminating separate sequences > that are not tree connected! > I believe the original Collatz tree is a good indicator that if there > ever was a counter example it would have to involve another Collatz > like tree where its trunk would be an endless loop of terminating > integers whose values > 1.2 * 10^12 which is highly unlikely. > Just thoughts from a laymen. > > Any comments welcome! > Dan But there is no reason to think a loop of 1,000,000 iterations can't happen. The term Excursion is used to represent the maximum value of x in a Collatz sequence. T(E) would be the number of iterations needed to reach the Excursion. One can easily construct a number having an arbitrarily large T(E). Simply start with a binary number with T(E) 1's. In other words, 2^(T(E)) - 1. Certain binary patterns are preserved (although made shorter) during the sequence, in the above case, the block of contiguous least significant 1s decreases by 1 in each (3x+1) x/2 cycle (the x's represent carry bits, which I don't know off the top of my head): 1111111111 x111111111 xx11111111 xxxx1111111 xxxxx111111 xxxxxxx11111 xxxxxxxxx1111 xxxxxxxxxx111 xxxxxxxxxxxx11 xxxxxxxxxxxxxx1 xxxxxxxxxxxxxxx For this pattern, the number of x's will be 1.585 times the number of original 1's. That's why we can construct a number that has any arbitrarily chosen Excursion. And note that none of the numbers up to the Excursion (and for a considerable number of iterations past it) are smaller than the original number. So if our original number had 1,000,000 binary bits, none of the numbers in the sequence would be part of any of the small sequences documented so far. If a loop occured right at the end of the Excursion, you'll never find it by looking at small numbers. Other binary patterns have more interesting effects. There is a certain pattern that not only is preserved during the sequence, but is also regenerated in the carry bits. This could be a mechanism that produces a loop or expansion to infinity. Alas, four rep-units are consumed in order to regenerate one rep-unit, so the process is convergent. But to see this happen for, say, 6 generations, it takes 4098 bits or 10^1233. So I can guarantee you that those who have tested the numbers up to 10^12 haven't encountered a 6th Generation Quadracycle Pulsar with 4-bit Rep-units and 2-bit Resonator. And if such simple creatures as Pulsars can hide in the astronomically high numbers, then the elusive million iteration loop could be hiding up there also. === Subject: Re: Thoughts on the Collatz conjecture > Dan > Observing the Collatz conjecture you would logically have to believe > it is true. > The reason I make this statement is if a counter example was ever > found and the integer path terminated in an endless loop OTHER THAN > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > other integers in this new path(s) and (tree) would also be involved. > ... > With the Collatz tree structure the branches keep spreading and > growing as the integers increase in size thus making it more and more > unlikely for a counter example showing up @ an arbitray large integer > point. > ... > Some probabilistic arguments have been put forward about the Collatz deal. > The existence of a divergent trajectory looks especially unlikely. Here's > one bit: > www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html > Larry But the original post suggests a problem that's far from solved: If a nontrivial cycle exists in the Collatz problem, find an effective upper bound for its length.If we knew an effective answer for this question, we could effectively find all nontrivial cycles.Anyone have any ideas? Ray Steiner === Subject: Re: Thoughts on the Collatz conjecture > Dan > Observing the Collatz conjecture you would logically have to believe > it is true. > The reason I make this statement is if a counter example was ever > found and the integer path terminated in an endless loop OTHER THAN > 4,2,1,4,2,1,4,2,1,... that would mean an arbitrary large number of > other integers in this new path(s) and (tree) would also be involved. > ... > With the Collatz tree structure the branches keep spreading and > growing as the integers increase in size thus making it more and more > unlikely for a counter example showing up @ an arbitray large integer > point. > ... > Some probabilistic arguments have been put forward about the Collatz deal. > The existence of a divergent trajectory looks especially unlikely. Here's > one bit: > www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html > Larry > But the original post suggests a problem that's far > from solved: > If a nontrivial cycle exists in the Collatz problem, find > an effective upper bound for its length.If we knew an effective > answer for this question, we could effectively find > all nontrivial cycles.Anyone have any ideas? A couple months ago in another thread (subject: Feedback on 3x+y attractors sets wanted) I discussed hailstone functions, crossover points and attractors which why there would necessarily be an upper bound. > Ray Steiner === Subject: Re: Three-signed arithmetic : T space Another simple theorem in Y: For any y in Y: * y + y - y = 0. === Subject: Treewidth and pathwidth Following problem Sum over all non-isomorphic graphs on n vertices and m edges (a) their treewidth, (b) their pathwidth. Is anything known about the absolute value of these quantities? (crude approximation would be ok) Is anything known about the ratio of these quantities? (crude approximation would be ok) I guess no on both accounts, but I am not sure. In any case, does anybody know some software to compute path- and treewidth of non-trivial graphs (i.e. say for treewidth of 10, 100, ...) exactly or approximatively? Andre' PS: This is not a homework assignment and I've seen quite a few papers (Arnborg, Bodlaender and others) on the general topic of treewidth and pathwidth, but none with numerical evidence... === Subject: Tricky derivative problem Given a function: f(x)=[(1+y(x)]/x Calculate 2002th derivative of f(x) at x=0 === Subject: Re: Tricky derivative problem > Given a function: > f(x)=[(1+y(x)]/x > Calculate 2002th derivative of f(x) at x=0 Is y(x) the same as f(x)?? === Subject: Re: Tricky derivative problem > Given a function: > f(x)=[(1+y(x)]/x > Calculate 2002th derivative of f(x) at x=0 Are you sure this is tricky? Clive Tooth http://www.clivetooth.dk === Subject: Re: Tricky derivative problem > Given a function: > f(x)=[(1+y(x)]/x > Calculate 2002th derivative of f(x) at x=0 > Is y(x) the same as f(x)?? Clive Tooth http://www.clivetooth.dk === Subject: Re: Tricky derivative problem > Given a function: > f(x)=[(1+y(x)]/x > Calculate 2002th derivative of f(x) at x=0 > Are you sure this is tricky? With unbalanced parentheses, yes. === Subject: Re: Tricky derivative problem > Given a function: > f(x)=[(1+y(x)]/x > Calculate 2002th derivative of f(x) at x=0 Hint: x f(x) = 1 + y(x) ==> n f^{n-1](x) + x f^[n](x) = y^[n](x) where g^[k] denotes the k'th derivative of the function g ... === Subject: Re: Well-ordering R >Under AC, of course, does there exist an algortihm which develops a >well-ordering of the reals (not necessarily in finite time) and which >has the property that, given any two real numbers, it determines in >finite time which of the two is no greater than the other under this >ordering? I was thinking that you might be able to order the reals in (0,1) lexicographically using continued fractions. The smallest irrational would be i=[0;1,1,1,...]. Do (0,i) and (i,1) also have a smallest elements? But [0;1,2,1,1,...]<<[0;2,1,1,...], ect. so one problem is that sets like {[0;2,1,1,...],[0;1,2,1,1,...],[0;1,1,2,1,1,...],...} have no least element. So suppose r or s is not deg(2). Say rMy guess is no. Such an algorithm would need to take countably many >steps. Infinitely many of the steps must order an uncountable subset >of R in finite time. If one could do that, one could well-order R >without choice [not sure about this step]. Have I just answered my own >question? >-- >Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Well-ordering R >I was thinking that you might be able to order the reals in (0,1) >lexicographically using continued fractions. The smallest irrational >would be >i=[0;1,1,1,...]. Do (0,i) and (i,1) also have a smallest elements? But >[0;1,2,1,1,...]<<[0;2,1,1,...], ect. so one problem is that sets like >{[0;2,1,1,...],[0;1,2,1,1,...],[0;1,1,2,1,1,...],...} have no least >element. >So suppose r or s is not deg(2). Say rr>Under AC, of course, does there exist an algortihm which develops a >>well-ordering of the reals (not necessarily in finite time) and which >>has the property that, given any two real numbers, it determines in >>finite time which of the two is no greater than the other under this >>ordering? >I was thinking that you might be able to order the reals in (0,1) >lexicographically using continued fractions. The smallest irrational would >i=[0;1,1,1,...]. Do (0,i) and (i,1) also have a smallest elements? But >[0;1,2,1,1,...]<<[0;2,1,1,...], ect. so one problem is that sets like >{[0;2,1,1,...],[0;1,2,1,1,...],[0;1,1,2,1,1,...],...} have no least element. >So suppose r or s is not deg(2). Say rthen >rUnder AC, of course, does there exist an algortihm which develops a >well-ordering of the reals (not necessarily in finite time) and which >has the property that, given any two real numbers, it determines in >finite time which of the two is no greater than the other under this >ordering? > I was thinking that you might be able to order the reals in (0,1) > lexicographically using continued fractions. The smallest irrational would be > i=[0;1,1,1,...]. Do (0,i) and (i,1) also have a smallest elements? But > [0;1,2,1,1,...]<<[0;2,1,1,...], ect. so one problem is that sets like > {[0;2,1,1,...],[0;1,2,1,1,...],[0;1,1,2,1,1,...],...} have no least element. > So suppose r or s is not deg(2). Say r then > r Rich Burge You're right, it doesn't quite work. But there is something similar that does. The set of irrationals in (0,1) homeomorphic to N^N (by continues fractions, for example)... ordered lexicographically. Every CLOSED subset has a least element. (Not every subset, wo it is not well-ordered.) This is very useful in descriptive set theory. For measurable cross-sections and such. This goes back to von Neumann, I think. G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: What is an algebraic integer? Mail-To-News-Contact: abuse@dizum.com >> I've been seeing this term a lot in James Harris's proofs. Can someone >> please give me the definition, so that I can check those proofs for >> myself? >For a complex number to be an algebraic integer, it is necessary and >sufficient that it be a zero of an irriducible monic polynomial in >one variable with integer coefficients. If I understand correctly from the crumbs that I've picked up, does monic mean coefficient of the highest-order term is unity? What does irreducible mean? It seems to me that showing closure of multiplication of the roots of polynomials with integer coefficients and highest-order coefficient unity will be easy enough. However, not knowing what irreducible means is a minor flaw :-> I'm not sure that showing closure of addition will be as simple as closure of multiplication. I'll have to think about that one. Michael F. Stemper #include It's Ensign Schrodinger! He's half-dead, Jim! === Subject: Re: What is an algebraic integer? > If I understand correctly from the crumbs that I've picked up, does monic > mean coefficient of the highest-order term is unity? Or negative unity. > What does > irreducible mean? Not factorable over the ring of coefficients. E.g., x^2 + 1 is irreducible over any subring of the reals, but is reducible over any superring of the Gaussian integers. > It seems to me that showing closure of multiplication of the roots of > polynomials with integer coefficients and highest-order coefficient unity > will be easy enough. However, not knowing what irreducible means is a > minor flaw :-> === Subject: Re: What is an algebraic integer? Mail-To-News-Contact: abuse@dizum.com >> If I understand correctly from the crumbs that I've picked up, does monic >> mean coefficient of the highest-order term is unity? >Or negative unity. OK. >> What does irreducible mean? >Not factorable over the ring of coefficients. So, a polynomial such as x^2 - 2x -15, which factors to (x+3)(x-5) would not be called an irreducible polynomial (and, presumably, would be called a reducible polynomial)? Or, doesn't ring of coefficients refer to the integers? Does it refer to some ring that is defined for each polynomial, based upon its specific coefficients? What about x+3 and x-5, taken as individual polynomials? Are they considered irreducible? All of a sudden, proving the closure properties to myself seems like it might be more challenging than I'd expected. Michael F. Stemper #include If it's tourist season, where do I get my license? === Subject: Re: What is an algebraic integer? > If I understand correctly from the crumbs that I've picked up, does monic > mean coefficient of the highest-order term is unity? > Or negative unity. News to me. One wants to refer to *the* monic irreducible polynomial for a given element of an algebraic extension field - can't do this if -1 is allowed as lead coefficient in the definition of monic. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: What is an algebraic integer? > What about x+3 and x-5, taken as individual polynomials? Are they > considered irreducible? They're not just considered irreducible, they *are* irreducible. Reducible (over a given domain) means you can write it as a product of things of strictly smaller degree. You can't write a polynomial of degree 1 as a product of polynomials of strictly smaller degree - can you? - so it's irreducible. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: What is an algebraic integer? >> What about x+3 and x-5, taken as individual polynomials? Are they >> considered irreducible? >They're not just considered irreducible, they *are* irreducible. >Reducible (over a given domain) means you can write it as a product >of things of strictly smaller degree. You can't write a polynomial >of degree 1 as a product of polynomials of strictly smaller degree >- can you? - so it's irreducible. >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) (15x + 3)*(7x + 1) = 105x^2 + 36x + 3 = x + 3 in Z/35Z. Here a polynomial of degree 1 is a product of two other degree-1 polynomials. Wanted: Experts at choosing the best of 100+ applicants for a position. Register as a California voter by September 22, and vote on October 7. Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California Microsoft Research and CWI === Subject: Re: What is an algebraic integer? >> What about x+3 and x-5, taken as individual polynomials? Are they >> considered irreducible? >They're not just considered irreducible, they *are* irreducible. >Reducible (over a given domain) means you can write it as a product >of things of strictly smaller degree. You can't write a polynomial >of degree 1 as a product of polynomials of strictly smaller degree >- can you? - so it's irreducible. >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) > (15x + 3)*(7x + 1) = 105x^2 + 36x + 3 = x + 3 > in Z/35Z. Here a polynomial of degree 1 is a product of > two other degree-1 polynomials. True. But, 1. Z/35Z isn't a domain, and 2. it works anyway for monic polynomials as in the original question. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: What is an algebraic integer? >> If I understand correctly from the crumbs that I've picked up, does >> monic >> mean coefficient of the highest-order term is unity? >Or negative unity. > OK. >> What does irreducible mean? >Not factorable over the ring of coefficients. > So, a polynomial such as x^2 - 2x -15, which factors to (x+3)(x-5) would > not be called an irreducible polynomial (and, presumably, would be called > a reducible polynomial)? Or, doesn't ring of coefficients refer to the > integers? Does it refer to some ring that is defined for each polynomial, > based upon its specific coefficients? > What about x+3 and x-5, taken as individual polynomials? Are they > considered irreducible? I think the irreducible bit in the definition is not really necessary. As you noticed, the fact that x^2 - 2x - 15 has roots -3 and 5 doesn't make -3 and 5 algebraic integers according to the definition, but because the polynomial is reducible you then get the irreducible polynomials x+3 and x-5, so -3 and 5 are algebraic integers after all. Same with the condition that the highest-order term can be -1. If it is -1, then just change the sign of the polynomial, and you get the same roots. You find a very nice proof for closure at Robin Chapman's web page at http://www.maths.ex.ac.uk/~rjc/rjc.html It is not very difficult, actually, but I would never have come up with the idea for the proof. === Subject: Re: What is counting? >It's not easy to get any definitions at all regarding counting. One I >found is this: counting is a technique for assessing the sizes of >sets. >My own definition is this: Counting is any procedure that correctly >determines the cardinality of a finite set. >Any comments on the topic? For what it's worth, in _Naive_Set_Theory_, Halmos calls the Counting Theorem that which states that any well-ordered set is order-isomorphic to a unique ordinal. I kind of agree with this more than with identifying the cardinality. That is, to count is to set up a (well-?)order of collection of objects and identify the collection with a another standard reference, usually a finite ordinal or aleph-0. After all, when you count out five cards, you might say 1, 2, 3, 4, 5, thus numbering them all, not just identifyting the cardinality 5. Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: What is counting? >It's not easy to get any definitions at all regarding counting. One I >found is this: counting is a technique for assessing the sizes of >sets. >My own definition is this: Counting is any procedure that correctly >determines the cardinality of a finite set. > >Any comments on the topic? > For what it's worth, in _Naive_Set_Theory_, Halmos calls the Counting > Theorem that which states that any well-ordered set is order-isomorphic > to a unique ordinal. I kind of agree with this more than with > identifying the cardinality. That is, to count is to set up a > (well-?)order of collection of objects and identify the collection with > a another standard reference, usually a finite ordinal or aleph-0. > After all, when you count out five cards, you might say 1, 2, 3, 4, 5, > thus numbering them all, not just identifyting the cardinality 5. Well, I must be referring to common or pre-naive set theory, because nobody in the real world expects to receive a specific ordering on a finite set when they ask for a count of the set. Counting is not used in the sense of ordinality in descrete math, for example. Maybe set theory has jargonized the notion of counting. For my money, I want a good definition of counting applicable to use by a student of combinatorics. Patrick === Subject: Re: What is counting? > My own definition is this: Counting is any procedure that correctly > determines the cardinality of a finite set. The cardinality of -any- set. Finite sets are interesting because we map ordinal types to arbitrarily ordered (or even unordered) finite sets to establish the cardinality. Bob Kolker === Subject: Re: What is counting? > This question came up in a discussion I had with colleague about > counting and permutations and combinations. > > It's not easy to get any definitions at all regarding counting. One I > found is this: counting is a technique for assessing the sizes of > sets. > > In the pdf paper at > > http://www.maths.qmw.ac.uk/~pjc/notes/counting.pdf > > we find some interesting characterizations of counting, but no clear > characterization of exactly what kinds of things can be counted. > > My own definition is this: Counting is any procedure that correctly > determines the cardinality of a finite set. > > Any comments on the topic? > > Patrick > > I'm not sure I agree with your definition, and also it does not seem > to answer your own question of what kind of things can be counted. > What does determine the cardinality exactly mean? > I know my definition doesn't seem to present an in your face meaning > of what kinds of things can be counted, but I believe that it is > accurate anyway: By implication ONLY, all finite sets are considered > to be countable. How they can be counted is really a whole other > matter. My definition treats the means of arriving at a finite set's > cardinality as primitive, i.e., as left unspecified. But to be > precise, my definition is only a definition, not an existence axiom. > All it says is that if you find a procedure that is able to arrive at > the cardinality of some finite set that procedure amounts to > 'counting' the elements of that set. If the answer to a counting question, in your mind, is a natural number, then the entity being counted is a finite set. What additional information do we gain from your definition? > The simplest question to ask regarding my definition is: What things > are countable that it leaves out, and what uncountable things does it > include? If you can come up with specific examples, then we have > something definite to go on. > A procedure to > determine the number of, say, the possible games of chess is certainly > available, but brute force enumeration is rarely what counting, in > the combinatorial sense, is about. > > In another direction, your definition misses the subtle and crucial > art of approximations, which is related to another ingredient you're > ignoring - the fact that, typically, one doesn't determine the size of > one given set, but rather looks for a formula to capture (or > approximate) the size of sets in some naturally defined infinite > family. > True, but I believe that counting is meant to produce an exact number, > not an approximation, though our common language often allows for this > abuse of meaning. Anyway, this is what I wanted for my definition of > counting. For example, the US census never really counts the > number of its citizens, though some may claim that it does. This is > one of those abuses of terminology I mentioned. Mathematicians should > do better, though. > > All of this is exaplained much more clearly in the first chapter (or > the introduction, I can't remember) of Stanley's superb Enumerative > Combinatoris, volume 1. > > Hope that helps, > > - EM > OK, then, what did Stanley offer as his definition of counting? > Patrick Stanley is interested in actual combinatorics, where the essence is what I described above - closed form formulas, explicit formulas involving (say) sums which are still more efficient than brute force counting, generating functions, approximations etc. Saying that to count a set is to determine its cardinality really isn't a very useful statement to me and, I humbly suspect, to Stanley. - EM === Subject: Re: What is counting? > > My own definition is this: Counting is any procedure that correctly > determines the cardinality of a finite set. > > Any comments on the topic? > > Patrick > > I'm not sure I agree with your definition, and also it does not seem > to answer your own question of what kind of things can be counted. > What does determine the cardinality exactly mean? > > I know my definition doesn't seem to present an in your face meaning > of what kinds of things can be counted, but I believe that it is > accurate anyway: By implication ONLY, all finite sets are considered > to be countable. How they can be counted is really a whole other > matter. My definition treats the means of arriving at a finite set's > cardinality as primitive, i.e., as left unspecified. But to be > precise, my definition is only a definition, not an existence axiom. > All it says is that if you find a procedure that is able to arrive at > the cardinality of some finite set that procedure amounts to > 'counting' the elements of that set. > If the answer to a counting question, in your mind, is a natural > number, then the entity being counted is a finite set. What additional > information do we gain from your definition? > > The simplest question to ask regarding my definition is: What things > are countable that it leaves out, and what uncountable things does it > include? If you can come up with specific examples, then we have > something definite to go on. > A procedure to > determine the number of, say, the possible games of chess is certainly > available, but brute force enumeration is rarely what counting, in > the combinatorial sense, is about. > > In another direction, your definition misses the subtle and crucial > art of approximations, which is related to another ingredient you're > ignoring - the fact that, typically, one doesn't determine the size of > one given set, but rather looks for a formula to capture (or > approximate) the size of sets in some naturally defined infinite > family. > > True, but I believe that counting is meant to produce an exact number, > not an approximation, though our common language often allows for this > abuse of meaning. Anyway, this is what I wanted for my definition of > counting. For example, the US census never really counts the > number of its citizens, though some may claim that it does. This is > one of those abuses of terminology I mentioned. Mathematicians should > do better, though. > > > All of this is exaplained much more clearly in the first chapter (or > the introduction, I can't remember) of Stanley's superb Enumerative > Combinatoris, volume 1. > > Hope that helps, > > - EM > > > OK, then, what did Stanley offer as his definition of counting? > > Patrick > Stanley is interested in actual combinatorics, where the essence is > what I described above - closed form formulas, explicit formulas > involving (say) sums which are still more efficient than brute force > counting, generating functions, approximations etc. Saying that to > count a set is to determine its cardinality really isn't a very > useful statement to me and, I humbly suspect, to Stanley. > - EM Perhaps I should have stated earlier that one issue I wanted to clear up to students of combinatorics is that counting is not always just simple addition. But it seems to me that authors of math texts should not use terms that they do not define at least to a level servicable for the course at hand. On the other hand, I don't think we need to appeal to the Axiom of Choice to teach counting by use of combinations and permutations. What's your definition of counting? Patrick === Subject: Re: What is counting? >>My own definition is this: Counting is any procedure that correctly >>determines the cardinality of a finite set. >(That sounds right to me ...) >> I definitely disagree. Counting is a procedure which >> assigns a distinct ordinal to every member of a finite >> set. >I think the procedure you describe would be called numbering, indexing, >or enumerating. As there's already at least three words that refer >precisely to this procedure, it would seem abusive to take the word >counting as another synonym, especially since counting ordinarily refers >to merely determining how many of something you have (i.e., the size of a >set), as the original poster suggested. I disagree; there are many other ways of determining how many. How many is a measure, not a count. The mint determines how many coins there are in a batch by weighing them; nothing is counted. 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: What is counting? >It's not easy to get any definitions at all regarding counting. One I >found is this: counting is a technique for assessing the sizes of >sets. >My own definition is this: Counting is any procedure that correctly >determines the cardinality of a finite set. > >Any comments on the topic? > For what it's worth, in _Naive_Set_Theory_, Halmos calls the Counting > Theorem that which states that any well-ordered set is order-isomorphic > to a unique ordinal. I kind of agree with this more than with > identifying the cardinality. That is, to count is to set up a > (well-?)order of collection of objects and identify the collection with > a another standard reference, usually a finite ordinal or aleph-0. > After all, when you count out five cards, you might say 1, 2, 3, 4, 5, > thus numbering them all, not just identifyting the cardinality 5. It's my fault! I should have stated right up front that my interest for the definition was for its use in discreet math and combinatorics classes, not a class on set theory per se. Patrick === Subject: Re: What is counting? ................. >> For what it's worth, in _Naive_Set_Theory_, Halmos calls the Counting >> Theorem that which states that any well-ordered set is order-isomorphic >> to a unique ordinal. I kind of agree with this more than with >> identifying the cardinality. That is, to count is to set up a >> (well-?)order of collection of objects and identify the collection with >> a another standard reference, usually a finite ordinal or aleph-0. >> After all, when you count out five cards, you might say 1, 2, 3, 4, 5, >> thus numbering them all, not just identifyting the cardinality 5. >Well, I must be referring to common or pre-naive set theory, because >nobody in the real world expects to receive a specific ordering on a >finite set when they ask for a count of the set. Counting is not used >in the sense of ordinality in descrete math, for example. Maybe set >theory has jargonized the notion of counting. For my money, I want a >good definition of counting applicable to use by a student of >combinatorics. A finite set is one which can be well-ordered in a manner corresponding to only one ordinal, so that it can be counted in exactly one way. Alternatively, it can be ordered in such a way that every non-empty subset has both a largest and smallest element. 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: What is counting? My favorite definition of to count is this: To count the number of elements of a set is to put the elements of a set into a 1-1 correspondence with an initial subset of the positive integers. I do not remember where this came from, and I know that I did not originate it. Naturally, it assumes the existence of the positive integers (aka natural numbers). We can play the Peano axioms to get them. Martin Cohen === Subject: Re: What is counting? >My favorite definition of to count is this: >To count the number of elements of a set is to put the >elements of a set into a 1-1 correspondence with >an initial subset of the positive integers. >I do not remember where this came from, and I know that >I did not originate it. >Naturally, it assumes the existence of the positive integers >(aka natural numbers). We can play the Peano axioms to get them. This is a much clearer statement than mine. 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: What is counting? > My own definition is this: Counting is any procedure that correctly > determines the cardinality of a finite set. > The cardinality of -any- set. Finite sets are interesting because we map > ordinal types to arbitrarily ordered (or even unordered) finite sets to > establish the cardinality. > Bob Kolker My fault. I should have specified up front that my interest was within topics in the area of undergraduate combinatorics, not set theory per se. Patrick === Subject: Re: What is counting? > My favorite definition of to count is this: > To count the number of elements of a set is to put the > elements of a set into a 1-1 correspondence with > an initial subset of the positive integers. > I do not remember where this came from, and I know that > I did not originate it. > Naturally, it assumes the existence of the positive integers > (aka natural numbers). We can play the Peano axioms to get them. > Martin Cohen What does this have to do with combinations and permutations? Parick === Subject: When Y is a variable dependent on X then - - (Part-1) WHEN X IS AN INDEPENDENT VARIABLE IT FOLLOWS NO RULE! But, if Y is dependent on X then what rule does it follow? A-1: If we want to know to what extent Y is dependent on X, then we have to first visualize how X and Y look like. We have two distinct options: A-1-1: X and Y are discrete, countable and different things like dollars and chairs. A-1-2: X and Y are continuous ideas like length and time or X and Y can be real fluid substances like water (W) and oil (O). The ratios Y/X, $/ chairs, L/T and W/O, all are useful in many calculations. B-1: When we have to use the ratio Y/X in the form of $ per chair then we have only one option: B-1-1: We have to place certain number of $S in front of each chair. It is wrong to show $-chair relation by a continuous line. Moreover we do not need system of coordinates. Here ratio Y /X (rate) does not fill space or time. C-1: When we have to use the ratio Y/X in the form of displacement/time or Water/oil then we can have three views, only one of which is correct: C-1-1: X fills Y, but Y does not fill X. (Is Y a container; X content? Cannot be.) C-1-2: X fills Y and Y also fills X. (like an emulsion or synthetic). This is the only possibility, because both X and Y are continuous and we cannot imagine one of them in the form of a container and the other as the content. Therefore linear speed L/T is analogous to a homogeneous synthetic made up of space and time; dL/dT is like a molecule of that synthetic and time gives quantity. On a line that shows speed, we cannot see/show time and displacement independent of each other, they are synthesized and are inseparable. C-2: We cannot communicate by numerical or symbolic expression the fact that L is completely filling T, and that T is also completely filling L. Therefore what we do is: C-2-1: We do the same thing that we did in case of $-chair relation: we place clock-readings in front of mile-stones in all communicated expressions. Let me repeat: we cannot show that time is filled by displacement and displacement is filled by time. D-1-1: The background we prepare in the form of Cartesian coordinates shows fixed length-scales on X and Y coordinates. Therefore coordinates also show time (scale) and each coordinate automatically also indicates a constant linear velocity. Thus X=V_1*T, Y=V_2T and Y/X=constant. This fact, whether anybody likes or not, cannot be denied. D-1-2: Therefore, in fact, with reference to Cartesian coordinate system we can only draw one straight line, not even two, and using it we cannot draw any curve. D-1-3: The problem in coordinate geometry is clear; Since an axis of coordinate must show length, it automatically shows time: we cannot avoid it or deny this fact. D-1-4: The alternative is disastrous: If an axis of coordinate does not show time it cannot show length! It means it cannot show position on that coordinate! or as the function of Y, or how Y/X varies as the function of space or time. F: If Y/X is a variable then it is not possible to develop universally applicable law (by convincing reasoning) that would correlate Y and X, when Y/X is variable. G: Therefore Y=F(X) should always give a straight line, no curvature! H: Y=F(X) seems to generate different curves in coordinate geometry but in fact the expression fails to prove that Y/X is a continuous variable the way the curve shows! I: If Y/X (or L/T) is a continuous variable we cannot develop any expression or formula to show Y/X varies as the function X or Y or as the function of distance or time. J: Whether X and Y are discrete-countable entities or continuous-uncountable things, in Y=F(X), we cannot designate one of them, say, X as independent variable and the other, Y, as dependent variable; both are equally dependent on each other. K: The fact is, in all Y=F(X) type of relation between X and Y, where X is supposedly independent variable and Y the dependent variable, if Y=F(X) is not a straight line then communicated expression can never fill the spatio-temporal gap between (x1,y1) and (x2,y2). This is the duration of state of change or duration of space and time where we hide all our ignorance! If Y is a variable dependent on X then Y only follows one rule: Y/X=constant. And if Y/X is a continuous variable then XY=constant! But it is impossible to demonstrate that both X and Y are continuous in XY=1. === Subject: Re: When Y is a variable dependent on X then - - (Part-1) You MUST stop smoking that stuff! >WHEN X IS AN INDEPENDENT VARIABLE IT FOLLOWS NO RULE! >But, if Y is dependent on X then what rule does it follow? >A-1: If we want to know to what extent Y is dependent on X, then >we have to first visualize how X and Y look like. >We have two distinct options: >A-1-1: X and Y are discrete, countable and different things >like dollars and chairs. >A-1-2: X and Y are continuous ideas like length and time or >X and Y can be real fluid substances like water (W) and oil (O). >The ratios Y/X, $/ chairs, L/T and W/O, all are useful >in many calculations. >B-1: When we have to use the ratio Y/X in the form of $ per chair >then we have only one option: >B-1-1: We have to place certain number of $S in front of each chair. >It is wrong to show $-chair relation by a continuous line. >Moreover we do not need system of coordinates. >Here ratio Y /X (rate) does not fill space or time. >C-1: When we have to use the ratio Y/X in the form of >displacement/time or Water/oil then we can have three views, >only one of which is correct: >C-1-1: X fills Y, but Y does not fill X. >(Is Y a container; X content? Cannot be.) >C-1-2: X fills Y and Y also fills X. (like an emulsion or synthetic). >This is the only possibility, because both X and Y are continuous >and we cannot imagine one of them in the form of a container and >the other as the content. Therefore linear speed L/T is analogous to a >homogeneous synthetic made up of space and time; dL/dT is like a >molecule of that synthetic and time gives quantity. >On a line that shows speed, we cannot see/show time and >displacement independent of each other, they are synthesized >and are inseparable. >C-2: We cannot communicate by numerical or symbolic expression the >fact that L is completely filling T, and that >T is also completely filling L. >Therefore what we do is: >C-2-1: We do the same thing that we did in case of $-chair relation: >we place clock-readings in front of mile-stones in all >communicated expressions. >Let me repeat: we cannot show that time is filled by displacement and >displacement is filled by time. >D-1-1: The background we prepare in the form of Cartesian coordinates >shows fixed length-scales on X and Y coordinates. >Therefore coordinates also show time (scale) and each coordinate >automatically also indicates a constant linear velocity. >Thus X=V_1*T, Y=V_2T and Y/X=constant. >This fact, whether anybody likes or not, cannot be denied. >D-1-2: Therefore, in fact, with reference to Cartesian >coordinate system we can only draw one straight line, >not even two, and using it we cannot draw any curve. >D-1-3: The problem in coordinate geometry is clear; >Since an axis of coordinate must show length, >it automatically shows time: we cannot avoid it or deny this fact. >D-1-4: The alternative is disastrous: If an axis of coordinate >does not show time it cannot show length! >It means it cannot show position on that coordinate! >or as the function of Y, or how Y/X varies as >the function of space or time. >F: If Y/X is a variable then it is not possible to develop >universally applicable law (by convincing reasoning) that would >correlate Y and X, when Y/X is variable. >G: Therefore Y=F(X) should always give a straight line, no curvature! >H: Y=F(X) seems to generate different curves in coordinate geometry >but in fact the expression fails to prove that Y/X is a continuous >variable the way the curve shows! >I: If Y/X (or L/T) is a continuous variable we cannot develop any >expression or formula to show Y/X varies as the function X or Y or >as the function of distance or time. >J: Whether X and Y are discrete-countable entities or >continuous-uncountable things, in Y=F(X), we cannot designate >one of them, say, X as independent variable and the other, >Y, as dependent variable; both are equally dependent on each other. >K: The fact is, in all Y=F(X) type of relation between X and Y, >where X is supposedly independent variable and Y the >dependent variable, if Y=F(X) is not a straight line then communicated >expression can never fill the spatio-temporal gap between >(x1,y1) and (x2,y2). This is the duration of state of change >or duration of space and time where we hide all our ignorance! >If Y is a variable dependent on X then Y only follows one rule: >Y/X=constant. >And if Y/X is a continuous variable then XY=constant! >But it is impossible to demonstrate that both X and Y >are continuous in XY=1. === Subject: Re: When Y is a variable dependent on X then - - (Part-1) > You MUST stop smoking that stuff! ...or share it with the inhabitants of THIS planet. -- 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: When Y is a variable dependent on X then - - (Part-1) Very good jokes ! The best is : > And if Y/X is a continuous variable then XY=constant! V.Gopal a .8ecrit dans le message de > WHEN X IS AN INDEPENDENT VARIABLE IT FOLLOWS NO RULE! > But, if Y is dependent on X then what rule does it follow? > A-1: If we want to know to what extent Y is dependent on X, then > we have to first visualize how X and Y look like. > We have two distinct options: > A-1-1: X and Y are discrete, countable and different things > like dollars and chairs. > A-1-2: X and Y are continuous ideas like length and time or > X and Y can be real fluid substances like water (W) and oil (O). > The ratios Y/X, $/ chairs, L/T and W/O, all are useful > in many calculations. > B-1: When we have to use the ratio Y/X in the form of $ per chair > then we have only one option: > B-1-1: We have to place certain number of $S in front of each chair. > It is wrong to show $-chair relation by a continuous line. > Moreover we do not need system of coordinates. > Here ratio Y /X (rate) does not fill space or time. > C-1: When we have to use the ratio Y/X in the form of > displacement/time or Water/oil then we can have three views, > only one of which is correct: > C-1-1: X fills Y, but Y does not fill X. > (Is Y a container; X content? Cannot be.) > C-1-2: X fills Y and Y also fills X. (like an emulsion or synthetic). > This is the only possibility, because both X and Y are continuous > and we cannot imagine one of them in the form of a container and > the other as the content. Therefore linear speed L/T is analogous to a > homogeneous synthetic made up of space and time; dL/dT is like a > molecule of that synthetic and time gives quantity. > On a line that shows speed, we cannot see/show time and > displacement independent of each other, they are synthesized > and are inseparable. > C-2: We cannot communicate by numerical or symbolic expression the > fact that L is completely filling T, and that > T is also completely filling L. > Therefore what we do is: > C-2-1: We do the same thing that we did in case of $-chair relation: > we place clock-readings in front of mile-stones in all > communicated expressions. > Let me repeat: we cannot show that time is filled by displacement and > displacement is filled by time. > D-1-1: The background we prepare in the form of Cartesian coordinates > shows fixed length-scales on X and Y coordinates. > Therefore coordinates also show time (scale) and each coordinate > automatically also indicates a constant linear velocity. > Thus X=V_1*T, Y=V_2T and Y/X=constant. > This fact, whether anybody likes or not, cannot be denied. > D-1-2: Therefore, in fact, with reference to Cartesian > coordinate system we can only draw one straight line, > not even two, and using it we cannot draw any curve. > D-1-3: The problem in coordinate geometry is clear; > Since an axis of coordinate must show length, > it automatically shows time: we cannot avoid it or deny this fact. > D-1-4: The alternative is disastrous: If an axis of coordinate > does not show time it cannot show length! > It means it cannot show position on that coordinate! > or as the function of Y, or how Y/X varies as > the function of space or time. > F: If Y/X is a variable then it is not possible to develop > universally applicable law (by convincing reasoning) that would > correlate Y and X, when Y/X is variable. > G: Therefore Y=F(X) should always give a straight line, no curvature! > H: Y=F(X) seems to generate different curves in coordinate geometry > but in fact the expression fails to prove that Y/X is a continuous > variable the way the curve shows! > I: If Y/X (or L/T) is a continuous variable we cannot develop any > expression or formula to show Y/X varies as the function X or Y or > as the function of distance or time. > J: Whether X and Y are discrete-countable entities or > continuous-uncountable things, in Y=F(X), we cannot designate > one of them, say, X as independent variable and the other, > Y, as dependent variable; both are equally dependent on each other. > K: The fact is, in all Y=F(X) type of relation between X and Y, > where X is supposedly independent variable and Y the > dependent variable, if Y=F(X) is not a straight line then communicated > expression can never fill the spatio-temporal gap between > (x1,y1) and (x2,y2). This is the duration of state of change > or duration of space and time where we hide all our ignorance! > If Y is a variable dependent on X then Y only follows one rule: > Y/X=constant. > And if Y/X is a continuous variable then XY=constant! > But it is impossible to demonstrate that both X and Y > are continuous in XY=1. === Subject: Re: When Y is a variable dependent on X then - - (Part-1) Taking drugs is illegal. But taking too many drugs is stupid. ;-) Friendly, Kostas. > You MUST stop smoking that stuff! >WHEN X IS AN INDEPENDENT VARIABLE IT FOLLOWS NO RULE! >But, if Y is dependent on X then what rule does it follow? >A-1: If we want to know to what extent Y is dependent on X, then >we have to first visualize how X and Y look like. >We have two distinct options: >A-1-1: X and Y are discrete, countable and different things >like dollars and chairs. >A-1-2: X and Y are continuous ideas like length and time or >X and Y can be real fluid substances like water (W) and oil (O). >The ratios Y/X, $/ chairs, L/T and W/O, all are useful >in many calculations. >B-1: When we have to use the ratio Y/X in the form of $ per chair >then we have only one option: >B-1-1: We have to place certain number of $S in front of each chair. >It is wrong to show $-chair relation by a continuous line. >Moreover we do not need system of coordinates. >Here ratio Y /X (rate) does not fill space or time. >C-1: When we have to use the ratio Y/X in the form of >displacement/time or Water/oil then we can have three views, >only one of which is correct: >C-1-1: X fills Y, but Y does not fill X. >(Is Y a container; X content? Cannot be.) >C-1-2: X fills Y and Y also fills X. (like an emulsion or synthetic). >This is the only possibility, because both X and Y are continuous >and we cannot imagine one of them in the form of a container and >the other as the content. Therefore linear speed L/T is analogous to a >homogeneous synthetic made up of space and time; dL/dT is like a >molecule of that synthetic and time gives quantity. >On a line that shows speed, we cannot see/show time and >displacement independent of each other, they are synthesized >and are inseparable. >C-2: We cannot communicate by numerical or symbolic expression the >fact that L is completely filling T, and that >T is also completely filling L. >Therefore what we do is: >C-2-1: We do the same thing that we did in case of $-chair relation: >we place clock-readings in front of mile-stones in all >communicated expressions. >Let me repeat: we cannot show that time is filled by displacement and >displacement is filled by time. >D-1-1: The background we prepare in the form of Cartesian coordinates >shows fixed length-scales on X and Y coordinates. >Therefore coordinates also show time (scale) and each coordinate >automatically also indicates a constant linear velocity. >Thus X=V_1*T, Y=V_2T and Y/X=constant. >This fact, whether anybody likes or not, cannot be denied. >D-1-2: Therefore, in fact, with reference to Cartesian >coordinate system we can only draw one straight line, >not even two, and using it we cannot draw any curve. >D-1-3: The problem in coordinate geometry is clear; >Since an axis of coordinate must show length, >it automatically shows time: we cannot avoid it or deny this fact. >D-1-4: The alternative is disastrous: If an axis of coordinate >does not show time it cannot show length! >It means it cannot show position on that coordinate! >or as the function of Y, or how Y/X varies as >the function of space or time. >F: If Y/X is a variable then it is not possible to develop >universally applicable law (by convincing reasoning) that would >correlate Y and X, when Y/X is variable. >G: Therefore Y=F(X) should always give a straight line, no curvature! >H: Y=F(X) seems to generate different curves in coordinate geometry >but in fact the expression fails to prove that Y/X is a continuous >variable the way the curve shows! >I: If Y/X (or L/T) is a continuous variable we cannot develop any >expression or formula to show Y/X varies as the function X or Y or >as the function of distance or time. >J: Whether X and Y are discrete-countable entities or >continuous-uncountable things, in Y=F(X), we cannot designate >one of them, say, X as independent variable and the other, >Y, as dependent variable; both are equally dependent on each other. >K: The fact is, in all Y=F(X) type of relation between X and Y, >where X is supposedly independent variable and Y the >dependent variable, if Y=F(X) is not a straight line then communicated >expression can never fill the spatio-temporal gap between >(x1,y1) and (x2,y2). This is the duration of state of change >or duration of space and time where we hide all our ignorance! >If Y is a variable dependent on X then Y only follows one rule: >Y/X=constant. >And if Y/X is a continuous variable then XY=constant! >But it is impossible to demonstrate that both X and Y >are continuous in XY=1. === Subject: Re: When Y is a variable dependent on X then - - (Part-1) > WHEN X IS AN INDEPENDENT VARIABLE IT FOLLOWS NO RULE! > But, if Y is dependent on X then what rule does it follow? Obviously Y is whatever function of X it is defined to be. That is Y=f(X) where f(X) is whatever is specified. Stating the obvious, are you posting while under the influence of booze or drugs, or simply very, very stupid? Either way, you really shouldn't be posting your babble in sci.physics. Harry C. === Subject: Re: When Y is a variable dependent on X then - - (Part-1) <3f3a8136.1682503@netnews.worldnet.att.net You MUST stop smoking that stuff! However if you had actually inhaled, then you would have realised that when Y is a variable dependent on X, that X may declare Y a dependent. Just inquire of Author-Anderson, they'll advise you to use such. === Subject: Why... I'm taking math classes again at U. of California at Berkeley, after a 12-year hiatus. One thing I noticed is that the pace of their math classes seems to have doubled -- when I took Math IB, we didn't cover differential equations at all. Now they cover half of differential equations in that class, and squeeze linear algebra/differential equations into one semester (used to be two separate courses). I haven't heard any reports of student IQ's doubling in the past decade -- have I missed something? Is there a similar phenomenon at other schools? === Subject: Re: Why... >I'm taking math classes again at U. of California at Berkeley, after a >12-year hiatus. One thing I noticed is that the pace of their math classes >seems to have doubled -- when I took Math IB, we didn't cover differential >equations at all. Now they cover half of differential equations in that >class, and squeeze linear algebra/differential equations into one semester >(used to be two separate courses). I haven't heard any reports of student >IQ's doubling in the past decade -- have I missed something? Is there a >similar phenomenon at other schools? There is constant pressure to pack extra material into courses. This is especially pronounced in courses we teach for engineering. The result is usually that you see more topics, but nothing is done in much depth. That being said, linear algebra and differential equations are rather natural candidates to be joined together: e.g. a very important application of eigenvalues and eigenvectors, which are often the climax of the first linear algebra course, is in differential equations. If you do these courses separately, the instructor of the DE course often has to spend a lot of time going over what the students should have learned in linear algebra. 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: Why... >>I'm taking math classes again at U. of California at Berkeley, after a >>12-year hiatus. One thing I noticed is that the pace of their math classes >>seems to have doubled -- when I took Math IB, we didn't cover differential >>equations at all. Now they cover half of differential equations in that >>class, and squeeze linear algebra/differential equations into one semester >>(used to be two separate courses). I haven't heard any reports of student >>IQ's doubling in the past decade -- have I missed something? Is there a >>similar phenomenon at other schools? >There is constant pressure to pack extra material into courses. Elsewhere (I regret to report) there is constant pressure to remove material from courses (because they're too hard with all that material in them). There's no way to win, it seems. Lee Rudolph === Subject: Re: Why... >I'm taking math classes again at U. of California at Berkeley, after a >12-year hiatus. One thing I noticed is that the pace of their math classes >seems to have doubled -- when I took Math IB, we didn't cover differential >equations at all. Now they cover half of differential equations in that >class, and squeeze linear algebra/differential equations into one semester >(used to be two separate courses). I haven't heard any reports of student >IQ's doubling in the past decade -- have I missed something? Is there a >similar phenomenon at other schools? >>There is constant pressure to pack extra material into courses. >Elsewhere (I regret to report) there is constant pressure to remove >material from courses (because they're too hard with all that >material in them). I know a place where there's constant pressure to do both. (One would _think_ that it would be pressure from different parties...) >There's no way to win, it seems. >Lee Rudolph ************************ David C. Ullrich === Subject: Re: Why... >Elsewhere (I regret to report) there is constant pressure to remove >material from courses (because they're too hard with all that >material in them). > I know a place where there's constant pressure to do both. Like that referee report I got the other day. It complimented me on keeping the paper short. Then he wanted more material, and the paper shortened. Right. V. mail me at lastname at cs utk edu === Subject: Re: Why... >>Elsewhere (I regret to report) there is constant pressure to remove >>material from courses (because they're too hard with all that >>material in them). >> I know a place where there's constant pressure to do both. >Like that referee report I got the other day. It complimented me on >keeping the paper short. Then he wanted more material, and the paper >shortened. Right. Lv t th vwls. Nd th vrbs. Rdndnt! LR === Subject: Re: Why... , abstract > I'm taking math classes again at U. of California at Berkeley, after a > 12-year hiatus. One thing I noticed is that the pace of their math classes > seems to have doubled -- when I took Math IB, we didn't cover differential > equations at all. Now they cover half of differential equations in that > class, and squeeze linear algebra/differential equations into one semester > (used to be two separate courses). I haven't heard any reports of student > IQ's doubling in the past decade -- have I missed something? Is there a > similar phenomenon at other schools? Well, perhaps this is dictated by engineering and science departments. They want their students to take more hours of their own courses, leaving less time for everything else. But, of course, they still want the math courses (fewer of them) to cover the same material as before.