mm-2629 === Subject: Re: Homotopic maps essential means not homotopic to a constant. The inclusion of X into U is not homotopic to a constant. This is a counter example to the original statement. === Subject: Re: Homotopic maps On Mon, Nov 21, 2005 2:45 PM, > essential means not homotopic to a constant. The > inclusion of X into U > is not homotopic to a constant. This is a counter > example to the > original statement. So (I) doesn't imply (II), O.K.! But it's quite evident that (II) implies (I). However, evidence is not a mathematical criterion of truth... === Subject: Re: Homotopic maps No that's false too. Let U = S^{n-1} x (0, infty) = R^n - {0} and f:S^{n-1} - S^{n-1} be an inessential map with image S^{n-1}, where all the S^{n-1}'s are the boundary of the (round) unit ball. Then the complementary domain of f(S^{n-1}) is open unit ball B, but B is not a subset of U. === Subject: Radical expression approach for solvable septics Let f(x) = x^7 + a6*x^6 + a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0 has rational coefficients and the roots of f(x) are r1, r2, r3, r4, r5, r6, r7 if f(x) is solvable then there exist 7 rational numbers u0, m1, m2, m3, n1, n2 and n3 such that r1 = 1/7*(u0 + u1 + u2 + u3 + u4 + u5 + u6) r2 = 1/7*(u0 + u1*w^1 + u2*w^2 + u3*w^3 + u4*w^4 + u5*w^5 + u6*w^6) r3 = 1/7*(u0 + u1*w^2 + u2*w^4 + u3*w^6 + u4*w^1 + u5*w^3 + u6*w^5) r4 = 1/7*(u0 + u1*w^3 + u2*w^6 + u3*w^2 + u4*w^5 + u5*w^1 + u6*w^4) r5 = 1/7*(u0 + u1*w^4 + u2*w^1 + u3*w^5 + u4*w^2 + u5*w^6 + u6*w^3) r6 = 1/7*(u0 + u1*w^5 + u2*w^3 + u3*w^1 + u4*w^6 + u5*w^4 + u6*w^2) r7 = 1/7*(u0 + u1*w^6 + u2*w^5 + u3*w^4 + u4*w^3 + u5*w^2 + u6*w^1) where w=exp(2*pi*i/7) u0 = -a6 u1^7 = k1/2+(k1^2/4-t1^7)^(1/2) u2^7 = k2/2+(k2^2/4-t2^7)^(1/2) u3^7 = k3/2+(k3^2/4-t3^7)^(1/2) u4^7 = k3/2-(k3^2/4-t3^7)^(1/2) u5^7 = k2/2-(k2^2/4-t2^7)^(1/2) u6^7 = k1/2-(k1^2/4-t1^7)^(1/2) k1 = m1 + (m2 + m3^(1/2))^(1/3) + (m2 - m3^(1/2))^(1/3) k2 = m1 + (m2 + m3^(1/2))^(1/3)*exp(2*pi*i/3) + (m2 - m3^(1/2))^(1/3)*exp(4*pi*i/3) k3 = m1 + (m2 + m3^(1/2))^(1/3)*exp(4*pi*i/3) + (m2 - m3^(1/2))^(1/3)*exp(2*pi*i/3) t1 = n1 + (n2 + n3^(1/2))^(1/3) + (n2 - n3^(1/2))^(1/3) t2 = n1 + (n2 + n3^(1/2))^(1/3)*exp(2*pi*i/3) + (n2 - n3^(1/2))^(1/3)*exp(4*pi*i/3) t3 = n1 + (n2 + n3^(1/2))^(1/3)*exp(4*pi*i/3) + (n2 - n3^(1/2))^(1/3)*exp(2*pi*i/3) Example ------- x^7-2*x^6-x^5+x^4+x^3+x^2-x-1=0 r1 = 1/7*(u0 + u1 + u2 + u3 + u4 + u5 + u6) r2 = 1/7*(u0 + u1*w^1 + u2*w^2 + u3*w^3 + u4*w^4 + u5*w^5 + u6*w^6) r3 = 1/7*(u0 + u1*w^2 + u2*w^4 + u3*w^6 + u4*w^1 + u5*w^3 + u6*w^5) r4 = 1/7*(u0 + u1*w^3 + u2*w^6 + u3*w^2 + u4*w^5 + u5*w^1 + u6*w^4) r5 = 1/7*(u0 + u1*w^4 + u2*w^1 + u3*w^5 + u4*w^2 + u5*w^6 + u6*w^3) r6 = 1/7*(u0 + u1*w^5 + u2*w^3 + u3*w^1 + u4*w^6 + u5*w^4 + u6*w^2) r7 = 1/7*(u0 + u1*w^6 + u2*w^5 + u3*w^4 + u4*w^3 + u5*w^2 + u6*w^1) where w=exp(2*pi*i/7) u0 = 2 u1 = (k1/2-(k1^2/4-t1^7)^(1/2))^(1/7) u2 = (k2/2+(k2^2/4-t2^7)^(1/2))^(1/7) u3 = -(-k3/2+(k3^2/4-t3^7)^(1/2))^(1/7) u4 = (k3/2+(k3^2/4-t3^7)^(1/2))^(1/7) u5 = (k2/2-(k2^2/4-t2^7)^(1/2))^(1/7) u6 = (k1/2+(k1^2/4-t1^7)^(1/2))^(1/7) k1, k2& k3 are the roots for k^3 - 120671*k^2 + 458101804*k +286230977129 = 0 t1, t2& t3 are the roots for t^3 -19*t^2 + 20*t + 461 = 0 k1 = m1 + (m2 + m3^(1/2))^(1/3) + (m2 - m3^(1/2))^(1/3) k2 = m1 + (m2 + m3^(1/2))^(1/3)*exp(4*pi*i/3) + (m2 - m3^(1/2))^(1/3)*exp(2*pi*i/3) k3 = m1 + (m2 + m3^(1/2))^(1/3)*exp(2*pi*i/3) + (m2 - m3^(1/2))^(1/3)*exp(4*pi*i/3) t1 = n1 + (n2 + n3^(1/2))^(1/3) + (n2 - n3^(1/2))^(1/3) t2 = n1 + (n2 + n3^(1/2))^(1/3)*exp(4*pi*i/3) + (n2 - n3^(1/2))^(1/3)*exp(2*pi*i/3) t3 = n1 + (n2 + n3^(1/2))^(1/3)*exp(2*pi*i/3) + (n2 - n3^(1/2))^(1/3)*exp(4*pi*i/3) m1 = 120671/3 m2 = 3009054516246583/54 m3 = -4396071811428172672945920121/108 n1 = 19/3 n2=-2149/54 n3=-3869089/108 Numeric results ---------------- k1 = 116725.380137767 k2 = 4491.57020604866 k3 =-545.950343815686 t1 = 15.9269194902174 t2 = 7.13168652832906 t3 =-4.05860601854649 w = 0.623489801858734 + 0.78183148246803*i u0 = 2 u1 = 3.01627004954521 u2 = 3.30111393735115 u3 = -2.48023785140106 u4 = 1.63637774347078 u5 = 2.16038787623659 u6 = 5.2803360536696 r1 = 2.1306068298389 r2 = 0.959691785305667 - 0.349160710012698*i r3 = -0.756135688040016 + 0.0737508631630883*i r4 = -0.2688595121851 - 0.841085774013299*i r5 = -0.268859512185099 + 0.841085774013298*i r6 =-0.756135688040017 - 0.0737508631630862*i r7 =0.959691785305668 + 0.349160710012695*i === Subject: Probability transition density -- just need a name I have a very simple question. Suppose we have a stationary Markov process X(t) where X(0) = x, with a transition density p(t,x,y)dy = P(X(t) in dy). As t -> infinity, assume p tends to a stationary (invariant) measure f(y). We can always write p(t,x,y) = f(y) + g(t,x,y), and this occurs naturally in eigenfunction expansions. What is a standard name for g( ), if any? alan === Subject: Re: Conjecture Related to Goldbach's Conjecture Levy's Conjecture isn't exactly the same as the one I described, since he apparently doesn't require that the twice a prime number be twice an odd prime. I also guess that his conjecture allows the twice a prime to be twice the prime that is added to it, to get the odd integer. I don't use that, because it is, of course, simply a multiple of three, which just repeats the odd number rather than being a total with new primes. My idea was to formulate a conjecture that used only the odd primes as factors of the twice a prime term, since the same primes would then be available for that conjecture as for Goldbach's--of course, it's impossible to use integer two as one of the primes in Goldbach's strong conjecture, but many odd integers are four more than a prime. Looks like quite a bit of research has been done on Levy's Conjecture, so maybe there is, in the research, some analytical method that I can use. Another interesting sums of primes investigation, is the problem of using only primes that are twin primes--which have some unusual properties--as pairs of primes that equal even integers. The two primes, of course, don't have to be the two successive twin primes of the pairs they belong to. I guess the proposition would be, that, at some magnitude, most even numbers above that magnitude are not the sums of two primes that are both twin primes. It's rather interesting that, among small even integers, only three even numbers between 90 and 100 are not the sum of two twin primes. Since twin primes thin out faster at large magnitudes than primes in general do, I have to think that, above a certain number, very few integers are the sum of two twin primes. Would be interesting to see how the diminishing of sums works out. === Subject: Re: This Week's Finds in Mathematical Physics (Week 223) Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Here's some more discussion about the problem of classifying fibrations in various contexts - my reply to a bemused comment by Jim Stasheff: > John and anyone else who cares to weigh in, > here are some comments from the purely topological > or rather homotopy theory side: > For both bundles and fibrations (e.g. over a paracompact base), your > last slogan is the oldest: > FIBRATIONS WITH FIBER F OVER THE SPACE B > MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) > `the same as' referring to homotopy classes. It's certainly old, but I mentioned another that may be older: COVERING SPACES OVER B WITH FIBER F although one usually sees this special case (which I didn't bother to mention): CONNECTED COVERING SPACES OVER B WITH FIBER F which is usually disguised as follows: CONNECTED COVERING SPACES OVER B Anyway, I wasn't trying to present things in historical order. I was trying present them roughly in order of increasing dimension, starting with extensions of groups, then going up to 2-groups, then expanding out to groupoids, then going up to n-groupoids, and finally omega-groupoids... which are the same as homotopy types! And here, as usual, the n-category theorists meet up with the topologists - and find that the topologists have already done everything there is to do with omega-groupoids ... but usually by thinking of them of them as *spaces*, rather than omega-groupoids! It's sort of like climbing a mountain, surmounting steep cliffs with the help of ropes and other equipment, and then finding a Holiday Inn on top and realizing there was a 4-lane highway going up the other side. So, as usual, the main point of calling homotopy types omega-groupoids instead of spaces is not to reinvent topology, but to see how ideas from topology generalize to n-category theory. Think of spaces as omega-groupoids but use those as a springboard for omega-categories - or at least n-categories, perhaps just for low values of n if one is feeling tired. In the case at hand, the omega-groupoidal slogan: FIBRATIONS OF OMEGA-GROUPOIDS WITH FIBER F AND BASE B WEAK OMEGA-FUNCTORS FROM B TO AUT(F) is just a reformulation of: FIBRATIONS WITH FIBER F OVER THE SPACE B MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) but it suggests a grandiose generalization: FIBRATIONS OF OMEGA-CATEGORIES WITH BASE B WEAK OMEGA-FUNCTORS FROM B^{op} TO THE OMEGA-CATEGORY OF OMEGA-CATEGORIES! I guess we can thank Grothendieck for making precise and proving a version of this with omega replaced by n = 1: FIBRATIONS OF CATEGORIES WITH BASE B WEAK 2-FUNCTORS FROM B^{op} TO THE 2-CATEGORY OF CATEGORIES. More recently people have been thinking about the n = 2 case, especially Claudio Hermida: Claudio Hermida, Descent on 2-fibrations and strongly 2-regular Also available at http://maggie.cs.queensu.ca/chermida/papers/2-descent.pdf He states something that hints at this: FIBRATIONS OF 2-CATEGORIES WITH BASE B WEAK 3-FUNCTORS FROM B^{op} TO THE WEAK 3-CATEGORY OF 2-CATEGORIES. where I'm using B^{op} to mean B with the directions of both 1-morphisms and 2-morphisms reversed. (Hermida follows tradition and calls this B^{coop} - op for reversing 1-morphisms and co for reversing 2-morphisms. But, it looks like we'll be needing to reverse all kinds of morphisms in n-category case, so we'll need a short name for that.) Best, jb === Subject: Re: Finite Projective Planes and Hilbert/Cohn-Vossen's Skew Line Triples The conjecture has an alternative form which is easier to visualize: ***** Conjecture: Let n**2 distinct lines be rulings of n distinct planes, n hyperbolic paraboloids (not necessarily distinct), and (n**2 - n) one-sheeted hyperboloids (not necessarily distinct) such that: 1) each of the n**2 lines is a ruling of just n+1 of the ( n**2 + n) surfaces; 2) each of the (n**2 + n) surfaces is ruled by just n of the n**2 lines; Then the n**2 non-planar surfaces are distinct if and only if n is the power of a prime. ******* It may also be worth noting that if we denote the n**2 lines of the affine plane as a square matrix, then the n planes and n hyperbolic paraboloids always have the following n-tuples of lines respectively: planes: 11 12 ... 1n 21 22 ... 2n ... n1 n2 ... nn paraboloids: 11 21 ... n1 12 22 ... n2 ... 1n 2n ... nn Furthermore, since n**2 - n is 2*(n choose 2), there is an obvious and simple way to find which n lines are rulings of each of the n**2 - n hyperboloids. === Subject: Re: Hilbert 16th Problem as a Pure Algebraic Problem >thank you >It seems there is a shorter proof(As I Heard from a member of Commalg.org) >let V and W are two linear subspace of a vector space Z with a countable base: >if V and W are infinit dimensional space and their codimension are infinit,then there is a linear bijective transformation of Z which send V to W,surjectively. No, that's all there is to it. Vector spaces have less structure, so linear maps are easy. Just map bases to bases. You simply have to insure the dimensions match. Since you want to have a linear bijective map from the full ring to itself carrying your subspace to some ideal, you only need to make sure the dimensions and codimensions of the subspace matches that of some ideal. quasi === Subject: How to show that if a group G has finitely many subgroups, then G is finite? How to show that if G has finitely many subgroups, then G is finite?