mm-2265 === Subject: Clarification to infinite matrix algebra automorphisms Epigone-thread: jywhahfrerl It has been pointed out to me that fixing K is ambiguous because the infinite matrix algebra lacks an identity element. By this, I meant that I am interested in the group of K-automorphisms, i.e. the automorphisms are K-linear. Chris === Subject: Re: Clarification to infinite matrix algebra automorphisms > It has been pointed out to me that fixing K is ambiguous because the > infinite matrix algebra lacks an identity element. By this, I meant > that I am interested in the group of K-automorphisms, i.e. the > automorphisms are K-linear. I was hasty in identifying your algebra with the algebra of ALL linear transformations on an arbitrary vector space. However, if you take a look at Jacobson's book mentioned in my previous posting and read Theorem 7 (Chapter IX) you MAY find what you need. I'm not sure of exactly what it says. But it applies to algebras other than the full algebra of linear transformations and I think it applies to the algebra you are concerned with. Edwin Clark === Subject: Smallest S_n in which a group G embeds. Consider the following: Given a finite group G, let s(G) be the smallest number n such that G embeds in the symmetric group on n letters, or equivalently, the cardinality of the smallest set on which G acts faithfully. Question: I have been looking for some references, even very basic, on this topic. Any ideas? Rex Butler rbutler@math.utah.edu RexButler@hotmail.com === Subject: Re: Smallest S_n in which a group G embeds. > Given a finite group G, let s(G) be the smallest number n such that G > embeds in the symmetric group on n letters, or equivalently, the > cardinality of the smallest set on which G acts faithfully. For an arbitatry G there is nothing like a uniform answer to this; it is equivalent to finding the maximal order of a subgroup of G that does not contain any proper normal subgroup... > Question: I have been looking for some references, even very basic, > on this topic. Any ideas? A basic reference would be P.Cameron's Permutation groups. In particular, the chapter on the O'Nan-Scott theorem. The latter would give one an idea of what is going on at least in the case of G a simple group. HTH, Dmitrii === Subject: Re: Smallest S_n in which a group G embeds. [...] > this; it is equivalent to finding the maximal order of a > subgroup of G that does not contain any proper normal > subgroup... I meant the maximum among the orders of subgroups of G that... (or indeed, the index of a largest subgroup not containing a proper normal subgroup, as suggested by J.McKay via email.) === Subject: Re: Is positve constant required in Gronwall's inequality? > In An Introduction to Partial Differential Equations by Renardy and Rogers > (Spring-Verlag, 1993), Gronwall's inequality is stated as: > Let u: [a,b] ->[0, inf) and v: [a,b] -> R be continuous functions and let C > be a constant. Then if > v(t) <= C + integral_from_a_to_t v(s)u(s) ds > for t in [a,b], it follows that > v(t) <= C exp(integral_from_a_to_t u(s) ds) > for t in [a,b]. > Is C > 0 required? == {REMARK 1:} We may omit the requirement that C be nonnegative . {REMARK 2:} Above remark is justified by following comparision result for linear differential inequalities : Lemma: ,, Let b(t) and f(t) be continuous for t>= a and suppose that v(t) , t>= a is a continuous differentiable function. If v'(t) =< b(t)v(t) + f(t) for all t>= a , and v(a) =< v_0 , then for t>= a we have v(t) =< v_0*exp(Integral_{s=a to s=t}b(s) ds) + + Integral_{s=a to s=t}f(s)*exp(Integral_{z=s to z=t}b(z)dz) ds . {REMARK 3:} Using Remark 2 it may be proved : THEOREM. ,, Let C:[a,b]-->R , U:[a,b]-->[0,infty) , V:[a,b]-->R be continuous . Suppose V(t) =< C(t)+ Integral_{s=a to s=t}V(s)U(s) ds for t in [a,b]. Then V(t)=R be a continuous function satisfying 0 =< v(t) =< Integral_{s=a to s=t}(A+B*v(s))ds , for t in [a,a+h] where A , B are nonegative constants. Then v(t) =< A*h*exp(B*h) for t in [a,a+h] . This result is the prototype for the study of many integral inequalities of Volterra type. The papers by R. Bellman [,,The stability of solutions of linear equations , Duke.Math.J., 10 (1943) 643-647] as well as by I. Bihari [,, A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations , Acta.MAth.Acad.Sci.Hungar., 7, 1 (1956) 71-94] are of great importance. Another interesting paper is G.I. Chandirov [,,On a generalization of Gronwall's inequality and its applications, (Russian), Ucen.Zap.Azerb.Univ.Ser.Fiz.-Mat.6(1956) 3-11] . {REMARK 6:} There is a nice book about Gronwall's type Inequalities. See Drumi Bainov and Pavel Simeonov , ,,Integral Inequalities and Applications, == === Subject: Re: Finite-dim linear transformations as simple compositions? Please ignore the question below. The statement contained therein is incorrect. What I confused it with is A linear transformation T of the plane R^2 into itself is invertible if and only if T consists of a finite sequence of: Reflections in the x-axis, the y-axis, or the line y=x; Vertical or horizontal expansions or contractions; and Vertical or horizontal shears. But now, with this more complicated characterization, the question of what happens in higher dimensions is not so charming anymore. GvZ > Hi > Any linear transformation from the plane to itself can be written as a > composition of three operations (reflection, rotation, scaling). > What happens in higher dimensions? > GvZ === Subject: Re: tensor product and Hodge decomposition for currents >[A complimentary Cc of this posting was sent to >Agust.92 Roig >> I've learned that, for distributions, this is part of the celebrated >> Laurent Schwartz Kernel Theorem. >Strange: I always thought that the Kernel Theorem is usually >associated with Grothendieck name... That's what they call it in the portrait of Laurent Schwartz in the *************************************** To round off the analogy with the finite-dimensional situation, it must be mentioned that this property is equivalent to the isomorphism of D'(M1[Times]M2) with the tensor product D'(M1) overhat{otimes} D'(M2), where the hat indicates completion [...] The Schwartz kernel theorem was Grothendieck.89s starting point in building his theory of nuclear spaces [...]. *************************************** >> So I can restate my question as: is there any Laurent Schwartz Kernel >> Theorem for currents? >Can you explain why do you find this problematic? Currents are just >tensors with generalized functions as coefficients. >In other words, going from generalized functions to currents is >tensoring with a finite-dimensional space; this can't spoil >anything... I believe so, but on the one hand, since it looks so easy, I am convinced that someone else has thought about it and written it down with details. On the other hand, I need the existence of the tensor product of currents as a simple step of a result which has nothing to do with Analysis. I suppose that I could say that the two steps (density and continuity) in Dieudonn.8e's book for defining the tensor product of distributions obviously extend to currents, but if someone else has written it down previously I think that it would be a better reference. Agust.92 Roig === Subject: Re: Eigenvalues of product > I'm looking for a good (reliable) reference for the following > matrix inequality. > Let A, B be positive definite hermitian n by n matrices, > with eigenvalues a_1 >= a_2 >= .... >= a_n and > b_1 >= b_2 >= .... >= b_n. Let c_1 >= ... >= c_n denote > the eigenvalues of the product AB. Then > c_1 >= a_i * b_{n+1-i} for all i=1...n. Probably in Horn -Johnson. Arnold Neumaier === Subject: Re: Automorphism group of infinite matrix algebra > Let K be a field and let R be the ring of all infinite matrices over K > with only finitely many non-zero entries. In other words, R is > generated by the matrix units e_ij with the regular matrix product, > but i and j run through all natural numbers. Does anybody know of > either a paper or a book where they classify all automorphisms of R > that fix K? If it is like the finite dimensional case, it should be > all the inner automorphisms generated by the general linear group. Isn't there an ambiguity in the term general linear group here? What vector space are you referring to? (I take it that general linear group means the group of invertible transformations of some vector space.) Eg it might be the space of all vectors (x_1,x_2,x_3,...) or it might be the subspace of such vectors having only a finite number of non-zero elements. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Automorphism group of infinite matrix algebra >Let K be a field and let R be the ring of all infinite matrices over K >with only finitely many non-zero entries. In other words, R is >generated by the matrix units e_ij with the regular matrix product, >but i and j run through all natural numbers. Does anybody know of >either a paper or a book where they classify all automorphisms of R >that fix K? If it is like the finite dimensional case, it should be >all the inner automorphisms generated by the general linear group. [ and Chris later clarified that he meant the K-linear automorphisms of R ] Well, these are certainly not inner, since the members of R are not invertible. For example, any permutation p of the natural numbers generates an automorphism of R taking e_{i,j} to e_{p(i),p(j)}. More generally: suppose V is a matrix over K with finitely many nonzero entries in each row, W a matrix with finitely many nonzero entries in each column, and VW = I. Then X -> WXV is a K-linear automorphism of R. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Automorphism group of infinite matrix algebra >>Let K be a field and let R be the ring of all infinite matrices over K >>with only finitely many non-zero entries. In other words, R is >>generated by the matrix units e_ij with the regular matrix product, >>but i and j run through all natural numbers. Does anybody know of >>either a paper or a book where they classify all automorphisms of R >>that fix K? If it is like the finite dimensional case, it should be >>all the inner automorphisms generated by the general linear group. >[ and Chris later clarified that he meant the K-linear automorphisms of R >More generally: suppose V is a matrix over K with finitely many >nonzero entries in each row, W a matrix with finitely many nonzero >entries in each column, and VW = I. Then X -> WXV is a K-linear >automorphism of R. Oops: these are K-linear injective homomorphisms, but are not necessarily surjective. For that I think you want V and W to have finitely many nonzeros in each row and column, and W = V^(-1) (i.e. both a left inverse and a right inverse). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Automorphism group of infinite matrix algebra > Let K be a field and let R be the ring of all infinite matrices over K > with only finitely many non-zero entries. In other words, R is > generated by the matrix units e_ij with the regular matrix product, > but i and j run through all natural numbers. Does anybody know of > either a paper or a book where they classify all automorphisms of R > that fix K? If it is like the finite dimensional case, it should be > all the inner automorphisms generated by the general linear group. In Jacobson's Lectures in Abstract Algebra, II Linear Algebra in Chapter IX, Infinite Dimesional Vector Spaces, on page 266 you will find: Corollary 2. Every automorphism of the ring of linear transformations of a vector space over a field which leaves the elements of the center fixed is inner. I believe this verifies your conjecture. --Edwin Clark === Subject: Re: Proof of the Fundamental Theorem of Algebra in R >Does anyone know of a proof of the fundamental theorem of algebra in the >form every polynomial over R splits into factors of degree at most 2 >that goes through working in R only (i.e., without using complex >numbers)? > Somebody told me that one of Gauss' proofs was like this. I've checked up on this and you are quite right: Gauss's 1799 proof of FTA is deliberately stated so as to avoid presupposing the existence of complex numbers. See: http://libraserv1.fsc.edu/proof/gauss.htm for an English translation of Gauss's doctoral dissertation presenting this proof, which alas is flawed - it relies on a property of real algebraic curves that (according to the book Numbers by Ebbinghaus et al.) wasn't proved satisfactorily until the 1920s, even despite Gauss's attempts to fix the proof up in his fourth proof in 1849 (but by that time he didn't feel the need to skirt around the complex numbers). Gauss was clearly aware of the deficiency: As far as I know, nobody has raised any doubts about this. However, should someone demand it then I will undertake to give a proof that is not subject to any doubt on some other occasion. > There's > a whole book on FTA from Springer. I suspect van der Waerden's algebra > book might be a good place to look. Rob. === Subject: Re: probabilistic approach to riemann hypothesis But there may be a fix afterall. I have heard that the density of numbers with mu(n)=0 approaches 6/pi^2. I think the idea of defining X_i as a random variable when mu(n)=0 as a random coin toss is invalid. If we eliminate this, then the variance approaches zero in your counterexample but a non-zero in this proposed approach. The law of the iterated logarithm is invalid when the variance approaches 0. What do you think? Are there any probabilists out there who can help? I withdrew it, but you can look at earlier versions. Craig > For what it's worth, didn't Norman Levinson write a paper showing > some big fraction of the zeros must be consistent with RH ? === Subject: Re: probabilistic approach to riemann hypothesis Charles, a good try but unfortunately wrong. (I don't know where it went wrong but it's clearly wrong and not fixable) I'll withdraw it from arxiv. Craig -------- Something is clearly incorrect in the argument. Here's why. The only facts about mu(N) used in the proof of the main theorem are - that mu(N) is always -1, 0 or 1 - that avg{1..N} mu(n) -> 0. is bounded by c.N^(1/2+h) for any positive h. But those facts alone *do not imply* this, as can be seen from the following counterexample. Define m(n) = { 1 if n is prime, else 0 }. Then m(n) is always -1, 0 or 1, and avg{1..N} m(n) = pi(n)/n which is of order 1/log N and therefore -> 0. But sum{1..N} m(n) = pi(N), which is certainly not bounded by c.N^(1/2+h) for any h smaller than 1/2. Any function m, always -1, 0 or 1, with sum{1..N} m(n) having order somewhere comfortably bigger than sqrt(N) and smaller than N will do equally well. I'm not a probabilist, and I don't know at what point in the claimed proof the argument goes astray[1]. But from the above counterexamples I *do* know that no argument that uses only the two facts about mu(N) listed above can possibly succeed. [1] My guess is that the appeal to the Glivenko-Cantelli theorem is incorrect. It certainly isn't clear to me that this appeal is correct. > For what it's worth, didn't Norman Levinson write a paper showing > some big fraction of the zeros must be consistent with RH ? === Subject: Re: probabilistic approach to riemann hypothesis I'm sorry Gareth, I meant to thank you. Craig > Charles, > a good try but unfortunately wrong. (I don't know where it went wrong > but it's clearly wrong and not fixable) I'll withdraw it from arxiv. > Craig > -------- > Something is clearly incorrect in the argument. Here's why. > The only facts about mu(N) used in the proof of the main > theorem are > - that mu(N) is always -1, 0 or 1 > - that avg{1..N} mu(n) -> 0. > is bounded by c.N^(1/2+h) for any positive h. But those > facts alone *do not imply* this, as can be seen from the > following counterexample. > Define m(n) = { 1 if n is prime, else 0 }. > Then m(n) is always -1, 0 or 1, and avg{1..N} m(n) = pi(n)/n > which is of order 1/log N and therefore -> 0. But > sum{1..N} m(n) = pi(N), which is certainly not bounded > by c.N^(1/2+h) for any h smaller than 1/2. > Any function m, always -1, 0 or 1, with sum{1..N} m(n) > having order somewhere comfortably bigger than sqrt(N) > and smaller than N will do equally well. > I'm not a probabilist, and I don't know at what point in > the claimed proof the argument goes astray[1]. But from > the above counterexamples I *do* know that no argument that > uses only the two facts about mu(N) listed above can possibly > succeed. > [1] My guess is that the appeal to the Glivenko-Cantelli > theorem is incorrect. It certainly isn't clear to me > that this appeal is correct. > For what it's worth, didn't Norman Levinson write a paper showing > some big fraction of the zeros must be consistent with RH ? === Subject: Re: probabilistic approach to riemann hypothesis > I'm sorry Gareth, I meant to thank you. That's OK. I knew it was me you meant :-). -- Gareth McCaughan .sig under construc === Subject: Re: probabilistic approach to riemann hypothesis >Charles, >a good try but unfortunately wrong. (I don't know where it went wrong >but it's clearly wrong and not fixable) I'll withdraw it from arxiv. >Craig >-------- >Something is clearly incorrect in the argument. Here's why. [probability analysis deleted] I don't know who did the probability analysis, but it was not me (Charles Blair). Perhaps some other Charles. All I did was make a probably irrelevant reference to a paper by Norman Levinson. === Subject: Paper published by Algebraic and Geometric Topology The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-30.abs.html Title: On the slice genus of links Author(s): Vincent Florens, Patrick M. Gilmer Abstract: We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding an annulus in the 4-ball. Secondary: 57M27 Keywords: Casson-Gordon invariants, link signatures Received: 23 October 2002 Author(s) address(es): Laboratoire I.R.M.A. Universite Louis Pasteur Strasbourg, France and Department of Mathematics, Louisiana State University Baton Rouge, LA 70803, USA Email: vincent.florens@irma.u-strasbg.fr, gilmer@math.lsu.edu === Subject: Re: Kuga surface The FAFA note has been published in a posthumous volume of Kuga's work edited by Satake. It is in Volume 3 in the Lectures in Mathematical Sciences of the University of Tokyo, pages 101-126. FAFA, by the way, stands for Fun Algebra and Fun Arithmetic, or something along those lines. Salman Abdulali > Hello -- > Can anyone give me a reference to a (published) construction of the fake > quadric with c_1^2=8, c_2=4, b_1=0 due to M. Kuga mentioned on p. 177 of > Barth/Peters/Van de Ven's _Compact Complex Surfaces_? The reference given > there is listed as an FAFA note. > Michael A. Van Opstall > Padelford C-113 > opstall@math.washington.edu > http://www.math.washington.edu/~opstall/ === Subject: Re: Factoring Polynomials > I am currently looking at the available literature > for factoring polynomials over finite fields of > characteristic $2$. I know some algorithms for the > field F_2, but I need to factor polynomials > of relatively small degree over fields of appreciable > degree. The degree might be bounded by 10, and the > extension degree might vary from about 30 to just a > few hundreds. Does anybody have useful pointer? Springer GTM 138, Henri Cohen: A Course in Computational ANT contains several chapters on polynomial factorization. I'm not sure whether your special topic is treated there. If it is GP-Pari might be helpful, too. hth Klaus > thank you in advance > Roberto > -- > _/_/ Roberto Maria Avanzi (a.k.a. Mocenigo) <>< > /_/ Institut fuer Experimentelle Mathematik / Uni Essen === Subject: Geometric Algebra for Pure Mathematicians Is there a good introduction to geometric algebra for pure mathematicians? I have started a number of introductions but I've been having a few problems: (1) Many seem to proselytise. It gets very tedious having to read long lectures on how we're all using the wrong types of geometric structures in our work and so on. I don't have to read this type of evangelical material when I read about other branches of mathematics. (2) Many treat you like a baby starting with the definition of a vector as a directed line or whatever. If a geometric structure is a Clifford algebra modulo some relation then I'd like it just to say so. As a mathematician I already know what a vector space, a Clifford algebra, a manifold, a Lie group, a Lie algebra and so on all are. I just don't know anything about geometric algebra. Does anyone have any recommendations?