mm-1729 I am writing a computer program to compute the smallest Primitive Root g of a positive integer n, g = spr(n). See http://mathworld.wolfram.com/PrimitiveRoot.html . In testing the program I have noticed that for p an odd prime spr(p^k) = spr(p) and spr(2*p^k) = spr(2*p) for all integers k > 0. For example spr(41) = spr(41^2) = spr(41^3) ... = 6. I check some values in Pari gp and got agreement: (06:14) gp > znprimroot(41) %14 = Mod(6, 41) (11:23) gp > znprimroot(41^2) %15 = Mod(6, 1681) (11:23) gp > znprimroot(41^3) %16 = Mod(6, 68921) (11:24) gp > znprimroot(41^4) %17 = Mod(6, 2825761) (11:24) gp > znprimroot(41^5) %18 = Mod(6, 115856201) Pari might not be the best checker for this since the Pari znprimroot function does not always return the smallest primitive root. Is this a known fact or can it be proven true or false? -Harry http://www.geocities.com/hjsmithh/ === Subject: Re: Smallest Primitive Roots > I am writing a computer program to compute the smallest Primitive > Root g of a positive integer n, g = spr(n). See > http://mathworld.wolfram.com/PrimitiveRoot.html . > In testing the program I have noticed that for p an odd prime > spr(p^k) = spr(p) > and > spr(2*p^k) = spr(2*p) > for all integers k > 0. For example > spr(41) = spr(41^2) = spr(41^3) ... = 6. > Is this a known fact or can it be proven true or false? See E. L. Litver, G. E. Judina, Primitive roots for the first million primes and their powers (Russian), Mathematical analysis and its applications, Vol. III (1971) 106--109. The review by J. B. Roberts (MR 49 #4915) says, ...the authors have shown that with the single exception of 40487 all primes up to 1001321 have a least positive primitive root that is also a primitive root of the square of the prime.... However 5, which is a primitive root of 40487, satisfies 5^{40486} = 1 mod{40487^2}. Since Roberts' review, other people have gone up to 4000000 without finding another example. On the other hand, I think it's easy to prove that if g is a primitive root for p and p^2 then t is a primitive root for p^k for all k. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Smallest Primitive Roots >> I am writing a computer program to compute the smallest >> Primitive >> Root g of a positive integer n, g = spr(n). See >> http://mathworld.wolfram.com/PrimitiveRoot.html . >> In testing the program I have noticed that for p an odd prime >> spr(p^k) = spr(p) >> and >> spr(2*p^k) = spr(2*p) >> for all integers k > 0. For example >> spr(41) = spr(41^2) = spr(41^3) ... = 6. >> Is this a known fact or can it be proven true or false? > See E. L. Litver, G. E. Judina, Primitive roots for the first > million > primes and their powers (Russian), Mathematical analysis and its > applications, Vol. III (1971) 106--109. > The review by J. B. Roberts (MR 49 #4915) says, > ...the authors have shown that with the single exception of > 40487 all > primes up to 1001321 have a least positive primitive root that > is also > a primitive root of the square of the prime.... However 5, which > is a > primitive root of 40487, satisfies 5^{40486} = 1 mod{40487^2}. > Since Roberts' review, other people have gone up to 4000000 > without finding another example. > On the other hand, I think it's easy to prove that if g is > a primitive root for p and p^2 then t is a primitive root > for p^k for all k. > -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) So it is proven false, that's what I needed. spr(40487) = 5 spr(40487^2) = 10 But spr(40487^3) = 10, spr(40487^4) = 10, spr(40487^5) = 10, spr(40487^6) = 10, spr(40487^7) = 10, spr(40487^8) = 10, spr(40487^9) = 10, spr(40487^10) = 10 So maybe spr(p^k) = spr(p^2) for k >= 2. If so this would speed up the calculation of spr(p^k) for k > 2. And maybe spr(2*p^k) = spr(2*p^2) for k >= 2. -Harry === Subject: finding the reason for why the integral gets bigger hi i used excel to find the area under a graph instead of integrating the function the analytical way. These are the different answers i got with different width of the x-values, the smaller the x-increment gets the more precise the area is when compared to the answer obtained the numerical way. I have trouble seeing why the answers i get is getting bigger and bigger and not smaller can someone plz explain? Area under curve (in area units): .8ax 1 11.2187436 0.5 11.4302568 0.3 11.5484789 0.25 11.6127904 0.1 11.7361791 0.05 11.7793645 0.01 11.8146327 0.005 11.8190861 0.001 11.8226559 0.0005 11.8231026 0.0001 11.8234600 0.00005 11.8235046 Please can someone help me soon! *-----------------------* www.GroupSrv.com *-----------------------* === Subject: Re: finding the reason for why the integral gets bigger >hi >i used excel to find the area under a graph instead of integrating the >function the analytical way. These are the different answers i got >with different width of the x-values, the smaller the x-increment >gets the more precise the area is when compared to the answer >obtained the numerical way. I have trouble seeing why the answers i >get is getting bigger and bigger and not smaller can someone plz >explain? Perhaps you are computing lower sum approximations to the integral, which increase toward the correct answer as your refinement gets better. --Lynn === Subject: Re: finding the reason for why the integral gets bigger It would help to know what the function is. === Subject: sigma-algebra generated by compact sets Hello I read (I didnt know)nthat, if a metric space S is separable and locally compact, them the sigma-algebra generated by its compact sets is its Borel sigma-algebra. My proof for this fact goes as follows: Since S is separable, it contains a set D = {x_1, x_2...x_n...} that is countable and dense. Since S is locally compact, to each x_n we can associate a neighborhood B_n whose closure B'_n is compact. Then, {B_n} is a countable topological basis for S and {B'_n} is a countable cover of S by compact sets. If C is aa closed subset of S, then C = Union(B'_n inter C). Since each B'_n is compact and C is closed, for every n (B'_n inter C) is compact. It follows every closed subset of S is a countable union of compact sets. According to the definition of sigma-algebra, this implies the sigma-algebra generated in S by its compact sets must contain the sigma-algebra generated by its closed sets, which is precisely its Borel sigma-algebra. On the other hand, every compact set is closed and, which implies the Borel sigma-algebra contains the one generated by the compact sets of S. Therefore, the sigma-algebra generated by the compact sets of S is precisely its Borel sigma-algebra. This is OK, right? Can we extend such conclusion to separable and locally compact Hausdorff spaces? The proof I gave doesn't work, because in non metric spaces separability doesn't imply the existence of a countable topological basis. Artur === Subject: You'd definatly appreciate this, sci.math What is the smallest prime greater than Grahm's number? Poly-poly man === Subject: Re: You'd definatly appreciate this, sci.math Poly-Poly Man a .8ecrit : > What is the smallest prime greater than Grahm's number? > Poly-poly man Well, the probability it is G+2 is *very* low, but I dont see offhand any way to prove it is not... === Subject: principal ideal domains of finite characteristic I've been greatly enjoying reading Milne's on--line notes on algebraic number theory. I've just read the proof that the ring of algebraic numbers is finitely generated over the ring of usual integers (Z). Milne proves this is a more general case, assuming instead of Z a integrally closed domain. As Milne point out, the proof requires the assumption that the extension is separable. This is of course not a problem for Z, but if instead of Z one has a ring A, one presumably could get a different result. So I would like to see a suitable ring A (an integrally closed domain and a principal ideal domain) over which there is an field of finite dimension (relative to the fraction field of A) but such that the ring of algebraic integers is not finitely generated as an A module. Thus A must be have finite characteristic (not 0). Further, if A=k[X], then Milne assures the reader that the ring of algebraic integers will indeed be finitely generated. Thus, where might I find examples of prinicipal idea rings of non-zero characteristic other than the fields of finite characteristic k and the polynomial ring k[X]? Mike Albert === Subject: Re: principal ideal domains of finite characteristic > I've been greatly enjoying reading Milne's on--line notes > on algebraic number theory. I've just read the proof > that the ring of algebraic numbers is finitely generated > over the ring of usual integers (Z). Milne proves this > is a more general case, assuming instead of Z a > integrally closed domain. As Milne point out, the > proof requires the assumption that the extension is separable. > This is of course not a problem for Z, but if instead > of Z one has a ring A, one presumably could get a different > result. > So I would like to see a suitable ring A (an integrally closed > domain and a principal ideal domain) over which there is > an field of finite dimension (relative to the fraction field of A) > but such that the ring of algebraic integers is not finitely > generated as an A module. > Thus A must be have finite characteristic (not 0). Further, > if A=k[X], then Milne assures the reader that the ring of algebraic > integers will indeed be finitely generated. > Thus, where might I find examples of prinicipal idea rings of > non-zero characteristic other than the fields > of finite characteristic k and the polynomial ring k[X]? See below ---------------------------------------------------------------------------- -- 13:113i 10.0X Artin, E. Questions de base minimale dans la theorie des nombres algebriques. Algebre et Theorie des Nombres. pp. 19-20. Colloques Internationaux du Centre National de la Recherche Scientifique, no. 24, Centre National de la Recherche Scientifique, Paris, 1950. ---------------------------------------------------------------------------- -- Let o be a Dedekind ring, F its field of quotients, E a finite algebraic extension field of F . Denote by Q the subring of E consisting of the elements which are integral over o . It is well known that Q is a Dedekind ring, that E is the field of quotients of Q , and that, if E/F is separable, Q is finitely generated as a o -module. It is known that the last property does not always obtain in the non-separable case. The author gives the following simple example for this: Let k be a perfect field of characteristic 2, and let y be a formal power series in one variable, x , with coefficients in k which is transcendental over the field k(x) of rational functions of x and all whose terms are squares. Let F = k(x,y) , and let o be the subring of F which consists of the integral power series in F . Then it is shown that if E = F(surd y) the corresponding Q is not finitely generated as an o -module. In the rest of the paper a very concise exposition of the Dedekind-Steinitz- Schur-Chevalley elementary divisor theory for Dedekind rings is sketched so as to yield the structure theorem for finitely generated regular o -modules. If E/F is separable, this can be applied to the (fractional) Q -ideals of E . If A is such an ideal, one obtains a representation A = a_1 e_1(+)... (+)a_n e_n , where the a_i are fractional ideals of F relative to o, where n = [E: F], and where the e_i are F-linearly independent elements of E. By modifying the e_i one can vary the a_i 's arbitrarily, subject only to the condition that the class modulo principal ideals of their product remain fixed. The ideal (a_1...a_n)^2|e_i^(j)|^2, where the e_i^(j) are the conjugates of the e_i relative to F , is then independent of the choice of the e_i and is called the discriminant D(A) of A . By localization one proves that D(A) = N_{E/F}(A)^2 D(Q), whence a_1...a_n = N_{E/F}(A)rootof{D(A)/|e_i^(j)|^2} . In particular, one sees from this that a necessary and sufficient condition for the existence of a minimal basis for Q over o (i.e., for Q to be a free o-module of rank n) is that the o-ideal rootof{D(Q)Delta^-1} of F be a principal ideal, where Delta is the discriminant of a defining equation for E/F . Reviewed by G. Hochschild ---------------------------------------------------------------------------- -- 53:386 13B20 (12F10) Butts, H. S.; Yeagy, R. W. Finite bases for integral closures. J. Reine Angew. Math. 282 (1976), 114-125. ---------------------------------------------------------------------------- -- Let R be an integrally closed commutative domain, Let an ordinary integral closure of R mean an integral closure of R in some finite separable algebraic extension of the quotient field of R . This paper discusses when an ordinary integral closure of R is a finite R-module. The final section gives a general criterion under which a finite basis exists, reminiscent of the classical situation in algebraic number theory. However, the bulk of the paper deals with the case when R is itself the integral closure of a domain R_1 in a countably infinite separable algebraic extension of the quotient field of R_1 . The main theorem states that for such R , if R_1 is a Dedekind domain all of whose residue fields are perfect, and if R' is an ordinary integral closure of R , then R' is a finite R-module if and only if every critical maximal ideal of R is unramified in R' . Here the maximal prime P< R is called critical if each finite subset of P is contained in the square of some maximal ideal of R . The authors use this theorem to give examples of two non-Noetherian almost Dedekind domains, one for which every ordinary integral closure is a finite R-module, and one which has an ordinary integral closure which is not a finite R -module. Reviewed by Stephen McAdam ---------------------------------------------------------------------------- -- Excerpted from I. Kaplansky: Commutative Rings, pp. 67-71 We consider the following setup: R is an integrally closed domain with quotient field K, L is an algebraic extension of K, and T is the integral closure of R in L (the ring of R-integers in L). It is an easy exercise that T has quotient field L. (A little more is true: any element of L is expressible as a quotient with numerator in T and denominator in R.) The following little picture may help visualize the relationships. R < T : : K < L The classical version of the problem is the case where R is the ring of integers, K the field of rational numbers, and L a finite algebraic extension of K (i. e. an algebraic number field). More generally, let R be a principal ideal domain and [L:K] finite. It was a great discovery of the 19th century that T need not be a principal ideal domain, but that it is a Dedekind ring. Having gone that far, we might as well let R be a Dedekind ring. THEOREM 98. Let R be a Dedekind ring with quotient field K. Let L be a field finite-dimensional over K, and let T be the integral closure of R in L. Then T is a Dedekind ring. Proof. We head for statement (2) of Theorem 96. T is integrally closed by Theorem 40, and has dimension <= 1 by Theorem 48. It remains to argue that T is Noetherian. We can find a vector space basis of L over K consisting of elements of T (first take any basis, then multiply by suitable elements of R to throw the basis into T). Say u1,..,un is the basis. Then T0 = R[ul,..,un] is Noetherian; indeed it is a finitely generated R-module. By another application of Theorem 48, dim(T0) = 1. Since T lies between T0 and its quotient field L, T is Noetherian by Theorem 93. Another method of proof is available if L is separable over K, and shows that T is a finitely generated R-module; see [51]. But it is not always true that T is a finitely generated R-module. We present,in a slightly recast version, a pertinent example of F. K. Schmidt (Theorem 100). The main change from Schmidt is to switch the point of view (a la Artin's Galois theory) from going up to going down. So we start with an integral domain T, its quotient field L, and a subfield K of L; we define R = T / K. At present we place no restrictions on the pair of fields K < L. For a number of facts that hold in this context, see Exs. 1-5. [...] THEOREM 100. Let k be a field of characteristic 2, and T = k[[x]], the power series ring in an indeterminate. With u in T, let K = k(x,u^2) and L = k(x,u) (field adjunction), R = T / K, S = T / L. Then [L:K] <= 2, R and S are discrete valuation rings with quotient fields K and L, and S is the integral closure of R in L. If [L:K] = 2, then S is not a finitely generated R-module. Remark. It is possible to arrange [L:K] = 2, for instance, by taking x and u to be algebraically independent over k, for which we need a transcendental power series u. We have a choice of a cardinal number argument when k is countable (then T has the power of the continuum) or the use of suitable gaps, a la Liouville. Proof. We illustrate the various rings and fields in the figure: R < S < T : : : k < K < L < M where M is the quotient field of T. The statement [L:K] <= 2 is obvious, since L is obtained from K by adjoining a square root of an element of K. Theorem 99 tells us that both R and S are DVR's, with quotient fields K and L, respectively. Since S contains R, has quotient field L, is integrally closed, and is integral over R (by characteristic 2), it follows that S is the integral closure of R. We now assume [L:K] = 2 and shall prove that S is not a finitely generated R-module. Suppose on the contrary that it is. Then S is spanned over R by elements ai + bi u (i=1,..,r) where ai,bi in K. (In fact, r can be 2, but there is no need for us to insist on this.) We note that any element of M can be thrown into T by multiplication by a sufficiently high power of x. It follows that any element of K can be thrown into R by multiplication by a suitable power of x. If we pick x^m to satisfy x^m ai in R and x^m bi in R for all i, we get x^m S < R + Ru. Suppose u = a0 + a1 x + a2 x^2 + ... We set v = (u - a0 - a1 x - ...) x^(-m-1) and note that v is again an element of T. (The minus signs might as well be plus signs since the characteristic is 2; however we thought the powers of x up to x^m.) Thus the expression for the element v takes the form v = (a{m+l} x^(m+l) + a{m+2} x^(m+2) + ...) x^(-m-1) making it apparent that v lies in T. Since v is also in L, we see that v in S. So x^m v in R + Ru. But the unique expression for x^m v in the form R + Ru has to have x^-1 for its coefficient of u, a contradiction since x^-1 is not in R (it is not even in T). ---------------------------------------------------------------------- Excerpted from H.C. Hutchins, Examples of Commutative Rings, pp. 92-93 Let K = Z/(p), where p is a prime integer. Let L = K(X1,X2,..).Notice that L^p = K(X1^p,X2^p,,..). Let G be the set of all fields L_a such that L^p < L_a < L and L_a is a finite algebraic extension of L^p. For each such field L_a, let R_a = L_a[[Y]]. Let T = L[[Y]]. (a) Each R_a is a Noetherian valuation domain, with maximal ideal M_a = (Y). (b) T is also a Noetherian valuation domain, with maximal ideal N = (Y). (e) Let R be the union of the domains R_a. Then R is a ring. (To add or multiply s,t in R, we have s in R_j, t in R_k; for some c, R_c contains R_j and R_k, so the sum or product of s and t lies in R_c < R.) Notice that R != T, since, for instance, the element r = X1 Y + X2 Y^2 + ... is in T but not in R. (d) In fact, R is a Noetherian valuation domain, with maximal ideal M = (Y) and residue field F = union of the fields L_a. To see this, observe that T^p < R_a < R < T, and T is obviously integral over T^p, so T is integral over R and over each ring R_a. Thus R is quasi-local and 1-dimensional, by inheritance from T. The maximal ideal M is the union of the maximal ideals M_a of the rings R_a; each M_a = Y R_a; so M = Y (the union of the rings R_a) = Y R = (Y). The other parts of the statement are similarly easy to see. (e) Let L1, be the quotient field of R, and let R1, be the integral closure of R in L1(r), for r as in (c). Notice that r in R1, so the quotient field of R1 is L1(r). L1(r) is a finite algebraic extension of L1. (f) In fact, R1 is a Noetherian valuation domain, with maximal ideal (Y). Why? Since T is integral over R and R < R1 < T, T is also integral over R1. Therefore, R1 is 1-dimensional and quasi-local. It is integrally closed by definition. Suppose M, is the maximal ideal of R1 . Then if g in M1, g in N, so g = Yf for some f in T. Then f = g/Y in L1(r). Since f in T, f is integral over R_1 so f in R1 since R1 is integrally closed. Hence M1 = Y R1, and R1 is actually Noetherian. (g) R1 is not a finitely generated R-module. To see this, notice that R1 = R + Y R1, and Y is in the Jacobson radical of R. If R, were a finitely generated R-module, Nakayama's lemma would imply that R = R1, which is a contradiction. (h) The whole point of this example is in (g); that R1 is the integral closure of R, a domain, in a finite algebraic extension of the quotient field of R, and yet R1 is not a finitely generated R-module. What makes this possible is the fact that L1(t) is not a separable extension of L1. It is also singularly nice that all of the domains involved are Noetherian valuation domains, which are rather well-behaved rings, in most ways. (i) This example is from vol. 1, pp. 88-90 of M. P. Murthy, 1976. Commutative Algebra, vols. 1 and 2, Chicago, Univ. of Chicago Math. Dept. --Bill Dubuque === Subject: movie of concave convex jitterbug dharmraj's latest work at synergeo: --- In synergeo@yahoogroups.com, swdharmraj > Here is a 400K movie using JPGanimator. > http://www.users.bigpond.com/dharmraj/SpVE/SpVe.html > I took the spherical triangle from the triangular > face of the VE and hooked it up so that it pumps back > and forth between convex and concave. Then I added each > one to the corner of a tet with the four triangles facing > convex faces towards one another in the center of the > tet. Eight of these tets I assembled together in a VE. > In the center of that VE we get a convex VE jitterbugging > down to a concave oct. Outside of the center sphere we > 12 potential spheres although there in my movie there is only > two spherical triangles for each one. > There is a zip file on the above page as well as two > vrml's. Dick Fischbeck East Belfast, Maine http://www.freewebtown.com/randome/ === Subject: Re: movie of concave convex jitterbug > dharmraj's latest work at synergeo: > --- In synergeo@yahoogroups.com, swdharmraj > Here is a 400K movie using JPGanimator. > http://www.users.bigpond.com/dharmraj/SpVE/SpVe.html > I took the spherical triangle from the triangular > face of the VE and hooked it up so that it pumps back > and forth between convex and concave. Then I added each > one to the corner of a tet with the four triangles facing > convex faces towards one another in the center of the > tet. Eight of these tets I assembled together in a VE. > In the center of that VE we get a convex VE jitterbugging > down to a concave oct. Outside of the center sphere we > 12 potential spheres although there in my movie there is only > two spherical triangles for each one. > There is a zip file on the above page as well as two > vrml's. What does VE mean? -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Why we cannot compute omega You seem to have bailed out of the thread countability of reals in which I asked you the following straightforward question: ------ Keep in mind that what we CAN OBSERVE (or prove) is only a small part of what IS THE CASE. We might not be able to prove whether a given TM+input ever halts or not, but (in the abstract) it either WILL eventually halt or it WON'T. So, your initial picture of what we know about the omega sequence turns out to be accurate after all: >0.000101X1X1X1XXXX100XXX011XXX101XX00 where the 1's are at indices of TMs that we know eventually halt, the 0's we know won't ever halt; and the X's we don't yet know. But, although OUR KNOWLEDGE is incomplete, WHAT IS THE CASE is not -- each of those X's corresponds to a TM that either will eventually halt or will not. That is, for a given enumeration of TM's, its omega sequence is a single well-defined (though imperfectly known) infinite binary sequence, Right? ------ Care to try and answer it now? >BUT THE SEQUENCE OF HALT VALUES *IS* WELL DEFINED you all say! >No it's not. The sequence of halt values of all programs except the program >generating that sequence >may be well defined, but Omega is only defined from a groundless platonic >perspective Are you saying here that the binary sequence for a TM-enumeration's omega is non-constructively defined, and is therefore inadmissible? If so, then when you've been saying on and on and on that all binary sequences are computable, is that your private shorthand for all (admissible) binary sequences are computable? If so, doesn't that then immediately reduce to the vacuous all constructible binary sequences are computable?? If so, you could haved saved all of us a lot of time... -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: Why we cannot compute omega > You seem to have bailed out of the thread countability of reals in which I > asked you the following straightforward question: I posted a question after you posted and you bailed. > ------ > Keep in mind that what we CAN OBSERVE (or prove) > is only a small part of what IS THE CASE. We might not be able to > prove whether a given TM+input ever halts or not, but (in the abstract) it > either WILL eventually halt or it WON'T. So, your initial picture of what > we know about the omega sequence turns out to be accurate after all: >0.000101X1X1X1XXXX100XXX011XXX101XX00 > where the 1's are at indices of TMs that we know eventually halt, the 0's we > know won't ever halt; and the X's we don't yet know. This is really TERRIBLE language Barb. Define a function that has a composite number as a built in constant, the number is composed of 2 1000 to 2000 digit primes. The function tests number from 1, 2, 3... incrementing until it solves the factors and outputs 1 if the last digit of either prime is 3, and starts over otherwise. Will you *ever know* the halt value of such a function? of those X's corresponds to a TM that either will eventually halt or will > not. That is, for a given enumeration of TM's, its omega sequence is a > single well-defined (though imperfectly known) infinite binary sequence, > Right? > ------ > Care to try and answer it now? well defined though imperfecty known I cannot lie Barb, being Adam, so I can't say right unless you can give some other examples of this. >BUT THE SEQUENCE OF HALT VALUES *IS* WELL DEFINED you all say! >No it's not. The sequence of halt values of all programs except the program >generating that sequence >may be well defined, but Omega is only defined from a groundless platonic >perspective > Are you saying here that the binary sequence for a TM-enumeration's omega is > non-constructively defined, and is therefore inadmissible? If so, then when > you've been saying on and on and on that all binary sequences are > computable, is that your private shorthand for all (admissible) binary > sequences are computable? If so, doesn't that then immediately reduce to > the vacuous all constructible binary sequences are computable?? > If so, you could haved saved all of us a lot of time... No, ALL sequences are computable 0000.... all sequences like this 0001.... all sequences like this 0010.... all sequences like this 0011.... all sequences like this 0100.... all sequences like this 0101.... all sequences like this 0110.... all sequences like this 0111.... all sequences like this 1000.... all sequences like this 1001.... all sequences like this 1010.... all sequences like this 1011.... all sequences like this 1110.... all sequences like this 1111.... all sequences like this ALL OF THEM. When you define Omega you put a cyclic reference into your semantics, making it impossible. Like most blackboards, this blackboard holds *any* number. ********** * 5 * 6 * Sum of all numbers on the blackboard ___ * ********** the sequence of halt values is just like the sum of all numbers on the blackboard Its defined to a platonic level, but the blackboard won't ever contain that number. just like all computable sequences wont contain that sequence Omega. Herc === Subject: Re: Why we cannot compute omega > Keep in mind that what we CAN OBSERVE (or prove) > is only a small part of what IS THE CASE. We might not be able to > prove whether a given TM+input ever halts or not, but (in the abstract) it > either WILL eventually halt or it WON'T. So, your initial picture of what > we know about the omega sequence turns out to be accurate after all: >>0.000101X1X1X1XXXX100XXX011XXX101XX00 > where the 1's are at indices of TMs that we know eventually halt, the 0's > we > know won't ever halt; and the X's we don't yet know. > But, although OUR KNOWLEDGE is incomplete, WHAT IS THE CASE is not -- each > of those X's corresponds to a TM that either will eventually halt or will > not. That is, for a given enumeration of TM's, its omega sequence is a > single well-defined (though imperfectly known) infinite binary sequence, > Right? In spite of the thread's use of the name omega for what is normally called a halting function, I think your question articulates a Platonist point of view beautifully (and I'm not trying to argue for that particular viewpoint). >> The sequence of halt values of all programs except the program >>generating that sequence may be well defined, but Omega is only >>defined from a groundless platonic perspective > Are you saying here that the binary sequence for a TM-enumeration's omega > is > non-constructively defined, and is therefore inadmissible? If so, then > when > you've been saying on and on and on that all binary sequences are > computable, is that your private shorthand for all (admissible) binary > sequences are computable? If so, doesn't that then immediately reduce to > the vacuous all constructible binary sequences are computable?? > If so, you could haved saved all of us a lot of time... Agreed. But just to dispel the idea that constuctivism necessarily Again I want to mention a viewpoint not as a supporter of it, but because it seems relevant to the discussion -- namely, Constructive Recursive Mathematics, in which (as far as I can tell) uncomputable halting functions are admissible because of the acceptance of potential (but not actual) infinity: A.A. Markov [72] formulated in 1948-[CapitalEth]49 the basic ideas of constructive recursive math[CapitalEth]ematics (CRM for short). They may be summarized as follows. 1. objects of constructive mathematics are constructive objects, concretely: words in various alphabets. 2. the abstraction of potential existence is admissible but the abstraction of actual infinity is not allowed. Potential realizability means e.g. that we may regard plus as a well[CapitalEth]- defined operation for all natural numbers, since we know how to complete it for arbitrarily large numbers. 3. a precise notion of algorithm is taken as a basis (Markov chose for this his own notion of `Markov[CapitalEth] algorithm'). 4. logically compound statements have to be interpreted so as to take the preceding points into account. -- from the paper by Troelstra ... History of Constructivism in the 20th Century http://staff.science.uva.nl/~anne/hhhist.ps --r.e.s. === Subject: Re: Why we cannot compute omega Again I want to mention a viewpoint not as a supporter of it, but > because it seems relevant to the discussion -- namely, Constructive > Recursive Mathematics, in which (as far as I can tell) uncomputable > halting functions are admissible because of the acceptance of > potential (but not actual) infinity: > A.A. Markov [72] formulated in 1948-[CapitalEth]49 the basic ideas of > constructive recursive math[CapitalEth]ematics (CRM for short). They > may be summarized as follows. > 1. objects of constructive mathematics are constructive objects, > concretely: words in various alphabets. > 2. the abstraction of potential existence is admissible but the > abstraction of actual infinity is not allowed. Potential > realizability means e.g. that we may regard plus as a well[CapitalEth]- > defined operation for all natural numbers, since we know how > to complete it for arbitrarily large numbers. > 3. a precise notion of algorithm is taken as a basis (Markov > chose for this his own notion of `Markov[CapitalEth] algorithm'). > 4. logically compound statements have to be interpreted so as > to take the preceding points into account. > -- from the paper by Troelstra ... > History of Constructivism in the 20th Century > http://staff.science.uva.nl/~anne/hhhist.ps constructivism. A slight point: no theorem or proof in algorithmic information theory or kolmogorov complexity requires more than CRM as above. Although the ordinary definition of Omega itself is a real number (and I don't see how positing a random real is compatible with 2. if it's indeed compatible a short explanation would be appreciated), there is no reason why you could not fix the definitions and proofs such that was correct). -- Eray Ozkural === Subject: Re: Why we cannot compute omega >A slight point: no theorem or proof in algorithmic information theory >or kolmogorov complexity requires more than CRM as above. Although the >ordinary definition of Omega itself is a real number (and I don't see >how positing a random real is compatible with 2. if it's indeed >compatible a short explanation would be appreciated), there is no >reason why you could not fix the definitions and proofs such that >was correct). (You're referring to a Chaitin Omega, as opposed to Herc's 'omega', right?) It would be interesting to know more about the reals that are admissible in CRM, but the question goes far beyond my abilities. (Troelstra does mention that in CRM there are numbers corresponding to 'intuitionistic reals'.) --r.e.s. === Subject: Re: Why we cannot compute omega > (You're referring to a Chaitin Omega, as opposed to Herc's > 'omega', right?) ..and Daryl's, and Barb's, and Ghost's Omega and everyone else who has posted about it. o = sum 2^(-|p|) (p halts) We've all been using sum 2^-p (p halts) |p| is the size in bits of program p. That's OK, that's a closer analogy to my blackboard problem! Note: if all programs halt.. programs of size 1, 2 programs of size 2, 4 programs of size 3, 8 omega = 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 .. omega = 1 + 1 + 1 .. If he actually meant to make a probability he should have used Herc's Omega = sum (p halts) 2^ -|2p| Herc === Subject: Re: Why we cannot compute omega >> (You're referring to a Chaitin Omega, as opposed to >> Herc's 'omega', right?) > ..and Daryl's, and Barb's, and Ghost's Omega > and everyone else who has posted about it. > o = sum 2^(-|p|) [1: Omega] > (p halts) > We've all been using > sum 2^-p [2: tau ] > (p halts) The form of [1] corresponds to a Chaitin Omega. The form of [2] corresponds to your original definition, and is sometimes named 'tau'. Omega and tau are related, and both are noncomputable, but they're *not* the same; significantly, Omega -- but not tau -- is algorithmically random. This is discussed in a paper by Ord & Kieu, but I can't seem to find it online anymore ... Representations of Omega in number theory: finitude versus parity >> omega = the sequence that represents Halt(TMn, n) In this form, your omega is usually called a halting function. This function (and tau, the number you get from it by [2]) depends on the ordering of the programs/TMs -- change the ordering, and your omega changes. Chaitin Omegas don't have that property. --r.e.s. === Subject: Re: Why we cannot compute omega > Herc's Omega = sum (p halts) 2^ -|2p| > Herc sum(p halts) 2^ -2|p| Since |2p| =/= 2|p| If all programs halt, you get a probability value to reflect this. Omega = 1/4 + 1/4 + 1/16 + 1/16 + 1/16 + 1/16 + 1/64 + 1/64 ... = 1/2 + 1/4 + 1/8 ... = 1 You saw it here 1st! Herc === Subject: Re: Why we cannot compute omega > Define a blackboard that holds all significant facts. > On the blackboard will be several numbers, most likely adding to a large sum. > A significant fact is the total of the numbers on the blackboard. > This number is 'well defined' to coin a sci.math phrase, but its impossible to put it > on the blackboard! > Is there always some numbers missing from the blackboard? Some integer that the board > cannot represent? Could the board hold all numbers? Maybe it could, we have NOT > proven that the board is incapable of storing a complete set of numbers just because > 'the sum of those numbers' is 'well defined' but missing from the board. For each fact the negation of its negation is a fact, so for every number would be its negation so the sum would be zero and you could easily add a zero, which is the sum, and it's still the sum because you just added zero. C-B PS Naive Set Theory is just fine. Since an expression certainly does represent a set (intuitively and formally), then there is no expression the sets that do not contain themselves because that would be the same as the expressions that are not true of themselves, and no such expression exists (by diagonalization.) ZF et. al. just threw the baby out with the bath water. For all the latest in self-reference, paradoxes and incompleteness in Logic and Computer Science, check out: > Herc > -- > If all the girls who attended the Yale prom were laid end to end, I > wouldn't be a bit surprised. - Dorothy Parker === Subject: Re: Why we cannot compute omega > Define a blackboard that holds all significant facts. > On the blackboard will be several numbers, most likely adding to a > large sum. > A significant fact is the total of the numbers on the blackboard. > This number is 'well defined' to coin a sci.math phrase, but its > impossible to put it > on the blackboard! > Is there always some numbers missing from the blackboard? Some > integer that the board > cannot represent? Could the board hold all numbers? Maybe it could, > we have NOT > proven that the board is incapable of storing a complete set of > numbers just because > 'the sum of those numbers' is 'well defined' but missing from the > board. > For each fact the negation of its negation is a fact, so for every > number would be its negation so the sum would be zero and you could > easily add a zero, which is the sum, and it's still the sum because you > just added zero. > C-B > PS Naive Set Theory is just fine. Since an expression certainly does > represent a set (intuitively and formally), then there is no expression > the sets that do not contain themselves because that would be the > same as the expressions that are not true of themselves, and no such > expression exists (by diagonalization.) > ZF et. al. just threw the baby out with the bath water. > For all the latest in self-reference, paradoxes and incompleteness in > Logic and Computer Science, check out: It doesn't matter, the negative population of the world is NOT significant, and the problem works for ANY blackboard of numbers that happens to add to sum amount. *********************************** * 133 * * 888 25 * * Sum of numbers on this board : ____ * ********************************** FILL IN THE ANSWER ALL YOU FOR BRAINS WHO CAN'T FOLLOW LOGIC Herc === Subject: Re: Why we cannot compute omega <35liu0F4o2td3U1@individual.net FILL IN THE ANSWER ALL YOU FOR BRAINS WHO CAN'T FOLLOW LOGIC > Herc hey Truman, you psychopath, have you checked lately then you still have a significant portion of a brain up there. maybe you don't have a lot left considering how nasa is constantly bombarding you with unseen laser beams. === Subject: Re: Why we cannot compute omega 10:41 ANSWER ALL YOU FOR BRAINS WHO CAN'T FOLLOW LOGIC 10:45 > hey Truman, you psychopath, have you checked lately then you still > have a significant portion of a brain up there. maybe you don't have a > lot left considering how nasa is constantly bombarding you with unseen > laser beams. Look who I summoned? Impressive hey? (most frequencies of lasers aren't visible btw) Herc === Subject: Re: Why we cannot compute omega <35liu0F4o2td3U1@individual.net> <35ljc8F4ndavsU1@individual.net> so what percentage of a brain is left? almost zilch? no wonder you are always kicking and screaming like a fool... === Subject: Re: Why we cannot compute omega > so what percentage of a brain is left? almost zilch? no wonder you are > always kicking and screaming like a fool... atleast I don't believe in hyperinfinity and join the biggest gloss job known to human kind. all I see here is sour quackery trying to ruin well thought out mathematical proof that exposes you and other sci.math lepers as the biggest conglomerate of idiots AND the biggest noted conglomerate of idiots in history. Herc === Subject: Re: Why we cannot compute omega <35liu0F4o2td3U1@individual.net> <35ljc8F4ndavsU1@individual.net> <35ln33F4nef68U1@individual.net> funny i think most people here would say YOU are the biggest idiot in the history of sci.math. now sit still, shut up, and let the lasers continue their bombardments. === Subject: Re: Why we cannot compute omega > funny i think most people here would say YOU are the biggest idiot in > the history of sci.math. now sit still, shut up, and let the lasers > continue their bombardments. I'm the one who makes commands around here ANSWER ALL YOU FOR BRAINS WHO CAN'T FOLLOW LOGIC Herc === Subject: Re: Why we cannot compute omega <35liu0F4o2td3U1@individual.net> <35ljc8F4ndavsU1@individual.net> <35ln33F4nef68U1@individual.net> <35lns8F4nqi0nU1@individual.net> nasa tells YOU to sit still and SHUT UP and to leave the many brilliant minds in sci.math, excluding YOURS, in peace. === Subject: Re: Why we cannot compute omega > Define a blackboard that holds all significant facts. Problem: significant facts is not well-defined. Even if it is, you have not demonstrated that a blackboard can (theoretically) exist which holds all significant facts. > On the blackboard will be several numbers, most likely adding to a large sum. The sum might not exist. If there are infinitely many positive numbers it is quite possible that the sum does not exist. > A significant fact is the total of the numbers on the blackboard. > This number is 'well defined' to coin a sci.math phrase, but its impossible to put it > on the blackboard! Which suggests, simply, that your blackboard cannot exist as stated. This corresponds to the idea that in ZFC you cannot have the set of all sets. > Is there always some numbers missing from the blackboard? Some integer that the board > cannot represent? Could the board hold all numbers? Maybe it could, we have NOT > proven that the board is incapable of storing a complete set of numbers just because > 'the sum of those numbers' is 'well defined' but missing from the board. Why do you believe that the sum of those numbers is well-defined? > ------------------------------------------ > Same drill with omega, all sequences are on a computable list, but not omega. You need to revisit your drill above. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Why we cannot compute omega > Define a blackboard that holds all significant facts. > Problem: significant facts is not well-defined. Even if it is, you > have not demonstrated that a blackboard can (theoretically) exist which > holds all significant facts. > On the blackboard will be several numbers, most likely adding to a large sum. > The sum might not exist. If there are infinitely many positive numbers > it is quite possible that the sum does not exist. > A significant fact is the total of the numbers on the blackboard. > This number is 'well defined' to coin a sci.math phrase, but its impossible to put it > on the blackboard! > Which suggests, simply, that your blackboard cannot exist as stated. > This corresponds to the idea that in ZFC you cannot have the set of all > sets. > Is there always some numbers missing from the blackboard? Some integer that the board > cannot represent? Could the board hold all numbers? Maybe it could, we have NOT > proven that the board is incapable of storing a complete set of numbers just because > 'the sum of those numbers' is 'well defined' but missing from the board. > Why do you believe that the sum of those numbers is well-defined? ITS A NUMBER. YOU CAN CALCULATE IT YOU MORON > ------------------------------------------ > Same drill with omega, all sequences are on a computable list, but not omega. > You need to revisit your drill above. DONT TELL ME WHAT TO DO YOU MORON PROVE YOUR CONSTANT WHINING FOR BRAINS COMPLAINTS STOP POSTING ON MY THEADS.. THEN IGNORING THE 100 TIMES I'VE CORRECTED YOU THEN DOING IT ALL OVER AGAIN.... YOUR A HEAD WILL THE WILL OF 20 HEAD MEN Herc === Subject: Re: Why we cannot compute omega >Define a blackboard that holds all significant facts. >>Problem: significant facts is not well-defined. Even if it is, you >>have not demonstrated that a blackboard can (theoretically) exist which >>holds all significant facts. >On the blackboard will be several numbers, most likely adding to a large sum. >>The sum might not exist. If there are infinitely many positive numbers >>it is quite possible that the sum does not exist. >A significant fact is the total of the numbers on the blackboard. >This number is 'well defined' to coin a sci.math phrase, but its impossible to put it >on the blackboard! >>Which suggests, simply, that your blackboard cannot exist as stated. >>This corresponds to the idea that in ZFC you cannot have the set of all >>sets. >Is there always some numbers missing from the blackboard? Some integer that the board >cannot represent? Could the board hold all numbers? Maybe it could, we have NOT >proven that the board is incapable of storing a complete set of numbers just because >'the sum of those numbers' is 'well defined' but missing from the board. >>Why do you believe that the sum of those numbers is well-defined? > ITS A NUMBER. YOU CAN CALCULATE IT YOU MORON I could make an argument that every natural number is representative of a significant fact. The sum of the natural numbers is not a number. Therefore, it cannot be calculated. Also, as has been pointed out repeatedly, a number can be defined but not be computable (in ZFC). Chaitin's Omega is one example. >------------------------------------------ >Same drill with omega, all sequences are on a computable list, but not omega. >>You need to revisit your drill above. > DONT TELL ME WHAT TO DO YOU MORON > PROVE YOUR CONSTANT WHINING FOR BRAINS COMPLAINTS Ok, I advise you to revisit your drill above, as there are some logical errors in it. I suspect it reveals some errors in your thinking that have been carried through into your work on computability. > STOP POSTING ON MY THEADS.. THEN IGNORING THE 100 TIMES > I'VE CORRECTED YOU suffice) and I'll respond. > THEN DOING IT ALL OVER AGAIN.... YOUR A HEAD WILL > THE WILL OF 20 HEAD MEN -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Why we cannot compute omega <35linrF4ni97jU1@individual.net> Mr. Twentyman is no moron. It seem the only person who shows strange behavior on this thread is yourself. Stop shouting and swearing will you? === Subject: Re: Why we cannot compute omega > Mr. Twentyman is no moron. It seem the only person who shows strange > behavior behaviour for non-americans. > on this thread is yourself. Who? > Stop shouting and swearing will > you? Who? -- In Z, are there integers with an infinite number of digits? http://tinyurl.com/3mokx === Subject: Re: Why we cannot compute omega <35linrF4ni97jU1@individual.net> Discussion, linux) >> Mr. Twentyman is no moron. It seem the only person who shows strange >> behavior > behaviour for non-americans. Your odd fascination with American spelling would go over better if you used standard rules for capitalization, don't you think? Anyway, keep up the good fight. -- Jesse F. Hughes And hey, if you're moping and miserable because mathematics tests you, then maybe, if you think you're a mathematician, you might want to try a different field. -- Another James S. Harris self-diagnosis. === Subject: Re: Why we cannot compute omega <35linrF4ni97jU1@individual.net > Could the board hold all numbers? No. > Maybe it could, No, really, under a very small number of simple, reasonable, normal assumptions, namely 1) that the board is finite, and 2) you can't write numbers on a blackboard via things SMALLER than atoms, It STANDS PROVEN that you cannot write an infinite number of numbers on the board, and therefore that you cannot write all of them (since there ARE an infinite number of natural numbers). > we have NOT > proven that the board is incapable of storing a complete > set of numbers Yes, we have. > just because > 'the sum of those numbers' is 'well defined' No, actually, it is NOT well-defined. > but missing from the board. Right. This has nothing to do with the fact that you can't put the sum on the board and EVERYthing to do with the fact that the board is finite. It also has to do with the fact that the sum of all the natural numbers, whatEVER that MIGHT be, is NOT itself a natural number. > Why do you believe that the sum of those numbers is well-defined? > ITS A NUMBER. No, actually, it isn't. > YOU CAN CALCULATE IT YOU MORON No, actually, you canNOT calculate the sum of all natural numbers. You'd be hard-pressed to even DEFINE it, since it is going to come up different depending on what order you add them in, yet addition is supposed to be commutatitve and associative (which implies that the order does NOT matter). > ------------------------------------------ Same drill with omega, Oh, shut up, dumbass. You don't have the first clue what omega EVEN MEANS. > DONT TELL ME WHAT TO DO YOU MORON Moron: Here's what we're telling you to do: 1) sit your dumb ass down. 2) shut the up. 3) go the hell away. You are a complete waste of everybody's time. > PROVE YOUR CONSTANT WHINING WE HAVE proofs of all our results, because WE HAVE axioms FROM WHICH to prove things. You, on the other hand, just have lame paraphrases in some pidgin dialect of British that even YOU can't understand and from which NObody can prove anything, except that Herc Is Really Stupid. > STOP POSTING ON MY THEADS.. Oooooohhh, SPANK me. > THEN IGNORING THE 100 TIMES > I'VE CORRECTED YOU You aren't smart enough to correct JACK, dumbass. It is hard to believe you are even over the age of 15, the way you phrase everything so childishly. === Subject: Re: Why we cannot compute omega > just because > 'the sum of those numbers' is 'well defined' > No, actually, it is NOT well-defined. *********************************** * 133 * * 888 25 * * Sum of numbers on this board : ____ * ********************************** I know its well defined because it equals 1040 in this example, dumbass! > but missing from the board. > Right. This has nothing to do with the fact that you can't > put the sum on the board and EVERYthing to do with the fact > that the board is finite. It also has to do with the fact that don't generalise you poxhead, if you have something to say put it into a clarifiable form. THIS IS NOTHING, THIS IS EVERYTHING is just moronic. > the sum of all the natural numbers, whatEVER that MIGHT be, is NOT > itself a natural number. Why do you believe that the sum of those numbers is well-defined? > ITS A NUMBER. > No, actually, it isn't. 1040? > YOU CAN CALCULATE IT YOU MORON > No, actually, you canNOT calculate the sum of > all natural numbers. You'd be hard-pressed to even DEFINE it, > since it is going to come up different depending on what order > you add them in, yet addition is supposed to be commutatitve > and associative (which implies that the order does NOT matter). if there's a tangent you'll be on it dumbass > ------------------------------------------ Same drill with omega, > Oh, shut up, dumbass. > You don't have the first clue what omega EVEN MEANS. That's because Ive been listening to you bunch describe it for 2 years. > DONT TELL ME WHAT TO DO YOU MORON > Moron: Here's what we're telling you to do: > 1) sit your dumb ass down. > 2) shut the up. > 3) go the hell away. > You are a complete waste of everybody's time. MIND YOUR OWN ASS FANTASY BUSINESS YOU POOFTER > PROVE YOUR CONSTANT WHINING > WE HAVE proofs of all our results, because WE HAVE > axioms FROM WHICH to prove things. You, on the other Which I asked for and got nothing. I've heard this 10,000 times its a load of crap, you've got a turd up your ass george and thats all you've got. put up your logic proof, lets have a laugh at your holy grail of lies in art form. the last one i saw was 100 lines of puff i put into 5 lines which you mistook me for agreeing with and shook my hand over it. YOU CANT PROVE ME THIS HAS NO PROOF IN ANY SYSTEM INFINITE COINS OF ANTIDIAG HAVE BEEN FLIPPED Its not hard to find contradictions to all your invented mathematics. > hand, just have lame paraphrases in some pidgin dialect > of British that even YOU can't understand and from which > NObody can prove anything, except that Herc Is Really Stupid. > STOP POSTING ON MY THEADS.. > Oooooohhh, SPANK me. GETS OFF ON PROFANITY SICK GEORGE > THEN IGNORING THE 100 TIMES > I'VE CORRECTED YOU > You aren't smart enough to correct JACK, > dumbass. > It is hard to believe you are even over the age of > 15, the way you phrase everything so childishly. NOW HE'S HARD. You're a sick george greene maybe you should watch your back at UNC. Herc === Subject: Re: Why we cannot compute omega >>just because >'the sum of those numbers' is 'well defined' >>No, actually, it is NOT well-defined. > *********************************** > * 133 > * 888 25 > * Sum of numbers on this board : ____ > ********************************** > I know its well defined because it equals 1040 in this example, dumbass! 1) this assumes a finite number of numbers. 2) When you write the sum: if it is not 0, it will change the sum of the numbers on the board making the statement incorrect. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Why we cannot compute omega >>just because >'the sum of those numbers' is 'well defined' >>No, actually, it is NOT well-defined. > *********************************** > * 133 > * 888 25 > * Sum of numbers on this board : ____ > ********************************** > I know its well defined because it equals 1040 in this example, dumbass! Here's my example: ********************* * * * Sum of numbers on * * this board: 1024 * * * ********************* Now, the sum of all numbers on this blackboard is 1024, and, yup, 1024 appears on the blackboard. Seeing as my result differs from your question (and actually appears to contradict it), and seeing as you are the self-professed genius of these newsgroups -- actually, why am I hedging -- the self-professed genius of Usenet, perhaps you can explain why my example is bad, and why your's isn't. __ Arthur === Subject: Re: Why we cannot compute omega You can prove in ZFC that omega exists. Which formal system would you prefer to use? === Subject: Re: Why we cannot compute omega > You can prove in ZFC that omega exists. > Which formal system would you prefer to use? Go ahead! Herc === Subject: Re: Why we cannot compute omega <35ja4rF4kp7n5U1@individual.net You can prove in ZFC that omega exists. > Which formal system would you prefer to use? > Go ahead! > Herc Right, well, a real number is defined to be a set of rationals, nonempty, bounded above, with no maximal elements, and such that if x is in the set then every rational y less than x is in the set. We need to prove the set corresponding to omega exists. Well, we can prove Q (the set of rationals) exists and then by the power set axiom and the axiom of separation we can prove that any set of the form {x in Q|phi(x)}, where phi is a formula in the first-order language of set theory, exists. And it's easy to show that the set corresponding to omega is of this form. That's an outline of how the proof would go. === Subject: Re: Why we cannot compute omega <35ja4rF4kp7n5U1@individual.net> Well, Omega is a random number by definition like any probability, e.g. it's in (0,1). If you believe that real numbers exist, then it wouldn't be so hard to believe that Omega, too, exists :) -- Eray === Subject: Re: Why we cannot compute omega In sci.logic, examachine@gmail.com it's in (0,1). If you believe that real numbers exist, then it wouldn't > be so hard to believe that Omega, too, exists :) Pedant point: Omega is not random. It's just not computable. :-) Though we should be able to get close via various heuristics. [.sigsnip] -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Why we cannot compute omega <35ja4rF4kp7n5U1@individual.net> the corrections. Good food for thought. > In sci.logic, examachine@gmail.com > it's in (0,1). If you believe that real numbers exist, then it wouldn't > be so hard to believe that Omega, too, exists :) > Pedant point: Omega is not random. It's just not computable. :-) > Though we should be able to get close via various heuristics. To be pedantic, Omega *is* random. A real number is random iff it is not compressible. You say it's uncomputable but not random. It might not matter that it is uncomputable (because as others pointed out there are random but compressible reals). What matters is that there is absolutely no redundancy in Omega. Then, it is uncomputable, incompressible and random at the same time. That a random number is definable in some mathematical framework like information theory does not mean that it is not random. (Although the total set of definable or nameable numbers itself would be countable according to Chaitin) That a random number is semi-computable (to use the terminology of an expert in Kolmogorov complexity: Paul Vitanyi) like Omega does not mean that it is not random. [If you somehow had in mind the fact that it can be approximated from below] Omega is random by any criterion of randomness, e.g. Solovay randomness, etc. There is a theorem for that. If you don't believe me, read Chaitin's monograph and find out the proofs yourself. -- Eray === Subject: Re: Why we cannot compute omega <35ja4rF4kp7n5U1@individual.net> compressible reals). Typo: random -> uncomputable Sorry Ghost, for the second mistake in a row. -- Eray === Subject: Re: Why we cannot compute omega >> uncomputable (because as others pointed out there are random but >> compressible reals). > Typo: random -> uncomputable What next? Typo: Napoleon -> Wellington? -- In Z, are there integers with an infinite number of digits? http://tinyurl.com/3mokx === Subject: Random reals are not computable! <35ja4rF4kp7n5U1@individual.net> it's in (0,1). If you believe that real numbers exist, then it wouldn't > be so hard to believe that Omega, too, exists :) > Pedant point: Omega is not random. It's just not computable. :-) > Though we should be able to get close via various heuristics. To be pedantic, Omega *is* random. A real number is random iff it is not computable. You say it's uncomputable but not random. There are no such numbers. There is a theorem for that!!!! That a random number is definable in some mathematical framework like information theory does not mean that it is not random. That a random number is semi-computable (to use the terminology of an expert in Kolmogorov complexity: Paul Vitanyi) does not mean that it is not random. Omega is random by any criterion of randomness, e.g. Solovay randomness, etc. If you don't believe me, read Chaitin's monograph and find out the proofs yourself. -- Eray OZkural === Subject: Re: Random reals are not computable! examachine@gmail.com says... >A real number is random iff it is not computable. You say it's >uncomputable but not random. There are no such numbers. There is a >theorem for that!!! No, there isn't. Chaitin's definition of a random bit-sequence is this: An infinite bit-sequence b is random if there is a constant C such that for all n, H(b|n) > n - C. where b|n means the first n bits of b, and H(x) is the Kolmogorov complexity of x. Roughly speaking, this means that if b is random, then it is incompressible. But not all noncomputable numbers are incompressible. For example, if you take a noncomputable real number, and modify it so that every other bit is 0, then it will be compressible by a factor of 2, but it will still be uncomputable. -- Daryl McCullough Ithaca, NY === Subject: Re: Random reals are not computable! Well, okay Daryl, you have a point. In reaction to Ghost's claim that Omega is not a random number, I think I went too far. I should have said: A real number is random iff all of the initial segments of its binary expansion are algorithmically incompressible. I should have left it at that to refute the Ghost. Omega is obviously incompresible and random. Strictly speaking, I do not think it's correct to say that the two numbers are Turing-equivalent, which kind of got me confused in my statements... --- Eray === Subject: Re: Random reals are not computable! > Strictly speaking, I do not think it's correct to say that the two > numbers are Turing-equivalent, which kind of got me confused in my > statements... So what's your private definition of Turing equivalent? === Subject: Re: Random reals are not computable! numbers are Turing-equivalent, which kind of got me confused in my > statements... > So what's your private definition of Turing equivalent? Torkel, as far as I know Turing equivalent means the equivalence of a formalization of computation to the Turing Machine formalization. See for example http://www.j-paine.org/students/lectures/lect7/node13.html I'll say that any computer which is equivalent in power to a Turing machine is Turing equivalent. This covers all digital computers. It is also an upper limit on connectionist systems. Some (e.g. the linear associator) may have less power than a Turing machine, in that they can't compute something it can. But none have more power. There is no such thing as the Turing-equivalence of two numbers that I know of, so I assumed he meant something roughly like computable by similar algorithms, or perhaps he meant reducibility, e.g. wrt to the way he defined the number with less entropy than Omega. Now, it's your turn to explain what it means for two numbers to be Turing-equivalent. I won't accept a 'private definition'. URL please. -- Eray Ozkural === Subject: Re: Random reals are not computable! > There is no such thing as the Turing-equivalence of two numbers that I > know of, It means they have the same Turing degree. === Subject: Re: Random reals are not computable! know of, > It means they have the same Turing degree. http://en.wikipedia.org/wiki/Turing_degree A real number is not a subset of natural numbers. Sure, it can be represented as some subset, but strictly speaking, you are abusing the term, and I don't like that. Again, I'm asking you: what does it mean for a real number a to be Turing equivalent to a real number b? What does it mean for a real number a to have a Turing degree? I honestly didn't hear such use, and I'm expecting a clarification. I asked you guys before: do you mean reduction between two problems when you said the two numbers are Turing equivalent? Let's be precise, please. You demand maximum precision from me, and I demand the same thing from you. -- Eray === Subject: Re: Random reals are not computable! examachine@gmail.com says... >A real number is not a subset of natural numbers. Sure, it can be >represented as some subset, but strictly speaking, you are abusing the >term, and I don't like that. >Again, I'm asking you: what does it mean for a real number a to be >Turing equivalent to a real number b? If A is a set of naturals, then an oracle for A is an infinite Turing machine tape such that the symbol in square n is 1 if n is in A, and 0 otherwise. If A and B are two sets of naturals, then we say that B is Turing reducible to A if there is a two-tape Turing machine program M such that if initially the first tape contains an oracle for A, and the second tape contains the unary representation for n, then M will halt if and only if n is in B. If A is reducible to B and vice-versa, then we say that A is Turing equivalent to B. To extend this definition to reals in the interval [0,1] note that each subset N' of the naturals can be mapped to a real in [0,1] by m(N') = sum j=0 to infinity of r_j 2^{-j-1} where r_j = 1 if j is in N', = 0 otherwise. Then we can say that two reals a and b in the range [0,1] are Turing-equivalent if there are two sets A and B such that m(A) = a and m(B) = b, and A and B are equivalent. Finally, for arbitrary reals a and b, we say that they are equivalent if their fractional parts are equivalent. -- Daryl McCullough Ithaca, NY === Subject: Re: Random reals are not computable! note that each subset N' of the naturals can be mapped to a real > in [0,1] by > m(N') = sum j=0 to infinity of r_j 2^{-j-1} > where r_j = 1 if j is in N', > = 0 otherwise. > Then we can say that two reals a and b in the range [0,1] are > Turing-equivalent if there are two sets A and B such that > m(A) = a and m(B) = b, and A and B are equivalent. > Finally, for arbitrary reals a and b, we say that they are equivalent > if their fractional parts are equivalent. Ok. My worry was whether there is an agreed-upon mapping. I'd gussed the above mapping, used quite often in ait. -- Eray Ozkural PS: I really didn't know this use for reals. === Subject: Re: Random reals are not computable! > Again, I'm asking you: what does it mean for a real number a to be > Turing equivalent to a real number b? It means they have the same Turing degree. > What does it mean for a real number a to have a Turing degree? I > honestly didn't hear such use, and I'm expecting a clarification. You can easily find information on the net, if you haven't studied the subject. === Subject: Re: Random reals are not computable! Turing equivalent to a real number b? > It means they have the same Turing degree. > What does it mean for a real number a to have a Turing degree? I > honestly didn't hear such use, and I'm expecting a clarification. > You can easily find information on the net, if you haven't studied > the subject. OK, then. -- Eray === Subject: Re: Random reals are not computable! >> Again, I'm asking you: what does it mean for a real number a to be >> Turing equivalent to a real number b? >> It means they have the same Turing degree. >> What does it mean for a real number a to have a Turing degree? I >> honestly didn't hear such use, and I'm expecting a clarification. >> You can easily find information on the net, if you haven't studied >> the subject. > OK, then. On the Turing Degrees of Weakly Computable Real Numbers, by Xizhong Zheng: IMO, is too infrequent in your postings. BTW^2: Although you may find it hard to believe (given our posting history), I am interested in much of what you write. The problem, for me, is in ignoring the condescending style you commonly adopt -- and I don't think this is just my private difficulty. Of course everyone is prone to the self-inflation of ego, and I too am guilty. For what it's worth, my opinion is that it also would be much more pleasant to read the content of your postings *minus* what Robin Chapman, as I recall, likes to call mathematical imperialism -- IMO, what you have to say about digital philosophy and physicalism in mathematics can be interesting without seeming so dogmatic. --r.e.s. === Subject: Re: Random reals are not computable! history), I am interested in much of what you write. The problem, > for me, is in ignoring the condescending style you commonly adopt -- > and I don't think this is just my private difficulty. Of course > everyone is prone to the self-inflation of ego, and I too am guilty. > For what it's worth, my opinion is that it also would be much more > pleasant to read the content of your postings *minus* what Robin > Chapman, as I recall, likes to call mathematical imperialism -- > IMO, what you have to say about digital philosophy and physicalism > in mathematics can be interesting without seeming so dogmatic. It's my silliness if I came across as a mathematical imperialist and dogmatic. I'm sorry. -- Eray Ozkural === Subject: Re: Random reals are not computable! > There is no such thing as the Turing-equivalence of two numbers that I > know of, > It means they have the same Turing degree. BSc wasn't he? Herc === Subject: Re: Random reals are not computable! >> It means they have the same Turing degree. >BSc wasn't he? Cambridge awards the BA only. --- Jeff === Subject: Re: Random reals are not computable! <35ot3lF4pll8rU1@individual.net> I have an Msc in CS and I'm working towards a Phd. What degree do *you* have to be able to make the huge mistakes that you consistently do, if we are now bringing academic credibility on the table? Which is surely the way of ad hominem argument of course, and not something that I approve, but since you asked... You certainly do not sound like you have ever been involved in any degree of academic discourse. Otherwise, why should you keep on swearing at everybody? === Subject: Re: Random reals are not computable! > I have an Msc in CS and I'm working towards a Phd. > What degree do *you* have to be able to make the huge mistakes that you > consistently do, if we are now bringing academic credibility on the > table? Which is surely the way of ad hominem argument of course, and > not something that I approve, but since you asked... > You certainly do not sound like you have ever been involved in any > degree of academic discourse. Otherwise, why should you keep on > swearing at everybody? what a waste of resources that monkeys can parrot learn the formula and get a PHD these days. BInfTech University Of Queensland (1991) Grad. Diploma Teaching (Secondary) Curtin University (2000) (genius who can become anyone he wants to be) and hundreds of other heros on your TV that portray me by a cross of prediction of my life's events throughout history and the Truman company passing on the recordings of my life to world media. is to sit for a PHD in mathematics? Herc === Subject: Re: Random reals are not computable! >>I have an Msc in CS and I'm working towards a Phd. >>What degree do *you* have to be able to make the huge mistakes that you >>consistently do, if we are now bringing academic credibility on the >>table? Which is surely the way of ad hominem argument of course, and >>not something that I approve, but since you asked... >>You certainly do not sound like you have ever been involved in any >>degree of academic discourse. Otherwise, why should you keep on >>swearing at everybody? > what a waste of resources that monkeys can parrot learn the formula and > get a PHD these days. > BInfTech University Of Queensland (1991) > Grad. Diploma Teaching (Secondary) Curtin University (2000) > (genius who can become anyone he wants to be) and hundreds of other > heros on your TV that portray me by a cross of prediction of my life's events > throughout history and the Truman company passing on the recordings of > my life to world media. > is to sit for a PHD in mathematics? Funny, I got the impression that he's working on a PhD in CS. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Random reals are not computable! > I have an Msc in CS and I'm working towards a Phd. I think you may have missed a joke, I said a joke, son. === Subject: Re: Random reals are not computable! <35ot3lF4pll8rU1@individual.net> <35qck4F4osh4gU1@individual.net I have an Msc in CS and I'm working towards a Phd. > I think you may have missed a joke, I said a joke, son. Ah, yes, what a fool I've made of myself. LOL === Subject: Re: Random reals are not computable! > I have an Msc in CS and I'm working towards a Phd. credentials, credentials, credentials .... > What degree do *you* have to be able to make the huge mistakes that you Who is you again? -- In Z, are there integers with an infinite number of digits? http://tinyurl.com/3mokx === Subject: Re: Random reals are not computable! >> What degree do *you* have [...] > Who is you again? I notice you've had to ask him that more than once. Eray seems not to care about the following, which was part of one of my replies to him in a recent comp.theory thread: >> unless a reader is using Google, it may >> very well be quite difficult to see what you're responding to. For >> various reasons, the previous post may not be available, or, if it >> is available, it may be a great inconvenience to find it (possibly >> buried in a jungle of other postings). >> IMO, it's likely that a lot of readers just ignore postings which, >> lacking any quoted context, are often effectively meaningless. --r.e.s. === Subject: Re: Random reals are not computable! >> Strictly speaking, I do not think it's correct to say that the two >> numbers are Turing-equivalent, which kind of got me confused in my >> statements... > So what's your private definition of Turing equivalent? What is your private definiton of private definition? It is apparently arbitrary. === Subject: Re: Random reals are not computable! > A real number is random iff it is not computable. On which definition of computable real number? === Subject: Re: Random reals are not computable! <35ja4rF4kp7n5U1@individual.net> On which definition of computable real number? Like it differs according to whose definition. What do you have in mind? -- Eray Ozkural === Subject: Re: Random reals are not computable! >A real number is random iff it is not computable. >> On which definition of computable real number? > Like it differs according to whose definition. What do you have in > mind? Rather than trading Socratic irony with Torkel, why not respond to my specific refutation? Unless you *like* trading Socratic irony with Torkel. Hey, none of my business. === Subject: Re: Random reals are not computable! > To be pedantic, Omega *is* random. > A real number is random iff it is not computable. You say it's > uncomputable but not random. There are no such numbers. There is a > theorem for that!!!! Either you have your own private definition of random, or you've simply made an error here. Consider the real number whose decimal expansion is obtained as follows: the first digit after the decimal point is 0. The next digit is the first digit of Omega. The next digit is 0. The next digit is the second digit of Omega. And so on. Is this number random? Not according to any standard definition--asymptotically, more than half of its decimal digits are 0. No random number has that property. And yet, it's just as uncomputable as Omega (the two numbers are trivially Turing equivalent). > That a random number is definable in some mathematical framework like > information theory does not mean that it is not random. > That a random number is semi-computable (to use the terminology of an > expert in Kolmogorov complexity: Paul Vitanyi) does not mean that it is > not random. This depends on definitions. I think Omega has a Sigma^0_2 definition, which ought to imply that it's not strongly 3-random (I've given myself a margin of error of one quantifier here; I actually suspect it's not strongly 2-random but I haven't gone through the calculation). === Subject: Re: Random reals are not computable! <35ja4rF4kp7n5U1@individual.net> <35n9rpF4oj711U1@individual.net Either you have your own private definition of random, or you've > simply made an error here. Consider the real number whose decimal > expansion is obtained as follows: the first digit after the decimal > point is 0. The next digit is the first digit of Omega. The next > digit is 0. The next digit is the second digit of Omega. And > so on. > Is this number random? Not according to any standard definition--asymptotically, > more than half of its decimal digits are 0. No random number has that > property. And yet, it's just as uncomputable as Omega (the two numbers > are trivially Turing equivalent). No, I do not have a private definition of random, an unneeded accusation of yours which kind of got me carried away in my previous reply. You are right, the sentence: A real number is random iff it's uncomputable is wrong. Trying to use natural language ended with disaster. It should have been: A real number is random if and only if it has no redundancy if I really had to give a short expression for Chaitin's def'n. The main point of my post stands however: Omega is random, and random reals are NOT computable. The Ghost In The Machine was obviously wrong. (Note however, my arguments in my previous reply are not affected by the mistake which you spotted. They happen to be independent of this particular confusion) -- Eray Ozkural === Subject: Re: Random reals are not computable! > No, I do not have a private definition of random, an unneeded > accusation of yours which kind of got me carried away in my previous > reply. I apologize--I could have chosen other words. FWIW it wasn't *entirely* an accusation. Random is a notoriously slippery concept; there isn't a single agreed definition. In particular whether Omega is random depends on how strongly you're using the term. E.g. a real is said to be strongly n-random if it misses every Pi^0_n null set. (A null set is a set of zero Lebesgue measure.) Or is it Sigma^0_n? Or Pi^0_{n+1} ? I don't remember exactly, which is why I won't pin myself down on the value of n such that Omega is not strongly n-random... but there is such an n, and it's pretty small. === Subject: Re: Random reals are not computable! <35ja4rF4kp7n5U1@individual.net> <35n9rpF4oj711U1@individual.net> <35nptgF4pdhspU1@individual.net> -- Eray === Subject: Re: Random reals are not computable! <35ja4rF4kp7n5U1@individual.net> <35n9rpF4oj711U1@individual.net To be pedantic, Omega *is* random. > A real number is random iff it is not computable. You say it's > uncomputable but not random. There are no such numbers. There is a > theorem for that!!!! > Either you have your own private definition of random, or you've > simply made an error here. Consider the real number whose decimal > expansion is obtained as follows: the first digit after the decimal > point is 0. The next digit is the first digit of Omega. The next > digit is 0. The next digit is the second digit of Omega. And > so on. > Is this number random? Not according to any standard definition--asymptotically, > more than half of its decimal digits are 0. No random number has that > property. And yet, it's just as uncomputable as Omega (the two numbers > are trivially Turing equivalent). I don't think you are right when you say they are Turing equivalent. To compute n initial bits of your number OmegaH (Omega slashed in Half), you need to know only n/2 + c scattered digits of Omega, but you do not need n + c digits. I think when you said Turing equivalent, you meant something like: The procedure that approximates Omega from below *can* be used to approximate OmegaH from below. If you think about it for a minute, you will see that this does not mean they have the same entropy. By definition, a random real's k-ary representation up to nth digit is incompressible, e.g. _all_ initial segments of its binary (or decimal, or whatever) representation. On the contrary, your number is quite compressible, every initial segment of it is. That is not a random real. But yes, it still has a lot of entropy, doesn't it? (If you don't believe me, pull out a large random string from your /dev/random, then change every even-th bit to 0, and then run it through gzip. gzip the original too. look at the difference in compressed size) The common sense answer to your question is: NO, it does not have as much entropy as Omega. (Now, I'm sure that fails your set theoretic intuition, but no worries) I think you might want to say something like but it has an infinite number of 'irreducible' digits which is of course not relevant to the definition. We define algorithmic randomness in terms of the randomness of the initial segments. The definition is _not_ in terms of a set theoretic cardinality, pi in the sky, or whatever. You might want to ask this question on kolmogorov list. Your question is semi-interesting, but not too interesting :) -- Eray Ozkural === Subject: Re: Random reals are not computable! <35ja4rF4kp7n5U1@individual.net> <35n9rpF4oj711U1@individual.net> Discussion, linux) > You might want to ask this question on kolmogorov list. Your question > is semi-interesting, but not too interesting :) He didn't have a question. He had a counterexample. He gave a number which was clearly uncomputable but not random. It is possible that you're not a good judge of semi-interesting questions. You seem to confuse them with obvious counterexamples. Are you seriously claiming that Mike's number (digit 2n = 0 and digit 2n + 1 is the nth digit of Omega) is not a counterexample? You've already said it is not random. Is it computable? If not, then it is obviously a non-computable real which is not random. Hence, a counterexample to your claim that a real is random iff it is not computable. (Or maybe we read Eray incorrectly. Perhaps he's offering his new personal definition of uncomputable real.) -- Meaningless movies on the screen behind the band that's blowing Waterboys, throwing shapes My Love is My Rock Half of the music is on tape in the Weary Land === Subject: Re: Random reals are not computable! > I don't think you are right when you say they are Turing > equivalent. They are trivially Turing equivalent, and Mike's number is uncomputable but not random. You're babbling. === Subject: Re: Random reals are not computable! >> I don't think you are right when you say they are Turing >> equivalent. > They are trivially Turing equivalent, and Mike's number is >uncomputable but not random. You're babbling. Eray is your intellectual superior, Torkel. You should listen to what he says instead of posting your own typical babbling. Larry === Subject: Re: Random reals are not computable! >> To be pedantic, Omega *is* random. >> A real number is random iff it is not computable. You say it's >> uncomputable but not random. There are no such numbers. There is a >> theorem for that!!!! >Either you have your own private definition of random, or you've >simply made an error here. Feh. It's far, far likelier that examachine has its own private definition of theorem. Lee Rudolph === Subject: Re: Random reals are not computable! > To be pedantic, Omega *is* random. > A real number is random iff it is not computable. So that's what random numbers are! Can those who bought the RAND corporation's list of random numbers now get their money back? (Serious point: why introduce a potentially confusing term like random when non-computable exists and suffices?) -- In Z, are there integers with an infinite number of digits? http://tinyurl.com/3mokx === Subject: Re: Why we cannot compute omega > Well, Omega is a random number by definition like any probability, e.g. > it's in (0,1). If you believe that real numbers exist, then it wouldn't > be so hard to believe that Omega, too, exists :) BLACKBOARD ********** * 5 * 399 * 6 * ********** The total of numbers on the blackboard exists too. I CAN CALCULATE THE NUMBER ITS SITTING ON MY CALULATOR But the blackboard can't have the number written on it can it? Did you understand that far? Herc expecting some dimwit... yes but its black... type remark === Subject: Re: Why we cannot compute omega <35ja4rF4kp7n5U1@individual.net> <35kfakF4kle0cU1@individual.net Well, Omega is a random number by definition like any probability, e.g. > it's in (0,1). If you believe that real numbers exist, then it wouldn't > be so hard to believe that Omega, too, exists :) > BLACKBOARD > ********** > * 5 > * 399 > * 6 > ********** Is this a finite blackboard with integers written on it? How would I know? Try to make your argument precise. -- Eray === Subject: Re: Why we cannot compute omega > Well, Omega is a random number A random number eh? Ain't that cool! -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: Why we cannot compute omega NOW he uses English!! Herc > You can prove in ZFC that omega exists. > Which formal system would you prefer to use? Go ahead! > Herc > Right, well, a real number is defined to be a set of rationals, > nonempty, bounded above, with no maximal elements, and such that if x > is in the set then every rational y less than x is in the set. We need > to prove the set corresponding to omega exists. Well, we can prove Q > (the set of rationals) exists and then by the power set axiom and the > axiom of separation we can prove that any set of the form {x in > Q|phi(x)}, where phi is a formula in the first-order language of set > theory, exists. And it's easy to show that the set corresponding to > omega is of this form. That's an outline of how the proof would go. === Subject: Re: Why we cannot compute omega I think you don't know what the halting probability is. -- Eray === Subject: Re: Why we cannot compute omega > I think you don't know what the halting probability is. > -- > Eray What *is* the halting probability? Herc === Subject: Re: Why we cannot compute omega <35j58cF4ju9snU1@individual.net> Omega. -- Eray === Subject: Re: Why we cannot compute omega > Omega. A highly amusing post from one that holds that all of mathematics can be understood from a digital perspective, e.g. computation. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: Why we cannot compute omega > Omega. > A highly amusing post from one that holds that > all of mathematics can be understood from a digital perspective, > e.g. computation. > -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html > Elegance is an algorithm > Iain M. Banks, _The Algebraist_ Is Omega's absense a syntactic weakness in countable systems, or is it semantics as part of Omega itself? Herc === Subject: Re: Why we cannot compute omega > Omega. > -- > Eray Is the halting probability different to the defn of Omega I gave? Herc === Subject: Re: Why we cannot compute omega <35j58cF4ju9snU1@individual.net> <35jaq5F4j5iplU1@individual.net> Yes, it is different! You said: omega = the sequence that represents Halt(TMn, n) Omega is not merely the sequence that represents Halt(TMn,n) whatever that means. What would the binary expansion of the ratio of the number of halting programs up to n bits (a number that can be approached from below) over the number of all programs up to n bits (2^n) be? Surely, each bit of that expansion summarizes information about many halting problem instances, not just one as you say (Does program n terminate in n steps?). The mathematics involved is a little harder than that :) -- Eray === Subject: Re: Why we cannot compute omega > Yes, it is different! > You said: > omega = the sequence that represents Halt(TMn, n) > that means. > What would the binary expansion of the ratio of the number of halting > programs up to n bits (a number that can be approached from below) over > the number of all programs up to n bits (2^n) be? > Surely, each bit of that expansion summarizes information about many > halting problem instances, not just one as you say (Does program n > terminate in n steps?). The mathematics involved is a little harder > than that :) then not one person in sci.math has a hope in hell of following any of it. and how is this related to missing countable sequences? Herc === Subject: Re: Why we cannot compute omega <35j58cF4ju9snU1@individual.net> <35jaq5F4j5iplU1@individual.net> <35kdvaF4o09seU1@individual.net Yes, it is different! > You said: > omega = the sequence that represents Halt(TMn, n) > that means. > What would the binary expansion of the ratio of the number of halting > programs up to n bits (a number that can be approached from below) over > the number of all programs up to n bits (2^n) be? > Surely, each bit of that expansion summarizes information about many > halting problem instances, not just one as you say (Does program n > terminate in n steps?). The mathematics involved is a little harder > than that :) > then not one person in sci.math has a hope in hell of following any of it. > and how is this related to missing countable sequences? You tell me. If you are talking about Cantor's theorem, then how does it help to talk about Omega? You can use any random real if that is the property you are interested in. -- Eray === Subject: Re: Why we cannot compute omega > Yes, it is different! You said: > omega = the sequence that represents Halt(TMn, n) > parser > Omega is not merely the sequence that represents Halt(TMn,n) > whatever > that means. What would the binary expansion of the ratio of the number of > halting > programs up to n bits (a number that can be approached from below) > over > the number of all programs up to n bits (2^n) be? Surely, each bit of that expansion summarizes information about > many > halting problem instances, not just one as you say (Does program n > terminate in n steps?). The mathematics involved is a little harder > than that :) then not one person in sci.math has a hope in hell of following any > of it. > and how is this related to missing countable sequences? > You tell me. > If you are talking about Cantor's theorem, then how does it help to > talk about Omega? You can use any random real if that is the property > you are interested in. off Herc === Subject: Re: Why we cannot compute omega <35j58cF4ju9snU1@individual.net> <35jaq5F4j5iplU1@individual.net> <35kdvaF4o09seU1@individual.net> <35kekaF4msjivU1@individual.net You tell me. > If you are talking about Cantor's theorem, then how does it help to > talk about Omega? You can use any random real if that is the property > you are interested in. > off What on earth makes you think you can go on swearing everybody on the group? You act like a simpleton. Question your motives before you post your self-righteous ignorant crap. === Subject: Re: Why we cannot compute omega > What on earth makes you think you can go on swearing everybody on the > group? You act like a simpleton. Question your motives before you post > your self-righteous ignorant crap. you hypocrite. the argument is still valid given the usual defn of Omega that 1000 people here have used already. i can modify it to use the ratio version but you won't follow that either. you can tell when sci.math is reaching and is going to be petty arseholes rather than follow a sequential proof about omega. find the erronous step you canniving WOMEN, when it takes 2 days for someone to speak up then you all join in like pack dogs you know you've got no recourse Herc === Subject: Re: Why we cannot compute omega <35j58cF4ju9snU1@individual.net> <35jaq5F4j5iplU1@individual.net> <35kdvaF4o09seU1@individual.net> <35kekaF4msjivU1@individual.net> <35lj6oF4q74ogU1@individual.net What on earth makes you think you can go on swearing everybody on the > group? You act like a simpleton. Question your motives before you post > your self-righteous ignorant crap. > you hypocrite. > the argument is still valid given the usual defn of Omega that 1000 people here > have used already. i can modify it to use the ratio version but you won't follow > that either. If you give a concrete argument, I promise to follow it. Some of the stuff you said with respect to DFAs and TMs were very much precise and true. Why don't you keep on with the same level of rigor? -- Eray === Subject: Re: Why we cannot compute omega > What on earth makes you think you can go on swearing everybody on > the > group? You act like a simpleton. Question your motives before you > post > your self-righteous ignorant crap. you hypocrite. > the argument is still valid given the usual defn of Omega that 1000 > people here > have used already. i can modify it to use the ratio version but you > won't follow > that either. > If you give a concrete argument, I promise to follow it. > Some of the stuff you said with respect to DFAs and TMs were very much > precise and true. Why don't you keep on with the same level of rigor? Rubbish, this entire group is a mob of morons with parrot learnt foirmula. whenever I write a sequence of English sentences that follow logically and prove a point, they pattern match the nearest formula that rebukes the message, nothing to do with any of my statements and hammer that. There should be an iq cuttoff for sci.math, usenet isn't entirely for application of standard theory and that's what 95% of people are limited to. Herc === Subject: Re: Why we cannot compute omega In sci.logic, |-|erc <35j58cF4ju9snU1@individual.net>: >> I think you don't know what the halting probability is. >> -- >> Eray > What *is* the halting probability? > Herc It's the probability that a given machine will halt, of course. Since there's a 1-1 and onto mapping from N to the set of machines (if a number doesn't correspond to a machine, close the ranks) there are some minor issues here, but the problems are the same as with Chaitin's Omega; it is not a computable number. However, it should be a reasonably well-defined one. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Why we cannot compute omega > Since there's a 1-1 and onto mapping from N to the set of machines > (if a number doesn't correspond to a machine, close the ranks) > there are some minor issues here There are indeed issues in any attempt to define a probability measure over a countable space. Some of these programs are going to have to count for more than others. In that sense, I think it would be more accurate to talk about *a* halting probability, depending upon your choice of which ones count for how much. - Tim === Subject: Bohr Complementarity in General Relativity The Question is: What is The Question? J.A. Wheeler Meaning of the Brazilian paper below is the Gestalt Shift http://www.neurosemantics.com/Stuttering/foreground-background.htm Bohr Complementarity between teleparallel gauge force and geometrodynamical curvature alternatives. Jack, There is a good summary of tests of GR in Clifford M. Will's paper Relativity at the Century, pp. 27-32, Physics World, Jan. 2005 (a special issue on Einstein). He does not mention the Mercury perihelion advance. [BTW this is covered in great detail in *Gravitation & Inertia* by Ciufolini & Wheeler (1995)]. But the many other tests leave GR looking very good. In particular the Lunar-Laser Ranging measurements (over the last two decades and more) have confirmed not only the weak equivalence principle, but also the strong equivalence principle. To quote from Will's paper: Lunar laser-ranging measurements actually test the strong equivalence principle because they are sensitive to both the mass and the gravitational self-energy of the Earth and the Moon. The bottom line of these experiments is that bodies fall with the same acceleration to a few parts in 10^13 (p. 29). Will also points out that the Brans-Dicke Scalar-Tensor theory (and many other alternative theories) are ruled out by these measurements (p. 30). Cliff told me personally at GR 17 Dublin that he thought Hal Puthoff's PV theory is a non-starter not of interest to serious physicists in the field. So did several others including Matt Visser and Bill Unruh and Professor X. Therefore, Eric Davis, Nick Cook et-al should stop trying to sell Hal's theory to USAF and Aerospace Companies as a serious contender for metric engineering unconventional propulsion systems. That only hurts the field. BTW( to show the irrelevancy of Paul Zielinski's thesis (LC) = GCT non-tensor inertial force - GCT tensor real gravity force, i .e. inertial forces in inertial frames is the contradiction in Z's proposal!) for the record from Wheeler & Ciufolini Gravitation and Inertia Princeton 1995: Weak equivalence principle uniqueness of free fall AKA Galilei to be electrically neutral, to have negligible gravitational binding energy compared to its rest mass, to have negligible angular momentum, and to be small enough that inhomogeneities of the gravitational field within its volume have negligible effect on its motion. ... the ratio of the inertial mass to the gravitational - passive- mass is the same for all bodies ... in every local, nonrotating, freely falling frame the agreement with special relativity. Einstein generalized the weak equivalence principle to all the laws of special relativity .. that in no local freely falling frame can we detect the existence of a weak equivalence principle, or from ANY OTHER SPECIAL RELATIVISTIC PHYSICAL PHENOMENON p. 14 physical phenomenon. *The LIF is taken so small that the geodesic deviation is below the resolution of the stretch-squeeze tidal tensor curvature detector! This restriction would seem to exclude the detection of gravity waves of course. All viable theories of gravity obey this weak equivalence principle, AKA, WEP. Medium strong equivalence principle at the base of METRIC theories of gravity, AKA EEP or Einstein Equivalence Principle: for every POINTLIKE event of spacetime, there exists a sufficiently small neighborhood such that in every local freely falling frame in that neighborhood, all the non-gravitational laws of physics obey the laws of special relativity. Note EEP restricts itself to non-gravitational laws of physics. If we replace all the nongravitational laws of physics with all the laws of physics we get the very strong equivalence principle AKA, SEP which is at the base of Einstein's electrodynamics. The medium strong and the very strong form of the equivalence principle differ: the former applies to all phenomena except gravitation itself whereas the latter applies to all phenomena of nature. This means that according to the medium strong form, the existence of a gravitational field might be detected in a freely falling frame by the influence of the gravitational field on local gravitational phenomena. For example, the gravitational binding energy of a body might be imagined to contribute differently to the inertial mass and to the passive gravitational mass ... This is ... the Nordtvedt effect ... However, Lunar Laser Ranging experiment has put strong limits on the existence of any such violation of the very strong equivalence principle. The EEP (medium strong) with a locally Minkowski space time is the physical meaning of the tangent fiber of differential geometry. Therefore, you cannot do differential geometry and also claim to VIOLATE EEP. See Roger Penrose's The Road to Reality for more details on that idea. The LOCAL tetrad map guv(LNIF) = eu^aeu^bnab(LIF) is part of differential geometry formally AND of EEP informally i.e, interpretively or physically. First, the equivalence between a gravitational field and an accelerated frame in the absence of gravity, and the equivalence between a flat region of spacetime an a freely falling frame in a gravity field, has to be considered valid only locally and not globally. Note that universal inertial forces (independent of mass) like the Coriolis and the centrifugal forces only are detected in NON-INERTIAL It is meaningless in Einstein's GR to claim that in a Local Inertial inertial force (GCT non-tensor). This is what Zielinski claims and it is false. You CAN make such a claim in Newton's force picture because Newton's idea of inertial frame is not the same as Einstein's idea of inertial frame. In Newton's paradigm you CAN think of a gravity force field in a global free-falling frame that is a NON-INERTIAL frame in Newton's picture! That is, the distinction inertial/non-inertial is SWITCHED in the transition from Newton to Einstein. This is Zielinski's error. He tries to force Newton's picture INTO Einstein's. The Brazilians have shown that there is a kind of Bohr complementarity between the two pictures so long as one does not try to force the square peg of one picture into the small round hole of the dual alternative picture. Wheeler address's the issue of tidal curvature that Zielinski misinterprets as some kind of violation of Einstein's key idea for GR. Zielinski then gets conspiratorial and psychoceramic that Wheeler & Co are somehow trying to pull the wool over our eyes. Wheeler addresses the long line of wrong critics of Einstein of which Zielinski is the latest: (p. 15) Note the EEP is immune from the spherical drop issue below. What is at stake is the SEP. The SEP has been the subject of ... criticisms over the years ... the content of the strong equivalence principle has been criticized even 'locally'/ It has been argued that if one puts a spherical drop of liquid in a gravity field, after some time one would observe a tidal deformation from sphericity of the drop. Of course, this deformation does not arise in a flat region of spacetime ... No matter if we are freely falling or not, the gradiometer will eventually detect the gravity field and thus allow us to distinguish between the freely falling cabin of a spacecraft in the gravity field of a central mass and the cabin of a spacedraft away from any mass, in a region of spacetime essentially flat. Then, may we still consider the STRONG EQUIVALENCE PRINCIPLE (SEP) to be valid? Wheeler then gives the Taylor expansion of the metric in the NEIGHBORHOOD {P'} of a spacetime event P that I have given many times before. The simple solution to the problem that Zielinski has magnified to excess is The Riemann curvature tensor represents at each point, the INTRINSIC CURVATURE of the manifold, and, since it is a tensor, one cannon transform it to zero in one coordinate system if it is non-zero in some other coordinate system. ... The metric tensor can indeed be written using the Riemann tensor Rijkl, in a NEIGHBORHOOD of a spacetime event, in a freely falling, nonrotating, local inertial frame to SECOND ORDER in the separation (P' - P) goo ~ - 1 - R0i0j(P'-P)^i(P'-P)^j Note that in the weak curvature slow speed Newtonian force limit of GR goo ~ -[ 1 + V(Newton)/c^2] V(Newton) = Universal Newtonian Gravity Potential Energy per unit mass Therefore in the LIF: V(Newton)/c^2 ~ R0i0j(P'-P)^i(P'-P)^j Note Rijkl has dimension 1/Area (P' - P) has dimension Length Furthermore, the post-Newtonian GRAVIMAGNETIC FIELD g0k of NASA experiments and used by Ray Chiao of UCB in his gravity radio idea for submarine warfare C^3 and efficent superconducting gravity wave detectors, is in Fermi Normal Coordinates g0k = -(2/3)Roikj(P'-P)^i(P'-P)^j k = 1,2,3 with electro-gravitic coupling ~ g0kA^k A^k is EM vector potential in p = mv - (e/c)A (3-vector) In NEAR FIELD this may allow a geodesic glider WEIGHTLESS WARP DRIVE for metric engineering! Ray Chiao only uses FAR FIELD. And finally, i,j,k,l = 1,2,3 gkl = (Kronecker Delta)kl - (1/3)Rkilj(P'-P)^i(P'-P)^j Zielinski ignores the levels of approximation of perturbation theory and, therefore, formulates a pseudo-problem. They do point out that the gauge representation is able to handle theories in which certain aspects of EP are violated i.e. mg =/= mi, [Z] That is not the point. They allude to the dissident literature on the equivalence principle in support of their teleparallel alternative to the standard theory. This amounts to a critical argument against the orthodox view. So you still don't understand what is meant by a mathematical decomposition of the LC connection into tensor and non-tensor parts? [S] I understand what it means. It is false in Einstein's 1916 GR geometrodynamic representation where {LC} = non-GCT tensor It has no tensor part at all. [Z] I have simply pointed out that the Einstein equivalence hypothesis, as classically stated by Einstein himself, is not necessary for Einstein geometrodynamics. [S] Be specific. What words by Einstein? [Z] I have given you direct quotes from Einstein any number of times. [S] All you do is cite a possibly BAD English translation. You do not read the original German and you do not know what Einstein really said here at all! Have you checked with a German speaking physicist in the field? NO! Your theory is based on quicksand. [Z] Except that the Brazilian paper shows how, mathematically, you can do exactly what I have been proposing within the teleparallel framework. [S] I deny that, nor do the Brazilians claim there is any conclusive experimental evidence for such theories that violate SEP. [Z] But at least the differences are testable in principle. [S] Yes, but they have all so far been falsified as mentioned by Sirag above. Puthoff's PV is such a theory and it does not agree with experiment beyond the 3 trivial weak field limit classic tests. PV fails to predict gravimagnetism now observed, it has no event horizons pretty much now observed, it fails to give correct 1913-16 pulsar curve off by what 1/3? When 1916 GR is on target to 10^-14 for which a Nobel Prize was given. At least Hal has a real theory. It happens to be wrong and confused in its foundations, but at least it is definite enough to be falsified and it has. Hal can at least calculate as you so far have not been able to do. [Z] So you wanted to talk about the Schwarzschild solution instead? [S] Yes, until you can solve that you have nothing of interest. Vilenken's vacuum wall has nothing to do with your claims. It is an entirely different problem. It has to do with a NONLOCAL Bohm-Aharonov closed non-exact 1-form in the connection from non-trivial topology of the vacuum coherence order parameter that is the fabric of spacetime. This also explains the NASA Pioneer 10&11 anomaly a_g = - cH as a hedgehog defect centered on Sun (and maybe ALL STARS). What is important in the Brazilian model for me, so far, is that their gauge potential theory representation is close to my macro-quantum theory for emergence of gravity from the COHERING of the ZPF of the false pre-inflation vacuum. They map their gauge force freedom into GCTs, which is key to what I do. [Z] This gauge freedom is intimately tied up with general covariance -- which is closely parallel to what I have been saying. [S] Really? Where? Show your equations for that. They do not seem to claim a LOCAL gravity energy tensor in the geometrodynamic representation. [Z] In the geometric model. They say only that there seems to be no such decomposition in the standard formalism, based on curvature. The reason they say this is because they are aware that there is no proof. [S] It's like p & x in Heisenberg. A sharp local energy density in the gauge force Newtonian rep is NONLOCAL in the geometrodynamic rep - like wave packet Fourier transforms now we have, in analogy [Z] That is not the Brazilians' theory -- it's your gloss. [S] Yes. [Z] You are trying to do complementarity -- but they are talking *replacement* of curved-vacuum geometrodynamics with a Newtonian-type force described by contortion. [S] Paul you do not seem to know that the Brazilians write GCT tidal stretch-squeeze geometrodynamic 1916 GR curvature tensor ~ teleparallel contortion type term which translates to teleparallel curvature = 0 The Brazilians do not claim that Riemann curvature is zero. The Brazilian gauge potential Bu^a is the non-trivial part of the dimensionless TETRAD eu^a = (Kronecker Delta)u^a + Bu^a that I call Bu ~ Lp^2(Goldstone Phase),u Bu = Bu^adx^a Bu has dimensions of length. [Z] OK. [S] But this is very important and very new. Indeed it's completely original. No one has done this before me I am pretty certain. Sakharov in 1967 did not realize that it is the cohering of the random ZPF that gives emergent gravity. PW Anderson was getting the needed idea of More is different simultaneously also in 1967 and they did not know of each other's work or how they might be connected. Note that when Lp^2 = hG/c^3 -> 0 MACRO-QUANTUM GRAVITY VANISHES! guv(LNIF) = eu^aeu^bnab Note that guv(LNIF) is the Einstein geometrodynamic representation where nab is the metric in the gauge force rep. [Z] OK. [S] Bu^a is the Newtonian non-geometrical gauge force representation. guv(LNIF) has ELASTIC terms LINEAR in Bu^a and PLASTIC terms NONLINEAR in Bu^a. The VANISHING gauge force curvature is NOT the same as the geometrodynamic tidal stretch-squeeze GCT tensor curvature! You garbled that Paul by simply looking at the spelling of the same word with two different meanings in complementary contexts! [Z] You are saying that they don't replace curvature with contortion? That they still need Riemann curvature to describe the tidal aspects of the gravitational field? I say this is FALSE. You are simply projecting your own prejudices into their paper. They say they use contortion to describe the gravitational field in its entirety, as an *alternative* to the usual geometric model. [S] No Paul, you obviously have not understood their equation (49) on p.7 R* = R + Q = 0 R* is the teleparallel gauge force curvature R is Einstein's 1916 GR tidal stretch-squeeze (LC) geodesic deviation GCT tensor curvature Q = teleparallel covariant curl of the teleparallel CONTORTION K. R(Einstein Geometrodynamics) = - Q(Teleparallel Gauge Force) =/= 0 This is not Einstein-Cartan-Shipov type theory agreed. However, it is not contradictory to the latter which would be an extension. You can have an Einstein-Cartan-Shipov extension of this Brazilian-Weitzenbock theory. [Z] No math that you were able to understand. :-) [S] No math you were ever able to write down. I cannot understand what you cannot manifest in formal language. [Z] What I gave you was formal: (LC) = A_ - Q Then you complained that it was too formal. It's either too formal, or not formal enough. What kind of a game is this? [S] Paul you need to define A_ and Q. You have not done that. Also it's not good enough to say Alex does it. You need to do it in TWO ways mathematical and physical. Otherwise it is too vague and useless. [Z] In the Brazilian paper, the intrinsic geometry of the spacetime manifold is implicitly defined by the teleparallel connection, and it is therefore no longer the curved Riemannian manifold of standard GR. As I read the paper, there is no need for Riemann curvature in this teleparallel approach. [S] That's not what eq. 49 p. 7 says. [Z] If you don't see this then you haven't yet understood their thesis IMO. I am just trying to find a way to do the same thing within the framework of standard GR. [S] Exactly, which shows you do not understand at all what the Brazilians have done! [Z] More likely that you haven't, since they actually spell it out: The definition of an energy-momentum density for the gravitational field is one of the oldest and most controversial problems of gravitation. [S] So? That spells out nothing. As a true field, it would be natural to expect that gravity should have its own local energy-momentum density. (Brazilians) [Z] Exactly. [S] IN WHAT CONTEXT? In what representation! Relative to what connection? Measured how? Loud silence by Zielinski on this key point. However, it is usually asserted that such a density cannot be locally defined because of the equivalence principle. [Z] Note the phraseology: ...it is usually *asserted* that such a density cannot be locally defined.... [S] So? What of it? What unwarranted inference do you make of that? As a consequence, any attempt to identify an energy-momentum density for the gravitational field leads to complexes that are not true tensors. [Z] Uh huh. [S] Hardly news Paul. [Z] Right. But wait, there's more.... The first of such attempt was made by Einstein who proposed an expression for the energy-momentum density of the gravitational field which was nothing but the canonical expression obtained from NoetherÍs theorem. Indeed, this quantity is a pseudotensor, an object that depends on the coordinate system. Several other attempts have been made, leading to different expressions for the energy-momentum pseudotensor for the gravitational field. [Z] As I have been saying. [S] As found in every text book. So far nothing has been solved. [Z] This is all direct from Arcos and Pereira. [S] So what? It's common knowledge defining The Question. It is not The Answer you yearn for Paul. Despite the existence of some controversial points related to the formulation of the equivalence principle, it seems true that, in the context of general relativity, no tensorial expression for the gravitational energy-momentum density can exist. [Z] Note the phraseology: ...it *seems* true that.... Not is true, but seems true. [S] And it IS true! [Z] As they write, it is usually asserted. I agree that this is simply an assertion. There is no proof. And the Brazilians know that. That's why they write, it *seems* true that.... Get it? [S] I claim it is true. For example see Penrose The Road to Reality 19.8 Gravitational Field Energy there is no LOCAL pure gravity stress-energy density GCT tensor vacuum field apart from the trivial one tuv(vacuum) = (c^4/8piG)(Guv + /zpfguv) Note, when the exotic dark energy vanishes, i.e. /zpf = 0, then Guv -> Ruv = R = 0 and so tuv(non-exotic vacuum) = 0 However, in the presence of either dark energy (negative pressure) or dark matter (positive pressure), both have w = -1 but positive pressure clumps look like w = 0 to distant observers (us), tuv(exotic vacua) = (c^4/8piG)/zpfguv =/= 0 Not in context of the geometrodynamic representation, but possibly in context of a Bohr complementary Newtonian gauge force representation! A VERY DIFFERENT STORY FROM YOURS PAUL! [Z] That is merely your personal hallucination. That is not in their paper. What is in their paper is the statement that the teleparallel treatment is an alternative to the standard formal description of the gravitational field in terms of spacetime curvature. You are trying to impose your own eccentric interpretation on their paper -- but your interpretation contradicts their actual remarks. [S] I am imposing my interpretation on their eccentric paper that is true. I do not see the contradiction because you do not understand what alternative means. [Z] I note that the Brazilians only say that there seems to be no such solution within the standard 1916 curved-manifold framework -- which they offer as a sales point for their flavor of teleparallelism (which effectively returns gravitational physics to a flat spacetime manifold and forces). [S] You completely misunderstand the paper! They never say that the 1916 GR tidal curvature is zero. As already discussed, in general relativity torsion is assumed to vanish from the very beginning, whereas in teleparallel gravity curvature is assumed to vanish. - p 2 Gauge force curvature = Geometrodynamic Einstein Curvature - Torsion type term = 0 [Z] But note that the intrinsic geometry of the manifold is now *defined* by the torsion connection. There is no longer any underlying curvature, and we are no longer dealing with the Riemannian manifold of standard GR. This is the subtlety that has evidently escaped you. [S] False. Again eq. 49 p. 7 you have over-extrapolated. In the present work, we will separate the notions of space and fact properties of a connection. [S] Yes. Strictly speaking, there is no such a thing as curvature or torsion of spacetime, but only curvature or torsion of connections. This becomes evident if we remember that many different connections are allowed to exist in the same spacetime. Of course, when restricted to the specific case of general relativity, universality of gravitation allows the Levi[CapitalEth]Civita connection to be interpreted as part of the spacetime [S] Yes However, when considering several connections with different curvature and torsion, *it seems far wiser and convenient to take spacetime simply as a manifold*, and connections (with their curvatures and torsions) as additional structures. - Arcos & Pereira, p 4 [S] Yes. Standard orthodox same as in Penrose's The Road to Reality so what? We may then say that the gravitational interaction can be described in terms of curvature, as is usually done in general relativity, or *alternatively* in terms of torsion, in which case we have the so called teleparallel gravity. - p 2 [S] YES EXACTLY! That simply means STRETCH-SQUEEZE in two qualitatively different pictures. [Z] It means that the torsion formalism *completely replaces* the geometric model as an alternative exhaustive description of the [S] It means nothing of the sort. That's you over-interpreting again. Just look at eq. (49) p. 7 ...whereas in general relativity gravitation is described in terms of the curvature tensor, in teleparallel gravity it is described in terms of torsion. - p 20 [S] Yes, if A ~ B - C and if A ~ 0 Therefore, B ~ C I use ~ not = in sense of a EQUIVALENCE RELATION that is the MAPPING between the TWO Bohr COMPLEMENTARITY pictures of teleparallelism with torsion* and zero curvature* ~ Einstein geometrodynamics without torsion but with curvature. Note curvature* =/= curvature AND torsion* =/= torsion Paul you are confused by SURFACE LABELS! You have found FOOL'S GOLD and the END OF THE RAINBOW. Again loud silence from Paul in this devastatingly accurate remark - checkmate, As I end the refrain, thrust home. Cyrano De Bergerac (Rostand) ...whereas in general relativity gravitation is described in terms of the curvature tensor, in teleparallel gravity it is described in terms of torsion. - p 20 [Z] Plain as day. Where is the curvature here? [S] Eq. 49 p.7 Note, this is not same idea as Shipov's theory. [Z] Duly noted. ...the classical equivalence between teleparallel gravity and general relativity implies that curvature and torsion might be simply *alternative ways* of describing *the gravitational field*, and consequently related to the *same degrees of freedom of gravity*. - p 32. A further consequence that emerges from the conceptual differences between general relativity and teleparallel gravity is that, whereas in the former curvature is used to geometrize the gravitational describes the gravitational interaction by acting as a force.84trajectories are not given by geodesics, but by force equations. [S] SO WHAT? You can write or and you can even write <> as a Wigner phase space density. = Integral of <>dx etc But you CANNOT write WHICH IS YOUR ERROR (by analogy). According to the teleparallel approach, therefore, the role played by torsion is quite well defined: it appears as an *alternative* to curvature in the description of the gravitational field.... - p 32 But you DID NOT UNDERSTAND the paper's real meaning! [Z] Either I'm hallucinating or you are. [S] Agreed. [Z] I think it's you. [S] Which proves it's you! ;-) ...the role played by torsion is quite well defined: it appears as an *alternative* to curvature in the description of the gravitational field... [Z] Jack, what part of alternative didn't you understand? [S] I understand the math that they actually DO. [Z] That wasn't the question. What do you think they mean here by [S] Yes, you can look at it EITHER WAY! Gestalt Shift! http://www.neurosemantics.com/Stuttering/foreground-background.htm http://online.sfsu.edu/~psych200/unit6/68.htm You really understand NOTHING of what they do because you are stuck on the surface of the words like curvature and torsion without understanding their very different meanings in the two alternative pictures! [Z] Really? [S] Really. You read the words not not the equations. [Z] I read the words and looked at the equations. [S] WITHOUT UNDERSTANDING ANY OF IT. [Z] Or perhaps I do understand the author's purposes and intentions, while you are hallucinating? [S] Am I Jack Sarfatti thinking he is Robin Williams, or am I Robin Williams thinking he is Jack Sarfatti? Let's leave Buddhism out of this. The curvature that vanishes is a DIFFERENT curvature from Einstein's 1916 which is battle-tested based on geodesic deviation - all relative to the LC connection - not a bigger one! [Z] You are trying to have your cake *and* eat it. The authors are quite clear in this point: ...theoretical speculations have since the early days of general relativity discussed the necessity of including torsion, in addition to curvature, in the description of the gravitational interaction. [S] Again the words torsion and curvature are being used too sloppily EVEN BY THE BRAZILIANS. [Z] Because their actual use of these terms in the paper doesn't support your eccentric complementarity gloss on their treatment? {s} Look Paul given any connection C with a covariant derivative D TORSION means [Du,Dv]SCALAR = TORSIONuv^w(Scalar),w ,w is ordinary partial derivative CURVATURE means [Du,Dv]Aw = (CURVATURE)uvw^lAl Where TORSION and CURVATURE are BOTH tensors relative to the given group e.g. GCT if we are doing 1916 GR. [Z] They say that in GR, the gravitational field is described by a curvature tensor -- which obvious means the Riemann tensor R^u_vwl. They say that *this* curvature is absent in their treatment. [S] Hogwash Paul. Again you have not understood their eq. 49 p. 7. [Z] The point is that the intrinsic geometry of the manifold is now defined entirely by the teleparallel connection. Are you saying that this teleparallel connection must have non-vanishing Riemann curvature in their treatment? [S] Yes, that's eq. 49 p. 7 Now in the Brazilian paper the gauge force connection C* is NOT same as the 1916 GR (LC) connection. [Z] Right. [S] Indeed Bu is the Gauge Force Connection C* = Bu Du* = ,u - Bu Just like in EM! [Z] Exactly. [S] In contrast, in the geometrodynamic representation C = (LC) Du = ,u - (LC) Therefore Paul you have been comparing apples with oranges. Your head is in the clouds, but your feet are not on the ground. You are a LUFTMENSCH right out of Thomas Mann's Felix Krull - or maybe I am? ;-) === Subject: Re: another moron Re: John misses point Re: ING RUDE CHAP Re: Idiocy of Muckenheim was Re: countability of reals <34jeejF4da72sU1@individual.net> <41e49ae3.2013035@news.ecn.ab.ca> <34jstgF49k7q4U1@individual.net> <34m94vF4b9ljeU1@individual.net> <34ma36F4ebngdU1@individual.net> <34mblnF4d4j04U1@individual.net> <34mchmF4d7a9tU1@individual.net> <34mdopF4d4avgU1@individual.net Here's the list of all real numbers START > n = 1 > FOR y = 1 to n > Real(index, digit) = UTM(n-y+1, y) mod 10 > n ++ > GOTO START Now DO TRY TO TELL ME A NUMBER that you think is missing Herc > i did moron. i said show me pi and sqrt(pi). now since you claim you > can map (index,digit) to pi why don't u get ur computer to print out > all the digits of pi for me and post it here? i see that u skipped > reading all my remarks, don't know anything about measure theory huh? > its Real( npi , Digit ) = the Digit th digit of the npi th program. npi e N > eg. Real ( 20000, 1 ) = 3 > Real (20000, 2) = 1 > since En, n e N, UTM( n, d ) = the dth digit of pi=3.14159.. > Say you have a turning machine that calculates the arctangent of 1 then multiplies turning machine? ouch > by 4, and given an input d, it will output the dth digit of that expression. > Herc === Subject: Algebraic int's subfield of C. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0R1ikZ14544; Hello I was just looking at some questions, and I got into a little difficulty. The question says. Let A be the set of all complex numbers which are algebraic over C. 1. Show that A is a subfield of C. 2. Prove that A is not a finite extension of Q. 3. Let B be an algebraic extension of A. Prove that B=A. 4. Deduce that every non-constant monic poly in A[x] has a root in A, and hence can be written as a product of linear factors over A. I can show part 1, using the tower law and various other little facts. But I am having trouble with the remaining questions. My thinking so far is for 3, if B is an algebraic extension of A, then it is A adjoined with some element already in A, as A contains all numbers algebraic over C. And by the primitive element theorem we can write it as a linear sum of the previous adjoined elements, but this is just going to be equal to the set we already have. I don't think that is formal enough for a proof, could someone help? For part 2, I am not sure how to show it is not a finite extension. I was thinking that the number of all algebraic numbers over Q is not finite, so it can not be a finite extension of Q. But I don't know if it is correc that the number of algebraic numbers is infinite. And well again to deduce 4, I am not completely sure either. I think these are pretty standard text problems, but I am not seeing something obvious. If someone could help it would be hugely appreciated. === Subject: Re: Algebraic int's subfield of C. Steve, > 2. Prove that A is not a finite extension of Q. Suppose A is a finite extension of Q, say of dimension D, and achieve a contradiction. (Hint: Are there any irreducible polynomials in Q[x] of degree D+1?) > 3. Let B be an algebraic extension of A. Prove that B=A. Pick an arbitrary element b of B. (What does such an element look like?) Now, show that b is in A. Travis === Subject: Re: Algebraic int's subfield of C. > Let A be the set of all complex numbers which are algebraic over C. I think you mean, algebraic over the rationals. Every complex number is algebraic over C, the complex numbers - pi, for example, satisfies the algebraic equation x - pi = 0. > 1. Show that A is a subfield of C. > 2. Prove that A is not a finite extension of Q. > 3. Let B be an algebraic extension of A. Prove that B=A. > 4. Deduce that every non-constant monic poly in A[x] has a root in A, > and hence can be written as a product of linear factors over A. > I can show part 1, using the tower law and various other little facts. > But I am having trouble with the remaining questions. > My thinking so far is for 3, if B is an algebraic extension of A, > then it is A adjoined with some element already in A, as A contains > all numbers algebraic over C. But you have to show that an element of C algebraic over A is algebraic over Q. > For part 2, I am not sure how to show it is not a finite extension. > I was thinking that the number of all algebraic numbers over Q is not > finite, so it can not be a finite extension of Q. But I don't know > if it is correc that the number of algebraic numbers is infinite. Can you show that A has elements of arbitrarily high degree over Q? Can you see how that helps? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Galois theory question by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0R1ik014548; Hey guys, I am having trouble with this question I found, I was hoping someone here could help me please. 1. Let E be an extension of F, let a be an element of E and let pi be an element of the Gal(E/F) (or Gal(E:F) if you prefer). Prove that pi(a) is a conjugate of a and lies in E. === Subject: Re: Galois theory question > Hey guys, I am having trouble with this question I found, I was > hoping someone here could help me please. > 1. Let E be an extension of F, let a be an element of E and let pi be an element of the Gal(E/F) (or Gal(E:F) if you prefer). > Prove that pi(a) is a conjugate of a and lies in E. I don't mean to seem insulting, but my response is this: Do you know the meanings of the words in the problem? If so, we can continue. Dale. === Subject: Re: Galois theory question > Hey guys, I am having trouble with this question I found, I was > hoping someone here could help me please. > > 1. Let E be an extension of F, let a be an element of E and let pi be an > element of the Gal(E/F) (or Gal(E:F) if you prefer). > > Prove that pi(a) is a conjugate of a and lies in E. > > > I don't mean to seem insulting, but my response is this: > Do you know the meanings of the words in the problem? > If so, we can continue. To expand a little on Dale's response: Every definition of the Galois group that I've ever seen says that its elements are (among other things) functions from E to E. So to ask this group why pi(a) lies in E you must a) be using a hugely non-standard definition of the Galois group, which you had better share with us if we are to be of any assistance, or b) you don't know the meanings of the words in the problem, or c) you had a massive attack of math blindness, a condition which affects most of us from time to time. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Need help on a group theory problem desperately. ------------------------------------------------------- Problem: Is GL+(n,R), general linear group with positive determinant over the real number field, homeomorphic to a Euclidean space in general. What is the simpliest form that GL+(n,R) can be reduced to? The concern is whether GL+(n,R) can be simplified further, or be parameterize by a simply subset of a Euclidean space. -------------------------------------------------------- Some time ago someone suggested the following argument: GL+(n,R) is not homeomorphic to a Euclidean space in general. For instance, at n=2. [1]: GL+(2,R) = SL(2,R) x R+, [2]: SL(2,R) is homeomorphic to S^1 x R x R+ [3]: S^1, a circle is not heomeomorphic to a Euclidean space, thus GL+(2,R) is not homeomorphic to a Euclidean space. I want to check with you to see if this argument is solid. In particular, due to my limited knowledge on this subject, I need help to understand line [2] in particular. Also, it'll be greatly appreciated if you can comment on the problem for all n. Kai === Subject: Re: Need help on a group theory problem > Is GL+(n,R), general linear group with positive determinant over the real > number field, homeomorphic to a Euclidean space in general. No. det:GL(n,R) -> R is continuous. Thus GL+(n,R) = det^-1((0,oo)) is an open subset of GL(n,R). Now as GL(n,R) homeomorphic R^2n, GL+(n,R) homeomorphic open subspace of Euclidean space R^2n > Some time ago someone suggested the following argument: > GL+(n,R) is not homeomorphic to a Euclidean space in general. > For instance, at n=2. > [1]: GL+(2,R) = SL(2,R) x R+, > [2]: SL(2,R) is homeomorphic to S^1 x R x R+ > [3]: S^1, a circle is not heomeomorphic to a Euclidean space, It's a subspace of R^2. > thus GL+(2,R) is not homeomorphic to a Euclidean space. It concurs it's homeomorphic to a subspace of R^4 > I want to check with you to see if this argument is solid. In particular, due > to my limited knowledge on this subject, I need help to understand line [2] > in particular. Me too, S^1 x R x R+ is not open. What's R+ to you? (0,oo) or the usual ordered group theory [0,oo)? > Also, it'll be greatly appreciated if you can comment on the problem for all > n. I did somewhat. === Subject: Correction on Bu The Brazilian gauge potential Bu^a is the non-trivial part of the dimensionless TETRAD eu^a = (Kronecker Delta)u^a + Bu^a that I call Bu ~ Lp^2(Goldstone Phase),u Bu = Bu^aLp^2Pa/h Bu^aPa/h = (Macro-Quantum Vacuum Coherent Goldstone Phase),u {Pa} is the mom-energy Lie algebra generating the translation Lie group T4 Bu has dimensions of length. The non-dynamical torsion-free 1916 GR Ricci Rotation coefficients are A^abu = e^aweb^w;u where ;u is the {LC} covariant derivative The Lorentz group is not yet locally gauged in 1916 GR only T4 is locally gauged. ;u = ,u - i/2A^abuSab Where ,u is ordinary partial derivative and {Sab} is the Lie algebra of the Lorentz Lie group. Sab = (i/4)[&a,&b] &a are the 4x4 Dirac gamma matrices that generate the Dirac Clifford Algebra This GCT {LC} covariant derivative operates on Dirac spinor fields. === Subject: Question #1: CP^n Hi all, I am having trouble understanding CP^n, complex projective n-space. Firstly, let me define RP^n and then try to make the transition. RP^n is the space of all lines through 0 in R^(n+1). In other words, it is the quotient of R^(n+1) - {0} by the equivalence relation ~ where (x_0,...,x_(n+1)) ~ (Lx_0,...,Lx_(n+1)) where L is any real number. So two points are equivalent if one is a scalar multiple of the other. But also we can think of RP^n as restricting this quotient to the sphere. So now, we look at the vectors x in R^(n+1) such that summation (x_i)^2 = 1 (that is the sum of the squares of the vector components equals 1) Now, since in the equivalence relation we want Lx = x for all real numbers L, we have summation (Lx_i)^2 = 1, so L^2 summation (x_i)^2 = 1, so L^2 = 1, so L = + or - 1. Thus, anitpodal points get identified, so RP^n is the quotient of the sphere by ~ where antipodal points are equivalent. Now, to CP^n. We use the same method : CP^n is the space of lines through the origin in C^(n+1), or the quotient of C^(n+1) - {0} by the equivalence relation that two vectors are equivalent if one is a scalar multiple of the other where the scalar is now an element of C. So let's restrict now this quotient to the complex sphere {z in C such that summation |z_i|^2 = 1, where z = (z_0, ..., z_(n+1))}. Then now we look at summation |z_i|^2 = 1 Now, since in the equivalence relation we want Lx = x for all complex numbers L, we have summation |Lx_i|^2 = 1, so |L|^2 summation |x_i|^2 = 1, so |L|^2 = 1, so |L| = 1, so L is on the unit circle. Ok, this is great. But I have no intuition as to what this means, that L is on the unit circle. In RP^n, we have the sphere modulo the relation where antipodal points are identified. Here, we have a complex sphere where two vectors are equivalent if one is a complex number of modulus 1 times the other. But what does that really mean? Can someone take this further and assist? Tony === Subject: Re: Question #1: CP^n >I am having trouble understanding CP^n, complex projective n-space. ... >Now, to CP^n. We use the same method : CP^n is the space of lines through >the origin in C^(n+1), or the quotient of C^(n+1) - {0} by the equivalence >relation that two vectors are equivalent if one is a scalar multiple of the >other where the scalar is now an element of C. So let's restrict now this >quotient to the complex sphere {z in C such that summation |z_i|^2 = 1, >where z = (z_0, ..., z_(n+1))}. ... >Ok, this is great. But I have no intuition as to what this means, that L is >on the unit circle. In RP^n, we have the sphere modulo the relation where >antipodal points are identified. Here, we have a complex sphere where two >vectors are equivalent if one is a complex number of modulus 1 times the >other. But what does that really mean? Can someone take this further and >assist? One way to rephrase your description of the relationship between S^n and RP^n is that S^n is a covering space of RP^n (specifically, a covering of degree 2). A covering space of any space X is a special case of a fiber bundle over X (namely, a fiber bundle with discrete fibers); in the case of S^n over RP^n, each fiber is a 2-point space (a 0-sphere). Analogous to all that, one way (and a very good way) to describe the relationship between S^{2n+1} (which is the unit sphere in C^{n+1}; calling it a complex sphere is not a good idea, misleading at best and wrong at worst) and CP^n) is to say that S^{2n+1} is a fiber bundle over CP^n, each fiber of which is a 1-sphere. Simplifying to the simplest non-trivial case, S^3 is a fiber bundle over CP^1, CP^1 is a 2-sphere (the Riemann sphere), and this fiber bundle is the Hopf fibration. I seem to remember that in some earlier post you were asking about the Hopf fibration; whether or not that was you, someone did, and you can find various answers by searching an archive of sci.math (and elsewhere). Once you have managed to see how S^3 (which can be identified by stereographic projection with our familiar R^3 supplemented by a single point at infinity) is partitioned into copies of S^1 (which can all but one actually be taken, in the R^3 model, to be *round* circles; the one exception being a straight line plus the point at infinity) by the fibers of the Hopf fibration, you should be well on your way to some kind of understanding of the (generalized) Hopf fibration of S^{2n+1} over CP^n. Lee Rudolph === Subject: Re: Question #1: CP^n >>I am having trouble understanding CP^n, complex projective n-space. > ... >>Now, to CP^n. We use the same method : CP^n is the space of lines through >>the origin in C^(n+1), or the quotient of C^(n+1) - {0} by the equivalence >>relation that two vectors are equivalent if one is a scalar multiple of >>the >>other where the scalar is now an element of C. So let's restrict now this >>quotient to the complex sphere {z in C such that summation |z_i|^2 = 1, >>where z = (z_0, ..., z_(n+1))}. > ... >>Ok, this is great. But I have no intuition as to what this means, that L >>is >>on the unit circle. In RP^n, we have the sphere modulo the relation where >>antipodal points are identified. Here, we have a complex sphere where two >>vectors are equivalent if one is a complex number of modulus 1 times the >>other. But what does that really mean? Can someone take this further and >>assist? > One way to rephrase your description of the relationship between S^n > and RP^n is that S^n is a covering space of RP^n (specifically, > a covering of degree 2). A covering space of any space X is > a special case of a fiber bundle over X (namely, a fiber bundle > with discrete fibers); in the case of S^n over RP^n, each fiber is > a 2-point space (a 0-sphere). Analogous to all that, one way (and > a very good way) to describe the relationship between S^{2n+1} > (which is the unit sphere in C^{n+1}; calling it a complex sphere > is not a good idea, misleading at best and wrong at worst) and > CP^n) is to say that S^{2n+1} is a fiber bundle over CP^n, I see. So you are saying that {z in C such that summation |z_i|^2 = 1, where z = (z_0, ..., z_(n+1))}, where I call it the complex sphere, it is really isomorphic to S^(2n+1) since we can identify C^(n+1) with R^(2n+2)? each > fiber of which is a 1-sphere. Simplifying to the simplest non-trivial > case, S^3 is a fiber bundle over CP^1, CP^1 is a 2-sphere (the > Riemann sphere), and this fiber bundle is the Hopf fibration. > I seem to remember that in some earlier post you were asking about > the Hopf fibration; whether or not that was you, someone did, and > you can find various answers by searching an archive of sci.math > (and elsewhere). Once you have managed to see how S^3 (which > can be identified by stereographic projection with our familiar > R^3 supplemented by a single point at infinity) is partitioned > into copies of S^1 (which can all but one actually be taken, in > the R^3 model, to be *round* circles; the one exception being a > straight line plus the point at infinity) by the fibers of the > Hopf fibration, you should be well on your way to some kind of > understanding of the (generalized) Hopf fibration of S^{2n+1} > over CP^n. > Lee Rudolph === Subject: Question #1: CP^n Hi all, I am having trouble understanding CP^n, complex projective n-space. Firstly, let me define RP^n and then try to make the transition. RP^n is the space of all lines through 0 in R^(n+1). In other words, it is the quotient of R^(n+1) - {0} by the equivalence relation ~ where (x_0,...,x_(n+1)) ~ (Lx_0,...,Lx_(n+1)) where L is any real number. So two points are equivalent if one is a scalar multiple of the other. But also we can think of RP^n as restricting this quotient to the sphere. So now, we look at the vectors x in R^(n+1) such that summation (x_i)^2 = 1 (that is the sum of the squares of the vector components equals 1) Now, since in the equivalence relation we want Lx = x for all real numbers L, we have summation (Lx_i)^2 = 1, so L^2 summation (x_i)^2 = 1, so L^2 = 1, so L = + or - 1. Thus, anitpodal points get identified, so RP^n is the quotient of the sphere by ~ where antipodal points are equivalent. Now, to CP^n. We use the same method : CP^n is the space of lines through the origin in C^(n+1), or the quotient of C^(n+1) - {0} by the equivalence relation that two vectors are equivalent if one is a scalar multiple of the other where the scalar is now an element of C. So let's restrict now this quotient to the complex sphere {z in C such that summation |z_i|^2 = 1, where z = (z_0, ..., z_(n+1))}. Then now we look at summation |z_i|^2 = 1 Now, since in the equivalence relation we want Lx = x for all complex numbers L, we have summation |Lx_i|^2 = 1, so |L|^2 summation |x_i|^2 = 1, so |L|^2 = 1, so |L| = 1, so L is on the unit circle. Ok, this is great. But I have no intuition as to what this means, that L is on the unit circle. In RP^n, we have the sphere modulo the relation where antipodal points are identified. Here, we have a complex sphere where two vectors are equivalent if one is a complex number of modulus 1 times the other. But what does that really mean? Can someone take this further and assist? Tony === Subject: Question #2 : Ordinals Let Y be the set of ordinals less than the first uncountable ordinal. I want to show that every countable subset E of Y has an upper bound in Y and hence a least upper bound. How can I do this? I feel that this is a sort of follow your nose problem. So, Y = {x in X such that x < omega}. An element b is an upper bound for E if x <= b for all x in E, and it's a least upper bound if b <= c for each upper bound c. How can I conclude this problem? Tony === Subject: Re: Question #2 : Ordinals > Let Y be the set of ordinals less than the first uncountable ordinal. I > want to show that every countable subset E of Y has an upper bound in Y and > hence a least upper bound. How can I do this? I feel that this is a sort > of follow your nose problem. Basically, there are just a couple things you need to see: (1) How do you define the supremum of a nonempty set of ordinals? (I don't mean just it's the least upper bound, but there is a simple formula which will always give you the supremum of a nonempty set of ordinals.) (2) Using this formula for the set E, and remembering how set-theoretic operations relate to cardinalities, it should be easy to calculate the cardinality of its supremum. (3) Note that if the cardinality of an ordinal is countable, then the ordinal itself is a countable ordinal (of course). (There are other ways of attacking, which would just require looking at the cofinality of omega_1, though the above would be the most elementary technique.) > So, Y = {x in X such that x < omega}. An element b is an upper bound for E > if x <= b for all x in E, and it's a least upper bound if b <= c for each > upper bound c. > How can I conclude this problem? As I've kind of mentioned above, this definition of supremum may not be the most useful. You should have a formula which will give you the supremum, and it is this formula that will be most helpful. Hope this helps, at least a bit. Don't really want to give a full solution, though perhaps others will... __ Arthur === Subject: Re: Question #2 : Ordinals > Let Y be the set of ordinals less than the first uncountable ordinal. I > want to show that every countable subset E of Y has an upper bound in Y and > hence a least upper bound. How can I do this? I feel that this is a sort > of follow your nose problem. According to standard notation, Y = w_1, the first uncountable ordinal. > So, Y = {x in X such that x < omega}. An element b is an upper bound for E > if x <= b for all x in E, and it's a least upper bound if b <= c for each > upper bound c. > How can I conclude this problem? The least upper bound of a set of ordinals is simply the union of that set. It remains to show that the union is a member of w_1. That is, you need to show that the union of a countable collection of countable ordinals is itself a countable ordinal. That's easy enough if we assume the axiom of choice. It amounts to saying that aleph_1 is a regular cardinal, which I think requires AC. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: proving continuous analog of E(X) = sum P(X>=n) I haven't seen it yet but am sure that there must be a continuous version. Can someone supply the proof? In particular, For X >=0 discrete, one can find the proof in many texts for E(X) = sum P(X>=n). The continuous version should be E(X) = integral P(X>x) dx but I can't find it in my small collection of books. === Subject: Re: proving continuous analog of E(X) = sum P(X>=n) Yes, if random variable X is nonegative, then EX = integral(x=0..infty, P{X > x}. This is usually proved in a book on stochastic processes (e.g., Ross or Wolff). Contrary to what another poster said, there is no need to consider a density (or for X to have one): Let I(x) = the indicator variable for the event {X > x}. Then X = integral(x=0..infty, I(x)), so EX = E integral(x=0..infty, I(x)) = integral(x=0..infty, EI(x)) = integral(x=0..infty, P{X > x}). The interchange of expectation and integral is allowed by Tonnelli's theorem (i.e., the integrand is nonnegative). -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: proving continuous analog of E(X) = sum P(X>=n) > I haven't seen it yet but am sure that there must be a > continuous version. Can someone supply the proof? > In particular, For X >=0 discrete, one can find the > proof in many texts for E(X) = sum P(X>=n). The > continuous version should be E(X) = integral P(X>x) dx > but I can't find it in my small collection of books. Assume X has a density f. Write P(X > x) as an integral, then then change the order of integration. (Why is this allowed?) Mike === Subject: random walk(Markov Chain) on unidirectional graph? Hi all, I suspect that all information about the Markov Chain can be obtained from the graph directly. Here I want to ask about the random walk on graph: If the graph is bi-directional, assuming all edges have unit weight, 1- if there is an edge between i and j; 0 - if there is no edge between i and j; then the random walk on the graph is associated with a Markov Chain P_ij= 1 / degree of node i, if there is an edge between i and j; 0, if there is no edge between i and j. Here the degree of node i = the number of edges connected with node i. For such bi-directional graph, it can be proved that the stationary distribution for node i= (degree of node i) / (2*the total edges in the graph). I want to know if similar results can be obtained for uni-directional graph. Can the stationary distribution for the Markov chain be analytically found? === Subject: off-diagonal elements of a 3x3 inertia tensor listed above. I'm working on a 3d physics engine for solid bodies. My engineering handbook (J.J.Tuma) says the moments of inertia are Ixx = integral-over-volume (y^2 + z^2) dV Iyy = integral-over-volume (x^2 + z^2) dV Izz = integral-over-volume (y^2 + x^2) dV which is sensible, and the products of inertia are Ixy = Iyx = integral-over-volume (xy) dV Izy = Iyz = integral-over-volume (zy) dV Ixz = Izx = integral-over-volume (xz) dV However, some other code I was looking at, says these should be negative, so Ixy = -integral-over-volume (xy) dV etc. So I tried it by putting the products and moments into a 3x3 matrix and finding the eigenvalues. And the negative seems to be correct. Can anyone explain why the negative sign is necessary? tia === Subject: Reason for the Center of the Circle?? Ok, I have a question regarding the Center of a Circle. Now, in math classes around the world, a circle formula is taught to students. This formula is derived from the distance formula. (x-h)^2 + (y-k)^2 = r^2 Great. Now here's my million dollar question: WHY IS THE CENTER OF THE CIRCLE (H,K)??? Who picked those coordinates and WHY??? Please help because my class asked me this today, and I'm looking for an answer to give my students. === Subject: Re: Reason for the Center of the Circle?? > Ok, I have a question regarding the Center of a Circle. Now, in math > classes around the world, a circle formula is taught to students. This > formula is derived from the distance formula. > (x-h)^2 + (y-k)^2 = r^2 > Great. Now here's my million dollar question: > WHY IS THE CENTER OF THE CIRCLE (H,K)??? > Who picked those coordinates and WHY??? By definition a circle is the set of points having some constant distance from a given point, which is called the center. Given h,k,r, the equation (x-h)^2 + (y-k)^2 = r^2 simply says that (x,y) has distance r from (h,k). That means h,k is the center, or rather _a_ center. It remains to show that 3 or more points cannot be equally distant from more than one other point, which is easy. === Subject: Is O(2) homeomorphic to anything? Hi all, Is the orthogonal group O(2), which has the form [ cos(t) & sin(t) ] [ -sin(t) & cos(t) ], homeomorphic to anything simplier? Is it homemorphic to a circle S^1, and how can you show that it is not homeomorphic to [0,1), or any simple subset in R? It is classic to show that when f : [0, 2.b9) .81¬ S^1 is defined by f(.83î) = (cos(.83î), sin(.83î)), then f is not the homeomorphism required because inv(f) is not continuous. But how do you know (or show) that two spaces are not homeomorphic? Your comments are greatly appreciated. Kai === Subject: Re: Is O(2) homeomorphic to anything? On 26-Jan-2005, Kai Is the orthogonal group O(2), which has the form > [ cos(t) & sin(t) ] > [ -sin(t) & cos(t) ], > homeomorphic to anything simplier? Just a nitpick that I haven't seen anyone else mention yet: What you've written above is actually just SO(2), the special (det = +1) orthogonal group, the connected identity component of O(2). The latter also contains another connected component with det = -1, namely the matrices of the form: [ cos(t) sin(t) ] [ sin(t) -cos(t) ] [...] -- Jim Heckman === Subject: Re: Is O(2) homeomorphic to anything? > On 26-Jan-2005, Kai Hi all, >> Is the orthogonal group O(2), which has the form >> [ cos(t) & sin(t) ] >> [ -sin(t) & cos(t) ], >> homeomorphic to anything simplier? > Just a nitpick that I haven't seen anyone else mention yet: > What you've written above is actually just SO(2), the special > (det = +1) orthogonal group, the connected identity component of > O(2). the generic element of O(2): O(2) is the *set* of such matrices. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: Is O(2) homeomorphic to anything? Kai, You're right; it's homeomorphic to S^1. Consider the standard embedding of S^1 in C (that is, the unit circle {e^it}, where t varies over an appropriate domain). Consider f: S^1 -> O(2) defined by f(t) = [ Re(e^it) & Im(e^it) ] [ -Im(e^it) & Re(e^it) ] Then, the inverse function is given by f^(-1): O(2) -> S^2, defined by f([ a & b ] [ -b & a ]) = a+ib (Checking the required conditions that this bijection constitutes a homeomorphism is fairly easy.) O(2) is not homeomorphic to [0,1) because O(2) is compact and [0,1) is not. It is not homeomorphic to (any) interval of R, because removing one point of the interior of the interval leaves a space that isn't simply connected, whereas removing a point from S^1 ~= O(2) leaves a space that *is* simply connected. Proving that two spaces are not homeomorphic can be tricky. Generally, one looks for topological invariants (like compactness), and shows that the two spaces don't share a particular invariant. Travis > Hi all, > Is the orthogonal group O(2), which has the form > [ cos(t) & sin(t) ] > [ -sin(t) & cos(t) ], > homeomorphic to anything simplier? > Is it homemorphic to a circle S^1, and how can you show that it is not > homeomorphic to [0,1), or any simple subset in R? It is classic to show > that when > f : [0, 2.b9) .81¬ S^1 is defined by > f(.83î) = (cos(.83î), sin(.83î)), > then f is not the homeomorphism required because inv(f) is not continuous. > But how do you know (or show) that two spaces are not homeomorphic? > Your comments are greatly appreciated. > Kai === Subject: Re: Is O(2) homeomorphic to anything? --------------------------------------------------------------------- > Is the orthogonal group O(2), which has the form > [ cos(t) & sin(t) ] > [ -sin(t) & cos(t) ], > homeomorphic to anything simplier? > Is it homemorphic to a circle S^1, and how can you show that it is not > homeomorphic to [0,1), or any simple subset in R? It is classic to show that Yes it is. It's not like R because remove single point from R and R is disconnected but remove single point from circle, it is not disconnected. [0,1) is different that R, because removal of 1, doesn't disconnect it while the removal of any point of R will disconnect it. [0,1) is different than circle because remove any point from circle, it's still connected while the same cannot be said of [0,1) > when > f : [0, 2#?) %.86.92 S^1 is defined by > f(#.86) = (cos(#.86), sin(#.86)), Please use plain low bit ascii, special characters show differently > then f is not the homeomorphism required because inv(f) is not continuous. > But how do you know (or show) that two spaces are not homeomorphic? > Your comments are greatly appreciated. > Kai === Subject: Re: Torah Matrix Code Prediction by Smart1234 [Smart1234] A letter I sent to the Pope. ---------------------------------------------------------------------------- ----------- I'm sorry Jesus Christ wasn't crucified... . I even mentioned this to you before. Well, here is more proof. How the Gospel of Barnabas Survived http://www.barnabas.net/how_survived.htm 220: Jesus and the Four Angels http://www.barnabas.net/barnabasP220.html 215: Divine Rescue of Jesus http://www.barnabas.net/barnabasP215.html Chapter Index http://www.barnabas.net/chapter_index.htm Like I said before, wasn't His word enough for you people, that you might be saved by just believing in Him? He made a living sacrifice not a dead one. Jim Smith === Subject: Corrective interpretation of real numbers Do not get me wrong. If I mention that WeierstrassÍs notion of a limit does never permit delta to be zero, I am fully aware of the fact that this notion is the decisive basis not merely for a most rigorous picture but rather for something fundamentally different from PeirceÍs description: ñA continuum is precisely that every part of which has partsî. So I do not share the widespread lack of understanding. I just would like to suggest a corrective interpretation of real numbers. Let me exemplary explain why. As long as we neglect the potentiality of infinity, and we do so with great success, we cannot avoid some unreasonable consequences. - Let e.g. any number x cut IR. Then there is no consensus whether x belongs to the smaller or the larger numbers. For x=0, both IR+ and IR- need a neutral element of addition and should offer the option of reunification. - BuridanÍs donkey is suffering starvation between two full mangers because of lacking preference for the left or the right one. - ñCorrectnessî demands to graphically represent |sign(x)|=0 like a singular point. - Practice would appreciate to be released from obligation to always carefully distinguish in IR between open and closed intervals just for unspecified ñmathematicalî reasons even if such distinction does obviously not make any sense. Common sense provides the only reasonable elucidation and corrective interpretation of real numbers if applied before or after calculating with mathematics based on WeierstrassÍs notion as usual: Imagine delta equal to zero: Now, any single number x does not matter any more. Infinitely many are required as to change a function f(x) by addition or removal of numbers. Singularities only belong to distributions. Equality of two irrational numbers tends to evade numerical examination. When mathematicians like Stifel and Weyl used profane terminology like fog or sauce as to express the essence of an untamed continuum, then perhaps they did know why. The alluded simple ñexternalî reinterpretation can be used without any scruples but with much ease and success as compared to hyperreal or surreal numbers in order to correct for unreasonable consequences of the standard analysis. === Subject: Re: Corrective interpretation of real numbers > but rather for something fundamentally different from PeirceÍs > description: ñA continuum is precisely that every part of which has > partsî. That characterization fails to distinguish between the reals and the rationals, for example. === Subject: Re: Corrective interpretation of real numbers it seems that the german team members WM and EB are fighting now for the championship as the greatest (or whatever) cranks of the world. Alois -- Alois Steindl, Tel.: +43 (1) 58801 / 32558 Inst. for Mechanics II, Fax.: +43 (1) 58801 / 32598 Vienna University of Technology, A-1040 Wiedner Hauptstr. 8-10 === Subject: Re: Corrective interpretation of real numbers Alois Steindl a .8ecrit : > it seems that the german team members WM and EB are fighting now for the > championship as the greatest (or whatever) cranks of the world. Of course, all-time champs like JSH or AP are ineligible to compete, I presume ? > Alois === Subject: Re: Corrective interpretation of real numbers > Of course, all-time champs like JSH or AP are ineligible to compete, I > presume ? I guess, I know what AP stands for: Archimedes plutonium JSH might be Jim or James Harris, right? Now it is your turn. Please notice my practical arguments and come up with something substantial. I sadly guess you are not in position for that, perhaps you are closer to your favorites than we. E. === Subject: Re: Corrective interpretation of real numbers Given you are unable to grasp something, you should nonetheless be able to either refute or ignore it. Just uttering disfavor could be understood as a signal for other lurking jolly poor fellows. === Subject: Where's respect? was Re: Corrective interpretation of real numbers > Given you are unable to grasp something, you should nonetheless be able > to either refute or ignore it. Just uttering disfavor could be > understood as a signal for other lurking jolly poor fellows. What do you expect? Prof. Mueckenheim (and you?) show no respect for generations of mathematicans and rank himself at least at the level of Cantor. This is almost unbearable... You're an electrical engineer. What would you say about this: All current and former physicists are fools: Sonic speed is faster then light speed because any time I switch on my TV, i'll hear before i'll see. With all due respect, this is the level of your mathematical arguments. Buying a goldfish, putting them into a bird cage, yelling out that a goldfish is a great fault of nature with respect to the rapid death of the fish in the bird cage, ignoring the millions of goldfishs in ponds at age of thirty or more and laughing about all the foolish biologists depicts it too. Andreas === Subject: Re: Where's respect? was Re: Corrective interpretation of real numbers >> Given you are unable to grasp something, you should nonetheless be able >> to either refute or ignore it. Just uttering disfavor could be >> understood as a signal for other lurking jolly poor fellows. > What do you expect? > Prof. Mueckenheim (and you?) show no respect for generations of > mathematicans On the contrary. I am perhaps among the few ones who even admire Charles S. Peirce. I just missed Cantor's respect towards Gauss and many others. > and rank himself at least at the level of Cantor. Hopefully we will not share his fate. > This is almost unbearable... Chill down. Turn to the subject. > You're an electrical engineer. What would you say about > this: > All current and former physicists are fools: Do you have reasonable arguments. Is your suspicion conform with physicists who lived before let's say Archimedes Homringhausen? I feel myself insired and cofirmed by many many mathematicians and physicists including Aristotele, Euklid, Archimedes (the other one) Newton, Leibniz, Galilei, Cauchy, Kronecker, v. Helmholtz, Poincar.8f, Brouwer, Borel, Weyl, and even some prudent words of Hilbert, ... > Sonic speed is faster then light speed because > any time I switch on my TV, i'll hear before i'll see. This would fail any double blind test. > With all due respect, this is the level of your mathematical > arguments. It is rather the level of your understanding. I do not accept mocking. Eckard === Subject: Re: Corrective interpretation of real numbers >Given you are unable to grasp something, you should nonetheless be able >to either refute or ignore it. Just uttering disfavor could be >understood as a signal for other lurking jolly poor fellows. Possibly, but in this case there's no need to worry, your original post is signal enough. ************************ David C. Ullrich === Subject: Re: Corrective interpretation of real numbers > your original post is signal enough. Perhaps, I intended a too concise attack on Robinso(h)n. Compare his hyperreal whish-list with the partially quite opposite practical corrections I am suggesting. Eckard Blumschein === Subject: Re: Corrective interpretation of real numbers it seems that the german team WM and EB are fighting now for the championship as the greatest (or whatever) cranks of the world. Alois -- Alois Steindl, Tel.: +43 (1) 58801 / 32558 Inst. for Mechanics II, Fax.: +43 (1) 58801 / 32598 Vienna University of Technology, A-1040 Wiedner Hauptstr. 8-10 === Subject: Re: Inflationary Theory ; I'm confused Hi Jim, I read a lot more about cosmology now, and I have some new notes below. Rob [...] > I think you are saying that since the time ran much slower in the > very early Universe than it does now, that therefor objects seem > to move away from us faster than light, but in reality we just see > them in their pre-birth state (right after BB). Right ? > No, that's not what he's saying. I don't even think it's meaningful to > compare the rate of time passage at two different times in history. > Such a comparison is only meaningful for two objects that go different > places and come back together. If you carefully define what it means > for two events at different places to be simultaneous, you can also > compare the time rates at the two places, but the answer you get can > depend on the definition of simultaneity chosen. Mmm. BB theory clearly states that the Universe started out microscopically small. So all matter / energy in the Universe was once (in) one 'place' and 'time'. The CMB that we observe now brings us photons back from that 'moment'. (albeit due to rapid expansion on now make it back to our neck of the woods). So I think we SHOULD be able to talk about RATE of time passage at two different times in history.... > Special relativity defines distance as it would be measured by rulers > at rest relative to an observer, and time as it would be measured by > clocks at rest relative to an observer. It defines an event near an > observer to be simultaneous with a distant event if the nearby event > occurs halfway between the time light needed to have left the observer > to reach the distant event and the time light from the distant event > arrives. I needed to draw a picture for that, but I think I understand you talk about the event horizon here. I understand that. SR also finds that if the same two events are measured by a second, moving observer, that they will not seem to be simultaneous. > In cosmology, distance is usually defined as it would be measured by > rulers at rest relative to the expanding universe (i.e. at rest > relative to most galaxies, the average momentum of the cosmic > background radiation, etc.). Time is defined as it would be measured > by clocks at rest relative to the expanding universe. Two events are > defined to be simultaneous if the age of the universe as measured by > such clocks was the same at both events. Makes sense. But this does not say anything about the RATE of time in the past. This is what I'm thinking of : If observers at-rest 7 billion years ago would measure the age of the universe, would they measure 14-7=7billion years, or would they maybe measure something much longer ? Since the density of the Universe was double back then from where it is now, GR predicts that their time rate must have run slower than ours today. At least as observed from our point of view. So we would observe then with a red-shift. Which, incidentally, happens to look like they are moving away from us... If this is true, then these old observers would have measured the time age of the Universe to be longer than 7 billion years. I am not very strong in GR, but I can imagine that they would observe the Universe to be anywhere from 7 to 14billion years old. If this effect is real, even to a certain extend, then it might make the horizon problem and the age-of-the-universe problem go away completely, so that there is no need for an inflationary theory... > The definitions of distance, time, and simultaneity are all different. > All of these concepts factor in to the determination of the velocity of > an object. > Special relativity assumes flat space-time. It assumes that lines that > are parallel at one place are parallel along their entire length, and > that non-accelerating objects at rest relative to one another are at > rest relative to each other forever. For down-to-earth purposes, the > first assumption is almost exactly true. The second is nowhere near > true if we use general relativity's definition of proper acceleration, > but it can be made almost exactly true by defining freely falling > objects to be accelerating downward. But over cosmological distances, > both assumptions fail miserably. > The consequence of this is that one must be careful when using special > relativity in cosmology. Does special relativity fail? In general, > yes. But along a narrow path at a specific time, or in a small volume, > the errors introduced are insignificant. One can still compare > velocities of distant objects using the special relativistic > definitions, but the answer you get depends on the path between the two > objects on which you apply the assumptions and definitions of special > relativity, and the specific time you are applying them at each point > along the path. > Thus, you can and will get multiple answers from special relativity > when applied to cosmology. However, none of the velocities you can > calculate using its definitions will be greater than the speed of > light. All laws of physics, including the constants seem not to have changed over time. So there is no reason to believe that SR and GR all of a sudden would fail. You are right that we have to be very carefull when applying them to cosmology, since it gets so confusing in a expanding Universe. But I tend to believe that if we apply SR and GR correctly that there would not be a need for a sort of 'out-of-the-blue' theory like the inflationary theory. > But using the common cosmological definitions of distance, time, and > simultaneity I mentioned, the rate of change on the distance to a > far-away object can exceed the speed of light. The reasons for the > discrepancy between this speed and the object's velocity relative to us > according to special relativity are: > 1. By the special relativistic definitions, those rulers at rest > relative to the expanding universe are moving away from us and thus > undergo length contraction. This works to reduce the change of the > distance from us to a far-away object and us as defined by special > relativity. > 2. By the special relativistic definitions, the clocks at rest > relative to the expanding universe are moving away from us and thus > undergo time dilation. This can make the time the object was moving, > as defined by special relativity, longer than it is when using the > cosmological definition. > 3. It might seem that effect #2 could be circumvented by measuring the > time at our location instead of the moving objects'. But the events > cosmology would define as happening in our present, would, by the > special relativistic definitions, be happening in the future. This > discrepancy increases as objects become more distant. Thus, distant > events that occur a short time apart as calculated using our clocks and > the cosmological definition of simultaneity, occur later and further > apart as measured by our clocks using the special relativistic > definition. I understand the relativistic effects of matter moving away from us, but I don't see why we need to explain that/if anything moves faster than light. Did we observe apparent faster-than-light movement that can not be explained by SR effects ? Also, was this related to the theory of inflation ? I thought inflation happened only in the first 10^23 sec or so after BB, when the universe was only of grapefruit size, so there are not a lot of speed measurements we can do... > If you're interested, I would recommend looking over Ned Wright's > tutorial, since he explains a lot of this stuff well in simple terms, > and he has some good illustrations, which are worth much more than a > thousand words. The tutorial website is here: > http://www.astro.ucla.edu/~wright/cosmo_01.htm > You should especially look at the diagram in part 2, where he shows how > a space-time diagram of a nearly empty expanding universe using special > relativistic coordinates differs from the previous diagram using > cosmological coordinates: > http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH > This illustrates what's going on pretty well, and does so much better > than I could ever try to do with words. I do not understand the horizon problem that was presented though. Seems to me there is some strange assumption there that something must be outside of the horizon, but I don't get why that assumption had to be made. I'll read it again. === Subject: Re: Inflationary Theory ; I'm confused I have to shake my head. 15 billion (to infinity) times 9.8 trillion kilometers of space to (per) 15 billion (to infinity) years of time equals a specified magnitude of 300,000 kilometers of space constant to (per) one second of time constant, just as the equivalent spatial unit length to (per) a trillionth of a trillionth of a trillionth of a trillionth of a trillionth of a trillionth of a trillionth of a trillionth of a trillionth....(to infinity of a trillionth) of one second of time also, and still, and always, equals a specified magnitude of 300,000 kilometers of space constant to (per) one second of time constant. 300,000 kilometers of space constant to (constant per) one second of time constant (macro-Universe) = 300,000 kilometers of space constant to (constant per) one second of time constant (micro-universe). 300,000 kps = 300,000 kps 1 second = 1 second 1c = 1c Velocity x = velocity x Acceleration y = acceleration y Motion z = motion z Cosmological constant zero (instantaneous moment) = cosmological constant zero (instantaneous moment) (Universal Real Time zero (instantaneous moment) = universal real time zero (instantaneous moment)). Frame = frame. Only the elastically subjective term relative to the observer causes the plural effect 'magnitudes' (objective shattering of 1c into infinities of 'magnitudes' of 1c). Tunnel vision's elasticity is a real but strictly compartmented physic. Any traveler in motion, as such, whatever his velocity or acceleration in velocity is will have its magnitude reset to match a matching magnitude of 1c, both as to the spatial unit length (1) and as to the unit length of time (1) (1:1). There is no such as observing space. Space is the very essence of simultaneously existing, parallelism, parallel sequential, or parallel universe. So you will observe an extending ruler of light-time frame event histories, the light physic itself, and magnitudes going away of 1c in the light physic itself. These are all you will ever physically observe as physical observers. All told, a seamless space-time continuum VIRTUALITY of space and time. Never the real things. The above is strictly an expression of my point of view of one or more of the realities of the Universe as I see them to be the realities (a point of view strictly relative to this observer). Any similarities to other modeling is either intentional or coincidental. Any dis-similarity is hyper-intentional. Brad I make many mistakes (butler in The Big Sleep). === Subject: Re: Inflationary Theory ; I'm confused <7u0Kd.16248$5R.10745@newssvr21.news.prodigy.com I do not understand the horizon problem that was presented though. > Seems to me there is some strange assumption there that something > must be outside of the horizon, but I don't get why that assumption > had to be made. I'll read it again. These days enlightened Bigbangers claim that all places are equally the center of the universe. So it must follow that an observer at what we perceive to be the horizon must see an equal distance beyond, and back to us, for him to be at his center. Jim G c'=c+v === Subject: Re: Inflationary Theory ; I'm confused > I do not understand the horizon problem that was presented though. > Seems to me there is some strange assumption there that something > must be outside of the horizon, but I don't get why that assumption > had to be made. I'll read it again. > These days enlightened Bigbangers claim that all places are > equally the center of the universe. So it must follow that an observer > at what we perceive to be the horizon > must see an equal distance beyond, and back to us, for him to be at his > center. These days enlightened people claim that all places are equally the center of the surface of the Earth. So it must follow that an observer at what we perceive to be the horizon must see an equal distance beyond, and back to us, for him to be at his center. Dirk Vdm === Subject: Re: Inflationary Theory ; I'm confused > And in a deep gravity well > time moves much slower than outside. Somehow I feel that that effect > might the 'inflation' theory of BB. So inflation theory would not > have to be > postulated 'out-of-the-blue', but could be explained with > relativistic > effects in early dense and expanding Universe.. > Rob, gravity is different at the earth's center than at the surface. > Have you noticed the core circling the sun at a different velocity than > the surface? No. Have you? Before answering, do take into account that there is no photon at the core of the sun which can escape to the surface. ergo, > as v=d/t gravity has NO EFFECT on time. Please show us your calculation. > It only effects the velocity > information. > The marvellous mish-mash of mathemagics conjured up to support GR and > BB are necessary for supporters to claim proofs, but 1c+1c=2c , and > the theories are therefore wrong. What is this marvellous mish-mash of which you speak? Please give us a small quantitative selection of that. Franz === Subject: Re: Inflationary Theory ; I'm confused > If you're interested, I would recommend looking over Ned Wright's > tutorial, since he explains a lot of this stuff well in simple terms, > and he has some good illustrations, which are worth much more than a > thousand words. The tutorial website is here: > http://www.astro.ucla.edu/~wright/cosmo_01.htm > .....and the cosmological pictures which bury SR are those which show > material streams (sic trains) which have been measured to be travelling > at 98%c, and which ARE NOT contracted. How do you know they are not foreshortened? In what direction does this motion take place? > Those who think that looking > ALONG the path of that material ACTUALLY shortens it, are victims of > ILLUSION, and should apply their intelligence into understanding that > distance and time are invariant. Don't become another deluded victim of > information transfer delay! You are drivelling. Now let me tell you of a real observation of relativistic foreshortening: The electric field of a stationary electron is spherically symmetrical. According to SR the field lines of a moving electron become foreshortened in such a way that almost all the field lies in a directon transverse to the direction of motion. The field lines so to speak gt squashed flat, so that at speeds of, say, 99.99% of that of light, which are dead easy to obtain in a high energy lab., the field lines in fact look flatter than the spokes of a bicycle wheel. The reality of this has beem proved as follows: There is a calculation, called the Weiszacker-Williams approximation, which says that these field lines are all so nearly transverse to the direction of motion of the electron, and they move so nearly with the speed of light, that they will behave very nearly like a collection of photons. On this basis they calculated how these photons would be deflected by a Compton scattering process as they interacted with the charge of a nearby nucleus. They ended with a formula for the production cross section for Bremsstrahlung. The calculated value and the experimental values agreed for all the nucleii and all the electron energies for which data was available. Needless to say, if this foreshortening is not taken into account, the predictions turn out to be crap. Franz === Subject: Re: Inflationary Theory ; I'm confused > And in a deep gravity well > time moves much slower than outside. Somehow I feel that that effect > might the 'inflation' theory of BB. So inflation theory would not > have to be > postulated 'out-of-the-blue', but could be explained with > relativistic > effects in early dense and expanding Universe.. > Rob, gravity is different at the earth's center than at the surface. > Have you noticed the core circling the sun at a different velocity than > as v=d/t gravity has NO EFFECT on time. But then, how come that clocks tick slower in space than they do here on Earth ? > It only effects the velocity > information. How does that differ from the rate of time ? > The marvellous mish-mash of mathemagics conjured up to support GR and > BB are necessary for supporters to claim proofs, but 1c+1c=2c , and > the theories are therefore wrong. Sorry, you lost me. > Jim G > c'=c+v === Subject: Re: Inflationary Theory ; I'm confused <7lHHd.13349$5R.2174@newssvr21.news.prodigy.com> time moves much slower than outside. Somehow I feel that that effect > might the 'inflation' theory of BB. So inflation theory would not > have to be > postulated 'out-of-the-blue', but could be explained with > relativistic > effects in early dense and expanding Universe.. > Rob, gravity is different at the earth's center than at the surface. > Have you noticed the core circling the sun at a different velocity than ergo, > as v=d/t gravity has NO EFFECT on time. > But then, how come that clocks tick slower in space than > they do here on Earth ? Because the tick of a clock is not chained to absolute time- the clock in space malfunctions. > It only effects the velocity > information. > How does that differ from the rate of time ? Wrong question: rate of time NEVER alters- only our perception of it, as in running a movie slower, or playing an old one. > The marvellous mish-mash of mathemagics conjured up to support GR and > BB are necessary for supporters to claim proofs, but 1c+1c=2c , and > the theories are therefore wrong. > Sorry, you lost me. Well, if you do think that velocity adding as proposed by SR contradicts the above, then you are indeed on a different position ref earth's orbit than I am. (But I am satisfied we are co-travellers; the fact we live in different gravity fields has F*A to do with time/velocity) > Jim G > c'=c+v === Subject: Re: Inflationary Theory ; I'm confused > as v=d/t gravity has NO EFFECT on time. > But then, how come that clocks tick slower in space than > they do here on Earth ? I'm sorry. That should be : How come that clocks tick slower here on Earth than they do in (free-fall) space ? === Subject: Re: Inflationary Theory ; I'm confused >> You don't believe in - absolute - truth KPD? >> Are you absolutely sure? >Yes. >I have breathed air once breathed by Heisenberg, i.e., all problems are equivalent to the problem of ascertaining the position & momentum of an >used numbers once used by Goedel, suffered >despair and the collapse of all my hopes as >Frege once did, and thus achieved enlightenment. >Others also have this option. Endeavor to persevere === Subject: Re: Inflationary Theory ; I'm confused >>Thesis: >>Space can't expand. > Proof: >>If space would be able to expand, it would be >>bigger than it is, a contradiction. > Wrong. Put a number in a spreadsheet cell. Allow iteration. Allow the > cell to increase its value by a tiny bit on each pass. What is the value > expanding into? Nothing, it is not expanding, the value is changing that's all. The space used by the spreadsheet cell doesn't change with the cell value unless the spredsheet resizes. What this has to do with space expansion is not at all clear. Can you elaborate? === Subject: Re: Inflationary Theory ; I'm confused >Thesis: >Space can't expand. >> Proof: >If space would be able to expand, it would be >bigger than it is, a contradiction. >> Wrong. Put a number in a spreadsheet cell. Allow iteration. Allow the >> cell to increase its value by a tiny bit on each pass. What is the >> value expanding into? > Nothing, it is not expanding, the value is changing that's all. > The space used by the spreadsheet cell doesn't change with > the cell value unless the spredsheet resizes. > What this has to do with space expansion is not at all > clear. > Can you elaborate? The distance between A and B is increasing. For all A and B sets. It is a trivial readjustment in the relationship between these points (ostensibly marked by physical bodies, perhaps). Since it appears that space is the product of mass and energy, then expansion is simply an adjustment in this spreadsheet that was created for this purpose. There is no evidence of an absolute. Even an absolute space for relative space to expand into. David A. Smith === Subject: Re: Inflationary Theory ; I'm confused >>Thesis: >>Space can't expand. > Proof: >>If space would be able to expand, it would be >>bigger than it is, a contradiction. > Wrong. Put a number in a spreadsheet cell. Allow iteration. Allow the > cell to increase its value by a tiny bit on each pass. What is the value > expanding into? >>Into what would >>space itself expand? > An alternative view, is that all of *now* is contracting. >>Into more space. > Wrong. Since space is not contained, then it is free to be merely a > relationship between all the matter in the Unvierse. Agreed. The subject matter here is whether the relationship is a *constant* one, i.e. if a constant yardstick can be used to measure the distance betwee two points A and B. Of course space is a relationship between all matter. For any points A and B, there is a relationship called distance! This is trivial. >>The limit of the incredible is reached with a >>theory that says that space expands faster than >>light. > Depends on how you define incredible, doesn't it? What is incredible to > me is how people don't use a search engine, and find good information sites > like: > URL:http://www.astro.ucla.edu/~wright/cosmolog.htm Your answer doesn't address any issue. You just imply that I should go on reading sites that speak about space expansion as it was a matter of fact. The last year brought new data from our scopes, specially from Chandra that discovered a huge black hole at 13.5 billion years, i.e. almost at the supposed big bang. This is an observation, not a theory. Now how can an active galaxy nucleus exist, i.e. having been created, formed, evolved into a black hole all in just 500 million years? > ... Here you cut without answering the difference between the observable universe (the subject of science) and the universe (the subject of meta-physics). >>Science can't offer any further explanation. With this I was resuming the argument you cut off. The subject of science is the observable universe and science can't go beyond it. > Armchair philosophers run out of explanations a lot faster than science > does. > David A. Smith No arguments, just an arrogant sentence to hide the fact that you presented no data or arguments to substantiate your opinion. Is the distinction between the observable universe and the universe clear to you? === Subject: Re: Inflationary Theory ; I'm confused >Thesis: >Space can't expand. >> Proof: >If space would be able to expand, it would be >bigger than it is, a contradiction. >> Wrong. Put a number in a spreadsheet cell. Allow iteration. Allow the >> cell to increase its value by a tiny bit on each pass. What is the >> value expanding into? >Into what would >space itself expand? >> An alternative view, is that all of *now* is contracting. >Into more space. >> Wrong. Since space is not contained, then it is free to be merely a >> relationship between all the matter in the Unvierse. > Agreed. The subject matter here is whether the relationship > is a *constant* one, i.e. if a constant yardstick can be > used to measure the distance betwee two points A and B. No. Physical objects accumulate damage. The alloy rod used in Paris expanded at the rate of 1 part in 10^8 per year, every year, since at least the 1940s. Similarly, a steel rod was found to shrink by 1 part in 10^8, for some fairly long period of time. The yardstick is light and time. And this shows expansion. Maybe not in the Solar System... > Of course space is a relationship between all matter. For any > points A and B, there is a relationship called distance! This > is trivial. here and not here, distance is forged... >The limit of the incredible is reached with a >theory that says that space expands faster than >light. >> Depends on how you define incredible, doesn't it? What is incredible >> to me is how people don't use a search engine, and find good information >> sites like: >> URL:http://www.astro.ucla.edu/~wright/cosmolog.htm > Your answer doesn't address any issue. You just imply that I should go > on reading sites that speak about space expansion as it was > a matter of fact. It fits the facts. > The last year brought new data from our scopes, specially from > Chandra that discovered a huge black hole at 13.5 billion years, > i.e. almost at the supposed big bang. I wouldn't be surprised if we found structures that survived the event horizon that we call the Big Bang. Revel in the truth. > This is an observation, not a theory. Now how can an active galaxy > nucleus exist, i.e. having been created, formed, evolved into a black > hole all in just 500 million years? Let's see, observation that is *validated* by a theory you spurn is... what exactly? >> ... > Here you cut without answering the difference between the observable > universe (the subject of science) and the universe (the subject of > meta-physics). How long do you wish this thread to be? >Science can't offer any further explanation. > With this I was resuming the argument you cut off. The subject of > science is the observable universe and science can't go beyond > it. And this cannot go anywhere good, since I do not wish to debate this. >> Armchair philosophers run out of explanations a lot faster than science >> does. > No arguments, just an arrogant sentence to hide the fact that you > presented no data or arguments to substantiate your opinion. Pot. Kettle. Black. > Is the distinction between the observable universe and the universe > clear to you? As clear as the difference between any observable, and the construct the human mind fabricates to provide mechanism to *justify* the observations. We cannot impress our flatlander belief system on the Universe, and expect it not to laugh. But we can use science to make predictions. And GR makes predictions that work. If space expands, Hubble redshift (of both light frequency and duration of events) is a result. David A. Smith === Subject: Re: Inflationary Theory ; I'm confused proposed by Groth, which states that the early Universe went > through a period of very rapid expansion called 'inflation'. > I understand that the inflation theory was > invented to explain the 'flat-ness' of space and the > MBR isotropy (microwave background indicates that > the early Universe had a very consistent high temperature). > I am confused about this inflationary period. > I've read somewhere that the Universe expanded to > a size of 40 million light years within a miniscule time. > This would clearly violate the limitation of the speed > of light. > IANACosmologist, but I think the speed limit applies to moving stuff > around in space, wheras the expansion of the universe is an expansion > of space itself, not subject to the speed limit. I am wondering then what constitutes space if there is little stuff occupying it? I thought space/time were aspects of the physical universe and that moving one directly moves the other so to speak and that the properties of the space WAS defined by the stuff in it. > If the universe is big enough, some of it will be receding from us > faster than the speed of light even today, due to the expansion of > space. What is the Universe defined by then if not the material in it that is moving around? Wouldn't we have to have something OTHER than what is typically known as empty space to differentiate it from the thing it was expanding in or expanding to become? According to general relativity, the concept of space detached from any physical content does not exist. The physical reality of space is represented by a field whose components are continuous functions of four independent variables-the coordinates of space and time. It is just this particular kind of dependence that expresses the spatial character of physical reality. Since the theory of general relativity implies the representation of material points cannot play a fundamental part, nor can the concept of motion. Just as Maxwell and Faraday assumed that a magnet creates certain properties in surrounding space, so Einstein concluded that stars, moons, and other celestial objects individually determine the properties of the space around them. And just as the movement of a piece of iron in a magnetic field is guided by the structure of the field, so is the path of any body in a gravitational field determined by the geometry of that field. Lincoln Barnett in The Universe and Dr. Einstein Edmond H. Wollmann P.M.A.F.A. © 2005 Altair Publications, SAN 299-5603 Astrological Consulting http://www.astroconsulting.com/ Artworks http://www.astroconsulting.com/personal/ http://www.astroconsulting.com/SDSU === Subject: Re: Inflationary Theory ; I'm confused > I thought space/time were aspects of the physical > universe and that moving one directly moves the other so to speak and > that the properties of the space WAS defined by the stuff in it. Then again, you also thought that dinosaurs were the size they were because of different gravity back then, so you kind of up any bell curve you're associated with. -- Dan Baldwin, unethical *by design* I am a minion of Satan, but my powers are mainly administrative. Hail the un-alive === Subject: Wollmannizer nocem 03344 @@NCM http://www.smbtech.com/ed/ http://www.nocem.org/ http://www.rahul.net/falk/quickrefs.html#W -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 http://www.smbtech.com/ed/ @@BEGIN NCM HEADERS Version: 0.9 Type: spew Action: hide Count: 1 Notice-ID: Wollmannizer03344 @BEGIN NCM BODY sci.astro sci.math sci.physics sci.skeptic @END NCM BODY -----BEGIN PGP SIGNATURE----- Version: PGP 6.5.8 iQA/AwUBQfU7C1IcW5ONdL49EQLQrwCgvFyN/rb3r27PYAaG5ZGY+P238CYAn320 uMxOSkxcKNI4hcRA2IwQfVtJ =sF3P -----END PGP SIGNATURE----- === Subject: Re: Inflationary Theory ; I'm confused http://www3.primushost.com/~a/raymurphy/rectification.html >> about the Big Bang Theory. Part of that is a theory >> proposed by Groth, which states that the early Universe went >> through a period of very rapid expansion called 'inflation'. >> I understand that the inflation theory was >> invented to explain the 'flat-ness' of space and the >> MBR isotropy (microwave background indicates that >> the early Universe had a very consistent high temperature). >> I am confused about this inflationary period. >> I've read somewhere that the Universe expanded to >> a size of 40 million light years within a miniscule time. >> This would clearly violate the limitation of the speed >> of light. >> IANACosmologist, but I think the speed limit applies to moving stuff >> around _in_ space, wheras the expansion of the universe is an > expansion >> of space itself, not subject to the speed limit. > I am wondering then what constitutes space if there is little stuff > occupying it? Jeez, more sig material from the kook that thought Tau Ceti was in Taurus, Orion is a star and dinosaurs were so big because of lessened gravity. > I thought space/time were aspects of the physical > universe and that moving one directly moves the other so to speak and > that the properties of the space WAS defined by the stuff in it. What happened to all that claptrap about a Multiverse, Pantihead? As usual, you got cause and effect backwards. >> If the universe is big enough, some of it will be receding from us >> faster than the speed of light even today, due to the expansion of >> space. > What is the Universe defined by then if not the material in it that is > moving around? Wouldn't we have to have something OTHER than what is > typically known as empty space to differentiate it from the thing it > was expanding in or expanding to become? Ever hear of black holes, Ed? There's one right between your ears in case you need a comparison. Intelligent words go in, and gibberish, screed and kookiness exit. It's positively amazing! -- Cujo - The Official Overseer of Kooks and Trolls in dfw.*, alt.paranormal, alt.astrology and alt.astrology.metapsych. Colonel of the Fanatic Legion. FL# 555-PLNTY Motto: ABUNDANCE!. Charter Member - Digital Brownshirts and Library Gestapo. focusing with laserlike precision and crystal clarity. === Subject: Re: Inflationary Theory ; I'm confused >about the Big Bang Theory. Part of that is a theory >proposed by Groth, which states that the early Universe went >through a period of very rapid expansion called 'inflation'. >I understand that the inflation theory was >invented to explain the 'flat-ness' of space and the >MBR isotropy (microwave background indicates that >the early Universe had a very consistent high temperature). >I am confused about this inflationary period. >I've read somewhere that the Universe expanded to >a size of 40 million light years within a miniscule time. >This would clearly violate the limitation of the speed >of light. >>IANACosmologist, but I think the speed limit applies to moving stuff >>around _in_ space, wheras the expansion of the universe is an > expansion >>of space itself, not subject to the speed limit. > I am wondering then what constitutes space if there is little stuff > occupying it? The same thing that occupies your intercranial compartment, dumb. === Subject: Re: the monkeyhouse is back... I am very interested in the new monkey house site. Do you need still several time to update ? Biagio Dave Anderson ha scritto nel messaggio > Check out the resurrected geodesic monkey house... geodesic mathematics, > inventions like the warped octet truss and blending functions, the > generalization of minimal waste dome coverings, and more... > http://www.tekcad.com/monkeyhouse === Subject: Re: the monkeyhouse is back... > Check out the resurrected Sort of resurrected. please be patient as the papers are revised and added back to the net! === Subject: Re: The untestability of ESP > When you're driving, and you > know the guy in the next lane wants to get over in your lane, even > though > he's not drifting over and he doesn't have his signal on. So you slow > down, and sure enough, there he goes, zip right in front of you. I > don't > think these things are always just coincidences. Well Michelle, like yourself I also believe in ESP but I think you might > want to listen to the skeptics on this one. If you slow down and the guy > in > the next lane has a car in front of him then maybe he's just thinking > (correctly) that you're letting him through and so taking you up on your > kind offer? ~ > Steve Could be. It was just an example of when you apparently know what someone > else is about to do. Another explanation is that you saw the guy's face in > the rearview mirror, and he was looking over at your lane, but you didn't > consciously realize it at the time. I don't believe in ESP with any fervor. It's just one of those things that > makes intuitive sense, even if not scientific sense. And so far I haven't > read anything scientific that would make ESP impossible. I know, I know, > the burden of proof is on the believer. But it just seems like almost > everyone has these experiences, and having to come up with a different > alternative explanation for every so-called psychic experience seems to > violate Occam's Razor. Maybe the simplest explanation is that the human > brain, which we don't yet fully understand, has some sensory abilities > beyond what we've discovered so far. Given how far we've come just in the > last 50 years in understanding the brain and its processes, it wouldn't > surprise me if it turns out ESP actually does exist. I don't think it's > reliable and shouldn't be used to make a living. But then again, memory > isn't all that reliable either. -- > Michelle Levin > http://www.mindspring.com/~lunachick I have only 3 flaws. My first flaw is thinking that I only have 3 flaws. > Occams razor on the side of psychics? Makes sense, its usually the other way > around. You know 100 out of 105 people in sci.skeptic all believe that > every believer of esp is mentally disturbed. > Like so: > You think there's a magical ghost helping you giving you answers? or > your just a looney? the simplest explanation is you're a looney! > alt.kook.watcher.central, full of 1000s of predators wanting to abuse > people for being frauds and such. > Anecdotal evidence requires a LOT of study to make facts out of, > and skeptics don't have the memory capacity or the attnetion span. > sci.skeptics doesn't have any scientific skeptical enquirers, its purely HARD > skeptics, > they think you have no right to talk about supernatural, and they'll prove > your > a fraud by your very spelling on your 1st post. > Herc > Um, ok. I don't think psychic abilities have anything to do with ghosts or > the supernatural. I think it's entirely natural, and perhaps has something > in the first brain. But, my guess is that the process is totally random and > uncontrollable. We know that thoughts affect the physical reality of the > body, but suggesting that maybe thoughts can affect the physical reality > _outside_ the body gets you labeled as a kook? Maybe only by mean people. > ;o) To some hard skeptics, one-2-one telepathy is on the same scale as one-2-many superbeings, but there is some interest here, as long as there's no external force at play. I should've read your claims more carefully, detecting the phone ring would just be your sister thinking about ringing you at the time.. cause and effect. Well here's my take on your molecule level theory. Molecules DO have that property, you can seperate the 2 electrons in the low level shell around the nucleas and they still 'communicate' with each other. But its VERY limited, there is a single reduction of the pairing of the electrons into seperate entities, but it can happen any time after they are seperated. You could take one part of a quantum entangled atom to the moon, and leave the other half on Earth (one electron each), and depending on what you do to one it will effect the other. These kinds of experiments, called non-locality are very active research right now. Sounds promising, and of course you are correct that is does make the underlying principle of esp, however the 2 seperate entangled electrons, (now in new 2 atoms) can't *send information*. There's some strange distiction between 'influence', i.e. change what it does, and realising that change. So you can't build a walkie talkie out of it! But, if the phenomenum is widespread, on a universal scale, then it could be affecting us is myriads of ways we don't even realise. Say that car in front of you held a child genius who would push the frontiers of science 1000 years. The 'collective' quantum effect could be strong enough to 'optimise' his surrounds and signal for you to give way! (the last bits my own theory, to bridge the gap from quantum entanglement to telepathy you have to introduce a superior being, no information can be transferred). Herc === Subject: put/call parity equation I think this is a simplified version, but we are using: C_E(S) + E*exp(-r(T-t) ) = S + P_E(S) The question is to find a lower bound for the call and put price. C_E(S) = -E*exp(-r(T-t) ) + S + P_E(S) Making the assumption that P_E(S) > 0 C_E(S) > -E*exp(-r(T-t) ) + S Similarly with the put: P_E(S) > E*exp(-r(T-t) ) - S However, when I plug in some given numbers, I get something reasonable for C_E(S), but I am getting negative numbers for P_E(S). It is clear why--the numbers I'm using S > E*exp(-r(T-t) ). Anyone know what I'm doing wrong? === Subject: evaluation of definite integrals I know the value of the definite intgral Int(cosA.dt, 0->t) = kt where A is also a function fo t say A = f(t). The exact behaviour of f(t) is not known. Only with this information is it possible to calculate Int(sinA.dt, 0->t) ? Harsha === Subject: Re: evaluation of definite integrals > I know the value of the definite intgral > Int(cosA.dt, 0->t) = kt > where A is also a function fo t say A = f(t). The exact behaviour of >f(t) is not known. > Only with this information is it possible to calculate > Int(sinA.dt, 0->t) ? There are various problems with your notation, but if you mean what I think you mean then Int(cosA.dt, 0->t) = kt for all t implies that cos(A) = k. Assuming A is continuous it follows that sin(A) is also constant, equal to plus or minus sqrt(1-k^2). > Harsha ************************ David C. Ullrich === Subject: Re: evaluation of definite integrals no, it is not possible with only that information. === Subject: Re: evaluation of definite integrals >no, it is not possible with only that information. Many readers of this Newsgroup will not have a clue what you are talking about. What is not possible, and with only what information? How do you expect anybody to know which question you are replying to? Derek Holt. === Subject: Re: evaluation of definite integrals >no, it is not possible with only that information. > Many readers of this Newsgroup will not have a clue what you are talking > about. What is not possible, and with only what information? How do you > expect anybody to know which question you are replying to? > Derek Holt. Bryant's message has the header which shows which message he is replying to. So news readers can show this in a threaded display. If yours didn't maybe the first message is missing from your news server; or maybe your news reader is very primitive. === Subject: EC-funded Research Visits: Application Deadline 28/02/2005 Are you a scientist of postgraduate level or above, working in EU or an Associated State? Do you require large computing power to improve your research? Would you like to visit a similar research group in Italy, Spain, UK, The Netherlands, France, or Germany? ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + + + EC-funded HPC-Europa Visitor programme is the answer + + + ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ See what we can offer you: - Access to the best high performance computing systems in Europe - Technical support and consultancy - Scientific collaboration with a host researcher - Travel and living expenses - Logistical and administrative support See http://www.hpc-europa.org/ta.html for more information and online application form. The Programme runs until the end of 2007. ++++++++++++++++++++++++++++++++++++++++++++++++++++++ + Current deadline for applications + + is 28 February 2005 + ++++++++++++++++++++++++++++++++++++++++++++++++++++++ Appications for visits can be made to: CINECA (Italy), CEPBA (Spain), EPCC (UK), HLRS (Germany), IDRIS (France), SARA (The Netherlands) === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <3552smF4ek6noU1@individual.net> <1xdHd.1366$Ju1.277@newsread3.news.pas.earthlink.net> <41efff44$11$fuzhry+tra$mr2ice@news.patriot.net> <41f3ed34$7$fuzhry+tra$mr2ice@news.patriot.net ROTF,LMAO! TYVM. ??? Can't understand. If you want me to understand, please explain. >Now I state in public: You have been wrong >unwillingly, because you did not know better either. But meanwhile >you are willfully cheating. > Now I state an public: you are a fool or a liar. That does not improve your position. The fact that someone > disagrees with you does not mean that he is wilfully cheating. Of course, not. But if he claims something which has been proven wrong, he has to adjust his position. In case of epsilon it is not a matter of time but a matter of priciple. There is not enough storage space available in the universe to have simultaneously available more than 10^100 digits or bits. Therefore irrational numbers are irrational in fact. === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <4UvHd.111$r27.100@newsread1.news.pas.earthlink.net> (Int(Pi*10^10^100))/3 is the number in question. > Okay ... We know, from what you've revealed to us, that > no irrationals exist; that although 10^10^10^10 exists, > Floor(pi*10^10^100)/3 does not ... My concern is that > too many Right Thinking mathematicians will be pestering > you to help them identify which are the relatively few > Extant Numbers. Would it possible to publish a list? I would like to return to the proof which would convince all thinkers, some earlier, some later, that not all numbers exist. Here is a slightly improved version: Consider the set Q of all rationals of (-oo, oo). Add Pi to each of them. This gives an equinumerous set X. In the intervall (0,1) we have equinumerous subsets IQ and IX. By reasons of symmetry, the elements of IQ U IX, when ordered by magnitude (which is not possible for us to do, but which exists, if these numbers do exist), form a set which is alternating between rationals and irrationals. This set contains all rational, but we can include some more irrationals. The result is a set which does not have a rational number between any pair of irrational numbers. Regrads, WM === Subject: Re: abundance of irrationals [correction -- my apology for misspelling Mueckenheim] >> Okay ... We know, from what you've revealed to us, that >> no irrationals exist; that although 10^10^10^10 exists, >> Floor(pi*10^10^100)/3 does not ... My concern is that >> too many Right Thinking mathematicians will be pestering >> you to help them identify which are the relatively few >> Extant Numbers. Would it possible to publish a list? -snip discussion of irrationals- About those natural numbers that you say don't exist ... You say that the number represented by Floor(pi*10^10^100) does not exist, arguing that it will never be available and that one can't calculate with it. So (1) you say the number T represented by 10^10^10^10 *exists*, and (2) you say some natural numbers less than T do *not* exist (e.g., the number represented by Floor(pi*10^10^100)). Consider this finite list of consecutive numbers, which starts and ends with numbers that *you* say exist, but which also contains numbers which *you* say do not exist: Representation of n Does n Mueckenheim-exist? ------------------- ------------------------- 10^10^10^10 Yes 10^10^10^10 - 1 ? 10^10^10^10 - 2 ? 10^10^10^10 - 3 ? ... ... floor(pi*10^10^100) No floor(pi*10^10^100) - 1 ? floor(pi*10^10^100) - 2 ? floor(pi*10^10^100) - 3 ? ... ... 3 Yes 2 Yes 1 Yes 0 Yes Where in the list does the mysterious disappearing act begin? What's the least natural number with no Mueckenheim-successor? --r.e.s. === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <4UvHd.111$r27.100@newsread1.news.pas.earthlink.net> > Okay ... We know, from what you've revealed to us, that >> no irrationals exist; that although 10^10^10^10 exists, >> Floor(pi*10^10^100)/3 does not ... My concern is that >> too many Right Thinking mathematicians will be pestering >> you to help them identify which are the relatively few >> Extant Numbers. Would it possible to publish a list? -snip discussion of irrationals- About those natural numbers that you say don't exist ... You say that the number represented by Floor(pi*10^10^100) does not exist, arguing that it will never be available and that one can't calculate with it. So (1) you say the number T represented by 10^10^10^10 *exists*, and (2) you say some natural numbers less than T do *not* exist (e.g., the number represented by Floor(pi*10^10^100)). Consider this finite list of consecutive numbers, which starts and ends with numbers that *you* say exist, but which also contains numbers which *you* say do not exist: Representation of n Does n Muckenheim-exist? ------------------- ------------------------ 10^10^10^10 Yes 10^10^10^10 - 1 ? 10^10^10^10 - 2 ? 10^10^10^10 - 3 ? ... ... floor(pi*10^10^100) No floor(pi*10^10^100) - 1 ? floor(pi*10^10^100) - 2 ? floor(pi*10^10^100) - 3 ? ... ... 3 Yes 2 Yes 1 Yes 0 Yes Where in the list does the mysterious disappearing act begin? What's the least natural number with no Muckenheim-successor? --r.e.s. === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <4UvHd.111$r27.100@newsread1.news.pas.earthlink.net> You say that the number represented by Floor(pi*10^10^100) > does not exist, arguing that it will never be available > and that one can't calculate with it. > So > (1) you say the number T represented by 10^10^10^10 *exists*, > and > (2) you say some natural numbers less than T do *not* exist > (e.g., the number represented by Floor(pi*10^10^100)). > Consider this finite list of consecutive numbers, which > starts and ends with numbers that *you* say exist, but > which also contains numbers which *you* say do not exist: > Representation of n Does n Muckenheim-exist? > ------------------- ------------------------ > 10^10^10^10 Yes > 10^10^10^10 - 1 ? > 10^10^10^10 - 2 ? > 10^10^10^10 - 3 ? > ... ... > floor(pi*10^10^100) No > floor(pi*10^10^100) - 1 ? > floor(pi*10^10^100) - 2 ? > floor(pi*10^10^100) - 3 ? > ... ... > 3 Yes > 2 Yes > 1 Yes > 0 Yes > Where in the list does the mysterious disappearing act begin? > What's the least natural number with no Muckenheim-successor? There is no mysterious disappearing. If you have accomplished to find out the fraction x/1, i.e. if you have determined how many units x contains, then a number does exist for you. The number 3.14 does exist, because we know that 3 units, 1 tenth of it and four hundredths make up this number. This can be calculated for all numbers of the above list except > floor(pi*10^10^100) No > floor(pi*10^10^100) - 1 ? > floor(pi*10^10^100) - 2 ? > floor(pi*10^10^100) - 3 ? > ... .. You see, there are some numbers, which may possibly be raised into existence, like floor(pi*10^10^50), which however do not yet exist for us humans (and I am sure nor for anybody else). And there are some numbers which never will come into being because all of their information cannot be contained simultaneously in the universe. There is no strict limit to define. === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <4UvHd.111$r27.100@newsread1.news.pas.earthlink.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> You say that the number represented by Floor(pi*10^10^100) >> does not exist, arguing that it will never be available >> and that one can't calculate with it. > You see, there are some numbers, which may possibly be raised into > existence, like floor(pi*10^10^50), which however do not yet exist for > us humans (and I am sure nor for anybody else). Not yet exist does not make sense in mathematics. The existence is a consequence of axioms, and those do not come into being slowly. > And there are some numbers which never will come into being because > all of their information cannot be contained simultaneously in the > universe. The universe is not involved in axioms. Anyway, of course, _all_ of the information of floor(pi*10^10^100) is easily contained in the universe: I can write it down with about 20 characters. 3.14 is a recipe for computation, and so is floor(pi*10^10^100) (because pi can again be split into a recipe). So you are just talking nonsense. If you want to get into some more serious philosophical domain, all you have to notice that _every_ number for which you can write a symbolic definition down is a computable number. And there are only countably many of those. But the set of irrational numbers is much larger: the majority of its members are not computable: still nobody can specify a _single_ noncomputable member of it. Now _that_ is a philosophical problem: talking about properties of numbers that one can't even specify singly in any manner: you only encounter them as unnamed entities in proofs. And one works with the properties of the continuum consisting of them. And this _is_ an interesting question for mathematical philosophy: should one create and work with models that give one entities one cannot ever specify? floor(pi*10^10^100) _is_ a specification. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: abundance of irrationals > Here is a slightly improved version: Consider the set Q of all No scare quotes necessary. > rationals of (-oo, oo). Add Pi to each of them. This gives an > equinumerous set X. In the intervall (0,1) we have equinumerous subsets > IQ and IX. By reasons of symmetry, the elements of IQ U IX, when > ordered by magnitude (which is not possible for us to do, ? > but which > exists, if these numbers do exist), form a set which is alternating > between rationals and irrationals. He really does think that rationals and irrationals alternate! :-( What actually is the state of affairs with this set. We have Q and X = { pi + a: a in Q } and consider Y = X union Q. What is the case is that between any two distinct elements of Y there are countably infinitely many elements of Q and countably infinitely many elements of X. They do not alternate in any sense. > This set contains all rational, No scare quotes necessary. > but we can include some more irrationals. Does this mean that there is a set Y' such that Y' is a subset of R and Y is a proper subset of Y'. That's true, but so what? > The result is a set which > does not have a rational number between any pair of irrational numbers. No. Typical Muckenheimian non-sequitur. Although I can perhaps reconstruct Muckenheim's thought process: M supposes that in the set Y the numbers go ... rational, irrational, rational, irrational, rational ... (absurd, there is no immediate successor to any number in this ordered set of course!). He then puts in another irrational ... obviously this must go into one of the gaps (tee-hee!) between a rational and an adjacent (groan!) irrational yielding a sequence like ... rational, irrational, irrational, rational, ... and so there are two adjacent irrationals without a rational between them! Totally cretinous of course! -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <4UvHd.111$r27.100@newsread1.news.pas.earthlink.net> M supposes that in the set Y the numbers go ... rational, irrational, > rational, irrational, rational ... (absurd, there is no immediate > successor to any number in this ordered set of course!). He then > puts in another irrational ... obviously this must go into > one of the gaps (tee-hee!) between a rational and an adjacent > (groan!) irrational yielding a sequence like > ... rational, irrational, irrational, rational, ... and so there are > two adjacent irrationals without a rational between them! > Totally cretinous of course! Can you imagine another model in case these number would really have actual existence? But if you are unable to grasp it at one, try the following model. Imagine all the rational numbers. Add one irrationale number number like pi to this ordered (but not well ordered) set. Can you imagine this? === Subject: Re: abundance of irrationals >> Although I can perhaps reconstruct Muckenheim's thought process: >> M supposes that in the set Y the numbers go ... rational, irrational, >> rational, irrational, rational ... (absurd, there is no immediate >> successor to any number in this ordered set of course!). He then >> puts in another irrational ... obviously this must go into >> one of the gaps (tee-hee!) between a rational and an adjacent >> (groan!) irrational yielding a sequence like >> ... rational, irrational, irrational, rational, ... and so there are >> two adjacent irrationals without a rational between them! >> Totally cretinous of course! Well, does this represent accurately your thoughts? > Can you imagine another model in case these number would really have > actual existence? Imagination has nothing to do with it. > But if you are unable to grasp it at one, try the following model. > Imagine all the rational numbers. Add one irrationale number number > like pi to this ordered (but not well ordered) set. Can you imagine > this? Again imagination has nothing to do with it. Both Q and Q union {pi} are dense totally ordered sets. No element of either has an immediate successor: in between any two distinct elements there are infinitely many other elements. It's really a very simple situation to understand (that said it is still beyond the intellectual capacities of many as evidence here shows). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <4UvHd.111$r27.100@newsread1.news.pas.earthlink.net> Discussion, linux) >> Although I can perhaps reconstruct Muckenheim's thought process: >> M supposes that in the set Y the numbers go ... rational, irrational, >> rational, irrational, rational ... (absurd, there is no immediate >> successor to any number in this ordered set of course!). He then >> puts in another irrational ... obviously this must go into >> one of the gaps (tee-hee!) between a rational and an adjacent >> (groan!) irrational yielding a sequence like >> ... rational, irrational, irrational, rational, ... and so there are >> two adjacent irrationals without a rational between them! >> Totally cretinous of course! > Can you imagine another model in case these number would really have > actual existence? What does actual existence mean? Do some elements in a model really have actual existence and others not? > But if you are unable to grasp it at one, try the following model. > Imagine all the rational numbers. Add one irrationale number number > like pi to this ordered (but not well ordered) set. Can you imagine > this? I guess it depends. Am I also supposed to be pretending that irrational numbers don't exist? Or is that a different game of make-believe? If I am allowed to know that pi exists, then of course I can imagine the set Q u {pi}. So what? -- Now I realize that he got away with all of that because sci.math is not important, and the rest of the world doesn't pay attention. Like, no one is worried about football players reading sci.math postings! -- James S. Harris on jock reading habits === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <3552smF4ek6noU1@individual.net> <1xdHd.1366$Ju1.277@newsread3.news.pas.earthlink.net> <87d5w1l34w.fsf@phiwumbda.org> don't make sense in that context. The number of atoms, the number of > atomic states, the number of time units, the number of square > centimeters in the universe: all of those do not make sense to stuff > into the concept of an abstract number. Try two utter your thoughts other than by physical means, try to think without relying to the matter of your brain. You will fail. Therefore physics is the basic science fof all others. As physical constraints limit mathematics, it is nothing but hope in vain to see mathematics independent of physics. PS. You can write tall tales in a book, but not all of them. === Subject: Re: abundance of irrationals <2HZGd.9414$pZ4.1757@newsread1.news.pas.earthlink.net> <3552smF4ek6noU1@individual.net> <1xdHd.1366$Ju1.277@newsread3.news.pas.earthlink.net> <87d5w1l34w.fsf@phiwumbda.org> think without relying to the matter of your brain. > You are just pixels on my screen. Try telling me something without > relying on the pixels of my monitor. You can't think more than can be > expressed with 80x43 characters. > That is the kind of logic you employ. No, that is your mistaken perception of my Mathe-Realism. > PS. You can write tall tales in a book, but not all of them. > I can print all of them using the letters of the alphabet > corresponding to the language in question. I can print all proper > English books using the basic 26 letters. No. You must not mix up each and all (in both of the meanings). === Subject: Re: abundance of irrationals > Therefore > physics is the basic science fof all others. Imperialistic, as ever :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Banach valued functions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RCKd302330; I am rading a PDE book that says : It is clear C(I,X*) is contained in D'(I,X*) . Where X Banach space , X* its dual , I an interval in R and D(I,X) the space of test functions and D'(I,X*) the space of continuous linear functions on D(I,X). Now my question is given L in C(I,X*) and u in D(I,X) how does L act on u ?? Are we supposed to interpret = integral{ : t in I} ? Any comments would be greatly appreciated . Don === Subject: Re: Banach valued functions > I am rading a PDE book that says : > It is clear C(I,X*) is contained in D'(I,X*) . > Where X Banach space , X* its dual , I an interval in R > and D(I,X) the space of test functions and D'(I,X*) the > space of continuous linear functions on D(I,X). > Now my question is given L in C(I,X*) and u in D(I,X) > how does L act on u ?? > Are we supposed to interpret > = integral{ : t in I} ? yes === Subject: Re: Algebraic int's subfield of C. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RCKeI02359; > 2. Prove that A is not a finite extension of Q. >Suppose A is a finite extension of Q, say of dimension D, and >achieve a contradiction. (Hint: Are there any irreducible >polynomials in Q[x] of degree D+1?) But I am still not sure how we get that contradiction Travis, could you please give me a few more hints or ideas thank you! I know that it is possible to generate an irreducible polynomial but not sure how to write it down formally and tie it in with the question > 3. Let B be an algebraic extension of A. Prove that B=A. >Pick an arbitrary element b of B. (What does such an element look >like?) >Now, show that b is in A. And similarily I am not completely sure how to do this, my thinking is though an element of C which is algebraic over C has the form a+bi (a,b in Q) but i is contained in A and so is Q, and by the primitive element theorem in reverse (kinda) it contains a,b,i which is already in A so B=A. Is that correct? === Subject: Re: Complex: convergence in norm for f hol. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RCKgt02477; > > Could someone please help me prove that if f_n->f > uniformly, then |f_n|^2->|f| also uniformly? >>Presumably you meant that |f_n|^2->|f|^2 uniformly. This is precisely what I stated in the line above what you have just quoted. >>This is not true - in the situation where you're >>supposed to prove it you know something that you're >>not telling us. >>Counterexample: Define f_n : R -> R by f_n(x) = x + 1/n. >>Then f_n -> f uniformly on R, where f(x) = x. But >>f_n^2 does not converge uniformly to f^2 >>(because for example |f_n(n)^2 - f(n)^2| does not >>tend to zero.) The statement of the message is: Convergence in *norm* for f holomorphic. the function f->f^2 is not a norm But you are using |f_n(n)^2-f(n(^2)|, when I said ( read second line in this post) : ...then |f_n|^2->|f| also uniformly, yet you seem to intend to prove that (correct me if I am wrong) that f_n(n)^2->/f(n)^2 uniformly. >> Now, I forgot to bring my book with me, and I did >> not quote my problem in an accurate way, and I should >> not have done that,and I am sorry but you are one >> aggressive guy , and more embarrassingly, you have >> --yet again--(I have read several of your posts) >> misread , >You haven't _yet_ stated the problem correctly, but >you complain about me misreading it? Fascinating. >Look. You originally asked whether |f_n| -> |f| >uniformly implies that |f_n|^2 -> |f| uniformly. >_That_ is _obviously_ false. Exactly what did >you mean to ask? I think this is not false, if the convergence is considered in compact sets K: (if you would please humor me once again, or, for the record, anyway): Assume f_n->f uniformly in K , then, for all eps.>0 , for all z in K, there is unique n with: |(U_n(z)-u(z))^2+(V_n-v)^2|< eps. (can work with eps. to get rid of square in norm above) (U_n->u and V_n->v ) Will show that this implies that in K: |f_n|^2->|f|^2 uniformly: ||f_n|^2-|f|^2|= |(U_n^2+V_n)-(u^2+v^2)|= (difference of squares) |(U_n^2-u^2)+(V_n^2-v^2)| Now U_n->u => U_n^2->u^2 ( U_n is continuous, f(x)=x^2 is continuous, then U_n^2 is continuous, by continuity: U_n->u =>g(U_n)=U_n^2->g(u)=u^2. ( or use continuity=>sequential continuity) By compactness, continuity of g(u)convergence is uniform for U_n^2 , for V_n^2. Find N large enough so that max (|U_n^2-u^2|, |V_n-v|^2)I conjectured you meant to ask whether it followed >that |f_n|^2 -> |f|^2 uniformly. If that was not >what you meant you might say what you _did_ mean. But why are you then trying to use f_n(n)^2->/f(n)^2 as a counterexample? >> which is not the end of the world, unless >> you come off so aggressively. >Ah. If I come off aggresively, or appear so to >you, then that _is_ the end of the world? >No. Just checked, the world's still here. >> And this is what I am telling you: You are _not_ >> *norm* of f squared, this usually means the square >> root of f^2 , and *not* f^2. (note :||f+g||^2 not nec. >> < ||f||^2+||g||^2 , so >> f^2 is not even a norm.) >> You chose to use f^2 as the square norm of f, with >> f a complex function. If you substitute f_n(n)^2 for >> (f_n(n)^2)^1/2 in the alleged counterexample: >> |f_n(n)^2 - f(n)^2| >> Your counterexample would *not* work. >You didn't ask about |f_n(n)^2 - f(n)^2|. (*) But that is what you tried to show, that is that (*) is not true, when I never claimed that it is true. Or did I? You're >upset that I wasn't able to guess that that's what >you were asking about? Now, I am upset at the fact that I asked to prove ( taken from second line in my first post) ... then |f_n|^2->|f| also uniformly? and then you tried to disprove this by proving that |f_n(n)^2-f^2|->/0 , which is not a counterexample that I can tell. >You're being very lame here. Here's a few facts, >all of which would be obvious from things I've >said if you decided to think about it instead >of whining about people not answering questions >that you never asked. Define f_n : C -> C by >f_n(z) = z + 1/n. Let f(z) = z. Then the following >are all obviously true (I'm still not sure which >one is relevant to the question that you haven't >quite got around to stating correctly): >[i] f_n -> f uniformly. >[ii] |f_n| -> |f| uniformly. >[iii] |f_n - f| -> 0 uniformly. >And the following are all false - again, I'm not >sure which is the one you care about, The second line in my post read: ... then |f_n|^2->|f| also uniformly the post also reads , convergence in norm , for f hol. What other norm would I be using for a holomorphic function? but if you'd >relevant one is false: >[iv] f_n^2 -> f^2 uniformly. >[v] |f_n|^2 -> |f|^2 uniformly. >[vi] |f^n^2 - f^2| -> 0 uniformly. > >> So much for the mythical devoted proffessor trying to >> help students. >If you want you can get your money back. >>If I had done careless posts 3-4 or more >> times ,I could understand your being upset, BUT I HAVE NOT >> (prove me wrong if you can), so your aggressiveness is >> unprovoked. >What makes you think I was upset? >I'm supposed to wait until you repeat something 3 or more >times before pointing out that what you say you're trying >to prove is false? >I'm supposed to wait until you've stated something >wrong more than 3 times before pointing out that >before you try to solve a problem you have to _read_ >the problem correctly? >Believe it or not, when I said there's no point in >telling you what the solution is before you've even >read the problem correctly I was trying to be >helpful. It's a fact that students _do_ try to >solve problems before getting the statement >straight, and it's a fact that there's no hope >for them until they stop doing that. >> P.S, these are functions converging uniformly in >> compact subsets. >Ah, so you finally read the problem. But you still haven't >_stated_ it correctly (from the things you've said it >seems like you're asking whether [ii], uniformly on >compact sets, implies [v], uniformly on compact sets, >but I tend to doubt that that's what the problem >actually asked.) It is a question of my own, actually and not a problem, I wondered if for K compact, f_n->f uniformly implies |f_n|^2->|f|^2 uniformly. >No, I'm still not going to give you the statement of >the problem before giving you the solution. If you >ever do decide to simply post a correct statement >of the problem good luck getting someone to show >you the solution. > I am very sorry, I meant to say that f is a holomorphic > function. Does it the follow? > I can only think of something like: 1)|f_n-f| > with conjugation) We then multiply 1),2) above, and get: > 3) > ||f_n|^2-2ref_n*f+|f|^2|00 f_n->f , and substitute in #3 , get: > ||f_n|^2-2|f|^2+|f|^2 |=||f_n|^2-|f|^2 | Does this work? > At any rate, I appreciate your help, which-- you have a point--is free help. >>************************ >>David C. Ullrich >************************ >David C. Ullrich === Subject: Re: Complex: convergence in norm for f hol. > Could someone please help me prove that if f_n->f >> uniformly, then |f_n|^2->|f| also uniformly? Presumably you meant that |f_n|^2->|f|^2 uniformly. > This is precisely what I stated in the line > above what you have just quoted. |f_n|^2->|f|^2 is precisely the same as |f_n|^2->|f| ? >This is not true - in the situation where you're >supposed to prove it you know something that you're >not telling us. Counterexample: Define f_n : R -> R by f_n(x) = x + 1/n. >Then f_n -> f uniformly on R, where f(x) = x. But >f_n^2 does not converge uniformly to f^2 >(because for example |f_n(n)^2 - f(n)^2| does not >tend to zero.) > The statement of the message is: > Convergence in *norm* for f holomorphic. the > function f->f^2 is not a norm And |f| is also not a norm. > But you are using |f_n(n)^2-f(n(^2)|, when I said > ( read second line in this post) : > ...then |f_n|^2->|f| also uniformly, yet you > > seem to intend to prove that (correct me if I > am wrong) that f_n(n)^2->/f(n)^2 uniformly. Of course you're wrong. First, I have no idea what the / refers to. Second, since we're talking about limits as n -> infinity, saying something converges to something that contains an n makes no sense, regardless of what you actually mean by /f(n)^2. Finally, I have not intended to prove any sort of uniform convergence, I've been giving _counterexamples_ to uniform convergence. > Now, I forgot to bring my book with me, and I did > not quote my problem in an accurate way, and I should > not have done that,and I am sorry but you are one > aggressive guy , and more embarrassingly, you have > --yet again--(I have read several of your posts) > misread , >>You haven't _yet_ stated the problem correctly, but >>you complain about me misreading it? Fascinating. >>Look. You originally asked whether |f_n| -> |f| >>uniformly implies that |f_n|^2 -> |f| uniformly. >>_That_ is _obviously_ false. Exactly what did >>you mean to ask? > I think this is not false, if the convergence > is considered in compact sets K: > (if you would please humor > me once again, or, for the record, anyway): > Assume f_n->f uniformly in K , then, for all > eps.>0 , for all z in K, there is unique n with: > |(U_n(z)-u(z))^2+(V_n-v)^2|< eps. (can work with eps. There is a unique n with this property? That's clearly wrong - this holds for _every_ sufficiently large n. > to get rid of square in norm above) > (U_n->u and V_n->v ) > Will show that this implies that in K: > |f_n|^2->|f|^2 uniformly: > ||f_n|^2-|f|^2|= > |(U_n^2+V_n)-(u^2+v^2)|= (difference of squares) > |(U_n^2-u^2)+(V_n^2-v^2)| > > Now U_n->u => U_n^2->u^2 ( U_n is continuous, > f(x)=x^2 is continuous, then U_n^2 is continuous, > by continuity: U_n->u =>g(U_n)=U_n^2->g(u)=u^2. > ( or use continuity=>sequential continuity) > By compactness, continuity of g(u)convergence is uniform > for U_n^2 , for V_n^2. Find N large enough so that > max (|U_n^2-u^2|, |V_n-v|^2) (NOTE: I do now the above results hold for > functions of one real variable, I am not > sure they work for functions of 2 real > variables) > Now we come back to: > |(U_n-u)^2+(V_n-v)^2|< > |eps/2+eps./2| > > >>I conjectured you meant to ask whether it followed >>that |f_n|^2 -> |f|^2 uniformly. If that was not >>what you meant you might say what you _did_ mean. > But why are you then trying to use > f_n(n)^2->/f(n)^2 as a counterexample? > which is not the end of the world, unless > you come off so aggressively. >>Ah. If I come off aggresively, or appear so to >>you, then that _is_ the end of the world? > >>No. Just checked, the world's still here. > And this is what I am telling you: You are _not_ > *norm* of f squared, this usually means the square > root of f^2 , and *not* f^2. (note :||f+g||^2 not nec. > < ||f||^2+||g||^2 , so > f^2 is not even a norm.) > You chose to use f^2 as the square norm of f, with > f a complex function. If you substitute f_n(n)^2 for > (f_n(n)^2)^1/2 in the alleged counterexample: > |f_n(n)^2 - f(n)^2| > Your counterexample would *not* work. >>You didn't ask about |f_n(n)^2 - f(n)^2|. (*) > But that is what you tried to show, that is that (*) > is not true, when I never claimed that it is true. > Or did I? > > You're >>upset that I wasn't able to guess that that's what >>you were asking about? > Now, I am upset at the fact that I asked to prove > ( taken from second line in my first post) > ... then |f_n|^2->|f| also uniformly? > and then you tried to disprove this by proving that > |f_n(n)^2-f^2|->/0 , which is not a counterexample > that I can tell. Oh - ->/ is supposed to mean does not tend to, fine. Yes, it is a counterexample, or would be if you'd stated it properly: |f_n(n)^2-f(n)^2|->/0. Because for those f we have |f_n(x)| = f_n(x) for x > 0. So saying |f_n(n)^2-f(n)^2|->/0 says that | |f_n(n)^2|-|f(n)^2| |->/0, which shows that |f_n|^2 does not tend to |f|^2 uniformly. >>You're being very lame here. Here's a few facts, >>all of which would be obvious from things I've >>said if you decided to think about it instead >>of whining about people not answering questions >>that you never asked. Define f_n : C -> C by >>f_n(z) = z + 1/n. Let f(z) = z. Then the following >>are all obviously true (I'm still not sure which >>one is relevant to the question that you haven't >>quite got around to stating correctly): >>[i] f_n -> f uniformly. >>[ii] |f_n| -> |f| uniformly. >>[iii] |f_n - f| -> 0 uniformly. >>And the following are all false - again, I'm not >>sure which is the one you care about, > The second line in my post read: > ... then |f_n|^2->|f| also uniformly > the post also reads , convergence in norm > , for f hol. What other norm would I be using > for a holomorphic function? How am I supposed to know if you don't say? There are many other norms of great interest in studying holomorphic functions. >but if you'd >>relevant one is false: >>[iv] f_n^2 -> f^2 uniformly. >>[v] |f_n|^2 -> |f|^2 uniformly. >>[vi] |f^n^2 - f^2| -> 0 uniformly. >> > So much for the mythical devoted proffessor trying to > help students. >>If you want you can get your money back. >If I had done careless posts 3-4 or more > times ,I could understand your being upset, BUT I HAVE NOT > (prove me wrong if you can), so your aggressiveness is > unprovoked. >>What makes you think I was upset? >>I'm supposed to wait until you repeat something 3 or more >>times before pointing out that what you say you're trying >>to prove is false? >>I'm supposed to wait until you've stated something >>wrong more than 3 times before pointing out that >>before you try to solve a problem you have to _read_ >>the problem correctly? >>Believe it or not, when I said there's no point in >>telling you what the solution is before you've even >>read the problem correctly I was trying to be >>helpful. It's a fact that students _do_ try to >>solve problems before getting the statement >>straight, and it's a fact that there's no hope >>for them until they stop doing that. > P.S, these are functions converging uniformly in > compact subsets. >>Ah, so you finally read the problem. But you still haven't >>_stated_ it correctly (from the things you've said it >>seems like you're asking whether [ii], uniformly on >>compact sets, implies [v], uniformly on compact sets, >>but I tend to doubt that that's what the problem >>actually asked.) > It is a question of my own, actually and not a > problem, I wondered if for K compact, > f_n->f uniformly implies |f_n|^2->|f|^2 > uniformly. And this is the _first_ time you've said that that's the question you wondered about. Although you still haven't stated it quite correctly - you mean to ask about uniform convergence on K. The answer to _this_ question is yes. If you hadn't been so obnoxious in your last post I'd give you a hint how to prove it. >>No, I'm still not going to give you the statement of >>the problem before giving you the solution. If you >>ever do decide to simply post a correct statement >>of the problem good luck getting someone to show >>you the solution. >> I am very sorry, I meant to say that f is a holomorphic >> function. Does it the follow? > >> I can only think of something like: >> 1)|f_n-f|> 2)|f_n-f|^-=|(f_n^-)-f^-|> >> with conjugation) >> We then multiply 1),2) above, and get: >> >> 3) >> ||f_n|^2-2ref_n*f+|f|^2|> so that eps^2> Then, as n->00 f_n->f , and substitute in #3 , get: >> ||f_n|^2-2|f|^2+|f|^2 |=||f_n|^2-|f|^2 |> >> Does this work? > At any rate, I appreciate your help, which-- > you have a point--is free help. >************************ David C. Ullrich >>************************ >>David C. Ullrich ************************ David C. Ullrich === Subject: Re: Need help on a group theory problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RCKf702412; >desperately. >------------------------------------------------------- >Problem: >Is GL+(n,R), general linear group with positive determinant over the >real number field, homeomorphic to a Euclidean space in general. >What is the simpliest form that GL+(n,R) can be reduced to? The concern >is whether GL+(n,R) can be simplified further, or be parameterize by a >simply subset of a Euclidean space. >-------------------------------------------------------- >Some time ago someone suggested the following argument: >GL+(n,R) is not homeomorphic to a Euclidean space in general. >For instance, at n=2. >[1]: GL+(2,R) = SL(2,R) x R+, >[2]: SL(2,R) is homeomorphic to S^1 x R x R+ >[3]: S^1, a circle is not heomeomorphic to a Euclidean space, >thus GL+(2,R) is not homeomorphic to a Euclidean space. >I want to check with you to see if this argument is solid. In >particular, due to my limited knowledge on this subject, I need help to >understand line [2] in particular. You might want to contemplate the Gram-Schmidt process for producing an orthonormal basis from a basis. If u, v are column vectors of a determinant 1 matrix, then there is a unique (s, t) in R x R+ such that t s ( ) (u v) 0 1/t is in SO(2), which is homeomorphic to a circle. The Gram-Schmidt argument shows more generally that GL+(n, R) is homeomorphic to SO(n) x R^(n(n+1)/2). The factor SO(n) is not contractible (it has fundamental group Z_2 if n > 2) and so SO(n) x R^(n(n+1)/2) cannot be homeomorphic to R^(n^2). (Don't worry about R+ vs R; they are homeomorphic.) >Also, it'll be greatly appreciated if you can comment on the problem for >all n. >Kai Todd Trimble === Subject: Re: Algebraic int's subfield of C. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RCf3M04118; Sorry looking at it again, I am not sure how I missed the answer for part 2. If it was some finite extension of Q, [A:Q]=d, so the minimal polynomial has degree d over Q. But it is possible to construct irreducible polynomials of any degree inc d+1, taking dth and higher roots of non-trivial squarefree numbers. But I am still not sure about parts 3 and 4 exspecially, if someone could help I would appreciate it === Subject: Re: fractional iteration of functions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RDP6T08030; >-- NPC Hope you will later on discover the more significant results you've got about fractional iterations. Since Abel and Schroeder we know 'counting' functions. It exists pairs (phi,f) verifying phi(f(x))=phi(x) + 1 (1) { +1 increment} ,phi invertible ,defined nearly a constant , phi(f^[r](x))= phi(x) + r ,r real (2) so (f^[r](x))= phi^[-1](phi(x) + r) (3); Two examples: phi(x)=1/x f1(x)= x/(x+1) f1^[r](x)= x/(r*x+1), phi(x)=ln(x +b/(a-1))-b/(a-1) f2(x)=ax+b, f2^[r](x)=a^r*(x+b/(a-1))-b/(a-1). In (2) and (3) r is real ! (negative ,fractinal as well). equations and iterated functions (sci.mathresearch,scidiscrete,forum) Good luck, Alain. === Subject: Connectedness I'm having problem with this question. 1. If the point set S contains 2 points and S is connected, then each point of S is a limit point of S. This is what I have so far.... By contradiction i.e Assume the hypothesis with the negation of the conclusion. I don't know where to go from here. I know we start off with S containing 2 points and S is connected implies S cannot be written as a union of two disjoint non-empty subsets? Now I am stuck,please help. === Subject: Re: Connectedness > I'm having problem with this question. > 1. If the point set S contains 2 points and S is connected, then each > point of S is a limit point of S. Let S be Sierpinski space { a,b } with topology { nulset, {a}, {a,b} } S is connected, b is a limit point, a is not. > This is what I have so far.... > By contradiction i.e Assume the hypothesis with the negation of the > conclusion. > I don't know where to go from here. Then start with a point p that's not a limit point. What's it mean for p to not be a limit point? > I know we start off with S containing 2 points and S is connected > implies S cannot be written as a union of two disjoint non-empty > subsets? > Now I am stuck,please help. To finish contradiction, you want to produce two open, nonnul, disjoint sets. You also not want us mind read. Is your space for real, Euclidean, metrizable, 1st countable, what? === Subject: Re: Connectedness > I'm having problem with this question. > 1. If the point set S contains 2 points and S is connected, then each > point of S is a limit point of S. > This is what I have so far.... > By contradiction i.e Assume the hypothesis with the negation of the > conclusion. > I don't know where to go from here. > I know we start off with S containing 2 points and S is connected > implies S cannot be written as a union of two disjoint non-empty > subsets? Not true; [0,1] is connected and contains (at least) 2 points, but it can be written as the union [0,1/2) U [1/2,1], which is a union of two disjoint non-empty subsets. You need to make sure you understand the definition of connectedness first. > Now I am stuck,please help. Suppose s is a member of S but is not a limit point of S. Can you find a neighborhood of s that contains no other points of S? What does this tell you? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: How far can you move the bar? Let N, W, J, floor(), Q, and R have their conventional definitions. Let T_10 = {k/10^n: k,n in W, n >= 0, 0 <= k < 10^n}. Let L : N -> T_10 be a certain countable list of reals in [0,1). These reals are constructed in the following manner: L(0) = 0.000000... L(1) = 0.300000... L(2) = 0.330000... L(3) = 0.333000... L(4) = 0.333300... L(5) = 0.333330... L(6) = 0.333333... ... L(n) = (10^n - 1) / (3 * 10^n) for any integer n. Let the notation L(n,m) : N x W -> {0,1,2,3,4,5,6,7,8,9} be the value floor(10^m * L(n)) - 10*floor(10^(m-1) * L(n)) for any integers m, n, m > 0, m >= 0. Define L(n,0) = 0. More intuitively, L(n,m) = the m'th digit to the right of the decimal point of L(n). Let P(r,m) : [0,1) x N -> T_10 be floor(r * 10^m) / 10^m. Let D(L, n) : (: N->T_10) x N -> {0,1,2,3,4,5,6,7,8,9} be defined as follows: if(L(n,n) = 3) then D(L,n) = 0 else D(L,n) = 3 and let D(L) = sum(n in N) (D(L,n) * 10^(-n)). Note that we do not assume D(L) is in T_10; it is, however, in R. We can compute D(L) fairly readily, especially if we notice that L(n,n) = 0 for all n in N. Therefore D(L,n) is always 3 and D(L) = 0.333... . The bar in this case refers to a certain integer b. For b=2 this can be diagrammed: L(0) = 0.00|0000... L(1) = 0.30|0000... L(2) = 0.33|0000... L(3) = 0.33|3000... L(4) = 0.33|3300... L(5) = 0.33|3330... L(6) = 0.33|3333... ... D(L) = 0.33|3333... and can be formally defined as max({b: (Ea) (a in W . P(D(L),b) = P(L(a), b))}) It turns out that there is no such b; P(D(L),b) = P(L(b), b), and therefore a = b. Therefore any b in N will work; the bar can be set anywhere in N. So ... is D(L) in the list? Well... If we assume D(L) = L(n), then we note that L(n,n) = 0 whereas D(L,n) = 3. Therefore no such n can exist. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: algorithm for laplace transformation / Java!? Hi NG. I am going to develop a Java-Applet and in the need of an algorithm, that is able to do the Laplace transformation numerically - and the same for the inverse Laplace transformation... of course the performance has to be good... ;-) My Internet research was not good and i would be very thankful for hints, maybe code (Java or other) === Subject: Re: algorithm for laplace transformation / Java!? D. Jagerman has a paper in the BELL SYSTEM TECHNICAL JOURNAL circa 1981 that gives a good algorithm for numerical inversion of Laplace of Laplace transforms; I think that's also in the BSTJ. === Subject: Simple rings and their order [My newsgroups aren't working right, so I have to repost a reply] >I can't figure out this problem: >Let R be a simple ring. Show that all nonzero elements of R have equal >additive order. Show that this order either is a prime number p or is >infinite. For positive integer d, let I_d = {r in R : dr = 0} where dr means r+r+r+...+r (d summands). I_d is an ideal of R, because 1. 0 is in I_d. 2. If r and s are in I_d, then d(r+s) = dr + ds = 0+0 = 0. 3. If r is in I_d and x is in R, then d(xr) = xr+xr+...+xr = x(r+...+r) = x(dr) = x0 = 0 d(rx) = rx + rx + ... + rx = (r+...+r)x = (dr)x = 0x = 0. Since R is simple, that means that I_d = {0} or I_d = R for each d. Let S = {d in N: d>0, I_d = R}. Show that if S is nonempty, then its smallest element is a prime. Do it using the same argument one uses to show that the characteristic of a domain is either 0 or a prime. -------------------------------------------------------------------------- --------------------- My reply: Ok, so if there is no N such that n*1=0 then 1 has infinite order. Now, suppose 1 has order n and n=st where s,t are less than n. Then 0=n*1=(st)*1=(s*1)(t*1). So (s*1)=0 or t*1=0. Since n is the least positive integer suich that n*1=0, s=n or t=n. Hence n is prime. Am I off here? Does this show that all nonzero elements have the same additive order? === Subject: Re: Forces and Projectile by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RFXTc22105; > Got a question or actually want confirmation. > Suppose a projectile is launched at 45 degrees and a magnitude of 50. > I have a simple little program in basic that traces the path. Now >> suppose > if the projectile enters a zone where a force will act on it as >> long as > it is in this zone. Is it correct to find the current velocity of >> the > projectile when it first entered the zone and redo the calculation >> with the > new force factored in ? >> Yes, if I understand what you're asking. By working out >> the path you're probably doing something like this: >> At each interval dt: >> vx = vx + ax*dt >> vy = vy + ay*dt >> x = x + vx*dt >> y = y + vy*dt >> Which is a reasonable thing to do, subject to some numerical >> cautions. In that case, (vx,vy) at the moment you enter the >> zone are the correct velocity components at that moment, >> and you can continue to use these techniques with the >> new acceleration values to see how (vx,vy) and (x,y) >> evolve. >> The caveats: If you do a lot of successive additions of >> small amounts, roundoff error can creep in causing your >> path to deviate from the correct one. With dt chosen >> small enough to make a smooth path, but not so small >> that you have hundreds of thousands of steps, you're >> probably OK. >> - Randy >Are you hinting that there is a better way to skin this cat ? >The main problem I have is I guess what you are talking about, I probally >am going at it a little too hard.. maybe I am on the right track.. but >like you even said, it appears to be the reasonable thing to do. >I will post my results soon. (i.e. screen shots of the projectile path) there are two ways to do this problem, I think what you were doing orginally was to find the equation of motion for the projectile and plot the curve or b) you can do an iteriation where each step finds the next location. you could do either way, in case a stoping at each interface to recalculate new equations of motion in case b doing as R, Poe indicates. the case a was more widely used before the advent of computers Harold Schmelzer On seeing a house a liberal says no one should own a house like that unless everyone does a conservative says everyone should be able to earn the money to own a house like a reactionary says its my house and every one else should pay so much taxes so that they will never earn enough to own some thing as good: === Subject: lines on a pseudosphere? Lines on a pseudosphere do not behave as lines on a euclidean plane. The more I think about it, the more there is that puzzles me. For one example among many: Consider two points A and B on a pseudosphere. Stretch a string to find the shortest path between them. Now continue (on the pseudosphere) in the direction indicated by such a string, until at some great distance later, you get to point C. Now find the shortest distance from A to C. This path doesn't necessarily include point B. What's going on???? Apparently I have generated paths but not lines by this method, or else lines behave differently, or . . . ? Can someone point me to a website that discusses this and similar matters? Yes, I do believe in reading, googling, and research, before pestering people. That's been done, but not with the results I'd like. Ted Shoemaker === Subject: Would you mind solving : f(x , y+2) =3*f(x , y) -1 ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0RGN3p28200; A little clue for your sagacity... f:R*R->R x,y real variables . Friendly,Alain. === Subject: cantor diagonal argument blues Here's a copy/paste of the wolfram version of the generalized Cantor diagonal argument with some editing, but I think leaving the heart of the proof intact: To show there is no bijection from a set S to its power set T: Finding an injection is trivial, as can be seen by considering the function f from S to T which maps an element s of S to the singleton set {s}. Suppose there exists a bijection b from S to T. Consider the subset D of S consisting of the elements d of S such that b(d) does not contain d. Since b is a bijection, there must exist an element x of S such that b(x) = D. But by the definition of D, the set D contains x if and only if b(x)=D does not contain x. i.e.: If x is in D, then it is in b(x). Then it can't be in D. If x is not in D, then b(x) does not contain x. But then x must be in D. These contradictions show there cannot exist a bijection from S to T. The problem for me is: yes, there's a clear contradiction. But we're actually making TWO assumptions: the existence of the bijection, AND the existence of the set D. Can't we simply say that the bijection could still exist, but there would be no set D defined as above, for it? To put it another way: take the set of all injections from S to T for which the set D doesn't exist. How do we know one of them isn't a bijection? I hope this makes some sense. I have a feeling you either get it or you don't, and if you don't you won't. Ken === Subject: Cantor diagonal argument blues Here's a copy/paste of the wolfram version of the generalized Cantor diagonal argument with some editing, but I think leaving the heart of the proof intact: To show there is no bijection from a set S to its power set T: Finding an injection is trivial, as can be seen by considering the function f from S to T which maps an element s of S to the singleton set {s}. Suppose there exists a bijection b from S to T. Consider the subset D of S consisting of the elements d of S such that b(d) does not contain d. Since b is a bijection, there must exist an element x of S such that b(x) = D. But by the definition of D, the set D contains x if and only if b(x)=D does not contain x. i.e.: If x is in D, then it is in b(x). Then it can't be in D. If x is not in D, then b(x) does not contain x. But then x must be in D. These contradictions show there cannot exist a bijection from S to T. The problem for me is: yes, there's a clear contradiction. But we're actually making TWO assumptions: the existence of the bijection, AND the existence of the set D. Can't we simply say that the bijection could still exist, but there would be no set D defined as above, for it? To put it another way: take the set of all injections from S to T for which the set D doesn't exist. How do we know one of them isn't a bijection? I hope this makes some sense. I have a feeling you either get it or you don't, and if you don't you won't. Ken Reply === Subject: Tree Algorithm Hi folks, I'm after some help on this problem, which has been bugging me for the last couple of days. A tree structure (ie. starting from a Root node, propogating along branches to the leaves) is built. It is displayed on a monitor with the leftmost node to the far left of the display area. I wish to be able to minimise the amount of space it takes up on the screen by rearranging groups of nodes (branches/leaves), squashing together nodes that are unecessarily separated and moving nodes up or down a level to facilitate further squashing together. I think this is probably a Genetic Algorithm problem (hence I've posted there as well but with more GA-type language). However, I wondered if any of you knew of a Functional Algorithm that would help (maybe by optimising the tree as it is constructed ... dunno, just guessing) Any help would be greatly appreciated. Rob Small === Subject: Probability question: normal distribution Let Z denote the standardized, normal random variable (with mean 0 and variance 1). Then Z has probability density function g(t) = 1/sqrt(2*pi) * exp(-t^2/2). Let r be a fixed real number. Let X denote the random variable given by: Z if Z > r, 0 otherwise. Show that E(X) = g(r), where E(X) denotes the expectation of X. All help is appreciated. -Wassily === Subject: Re: Probability question: normal distribution > Let Z denote the standardized, normal random variable (with mean 0 and > variance 1). Then Z has probability density function g(t) = > 1/sqrt(2*pi) * exp(-t^2/2). > Let r be a fixed real number. > Let X denote the random variable given by: > Z if Z > r, > 0 otherwise. > Show that E(X) = g(r), where E(X) denotes the expectation of X. > All help is appreciated. > -Wassily Cute little problem. It looks suspiciously like homework. It is a straightforward application of the definition of expectation. Note that the expectation of X is the sum of two integrals. The first is from t = -infinity to t = r, where X has value 0. The second is from t = r to t = +infinity, where X has value t ... Nora B. === Subject: JIM BLACK... I'LL RIP YOUR HEAD OFF IF YOU DO THAT AGAIN repeating the post I just lost to a.u.k. Ghost analysed the algorithm, I proved the lists are equal. Q 2. Are the 2 lists (given they are parts of infinite lists) equal? Jim Black and Mike Terry are wrong. Here's a random binary list 0.010101001.. 0.111010101.. 0.101110000.. .. Can you rearrange the elements to make the 1st digits like so 0.0....... 0.1....... 0.0....... 0.1....... .. ? Herc === Subject: Re: JIM BLACK... I'LL RIP YOUR HEAD OFF IF YOU DO THAT AGAIN > repeating the post I just lost to a.u.k. > Ghost analysed the algorithm, I proved the lists are equal. > Q 2. Are the 2 lists (given they are parts of infinite lists) equal? > Jim Black and Mike Terry are wrong. Really? I'm truly devastated to hear that! I expect you'll tell me why one day... > Here's a random binary list > 0.010101001.. > 0.111010101.. > 0.101110000.. > .. > Can you rearrange the elements to make the 1st digits like so > 0.0....... > 0.1....... > 0.0....... > 0.1....... > .. Yes, if the original list has infinitely many 0s and infinitely many 1s in the first position. Using measure theory we could say that the set of binary lists for which we cannot do this has measure zero. (Or less formally, that we can almost certainly rearrange the list the way you ask...) Mike. === Subject: Re: JIM BLACK... I'LL RIP YOUR HEAD OFF IF YOU DO THAT AGAIN <427d699b$0$93757$ed2619ec@ptn-nntp-reader01.plus.net> measure theory is beyond the comprehension of our local clown Herc. === Subject: Re: JIM BLACK... I'LL RIP YOUR HEAD OFF IF YOU DO THAT AGAIN <427d699b$0$93757$ed2619ec@ptn-nntp-reader01.plus.net> wrong as a hemaphrodite at a brazillian wax shop that's HOW wrong, not WHY Herc === Subject: Re: Axioms >>*HILBERT'S AXIOMS FOR GEOMETRY* > Followed by some meaningless babble, being (I surmise) Albert Wagner's > mangling of Hilbert's axioms. Nothing is mangled. Undefined words and phrases are simply replaced with the strings identifying undefined words and phrases. A template, as it were. > Hmm. Since you show yourself incapable of > even the simplest abstract reasoning, perhaps you might contemplate > training as a typesetter or proof-reader. Then one day you might be > able to check one copy of Hilbert's actual axioms against another, to > make sure that all the letters, numbers, and punctuation are exactly > the same. That may be your most intimate contact with them. Transparent ad hominem. A logical fallacy. Ignored. > FWIW: > 1. For every two undefined things /A, B/ there exists an >>undefined thing /a/ that is in an undefined relationship with >>each of the undefined things /A, B/. > Yes, but there are (at least) two different sorts of undefined things > in the meaningful version of this. Only two. > You can replace the original terms > with any other terms you like - in one of his helpful posts to you, > Guenther suggested twinkies. However, as would probably be obvious to a > bright 11-year old, you can't just jumble the different things up by > making them indistinguishable. That's my point. Undefined things are indistinguishable. You and Hilbert are cheating when you insist that specific words/phrases with predefined meanings are undefined and at the same time insist that they be present with their own semantic burden. > Is this really the best you can do? Don't you feel at least slightly > stupid, or ashamed of yourself? Are you feeling stupid, or ashamed of yourself, that a non-mathematician has to point out that the emperor has no clothes? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Axioms > That's my point. Undefined things are indistinguishable. Wrong. Flat out wrong. Undefined terms have those properties given to them by the axioms. That is why the undefined terms have to specified. Bob Kolker === Subject: Re: Axioms undefined so that the the system spawned is equivalent to Hilbert's > you might even gain a lot more out of it than a good laugh. I wasn't aiming for a laugh. There is only *one* category of undefined. Otherwise, knowing the category takes you halfway to a definition. There can be no better definition of undefined than undefined. There is nothing to distinguish one undefined from another undefined. That was my point. > On the > other hand, defining several categories of undefined might get you > into trouble with Zick and Blumschein. Well it's your choice wagner, > learn something or side with those guys. After all that is what this > thread has been all about since you became involved. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Axioms >> *I. Axioms of Incidence*: >> 1. For every two undefined things /A, B/ there exists an undefined >> thing /a/ that is in an undefined relationship with each of the >> undefined things /A, B/. > Ill formed axiom. You said it, I didn't. > You fail to say which of the set of undefined things. > The specific undefined terms have to occur in the axioms. Why? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Axioms > You said it, I didn't. >> You fail to say which of the set of undefined things. The specific >> undefined terms have to occur in the axioms. > Why? Any meaning either operative or semantic that the undefined terms come from exactly one place: the axioms the use them. By leaving the particular undefined terms blank you your parody on axioms you have made perfectly meaningless assertions. Compare: Two points determine a straigh line to To x determine a y, without saying which primitive terms x and y stand for. The first is comprehensible the second is not. Compare x is Red where x is a variable over a domain of individuals to this rose in my hand is is Red. The first is not a well formed proposition, it is a proposition form. You have to plug in values for the variables to make it meaningful or definite. Bob Kolker === Subject: Re: Axioms > *HILBERT'S AXIOMS FOR GEOMETRY* Examine carefully what I have done. I have simply followed Guenther's suggestion, but instead of pixies or twinkies or whatever I simply replaced 'point', 'line' and 'plane' with 'undefined thing'; And similarly, replaced undefined relationships with 'undefined relationship', yielding the pure core of Hilbert's axioms without suggestive names. Guenther was absolutely correct: this substitution reveals the absurdity of assuming that the undefined words have any bearing at all on the core system. Everyone so far seems to think that I conflated some undefined things with some other undefined things. But, if those undefined things were in fact truly undefined, then that would be impossible since undefined word/phrases each inherently have no distinguishing characteristics. I assume that what is meant by the angry replies is that the undefined things and undefined relationships, were *not in fact undefined nor intended to be actually undefined*. I suspect you are outraged that this exercise very clearly illustrates that the 'undefined' terms were in fact 'minimally' defined in that they were intended to carry a certain semantic burden in their names. The absurdity and obtuseness of the axioms, as now revealed, shows the high level of dependence on implied, but hypocritically denied, definitions. You cannot insist simultaneously that a phrase is undefined and that it must nevertheless be distinguished from other undefined phrases; Especially, distinguished in a very suggestive manner that gives more than a strong clue as to what you intend but will not admit should be there. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: algebra How would i prove this anyone please, i just require hints. If M_1 AND M_2 are submodules of an R-module M such that M= M_1 direct sum M_2, prove that M_1 is isomorphic to M/M_2 and M_2 is isomorphic to M/M_2. === Subject: Re: Cardinality question > >The part that would allow me to generate constructions with >>things that are not the square roots of integers. >> Your problem not mine. You're the one who claims they lie on a >>real >> number line. The place where the curve y = x^3 - 4 crosses the real >number line is the cube root of 4. Which is where exactly, sport. >>I think it should be where Randy says it is, Lester. Do you have a >>problem with that? >> It should be where Randy says it is but Randy doesn't say where that >> is. I don't have a problem with that; Randy does. >No I don't. It's exactly where the graph of x^3 - 4 >crosses the x-axis. Which is where on the x axis? > That happens at a single point. I >can tell you what that point is to any precision you want, >in principle (calculating it to 10^100 digits may take a >little time). You can tell what that point is to any precision except exactly. >Are you saying that you don't know how to draw the graph >precisely enough to find that point graphically to arbitrary >precision? Constructions of integer-length segments and >right angles suffer from the same problem. No I'm saying you can't find that point exactly. >So what's the problem. Tell me how to point out a transcendental >on a curve and I'll tell you how to point out the cube >root of 4. Deal? So if I tell you, you'll tell me what I told you? Why is this an invitation I can refuse? I know what, why don't you go back to explaining to me why all problematic empirical observations aren't tautologically regressible. >So yes, I claim that this point lies on the real number line. Oh, I'm quite sure it does. It just doesn't lie on any real number > line together with rationals, irrationals, and other >transcendentals. >Really? You mean the cube root of 4 is yet a fourth class, >different from Zrational, Zirrationals, and Ztranscendentals? No I'm telling you you're a second class transcendental irrational. >>Ah, is this a Result? You say other transcendentals: would I be >>correct in deducing that cube root(4) is itself a transcendental? >Oh, I missed that. No, the cube root of 4 is not transcendental >though it may be Ztranscendental. Who can tell? Nobody, till >we know what pointing out looks like. What don't you understand about the Pythagorean Theorem? Probably similar to what you don't understand about tautologies. >> No, >>but hang, on, Randy says it's on a straight line, which you don't >deny, >> I've denied transcendentals lie on straight lines many times. >You just admitted it lies on a real number line. Since there's >only one real number line, I think that covers it. Since what, sport? Do you just make all this up as you go along? You're a real Looney tune. >Besides, it lies on the x-axis. Are you saying that the graph >of x^3-4 doesn't cross the x-axis? I'm saying you can't point out transcendentals on straight lines. Circles cross diameters. Hardly a big deal. That merely points out pi on the circular arc not the diameter. >Consider an arbitrary curve and an arbitrary line drawn through >it. Do they intersect at a point? Does that point not lie on >both the curve and the line? So which class of number does >it correspond to in Lester's Universe? It corresponds to some point defined by the intersection. === Subject: Re: Cardinality question >> So what? Lester tends to know only what he needs to know. >Lester tends to know Jack-t. Alias Bob Kolker. === Subject: Re: Cardinality question >> Which nonsensical rambling exactly did you have in mind? >Tautological regression for starters. What problem do you have with tautological regression, pray tell? It's a common enough subject in science. === Subject: Re: Cardinality question > What problem do you have with tautological regression, pray tell? It's > a common enough subject in science. Meaningless bafflegab. Bob Kolker === Subject: Re: Courage? :>: :>:> If the path is subjective, then it seems the destination must :>:> also be subjective. :>: As a mathematician dealing only with abstractions that exist only :>: in your head then it would seem that your observation above is :>: easily justified. :>Can you explain how subjective mechanics leads to objective :>truths? Do you believe that you can subjectively arrive at an :>objective truth? What objective evidence is there that the :>truth arrived at subjectively is objectively true? : I didn't say had to be just can be. In the case of mathematikers : truth is both arrived at subjectively and remains so as a personal : testament to their touching faith. I, and many others, are still waiting for an example of how your subjective mechanics arrives at universal truths. What evidence do you have that these are universal truths if it is all subjective? Stephen === Subject: Re: Courage? > I, and many others, are still waiting for an example of > how your subjective mechanics arrives at universal > truths. What evidence do you have that these are > universal truths if it is all subjective? Subjective Mechanics is oxymoronic. A scientific theory must be empirically testable which implies that it is objective. Bob Kolker === Subject: Re: Courage? :> :> :> Do you believe that you can subjectively arrive at an :> :> :> objective truth? :> :> :> :> : I know of no other way. :> :> :> :> :> What objective evidence is there that the :> :> :> truth arrived at subjectively is objectively true? :> :> :> :> : What objective evidence would satisfy you, knowing as you do, :> :> : that both objective evidence and objective truth are only :> :> : available to us subjectively? :> :> :> :> How about a repeatable demonstration that Lester's :> :> subjective mechanics leads to universal truth. If :> :> it can be shown that different people will always :> :> arrive at the same universal truths when they use :> :> Lester's subjective mechanics that would be very :> :> strong evidence there is something universal and :> :> objective about it. :> : Well, you would have to take that up with Lester. Personally, I'm :> : not sure that anything can be both universal /and/ objective. :> I was talking to Lester. You are the one who jumped in. : You were probably talking to Lester in some much earlier post. : But according to the last seven headers you have been talking to me. That does not change the fact that I was talking to Lester and you butted in. :> So it is your opinion that universal truths are not :> necessarily objective truths? What do you mean :> by 'universal truth'? : Are you talking to Lester or me? Yes. : I would think that universal is er...universal and objective is a : small subset of that. Just above you said :> : Personally, I'm :> : not sure that anything can be both universal /and/ objective. which means that you do not think that objective is a small subset of universal. If something cannot be both universal /and/ objective, then objective is not a subset of universal. I still do not know what you and Lester mean by subjective universal truths. It would seem that a universal truth would be always true, everywhere and for everyone. :> :> Of course, in order to do that Lester would have :> :> to explain how one applies his subjective mechanics, :> :> and we would then have to be able to determine if someone :> :> was using the subjective mechanics correctly. :> : How would you determine is someone else was is being properly :> : subjective? :> It does seem like a fatal flaw with Lester's subjective :> mechanics. : How so? Because there is no way to determine if someone is using it or if they are using it correctly. If two people apply his method and come up with different universal truths does that mean that the method is incorrect, or that someone applied it incorrectly, or what? :> I do not see how it can actually be demonstrated, :> and the fact he refuses to ever demonstrate it suggests :> that it cannot be demonstrated. : Nothing in mathematics can be demonstrated except with subjective : definitions, axioms and assumptions. Surely, you didn't think : that mathematical truths were either universal or objective. I have never said that math was about universal truths. Lester is the one claiming to have discovered universal truth. Why the strawman? Stephen === Subject: Re: Courage? > :> :> :> Do you believe that you can subjectively arrive at an > :> :> :> objective truth? > :> :> > :> :> : I know of no other way. > :> :> > :> :> :> What objective evidence is there that the > :> :> :> truth arrived at subjectively is objectively true? > :> :> > :> :> : What objective evidence would satisfy you, knowing as you do, > :> :> : that both objective evidence and objective truth are only > :> :> : available to us subjectively? > :> :> > :> :> How about a repeatable demonstration that Lester's > :> :> subjective mechanics leads to universal truth. If > :> :> it can be shown that different people will always > :> :> arrive at the same universal truths when they use > :> :> Lester's subjective mechanics that would be very > :> :> strong evidence there is something universal and > :> :> objective about it. > :> > :> : Well, you would have to take that up with Lester. Personally, I'm > :> : not sure that anything can be both universal /and/ objective. > :> > :> I was talking to Lester. You are the one who jumped in. > : You were probably talking to Lester in some much earlier post. > : But according to the last seven headers you have been talking to me. > That does not change the fact that I was talking to Lester > and you butted in. Well, exuuuuuuuse me. > :> So it is your opinion that universal truths are not > :> necessarily objective truths? What do you mean > :> by 'universal truth'? > : Are you talking to Lester or me? > Yes. > : I would think that universal is er...universal and objective is a > : small subset of that. > Just above you said > :> : Personally, I'm > :> : not sure that anything can be both universal /and/ objective. > which means that you do not think that objective is a small subset > of universal. I think I said, I'm not sure... Yep, there it is, what you quoted just above. > If something cannot be both universal /and/ > objective, then objective is not a subset of universal. Yes, it would seem so, wouldn't it. But I'm still not sure. > still do not know what you and Lester mean by subjective > universal truths. It would seem that a universal truth > would be always true, everywhere and for everyone. That is my understanding also. But, to keep assuming that I fully understand everything in Lester's mind is an error, even though much of what he says makes sense to me. I suspect he is on to something important. But I could be wrong. > :> > :> :> Of course, in order to do that Lester would have > :> :> to explain how one applies his subjective mechanics, > :> :> and we would then have to be able to determine if someone > :> :> was using the subjective mechanics correctly. > :> > :> : How would you determine is someone else was is being properly > :> : subjective? > :> > :> It does seem like a fatal flaw with Lester's subjective > :> mechanics. > : How so? > Because there is no way to determine if someone > is using it or if they are using it correctly. I'm afraid that is just an elaboration of your 'fatal flaw' objection above. *Why* do you see it as a problem. > If two > people apply his method and come up with different > universal truths does that mean that the method is incorrect, > or that someone applied it incorrectly, or what? It may mean that universal truth is best judged subjectively. > :> I do not see how it can actually be demonstrated, > :> and the fact he refuses to ever demonstrate it suggests > :> that it cannot be demonstrated. > : Nothing in mathematics can be demonstrated except with subjective > : definitions, axioms and assumptions. Surely, you didn't think > : that mathematical truths were either universal or objective. > I have never said that math was about universal truths. Of course not. You argue above that because Lester's theory has not been demonstrated that it cannot be demonstrated and must therefore be wrong. I merely pointed out that mathematics is also purely subjective and therefore cannot be demonstrated either. > Lester is the one claiming to have discovered universal truth. That wasn't my understanding of his claims. I thought he had found a way to regress to universal truth. > Why the strawman? What strawman? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: R-modules hello, please help me out a little here... Let M and N be left R-modules. Let f: M --> N and g: N --> M be left R-module homomorphisms s.t. fg(y)=y for all y belong to N. Show that M = ker(f)direct sum belong to g. any ideas? === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF [Herc] the diagonal DOESN'T EVEN DEPEND ON THE LIST. by definition its meaningless. [Lassy] You are wrong. The diagonal does depend on the list. No matter how you reorder your list, a number which differs from a number on the list in every digit will never be a diagonal. Will you please acknowledge this fact? [Herc] John if you set the diagonal to a number different to every digit of some element, when you shuffle the list indefinately that element is forever pushed down the queue since it never gets a digit match, that does not make an equivalent set. But, if you force a random diagonal on the list it tends to be a minor shuffling procedure. In base 3, every real has 1/3 chance of going across repeatedly until it does. Your FACT is wrong. Here is the proof. www.freewebs.com/namesort/linux.html I am TRYING to establish the fact that the sets are equal, your 'anit-diag element' does not affect the algorithm AT ALL. You can see the algorithm running! Its IMPOSSIBLE to miss an element using a random diagonal. You are thinking BACKWARDS about everything, just watch the algorithm run! The algorithm can be reduced down to this.. n=1 START IF RANDOM < 0.1 then LIST2+= LIST1(n); LIST1-= LIST1(n); n=0 n++ GOTO START IT CANT MISS! Herc === Subject: Re: abundance of irrationals!) ... > > Were is the self-contradiction? > > > > Which axiom results in a whole number aleph_0 which is lager than > > any natural? > > Where in the *definition* above is there any talk about aleph_0? > Where is the self-contradiction in the definition of cardinality? Where > does the definition use any concept of infinite sets? It only states that > if there *is* a bijection between two sets, those two sets have the > same cardinality. Period. > So the set of all numbers n and the set of all sequences {1,2,3,...,n} > have the same cardinality. The set of all numbers n is N. Period. Right. > A definition can not be self-contradictive. > Of course it can. A definition could say the set of finite natural > numbers has an infinite cardinality which is obviously a > self-contradiction. I would say that that is a theorem. Assuming that definitions for natural numbers and infinite cardinality do exist. Otherwise, what does it define? And as theorem it is true. > And if you cannot recognize that then certainly the > definition: Beleph is the finite cardinal number of the set N. That definition is not even wrong. It merely defines a name for a number that does not exist. If I define: a sheeeeeeep is an animal similar to a sheep but with nine legs, that definition is not wrong. If I define that a fumble is any integer smaller than 1 and larger than 0, the definition is not wrong. So no self-contradiction at all here. > It only tells us how we > define the words we use. You are getting now extremely close to JSH > and Archimedes Plutonium. They also both say some definitions are > wrong. But definitions are never wrong, they just tell us the > terminology used. > Def. 1. Ether is the medium which transports the light. Yup. Exactly. So you give a name to something that does not exist. > Def. 2. Let a be the smallest positive fraction. Yup. Exactly. So you give a name to something that does not exist. > Def. 3. Let M be the largest natural number. Yup. Exactly. So you give a name to something that does not exist. > Def. 4. Let b be the luminosity of the ether-brain-mass-index contained > in M/a. Yup. Exactly. So you give a name to something that does not exist. > Never wrong? What is wrong about those definitions? That they give a name to something that does not exist? Why is that wrong? They merely define an empty set of things, and as such are meaningless, but not wrong. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: abundance of irrationals!) ... > > This property means something like: An Em : n < m > > but not Em An : n < m > > Amongst other things. And nobody claims (except perhaps Cantor in a > paper over 100 years old) that Em An : n < m would be true. On the other > hand, I really doubt whether he claimed that aleph_0 was a number that > satisfied the Peano axioms. Does he claim that m and n in N in both > statements? > He claims ialeph_0 is a whole number larger than any natural number. > Of course he does not claim that it is a natural number itself. (It > must be larger than itself otherwise.) If you look at my sentences you > will find that I did neither claim m to be a natural, but a number > larger than all n. That is the problem with your statements, you do not make clear about what type of numbers you are talking. When you are *not* talking about natural numbers, the axiom does *not* mean An Em : n < m. It only implies this when n and m are natural numbers, because it only talks about natural numbers. So in this sense Cantor does *not* claim that Em An : n < m, as -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: abundance of irrationals!) > Indeed, if the last element is finite, then the set is finite. But > there is no last element. > If there is no last element, it cannot belong to the set. But we know > that every element of N is finite, whether last or not. Yes, so what? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: abundance of irrationals!) > ... > > Fortunately, mathematicians have axiom sysems which include > > axioms of infinity, which give then an infinite N as a set > > in their axiom system. > > > > The axiom says: > > An Em : n < m > > No. The axiom does not state anything like that. > It states for n here is n+1. That meaning is covered by my expression. Nope. It states that there is a set minimal set N where the other Peano axioms do hold. > > You say: > > Em An : n < m > > Neither does the axiom state this, nor does Virgil. > > The axiom does not talk about cardinal numbers. The only thing it > states that there is an infinite set N containing all natural numbers. > Where can I read that, please? Pray, let me know the source. All I know > is that {} must be contained and with m also {m}. No word is said about > actual infinity, not even about the existence of meaningful cardinal > numbers for infinite sets. also not talk about infinity. My rendering was incomplete. That N is infinite is a theorem. But what is the mathematical meaning of meaningful? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: abundance of irrationals!) >But definitions are never wrong, they just tell us the >terminology used. > That's going a bit far. To define infinity as 6 would be wrong. Is it wrong? What do you mean that it is wrong? If I write that definition I mean just that, I am only using a completely different definition of infinity from what is commonly understood. Moreover, in the integers modulo 6, the definition infinity = 6 might be seen as quite right. > And the reason, I think, that > we have so much trouble with infinity is that our intuitions about it > are rather unreliable. This is true. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: abundance of irrationals!) > Dik T. Winter said: > ... > > I don't tend to deal with these axioms much. The only reason I am dealing > > with them now is that mathematicians seem not to be able to do without > > them. > > Well, you know, without axioms, no mathematics... You have to start with > some basic truths, otherwise you can not logically reason to conclusions. > The basic truths are the axioms. > Axioms are not basic truths. The axioms are assertions which, within the > logic of the axiomatic system are atomic statements assumed to be true, and > treated in that system as basic truths. Yup, within an axiom system the axioms are basic truths. > eated in that system as basic truths. HOWEVER, the basic truths are NOT > the axioms. The axioms are statements that we agree are true, based on > reasons OUTSIDE of the particular axiomatic system. Not so. There are no reasons OUTSIDE an axiomatic system for either of the three possible variants of the parallel axiom. > This is true when dealing with logic in general. We start with a certain > set of given facts, each with a certain truth value (generally 1, or > 100% true), and a logical statement using those atomic facts, and evaluate > the truth value of the logical statement, by plugging in truth values and > evaluating the logical construction. Now, if all our facts are true, and > all our logical operations are correctly performed, we will get an correct > answer. IF, however, any of our assertions is FALSE, then no matter how > well we perform our oeprations on the truth values of those facts, we > are going to get an erroneous result. How do you measure whether an axiom is false or not? What does it *mean* that an axiom is false? > Axiomatic mathematics IS logical inference. ASSUMING all the axioms we have > asserted are CORRECT, then we can prove a given logical statement involving > them. If we get results that make no sense, then we can examine our logic to > see if we made a mistake. What does it mean that a result makes no sense? If a result is not in conflict with the axioms used, it makes sense, in that axiomatic system. I have no idea what is the case outside the axiomatic system. ... > The truth of axioms can be tested, as I said before, by their compatibility > with all other axioms and their compatibility with reality. What does that mean? How do you check compatibility with reality? Mathematics is not about reality, it is about ideas. > > Well f(0) is undefined, why? Because sin(1/0), or sin(oo), is > > undefined. But, we know that sin is always between 1 and -1 and > > so x sin(1/x) at x=0 is equivalent to 0*(some number between 1 > > and -1), which is always 0. > > Except when you go to the complex numbers, where sin(x) can be any value, > including values larger than 0... But what you are telling above seems a > lot like taking a limit... > It's not dissimilar, but doesn't require the full extent of limits. You are applying one of the common tests for whether a limit exists. And indeed, majoring x.sin(x) by x is one of such tests. But it is a theorem that that test indeed gives the limit. Moreover, you have a (mathematically) wrong statement: you state that at x = 0 x.sin(1/x) is equivalent to 0*(some number between 1 and -1). That is false; as you properly write just a little higher, sin(1/0) is undefined. So how can you conclude that it is a number between 1 and -1? > I was playing cards with a buddy, five card draw. Completely irrelevant. > > To me there is no confusion: f is not defined at x=0. On the other > > hand the limit of f, as x tends to zero, is indeed L = 0 because > > for every _nonzero_ x, |f(x)| <= |x|, which means that the values > > of f can be kept smaller than any pre-assigned epsilon simply by > > keeping x close to (but of course different from) epsilon. > > Somehow, the above looks indented like my statement, but I don't think it is. It is not. If you look closely you will see it is not indented as your statement. Count the number of > signs. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Letting it drop ... > The Magidin-Mckinnon paper appears in the May 2005 Amer Math Monthly: > http://www.maa.org/pubs/monthly_may05_toc.html > is not an easy journal to get into! Great. Already on its first instantiation in this newsgroup I thought it was important. Apparently it has been expanded from also its last public incarnation, but I think this is great. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Need help on calculating ATR for a mutual fund > I'll need to study your response some, but first there are 2 items that > confuse me: what is e and the rate r is not known. Can you help me a > litle on those 2 items? > Jim Can someone please help me with a calculation of ATR for a somewhat > complex > mutual fund? I have a fund, held for part of a year, with some of the dividends > reinvested and some not reinvested (withdrawn). Also some shares have > been > sold and some additional shares bought during the period. Also the NAV > has > changed during the period held. I need to prepare a formula which will calculate the Annualized Total > Return > of this fund. I plan to use this formula for calcs on a large number of > funds. > Jim. > Let Ak be the k'th (addition|withdrawal) with A0 the initial > investment > tk be the time between the Ak and A(k+1) in years > tn is time between last addition and end of period > r the rate (continuous compounded) > F = Final value at end of period > f(r) = (..(A0*e^(r*t0)+A1)*e^(r*t1)+A2) .. + An)*e^(r*tn) - F > multiply right hand side out and find f'(r) > Use Newton's method to find r where f(r) is zero. > The annualized growth rate in percent is then 100*(e^r - 1) > Note: reinvested interest is not considered an addition but is part of > the investment and contributes to the growth. 'r' is the continuously compounded growth rate and 'e' is the base of the natural logarithms. If there were no additions r could be found by solving this(t is in years): e^(r*t) = FINAL/START r = (1/t)*ln (FINAL/START) r expressed as a % is slightly lower than the effective annual rate. The above could be expressed using the annual growth directly, g=e^r f(g) = (..(A0*g^t0+A1)*g^t1+A2) .. + An)*g^tn) - F = A0*g^(t0+t1+..+tn) + A1*g^(t1+t2+..+tn) +..+ An*g^tn - F I use the f(r) instead of f(g) because the derivative seems easier/more natural to calculate, at least for me. === Subject: Re: Formal Generalization Problem - Solve it I also mistaken in the conditions of the problem (namely in definition of the operation -top->, which I call top-substitution). Corrected version of the problem: http://ex-code.com/~porton/math/news/2005-05-10-generalization-problem.html > Oh, a mistake, below is a corrected conjecture which is more likely to be > true: > Conjecture. The enough and necessary answer to the problem is the > following predicate: If any subformula X of A is specialization of any > subformula Y of A then X=Y. (That is it is required and necessary that no > subformula of A is a specialization of a different subformula of A.) This problem was raised during my math logic research (21 Century Math Method): http://ex-code.com/~porton/math/method.html) >> I think this can be proved by mapping formulas to multigraphs (maybe with >> colorized edges) and then specialization functions would become >> isomorphisms of one graph to a part of another graph. This would make the >> proof obvious, I think. -- Victor Porton (http://ex-code.com/~porton/) * Mathematics research, Christian revelations, software * === Subject: Need help with a probability problem Hi all, need help with a probability problem: In a game a metal disc with 4cm diameter is to be thrown on a table, which consist of (12x8) 96 squares, each 6x6 cm. To win the game the disc has to be inside any of the 96 squares and does not touch their perimeters. What is the probability of winning the game in one throw? (Suppose the disc will not be thrown outside of the 96 squares) [The answer is 1/9] I could not figure out the solution, any help will be greatly appreciated. === Subject: Re: Need help with a probability problem > Hi all, need help with a probability problem: In a game a metal disc > with 4cm diameter is > to be thrown on a table, which consist of (12x8) 96 squares, each 6x6 > cm. To win the game > the disc has to be inside any of the 96 squares and does not touch > their perimeters. What is the > probability of winning the game in one throw? (Suppose the disc will > not be thrown outside of > the 96 squares) [The answer is 1/9] > I could not figure out the solution, any help will be greatly > appreciated. I don't know the answer, but the way I would approach it is to first assume that the center of the disk lands on one of the inside squares. Find the area where the center can be to not touch a line. Then assume the center lands on an edge, but not corner, square and see what area it can be in while not touching a line. etc. Note, I think the probabilities of landing in edge square vs. and inside square are not equal since you can not go over the edge in an edge square, but you are allowed to be anywhere in an inside square. There may be an easier way to approach this. Bill === Subject: Re: Need help with a probability problem === Subject: Need help with a probability problem > In a game a metal disc with 4cm diameter is to be thrown on a table, > which consist of (12x8) 96 squares, each 6x6 cm. To win the game the > disc has to be inside any of the 96 squares and does not touch their > perimeters. What is the probability of winning the game in one throw? > (Suppose the disc will not be thrown outside of the 96 squares) > [The answer is 1/9] > I don't know the answer, but the way I would approach it is to first > assume that the center of the disk lands on one of the inside > squares. Find the area where the center can be to not touch a line. Center of disc falls in winning position inside 6cm square (6cm - 4cm)^2 / (6cm)^2 ---- === Subject: Where not to study science At al-Eman University in beautiful Sana'a: http://www.eman-univ.edu.ye/al-ema1.htm But before you pack to go there, see also http://travel.state.gov/travel/cis_pa_tw/tw/tw_936.html The founder and president of that school, Abdulmajid az-Zindani, is the subject of this interesting document from the United States Dept. of the Treasury: www.treas.gov/press/releases/js1190.htm === Subject: This Week's Finds in Mathematical Physics (Week 215) Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Originator: israel@math.ubc.ca (Robert Israel) Also available at http://math.ucr.edu/home/baez/week215.html April 15, 2005 This Week's Finds in Mathematical Physics - Week 215 John Baez This week I'd like to report on some cool things people have been explaining to me. The science fiction writer Greg Egan has been helping me understand Klein's quartic curve, and the mathematician Darin Brown has been explaining the analogy between geodesics and prime numbers. The two subjects even overlap slightly! Last week I talked about Klein's quartic curve. This led Gerard Westendorp and Mike Stay to draw some pictures of it, and their ideas helped Greg Egan create this really nice picture: 1) Greg Egan, Klein's quartic curve, http://math.ucr.edu/home/baez/KleinDual.gif It looks sort of tetrahedral at first glance, but if you look carefully you'll see that topologically speaking, it's a 3-holed torus. It's tiled by triangles, with 7 meeting at each vertex. So, it's the Klein quartic curve! Perhaps I should explain. Last week I talked about a tiling of the hyperbolic plane by regular heptagons with 3 heptagons meeting at each vertex. Dual to this is a tiling of the hyperbolic plane by equilateral triangles with 7 triangles meeting at each vertex. We can take a quotient space of this by a certain symmetry group and get a 3-holed torus tiled by 56 triangles with 7 meeting at each vertex. This is what Egan drew! With this picture you can almost *see* the 168 symmetries of Klein's quartic curve. First, you can take any vertex and twist it, causing the 7 triangles that meet at this vertex to cycle around. It's not obvious that this is a symmetry of the whole tiled surface, but it is. This gives a 7-element symmetry group. Second, the whole thing looks like a tetrahedron, so it inherits the rotational symmetries of a tetrahedron. This gives a more obvious 12-element symmetry group. 7 x 12 = 84, so how do we get a total of 168 symmetries? Well, there's also a 2-fold symmetry that corresponds to turning the tetrahedron inside out! And Egan made a wonderful *movie* of this. If a picture is worth a thousand words, this is worth about a million: 2) Greg Egan, Turning Klein's quartic curve inside out, http://math.ucr.edu/home/baez/KleinDualInsideOut.gif So, we get a total of 7 x 24 = 168 symmetries. Even better, if you watch carefully, you'll see that the tetrahedron in Egan's movie gets *reflected* as it turns inside out. More precisely, if you follow the four corners of the tetrahedron, you'll see that two come back to where they were, while the other two get switched. So, this symmetry acts as a reflection, or odd permutation, of the 4 corners. The rotations act as even permutations of the corners. This means that the Klein quartic has 24 symmetries forming a group isomorphic to the rotation/reflection symmetry group of a tetrahedron. Algebraically speaking, this group is S_4: the permutations of 4 things. This group is also the rotational symmetry group of a cube. In fact, Egan was able to spot a hidden cube lurking in his picture! Can you? If you look carefully, you'll see each corner of his tetrahedral gadget is made of a little triangular prism with one triangle facing out and one facing in. Since 4 x 2 = 8, there are 8 of these triangles. Abstractly, we can think of these as the 8 corners of a cube! They aren't really, but we can pretend. The way these 8 triangles come in pairs corresponds to how the vertices of a cube come in diagonally opposite pairs. Using this, you can see that the group S_4 acts on these 8 triangles in precisely the same way it acts via rotations on the vertices of a cube. In fact, you can even draw a PICTURE of a cube on the Klein quartic by drawing suitable curves that connect the centers of these 8 triangles! It's horribly distorted, but topologically correct. Part of the distortion is caused by embedding the Klein quartic in ordinary 3d Euclidean space. If we gave the Klein quartic the metric it inherits from the hyperbolic plane, the edges of the cube would be geodesics. This remark also helps us see something else. The Klein quartic is tiled by 56 triangles. 8 of them give the cube we've just been discussing. In Egan's picture these triangles look special, since they lie at the corners of his tetrahedral gadget. But this is just an illusion caused by embedding the Klein quartic in 3d space. In reality, the Klein quartic is perfectly symmetrical: every triangle is just like every other. So in fact there are lots of these cubes, and every triangle lies in some cube. But this is where it gets really cool. In fact, each triangle lies in just *one* cube. So, there's precisely one way to take the 56 triangles and divide them into 7 bunches of 8 so that each bunch forms a cube. So: the symmetry group of the Klein quartic acts on the set of cubes, which has 7 elements. But as I explained last week, this symmetry group also acts on the Fano plane, which has 7 points. This suggests that cubes in the Klein quartic naturally correspond to points of the Fano plane. And Egan showed this is true! He showed this by showing more. The Fano plane also has 7 lines. What 7 things in the Klein quartic do these lines correspond to? ANTICUBES! You see, the cubes in the Klein quartic have an inherent handedness to them. You can go between the 8 triangles of a given cube by following certain driving directions, but these driving directions involve some left and right turns. If you follow the mirror-image driving directions with left and right switched, you'll get an ANTICUBE. Apart from having the opposite handedness, anticubes are just like cubes. In particular, there's precisely one way to take the 56 triangles and divide them into 7 bunches of 8 so that each bunch forms an anticube. Here's a picture: 3) Greg Egan, Cubes and anticubes in the Klein quartic curve, http://math.ucr.edu/home/baez/KleinFigures.gif Each triangle has a colored circle and a colored square on it. There are 7 colors. The colored circle says which of the 7 *cubes* the triangle belongs to. The colored square says which of the 7 *anticubes* it belongs to. If you stare at this picture for a few hours, you'll see that each cube is completely disjoint from precisely 3 anticubes. Similarly, each anticube is completely disjoint from precisely 3 cubes. This is just like the Fano plane, where each point lies on 3 lines, and each line contains 3 points! So, we get a vivid way of seeing how every figure in the Fano plane corresponds to some figure in the Klein quartic curve. This is why they have the same symmetry group. This is an excellent example of Klein's Erlangen program for reducing geometry to group theory, which I discussed in week213. Here we are beginning to see how two superficially different geometries are secretly the same: FANO PLANE KLEIN'S QUARTIC CURVE 7 points 7 cubes 7 lines 7 anticubes incidence of points and lines disjointness of cubes and anticubes However, we're only half done! We've seen how to translate simple figures and indicence relations in the Fano plane to complicated ones in Klein's quartic curve. But, we haven't figured out translate back! KLEIN'S QUARTIC CURVE FANO PLANE 24 vertices ??? 84 edges ??? 56 triangular faces ??? incidence of vertices and edges ??? incidence of edges and faces ??? Here I'm talking about the tiling of Klein's quartic curve by 56 equilateral triangles. We could equally well talk about its tiling by 24 regular heptagons, which is the Poincare dual. Either way, the puzzle is to fill in the question marks. I don't know the answer! To conclude - at least for now - I want to give the driving directions that define a cube or an anticube in Klein's quartic curve. Say you're on some triangle and you want to get to a nearby triangle that belongs to the same cube. Here's what you do: hop across any edge, turn left, hop across the edge in front of you, turn right, then hop across the edge in front of you. Or, suppose you're on some triangle and you want to get to another that's in the same anticube. Here's what you do: hop across any edge, turn right, hop across the edge in front of you, turn left, then hop across the edge in front of you. (If you don't understand this stuff, look at the picture above and see how to get from any circle or square to any other circle or square of the same color.) You'll notice that these instructions are mirror-image versions of each other. They're also both 1/4 of the driving directions from hell that I described last time. In other words, if you go LRLRLRLR or RLRLRLRL, you wind up at the same triangle you started from. You'll have circled around one face of a cube or anticube! In fact, your path will be a closed geodesic on the Klein quartic curve... like the long dashed line in Klein and Fricke's original picture: 4) Klein and Fricke, Klein's quartic curve with geodesic, http://math.ucr.edu/home/baez/Klein168.gif Next, a little about geodesics and prime numbers. I've just been talking a little about geodesics in the Klein quartic, which is the quotient H/G of the hyperbolic plane H by a certain group G which I explained last week. This group, usually called Gamma(7), is a nice example of a Fuchsian group - that is, a discrete subgroup of the isometries of the hyperbolic plane. Darin Brown and his thesis advisor Jeff Stopple at U. C. Santa Barbara have been thinking about geodesics in H/G for other Fuchsian groups G, and their relation to number theory: 4) Jeff Stopple, A reciprocity law for prime geodesics, J. Number Theory 29 (1988), 224-230. 5) Darin Brown, Lifting properties of prime geodesics on hyperbolic I'd really like to learn about this, because it connects all sorts of stuff I dream of understanding someday, especially quantum chaos (week190), zeta functions in physics and number theory (week199), and Galois theory as a theory of covering spaces (week205). Also, it involves a big mysterious analogy, and I always like those! I don't understand this stuff well enough to try a full-fledged explanation yet, so I'll just give a vague sketch. A prime geodesic in a Riemannian manifold X is a closed geodesic f: S^1 -> X that cycles around just once. In other words, f should be one-to-one. We say a closed geodesic is the nth power of a prime one if it's just like the prime one but it cycles around n times. Every closed geodesic is the nth power of a prime one in a unique way. If we have a Fuchsian group G, H/G is a surface with a Riemannian metric. It looks locally like the hyperbolic plane, so it's called a hyperbolic surface. And, we can look at prime geodesics in it. If G' is a subgroup of G, we get a covering map H/G' -> H/G so we can ask about lifting prime geodesics in H/G to closed geodesics in H/G'. There can be a bunch of ways to do this, so we say a prime geodesic in H/G splits into powers of prime geodesics up in H/G'. If you know any number theory - reading week205 should be enough - this should remind you of how a prime ideal in some algebraic number field can split into prime ideals in an extension of this field, and/or ramify into powers of prime ideals. And indeed, Darin Brown has found a big mysterious analogy that goes like this: Number field K Hyperbolic surface H/G Field extension K' of K Covering p: H/G' -> H/G' Galois group Gal(K'/K) Deck transformation group Aut(p) Prime ideal Q of K Prime geodesic f in H/G Prime ideal Q' lying over Q Prime geodesic f' lying over Q Splitting of prime ideal Q in K' Lifting of prime geodesic f to H/G' Norm N(Q) of ideal Q Norm N(f) of closed geodesic f Frobenius conjugacy class of Q Frobenius conjugacy class of f Artin L-function Selberg zeta function But it's more than an analogy: there's even a way to associate number fields to certain hyperbolic surfaces! The reason is that often Fuchsian groups will consist of matrices whose entries lie in some number field. I would like to understand the Selberg zeta function and its relation to quantum mechanics. The Selberg zeta function is related to closed geodesics, which are periodic classical trajectories, while the zeta function of a Laplacian is related to periodic *quantum* trajectories (namely eigenfunctions of the Laplacian). So, the two are related. I know there's a lot of cool stuff going on here - especially since so quantum chaos rears its ugly head. But, I don't understand any of the details. The classical periodic orbits are a crucial stepping stone in the understanding of quantum mechanics, in particular when then classical system is chaotic. This situation is very satisfying when one thinks of Poincare who emphasized the importance of periodic orbits in classical mechanics, but could not have had any idea of what they could mean for quantum mechanics. The set of energy levels and the set of periodic orbits are complementary to each other since they are essentially related through a Fourier transform. Such a relation had been found earlier by the mathematicians in the study of the Laplacian operator on Riemannian surfaces with constant negative curvature. This led to Selberg's trace formula in 1956 which has exactly the same form, but happens to be exact. The mathematical proof, however, is based on the high degree of symmetry of these surfaces which can be compared to the sphere, although the negative curvature allows for many more different shapes. When I get serious, I'll read these: 6) M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, Berlin, 1990. 7) Predrag Cvitanovic, Roberto Artuso, Per Dahlqvist, Ronnie Mainieri, Gregor Tanner, Gabor Vattay, Niall Whelan and Andreas Wirzba, Chaos: Classical and Quantum, http://www.nbi.dk/ChaosBook/ 8) Svetlana Katok, Fuchsian Groups, U. Chicago Press, Chicago, 1992. 9) J. Elstrodt, F. Grunewald, and J. Mennicke, Groups Acting on Hyperbolic Space, Springer, Berlin, 1998. 10) Peter Sarnak, Quantum chaos, symmetry and zeta functions, in Current Developments in Mathematics, 1997, eds R. Bott et al., International Press, Boston, 1999, pp. 127-159. 11) C. Schmit, Quantum and classical properties of some billiards on the hyperbolic plane, in Chaos and Quantum Physics, eds. M.-J. Giannoni et al., Elsevier, New York, 1991, pp. 333-369. ----------------------------------------------------------------------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === Subject: Re: metrizable + zero-dimensional implies ultrametrizable ? Originator: israel@math.ubc.ca (Robert Israel) I looked at the paper you mentioned, but I could not find a solution to my problem in it. What Prabir proves is that there is a metrizable topological space S such that ind(S) neq dim(S). Did you mean some other paper? AG. >> I found the following conjecture in a paper: >> Conjecture: Every metrizable zero-dimensional topological space is >> ultrametrizable >> I was not able to find any references to the problem. Could anybody >> supply >> some? Is the problem still open? >> > This problem was solved in the 1960's by Prabir Roy: > http://www.ams.org/mathscinet-getitem?mr=37,3544 > there is a counterexample. > KP === Subject: Re: (Help)! Banach-Alaoglu theorem Originator: israel@math.ubc.ca (Robert Israel) >Can anybody explain the expression of Banach-Alaoglu theorem for >weighted L_p space ( L_{p,nu} )? I will be happy if you can send as >early as you can do. >sincerely. Well, the theorem says that the closed unit ball of the dual space is weak-* compact, just as for any other Banach space. What is it you want explained? (Not trying to be difficult, it's just not clear what you're asking for here.) ************************ David C. Ullrich [ Moderator's note: I'm also puzzled, but perhaps gilevgi will find the following facts useful: if 1 < p <= infinity, L_p(nu) is the dual of L_q(nu) where 1/p + 1/q = 1 (q = 1 in the case p = infinity). In many cases L^q(nu) is separable (e.g. if nu is a regular Borel measure on Euclidean space), and then the unit ball of L^p(nu) is weak-* metrizable, so in this case Banach-Alaoglu is equivalent to saying that every sequence in the unit ball of L^p(nu) has a weak-* convergent subsequence. - RI ] === Subject: Re: Cyclic Mobius Transform and roots of unity Originator: israel@math.ubc.ca (Robert Israel) It is not only a matter of 'Elegance' I do believe simplification in all domains is now a very need. I wish an understandable world and think lot of TOO IMPORTANT MASTERS are making things more cumbersome, complex,intricate than they are... In many domains:Algebra,quanta physics,logic,linguistic, econometrics (not playometrics!)breaking,cutting, and chiefly reordering/reorganizing seems a real present work to put both hands into.. Back to our subject, have you got an idea about continuous real or complex cycle number ? h^[sqrt(3)-I](z)=z any use ? Remarque: ->one fixed point 1/(h(z)-z1)=1/(z-z1)+ c and 1/(h[r](z)-z1)=1/(z-z1)+ r*c r iteration Counting function phi(x)=1/(c*(z-z1)), ->two fixed points Counting function phi(x)=ln((z-z1)/(z-z2))/ln(a); Counting function phi(z)or Abel verifies for uncyclic functions:phi(h[r](z))= phi(z) + r Amicalement,Alain. === Subject: Re: Cyclic Mobius Transform and roots of unity Originator: israel@math.ubc.ca (Robert Israel) > Back to our subject, have you got an idea about continuous real or complex > cycle number ? > h^[sqrt(3)-I](z)=z any use ? There are various one-dimensional Lie subgroups of SL(2,C). Consider {exp(tA): t in C} where A is a trace-zero matrix. This is a subgroup of SL(2,C). For most A the map t -> exp(tA) is injective. Then one can think of exp(tA) as B^t dor B = exp(A). Sometimes it isn't, and then one has the same sort of ambiguity for B^t as for exponentials in the complex plane. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: This week in the mathematics arXiv (25 Apr - 29 Apr) Originator: israel@math.ubc.ca (Robert Israel) Here are this week's titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modification. Titles in the mathematics arXiv (25 Apr - 29 Apr) ------------------------------------------------- AG: Algebraic Geometry ---------------------- math.AG/0504576 Vincenzo Di Gennaro: Hierarchical structure of the family of curves with maximal genus verifying flag conditions math.AG/0504575 Jonathan A. Cox: A presentation for the Chow ring of M_{0,2}(P^1,2) math.AG/0504563 Yunfeng Jiang: The Orbifold Cohomology Ring of Simplicial Toric Stack Bundles math.AG/0504561 Mark A. de Cataldo: Lectures on the Hodge theory of projective manifolds math.AG/0504560 Mark A. de Cataldo, Luca Migliorini: The Gysin map is compatible with mixed Hodge structures math.AG/0504554 Mark Andrea A. de Cataldo, Luca Migliorini: Intersection forms, topology of maps and motivic decomposition for maps of threefolds math.AG/0504542 Eckart Viehweg; Kang Zuo: Numerical bounds for semi-stable families of curves or of certain higher dimensional manifolds math.AG/0504523 Yuri G. Zarhin: Isogeny classes of abelian varieties over function fields math.AG/0504517 Vladimir L. Popov: Problems for problem session math.AG/0504500 Laurent Ducrohet: The action of the Frobenius map on rank 2 vector bundles over a supersingular genus 2 curve in characteristic 2 math.AG/0504492 Daniele Faenzi: Rank 2 arithmetically Cohen-Macaulay bundles on a nonsingular cubic surface math.AG/0504486 Mircea Mustata, Sam Payne: Ehrhart polynomials and stringy Betti numbers math.AG/0504482 Benoit Bertrand: Asymptotically maximal families of hypersurfaces in toric varieties math.AG/0504467 Peter Vermeire: The Cohomology of Reflexive Sheaves on Smooth Projective 3-folds math.AG/0504465 Peter Vermeire: Moduli of Reflexive Sheaves on Smooth Projective 3-folds math.AG/0504460 Jian Zhou: On a deformed topological vertex math.AG/0504457 Joaquim Ro'e: Maximal rank for planar singularities of multiplicity two math.AG/0504449 Arnaud Beauville: Orthogonal bundles on curves and theta functions math.AG/0504448 Alexander Polishchuk: Lie symmetries of the Chow group of a Jacobian and the tautological subring math.AG/0504443 Ulrich Goertz, Thomas J. Haines, Robert E. Kottwitz, Daniel C. Reuman: Dimensions of some affine Deligne-Lusztig varieties AP: Analysis of PDEs -------------------- math.AP/0504568 Xavier Carvajal Paredes: Sharp global well-posedness for a higher order Schrodinger equation math.AP/0504536 Elise Fouassier: High frequency analysis of Helmholtz equations: case of two point sources math.AP/0504502 Yuxiang Li, Youde Wang: Bubbling location for $F$-harmonic maps and Inhomogeneous Landau-Lifz equations math.AP/0504497 S. Gustafson, K. Kang, T.-P. Tsai: Schrodinger Flow Near Harmonic Maps math.AP/0504481 Jean-Philippe Nicolas: On Lars Hormander's remark on the characteristic Cauchy problem math.AP/0504455 Julie Clutterbuck: Parabolic equations with continuous initial data math.AP/0504451 Kei Morii: A Fourier restriction theorem for hypersurfaces which are graphs of certain real polynomials math.AP/0504450 Alberto Bressan, Massimo Fonte: An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation AT: Algebraic Topology ---------------------- math.AT/0504555 Gregory D. Landweber: K-theory and elliptic operators CA: Classical Analysis and ODEs ------------------------------- math.CA/0504580 Maria J. Cantero, Leandro Moral, Luis Velazquez: Measures on the unit circle and unitary truncations of unitary operators math.CA/0504567 Oliver C. Schnurer: Convex functions with unbounded gradient math.CA/0504545 P. Shmerkin, B. Solomyak: Zeros of {-1,0,1}-power series and connectedness loci for self-affine sets math.CA/0504540 Kouichi Takemura: On finite-gap potential math.CA/0504476 Diego Dominici: Some remarks on a paper by L. Carlitz math.CA/0504456 Jacob S. Christiansen, Erik Koelink: Self-adjoint difference operators and classical solutions to the Stieltjes--Wigert moment problem math.CA/0504454 Eiji Onodera: Bilinear estimates associated to the Schrodinger equation with a nonelliptic principal part CO: Combinatorics ----------------- math.CO/0504569 Victor J. W. Guo, Jiang Zeng: Some Arithmetic Properties of the q-Euler Numbers and q-Sali'e Numbers math.CO/0504564 Jeremy Alexander: Ringel's generalized earth-moon problem math.CO/0504528 Satoshi Murai: Gotzmann monomial ideals math.CO/0504525 William Y.C. Chen, Qing-Hu Hou, Yan-Ping Mu: A Telescoping Algorithm for Double Summations math.CO/0504522 Lars Eirik Danielsen, Matthew G. Parker: On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12 math.CO/0504488 Guo-Guang Yan, Arthur L. B. Yang, Joan J. Zhou: The Zrank Conjecture and Restricted Cauchy Matrices math.CO/0504487 Susan Y. J. Wu, Arthur L. B. Yang: Division and the Giambelli Identity math.CO/0504472 Terence Tao: Szemer'edi's regularity lemma revisited math.CO/0504444 Alexander Barvinok: Computing the Ehrhart quasi-polynomial of a rational simplex CV: Complex Variables --------------------- nlin.SI/0504053 Alexandre Eremenko: Meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation math.CV/0504521 Nicolas Eisen: On the Holomorphic Extension of CR Distributions from Non Generic CR Submanifolds of $C^L$ math.CV/0504489 Alexander Brudnyi: Hartogs type theorems on coverings of Stein manifolds math.CV/0504471 Finnur Larusson, Ragnar Sigurdsson: The Siciak-Zahariuta extremal function as the envelope of disc functionals DG: Differential Geometry ------------------------- math.DG/0504557 Marianty Ionel, Maung Min-Oo: Cohomogeneity One Special Lagrangian Submanifolds in the Deformed Conifold math.DG/0504556 Boris Khesin, Gerard Misiolek: Asymptotic directions, Monge-Ampere equations and the geometry of diffeomorphism groups math.DG/0504550 M. Castrillon Lopez, P. M. Gadea, A. F. Swann: Homogeneous quaternionic Kaehler structures and quaternionic hyperbolic space math.DG/0504535 Harish Seshadri: Weyl curvature and the Euler characteristic in dimension four math.DG/0504529 G.Sardanashvily: The variational bicomplex on graded manifolds and its cohomology math.DG/0504527 Xianzhe Dai, Xiaodong Wang, Guofang Wei: On the Stability of Kahler-Einstein Metrics math.DG/0504526 Huai-Dong Cao, Natasa Sesum: The compactness result for Kahler Ricci solitons math.DG/0504524 Jerome A. Jenquin: Classical Chern-Simons on manifolds with spin structure math.DG/0504504 Fuquan Fang: Finite isometry groups of 4-manifolds with positive sectional curvature math.DG/0504498 Novica Blazic, Peter Gilkey: Conformally Osserman manifolds and self-duality in Riemannian geometry math.DG/0504491 Valerii Dryuma: On geometrical properties of the spaces defined by the Pfaff equations math.DG/0504484 S. Tabachnikov, Yu. Tyurina: Existence and non-existence of skew branes math.DG/0504478 Bing-Long Chen, Xi-Ping Zhu: Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature math.DG/0504469 Andreas Cap, Vojtech Zadnik: On the Geometry of Chains math.DG/0504440 Pierre Bayard: Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space DS: Dynamical Systems --------------------- math.DS/0504490 S. Bezuglyi, J. Kwiatkowski, K. Medynets: Approximation in ergodic theory, Borel, and Cantor dynamics math.DS/0504442 M. Chugunova, D. Pelinovsky: Block-diagonalization of the linearized coupled-mode system FA: Functional Analysis ----------------------- math.FA/0504452 vector-valued random sums GM: General Mathematics ----------------------- math.GM/0504565 Elemer E Rosinger: Nel's category theory based differential and integral Calculus, or did Newton know category theory ? math.GM/0504549 Dhananjay P. Mehendale: On Isomorphism of Graphs and the k-clique Problem math.GM/0504520 Sukanto Bhattacharya, Kuldeep Kumar, Florentin Smarandache: Conditional probability of actually detecting a financial fraud - a neutrosophic extension to Benford's law GR: Group Theory ---------------- math.GR/0504577 Gregory C. Bell: Growth of the asymptotic dimension function for groups math.GR/0504574 Thomas Michael Keller: Fixed conjugacy classes of normal subgroups and the k(GV)-problem math.GR/0504566 Alexander Dranishnikov, Viktor Schroeder: Embedding of hyperbolic Coxeter groups into products of binary trees and aperiodic tilings math.GR/0504447 J. Smith: On asymptotic dimension of countable abelian groups math.GR/0504445 Ilya Kapovich, Tatiana Nagnibeda: The Patterson-Sullivan embedding and minimal volume entropy for outer space GT: Geometric Topology ---------------------- math.GT/0504578 D. Kotschick: Minimizing Euler characteristics of symplectic four-manifolds math.GT/0504546 Ken'ichi Ohshika: Realising end invariants by limits of minimally parabolic, geometrically finite groups math.GT/0504519 Erol Akbas: A Presentation For The Automorphisms Of The 3-Sphere That Preserve A Genus Two Heegaard Splitting math.GT/0504495 Justin Sawon: Perturbative expansion of Chern-Simons theory math.GT/0504479 Slavik Jablan, Radmila Sazdanovich: Braid Family Representatives math.GT/0504474 Juan Souto, Peter Storm: Dynamics of the mapping class group action on the variety of PSL(2,C) characters math.GT/0504473 Peter Storm: Rigidity of minimal volume Alexandrov spaces math.GT/0504446 Mikami Hirasawa, Masakazu Teragaito: Crosscap numbers of 2-bridge knots math.GT/0504441 Yair Glasner, Juan Souto, Peter Storm: Finitely generated subgroups of lattices in PSL(2,C) KT: K-Theory and Homology ------------------------- math.KT/0504548 Anwar A. Irmatov, Alexandr S. Mishchenko: On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators math.KT/0504458 Ulrich Bunke, Ingo Schroeder: Twisted K-Theory and TQFT LO: Logic --------- math.LO/0504553 David J. Foulis: Logic and Partially Ordered Abelian Groups MP: Mathematical Physics ------------------------ nucl-th/0504075 B.G. Giraud, A. Weiguny, L. Wilets: Coordinates, modes and maps for the density functional math-ph/0504083 C. Cacciapuoti, R. Figari, A. Posilicano: Point interactions in acoustics: one dimensional models math-ph/0504082 Kazuhiro Hikami: On the Quantum Invariant for the Spherical Seifert Manifold math-ph/0504081 M. Aslam Chaudhry, Amer Iqbal, Asghar Qadir: A Representation for the Anyon Integral Function cond-mat/0504674 C. Albert, L. Ferrari, J. Froehlich, B. Schlein: Magnetism and the Weiss Exchange Field - A Theoretical Analysis Inspired by Recent Experiments quant-ph/0504200 M. Blasone, P. Jizba, H. 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Heinonen, A. Toigo: On the coexistence of position and momentum observables physics/0503235 Ivan Todorov: Werner Heisenberg (1901-1976) math-ph/0504075 Eman Hamza, Alain Joye, Gunter Stolz: Localization for Random Unitary Operators math-ph/0504074 Benjamin Bahr: The Hot Bang state of massless fermions math-ph/0504073 Brice Camus: Spectral fluctuations of Schrodinger operators generated by equilibriums math-ph/0504072 Ramazan Koc, Hayriye Tutunculer, Mehmet Koca: Solution of two-mode bosonic Hamiltonians and related physical systems math-ph/0504071 I. Struchiner, M. Rosa: On Zeeman Topology in Kaluza-Klein and Gauge Theories math-ph/0504070 Nikos Kalogeropoulos: Entropy and curvature variations from effective potentials math-ph/0504069 S. Bernstein, R. Hielscher, H. Schaeben: The Generalized Spherical Radon Transform and Its Application in Texture Analysis math-ph/0504068 Teunis C. Dorlas, Philippe A. Martin, Joseph V. 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Saez: A new algorithm to search for small nonzero |x^3 - y^2| values math.NT/0504570 Jonathan Pila: Counting points on curves over families in polynomial time math.NT/0504552 Herbert Gangl, Alexander B. Goncharov, Andrey Levin: Multiple logarithms, algebraic cycles and trees math.NT/0504534 Alexander Schmidt: Circular sets of prime numbers and p-extension of the rationals math.NT/0504533 J. K. Canci: Cycles for rational maps with good reduction outside a prescribed set math.NT/0504483 Oleg German: Klein polyhedra and norm minima of lattices math.NT/0504453 Cristiana Bertolin: Realizations of Biextensions OA: Operator Algebras --------------------- math.OA/0504573 Christopher Hillar, Charles R. Johnson, Ilya M. Spitkovsky: Positive eigenvalues and two-letter generalized words math.OA/0504572 Jonathan M. Groves, Yonatan Harel, Christopher J. Hillar, Charles R. Johnson, Patrick X. Rault: Absolutely Flat Idempotents quant-ph/0504189 David W. 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Ionescu: Higher Derived Brackets and Deformation Theory I math.QA/0504494 Shilin Yang: Weak Hopf algebras corresponding to Cartan matrices math.QA/0504493 Hua-Lin Huang, Shilin Yang: Quantum groups and double quiver algebras math.QA/0504463 William J. Cook, Haisheng Li, Kailash C. Misra: Affine Lie algebras and multisum identities RA: Rings and Algebras ---------------------- math.RA/0504544 Jakob Palmkvist: A realization of the Lie algebra associated to a Kantor triple system math.RA/0504531 Ralf Holtkamp: On Hopf algebra structures over free operads math.RA/0504475 V. V. Bavula: Generators and defining relations for the ring of differential operators on a smooth affine algebraic variety math.RA/0504468 Peter Jorgensen: Finite Cohen-Macaulay type and smooth non-commutative schemes RT: Representation Theory ------------------------- math.RT/0504547 Christine Bessenrodt, Thorsten Holm: q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras math.RT/0504543 Ian M. Musson: Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes math.RT/0504539 Joerg Feldvoss: Injective Modules and Prime Ideals of Universal Enveloping Algebras math.RT/0504538 Sophie Morier-Genoud: Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties SG: Symplectic Geometry ----------------------- math.SG/0504537 Victor Guillemin, Reyer Sjamaar: Convexity theorems for varieties invariant under a Borel subgroup SP: Spectral Theory ------------------- math.SP/0504571 Emily B. Dryden, Alexander Strohmaier: Huber's theorem for hyperbolic orbisurfaces ST: Statistics -------------- physics/0504185 S.N. Dorogovtsev, J.F.F. Mendes, J.G. Oliveira: Frequency of occurrence of numbers in the World Wide Web math.ST/0504516 Yvonne H. S. Ho, Stephen M. S. Lee: Iterated smoothed bootstrap confidence intervals for population quantiles math.ST/0504515 Snigdhansu Chatterjee, Arup Bose: Generalized bootstrap for estimating equations math.ST/0504514 Yijun Zuo, Hengjian Cui: Depth weighted scatter estimators math.ST/0504513 Maria Teresa Gallegos, Gunter Ritter: A robust method for cluster analysis math.ST/0504512 Chris A. J. Klaassen, Hein Putter: Efficient estimation of Banach parameters in semiparametric models math.ST/0504511 Peter Hall, Kee-Hoon Kang: Bandwidth choice for nonparametric classification math.ST/0504510 Ibrahim Ahmad, Sittisak Leelahanon, Qi Li: Efficient estimation of a semiparametric partially linear varying coefficient model math.ST/0504509 Yannick Baraud, Sylvie Huet, Beatrice Laurent: Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function math.ST/0504508 T. Tony Cai, Mark G. Low: Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation math.ST/0504507 Kesar Singh, Minge Xie, William E. Strawderman: Combining information from independent sources through confidence distributions math.ST/0504506 Arthur Cohen, Harold B. Sackrowitz: Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure math.ST/0504505 Arthur Cohen, Harold B. Sackrowitz: Decision theory results for one-sided multiple comparison procedures math.ST/0504503 Harrison H. Zhou, J. T. Gene Hwang: Minimax estimation with thresholding and its application to wavelet analysis math.ST/0504501 Cun-Hui Zhang: General empirical Bayes wavelet methods and exactly adaptive minimax estimation math.ST/0504499 Andrew Gelman: Analysis of variance -- why it is more important than ever math.ST/0504477 Sergey Plyasunov: On hybrid simulation schemes for stochastic reaction dynamics -- / Greg Kuperberg (UC Davis) / Home page: http://www.math.ucdavis.edu/~greg/ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that's fit to e-print * === Subject: Eigenvalues of matrices in SL(n, Z) Originator: israel@math.ubc.ca (Robert Israel) Is anything general known about which algebraic integers are eigenvalues of matrices in SL(n, Z) for some n? The characteristic polynomial of such a matrix has constant term equal to the determinant, which is 1, so it would seem there is some restriction on the algebraic integers that could be eigenvalues. === Subject: Re: Eigenvalues of matrices in SL(n, Z) Originator: israel@math.ubc.ca (Robert Israel) > Is anything general known about which algebraic integers are > eigenvalues of matrices in SL(n, Z) for some n? The characteristic > polynomial of such a matrix has constant term equal to the determinant, > which is 1, so it would seem there is some restriction on the algebraic > integers that could be eigenvalues. They have to be units in the ring of algebraic integers. That is all. If u is a unit then u is a root of an integral polynomial f(X) X^n + a_1 X^{n-1} + ... +- 1 = 0. The companion matrix of f has u as an eigenvalue and determinant +-1. If it's -1, the consider instead (X+1)f(X). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: Eigenvalues of matrices in SL(n, Z) Originator: israel@math.ubc.ca (Robert Israel) === Subject: Re: Maximal order subgroups of Z(m;*) with m=2^n and n>3. Originator: israel@math.ubc.ca (Robert Israel) >I had found that all the subgroups of maximal order 2^(n-2) of the >multiplicative group of the ring of integers modulo m=2^n are the ones >generated by the elements congruent with 3 and 5 modulo 8 (for any >natural number n>3). >Per example, for n=4, all the subgroups of maximal order 4 are >represented by {3,9,11,1} and {5,9,13,1}, generated the first by the >powers of the elements 3 or 11, and generated the second by the powers >of the elements 5 or 13. Theorem 2.43 on pp. 105 of the fifth edition of An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery (Wiley & Sons) reads: Theorem 2.43: Suppose that a>=3. The order of 5 (mod 2^a) is 2^{a-2}. The numbers +/- 5, +/- 5^2, +/- 5^3, ..., +/- 5^{a-2} form a system of reduced residues (mod 2^a). If a is odd, then there exists i and j such that a = (-1)^i 5^j (mod 2^a). The values of i and j are uniquely determined (mod 2) and (mod 2^{a-2}), respectively. So the fact that one of the maximal subgroups is generated by the powers of 5 is well known. The fact that 5+8k will generate will probably follow from the binomial theorem. That 3 generates the other one would be equivalent to showing that 5b = 3 (mod 2^{a}) has a solution with b odd, b^2 = 1 (mod 2^a) and b not congruent to 5^{2^{a-3}} modulo 2^a. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: McDowell-Mansouri gravity Originator: israel@math.ubc.ca (Robert Israel) > ..... > Now, if you want to think about this in terms of group theory, the best > thing to do is think of the basis elements of the Clifford algebra as > Lie algebra generators. Then I think, for example, the Lie algebra > corresponding the 16 generators of Cl_4 is... u(4)? And I think the > Lie algebra corresponding to just the 10 Cl_4 vectors and bi-vectors is > sp(2), but I'm not at all sure of that, as my group theory is lacking. You may embed the Clifford algebra Cl(n) within Cl(n+1) by identifying the vectors (gamma_k) of Cl(n) with bivectors (gamma_k0) of Cl(n+1). Thus to answer your question the vectors + bivectors of Cl(4) generate the group Spin(5), as they will be all the bivectors of Cl(5). You'll have to check your sign convention for indefinite case to see if e.g. vectors + bivectors of Cl(3,1) are equivalent to bivectors of Cl(4,1) or Cl(3,2). E.g if the three spatial gamma's square to +1 then the Killing signature for vectors and bivectors is: (n+,n-)=(3+3,3+1) same as bivectors of Cl(3,2). Make the spatial gammas square to -1 and you will get (n+,n-)=(3+1,3+3) e.g. Cl(4,1) bivectors. James Baugh