mm-4219 === Subject: Re: Circumscribed spheres in Minkowski-spaces It is certainly well known that any three > non-collinear points in a Minkowski-plane X have a > unique circumscribed circle iff X is smooth, i.e. iff > the unit circle of X has a unique tangent line at > each of its points. Does anybody know of a stringent and published proof > of this fact in Minkowski-spaces X of arbitrary > dimension? Michael > Since nobody in this community responded ever since I'm tending to conclude that this is not well-known and was not yet published for Minkowski-spaces of arbitrary finite dimension. Any hints are still appreciated. Michael === Subject: Complemented ideals I have a question on complemented subspaces. Take the algebra L_1(G) on an abelian lcg and a closed subset F in G^ (the dual group). Let I(F) be the ideal of all f in L_1 such that f^ vanishes on F. Will I(F) be complemented as a subspace in L_1? Or among ideals with the zero set F will there be any complemented? What if we replace L_1 by a commutative semisimple normal (i.e. separating closed sets in its maximal ideals space) Banach algebra? === Subject: Re: Complemented ideals On 16 Jun 2007 06:30:49 -0400, Julia Kuznetsova subset F in G^ (the dual group). Let I(F) be the ideal of all >f in L_1 such that f^ vanishes on F. Will I(F) be complemented as a subspace in L_1? I'm pretty sure the answer is sometimes yes, sometimes no. If F is in the coset ring of G^, so that the characteristic function of F is the Fourier transform of a complex measure, then the answer is yes - convolution with that measure gives a bounded projection from L^1(G) onto I(F). Otoh, at least if G is compact and F is not in the coset ring then the answer is no (this is where I fear I may be remembering things wrong). If there were a bounded projection onto I(F) then an averaging argument shows that there is a bounded translation-invariant projection onto I(F). That would be convolution with some measure, and the Cohen idempotent theorem says then that F must be in the coset ring. >Or among ideals with the zero set F will there be any >complemented? What if we replace L_1 by a commutative semisimple >normal (i.e. separating closed sets in its maximal ideals >space) Banach algebra? ************************ David C. Ullrich === Subject: Re: Complemented ideals I learned even more meanwhile: that F belongs to the coset ring is a criterion for I(F) (or other closed ideal with zero set F) to be complemented in L_1(G) on any lcag, it was proved by Rosenthal, Mem. Am. Math. Soc. 63, 84 p. (1966). Can't help citing another close result: characteristic function of a set F in R^n is an L_p- multiplier (1I learned even more meanwhile: that F belongs to the >coset ring is a criterion for I(F) (or other closed ideal >with zero set F) to be complemented in L_1(G) on any lcag, >it was proved by Rosenthal, Mem. Am. Math. Soc. 63, 84 p. (1966). Took me a second to see why this was not entirely clear by the argument I gave - oh, you're talking about G possibly not compact. >Can't help citing another close result: >characteristic function of a set F in R^n is an L_p- >multiplier (1open set by a measure zero set. It's by Lebedev and Olevskii, >Geom. Funct. Anal. 4, No.5, 539-544 (1994). That's neat. ************************ David C. Ullrich === Subject: Re: Complemented ideals by the argument I gave - oh, you're talking about G possibly > not compact. Yes. I know approximately how Rosenthal does it: take the set of all projections onto I, it's compact in some (don't know what) topology, then conjugations with all translations P-> T_a P T_{-a}, a in G, map this set onto itself and must have a common fixed point by Kakutani's theorem. >characteristic function of a set F in R^n F is of course closed. === Subject: The extreme solution to Catalan conjecture Do you think Catalan conjecture has been solved clearly and thoroughly? If we still depend on so called Double wieferich primes below 10^17, depend on computation, that means the problem has already gone to dead end. Why? Because we can never do it completely, maybe it is God's job. My basic thinking: in fact, Catalan conjecture is an extreme value problem concerning integers. If you get interested in this question, I can send you my paper, only 3 pages in PDF, it is very concise, natural and smooth. Just give me your email address, I can send you my paper immediately. Any questions, criticism and information are always welcome. Kexiong Li June 17, 2007. likexiong@hotmail.com === Subject: Re: request reference ... fractional iteration of series > I am seeking a reference for something, which is probably 19th > century material. Power series of the form f(x) = x + ... > have fractional iterates (not only integer number of times, > but also fractional), the coefficients are polynomials > in the relevant items. It seems the ultimate reference I wanted is: Arthur Cayley, Quarterly Journal 3 (1860) 366--369 . It is notable that I can find this in our library after 147 years, while certain on-line references from just 1995 are no longer available... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: request reference ... fractional iteration of series %%% Somehow this post didnt come through, so I post a recovered %%% version again: Are you interested in fractional iteration of formal or of converging power series? FORMAL POWERSERIES x + c1 x^2 + c2 x^3 + ... it is clear that they have a unique fractional iteration with respect to formal composition. The method used by Jabotinsky to compute the coefficients is based on Matrix multiplication. Unfortunately the probably most detailed explanation is in Jabotinsky's thesis Iteration, which is in Hebrew. However there are also some explanations in the paper Erdoes & Jabotinsky: On analytic iteration, 1960 If we write a power series (without constant) as a (infinite) matrix that has in the nth line the coefficients of the nth (multiplicative) power of that series, then we get a triangular matrix and the multiplication of such matrices corresponds to the composition of their series. In case of the coefficient of x is 1, the corresponding matrix has 1's at their diognal. somehow is the expansion (x+1)^t = sum_k=0^infty (t over k) x^k for a real t also applicable for matrices. And we apply for A being the matrix of a powerseries p (A-I+I)^t = sum_k=0^infty (t over k) (A-I)^k Now the nth element of the first line of A is the nth coefficient of the corresponding powerseries a_1 x + a2 x^2 + ... Because (A-I) has 0's on the diagonal for a1=1, the nth element of the first line of (A-I)^k is 0 for k>=n. So to determine the nth coefficient we merely need to sum up to n-1 instead of to infinity. If we would again expand (A-I)^k, I come to the following formula for the nth coefficient of the fractional iteration of the formal powerseries f to the real number s, (denoted by f^s_n) f^s_n = sum_{i=0}^{n-1} f^i_n (-1)^(n-1-i) (s over i) (s-1-i over n-1- i) I dont know whether Jabotinsky this already mentioned in his thesis. CONVERGENCE OF POWERSERIES Unfortunately the formal fractional iteration of a converging powerseries does not necessarily converge. For example only the integer-iterations of e^x-1 converge at 0. (Baker: Zusammensetzungen ganzer Funktionen, 1958) In the above mentioned paper by Jabotinsky and Szekeres there is also a criterion for convergence given (by compositional logarithm). This is a quite unsatisfactory statement and it is possible to push the limits. There are two papers about fractional iteration of *asymptotic* series. An analytic function can have an asymptotic development at a point where it is not defined (say 0) but which is on the boundary of an open area where it is defined. The asymptotic development of f at 0 is a formal powerseries such that the partial sum for each n are f(z)+O(z^{n+1}) for z->0. The result is that for each such real-analytic f with asymptotic development in 0 with respect to a certain adjacent region in the complex plane, there is a unique fractional iteration of f that has as asymptotic development the fractional iteration of the asymptotic development of f (which is a formal powerseries). This is shown for a certain trumpet like area in Szekeres: Regular iteration of real and complex functions,1958 and for sectors with tip at 0 Ecalle: Theorie des Invariants Holomorphes, 1974 I guess this can be applied to for example the fractional iterates of e^x-1 which then would have in 0 merely an asymptotic development but are defined for x>=0. === Subject: Re: request reference ... fractional iteration of series > I am seeking a reference for something, which is probably 19th > century material. Power series of the form f(x) = x + ... > have fractional iterates (not only integer number of times, > but also fractional), the coefficients are polynomials > in the relevant items. One key word to search for is 'Bell matrix'. You can let the entries in the Bell matrix correspond to coefficients in the series. Then, the group formed by the matrices is isomorphic to the group of compositions of functions described by the series. Fractional iterates are roots of the matrices. Gerard === Subject: Re: Eulerian trigonometric identity? Euler's writings. I doubt that, because your formula is a simple generalization of sin(x_1 + x_2) = sinx_1 cosx_2 + sinx_2 cosx_1. But if Euler really published it, then you will certainly find it in one of the 834 original documents available at http://www.math.dartmouth.edu/~euler/available.html Here are some translations by Ian Bruce http://www.17centurymaths.com/contents/eulercontents.html Here are Euler's greatest hits by Ed Sandifer on http://www.maa.org/news/howeulerdidit.html By the way, the sign for |A| = n = 1 (one single sinus, rest cosinus), for instance sinx_1 cosx_2 cosx_3 cosx_4 ..., must be positive while in your formula it is negative. === Subject: Re: Eulerian trigonometric identity? > The identity says sin(x_1 + x_2 + x_3 + ...) = SUM_{positive odd n} (-1)^{(n + 1)/2} ( SUM_{A subset of {1, 2, 3, 4, ...}, |A| = n} PRODUCT_{i in A} sin(x_i) PRODUCT_{i not in A} cos(x_i).) The question is whether this identity is stated anywhere in > Euler's writings. I doubt that, because your formula is a simple generalization of > sin(x_1 + x_2) = sinx_1 cosx_2 + sinx_2 cosx_1. Yes, it is a simple generalization, but I don't see why that's the identity expressing sin(nx) as a polynomial in sin(x) and cos(x). > But if Euler really published it, then you will certainly find it in > one of the 834 original documents available at > http://www.math.dartmouth.edu/~euler/available.html > Here are some translations by Ian Bruce > http://www.17centurymaths.com/contents/eulercontents.html > Here are Euler's greatest hits by Ed Sandifer on > http://www.maa.org/news/howeulerdidit.html > By the way, the sign for |A| = n = 1 (one single sinus, rest cosinus), > for instance > sinx_1 cosx_2 cosx_3 cosx_4 ..., > must be positive while in your formula it is negative. Correct. A typo, I guess. -- Mike Hardy === Subject: More general Near-fields or Semi-fields are there algebraic structures (A,1,+,*,~) considered in literature (1 is a constant, + and * are binary operations, ~ is a unary operation) where (A,1,*,~) is a group and * is right- (or left-) distrivutive over +? I know that a near field is such a structure but additionally + forms a group. And I know that a semi-field is such a structure but additionally + is associative and the second distributivity has to be satisfied. === Subject: Re: More general Near-fields or Semi-fields are there algebraic structures (A,1,+,*,~) considered in literature (1 > is a constant, + and * are binary operations, ~ is a unary operation) > where (A,1,*,~) is a group and * is right- (or left-) distrivutive > over +? I know that a near field is such a structure but additionally + forms > a group. > And I know that a semi-field is such a structure but additionally + is > associative and the second distributivity has to be satisfied. > There is a pretty considerable amount of literature about near-rings. Perhaps some of those papers and books might lead you to the right place? See: http://www.algebra.uni-linz.ac.at/Nearrings/ What properties do you want + to have? It sounds like you want + to be a binary operation and nothing else---not associative, no additive identity. Is that right? In such a case I can't imagine it's been studied very much. But perhaps if you contact some of the Nearrings experts mentioned on the above web page they can tell you more. Good luck, Zach Teitler === Subject: Re: Infinities and infinitesimals On Jun 8, 4:00 pm, Norman Wildberger this purely finite, computer friendly form, and how nonstandard analysis may > be initiated in a much simpler way than using the axiom of choice and > ultrafilters. It also suggests how this approach allows portions of calculus > to be done purely over the rational numbers. That's a misconception. The point of the usual construction in terms of ultrafilters is *not* to define non-standard analysis, but merely to establish its equipollence; particularly that it can be constructed with the Axiom of Choice, but appears to require something less than it. So, on the equipollence hierarchy, it (or any other means of explicating non-standard analysis) lies somewhere between the full power of Choice and the lesser power of Ultrafilters. All an ultrafilter is, is just a fancy way of saying that you're assigning a Boolean value to infinite Boolean sequences that respects (1) constant sequences, (2) bit-wise AND, OR and NOT. Once you understand that, then everything you're doing is just a transparent restatement of the standard construction. How to Make Your Own Non-Standard Model sci.logic, 1993 February 21 === Subject: Completion of integer power series I am not sure whether I am following a fata morgana, but I have the following question: We know that the formal power series of the form x + a1 x^2 + a2 x^3 + .... with ai being integer numbers form a group with respect to composition, call it (S,o). We can linearly order them by the lexicographic order on the sequences (0,1,a1,a2,....). This order is dense and its order topology is Hausdorff. Further the identity element e=(0,1,0,...) has a countable neighbourhood filter base, for example the intervals with (0,1,1,..), (0,1,0,1,..), (0,1,0,0,1,...),.... as upper bounds and their inverses as lower bounds. So we can work with sequences instead of filters for convergence and continuity questions. We can define LR-Cauchy sequences (s_i) as that for each eps > e, there exist M such that for all i,k>M eps^{-1} < s_i^{-1} o s_k < eps and eps^{-1} < s_i o s_k^{-1} < eps. (LR because we consider both last conditions not just the left or the right condition). Are there Cauchy sequences of powerseries of (S,o) that do not converge to a powerseries of (S,o)? How does the LR-completion look? (By some theorem the LR-completion should exist for each Hausdorff group.) Is it embeddable into Q[[x]] or perhaps R[[x]]? So at all: is this already somewhere considered or are my considerations somehow faulty or trivial? === Subject: Re: Completion of integer power series Oh I see that (S,o) is already Cauchy-complete: The Cauchy-sequence condition can be rewritten as s_i < s_k o eps < s_i o eps o eps and s_i < eps o s_k < eps o eps o s_i If we use the following sequence eps_n representing a neighborhood base: eps_n = 0,1,0,0 .... ,1,0, .... ^ nth place after 1st 1 then one can compute that eps_n o a = 0,1,a_1,a_2,...,a_n + 1,x_i ... a o eps_n = 0,1,a_1,a_2,...,a_n + 1,y_i ... So both above Cauchy conditions (let a=s_i and b=s_k) look like 0,1,a_1,...,a_n,... < 0,1,b_1,...,b_n + 1,... < 0,1,a_1,...,a_n + 2,... which means that they have to be identical up to n-1. But this in turn means that in a Cauchy sequence of series the first n Elements of the series become fixed behind the Mth series. So the limit is again a series of (S,o). === Subject: Nine papers published by Geometry & Topology Publications Seven papers have been published by Algebraic & Geometric Topology (1) Algebraic & Geometric Topology 7 (2007) 797-828 Connective Im(J)-theory for cyclic groups by Karlheinz Knapp URL: http://www.msp.warwick.ac.uk/agt/2007/07/p033.xhtml DOI: 10.2140/agt.2007.7.797 (2) Algebraic & Geometric Topology 7 (2007) 829-843 Some results on vector bundle monomorphisms by Daciberg L Goncalves, Alice K M Libardi and Oziride Manzoli Neto URL: http://www.msp.warwick.ac.uk/agt/2007/07/p034.xhtml DOI: 10.2140/agt.2007.7.829 (3) Algebraic & Geometric Topology 7 (2007) 845-917 Quantum hyperbolic geometry by Stephane Baseilhac and Riccardo Benedetti URL: http://www.msp.warwick.ac.uk/agt/2007/07/p035.xhtml DOI: 10.2140/agt.2007.7.845 (4) Algebraic & Geometric Topology 7 (2007) 919-956 Matching theorems for systems of a finitely generated Coxeter group by Michael L Mihalik, John G Ratcliffe and Steven T Tschantz URL: http://www.msp.warwick.ac.uk/agt/2007/07/p036.xhtml DOI: 10.2140/agt.2007.7.919 (5) Algebraic & Geometric Topology 7 (2007) 957-1006 Multiple bridge surfaces restrict knot distance by Maggy Tomova URL: http://www.msp.warwick.ac.uk/agt/2007/07/p037.xhtml DOI: 10.2140/agt.2007.7.957 (6) Algebraic & Geometric Topology 7 (2007) 1007-1062 Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes by Ulrich Bunke, Thomas Schick and Markus Spitzweck URL: http://www.msp.warwick.ac.uk/agt/2007/07/p038.xhtml DOI: 10.2140/agt.2007.7.1007 (7) Algebraic & Geometric Topology 7 (2007) 1063-1070 A note on spaces of asymptotic dimension one by Koji Fujiwara and Kevin Whyte URL: http://www.msp.warwick.ac.uk/agt/2007/07/p039.xhtml DOI: 10.2140/agt.2007.7.1063 Two papers have been published by Geometry & Topology (8) Geometry & Topology 11 (2007) 1225-1254 Shapes of geodesic nets by Alexander Nabutovsky and Regina Rotman URL: http://www.msp.warwick.ac.uk/gt/2007/11/p023.xhtml DOI: 10.2140/gt.2007.11.1225 (9) Geometry & Topology 11 (2007) 1255-1288 Cohomological estimates for cat(X,xi) by Michael Farber and Dirk Schütz URL: http://www.msp.warwick.ac.uk/gt/2007/11/p024.xhtml DOI: 10.2140/gt.2007.11.1255 Abstracts follow (1) Connective Im(J)-theory for cyclic groups by Karlheinz Knapp We study connective Im(J)-theory for the classifying space BZ/p^a of a finite cyclic p-group and compute the Im(J)-cohomology groups completely. We also compute the Im(J)-homology groups, with the exception of a finite range of dimensions. (2) Some results on vector bundle monomorphisms by Daciberg L Goncalves, Alice K M Libardi and Oziride Manzoli Neto In this paper we use the singularity method of Koschorke [Lecture Notes in Math. 847 (1981)] to study the question of how many different nonstable homotopy classes of monomorphisms of vector bundles lie in a stable class and the percentage of stable monomorphisms which are not homotopic to stabilized nonstable monomorphisms. Particular attention is paid to tangent vector fields. This work complements some results of Koschorke [Lecture Notes in Math. 1350 (1988), Topology Appl. 75 (1997)], Libardi-Rossini [Proc. of the XI Brazil. Top. Meeting 2000] and Libardi-do (3) Quantum hyperbolic geometry by Stephane Baseilhac and Riccardo Benedetti We construct a new family, indexed by the odd positive integers N, of (2+1)-dimensional quantum field theories that we call quantum hyperbolic field theories (QHFT), and we study its main structural properties. The QHFT are defined for marked (2+1)-bordisms supported by compact oriented $3$--manifolds Y with a properly embedded framed tangle L_F and an arbitrary PSL(2,C)-character rho of Y-L_F (covering, for example, the case of hyperbolic cone manifolds). The marking of QHFT bordisms includes a specific set of parameters for the space of pleated hyperbolic structures on punctured surfaces. Each QHFT associates in a constructive way to any triple (Y,L_F,rho) with marked boundary components a tensor built on the matrix dilogarithms, which is holomorphic in the boundary parameters. When N=1 the QHFT tensors are scalar-valued, and coincide with the Cheeger--Chern--Simons invariants of PSL(2,C)-characters on closed manifolds or cusped hyperbolic manifolds. We establish surgery formulas for QHFT partitions functions and describe their relations with the quantum hyperbolic invariants of Baseilhac and Benedetti defined for unframed links in closed manifolds and characters trivial at the link meridians, or cusped hyperbolic $3$--manifolds). For every PSL(2,C)-character of a punctured surface, we produce new families of conjugacy classes of ``moderately projective representations of the mapping class groups. (4) Matching theorems for systems of a finitely generated Coxeter group by Michael L Mihalik, John G Ratcliffe and Steven T Tschantz We study the relationship between two sets S and S' of Coxeter generators of a finitely generated Coxeter group W by proving a series of theorems that identify common features of S and S'. We describe an algorithm for constructing from any set of Coxeter generators S of W a set of Coxeter generators R of maximum rank for W. A subset C of S is called complete if any two elements of C generate a finite group. We prove that if S and S' have maximum rank, then there is a bijection between the complete subsets of S and the complete subsets of S' so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of (W,S) and (W,S') have the same multiset of entries. (5) Multiple bridge surfaces restrict knot distance by Maggy Tomova Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M, P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or d(K,P) <= 2-chi(Q-K). If K is not a 2-bridge knot, then the result holds even if K is removable with respect to Q. As a corollary we show that if a knot in S^3 has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position. (6) Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes by Ulrich Bunke, Thomas Schick and Markus Spitzweck In this paper we give a sheaf theory interpretation of the twisted cohomology of manifolds. To this end we develop a sheaf theory on smooth stacks. The derived push-forward of the constant sheaf with value R along the structure map of a U(1) gerbe over a smooth manifold X is an object of the derived category of sheaves on X. Our main result shows that it is isomorphic in this derived category to a sheaf of twisted de Rham complexes. (7) A note on spaces of asymptotic dimension one by Koji Fujiwara and Kevin Whyte Let X be a geodesic metric space with H_1(X) uniformly generated. If X has asymptotic dimension one then X is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus g >= 2 and one boundary component is at least two. (8) Shapes of geodesic nets by Alexander Nabutovsky and Regina Rotman Let M^n be a closed Riemannian manifold of dimension n. In this paper we will show that either the length of a shortest periodic geodesic on M^n does not exceed (n+1)d, where d is the diameter of M^n or there exist infinitely many geometrically distinct stationary closed geodesic nets on this manifold. We will also show that either the length of a shortest periodic geodesic is, similarly, bounded in terms of the volume of a manifold M^n, or there exist infinitely many geometrically distinct stationary closed geodesic nets on M^n. (9) Cohomological estimates for cat(X,xi) by Michael Farber and Dirk Schütz This paper studies the homotopy invariant cat(X,xi) introduced in [1: Michael Farber, `Zeros of closed 1-forms, homoclinic orbits and Lusternik--Schnirelman theory', Topol. Methods Nonlinear Anal. 19 (2002) 123--152]. Given a finite cell-complex X, we study the function xi --> cat(X,xi) where xi varies in the cohomology space H^1(X;R). Note that cat(X,xi) turns into the classical Lusternik--Schnirelmann category cat(X) in the case xi=0. Interest in cat(X,xi) is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1-forms, see [1] and [2: Michael Farber, `Topology of closed one-forms', Mathematical Surveys In this paper we significantly improve earlier cohomological lower bounds for cat(X,xi) suggested in [1] and [2]. The advantages of the current results are twofold: firstly, we allow cohomology classes xi of arbitrary rank (while in [1] the case of rank one classes was studied), and secondly, the theorems of the present paper are based on a different principle and give slightly better estimates even in the case of rank one classes. We introduce in this paper a new controlled version of cat(X,xi) and find upper bounds for it. We apply these upper and lower bounds in a number of specific examples where we explicitly compute cat(X,xi) as a function of the cohomology class xi in H^1(X;R). === Subject: Classical injectivity radii *** Classical injectivity radii *** Hi All, Where can one find a table/survey of the injectivity radii of the classical simply connected compact homogeneous riemannian manifolds? (To give this meaning let each be normalized to volume 1). The injectivity radius of a closed riemannian m-manifold M is the largest real number R such that, for every real number r < R, and every point x in M, the geodesic spray embeds into M the radius r ball about x in the tangent m-plane to M at x. Similar question for the spectrum of the Laplacian? Laurent S. === Subject: E6 matrix representation I am looking for matrix representation of exceptional Lie algebras. Has anyone knows where I can find construction of E6 Lie group or Lie algebra ? This exceptional Lie group can be embedded in SO(27). There are useful notes on page http://www.7stones.com/Homepage/octotut15.html but this is too little for me to find the base of the lie algebra. I was able to construct base of the f4 lie algebra using 1) Dixon's page mentioned above 2) Matsushima work from 1952 I has also used CLICAL of Pertti Lounesto and self-made program for commuting elements in so(n). Algebra f4 is subalgebra of so(26). Algebra e6 is subalgebra of so(27). Algebra e7 is subalgebra of so(56). Finally e8 is subalgebra of so(248). I have already achieved following. Definition: h3(C*O) = {x belongs to M3(C*O): x*=x} sa3(C*O) = {x belongs to M3(C*O): x*=-x;tr x = 0} Define e6 as derivation of algebra h3(C*O). This means group E6 is automorfizms of h3(C*O). In Baez work we have: e6 = Der O + sa3(C*O) For element x from sa3(C*O) we define derivation Dx acting on h3 by formula Dx(y) = xy - yx I can find su3 and so8 this way but in this presentation h3(C*O) is 16*3+3*8 = 72-dimensional. I don't know how to map su3 found into so(27). Marek Mitros === Subject: Re: deriative > [S.A] f is describle, continious, derative in R > f(x+y)=f(x)+f(y)+xy and h>0( {lim[f(h)/h] }=3 and f ' (1)=? Only determining f'(1) is too easy, determine all of f! And assuming that f' exists is giving away too much! Assume f:R->R is continuous (but not necessarily differentiable!) and f(x+y)=f(x)+f(y)+xy holds for all x,y, in R. Consider g(x) := f(x) -x^2/2. Then g is continuous and we have g(x+y) = f(x+y)-(x+y)^2/2 = f(x)+f(y) + xy - (x+y)^2/2 = f(x)-x^2/2 + f(y)-y^2/2 = g(x) + g(y). It follows that 1) g(0) = 0 because g(0+0)=g(0)+g(0). 2) g(n*x) = n*g(x) for x in R and n in N by induction as g((n +1)x)=g(nx)+g(x) 3) g(-x) = -g(x) as 0=g(0)=g(x)+g(-x) 4) hence g(n*x) = n*g(x) for all n in Z 5) g(q*x) = q*g(x) for all rational q=a/b, because b*g(q*x)=g(b*q*x)=g(a*x)=a*g(x) 6) g(x*y) = x*g(y) for all x,y in R by continuity 7) Especially g(x) = c*x with c:=g(1) We conclude that f has the form f(x) = g(x) + x^2/2 = c*x + x^2/2. Hence f is differentiable with f'(x) = c + x === Subject: Re: Simplify > Can you simplify this expression in less than 10 steps. { cosA + (sin2A/secA)} * sec 2A * (1 - tan A) ---------------------------------------------------------------------------? - ------ > Tarun Ayitam > FIITJEE Junior College > India In[3]:= FullSimplify[{Cos[A] + (Sin[2 A]/Sec[A])}*Sec[ 2A]*(1 - Tan[ A]), Trig -> True] Out[3]= {Cos[A] + Sin[A]} === Subject: Re: groups, rings - books >I'm looking books with examples and solved problems in group and ring >theory I'm pretty sure there's a Schaum's Outline book on abstract algebra. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Comprehensive Solution Manual for Textbooks <8074295.1181270484324.JavaMail.jakarta@nitrogen.mathforum.org> All msg replied === Subject: Re: how do I find branching cut of a complex-valued function? On 2007-06-21 07:31:34 -0400, Vista said: > Hi all, I have a complex-valued function F(z)=(f(z))^r, where r is a negative irational number. f(z) is a function of complex number z. f(z) itself is analytical in z. I was able to find the singularities of F(z) which is the roots of f(z)=0. My understanding is that I have to connect the singularities in such a way > that they consistute branch cuts. How to do that? I have a limited number of books on complex analysis on my hand, and I > didn't find a good chapter on this topic. Could you please give me some detailed help? Pointers to online books/notes/resources are also mostly welcome! > I'm curious as to how is a branch cut precisely defined? Can I say even for an analytic function that I can declare a branch cut from any two points, even though it is not necessary? Can we say one goal of a branch cut is to declare a domain D = the Riemann sphere minus the branch points and branch cuts, so that any line integral of a close loop in D is zero? If so, then wouldn't any connected graph for the set of all branch points be sufficient to have the above said property? Of course, this branch cut may have more lines than necessary. -- -kira === Subject: Re: PARADISE LOST: Debunking Cantor's theory > 0 / 0 1 / / 0 1 0 1 ... > I don't understand this tree. It looks like you are working in base two but I don't know. Is your sequence this one: > 0.000... 0.1000... 0.01000... 0.11000... 0.001000... 0.011000... 0.111000... etc.? > I'm trying to figure out just how you place your sequence in a one to one correspondence with the naturals so as to call them countable. There is no bijection, but there is a proof that the set of paths is > not larger than the set of nodes because there cannot be more paths > than separations are available. Would you show me that proof? I don't believe the assertion. In every element of the tree | o / the number of separated path bunches leaving a node minus the number of separated path bunches arriving at the node is the number of nodes: 2 - 1 = 1. Therefore, for the whole tree and down to every level n, the difference in separated paths bunches and nodes is (2-1-1+2-1-1+2-1-1+...) < 3 Thsi holds as far as the tree stretches. Suppose I want to find a specific path. What procedure would I > follow? > It seem like there is an unbounded set of paths flowing from any > particular node. Correct. These latter are the nodes. The set of > nodes is countable by simply using > 0 > 1,2 > 3,4,5,6 > 7,8,... One might think this sufficient but for any means of counting these > nodes I can give a simple procedure, using Cantor's method to specify > a path you cannot reach because at every node reached by your method > of counting them it differs in at least one digit. But obviously a path cannot exists where no nodes are. Or can a path reach further than any node? > The problem seems to come in with your definition of countable. It seems maliable. Just how many zeros does the ... represent? The tree is infinite, i.e., it has infinitely many levels. Therefore > all sequences of bits with infinitely many bits are in the tree. That doesn't make them countable. It proves their cardinality C being not less than the cardinality of the set of nodes. > Since you deny certain natural numbers exist When using this argument, I accept all natural numbers. I assume an > infiite model here, showing that set theory would even be false if > infinity would exist. Denial of infinite sets obviously leads to > exclusion of uncountable sets. I don't have a problem with infinite sets. I have a problem with > labeling a set countably infinite when I can't assign a counting > number to each member. The constructible numbers cannot be counted in this way, because this would be equivalent to constructing a list of such numbers. If such a list could be constructed, however, then a diagonal number could be constructed, proving the constructible numbers uncountable. But there is another argument which proves the constructible numbers uncountable. It is the same state of affairs which we face in the tree. === Subject: Re: Joint distribution between a continuous distribution RV and a discrete distribution RV ?? >Hi all, assume N is a discrete distribution random variable and X is >uniformally distributed in [0 1], known P(N=n | X=x)=x^n(n+1). Now I need to find P(N=n). this is the case which is not covered in any textbooks I read. Can you >please help or give me any references discussing about this issue. For any two random variables*, EX = EE[X | Y]. For probabilities, let X be an indicator variable. Then PA = EP(A | Y). This is sometimes called the Law of Total Probability. If X is continuous with density f, this becomes PA = integral(x=-infty..infty, P(A | X = x) f(x)). *Technically, the two random variables need be defined on the same probability space. I guess the solution may be that we need to find P(N=n and X=x) i.e a >jointly distribution between N and X, then integral it with respect to >x over [0 1] to get P(N=n). There is no joint mass function or joint density here. (I suppose if you want to use Dirac delta functions, you could fudge your way to a density.) If you really want to go the way of joint distribution, you need to specify the joint cumulative distribution function. However, it is easiest to specifiy this in terms of conditional distributions. >However finding P(N=n and X=x) is not simple at all since P(X=x) >always is zero since X is a continuous distrubtion RV. I guess we >should do something with lim such as P(X=x)=lim delta->0 of P(x < X < x+delta). Though I still get stuck. > -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Joint distribution between a continuous distribution RV and a discrete distribution RV ?? > Hi all, assume N is a discrete distribution random variable and X is > uniformally distributed in [0 1], known P(N=n | X=x)=x^n(n+1). Now I need to find P(N=n). this is the case which is not covered in any textbooks I read. Can you > please help or give me any references discussing about this issue. I guess the solution may be that we need to find P(N=n and X=x) i.e a > jointly distribution between N and X, then integral it with respect to > x over [0 1] to get P(N=n). However finding P(N=n and X=x) is not simple at all since P(X=x) > always is zero since X is a continuous distrubtion RV. I guess we > should do something with lim such as P(X=x)=lim delta->0 of P(x < X < x+delta). Though I still get stuck. Although it is not rigorous, you can think of P{N=n|X=x} as a limit of P{N=n|x < X x x+h} as h --> 0. Now the latter is P{N=n & x < X < x+h}/ P{x < X < x+h}, so P{N=n & x < X < x+h} = x^(n(n+1))*f(x)*h, where f is the density function of X. To get P{N=n} you need to sum over all x. You can imagine doing this as a Riemann sum then going to the limit on a rigorous basis, it helps to have some measure theory, but introductory probability books (of the applied variety) usually do not have such material. R.G. Vickson > Please help!!!!! > I assume that x^n(n+1) means x^(n(n+1)), rather than (n+1)x^n. P{N=n} = int{P{N=n|X=x} f(x) dx, x=0..1} = int{x^(n(n+1)) dx,x=0..1} = 1/[n(n+1)]. R.G. Vickson === Subject: Re: Two dimensional formulas vs tautology checking <467A13EC.E5600657@gmail.com> Phobias, paranoia, phsychosis are all mental illnesses that must be looked into immediately by oneself. If you do not correct yourself, then u run the risk of harming ur health, others health, or running into trouble with the law. In my opinion, a healthy body is the best way to maintain a healthy mind. Pls read below, my tips to good health: 1.) Do not smoke or drink 2.) Do exercises for 5 days per week (30 mins jogging + other exercises). Rest the other 2 days by going to cinemas/fairs 3.) Eat lots of fruits & vegetables, and try to be vegetarian. Non-veg food not only causes Cancer, but also clogs up your brain with fat 4.) Sleep atleast 8 hrs a day, becos sleep makes u more intelligent 5.) Start working in a private company like me (I am working in www.ustri.com right now), the work stress will be so high that ur mental illnesses will run away automatically. Occupy ur time with practical real-world problems.....it is OK to spend time on theoretical problems, but try to limit that to 1 or 2 hrs per day (not as full-time). Example: u could spend time on theoretical problems like P=NP just before going to bed, or while resting after ur Jogging/workout. :) All the best !!! :) === Subject: Re: P(X=x) when X is a uniform distribution in (0,1) X, and N are two Random Variable while X is a RV which is uniformally >distributed in [0 1] and N is a descrete RV. KNown that the >conditional probability: P(N=n|X=x)=x^n(x+1). They ask to find P(N=n). > [...] > I just postsed a repsone to this on sci.math. In the future, please cross-post the same message to the two groups, rather than post two messages separately. > assume N is a discrete distribution random variable and X is > uniformally distributed in [0 1], known > P(N=n | X=x)=x^n(n+1). Now I need to find P(N=n). > this is the case which is not covered in any textbooks I read. Can you > please help or give me any references discussing about this issue. > For any two random variables*, EX = EE[X | Y]. For probabilities, > let X be an indicator variable. Then PA = EP(A | Y). This is > sometimes called the Law of Total Probability. If X is continuous with density f, this becomes PA = > integral(x=-infty..infty, P(A | X = x) f(x)). *Technically, the two random variables need be defined on the same > probability space. > I guess the solution may be that we need to find P(N=n and X=x) i.e a > jointly distribution between N and X, then integral it with respect to > x over [0 1] to get P(N=n). > There is no joint mass function or joint density here. (I suppose if > you want to use Dirac delta functions, you could fudge your way to a > density.) If you really want to go the way of joint distribution, you > need to specify the joint cumulative distribution function. However, > it is easiest to specifiy this in terms of conditional distributions. > However finding P(N=n and X=x) is not simple at all since P(X=x) > always is zero since X is a continuous distrubtion RV. I guess we > should do something with lim such as > P(X=x)=lim delta->0 of P(x < X < x+delta). Though I still get stuck. > -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Twenty three months. > Well, all I get is ------------------------------------------------------ > ----------------- > Proxy Error > The proxy server received an invalid response from an > upstream server. > The proxy server could not handle the request GET > /kb/message.jspa. > Reason: Error reading from remote server > ------------------------------------------------------ > ----------------- so technically you are correct. I only get (A->A),that it is not a bad starting point. I have the hope to set case law without proxy statements. But I am not a sociologist. Not clear? Fernando. === Subject: Re: Composite factorization coupling I haven't followed these threads, so I'm unfamiliar with the any > aspects previously explained. However, if you can work towards specifying your method in a clearly > stated [rest deleted] Never happen. -- Clive Tooth http://www.shutterstock.com/cat.mhtml?gallery_id=61771