mm-1079 === Subject: question about Banach spaces Is there an accessible example of a linear space endowed with two non-equivalent complete norms? jenny === Subject: Re: question about Banach spaces > Is there an accessible example of a linear > space endowed with two non-equivalent > complete norms? > jenny Infinite-dimensional only. And AC required. How about this: Banach spaces l^2 and l^1 are isomorphic as linear spaces (both having Hamel dimension c). -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: question about Banach spaces > Is there an accessible example of a linear > space endowed with two non-equivalent > complete norms? > jenny > Infinite-dimensional only. And AC required. How about this: > Banach spaces l^2 and l^1 are isomorphic as linear spaces > (both having Hamel dimension c). one has to necessarily appeal to AC? In other words, is AC equivalent to that statement? I am asking that, since I have a somehow realated problem: It is well known that the dual of L^infty strictly contains L^1 (this can be directly argued from separability arguments; clearly, only in infinite dimension)). In spite of that, I am not aware af any example of a continuous linear functional on L^infty which is not L^1 without appealing to the Hahn-Banach theorem, which is to say, without appealing to AC. jenny === Subject: Re: question about Banach spaces > Is there an accessible example of a linear > space endowed with two non-equivalent > complete norms? > jenny > Infinite-dimensional only. And AC required. How about this: > Banach spaces l^2 and l^1 are isomorphic as linear spaces > (both having Hamel dimension c). > one has to necessarily appeal to AC? In other words, is AC > equivalent to that statement? Equivalent, probably not, but in fact it cannot be proved in ZF alone. In Solovays model where every set of reals (and every set in a Polish space) has the property of Baire, it follows (proved by Christensen, I guess) that any linear map of separable Banach spaces is automatically continuous. So, in particular, if a given vector space admits two complete separable norms, then the identity is a homeomorphism. (And since, in metric spaces, a map is continuous if and only if its restriction to every separable subspace is, we can get rid of the separable assumptions I made above.) Thats in Solovays model. But there is some principle that goes with this saying any spaces and maps THAT YOU CAN ACTUALLY WRITE DOWN EXPLICITLY also work like this, merely using ZF. So, depending on what accessible means in the original question, the answer may be no. > I am asking that, since I have a somehow realated problem: > It is well known that the dual of L^infty strictly contains L^1 > (this can be directly argued from separability arguments; clearly, only > in infinite dimension)). > In spite of that, I am not aware af any example of a continuous linear > functional on L^infty which is not L^1 without appealing to > the Hahn-Banach theorem, which is to say, without appealing to AC. Again, this existence cannot be proved in ZF. But it can be proved using HB. It is known that HB is strictly weaker than AC, so this answers the equivalent question above. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: question about Banach spaces > Is there an accessible example of a linear > space endowed with two non-equivalent > complete norms? > jenny > > Infinite-dimensional only. And AC required. How about this: > Banach spaces l^2 and l^1 are isomorphic as linear spaces > (both having Hamel dimension c). > one has to necessarily appeal to AC? In other words, is AC > equivalent to that statement? > Equivalent, probably not, but in fact it cannot be proved in ZF alone. (also for David C. Ullrich) When you say that it cannot be proved what do you exactly mean? I mean, if it is provable with AC but not provable without AC why it is not automatically equivalent to AC. My point is: one cannot prove this within ZF because he/she is not able to, but in principle he/she could, or there is a way to show this impossibility? jenny === Subject: Re: question about Banach spaces > Is there an accessible example of a linear >> space endowed with two non-equivalent >> complete norms? >> jenny >> Infinite-dimensional only. And AC required. How about this: > Banach spaces l^2 and l^1 are isomorphic as linear spaces >> (both having Hamel dimension c). > one has to necessarily appeal to AC? In other words, is AC >> equivalent to that statement? >> Equivalent, probably not, but in fact it cannot be proved in ZF alone. >(also for David C. Ullrich) >When you say that it cannot be proved what do you exactly mean? >I mean, if it is provable with AC but not provable without AC >why it is not automatically equivalent to AC. >My point is: one cannot prove this within ZF because he/she is >not able to, but in principle he/she could, or there is a way to >show this impossibility? There is a way to show this impossibility. There was a big hint how that works in the paragraph you snipped, where Edgar said something about a certain model of ZF. Any statement that can be proved from the axioms of ZF will be true in every model of ZF, so if there is a model of ZF in which P is false it follows that P _cannot_ be proved in ZF. You need to study a small bit of mathematical logic to really know exactly what that means. For now an analogy: Say GT is the axioms for a group: (ab)c = a(bc), etc. So for example (ab(cd) = (a(bc))d is a theorem of GT - it can be proved from the axioms. Now a model of GT is precisely the same thing as a _group_. There exists a group in which it is not true that ab = ba for all a, b, and the existence of such a group shows that ab = ba for all a, b cannot be proved from the axioms of GT. >jenny ************************ David C. Ullrich === Subject: Re: question about Banach spaces > (also for David C. Ullrich) > When you say that it cannot be proved what do you exactly mean? > I mean, if it is provable with AC but not provable without AC > why it is not automatically equivalent to AC. > My point is: one cannot prove this within ZF because he/she is > not able to, but in principle he/she could, or there is a way to > show this impossibility? An example is the Hahn-Banach theorem, HB. Assuming consistency when needed, it has been shown that: (1) ZF does not prove HB, (2) ZF+AC proves HB, (3) ZF+HB does not prove AC. In that sense, HB is beyond ZF, but is not as strong as AC. On the other hand, Zorns Lemma, ZL, is equivalent to AC, meaning: (4) ZF+AC proves ZL (5) ZF+ZL proves AC In most cases, a cannot be proved result is shown by exhibiting a model. For example, a model of ZF+HB where AC fails will show (3), above. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: question about Banach spaces >> Is there an accessible example of a linear >> space endowed with two non-equivalent >> complete norms? >> jenny > Infinite-dimensional only. And AC required. How about this: >> Banach spaces l^2 and l^1 are isomorphic as linear spaces >> (both having Hamel dimension c). >one has to necessarily appeal to AC? In other words, is AC >equivalent to that statement? Not that I know the answer to either question, but Ill point out that the in other words isnt right - saying something requires AC is not the same as saying its equivalent to AC. If ZF does not prove P but ZFC does then P requires AC; saying P is equivalent to AC is a much stronger statement. (For example: If ZFC proves P but, say, ZF + AD (Axiom of Determinacy) proves not P then AC is required for P, although this doesnt say that P is equivalent to AC. This actually happens if P is every set of reals is Lebesgue measurable.) >I am asking that, since I have a somehow realated problem: >It is well known that the dual of L^infty strictly contains L^1 >(this can be directly argued from separability arguments; clearly, only >in infinite dimension)). >In spite of that, I am not aware af any example of a continuous linear >functional on L^infty which is not L^1 without appealing to >the Hahn-Banach theorem, which is to say, without appealing to AC. >jenny ************************ David C. Ullrich === Subject: toplogically equevalent... hello.....doctor~ Let (X,d) be a metric space. Define a function d :XxX -> (R+) U {0} by d(x,y) = min {1, d(x,y)}, the minumum of 1 and d(x,y), for all x,y in X. show that (1) d is a bounded metric for X. (2) The metric space (X,d) is topologically equivalent to the bounded metric space (X,d) ---------------------------------------------------- i can do (1) by metric conditions. but...in the (2) let x in X and e>0 then y in B_d_(x,e) => d(x,y) < e => d(x,y) <= d(x,y) < e => y in B_d_(x,e) y in B_d_(x,e) => d(x,y) < e =>min {1, d(x,y)} < e => 1 < e or d(x,y) < e if d(x,y) < e => y in B_d_(x,e) else 1 < e => (***) i cant deduce y in B_d_(x,e) in the (***) step. so, i need your advice. thank you very much for your advice. === Subject: Re: toplogically equevalent... > hello.....doctor~ > Let (X,d) be a metric space. > Define a function d :XxX -> (R+) U {0} by > d(x,y) = min {1, d(x,y)}, the minumum of 1 and d(x,y), > for all x,y in X. > show that > (1) d is a bounded metric for X. > (2) The metric space (X,d) is topologically equivalent to > the bounded metric space (X,d) > ---------------------------------------------------- > i can do (1) by metric conditions. > but...in the (2) > let x in X and e>0 > then y in B_d_(x,e) => d(x,y) < e > => d(x,y) <= d(x,y) < e > => y in B_d_(x,e) > y in B_d_(x,e) => d(x,y) < e > =>min {1, d(x,y)} < e > => 1 < e or d(x,y) < e > if d(x,y) < e => y in B_d_(x,e) > else 1 < e => (***) > i cant deduce y in B_d_(x,e) in the (***) step. > so, i need your advice. > thank you very much for your advice. Let N_e(x) = {y in X | d(x,y)0, B_L = { N_1/n(x) | x in X, n in N, 1/N Let (X,d) be a metric space. > Define a function d :XxX -> (R+) U {0} by > d(x,y) = min {1, d(x,y)}, the minumum of 1 and d(x,y), > for all x,y in X. > show that > (1) d is a bounded metric for X. > (2) The metric space (X,d) is topologically equivalent to > the bounded metric space (X,d) Exercise: Start with a metric d and f:R -> R, f(x) = 0 iff x = 0, f(x+y) <= f(x) + f(y) Show fd = f o d is a metric. Show when f is ascending and continuous at 0 that f is uniformly continuous and d,fd are equivalent metrics For your problem f(x) = min{ 1,x } and d = fd. > but...in the (2) > let x in X and e>0 > then y in B_d_(x,e) => d(x,y) < e > => d(x,y) <= d(x,y) < e > => y in B_d_(x,e) > y in B_d_(x,e) => d(x,y) < e > =>min {1, d(x,y)} < e > => 1 < e or d(x,y) < e > if d(x,y) < e => y in B_d_(x,e) > else 1 < e => (***) > i cant deduce y in B_d_(x,e) in the (***) step. I think you want to show y in B_d(x, min(1,r)) ==> y in B_d(x,r) Heres from my notes for the exercise: Let F(a,r) = { x | fd(a,x) < r } F(a,f(r)) subset B(a,r). If x in F(a,f(r)): fd(a,x) < f(r) if r <= d(a,x): f(r) <= fd(a,x) < f(r) which cannot be d(a,x) < r; x in B(a,r) some s > 0 with B(a,s) subset F(a,r). Some s > 0 with f(s) < r if x in B(a,s): d(a,x) < s; fd(a,x) <= f(s) < r; x in F(a,r) ---- === Subject: Re: Unstoppable Force vs Immovable Object > > Why? Whats wrong with the argument? > Force has a physical meaning, not a logical meaning. A force is some > kind of action that changes the momentum of a body. > Bob Kolker I dont think youve identified a problem with my reasoning. === Subject: Re: Unstoppable Force vs Immovable Object > I dont think youve identified a problem with my reasoning. Your reasoning addresses a physical concept in a non-physical way. As I said, you are playing word games. Bob Kolker === Subject: Re: Unstoppable Force vs Immovable Object > > I dont think youve identified a problem with my reasoning. > Your reasoning addresses a physical concept in a non-physical way. As > I said, you are playing word games. > Bob Kolker Whats that supposed to mean, I address a physical concept in a non-physical way? I conclude that its logically impossible for an unstoppable force to meet an immovable object. Do you disagree? Do you think its logically possible? === Subject: Re: Unstoppable Force vs Immovable Object > Whats that supposed to mean, I address a physical concept in a > non-physical way? > I conclude that its logically impossible for an unstoppable force to > meet an immovable object. Do you disagree? Do you think its logically > possible? If you consider what the word force means you would conclude there is no unstopable (i.e. infinite) force in the physical universe. Likewise, an immovable object would have infinite momentum (physically impossible). In either case you are talking about things which not only dont exist, but cant exist. Bob Kolker === Subject: compute normal how can I compute the normal of a plane when I have only three vectors which are all standing on the plane at the same point (cp trihedron startpoint of the vectors is the same) and I know the three (different) angles between the vectors and the plane. S. Nurbe === Subject: Re: compute normal >how can I compute the normal of a plane when I have only three vectors >which are all standing on the plane at the same point (cp trihedron >startpoint of the vectors is the same) and I know the three >(different) angles between the vectors and the plane. >S. Nurbe Why start a new thread when you have already received an answer? --Lynn === Subject: Residues, integrals I know about how to find some real valued integrals using Residue theorem and complex analysis, when the integrals would not be able to be solved using standard techniques. Well, can we generalize this method to higher dimensions? For example, we can find integrals on the real line by looking at a contour in R^2 (i.e. the complex plane) and integrating on the contour, where one piece of the contour is actually the part we need in order to get the original integral. Then we use residue theorem, etc etc. Can we now look at a SURFACE in R^3 and integrate on this, where one piece of the surface (i.e. a line) is the part we need in order to get the original real-valued single integral? In this case we dont use residues of course, but we can use integrals in R^3, right? Is there any work on this? Isaac === Subject: solving for the difference of two exponentials by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB42XUF02007; Im looking to solve or approximate the following equation for x: a^x -q*b^x = c where a,b,q are elements of (0,1). Any ideas? I have had little success..... Fred === Subject: Re: solving for the difference of two exponentials > I am trying to solve or approximate the following equation for x: > a^x - q*b^x = c > Here a,b,q are elements of (0,1). . After looking at that, if youd like to see another example, look at . HTH, David Cantrell === Subject: solving for the difference of two exponentials by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB42XUW02016; I am trying to solve or approximate the following equation for x: a^x - q*b^x = c Fred === Subject: Re: Question about numbers in arithmetic progression I. M. Davidson wote: >This gives minima which agree with your results. But doesnt prove these are the actual minima, right? >What values did you get for m(1001,31,7) etc. ? I get: m(1001,31,7)=87 m(10001,31,7)=1417 m(100001,31,7)=6462 m(1000001,31,7)=107548 m(10000001,31,7)=322585 and, curiously, m(10^12+1,31,7)=64516129040 m(10^22+1,31,7)=322580645161290322585. >What is the basis of your method ? Casting in a small pond full of big fish and hoping for the best? Seriously, I dont have a answer for your question. I will try to write a good description and post it here, though. Rich === Subject: Re: Question about numbers in arithmetic progression posting-account=YU8ZnAwAAABFplvVaRgMUYVYk_DRcg4s > I. M. Davidson wote: >This gives minima which agree with your results. > But doesnt prove these are the actual minima, right? Precisely. I was thinking that the CF method gives the smallest (in your sense) solution to AX + dB =1 and that the smallest solution to a_2 + d*a_2 + + (n-1)da_n must involve the a_j wth the largest coefficient and some other a_k =1. Bit Im not sure if this necessarily follows. >What values did you get for m(1001,31,7) etc. ? > I get: > m(1001,31,7)=87 I get 83 as the minimum here a_1 = 45, a_5 =-1,a_6 = -37 i.e 45*1001 +1125*(-1) +1187(-37) =1 So your method does not find the minima in all cases, even though the actual minimum may be less than 83. > m(10001,31,7)=1417 Here I get 1954. A considerable difference. What are the actual values you get for the as ? Whatever they are they must satisfy the Jacobi reduction. > m(100001,31,7)=6462 I also have a different value here. What were the as in this case ? === Subject: Re: Question about numbers in arithmetic progression >> I. M. Davidson wote: >This gives minima which agree with your results. >> But doesnt prove these are the actual minima, right? >Precisely. I was thinking that the CF method >gives the smallest (in your sense) solution >to AX + dB =1 and that the smallest solution >to a_2 + d*a_2 + + (n-1)da_n must involve the >a_j wth the largest coefficient and some >other a_k =1. Bit Im not sure if this >necessarily follows. >>What values did you get for m(1001,31,7) etc. ? >> I get: >> m(1001,31,7)=87 >I get 83 as the minimum here >a_1 = 45, a_5 =-1,a_6 = -37 >i.e 45*1001 +1125*(-1) +1187(-37) =1 >So your method does not find the minima >in all cases, even though the actual >minimum may be less than 83. Yes. This is not a real suprise as the problem is quite difficult. Ill have to work on this one a bit... >> m(10001,31,7)=1417 >Here I get 1954. A considerable >difference. >What are the actual values you >get for the as ? 1417: -715*10001+689*10187+5*10125+8*10156=1 1433: -723*10001+689*10187+13*10125+8*10032=1 1433: -715*10001+689*10187-8*10094+21*10125=1 1433: -715*10001+689*10187+13*10125+8*10094=1 1443: -728*10001+18*100063+689*100187+8*10094=1 >Whatever they are they must satisfy >the Jacobi reduction. >> m(100001,31,7)=6462 >I also have a different value here. >What were the as in this case ? 6462: 3234*100001-1*100094-3222*100187-5*100125=1 6472: 3223*100001+16*100063-3222*100187-11*100156 Rich === Subject: are C_r fields C_r? any progress? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Some time around Ô52, Serge Lang generalized Artins notion of quasi-algebraically closed fields (C_1) and defined C_r fields: A field k is said to satisfy property (C_r) when any homogeneous polynomial of degree d with coefficients in k in n+1 variables satisfying n >= d^r has a non-trivial zero. (Or, in arithmetic geometry terminology, any hypersurface in P^n over k with degree d has a k-point if n >= d^r.) (Note that a C_0 field is merely an algebraically closed field. Finite fields are C_1 by a theorem of Chevalley, the field of rational functions, or Laurent series, in r variables over an algebraically closed fields, are C_r by results of Tsen and Lang. The field Q_p of p-adic numbers is not C_2, contrary to what was believed for some time: Terjanian gave some counterexamples to that effect.) Under a certain technical assumption (viz., the existence of normic forms of order r and of all degrees), Lang proved that the existence of a non-trivial zero generalizes to families of polynomials, provided the inequality is satisfied for the sum of the degrees-to-the-r. Precisely, let us define: A field k is said to satisfy property (C_r) when any s homogeneous polynomials of degrees d_1,...,d_s with coefficients in k in n+1 variables satisfying n >= d_1^r + ... + d_s^r have a common non-trivial zero. (Or, in arithmetic geometry terminology, any intersection of s hypersurfaces in P^n over k with degrees d_1,...,d_s has a k-point if n >= d_1^r + ... + d_s^r.) (Im not sure that my notation is perfectly standard. Perhaps what other people call C_r is not exactly what is written above. But let us continue with this notation.) So Lang proves that, under a certain technical assumption which I wont recall, a C_r field is actually C_r. Later (around Ô57), Nagata showed that the technical assumption is not necessary provided all the d_j are equal. However, the question of whether the technical assumption is necessary in full generality remained open (as far as I know). My question is: has any progress been made on this question since then? Is there now a known example of a C_r field which is not C_r, or a proof that all C_r fields are C_r? Perhaps if we restrict to the (most interesting) r=1 case? At any rate, what would the educated guess be? Does it help in any way if we assume the (hypersurfaces defined by the) f_j to be in complete intersection (I cant see a way to reduce the general problem to that case, but it seems like a reasonable assumption to make)? (Basically, Id like to formulate the conjecture that, over a C_1 field k, any smooth projective (geometrically) separably rationally connected variety has a k-point: this is known when k is finite or when it is the field of rational functions over an algebraically closed field, by results of Esnault on the one hand, and Graber, Harris, de Jong and Starr on the other. At the very least, it is necessary, for the conjecture to be sensible, for the field to be C_1, so it would be embarrassing if there were already a known example of a C_1 field that is not C_1.) of Maths) and Nagata (in Kyoto University something-or-other) if necessary. -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Paper published by Geometry and Topology Received-SPF: Received-SPF: pass (mailbox2.ucsd.edu: domain of gt@maths.warwick.ac.uk designates 137.205.233.100 as permitted sender) receiver=mailbox2.ucsd.edu; client_ip=137.205.233.100; envelope-from=gt@maths.warwick.ac.uk; Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.html Title: Commensurations of the Johnson kernel Author(s): Tara E Brendle, Dan Margalit Abstract: Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes. Secondary: 20F38, 20F36 Keywords: Torelli group, mapping class group, Dehn twist Proposed: Walter Neumann Seconded: Shigeyuki Morita, Joan Birman Author(s) address(es): Department of Mathematics, Cornell University 310 Malott Hall, Ithaca, NY 14853, USA and Department of Mathematics, University of Utah 155 S 1440 East, Salt Lake City, UT 84112, USA Email: brendle@math.cornell.edu, margalit@math.utah.edu === Subject: Teiji Takagi and principalization Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I couldnt get an answer on sci.math, so I hope this is suitable for here: Some sources say that Takagi proved not only that the maximal unramified extension of a L/K number field K has a Galois group corresponding to the class group, but that principalization occurs in this field--all the ideals of K extend to principal ideals of L. Others say it had to wait for Artin and Furtwangler to get the proof of this. Does anyone have the straight dope? === Subject: Re: Teiji Takagi and principalization Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I couldnt get an answer on sci.math, so I hope this is suitable for here: > Some sources say that Takagi proved not only that the maximal > unramified extension of a L/K number field K has a Galois group > corresponding to the class group, but that principalization occurs in > this field--all the ideals of K extend to principal ideals of L. > Others say it had to wait for Artin and Furtwangler to get the proof > of this. Does anyone have the straight dope? in Cassells and Froelich, _Algebraic Number Theory_ (Thompson, 1967). He says (p. 273): With the help of this [general reciprocity] law, Artin could also reduce the principal divisor theorem, enunciated by Hilbert and not yet proved by Takagi, to a pure group-theoretical proposition, which was then proved by Furtwaengler. Ive looked at Artins paper, Idealklassen in Oberkoerpern und allgemeines Reziprozitaetsgesetz (Collected Papers 159-164), and its clear that he did not know of any earlier proof. William C. Waterhouse Penn State === Subject: Normal subgroups of surface groups? Epigone-thread: thulgonstrix Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let G be a surface group, and H be a non-trivial normal subgroup of G. How can I prove that H is of finite index in G ? (This is a conjecturally true for any one-relator group) === Subject: Re: Normal subgroups of surface groups? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Let G be a surface group, and H be a non-trivial normal subgroup of G. > How can I prove that H is of finite index in G ? > (This is a conjecturally true for any one-relator group) You cant, because this is not true. For instance, any surface group (other than the 2-sphere or projective plane) has a surjection to the integers, whose kernel is of infinite index. The simplest of course would be the torus, where the obvious Z subgroup is normal, and of infinite index. DR === Subject: Re: Normal subgroups of surface groups? Epigone-thread: thulgonstrix Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Youre right. Here is the restatement of the question: Show that every finitely generated normal subgroup of a non-abelian surface group (with or without boundary) is of finite index. JJ >> Let G be a surface group, and H be a non-trivial normal subgroup of G. >> How can I prove that H is of finite index in G ? >> (This is a conjecturally true for any one-relator group) >You cant, because this is not true. For instance, any surface group >(other than the 2-sphere or projective plane) has a surjection to the >integers, whose kernel is of infinite index. The simplest of course >would be the torus, where the obvious Z subgroup is normal, and of >infinite index. === Subject: rapidly converging rational sqrt Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Below is a description of an algorithm which, with each iteration, will double the number of significant digits in the computation of a rational square root approximation. I do not know if this algorithm is new, but I found it interesting nonetheless. Lisp code for implementing this algorithm can be found at: http://thegreves.com/david/sqrt/sqrt.html If by convention we say that: s = isqrt(C) d = C - s^2 Then the square root of C can be expressed as as the infinite continued fraction: d s + ----------------- d 2s + ------------ d 2s + ------- 2s + .. We designate the tail of this continued fraction using e(n) = nth error term and we say that the nth error term of the continued fraction representation has the form: d e(n) = ---------- 2s + e(n+1) Although we sometimes drop the subscript on the error term for notational convenience. A pretty good first order approximation for a square root can be computed as follows (even allowing e(1) to be zero): d sqrt(C) ~= s + -------- 2s + e(1) Without proof, we claim that a generalized expression for a partial evaluation of our continued fraction can be represented as: A + Be sqrt(C) = s + ------- C + De It is easy to see that the first order approximation given above is an instance of this expression when A = d, B = 0, C = 2s, D = 1. The generalized representation is useful for representing the result of evaluating some number of sucessive terms in the continued fraction representation. Assuming that the above representation is the result of evaluating n terms of the continued fraction for sqrt(C), then the n+1 term would be computed by substituting the next error term into the error expression in the representation. A + B(d/(2s + e)) s + --------------- C + D(d/(2s + e)) A(2s + e) + Bd s + -------------- C(2s + e) + Dd (A2s + Bd) + Ae s + ----------------- (C2s + Dd) + Ce Which, we observe, is once again in the general representational form. If the evaluation of the first n terms produced A + Be ------ C + De and the evaluation of the next m terms were to produce W + Xe ------ Y + Ze Then the evaluation of the first n+m terms would be: W + Xe A + B(------) Y + Ze A(Y + Ze) + B(W + Xe) ------------ = --------------------- W + Xe C(Y + Ze) + D(W + Xe) C + D(------) Y + Ze (AY + BW) + (AZ + BX)e ---------------------- (CY + DW) + (CZ + DX)e Because the continued fraction representation of the square root is uniform, the evaluation of any n sucessive terms will always produce the same result. We can take advantage of this fact to refine our first order approximation by substituting A,B,C, and D in for W,X,Y and Z in the above expression. The will double the number of terms we have evaluated. Of course, this procedure can be repeated again and again, with each iteration of the algorithm doubling the number of significant digits in our representation. Here is an example run computing the sqrt of 1973 for 1 to 10 iterations of the algorithm. Note that after iteration 6 we have more than 100 significant digits. 1 : 44.41845521141241485670222336460609176198432078139056676519727 541447114766739 49363834982650044981364863 2 : 44.41846462881467219505665982727838995343272680250852496023827 788068578570854 49370627285274658896791048 3 : 44.41846462902561876427524312325572040240985914842970903422305 486056596616470 16608600579504981095676558 4 : 44.41846462902561876438107965740906053956833272941354961112468 508573375443791 08155113103260155775597834 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 26805283703947239542091268 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 7 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 8 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 9 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 10 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 Dave === Subject: Re: rapidly converging rational sqrt Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Below is a description of an algorithm which, with > each iteration, will double the number of significant > digits in the computation of a rational square root > approximation. > Here is an example run computing the sqrt of 1973 for 1 to 10 > iterations of the algorithm. Note that after iteration 6 we have > more than 100 significant digits. > 1 : 44.4184 | 55211412414856702223364606091761984320781390566765197275414471 147667394936383 4982650044981364863 > 2 : 44.41846462 | 88146721950566598272783899534327268025085249602382778806857857 085449370627285 274658896791048 > 3 : 44.418464629025618764 | 27524312325572040240985914842970903422305486056596616470166086 005795049810956 76558 > 4 : 44.4184646290256187643810796574090605395 | 68332729413549611124685085733754437910815511310326015577559783 4 > 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 | 26805283703947239542091268 > 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 7 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 8 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 9 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 10 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 I dont wish to denigrate your algorithm unduly, but square root algorithms with this rate of convergence have been known for millennia. The iteration x_{n+1} = 1/2 (x_n + C/x_n), which is believed to have been known to the ancient Babylonians, yields the following output if we take C = 1973 and start it at 44 (since your algorithm presupposes we know isqrt(C) this seems a fair comparison.) I have added vertical bars to indicate the correct portion of each decimal expansion; I have done the same for the quoted output from your algorithm above. 1 : 44.4 | 20454545454545454545454545454545454545454545454545454545454545 454545454545454 54545454545454545455 2 : 44.4184646 | 73597060396753412870066745738272983092630061164213121235377566 920160933975208 72578432056559 3 : 44.418464629025618786 | 74355237890368384233529296754366053971048032852669727540929455 186628564646861 810 4 : 44.4184646290256187643810796574090605 | 45224167043182294394699285292428539812171115734428794096905254 72 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066 | 216106505407447827420345844 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 436869171473600589118301... So like your algorithm this also produces 100 digits in 6 iterations, and it is easy to prove (since this is a special case of Newtons method for a general function) that convergence is quadratic in general. In short, your algorithm is interesting but it doesnt outperform the standard algorithms for square roots. Yours, David Loefßer (student, Trinity College, University of Cambridge, UK) === Subject: Laplaces method and a Double Integral Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hello all, I wonder if anyone can show me how to evaluate the asymptotics (as N) gets large of this integral: int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). If we define f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr where f takes on its mininum value of 0 at the pt (0,0). Unfortunately the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the explicit expansions I have seen require that the Hessian be non-zero. Does anyone know of another reference to try that might have this worked out? Jim PS: Im not a mathematician....just a humble plodding engineer! === Subject: Re: Laplaces method and a Double Integral Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Hello all, >I wonder if anyone can show me how to evaluate the asymptotics (as N) >gets large of this integral: >int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr >Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). >If we define >f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] >Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr >where f takes on its mininum value of 0 at the pt (0,0). Unfortunately >the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the >explicit expansions I have seen require that the Hessian be non-zero. Do the substitutions t=r^2, s=-n help? The integral becomes (1/2)int_0^{R^2} int_0^u t [1 - c_1(c_2 s + c_3 t)^2]^N ds dt For the corresponding f(s,t), a quick calculation gave me f_{tt}(0,0) not= 0, but you should check that yourself. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 To reply by email, change Look-In-Sig to luecking === Subject: Re: Laplaces method and a Double Integral Originator: israel@math.ubc.ca (Robert Israel) Dans substitutions are OK, a nice trick, but I wonder if the origin (0,0) is the main contributing point. I expect also contributions from the boundaries, but this depends on the values of c_1, c_2, c_3, u and R. Nico M. Temme, http://homepages.cwi.nl/~nicot/ C W I: Centrum voor Wiskunde en Informatica Kruislaan 413, NL-1098 SJ Amsterdam Tel +31 20 592 4240 P.O. Box 94079, NL-1090 GB Amsterdam Fax +31 20 592 4199 === > Subject: Re: Laplaces method and a Double Integral >Hello all, >I wonder if anyone can show me how to evaluate the asymptotics (as N) >gets large of this integral: >int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr >Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). >If we define >f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] >Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr >where f takes on its mininum value of 0 at the pt (0,0). Unfortunately >the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the >explicit expansions I have seen require that the Hessian be non-zero. > Do the substitutions t=r^2, s=-n help? The integral becomes > (1/2)int_0^{R^2} int_0^u t [1 - c_1(c_2 s + c_3 t)^2]^N ds dt > For the corresponding f(s,t), a quick calculation gave me > f_{tt}(0,0) not= 0, but you should check that yourself. > Dan > -- > Dan Luecking Department of Mathematical Sciences > University of Arkansas Fayetteville, Arkansas 72701 > To reply by email, change Look-In-Sig to luecking === Subject: Re: Laplaces method and a Double Integral Originator: israel@math.ubc.ca (Robert Israel) Re: Laplaces method and a Double Integral Since Nico thinks that the asymptotic form of the integral has also contributions not only from the neighborhood of (0,0), I derived an asymptotic approximation given in the following. It shows that only the neighborhood of (0,0) is important. Ciao Karl Breitung Schellingstr. 21 D-80799 Munich, Germany AN ASYMPTOTIC APPROXIMATION FOR THE TWO-DIMENSIONAL INTEGRAL: ,- R ,- 0 3 [ 2 2 ]N I(N)= | | r [ 1-a(bn-cr ) ] dn dr, N --> oo - 0 - -u Here we write instead of the original form: a=c , b=c , c=c . 1 2 3 1/2 2 1/2 Making the substitutions r --> v=a cr and n --> z=-a bn transforms this into: 2 ,- cR ,- bu / v 3/2 [ 2 ]N d r d n I(N)=- | | | ----- | [ 1-(z+v) ] --- ---- dz dv - 0 - 0 | 1/2 | d v d z a c / 2 ,- cR ,- bu / v 3/2 [ 2 ]N 1 1/2 -1/2 1 = | | | ----- | [ 1-(z+v) ] -(a cv) ----- dz dv= - 0 - 0 | 1/2 | 2 1/2 a c / a b 1/2 2 1/2 ,- a cR ,- a bu [ 2 ]N K | | v [ 1-(z+v) ] dz dv - 0 - 0 3/2 2 -1 with K=(2a bc ) . Now we will consider only the integral over a triangle 1/2 2 1/2 (0,0), (d,0) and (0,d) with 00 and K >0 are constants. Then: 1 2 ,- d ,- d-v [ 2 ]N I(N) sim K | | v [ 1-(z+v) ] dz dv - 0 - 0 In this triangle, we make the variable transformation (v,z) --> (w,y) with w=z+v, y=z-v. Then: ,- d [ ,- w w-y [ 2 ]N ] I(N) sim K | | | --- [ 1-w ] |det (J(w,y))| dy | dw= - 0 [ - -w 2 ] ,- d [ ,- w w-y [ 2 ]N ] K | | | --- [ 1-w ] dy | dw (*) - 0 [ - -w 2 ] J(w,y) is the Jacobian of the inverse transformation with its determinant equal to 1/2. The integral in the brackets is: ,- w w-y [ 2 ]N [ 2 ]N ,- w / w y [ 2 ]N ,- w w | --- [ 1-w ] dy=[ 1-w ] | | - - - | dy=[ 1-w ] | - dy= - -w 2 - -w 2 2 / - -w 2 [ 2 ]N 2 [ 1-w ] w 2 Inserting this into equ. (*) and writing f(w)=log (1-w ) gives: ,- d 2 I(N) sim K | w exp (Nf(w)) dw - 0 This we can evaluate using the generalized Laplace method (derived in [2], p. 37, see also [1], p. 48). Here we derive the result directly by approximating 2 2 f(w) by its second order Taylor expansion at zero, i.e. f(w)=-2w +o(w ) and 2 then making the substitution w --> x=2N w . This gives: ,- dN x -x dw ,- dN x -x 1 -1/2 I(N)sim K | -- e --dx= K | -- e ------ x dx = - 0 2N dx - 0 2N +--+ 2|2N -3/2 K ,- dN 1/2 -x N ----- | x e dx +-+ - 0 4|2 For this we get the asymptotic form replacing dN by oo: -3/2 K ,- oo 1/2 -x -3/2 K I(N)sim N ----- | x e dx sim N ----- Gamma(3/2)= +-+ - 0 +-+ 4|2 4|2 +---+ +---+ -3/2 K |pi -3/2 |pi N --- ----- = N ---------------- , N --> oo +-+ 2 +-+ 3/2 2 4|2 16|2 a bc If necessary this approximation can be refined by deriving a second term in the asymptotic expansion of I(N). Bibliography: [1] K. Breitung. Asymptotic Approximations for Probability Integrals. Springer, Berlin, 1994. Lecture Notes in Mathematics, Nr.1592. [2] A. Erdelyi. Asymptotic Expansions. Dover, New York, 1956. === Subject: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an arbitrary metric space, such that all orbits of $G$ are finite. We suppose that there exists $pin N$ such that for all $gin G$ we have $g^p = 1$. The group $G$ is it finite? === Subject: Re: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an >arbitrary metric space, such that all orbits of $G$ are finite. >We suppose that there exists $pin N$ such that for all $gin G$ we >have $g^p = 1$. The group $G$ is it finite? The answer is no. This was essentially known as the Burnside problem, and solved in the negative by Novikov and Adjan in 1968: there exist infinite two-generator groups identically satisfying x^n=1 with n any sufficiently large odd integer. (The additional condition stipulating that G should be a subgroup of Homeo(E) adds nothing to the problem, for every group can be so represented: choose E = G, with the discrete metric where distinct points always have distance 1, and let G operate on itself by left translations; these, of course, are homeomorphisms in the given metric.) === Subject: Re: groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is >>an arbitrary metric space, such that all orbits of $G$ are finite. >>We suppose that there exists $pin N$ such that for all $gin G$ we >>have $g^p = 1$. The group $G$ is it finite? >The answer is no. This was essentially known as the Burnside >problem, and solved in the negative by Novikov and Adjan in 1968: >there exist infinite two-generator groups identically satisfying >x^n=1 with n any sufficiently large odd integer. (The additional >condition stipulating that G should be a subgroup of Homeo(E) adds >nothing to the problem, for every group can be so represented: choose >E = G, with the discrete metric where distinct points always have >distance 1, and let G operate on itself by left translations; these, >of course, are homeomorphisms in the given metric.) But among the hypotheses you have that the orbits of G are finite. This implies that G is residually finite and thus, by Zelmanovs positive solution to the restricted Burnside problem, G is indeed finite. Andreas === Subject: Re: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Sorry for my oversight! Of course, Andreas is right; I overlooked the requirement that the orbits should be finite. Regretfully, Peter === Subject: Re: groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an >>arbitrary metric space, such that all orbits of $G$ are finite. >>We suppose that there exists $pin N$ such that for all $gin G$ we >>have $g^p = 1$. The group $G$ is it finite? >The answer is no. This was essentially known as the Burnside >problem, and solved in the negative by Novikov and Adjan in 1968: >there exist infinite two-generator groups identically satisfying x^n=1 >with n any sufficiently large odd integer. (The additional condition >stipulating that G should be a subgroup of Homeo(E) adds nothing to >the problem, for every group can be so represented: choose E = G, with >the discrete metric where distinct points always have distance 1, and >let G operate on itself by left translations; these, of course, are >homeomorphisms in the given metric.) But the orbits of G will not be finite in this example. G having a faithful representation as mappings of a set with all orbits finite means that G has a collection of normal subgroups of finite index whose intersection is the identity. I dont know if this is true for any of the known examples in the Burnside problem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Fall Pacific NW Geometry Seminar at U of Oregon Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Final announcement: PACIFIC NORTHWEST GEOMETRY SEMINAR University of Oregon Eugene, OR SCHEDULE Saturday, November 6 10:30 - 11:00 Morning Reception 11:00 - 12:15 Charles Doran (University of Washington) Mirror Symmetry, K-Theory, and Toric Geometry 12:15 - 2:00 Lunch 2:00 - 3:15 Mutao Wang (Columbia) Mean curvature ßows of Lagrangian submanifolds 3:15 - 4:00 Break 4:00 - 5:15 Lei Ni (UCSD) Ancient Solutions of the K.8ahler-Ricci Flow 7:00 Party at Botvinniks Sunday, November 7 8:30 - 9:00 Morning Reception 9:00 - 10:15 John Lott (University of Michigan) Ricci curvature for metric-measure spaces 10:15 - 10:45 Break 10:45 - 12:00 David Auckly (Kansas State University) The structure of maps into homogenous spaces and the Faddeev and Skyrme models Note: Each speakers time allotment includes 15 minutes for a discussion of Open Problems related to his topic. The talks will be in 110 Fenton Hall (D-7 on the campus map). The receptions and breaks will be right outside 110 Fenton. ----------------------------------------------------------- For general information about the PNGS, visit the PNGS web site: http://www.math.washington.edu/~lee/PNGS It contains up-to-date information about this meeting, travel and lodging information, general information about the PNGS, and a historical record of all PNGS meetings and speakers. ----------------------------------------------------------- For more information about this meeting, contact the organizers: Boris Botvinnik (botvinn@math.uoregon.edu) Jim Isenberg (jim@newton.uoregon.edu) === Subject: Non linear hyperbolic PDEs : ill posedness ? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi everybody I am studying the well-posedness of the Cauchy problem for systems of PDE of the form : d U/dt = A d U/dx, with unknown vector U(x,t) In the linear case (A is a matrix function of (x,t)), it is well known that the problem is well-posed (existence of a unique solution depending continuously on the initial data) iff A is diagonalisable with real eigenvalues for all x and t. Here is my question : In the non linear case (A is a function of U, x and t) does one know such a system where the matrix A is not always diagonalisable but wich is still well-posed ? Michael === Subject: Partitioning 4 space with ultraskew lines, and the three body problem. Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Suppose we are in 4 space. Two lines are skew if they do not lie in the same plane(2 space). Skewness is a binary relation on lines. Now two skew lines will lie in the same 3 space. But it is possible for three lines to not lie in the same 3 space. This is a ternary relation on lines. I dont know if theres a name for this relation, so lets call it ultraskewness until we find out. 1) Can one foliate, or at least partition, 4 space with lines that are tripletwise ultraskew? I mean EACH 3 line subset of the partition must not lie in the same 3 space. 2) In the three-body problem, one could approximate the trajectory of each of the 3 masses as a straight lines until they became close enough to each other for their gravitation to have an appreciable effect. I think much of the work on this problem assumes all three trajectories lie in the same plane.(The restricted 3-body problem.) But some work has been where the trajectories are not coplanar. Wouldnt it be fun to explore the three-body problem when the trajectories are not cospatial? Richard Peterson, CSU Sacramento === Subject: Re: Partitioning 4 space with ultraskew lines, and the three body problem. ath: nntpswitch.com Originator: israel@math.ubc.ca (Robert Israel) > Suppose we are in 4 space. Two lines are skew if they do not lie in > the same plane(2 space). Skewness is a binary relation on lines. Now > two skew lines will lie in the same 3 space. But it is possible for > three lines to not lie in the same 3 space. This is a ternary relation > on lines. I dont know if theres a name for this relation, so lets > call it ultraskewness until we find out. > 1) Can one foliate, or at least partition, 4 space with lines that > are tripletwise ultraskew? I mean EACH 3 line subset of the partition > must not lie in the same 3 space. > 2) In the three-body problem, one could approximate the trajectory > of each of the 3 masses as a straight lines until they became close > enough to each other for their gravitation to have an appreciable > effect. I think much of the work on this problem assumes all three > trajectories lie in the same plane.(The restricted 3-body problem.) > But some work has been where the trajectories are not coplanar. > Wouldnt it be fun to explore the three-body problem when the > trajectories are not cospatial? > Richard Peterson, CSU Sacramento I dont know about the three-body problem, or about foliations, but I think one can partition 4-space into ultraskew lines without much trouble. The proof is via a transfinite induction of c (the cardinality of the continuum) many steps; at step k one considers the k-th point p in a fixed enumeration of 4-space, and one has already constructed a collection L of |k|-many (fewer than c) lines. If p is in the union of L there is nothing to do. Otherwise one need only find a unit tangent vector u at p so that the line through p in direction u is (a) disjoint from each line in L and (b) ultraskew to every pair of lines in L. Since p lies on no line in L (a) is satisfied so long as u does not lie in any plane containing both p and a line in L. Also, (b) is satisfied so long as u does not lie in any translate containing p of an (affine) 3-space generated by a pair of lines in L. So we need a point on the unit 3-sphere in R^4 not lying in any of a collection of fewer than c many subspaces of R^4 of dimension at most 3. Without loss of generality we may assume all the subspaces have dimension 3, so each has a perp that meets the 3-sphere in at most 2 points. As we have fewer than c subspaces, there is a point w on the 3-sphere not in the perp of any of them. Then the perp P of the line spanned by w is a 3-dimensional space meeting each of the spaces we want to avoid in a subspace of dimension at most 2. Thus it suffices to find a point on the intersection of the unit 3-sphere with P (i.e., a unit 2-sphere) not in any of a collection of fewer than c subspaces of P of dimension at most 2. By a similar argument we can drop the dimension once again, and then we need to find a point on the unit circle avoiding fewer than c lines through the origin. As the circle has c points, and each line meets it in two point, that is easy. Well, its late and Im hurrying, but I think this argument holds water. Bob Beaudoin === Subject: Integral recurrence relation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I have encountered the following integral in some research in the physical sciences int |u-A|^(2a) |u-B|^(2b) Exp[-|u|^2] du where u, A and B are cartesian vectors in 3 dimensions and the integral is to performed over all space. This seems like quite a straightforward integral but the best I can do is to write it as a triple infinite series in A^2, B^2 and |A-B|^2 (which quickly truncates, depending on the values of a and b). I was wondering if anyone has any suggestions as how I might produce a more useful formulation. Even more useful would be a suggestion as to how I might derive a recurrence relation to generate integrals of higher values of a and b or if it is possible to prove or disprove the existence of such a relation. Darragh === Subject: Re: Integral recurrence relation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >I have encountered the following integral in some research in the >physical sciences > int |u-A|^(2a) |u-B|^(2b) Exp[-|u|^2] du >where u, A and B are cartesian vectors in 3 dimensions and the >integral is to performed over all space. This seems like quite a >straightforward integral but the best I can do is to write it as a >triple infinite series in A^2, B^2 and |A-B|^2 (which quickly >truncates, depending on the values of a and b). I was wondering if >anyone has any suggestions as how I might produce a more useful >formulation. Let your integral be F(a,b) (for nonnegative integers a,b). Consider the exponential generating function f(s,t) = sum_{a=0}^infinity sum_{b=0}^infinity F(a,b) s^a t^b/(a! b!) = int_{R^3} exp(s |u-A|^2) exp(t |u-B|^2) exp(-|u|^2) du = int_{R^3} exp(-(1-s-t) |u|^2 - 2 u.(sA+tB) + s|A|^2 + t|B|^2) du = exp(s|A|^2 + t|B|^2 + |sA+tB|^2/(1-s-t)) int_{R^3} exp(-(1-s-t) |u-(sA+tB)/sqrt(1-s-t)|^2) du = exp(s|A|^2 + t|B|^2 + |sA+tB|^2/(1-s-t)) (pi/(1-s-t))^(3/2) for |s|+|t| < 1. Then F(a,b) can be obtained from the coefficients of the bivariate Taylor series for f(s,t) around (0,0). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Paper published by Geometry and Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper38.abs.html Title: Noncommutative localisation in algebraic K-theory I Author(s): Amnon Neeman, Andrew Ranicki Abstract: associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A-->B be the localisation with respect to a set sigma of maps between finitely generated projective A-modules. Suppose that Tor_n^A(B,B) vanishes for all n>0. View each map in sigma as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D^perf(A). Denote by the thick subcategory generated by these complexes. Then the canonical functor D^perf(A)-->D^perf(B) induces (up to direct factors) an equivalence D^perf(A)/--> D^perf(B). As a consequence, one obtains a homotopy fibre sequence K(A,sigma)-->K(A)-->K(B) (up to surjectivity of K_0(A)-->K_0(B)) of Waldhausen K-theory spectra. consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor_n^A(B,B), we also assume that every map in sigma is a monomorphism, then there is a description of the homotopy fiber of the map K(A)-->K(B) as the Quillen K-theory of a suitable exact category of torsion modules. Secondary: 19D10, 55P60 Keywords: Noncommutative localisation, $K$--theory, triangulated category Proposed: Bill Dwyer Seconded: Thomas Goodwillie, Gunnar Carlsson Author(s) address(es): Centre for Mathematics and its Applications The Australian National University Canberra, ACT 0200, Australia and School of Mathematics, University of Edinburgh Edinburgh EH9 3JZ, Scotland, UK Email: Amnon.Neeman@anu.edu.au, a.ranicki@ed.ac.uk === Subject: monoidal enriched natural transformations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi. I have a problem which, it seems to me, requires the notion of enriched natural transformation between enriched monoidal functors, but I havent been able to find a good reference for it. Ive taken a look at the books of Borceux (Handbook of categorical algebra), Kelly (Basic notions of enriched category) and the appendix in Levines Mixed motives, but they all end just where I need them, or even long before that point. More precisely, the situation Ive encountered seems to be the following. I have: - Two monoidal (symmetric) categories: V and W . - A couple of monoidal (symmetric) functors between them: S, T : V ---> W - A monoidal natural transformation between S and T : And here is where my problems begin. There is a well-known notion of what is a V-functor between V-categories and what a V-natural transformation is. My first need is to understand what an S-functor between a V-category C and a W-category D should be: F : C ---> D I didnt find this thing in the literature, but I expect it ought to be something like a V-functor, but with a family of morphisms in W lambda_{XY} : S[X,Y] ---> [FX,FY] for every pair of objects X, Y in C . (Here the square brackets [,] stand for the objects in V and W of morphisms of C and D , and Im leaving aside units and commutative isomorphisms for the moment.) This seems reasonable to me, since (a) is the situation I have in the real world and (b) if I put S = id_V , I find the definition of a V-functor. Next, I would need the notion of an omega - natural transformation and I think this should be something like a V-natural transformation between an S-functor F : C ---> D and a T-functor G: C ---> D, but placing at the beginning of the commutative diagram which defines a V-natural transformation an arrow like omega_{[X,Y]} : S[X,Y] ---> T[X,Y] . Assuming that this is ok, I should also need to understand what might be the definition of a monoidal omega - natural transformation. That is to say, C is a monoidal V-category, D is a monoidal W-category, F is a monoidal S-functor and G a monoidal T-functor: what is a monoidal natural transformation between F and G , over Ive drawn a couple of commutative diagrams that should appear in the definition of such a construct, but I feel I could be forgetting a dozen more. Any references for it? Unfortunately for me, Kellys book ends before this point: it explicitely says: is our decision not to discuss the Ôchange of base-category given by a symmetric monoidal functor V ---> W. Has someone else done the job after Kellys book? Agust.92 Roig === Subject: Re: monoidal enriched natural transformations Epigone-thread: tixslongcax Content-Length: 6836 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Hi. >I have a problem which, it seems to me, requires the notion of >enriched natural transformation between enriched monoidal functors, >but I havent been able to find a good reference for it. >Ive taken a look at the books of Borceux (Handbook of categorical >algebra), Kelly (Basic notions of enriched category) and the appendix >in Levines Mixed motives, but they all end just where I need them, >or even long before that point. >More precisely, the situation Ive encountered seems to be the >following. I have: >- Two monoidal (symmetric) categories: > V and W . >- A couple of monoidal (symmetric) functors between them: > S, T : V ---> W >- A monoidal natural transformation between S and T : >And here is where my problems begin. >There is a well-known notion of what is a V-functor between >V-categories and what a V-natural transformation is. >My first need is to understand what an S-functor between a >V-category C and a W-category D should be: > F : C ---> D >I didnt find this thing in the literature, but I expect it ought to >be something like a V-functor, but with a family of morphisms in W > lambda_{XY} : S[X,Y] ---> [FX,FY] >for every pair of objects X, Y in C . (Here the square brackets >[,] stand for the objects in V and W of morphisms of C and D >, and Im leaving aside units and commutative isomorphisms for the >moment.) >This seems reasonable to me, since (a) is the situation I have in the >real world and (b) if I put S = id_V , I find the definition of a >V-functor. >Next, I would need the notion of an omega - natural transformation >and I think this should be something like a V-natural transformation >between an S-functor F : C ---> D and a T-functor G: C ---> D, >but placing at the beginning of the commutative diagram which defines >a V-natural transformation an arrow like > omega_{[X,Y]} : S[X,Y] ---> T[X,Y] . >Assuming that this is ok, I should also need to understand what might >be the definition of a monoidal omega - natural transformation. >That is to say, C is a monoidal V-category, D is a monoidal >W-category, F is a monoidal S-functor and G a monoidal T-functor: >what is a monoidal natural transformation between F and G , over >Ive drawn a couple of commutative diagrams that should appear in the >definition of such a construct, but I feel I could be forgetting a >dozen more. Any references for it? >Unfortunately for me, Kellys book ends before this point: it >explicitely says: is our decision not to discuss the Ôchange of >base-category given by a symmetric monoidal functor V ---> W. Has >someone else done the job after Kellys book? If you havent done so already, I recommend that you get on the categories mailing list: categories@mta.ca where you would surely get a reply and advice about the literature. Alternatively, you might write Max Kelly (at the University of Sydney) or Ross Street (Macquarie University) directly. I wish I had suitable references at hand, but here are some remarks on your query. First, a monoidal functor S: V --> W induces a 2-functor S_{*}: V-Cat --> W-Cat, making straightforward use of the monoidal structure on S. If C is a V-category and D is a W-category, then what you call an S-functor is undoubtably the same as a W-functor of the form F: S_{*}C --> D. Next, a monoidal natural transformation omega: S --> T induces a 2-natural transformation between 2-functors omega_{*}: S_{*} --> T_{*} and in particular provides, for each V-category C, a W-functor of the form omega_{*}(C): S_{*}C --> T_{*}C This too is straightforward, using just the data and equations for an m.n.t. Then what you call an omega-natural transformation from F to G is undoubtably the same as a W-transformation of the form F --> G(omega_{*}(C)) where F: S_{*}C --> D and G: T_{*}C --> D are W-functors, and the target on the right is a composite of W-functors. Now to define a monoidal omega-natural transformation, you want to do a jazzed-up version of the above definitions. Heres what youll need (minimally): -- V, W braided monoidal categories -- S, T braided monoidal functors of the form V --> W -- omega a m.n.t. of the form S --> T (You can of course replace braided by symmetric, but you lose some generality in doing so.) Since V is braided monoidal, V-Cat is a monoidal 2-category, and monoidal V-categories are the same as (pseudo-)monoids in V-Cat as a monoidal 2-category. Indeed, the 2-category Mon(V-Cat) whose objects are monoidal V-categories, whose objects are monoidal V-functors, and whose 2-cells are monoidal V-transformations, is definable purely in terms of the monoidal 2-category structure on V-Cat, and therefore, the desired change of base induced by S: V --> W, Mon(V-Cat) --> Mon(W-Cat), requires only a monoidal 2-functor S_{*}: V-Cat --> W-Cat to get off the ground. The point of demanding that S be braided monoidal is so that the 2-functor S_{*} is in fact monoidal. So: under these hypotheses, a monoidal S-functor (to use your terminology) should be the same as a monoidal W-functor of the form F: S_{*}C --> D where C is a monoidal V-category and D is a monoidal W-category. Finally, if omega: S --> T is an m.n.t., there is a monoidal W-functor omega_{*}(C): S_{*}C --> T_{*}D and what you call a monoidal omega-natural transformation should just be a monoidal W-transformation of the form F --> G(omega_{*}(C)). Notice that it is unnecessary to write down a whole bunch of commutative diagrams to define these constructs: the data and axioms inherent in ordinary enriched notions and in braided monoidal notions do the work for you. However, this approach does involve some machinery of higher-dimensional categories (monoidal 2-categories, monoidal 2-functors), which is probably why Kelly didnt touch this in his book -- the relevant notions hadnt yet been formulated properly. If you want to follow up on this machinery, you might want to look at Coherence for Tricategories by Gordon, Power & Street. The full-ßedged definition of monoidal 2-category can be found there, and is shown to be equivalent in an appropriate sense to so-called Gray-monoids, which are much simpler (indeed, for V braided monoidal, V-Cat *is* a Gray-monoid). Also look at the havent already done so. Todd Trimble === Subject: Re: monoidal enriched natural transformations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) vas dir: >>Assuming that this is ok, I should also need to understand what might >>be the definition of a monoidal omega - natural transformation. [...] >>Unfortunately for me, Kellys book ends before this point: it >>explicitely says: is our decision not to discuss the Ôchange of >>base-category given by a symmetric monoidal functor V ---> W. Has >>someone else done the job after Kellys book? >If you havent done so already, I recommend that you get on the >categories mailing list: > categories@mta.ca >where you would surely get a reply and advice about the literature. >Alternatively, you might write Max Kelly (at the University of >Sydney) or Ross Street (Macquarie University) directly. >I wish I had suitable references at hand, but here are some remarks >on your query. [...] >Todd Trimble Agust.92 Roig === Subject: weighted tree generation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For a given undirected graph G(V,E) I would like to generate all possible rooted trees. How many will there be? If there are a lot, at least I would like to generate a set of different ones or make a single change to a given tree. Secondly, is there any standard method to assign weights to an undirected graph such that for a given root, the all shortest paths algorithm (e.g. dijkstra) will yield a given tree? (I guess that you could assign low values to the links that are elements of the tree and high values for all others, but can it be proven that it will always yield the given tree?) Diego diego at aulignac dot com www.aulignac.com === Subject: fundamental bounds on the elements of covariance matrices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let x and y be n-tuplets of normally distributed random variables, and let C = cov(x,y) be the nxn covariance matrix of x and y. Let z be some fixed n-tuplet whose elements are all strictly positive. 1) Are all the elements of the n-vector, Cz, non-negative? 2) Is zCz > 0? Where can I read more about bounds on the elements of covariance matrices? === Subject: Re: fundamental bounds on the elements of covariance matrices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) See Morris L. Eatons book with multivariate in the title. I was about to answer that any matrix whatsoever could be in the role of C, but I find Im a bit rusty. -- Mike Hardy > Let x and y be n-tuplets of normally distributed random variables, and let > C = cov(x,y) be the nxn covariance matrix of x and y. > Let z be some fixed n-tuplet whose elements are all strictly positive. > 1) Are all the elements of the n-vector, Cz, non-negative? > 2) Is zCz > 0? > Where can I read more about bounds on the elements of covariance matrices? === Subject: Re: binary vector packings Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Could someone please help me identify the following problem? > Consider binary arrays {u_i} of length n with k 1s (and n-k 0s). How > many can we choose such that all pairwise inner products > sum_i u_i v_i < t ? > Equivalently, what is the maximum number of k-subsets of the n-set with > pairwise intersections less than t elements? > Does this problem, or some equivalent, have a name? Any references? Looks like your problem is more or less equivalent to finding the maximum size of a certain constant weight code (aka fixed weight code). Searching with those buzzwords should lead you to sources of known results, tables of upper bounds etc. Jyrki Lahtonen, Turku, Finland === Subject: two fibrations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hallo, consider an isometric action of a compact Lie group G on a connected Riemannian manifold M with a fixed point p. I am interested in the topology (especially the homology) of the space of paths in M, starting at p and ending in some fixed orbit N=Gq, denoted by P(M,ptimes N). Even for arbitrary p and submanifolds N, the end point map P(M,ptimes N)to N; c mapsto c(1) is a fibration with fibre P(M,ptimes q), the space of paths from p to q (which ist homotopy equivalent to the space of loops on M). So I can deduce some information on the homology from this fibration, e.g. by using the Leray-Serre spectral sequence. Furthermore, now restricting to the case of N being some orbit, we have P(M,ptimes Gq)=P(M,ptimes q)times_{G_q} G (twisted product - G_q is the isotropy group at q), which can be easily seen by regarding the mapping P(M,ptimes q)times Gto P(M,ptimes Gq); (c,g)mapsto gc (well-defined since p is fixed). Summarizing, we have two fibrations: i) P(M,ptimes q)to P(M,ptimes Gq)to Gq and ii) G_qto P(M,ptimes q)times G to P(M,ptimes Gq). Now my question: Does the existence of such two fibrations give any new relation between the topology of these spaces? Is there some method of extracting information from such two similar-looking fibrations? Oliver Goertsches === Subject: RA Positions at UNR Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The Computer Vision Laboratory (CVL) at the University of Nevada, Reno (UNR) invites applications for research assistant positions starting in Spring2005/Fall 2005. Preference will be given to students who want to pursue a PhD degree in Computer Vision. Active research areas within CVL include object recognition, visual motion analysis, face detection and recognition, biometrics, tracking and pose estimation of human body/head/hand/eye-gaze, surveillance and activity recognition. CVL is currently funded by NSF, NASA, ONR, and Ford Motor Company. We are also collaborating with several government and industry laboratories. For more information, please visit http://www.cs.unr.edu/CVL Requirements: You must have a first degree in either an Engineering subject, in Mathematics, in Physics, or in Computer Science. Good Mathematical background, programming skills in C or C++, and familiarity with Unix/Linux/Windows are necessary. Prior familiarity with Image Processing, Computer Vision, Pattern Recognition, and Machine Learning is desirable. Good communication and writing skills in English are essential. Interested students should send their CV by regular mail, e-mail, or fax to Dr. George Bebis (bebis@cs.unr.edu) or Dr. Mircea Nicolescu (mircea@cs.unr.edu) Dr. George Bebis Department of Computer Science & Engineering University of Nevada Reno, NV 89557, USA phone: (775) 784-6463 email: bebis@cs.unr.edu http://www.cs.unr.edu/~bebis Dr. Mircea Nicolescu Department of Computer Science & Engineering University of Nevada Reno, NV 89557, USA phone: (775) 784-4356 email: mircea@cs.unr .edu http://www.cs.unr.edu/~mircea === Subject: A type of regular graph Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For a graph with verticies the integers mod n, we may draw an edge between a and b iff b=a+i for some i in a set of residues mod n. Clearly such a graph is a regular graph, is it possible to characterize it further with known graph-theoretic properties? === Subject: Re: A type of regular graph 3QLpj-NoP*NzsIC,boYU]bQ]Hy<#4ga3$21: Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > For a graph with verticies the integers mod n, we may draw an edge > between a and b iff b=a+i for some i in a set of residues mod n. > Clearly such a graph is a regular graph, is it possible to > characterize it further with known graph-theoretic properties? These graphs are known as circulants, e.g. see . That doesnt answer your question, but it should at least help in searching for an answer. -- David Eppstein Computer Science Dept., Univ. of California, Irvine http://www.ics.uci.edu/~eppstein/ === Subject: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) hi world, im looking for toolkits of segments / magnets / plastic caps to build 3d shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. i want to play around with the cell decomposition of the lattice a_3, so ill need quite a bunch of these. i quickly searched on the net for some way of buying these online, with no success. does anybody know where i can find them? tia, laurent -- Laurent Bartholdi laurent.bartholdiepßch EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 === Subject: Re: building 3d shapes Originator: israel@math.ubc.ca (Robert Israel) > hi world, > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? One possibility: http://www.zometool.com/ -- http://hertzlinger.blogspot.com === Subject: Re: building 3d shapes Originator: israel@math.ubc.ca (Robert Israel) > hi world, > im looking for toolkits of segments / magnets / plastic caps to > build 3d shapes like the 1- or 2-skeleta of tetrahedra, octahedra, > etc. > i want to play around with the cell decomposition of the lattice a_3, > so ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, > with no success. does anybody know where i can find them? > tia, laurent Or try Geomag and Supermag (dont know if these are the same), e.g. at http://www.toymagnets.com/geomag/index.cfm or http://www.geomags.com/. Ive seen them sold in Swiss toyshops as well. Christian Graf === Subject: Re: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > hi world, > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? > tia, laurent > -- > Laurent Bartholdi laurent.bartholdiepßch > EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 > CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 For magnets look at http://www.supermagnete.ch/magnets.php?at=Z Hugo Pfoertner === Subject: Re: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > .... > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? .... Is http://www.korthalsaltes.com/ any use to you? Ken Pledger. === Subject: natural numbers as coequalizer (Re: Those Naughty Category Theorists) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) begin{quote} Incidentally for those who think that by defining numbers as lengths of strings Im making some sort of obscure automata theoretic point with my computer science hat on, let me just point out that not only can this representation be made entirely respectable mathematically, but via a slicker mathematical name than any other Ive seen proposed in this thread so far: the free monoid (N,+,0) on one generator. Just as punchy, and as clear to a category theorist as free monoid is to an algebraist: the coequalizer of the elements of (the ordinal) *2. The set-theoretic explanation of this is a bit clumsy, but here goes. The scene is Cat, the category of all small categories. The diagram to be coequalized is the left half of 0 ----> F *1 *2 ----> Coeq(0,1) ----> 1 with the coequalizer Coeq(0,1) and its coequalizing arrow F shown on the right. The ordinal *2 is the one-nonidentity-arrow category {0->1}, the ordinal *1 is the evident {0}, the elements 0,1 form the set Hom(*1,*2). The coequalizer of 0 and 1 creates Coeq(0,1), a copy of *2 which identifies 0 and 1. This has the side effect of looping the nonidentity arrow back on itself. Since we are in Cat, we now have to specify a composition law for this arrow with itself in the least constraining way, i.e. Coeq(0,1) has to be universal. Clearly we need all composites f, ff, fff, etc. Identifying any two of these is an unwanted constraint, so we leave them all unidentified. We now have a monoid whose arrows 1,f,ff,fff,... represent the natural numbers 0,1,2,3,..., composition represents addition, and the identity arrow represents 0. (Represent is meaningful only in this set-theoretic view.) The functor F takes both objects of *2 to the object of Coeq(0,1), and takes the nonidentity arrow of *2 to f or 1, the generator of (N,+,0). This construction may seem a bit contrived until you look at how category theoretic foundations are typically organized. (Good reading: McLarty, Axiomatizing a Category of Categories, J.Symbolic Logic, 56:4(Dec91).) Ordinal constructions involving the four ordinals up to *3, along with the product *2 x *2, are at the heart of this organization, and the above construction of the natural numbers as a monoid in Cat is not only slick but very natural and in that setting. end{quote} Can we construct in a similar manner integers, rational, reals and complex numbers? David, === Subject: Re: Schlomilchs series Epigone-thread: yendwholgrimp Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi--Im a newbie to special functions, I hope somebody can help me out. Im doing a problem in scattering, and it would be really helpful if I could sum the following series: sum_{n=1}^inf J_0(n x + b) where b is a real number. If b=0 then this is a bread & butter Schlomilch series (see e.g. Gradshteyn and Ryzhik 5th ed. 8.521). I tried expanding J_0(n x + b) via the addition theorem. But I didnt get too far. I was wondering if this was a known series...hoping the experts have some help! :) === Subject: Re: Schlomilchs series Epigone-thread: yendwholgrimp Content-Length: 782 Originator: rusin@vesuvius There is a sort of obvious way to do it if b is an integer...take your original sum, replace x by 2x and then subtract the two infinite sums, etc. For arbitrary b Im really not sure. >Hi--Im a newbie to special functions, I hope somebody can help me >out. Im doing a problem in scattering, and it would be really >helpful if I could sum the following series: > sum_{n=1}^inf J_0(n x + b) >where b is a real number. >If b=0 then this is a bread & butter Schlomilch series (see e.g. >Gradshteyn and Ryzhik 5th ed. 8.521). >I tried expanding J_0(n x + b) via the addition theorem. But I didnt >get too far. >I was wondering if this was a known series...hoping the experts have >some help! :) === Subject: Re: Schlomilchs series Content-Length: 1302 Originator: rusin@vesuvius Please dont top-post. Im putting the original question here where it belongs: >>Hi--Im a newbie to special functions, I hope somebody can help me >>out. Im doing a problem in scattering, and it would be really >>helpful if I could sum the following series: >> sum_{n=1}^inf J_0(n x + b) >There is a sort of obvious way to do it if b is an integer...take >your original sum, replace x by 2x and then subtract the two infinite >sums, etc. Sorry, I dont understand this. You seem to be saying take the original sum, J_0(x+b) + J_0(2x+b) + J_0(3x+b) + ... and subtract J_0(2x+b) + J_0(4x+b) + J_0(6x+b) + .... But that will just give you J_0(x+b) + J_0(3x+b) + J_0(5x+b) + ... and I dont see how that helps, or what difference b being an integer makes. On the other hand, if b is an integer multiple of x, you can say something: if F(x,b) is the sum, F(x,b+x) = F(x,b) - J_0(x+b) so F(x, kx) = F(x,0) - sum_{j=1}^k J_0(jx) for positive integers k Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: On a property of Bernoulli numbers Epigone-thread: clephoystald Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Let A(n,k)=(n^(2k)-1)*B(2k) where B(2k) denotes the 2k-th Bernoulli >number. Then I suspect the existence of a minimal positive rational >value, depending on n, r(n)=P(n)/Q(n) with (P(n),Q(n))=1 and such >that : >for any k>0 r(n)*A(n,k) is an integer value. >r(2)=2, r(3)=3/4 .... >P(n) appears to be the largest square-free divisor of n but I didnt >observation when n is a power of 2 : >for p prime, if 2^p-1 and (2^p+1)/3 are both primes then >Q(2^p)=(4^p-1)/3 (converse doesnt hold). >Can anyone confirm theorically the existence of r(n) and the formula >for P(n)? If so, what is the formula for Q(n)? Update : I found that r(n)=rad(n^3-n)/(n^2-1) where rad(n) is the square-free kernel of n, the largest square-free divisor of n. This explains why P(n)=rad(n) and we have Q(n)=(n^2-1)/rad(n^2-1). I cant say if this property of Bernoullis numbers is known. Studying r(n) I came across something looking as an integer formulation of Agohs conjecture : p is prime iff p divides (p^p-p)*B(p-1)-1 and I unearthed this amusing connection with 3-smooth numbers (numbers of form 2^i*3^j i,j>=0) : fractional part of ((n^(2k)-1)*B(2k)) is constant for any k>0 iff n is a 3-smooth number. Benoit Cloitre === Subject: Paper published by Algebraic and Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-42.abs.html Title: A class of tight contact structures on Sigma_2 x I Author(s): Tanya Cofer Abstract: We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case. Secondary: 53C15 Keywords: Tight, contact structure, genus-2 surface Author(s) address(es): Department of Mathematics, Northeastern Illinois University 5500 North St Louis Avenue, Chicago, IL 60625-4699, USA Email: T-Cofer@neiu.edu URL: http://www.neiu.edu/~tcofer/ === Subject: This week in the mathematics arXiv (18 Oct - 22 Oct) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Here are this weeks 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 (18 Oct - 22 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410375 Kiran S. Kedlaya: Finite automata and algebraic extensions of function fields math.AC/0410340 Claudia Polini, Bernd Ulrich: A formula for the core of an ideal AG: Algebraic Geometry ---------------------- math.AG/0410469 Frederic Campana: Fibres multiples des surfaces math.AG/0410458 Samuel Boissiere: Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane math.AG/0410444 S. Kaplan, E. Liberman, M. Teicher: Braid Monodromy Computation of Real Singular Curves math.AG/0410442 Nickolas Michelacakis, Apostolos Thoma: On the geometry of complete intersection toric varieties math.AG/0410432 Qi Zhang: On projective varieties with nef anticanonical divisors hep-th/0410055 Volker Braun, Burt A. Ovrut, Tony Pantev, Rene Reinbacher: Elliptic Calabi-Yau Threefolds with Z_3 x Z_3 Wilson Lines math.AG/0410408 Ivan Cheltsov: Double cubics and double quartics hep-th/0410018 A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger: Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections math.AG/0410401 Gabor Szekelyhidi: Extremal metrics and K-stability math.AG/0410394 Ana Cristina Lopez: Relative Jacobians of elliptic fibrations with reducible fibers math.AG/0410393 Ana Cristina Lopez: Simpson Jacobians of reducible curves math.AG/0410392 variety math.AG/0410388 M. E. Kazaryan, S. K. Lando: Towards the Intersection Theory on Hurwitz Spaces math.AG/0410383 Philibert Nang, Kiyoshi Takeuchi: Addendum to the paper Characteristic Cycles of Perverse Sheaves and Milnor Fibers math.AG/0410379 Seongchun Kwon: Transversality properties on the moduli space of genus 0 stable maps to a smooth rational projective surface and their real enumerative implications math.AG/0410378 Silvano Baggio: Equivariant K-Theory of Smooth Toric Varieties math.AG/0410360 Tyler J. Jarvis, William E. Lang, Nansen Petrosyan, Gretchen Rimmasch, Julie Rogers, Erin D. Summers: Classification of Singular Fibres on Rational Elliptic Surfaces in Characteristic Three hep-th/0410170 Bjorn Andreas, Daniel Hernandez Ruiperez: U(n) Vector Bundles on Calabi-Yau Threefolds for String Theory Compactifications math.AG/0410349 Igor Burban, Bernd Kreussler: On a relative Fourier-Mukai transform on genus one fibrations math.AG/0410346 Oliver Lorscheid: Completeness and compactness for varieties over local fields AP: Analysis of PDEs -------------------- math.AP/0410475 Zhongwei Shen: Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators math.AP/0410462 Fernando Cardoso, Georgi Vodev: Weighted L^p decay estimates of solutions to the wave equation with a potential math.AP/0410452 A.G.Ramm: Existence of a solution to a nonlinear equation math.AP/0410451 A.G.Ramm: A singular perturbation problem math.AP/0410443 Arnaud Debussche, Cyril Odasso: Ergodicity for the weakly damped stochastic non-linear Schrodinger equations math.AP/0410441 Giuseppe Da Prato, Arnaud Debussche, Luciano Tubaro: Coupling for some partial differential equations driven by white noise math.AP/0410431 Burak Erdogan, Wilhelm Schlag: Dispersive estimates for Schr{o}dinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I math.AP/0410416 Dian Palagachev, Lubomira Softova: Fine regularity for elliptic systems with discontinuous ingredients math.AP/0410415 Dian K. Palagachev, Lubomira G. Softova: Apriori estimates and precise regularity for parabolic systems with discontinuous data math.AP/0410380 Fabian Waleffe: On some dyadic models of the Euler equations math.AP/0410344 Isabelle Gallagher, Thierry Gallay, Pierre-Louis Lions: On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity AT: Algebraic Topology ---------------------- math.AT/0410405 Scott O. Wilson: Partial Algebras Over Operads of Complexes and Applications math.AT/0410398 R. Brown, H.K. Kamps, T. Porter: A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem math.AT/0410374 Norio Iwase, Donald Stanley, Jeffrey Strom: Implications of the Ganea Condition math.AT/0410367 Z. Fiedorowicz, R. M. Vogt: Topological Hochschild Homology of $E_n$-Ring Spectra math.AT/0410363 A.D.R. Choudary, A. Dimca, S. Papadima: Some Analogs of Zariski Theorem on Nodal Line Arrangements math.AT/0410342 Nicholas J. Kuhn: Goodwillie towers and chromatic homotopy: an overview CA: Classical Analysis and ODEs ------------------------------- math.CA/0410439 Jose L. Lopez, Nico M. Temme: Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials math.CA/0410436 Jose L. Lopez, Nico M. Temme: Multi-point Taylor Expansions of Analytic Functions math.CA/0410395 Sever Silvestru Dragomir: Some Inequalities for Functions of Bounded Variation with Applications to Landau Type Results CO: Combinatorics ----------------- math.CO/0410471 Michiel Hazewinkel: Word Hopf algebras math.CO/0410466 Charles F. Dunkl: Hook-lengths and Pairs of Compositions math.CO/0410455 David E Speyer: Tropical Linear Spaces math.CO/0410429 Jens Christian Claussen: Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration math.CO/0410425 Joseph E. Bonin, Omer Gimenez: Multi-Path Matroids math.CO/0410410 Annalies Vuong, M. Ian Wyckoff: Conditions for Weighted Cover Pebbling of Graphs math.CO/0410404 F. Bonetto, H. Matzinger: Fluctuations of the Longest Common Subsequence in the Asymmetric Case of 2- and 3-Letter Alphabets math.CO/0410382 B.M. Kim, Y. Rho: Van der Waerdens Theorem on Homothetic copies of {1,1+s, 1+s+t} math.CO/0410373 Ira M. Gessel, Louis H. Kalikow: Hypergraphs and a functional equation of Bouwkamp and de Bruijn math.CO/0410366 Michiel Hazewinkel: Explicit polynomial generators for the ring of quasi-symmetric functions over the integers math.CO/0410361 Howard Kleiman: The Floyd-WarshallAlgorithm and the Asymmetric TSP nlin.AO/0407024 Fatihcan M. Atay, Tuerker Biyikoglu, Juergen Jost: On the synchronization of networks with prescribed degree distributions math.CO/0410347 Svante Linusson, Johan Waestlund: Completing a k-1 assignment math.CO/0410345 Svante Linusson, John Shareshian, Volkmar Welker: Complexes of graphs with bounded matching size CT: Category Theory ------------------- math.CT/0410412 Dominic Verity: Complicial Sets CV: Complex Variables --------------------- math.CV/0410445 P. Ebenfelt, L. P. Rothschild: Transversality of CR mappings math.CV/0410420 Rostyslav O. Hryniv, Yaroslav V. Mykytyuk: Asymptotics of zeros for some entire functions math.CV/0410399 Vladimir V. Kisil, Debapriya Biswas: Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains math.CV/0410390 Franc Forstneric, Joerg Winkelmann: Holomorphic discs with dense images math.CV/0410386 Franc Forstneric, Christine Laurent-Thiebaut: Stein compacts in Levi-ßat hypersurfaces math.CV/0410376 Bertrand Deroin: Laminations dans les esapces projectifs complexes math.CV/0410362 Young-Heon Kim: Holomorphic extensions of determinants of Laplacians math.CV/0410353 J. J. Kohn: Superlogarithmic estimates on pseudoconvex domains and CR manifolds math.CV/0410343 Mikhail Sodin: Zeroes of Gaussian analytic functions math.CV/0410341 Fedor Nazarov, Mikhail Sodin: Coarse equidistribution of the argument of entire functions of finite order DG: Differential Geometry ------------------------- math.DG/0410461 Josef Janyv{s}ka: Natural connections given by general linear and classical connections math.DG/0410460 Tom Mestdag, Bavo Langerock: A Lie algebroid framework for non-holonomic systems math.DG/0410456 Mikhail G. Katz, Yuli B. Rudyak: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds nlin.SI/0407057 Paolo Lorenzoni, Marco Pedroni: On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations math.DG/0410435 Isabel Fernandez, Francisco J. Lopez: Relative parabolicity of zero mean curvature surfaces in $R^3$ and $R_1^3$ math.DG/0410434 Michael Schulze: On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry math.DG/0410418 Jian Song, Ben Weinkove: On the convergence and singularities of the J-ßow with applications to the Mabuchi energy math.DG/0410413 2-surfaces of prescribed mean curvature DS: Dynamical Systems --------------------- math.DS/0410464 I.Dynnikov, S.Novikov: Topology of quasiperiodic functions on the plane physics/0410160 R. Ball: The case of the trapped singularities math.DS/0410417 Charles Favre, Mattias Jonsson: Eigenvaluations math.DS/0410384 N. Haydn, Y. Lacroix & S. Vaienti: Hitting and return times in ergodic dynamical systems math.DS/0410355 Marco Lenci: Typicality of recurrence for Lorentz gases nlin.CD/0410019 Sylvie Oliffson Kamphorst, Sonia Pinto de Carvalho: The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on Billiards FA: Functional Analysis ----------------------- math.FA/0410427 W. B. Johnson, N. L. Randrianarivony: $ell_p$ (p>2) does not coarsely embed into a Hilbert space math.FA/0410422 Assaf Naor, Yuval Peres, Oded Schramm, Scott Sheffield: Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces math.FA/0410403 Stefan Bildea, Dorin Ervin Dutkay, Gabriel Picioroaga: MRA Super-wavelets math.FA/0410391 Roberto Giambo, Fabio Giannoni, Paolo Piccione: Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds math.FA/0410351 Frederic Bayart, Catherine Finet, Daniel Li, Herve Queffelec: Composition operators on the Wiener-Dirichlet algebra math.FA/0410348 Wojciech Czaja: Remarks on Naimarks duality GM: General Mathematics ----------------------- math.GM/0410377 Jacky Cresson: Non-differentiable variational principles GT: Geometric Topology ---------------------- math.GT/0410476 J.-F. Lafont: Strong Jordan separation and applications to rigidity math.GT/0410474 Brent Everitt, John Ratcliffe, Steven Tschantz: The smallest hyperbolic 6-manifolds math.GT/0410433 Carlo Petronio: Complexity of 3-orbifolds math.GT/0410381 Suhyoung Choi: Drilling cores of hyperbolic 3-manifolds to prove tameness math.GT/0410370 Jerzy Dydak, Michael Levin: Extension of maps to the projective plane math.GT/0410369 Michael Levin: Rational acyclic resolutions math.GT/0410368 Michael Levin: Universal acyclic resolutions for arbitrary coefficient groups math.GT/0410358 Robion Kirby, Paul Melvin: Local surgery formulas for quantum invariants and the Arf invariant math.GT/0410356 Eaman Eftekhary: Filtration of Heegaard Floer homology and gluing formulas HO: History and Overview ------------------------ math.HO/0410411 Tommaso Toffoli: Maxwells daemon, the Turing machine, and Jaynes robot math.HO/0410397 V.G.Gurzadyan: Kolmogorov and Aleksandrov in Sevan Monastery, Armenia, 1929 MG: Metric Geometry ------------------- math.MG/0410440 Andreas Balser, Alexander Lytchak: Centers of convex subsets of buildings math.MG/0410437 Andreas Balser, Alexander Lytchak: Building-like spaces math.MG/0410421 Alexander Lytschak, Viktor Schroeder: Affine functions on CAT(kappa) spaces MP: Mathematical Physics ------------------------ quant-ph/0410131 Xiong-Jun Liu, Hui Jing, Xin Liu, Mo-Lin Ge: Dynamical Symmetry and Its Applications In Electromagnetically Induced Transparency math-ph/0410046 Michiel Hazewinkel, Hugo H Torriani: Coherence and uniqueness theorems for averaging processes in statistical mechanics hep-th/0410199 A.P. Balachandran, A. Pinzul: On Time-Space Noncommutativity for Transition Processes and Noncommutative Symmetries hep-th/0008117 M. Hssaini, M. Kessabi, B. Maroufi, M.B.Sedra: Central extended D=2 N=4 SU(2) Liouville self interacting model and explicit hyperkahler metric math-ph/0410045 Alexei F. Cheviakov: Plasma equilibrium equations in coordinates connected with magnetic surfaces. Exact equilibrium solutions gr-qc/0410069 Antonio Lopez-Pinto: Nonstandard spin 2 field theory physics/0410127 A. Figotin, J. H. Schenker: Hamiltonian treatment of time dispersive and dissipative media within the linear response theory math-ph/0410044 Daniel Peralta-Salas: A geometric approach to the equilibrium shapes of self-gravitating ßuids hep-th/0410013 Patrick Dorey, Adam Millican-Slater, Roberto Tateo: Beyond the WKB approximation in PT-symmetric quantum mechanics cond-mat/0410435 F. Guerra: Mathematical aspects of mean field spin glass theory math-ph/0410043 Volodymyr Sushch: On some discrete model of the magnetic Laplacian math-ph/0410042 Jochen Bruening, Vladimir Geyler, Konstantin Pankrashkin: Continuity of integral kernels related to Schrodinger operators on manifolds math-ph/0410041 O.M. Kiselev, S.G. Glebov, V.A. Lazarev: Resonant pumping in nonlinear Klein-Gordon equation and solitary packets of waves math-ph/0410040 G.Giachetta, L.Mangiarotti, G.Sardanashvily: Geometric and Algebraic Topological Methods in Quantum Mechanics hep-th/0408241 S. Meljanac, A. Samsarov: Matrix oscillator and Calogero-type models math-ph/0410039 Nasser Saad, Richard L. Hall, Qutaibeh D. Katatbeh: Study of anharmonic singular potentials NT: Number Theory ----------------- math.NT/0410428 L.A.Gutnik: On the difference equation of the Poincare type math.NT/0410409 A. Agboola: Galois modules and p-adic representations math.NT/0410387 C. S. Rajan: Recovering modular forms and representations from tensor and symmetric powers math.NT/0410372 Mark van Hoeij: Solving conics over Q(t1,..,tk) OA: Operator Algebras --------------------- math.OA/0410449 Kenneth Davidson, Elias Katsoulis: Nest representations of directed graph algebras math.OA/0410426 Benjam{i}n Itza-Ortiz: Eigenvalues, K-theory and Minimal Flows math.OA/0410400 Marius Dadarlat: On the topology of the Kasparov groups and its applications OC: Optimization and Control ---------------------------- math.OC/0410467 Antonios Armaou, Ioannis G. Kevrekidis: Equation-free optimal switching policies for bistable reacting systems using coarse time-steppers PR: Probability --------------- math.PR/0410465 Federico Camia: Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation math.PR/0410459 Florent Benaych-Georges: Taylor expansions of R-transforms, application to supports and moments math.PR/0410457 Catherine Donati-Martin: Large deviations for Wishart processes math.PR/0410453 Patrick Cheridito, Freddy Delbaen, Michael Kupper: Dynamic monetary risk measures for bounded discrete-time processes math.PR/0410447 Michail Loulakis: On the Symmetry of the Diffusion Coefficient in Asymmetric Simple Exclusion math.PR/0410430 Yuval Peres, David Revelle: Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs math.PR/0410414 Robert C. Dalang, Carl Mueller, Lorenzo Zambotti: Hitting properties of s.p.d.e.s with reßection math.PR/0410402 David J. Aldous, Lea Popovic: A critical branching process model for biodiversity math.PR/0410371 Harry Kesten, Vladas Sidoravicius: A phase transition in a model for the spread of an infection math.PR/0410359 Bela Bollobas, Oliver Riordan: A short proof of the Harris-Kesten Theorem cond-mat/0410309 V. Sood, S. Redner, D. ben-Avraham: First Passage Properties of the Erdos-Renyi Random Graph QA: Quantum Algebra ------------------- math.QA/0410470 Michiel Hazewinkel: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions II math.QA/0410468 Michiel Hazewinkel: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions math.QA/0410463 S. Sinelshchikov, A. Stolin, L. Vaksman: A Quantum Analogue of the Bernstein Functor math.QA/0410450 J.E. McClure: On the chain-level intersection pairing for PL manifolds math.QA/0410448 Christian Blohmann: Reconstruction of universal Drinfeld twists from representations math.QA/0410446 Haisheng Li, Gaywalee Yamskulna: On certain vertex algebras and their modules associated with vertex algebroids math.QA/0410407 K. Szlachanyi: Monoidal Morita equivalence math.QA/0410396 S. Sinelshchikov, L. Vaksman: Quantum groups and non-commutative complex analysis math.QA/0410389 Harald Grosse, Stefan Schraml: The Eigenfunctions of the q-Harmonic Oscillator on the Quantum Line math.QA/0410365 Michiel Hazewinkel: The primitives of the Hopf algebra of noncommutative symmetric functions math.QA/0410364 Michiel Hazewinkel: Hopf algebras of endomorphisms of Hopf algebras math.QA/0410350 Henrique Bursztyn, Stefan Waldmann: Hermitian star products are completely positive deformations RA: Rings and Algebras ---------------------- math.RA/0410473 Gizem Karaali: A New Lie Bialgebra Structure on sl(2,1) math.RA/0410406 Michael Pinsker: The number of unary clones containing the permutations on an infinite set RT: Representation Theory ------------------------- math.RT/0410472 Paolo Bravi, Guido Pezzini: Wonderful varieties of type D math.RT/0410454 Francois Digne, Jean Michel, Raphael Rouquier: Cohomologie des varietes de Deligne-Lusztig math.RT/0410423 Calin Chindris: Quivers, long exact sequences and Horn type inequalities math.RT/0410357 Helmer Aslaksen, Mong Lung Lang: Extending $pi$-systems to bases of root systems SG: Symplectic Geometry ----------------------- math.SG/0410352 Eaman Eftekhary: Embedded curves and Gromov-Witten invariants of three-folds SP: Spectral Theory ------------------- math.SP/0410438 M.A. Kaashoek, A.L. Sakhnovich: Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model ST: Statistics -------------- math.ST/0410424 George Kahrimanis, Daniel Berleant: Direct pivotal predictive inference math.ST/0410419 Grace Wahba: An introduction to (smoothing spline) ANOVA models in RKHS with examples in geographical data, medicine, atmospheric science and machine learning math.ST/0410385 Mario Ruetti, Matthias Troyer, Wesley P. Petersen: A Generic Random Number Generator Test Suite math.ST/0410354 Stephane Gaiffas: Rates of convergence for pointwise curve estimation with a degenerate design -- / 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 thats fit to e-print * === Subject: Question about Gomory-Cut Originator: israel@math.ubc.ca (Robert Israel) for an application I have to find an integer solution. I use the simplex-algorithm and the Gomory-cut. It works fine, but it is to slowly. There are multiply opportunities to make a Gomory-cut. Which Gomory-cuts should I take, to get a solution in shortest time ? Where can I get information about that (in the internet) ? Ulrich === Subject: Combinatorial inequality Epigone-thread: wunloifo Originator: israel@math.ubc.ca (Robert Israel) Can anyone help me in deriving this inequality which appears to be true? {C(n,k)/C(a,k)}*(m/n)^k <={((n-m)/(n-a))^(n-a)}*{(n/a)^a} where k=(a-m)/(1-m/n); and 0Can anyone help me in deriving this inequality which appears to be >true? > {C(n,k)/C(a,k)}*(m/n)^k <={((n-m)/(n-a))^(n-a)}*{(n/a)^a} > where k=(a-m)/(1-m/n); and 1 Here C(n,k) means the binomial coefficient and ^ means >exponentiation,* denotes ordinary multiplication. === Subject: Conformal Mapping Question Originator: israel@math.ubc.ca (Robert Israel) Id like to find the explicit formula of a bijective conformal mapping from the following region, say K, K={ x+iy in C mid x>0, y > arccos(e^{-x}) } to the interior of the unit disk (K is just the subregion in the first quadrant of the complex plane bounded below by the graph of e^{x}cos(y)=1). Ive tried basic ones and looked into Dictionary of Conformal Representations by H. Kober, but so far nothing works for me. Any suggestions are greatly appreciated. -- So Okada Ph.D. Student of Math at UMass Amherst okada@math.umass.edu === Subject: This week in the mathematics arXiv (25 Oct - 29 Oct) Originator: israel@math.ubc.ca (Robert Israel) Here are this weeks 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 Oct - 29 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410598 Kamran Divaani-Aazar, Amir Mafi: Associated primes of local cohomology module math.AC/0410585 Nicholas Baeth: A Krull-Schmidt Theorem for One-dimensional Rings of Finite Cohen-Macaulay Type math.AC/0410535 Anurag K. Singh, Uli Walther: On the arithmetic rank of certain Segre products math.AC/0410497 Juan C. Migliore, Uwe Nagel, Tim Romer: The Multiplicity Conjecture in low codimensions math.AC/0410478 Carlos DAndrea, Laurent Buse: Properness and inversion problems by means of matrices AG: Algebraic Geometry ---------------------- math.AG/0410604 Elizabeth S. Allman, John A. Rhodes: Phylogenetic ideals and varieties for the general Markov model math.AG/0410602 E. Carlini: Codimension one decompositions and Chow varieties math.AG/0410600 J. C. Sierra, L. Ugaglia: On double Veronese embeddings in the Grassmannian G(1,N) math.AG/0410584 Carolina Araujo: Rational curves of minimal degree and characterizations of ${mathbb P}^n$ math.AG/0410572 Israel Moreno Mej{i}a: The trace of an automorphism on H^0(J,O(nTheta)) math.AG/0410558 Ivan Cheltsov: Birationally superrigid cyclic triple spaces math.AG/0410554 Meirav Amram, David Goldberg: Higher degree Galois covers of CP^1 x T math.AG/0410547 D. A. Stepanov: Non-rational divisors over non-Gorenstein terminal singularities math.AG/0410540 Pan Peng: A simple proof of Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds math.AG/0410537 Eduardo Esteves, Steven Kleiman: The compactified Picard scheme of the compactified Jacobian math.AG/0410527 Cristiano Bocci: Special effect varieties and (-1)-curves math.AG/0410526 Shihoko Ishii: Arcs, valuations and the Nash map math.AG/0410524 Boris E. Kunyavskii, Louis H. Rowen, Sergey V. Tikhonov, Vyacheslav I. Yanchevskii: Division algebras that ramify only on a plane quartic curve math.AG/0410520 Laurent Manivel, Emilia Mezzetti: On linear spaces of skew-symmetric matrices of constant rank math.AG/0410518 Elena Drozd: Curves on a nonsingular Del Pezzo Surface in $P^4_k$ math.AG/0410513 Kalle Karu: The cd-index of fans and lattices AP: Analysis of PDEs -------------------- math.AP/0410581 G. Olafsson, A. Pasquale: Support properties and Holmgrens uniqueness theorem for differential operators with hyperplane singularities math.AP/0410564 James Nolen, Jack Xin: A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears math.AP/0410546 Plamen Stefanov, Gunther Uhlmann: Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map math.AP/0410538 J. Colliander, W. Staubach: $L^2$ blowup solutions of cubic NLS on $R^2$ concentrate a fixed amount of mass math.AP/0410525 Gianni Dal Maso, Rodica Toader: On a notion of unilateral slope for the Mumford-Shah functional math.AP/0410499 Hans Lindblad, Jacob Sterbenz: Global Stability for Charge Scalar Fields on Minkowski Space AT: Algebraic Topology ---------------------- math.AT/0410589 Nora Ganter: Smash products of E(1)-local spectra at an odd prime math.AT/0410552 Javier Turiel: Polynomials Maps and Even Dimensional Spheres math.AT/0410503 Kathryn Hess, Ran Levi: An algebraic model for the loop homology of a homotopy fiber CA: Classical Analysis and ODEs ------------------------------- math.CA/0410548 Marc Artzrouni: A new family of periodic functions as explicit roots of a class of polynomial equations math.CA/0410542 Projections And Universal Encoding Strategies math.CA/0410508 Stephen Semmes: Potpourri, 8 math.CA/0410490 Stephen Semmes: Potpourri, 7 math.CA/0410489 Stephen Semmes: Potpourri, 6 math.CA/0410483 A. A. Bolibruch, S. Malek, C. Mitschi: On the generalized Riemann-Hilbert problem with irregular singularities CO: Combinatorics ----------------- math.CO/0410592 S. Ole Warnaar: Hall--Littlewood functions and the A_2 Rogers--Ramanujan identities quant-ph/0410226 P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon: Deformed Bosons: Combinatorics of Normal Ordering math.CO/0410550 Ewa Krot: Further develpoements in finite fibonomial calculus math.CO/0410529 math.CO/0410482 Michael Anshelevich: Orthogonal polynomials with a resolvent-type generating function CT: Category Theory ------------------- math.CT/0410555 Alan Robinson: Partition complexes, duality and integral tree representations gr-qc/0410104 J. Daniel Christensen, Louis Crane: Causal sites as quantum geometry CV: Complex Variables --------------------- math.CV/0410599 Laurent Gendre: Inegalites de Markov tangentielles locales sur les courbes algebriques singulieres de R^n math.CV/0410578 Dmitri Prokhorov, Alexander Vasilev: Optimal control in Bombieris and Tammis conjectures math.CV/0410509 David E. Barrett: A ßoating body approach to Feffermans hypersurface measure DG: Differential Geometry ------------------------- math.DG/0410610 Francisco Martin Cabrera: SU(3)-structures on hypersurfaces of manifolds with $G_2$-structure hep-th/0410183 Anton Alekseev, Thomas Strobl: Current Algebras and Differential Geometry math.DG/0410579 Jorge Lauret: A canonical compatible metric for geometric structures on nilmanifolds math.DG/0410575 Joseph H.G. Fu: Structure of the unitary valuation algebra math.DG/0410561 math.DG/0410559 Tomasz S. Mrowka, Yann Rollin: Legendrian knots and monopoles math.DG/0410557 A. V. Kiselev, G. Manno: On the symmetry structure of the minimal surface equation math.DG/0410553 Anton Deitmar: A prime geodesic theorem for higher rank II: singular geodesics math.DG/0410551 Eduardo Martinez: Classical field theory on Lie algebroids: Variational aspects math.DG/0410512 Maks A. Akivis, Vladislav V. Goldberg, Arto V. Chakmazyan: Induced connections on submanifolds in spaces with fundamental groups math.DG/0410511 Maks A. Akivis, Vladislav V. Goldberg: Dually degenerate varieties and the generalization of a theorem of Griffiths--Harris math.DG/0410498 Boris S. Kruglikov, Vladimir S. Matveev: Strictly non-proportional geodesically equivalent metrics have $h_text{top}(g)=0$ math.DG/0410494 George Papadopoulos: Spin Cohomology math.DG/0410493 Abdenago Barros G. Pacelli Bessa: Estimates of the first eigenvalue of minimal hypersurfaces of $mathbb{S}^{n+1} math.DG/0410487 Hiroshi Iritani: Quantum D-modules and equivariant Floer theory for free loop spaces math.DG/0410484 Aleksis Raza: An application of Guillemin-Abreu theory to a non-abelian group action DS: Dynamical Systems --------------------- math.DS/0410580 I. Binder, M. Braverman, M. Yampolsky: Filled Julia sets with empty interior are computable math.DS/0410517 Le Van Hien: Stability of Solutions of Fuzzy Differential Equations math.DS/0410507 Topologies on the group of homeomorphisms of a Cantor set math.DS/0410506 Topologies on the group of Borel automorphisms of a standard Borel space math.DS/0410505 Sergey Bezuglyi, Anthoni H. Dooley, Konstantin Medynets: The Rokhlin lemma for homeomorphisms of a Cantor set math.DS/0410504 Sergey Bezuglyi, Konstantin Medynets: Smooth automorphisms and path-connectedness in Borel dynamics math.DS/0410500 Ara Basmajian, Mahmoud Zeinalian: Maximal Convergence Groups and Rank One Symmetric Spaces math.DS/0410481 Pavlos B. Konstadinidis: The Real 3x+1 Problem FA: Functional Analysis ----------------------- math.FA/0410596 Ralf Meyer: Embeddings of derived categories of bornological modules math.FA/0410573 Jorge Antezana, Gustavo Corach, Demetrio Stojanoff: Spectral shorted operators math.FA/0410571 Massimo Fornasier, Holger Rauhut: Continuous Frames, Function Spaces, and the Discretization Problem math.FA/0410567 A. Brudnyi: Contractibility of Maximal Ideal Spaces of Certain Algebras of Almost Periodic Functions math.FA/0410549 Massimo Fornasier: Banach frames for alpha-modulation spaces math.FA/0410501 V.Yaskin: The Busemann-Petty problem in hyperbolic and spherical spaces math.FA/0410496 A.Koldobsky, V.Yaskin, M.Yaskina: Modified Busemann-Petty problem on sections of convex bodies math.FA/0410491 T. Banks, T. Constantinescu, Nermine El-Sissi: Tensor algebras and displacement structure. IV. Invariant kernels math.FA/0410479 A.G.Ramm: Dynamical systems method (DSM) for nonlinear equations in Banach spaces GM: General Mathematics ----------------------- math.GM/0410556 Joao R. Cardoso: An Explicit Formula for the Matrix Logarithm GR: Group Theory ---------------- math.GR/0410593 Henrik Baarnhielm: The Schreier-Sims algorithm for matrix groups math.GR/0410590 Edith Adan-Bante: Products of characters with few irreducible constituents math.GR/0410583 Edith Adan-Bante: Products of characters and derived length II math.GR/0410582 Edith Adan-Bante: Squares of characters and groups of odd order math.GR/0410539 Daniel Farley, Lucas Sabalka: Discrete Morse theory and graph braid groups math.GR/0410533 Stephen DeBacker: Parametrizing nilpotent orbits via Bruhat-Tits theory math.GR/0410516 J.Mostovoy, J.M. Perez-Izquierdo: Dimension filtration on loops math.GR/0410515 Jacob Mostovoy: On the notion of lower central series for loops GT: Geometric Topology ---------------------- math.GT/0410606 Greg Friedman: Knot spinning math.GT/0410603 R. C. Penner, Dennis Sullivan: The Structure and Singularities of Arc Complexes math.GT/0410595 Pascal Hubert, Samuel Lelievre: Noncongruence subgroups in H(2) math.GT/0410570 Andras Nemethi: On the Heegaard Floer homology of S^3_{-p/q}(K) math.GT/0410565 Brooke Brennan, Thomas W. Mattman, Roberto Raya, Dan Tating: Ribbonlength of torus knots math.GT/0410541 Ensil Kang, J. Hyam Rubinstein: Ideal triangulations of 3--manifolds I: spun normal surface theory math.GT/0410495 Dror Bar-Natan: Khovanovs Homology for Tangles and Cobordisms KT: K-Theory and Homology ------------------------- math.KT/0410597 Ralf Meyer: Combable groups have group cohomology of polynomial growth LO: Logic --------- math.LO/0410523 Fredrik Engstrom: Expansions, omitting types, and standard systems MG: Metric Geometry ------------------- math.MG/0410566 Piotr W. Nowak: On coarse embeddability into $ell_p$-spaces and a conjecture of Dranishnikov MP: Mathematical Physics ------------------------ quant-ph/0410201 Kazuyuki Fujii: Jaynes-Cummings Model and a Non-Commutative Geometry : A Few Problems Noted math-ph/0410062 David Damanik, Daniel Lenz, Gunter Stolz: Lower Transport Bounds for One-Dimensional Continuum Schrodinger Operators math-ph/0410061 P. Di Francesco, P. Zinn-Justin: Razumov-Stroganov sum rule: a proof based on multi-parameter generalizations math-ph/0410060 A.W.Beckwith: How false vacuum synthesis of a universe sets initial conditions which permit the onset of variations of a nucleation rate per Hubble volume per Hubble time math-ph/0410059 Manfred Requardt: Supersymmetry on Graphs and Networks math-ph/0410058 Vitaly V. Bulatov, Yuriy V. Vladimirov, Vasily A. Vakorin: Weak Singularity for Two-Dimensional Nonlinear Equations of Hydrodynamics and Propagation of Shock Waves math-ph/0410057 Joseph V. Pule, Andre F. Verbeure, Valentin A. Zagrebnov: Models with Recoil for Bose-Einstein Condensation and Superradiance math-ph/0410056 Petko Nikolov, Tihomir Valchev: Description of all conformally invariant differential operators, acting on scalar functions astro-ph/0404408 Ing-Guey Jiang, Li-Chin Yeh: On the Chaotic Orbits of Disc-Star-Planet Systems math-ph/0410055 A. A. Hujeirat: Problem-orientable numerical algorithm for modelling multi-dimensional radiative MHD ßows in astrophysics -- the hierarchical solution scenario math-ph/0410054 A.A. Hujeirat: A method for enhancing the stability and robustness of explicit schemes in astrophysical ßuid dynamics hep-th/0410172 Beatriz Gato-Rivera: The Adapted Ordering Method in Representation Theory math-ph/0410053 Alexander Rybko, Senya Shlosman: Poisson Hypothesis for Information Networks II. Cases of Violations and Phase Transitions math-ph/0410052 A.C.D.van Enter, E.A.Verbitskiy: On the Variational Principle for Generalized Gibbs Measures math-ph/0410051 Xavier Gracia, Ruben Martin: Time-dependent singular differential equations math-ph/0410050 Nicolae Cotfas: Systems of orthogonal polynomials defined by hypergeometric type equations math-ph/0410049 Joachim Kupsch, Subhashish Banerjee: Ultracoherence and Canonical Transformations math-ph/0410048 Sebastian Bauer: Post-Newtonian approximation of the Vlasov-Nordstrom system hep-th/0410212 C. Chryssomalakos, E. Okon: Generalized Quantum Relativistic Kinematics: a Stability Point of View cond-mat/0410424 Sandeep Tyagi: New series representation for Madelung constant quant-ph/0410151 S. Twareque Ali, F. Bagarello: Some Physical Appearances of Vector Coherent States and CS Related to Degenerate Hamiltonians math-ph/0410047 Volodymyr Sushch: Discrete model of Yang-Mills equations in Minkowski space hep-th/0410109 S. Odake, R. Sasaki: Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials hep-th/0410102 S. Odake, R. Sasaki: Shape Invariant Potentials in Discrete Quantum Mechanics NA: Numerical Analysis ---------------------- math.NA/0410488 Paul Sablonniere: Recent Results on Near-Best Spline Quasi-Interpolants NT: Number Theory ----------------- math.NT/0410563 Dragos Ghioca: The Mordell-Lang Theorem for Drinfeld modules math.NT/0410536 construction of some Galois modules math.NT/0410531 Takashi Taniguchi: A mean value theorem for the square of class numbers of quadratic fields math.NT/0410522 A. Ivic, E. Kratzel, M. Kuhleitner, W.G. Nowak: Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic math.NT/0410519 Howard Kleiman: Bounds for the Solutions of Cubic Diophantine Equations math.NT/0410502 Avner Ash, David Pollack, Warren Sinnott: A_6-extensions of Q and the mod p cohomology of GL(3,Z) math.NT/0410477 Bernd C. Kellner: Some remarks on Kurepas left factorial OA: Operator Algebras --------------------- math.OA/0410607 P. S. Muhly, M. Skeide, B. 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Shklyarov, S. Sinelshchikov, L. Vaksman: Fock Representations and Quantum Matrices math.QA/0410562 Vasiliy Dolgushev, Pavel Etingof: Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology math.QA/0410530 S. Sinelshchikov, L. Vaksman: Quantum Groups and Bounded Symmetric Domains math.QA/0410528 Michel Van den Bergh: Double Poisson algebras math.QA/0410486 Vladimir D. Lyakhovsky: On a class of skew classical r-matrices with large carrier RA: Rings and Algebras ---------------------- math.RA/0410591 Aaron Lauve: NSym into Q_{infty} is not a Hopf Map math.RA/0410576 Friedrich Wehrung, Jiri Tuma: Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties math.RA/0410521 Jerzy Matczuk: Ore Extensions over Duo Rings RT: Representation Theory ------------------------- math.RT/0410588 Konstantin Styrkas: Regular representation on the big cell and big projective modules in the category O SG: Symplectic Geometry ----------------------- math.SG/0410609 Joa Weber: Noncontractible periodic orbits in cotangent bundles and Floer homology math.SG/0410608 Weimin Chen: Pseudoholomorphic curves in four-orbifolds and some applications math.SG/0410568 Eugene Lerman: Gradient ßow of the norm squared of a moment map SP: Spectral Theory ------------------- math.SP/0410577 Sergio Albeverio, Alexander K. Motovilov: Operator integrals with respect to a spectral measure and solutions to some operator equations ST: Statistics -------------- math.ST/0410586 Rasa Karapandza, Milos Bozovic: You Can Fool Some People Sometimes math.ST/0410574 Igor Podlubny: A note on comparison of scientific impact expressed by number of citations in different fields of science -- / 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 thats fit to e-print * === Subject: 3rd cohomology of loop group Originator: israel@math.ubc.ca (Robert Israel) Id like to know the 3rd cohomology group of the loop group OmegaE_8 over E_8, H^3(OmegaE_8,H) , with H some abelian group. Does anyone know where I could find respective information? === Subject: Re: Open questions related to periodic continued fractions Originator: israel@math.ubc.ca (Robert Israel) Diana Mecum asked: > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? If n is not a perfect square, let p(n) be the length of the period of the s.c.f. of sqrt(n). In terms of n, how large can p(n) be? Its been shown that p(n) = O(sqrt(n) log(n)). It follows from some generalization of the Riemann hypothesis that p(n) = O(sqrt(n) log(log(n))). It seems likely that p(n)/sqrt(n) is unbounded, but I dont think its even been shown that it doesnt tend to 0. The first 23 record-setting values of p(n)/sqrt(n) are shown below: n p(n) p(n)/sqrt(n) 2 1 0.70711 3 2 1.15470 7 4 1.51186 43 10 1.52499 46 12 1.76930 211 26 1.78991 331 34 1.86881 631 48 1.91085 919 60 1.97922 1726 88 2.11818 4846 152 2.18349 7606 194 2.22445 10399 228 2.23583 10651 234 2.26736 10774 238 2.29292 18379 322 2.37517 19231 332 2.39407 32971 438 2.41217 48799 544 2.46260 61051 614 2.48497 78439 696 2.48510 82471 716 2.49323 111094 834 2.50219 See http://www.research.att.com/projects/OEIS?Anum=A003285 for some references. Dean Hickerson dean@math.ucdavis.edu === Subject: Re: Open questions related to periodic continued fractions > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? As you may know, the outstanding problem in this area is to improve on known conditions for the length of the period of the continued fraction expansion of sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having solutions). The following might be a current summary of known conditions: B. D. Beach and H. C. Williams, A Numerical Investigation of the Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Mathematica Publishing Inc., Winnipeg, Canada, 1972, pages 37 to 52. A less well known problem is as follows. As far as I know, this is an open problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo 4, let L1(D) and L4(D) denote the lengths of the periods of the continued fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). As far as I can tell, this is not prohibited by results in the literature. I have empirical evidence that it is not possible based on testing those D up to 30 billion that have L4(D) <= 255. It is not hard to show this for one particular case, namely if L4(D) = 3 then L1(D) cannot be 7. See discussion under the heading ``Periods of Continued Fractions in April and May of 2000 in the archives of the Number Theory Listserver at http://listserv.nodak.edu/archives/nmbrthry.html for related comments and some references that might be of interest. John Robertson === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox4.ucsd.edu: domain of news@newsread1.news.pas.earthlink.net does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? > As you may know, the outstanding problem in this area is to improve on known > conditions for the length of the period of the continued fraction expansion of > sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having > solutions). The following might be a current summary of known conditions: > B. D. Beach and H. C. Williams, A Numerical Investigation of the > Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, > Proceedings of the Third Southeastern Conference on Combinatorics, > Graph Theory and Computing, Utilitas Mathematica Publishing Inc., > Winnipeg, Canada, 1972, pages 37 to 52. > A less well known problem is as follows. As far as I know, this is an open > problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo > 4, let L1(D) and L4(D) denote the lengths of the periods of the continued > fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question > is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). > As far as I can tell, this is not prohibited by results in the literature. I > have empirical evidence that it is not possible based on testing those D up to > 30 billion that have L4(D) <= 255. It is not hard to show this for one > particular case, namely if L4(D) = 3 then L1(D) cannot be 7. > See discussion under the heading ``Periods of Continued Fractions in April > and May of 2000 in the archives of the Number Theory Listserver at > http://listserv.nodak.edu/archives/nmbrthry.html > for related comments and some references that might be of interest. > John Robertson === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Regarding the maximal element in the CF of sqrt(n) (say M(n)) . Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this behaviour : sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 and = 0.8... I dont know references on this subject. Should exist some. B. Cloitre === Subject: Re: Open questions related to periodic continued fractions Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Regarding the maximal element in the CF of sqrt(n) (say M(n)) . > Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this > behaviour : > sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 > and = 0.8... > I dont know references on this subject. Should exist some. Im not sure what ``sqrtint(n) is. The maximum partial quotient in the continued fraction expansion of sqrt(n) (for n not a square) is 2 times the integer part of the square root of n. This is proved in most references that consider the continued fraction expansion of sqrt(n), e.g., Mollins Fundamental Number Theory with Applications, or Niven, Zuckerman, and Montgomery. Also, (1/n) Sum_{i=1}^{n} [sqrt(i) - int(sqrt(i))] tends to 1/2. John Robertson === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Regarding the maximal element in the CF of sqrt(n) (say M(n)) . >Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this >behaviour : >sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 and = >0.8... >I dont know references on this subject. Should exist some. >B. Cloitre May be the following remarks are all well known: observe for instance sqrt(41) giving sequence (6;1/2,1/2,1/12,...) the period (1/2,1/2,1/12 ..)is linked to function (1/2,1/2,1/12,x) or (5x+62)/(2x+25) with two fixed points ,the positive -6+sqrt(41) is related to our continuous fraction. f(x)=(5x+62)/(2x+25) is easily iterated (sci.math 20 oct), Friendly yours,Alain. === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Content-Length: 2687 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Diana, in case it can help. Is anything known about the continued fraction of log_2(3)? (that is, of the logarithm of 3, base 2) Is it bounded,...!?? Maybe there are some results on the continued fractions of logarithms.. I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, for that number. (One more thing, is log_2(3) algebraic or trascendent?..!) Jose >> I am starting research for a thesis on continued fractions, and want >> to look at open questions related to periodic continued fractions. Is >> anyone aware of current open questions of interest? > As you may know, the outstanding problem in this area is to improve on >known >> conditions for the length of the period of the continued fraction >expansion of >> sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 >having >> solutions). The following might be a current summary of known conditions: >> B. D. Beach and H. C. Williams, A Numerical Investigation of the >> Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, >> Proceedings of the Third Southeastern Conference on Combinatorics, >> Graph Theory and Computing, Utilitas Mathematica Publishing Inc., >> Winnipeg, Canada, 1972, pages 37 to 52. >> A less well known problem is as follows. As far as I know, this is an >open >> problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 >modulo >> 4, let L1(D) and L4(D) denote the lengths of the periods of the continued >> fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The >question >> is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod >6). >> As far as I can tell, this is not prohibited by results in the literature. >> have empirical evidence that it is not possible based on testing those D >up to >> 30 billion that have L4(D) <= 255. It is not hard to show this for one >> particular case, namely if L4(D) = 3 then L1(D) cannot be 7. >> See discussion under the heading ``Periods of Continued Fractions in >April >> and May of 2000 in the archives of the Number Theory Listserver at >> http:// listserv.no dak.edu/archives/nmbrthry.html> for related comments and some references that might be of interest. >> John Robertson === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox3.ucsd.edu: domain of news@nntp.itservices.ubc.ca does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Diana, > in case it can help. > Is anything known about the continued fraction of log_2(3)? (that >is, of the logarithm of 3, base 2) > Is it bounded,...!?? > Maybe there are some results on the continued fractions of >logarithms.. I doubt that this will help. Very little is known about the continued fractions of closed-form numbers apart from rationals and quadratic irrationals, and AFAIK there is no prospect, with currently available mathematical techniques, of being able to prove whether the continued fraction of a number such as this has bounded elements. IMHO this is not a problem to give to a student starting her thesis. > I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, >for that number. > (One more thing, is log_2(3) algebraic or trascendent?..!) Transcendental, by the Gelfond-Schneider Theorem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Open questions related to periodic continued fractions Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Diana, > in case it can help. > Is anything known about the continued fraction of log_2(3)? (that > is, of the logarithm of 3, base 2) > Is it bounded,...!?? No, its unbounded. But, so far, nobody knows how to prove that. > Maybe there are some results on the continued fractions of > logarithms.. > I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, > for that number. How does that compare to almost all continued fractions? > (One more thing, is log_2(3) algebraic or trascendent?..!) It is transcendental. -- G. A. Edgar edgar at math.ohio-state.edu === Subject: Re: Galois group of a given quartic equation Epigone-thread: twomprendlex Originator: israel@math.ubc.ca (Robert Israel) The pending issue is whether the quartic Q(x) = 0 may have the Galois group isomorphic to Z4 for a and b different. No progress has been posted. One way to rule out D4 in the Z4/D4 case is given below. (If Q(x) is irreducible over Z, the discriminant is not a square and the cubic resolvent R(t) = 0 of the quartic has one and only one integral root t0, then the Galois group G is either Z4 or D4.) Let R(t) = (t - t0) r(t) where r(t) is a monic irreducible quadratic with integral coefficients. Further, let the discriminant of r(t) be D. Let d be the squarefree part of D and E the splitting field of R(t) = 0. Then E = Q[Sqrt[d]] and therefore easy to determine. Then put E to use as follows: If Q(x) is reducible over E then G is Z4; otherwise G is D4. May be this can be used to settle the case whether G may be Z4 or not. Kent Holing === Subject: matrix derivative Epigone-thread: zoypryrwhou Originator: israel@math.ubc.ca (Robert Israel) I am currently having a matrix derivative problem What is derivative of trace{(A+F*B*F)^(-1)} with respect to matrix F where () is the transpose of a matrix. A and B is diagonal matrices. I searched online and was only able to find the derivative of d trace{(F*B*F)^(-1)}/d F =-2*B*F*(F*B*F)^(-2) without knowing how they get it. Moreover, all these matrix derivative problems seem to be difficult forme. Could anyone be kind enough to give me some good references on this topic. === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) > I am currently having a matrix derivative problem > What is derivative of > trace{(A+F*B*F)^(-1)} > with respect to matrix F > where () is the transpose of a matrix. > A and B is diagonal matrices. > I searched online and was only able to find the derivative of > d trace{(F*B*F)^(-1)}/d F > =-2*B*F*(F*B*F)^(-2) > without knowing how they get it. > Moreover, all these matrix derivative problems seem to be > difficult forme. Could anyone be kind enough to give me > some good references on this topic. d A denotes the matrix differential for the matrix A. x = trace{ (A+F*B*F)^(-1) } d x = trace{ d ( (A+F*B*F)^(-1) ) } = trace{ - (A+F*B*F)^(-1) ( d (A+F*B*F) ) (A+F*B*F)^(-1) } = trace{ - (A+F*B*F)^(-2) ( d (F*B*F) ) } = trace{ - (A+F*B*F)^(-2) ( d F *B*F + F*B* d F ) } = - ( trace{ (A+F*B*F)^(-2) d F *B*F } + trace{ (A+F*B*F)^(-2) F*B* d F } ) = - trace{ B * F * (A+F*B*F)^(-2) d F } - trace{ (A+F*B*F)^(-2) F*B* d F } = - trace{ (A+F*B*F)^(-2) * F * B * d F} - trace{ (A+F*B*F)^(-2) F*B* d F } (Note that B is symmetric, and B=B; also, (A+F*B*F) is also symmetric) = -2 trace{ (A+F*B*F)^(-2) * F * B * d F} Therefore, by the first identification theorem, the differential is as given above. Recall that trace( (d x / d F) * d F ) = d x I find working with matrix differential much easier than other alternatives. The book recommended by Peter has a great account on how to manipulate matrix differential. It also provides the theorectical justification. Hope that helps. === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) >I am currently having a matrix derivative problem >What is derivative of >trace{(A+F*B*F)^(-1)} >with respect to matrix F >where () is the transpose of a matrix. >A and B is diagonal matrices. >I searched online and was only able to find the derivative of >d trace{(F*B*F)^(-1)}/d F >=-2*B*F*(F*B*F)^(-2) >without knowing how they get it. >Moreover, all these matrix derivative problems seem to be >difficult forme. Could anyone be kind enough to give me >some good references on this topic. Derivatives are not appropriate for functions of more than one variable, and you are asking for derivatives with respect to a matrix. Differentials satisfy the usual properties, but the failure of commutativity is quite important. So if q is a differential function of an argument in a locally ßat space, the Frechet derivative is dq(x, m) = lim ((q(x+em) - q(x))/e) as e -> 0. Using this, and the result that d(X^(-1)) = - X^(-1) dX X^(-1), we get that the differential you seek is trace (-2*(A+F*B*F)^(-1)*F*B*dF*(A+F[CapitalOTilde ]*B*F)^(-1)); this uses the results about trace being invariant under permutation and transposition. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) >I am currently having a matrix derivative problem >What is derivative of >trace{(A+F*B*F)^(-1)} >with respect to matrix F >where () is the transpose of a matrix. >A and B is diagonal matrices. >I searched online and was only able to find the derivative of >d trace{(F*B*F)^(-1)}/d F >=-2*B*F*(F*B*F)^(-2) >without knowing how they get it. >Moreover, all these matrix derivative problems seem to be >difficult forme. Could anyone be kind enough to give me >some good references on this topic. some reference which might help: Magnus, J.R., Neudecker, H. Matrix Differential Calculus with Applications in Statistics and Econometrics. paul fackler has some notes at http://www4.ncsu.edu/~pfackler/MATCALC.ps hth peter === Subject: standard probability spaces Originator: israel@math.ubc.ca (Robert Israel) let $(Omega,F)$ be a standard probability space and $X:[0,t]times Omegato E$ a stochastic process with values in a Polish space $E$ and RCLL trajectories. Is it true that $(Omega,F^X)$ is standard where $F^X$ denotes the $sigma$-algebra generated by $X$? J. p.s.: Please reply to email as well. Thx. === Subject: Average number of vectors on a plane Originator: israel@math.ubc.ca (Robert Israel) Hi there, I try to solve the following problem: Given N points random (lets say with mean m) distributed on a plane(2D). Each of these points can be the beginning or the end of a vector of length less than R. Under the condition that the beginning of each vector must have distance greater than R from the end of all the other vectors, what is the average number of vectors I can have on the given plane? (e.g. if (a_i) is the beginning point and (b_i) is the end point of the i-th vector, what is the mean{i} under the constrains ||a_i-b_i||<=R and ||a_i-b_j||>R [for i != j] , where ||x-y|| denotes the Euclidean diatance) Any ideas on which direction I have to look or about any related work it would be very grateful. Thanos === Subject: Two papers published by AGT Originator: israel@math.ubc.ca (Robert Israel) The following two papers have been published: (1) The conjugacy problem for relatively hyperbolic groups by Inna Bumagin URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-43.abs.html (2) Mp-small summands increase knot width by Jacob Hendricks URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-44.abs.html Full details follow: (1) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-43.abs.html Title: The conjugacy problem for relatively hyperbolic groups Author(s): Inna Bumagin Abstract: Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups. Secondary: 20F10 Keywords: Negatively curved groups, algorithmic problems Received: 5 May 2002 Author(s) address(es): Department of Mathematics and Statistics, Carleton University 1125 Colonel By Drive, Herzberg Building Ottawa, Ontario, Canada K1S 5B6 Email: bumagin@math.carleton.ca (2) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-44.abs.html Title: Mp-small summands increase knot width Author(s): Jacob Hendricks Abstract: Scharlemann and Schultens have shown that for any pair of knots K_1 and K_2, w(K_1 # K_2) >= max{w(K_1),w(K_2)}. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann-Schultens, Rieck-Sedgwick and Thompson, it is shown that for K= #_{i=1}^n K_i a connected sum of mp-small knots and K any non-trivial knot, w(K # K)>w(K). Secondary: 57M27 Keywords: Thin position, knot width Author(s) address(es): Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA Email: jghendr@uark.edu === Subject: are C_r fields C_r? any progress? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Some time around Ô52, Serge Lang generalized Artins notion of quasi-algebraically closed fields (C_1) and defined C_r fields: A field k is said to satisfy property (C_r) when any homogeneous polynomial of degree d with coefficients in k in n+1 variables satisfying n >= d^r has a non-trivial zero. (Or, in arithmetic geometry terminology, any hypersurface in P^n over k with degree d has a k-point if n >= d^r.) (Note that a C_0 field is merely an algebraically closed field. Finite fields are C_1 by a theorem of Chevalley, the field of rational functions, or Laurent series, in r variables over an algebraically closed fields, are C_r by results of Tsen and Lang. The field Q_p of p-adic numbers is not C_2, contrary to what was believed for some time: Terjanian gave some counterexamples to that effect.) Under a certain technical assumption (viz., the existence of normic forms of order r and of all degrees), Lang proved that the existence of a non-trivial zero generalizes to families of polynomials, provided the inequality is satisfied for the sum of the degrees-to-the-r. Precisely, let us define: A field k is said to satisfy property (C_r) when any s homogeneous polynomials of degrees d_1,...,d_s with coefficients in k in n+1 variables satisfying n >= d_1^r + ... + d_s^r have a common non-trivial zero. (Or, in arithmetic geometry terminology, any intersection of s hypersurfaces in P^n over k with degrees d_1,...,d_s has a k-point if n >= d_1^r + ... + d_s^r.) (Im not sure that my notation is perfectly standard. Perhaps what other people call C_r is not exactly what is written above. But let us continue with this notation.) So Lang proves that, under a certain technical assumption which I wont recall, a C_r field is actually C_r. Later (around Ô57), Nagata showed that the technical assumption is not necessary provided all the d_j are equal. However, the question of whether the technical assumption is necessary in full generality remained open (as far as I know). My question is: has any progress been made on this question since then? Is there now a known example of a C_r field which is not C_r, or a proof that all C_r fields are C_r? Perhaps if we restrict to the (most interesting) r=1 case? At any rate, what would the educated guess be? Does it help in any way if we assume the (hypersurfaces defined by the) f_j to be in complete intersection (I cant see a way to reduce the general problem to that case, but it seems like a reasonable assumption to make)? (Basically, Id like to formulate the conjecture that, over a C_1 field k, any smooth projective (geometrically) separably rationally connected variety has a k-point: this is known when k is finite or when it is the field of rational functions over an algebraically closed field, by results of Esnault on the one hand, and Graber, Harris, de Jong and Starr on the other. At the very least, it is necessary, for the conjecture to be sensible, for the field to be C_1, so it would be embarrassing if there were already a known example of a C_1 field that is not C_1.) of Maths) and Nagata (in Kyoto University something-or-other) if necessary. -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Paper published by Geometry and Topology Received-SPF: Received-SPF: pass (mailbox2.ucsd.edu: domain of gt@maths.warwick.ac.uk designates 137.205.233.100 as permitted sender) receiver=mailbox2.ucsd.edu; client_ip=137.205.233.100; envelope-from=gt@maths.warwick.ac.uk; Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.html Title: Commensurations of the Johnson kernel Author(s): Tara E Brendle, Dan Margalit Abstract: Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes. Secondary: 20F38, 20F36 Keywords: Torelli group, mapping class group, Dehn twist Proposed: Walter Neumann Seconded: Shigeyuki Morita, Joan Birman Author(s) address(es): Department of Mathematics, Cornell University 310 Malott Hall, Ithaca, NY 14853, USA and Department of Mathematics, University of Utah 155 S 1440 East, Salt Lake City, UT 84112, USA Email: brendle@math.cornell.edu, margalit@math.utah.edu === Subject: Teiji Takagi and principalization Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I couldnt get an answer on sci.math, so I hope this is suitable for here: Some sources say that Takagi proved not only that the maximal unramified extension of a L/K number field K has a Galois group corresponding to the class group, but that principalization occurs in this field--all the ideals of K extend to principal ideals of L. Others say it had to wait for Artin and Furtwangler to get the proof of this. Does anyone have the straight dope? === Subject: Re: Teiji Takagi and principalization Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I couldnt get an answer on sci.math, so I hope this is suitable for here: > Some sources say that Takagi proved not only that the maximal > unramified extension of a L/K number field K has a Galois group > corresponding to the class group, but that principalization occurs in > this field--all the ideals of K extend to principal ideals of L. > Others say it had to wait for Artin and Furtwangler to get the proof > of this. Does anyone have the straight dope? in Cassells and Froelich, _Algebraic Number Theory_ (Thompson, 1967). He says (p. 273): With the help of this [general reciprocity] law, Artin could also reduce the principal divisor theorem, enunciated by Hilbert and not yet proved by Takagi, to a pure group-theoretical proposition, which was then proved by Furtwaengler. Ive looked at Artins paper, Idealklassen in Oberkoerpern und allgemeines Reziprozitaetsgesetz (Collected Papers 159-164), and its clear that he did not know of any earlier proof. William C. Waterhouse Penn State === Subject: Normal subgroups of surface groups? Epigone-thread: thulgonstrix Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let G be a surface group, and H be a non-trivial normal subgroup of G. How can I prove that H is of finite index in G ? (This is a conjecturally true for any one-relator group) === Subject: Re: Normal subgroups of surface groups? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Let G be a surface group, and H be a non-trivial normal subgroup of G. > How can I prove that H is of finite index in G ? > (This is a conjecturally true for any one-relator group) You cant, because this is not true. For instance, any surface group (other than the 2-sphere or projective plane) has a surjection to the integers, whose kernel is of infinite index. The simplest of course would be the torus, where the obvious Z subgroup is normal, and of infinite index. DR === Subject: Re: Normal subgroups of surface groups? Epigone-thread: thulgonstrix Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Youre right. Here is the restatement of the question: Show that every finitely generated normal subgroup of a non-abelian surface group (with or without boundary) is of finite index. JJ >> Let G be a surface group, and H be a non-trivial normal subgroup of G. >> How can I prove that H is of finite index in G ? >> (This is a conjecturally true for any one-relator group) >You cant, because this is not true. For instance, any surface group >(other than the 2-sphere or projective plane) has a surjection to the >integers, whose kernel is of infinite index. The simplest of course >would be the torus, where the obvious Z subgroup is normal, and of >infinite index. === Subject: rapidly converging rational sqrt Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Below is a description of an algorithm which, with each iteration, will double the number of significant digits in the computation of a rational square root approximation. I do not know if this algorithm is new, but I found it interesting nonetheless. Lisp code for implementing this algorithm can be found at: http://thegreves.com/david/sqrt/sqrt.html If by convention we say that: s = isqrt(C) d = C - s^2 Then the square root of C can be expressed as as the infinite continued fraction: d s + ----------------- d 2s + ------------ d 2s + ------- 2s + .. We designate the tail of this continued fraction using e(n) = nth error term and we say that the nth error term of the continued fraction representation has the form: d e(n) = ---------- 2s + e(n+1) Although we sometimes drop the subscript on the error term for notational convenience. A pretty good first order approximation for a square root can be computed as follows (even allowing e(1) to be zero): d sqrt(C) ~= s + -------- 2s + e(1) Without proof, we claim that a generalized expression for a partial evaluation of our continued fraction can be represented as: A + Be sqrt(C) = s + ------- C + De It is easy to see that the first order approximation given above is an instance of this expression when A = d, B = 0, C = 2s, D = 1. The generalized representation is useful for representing the result of evaluating some number of sucessive terms in the continued fraction representation. Assuming that the above representation is the result of evaluating n terms of the continued fraction for sqrt(C), then the n+1 term would be computed by substituting the next error term into the error expression in the representation. A + B(d/(2s + e)) s + --------------- C + D(d/(2s + e)) A(2s + e) + Bd s + -------------- C(2s + e) + Dd (A2s + Bd) + Ae s + ----------------- (C2s + Dd) + Ce Which, we observe, is once again in the general representational form. If the evaluation of the first n terms produced A + Be ------ C + De and the evaluation of the next m terms were to produce W + Xe ------ Y + Ze Then the evaluation of the first n+m terms would be: W + Xe A + B(------) Y + Ze A(Y + Ze) + B(W + Xe) ------------ = --------------------- W + Xe C(Y + Ze) + D(W + Xe) C + D(------) Y + Ze (AY + BW) + (AZ + BX)e ---------------------- (CY + DW) + (CZ + DX)e Because the continued fraction representation of the square root is uniform, the evaluation of any n sucessive terms will always produce the same result. We can take advantage of this fact to refine our first order approximation by substituting A,B,C, and D in for W,X,Y and Z in the above expression. The will double the number of terms we have evaluated. Of course, this procedure can be repeated again and again, with each iteration of the algorithm doubling the number of significant digits in our representation. Here is an example run computing the sqrt of 1973 for 1 to 10 iterations of the algorithm. Note that after iteration 6 we have more than 100 significant digits. 1 : 44.41845521141241485670222336460609176198432078139056676519727 541447114766739 49363834982650044981364863 2 : 44.41846462881467219505665982727838995343272680250852496023827 788068578570854 49370627285274658896791048 3 : 44.41846462902561876427524312325572040240985914842970903422305 486056596616470 16608600579504981095676558 4 : 44.41846462902561876438107965740906053956833272941354961112468 508573375443791 08155113103260155775597834 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 26805283703947239542091268 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 7 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 8 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 9 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 10 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 Dave === Subject: Re: rapidly converging rational sqrt Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Below is a description of an algorithm which, with > each iteration, will double the number of significant > digits in the computation of a rational square root > approximation. > Here is an example run computing the sqrt of 1973 for 1 to 10 > iterations of the algorithm. Note that after iteration 6 we have > more than 100 significant digits. > 1 : 44.4184 | 55211412414856702223364606091761984320781390566765197275414471 147667394936383 4982650044981364863 > 2 : 44.41846462 | 88146721950566598272783899534327268025085249602382778806857857 085449370627285 274658896791048 > 3 : 44.418464629025618764 | 27524312325572040240985914842970903422305486056596616470166086 005795049810956 76558 > 4 : 44.4184646290256187643810796574090605395 | 68332729413549611124685085733754437910815511310326015577559783 4 > 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 | 26805283703947239542091268 > 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 7 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 8 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 9 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 10 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 I dont wish to denigrate your algorithm unduly, but square root algorithms with this rate of convergence have been known for millennia. The iteration x_{n+1} = 1/2 (x_n + C/x_n), which is believed to have been known to the ancient Babylonians, yields the following output if we take C = 1973 and start it at 44 (since your algorithm presupposes we know isqrt(C) this seems a fair comparison.) I have added vertical bars to indicate the correct portion of each decimal expansion; I have done the same for the quoted output from your algorithm above. 1 : 44.4 | 20454545454545454545454545454545454545454545454545454545454545 454545454545454 54545454545454545455 2 : 44.4184646 | 73597060396753412870066745738272983092630061164213121235377566 920160933975208 72578432056559 3 : 44.418464629025618786 | 74355237890368384233529296754366053971048032852669727540929455 186628564646861 810 4 : 44.4184646290256187643810796574090605 | 45224167043182294394699285292428539812171115734428794096905254 72 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066 | 216106505407447827420345844 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 436869171473600589118301... So like your algorithm this also produces 100 digits in 6 iterations, and it is easy to prove (since this is a special case of Newtons method for a general function) that convergence is quadratic in general. In short, your algorithm is interesting but it doesnt outperform the standard algorithms for square roots. Yours, David Loefßer (student, Trinity College, University of Cambridge, UK) === Subject: Laplaces method and a Double Integral Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hello all, I wonder if anyone can show me how to evaluate the asymptotics (as N) gets large of this integral: int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). If we define f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr where f takes on its mininum value of 0 at the pt (0,0). Unfortunately the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the explicit expansions I have seen require that the Hessian be non-zero. Does anyone know of another reference to try that might have this worked out? Jim PS: Im not a mathematician....just a humble plodding engineer! === Subject: Re: Laplaces method and a Double Integral Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Hello all, >I wonder if anyone can show me how to evaluate the asymptotics (as N) >gets large of this integral: >int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr >Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). >If we define >f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] >Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr >where f takes on its mininum value of 0 at the pt (0,0). Unfortunately >the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the >explicit expansions I have seen require that the Hessian be non-zero. Do the substitutions t=r^2, s=-n help? The integral becomes (1/2)int_0^{R^2} int_0^u t [1 - c_1(c_2 s + c_3 t)^2]^N ds dt For the corresponding f(s,t), a quick calculation gave me f_{tt}(0,0) not= 0, but you should check that yourself. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 To reply by email, change Look-In-Sig to luecking === Subject: Re: Laplaces method and a Double Integral Originator: israel@math.ubc.ca (Robert Israel) Dans substitutions are OK, a nice trick, but I wonder if the origin (0,0) is the main contributing point. I expect also contributions from the boundaries, but this depends on the values of c_1, c_2, c_3, u and R. Nico M. Temme, http://homepages.cwi.nl/~nicot/ C W I: Centrum voor Wiskunde en Informatica Kruislaan 413, NL-1098 SJ Amsterdam Tel +31 20 592 4240 P.O. Box 94079, NL-1090 GB Amsterdam Fax +31 20 592 4199 === > Subject: Re: Laplaces method and a Double Integral >Hello all, >I wonder if anyone can show me how to evaluate the asymptotics (as N) >gets large of this integral: >int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr >Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). >If we define >f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] >Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr >where f takes on its mininum value of 0 at the pt (0,0). Unfortunately >the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the >explicit expansions I have seen require that the Hessian be non-zero. > Do the substitutions t=r^2, s=-n help? The integral becomes > (1/2)int_0^{R^2} int_0^u t [1 - c_1(c_2 s + c_3 t)^2]^N ds dt > For the corresponding f(s,t), a quick calculation gave me > f_{tt}(0,0) not= 0, but you should check that yourself. > Dan > -- > Dan Luecking Department of Mathematical Sciences > University of Arkansas Fayetteville, Arkansas 72701 > To reply by email, change Look-In-Sig to luecking === Subject: Re: Laplaces method and a Double Integral Originator: israel@math.ubc.ca (Robert Israel) Re: Laplaces method and a Double Integral Since Nico thinks that the asymptotic form of the integral has also contributions not only from the neighborhood of (0,0), I derived an asymptotic approximation given in the following. It shows that only the neighborhood of (0,0) is important. Ciao Karl Breitung Schellingstr. 21 D-80799 Munich, Germany AN ASYMPTOTIC APPROXIMATION FOR THE TWO-DIMENSIONAL INTEGRAL: ,- R ,- 0 3 [ 2 2 ]N I(N)= | | r [ 1-a(bn-cr ) ] dn dr, N --> oo - 0 - -u Here we write instead of the original form: a=c , b=c , c=c . 1 2 3 1/2 2 1/2 Making the substitutions r --> v=a cr and n --> z=-a bn transforms this into: 2 ,- cR ,- bu / v 3/2 [ 2 ]N d r d n I(N)=- | | | ----- | [ 1-(z+v) ] --- ---- dz dv - 0 - 0 | 1/2 | d v d z a c / 2 ,- cR ,- bu / v 3/2 [ 2 ]N 1 1/2 -1/2 1 = | | | ----- | [ 1-(z+v) ] -(a cv) ----- dz dv= - 0 - 0 | 1/2 | 2 1/2 a c / a b 1/2 2 1/2 ,- a cR ,- a bu [ 2 ]N K | | v [ 1-(z+v) ] dz dv - 0 - 0 3/2 2 -1 with K=(2a bc ) . Now we will consider only the integral over a triangle 1/2 2 1/2 (0,0), (d,0) and (0,d) with 00 and K >0 are constants. Then: 1 2 ,- d ,- d-v [ 2 ]N I(N) sim K | | v [ 1-(z+v) ] dz dv - 0 - 0 In this triangle, we make the variable transformation (v,z) --> (w,y) with w=z+v, y=z-v. Then: ,- d [ ,- w w-y [ 2 ]N ] I(N) sim K | | | --- [ 1-w ] |det (J(w,y))| dy | dw= - 0 [ - -w 2 ] ,- d [ ,- w w-y [ 2 ]N ] K | | | --- [ 1-w ] dy | dw (*) - 0 [ - -w 2 ] J(w,y) is the Jacobian of the inverse transformation with its determinant equal to 1/2. The integral in the brackets is: ,- w w-y [ 2 ]N [ 2 ]N ,- w / w y [ 2 ]N ,- w w | --- [ 1-w ] dy=[ 1-w ] | | - - - | dy=[ 1-w ] | - dy= - -w 2 - -w 2 2 / - -w 2 [ 2 ]N 2 [ 1-w ] w 2 Inserting this into equ. (*) and writing f(w)=log (1-w ) gives: ,- d 2 I(N) sim K | w exp (Nf(w)) dw - 0 This we can evaluate using the generalized Laplace method (derived in [2], p. 37, see also [1], p. 48). Here we derive the result directly by approximating 2 2 f(w) by its second order Taylor expansion at zero, i.e. f(w)=-2w +o(w ) and 2 then making the substitution w --> x=2N w . This gives: ,- dN x -x dw ,- dN x -x 1 -1/2 I(N)sim K | -- e --dx= K | -- e ------ x dx = - 0 2N dx - 0 2N +--+ 2|2N -3/2 K ,- dN 1/2 -x N ----- | x e dx +-+ - 0 4|2 For this we get the asymptotic form replacing dN by oo: -3/2 K ,- oo 1/2 -x -3/2 K I(N)sim N ----- | x e dx sim N ----- Gamma(3/2)= +-+ - 0 +-+ 4|2 4|2 +---+ +---+ -3/2 K |pi -3/2 |pi N --- ----- = N ---------------- , N --> oo +-+ 2 +-+ 3/2 2 4|2 16|2 a bc If necessary this approximation can be refined by deriving a second term in the asymptotic expansion of I(N). Bibliography: [1] K. Breitung. Asymptotic Approximations for Probability Integrals. Springer, Berlin, 1994. Lecture Notes in Mathematics, Nr.1592. [2] A. Erdelyi. Asymptotic Expansions. Dover, New York, 1956. === Subject: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an arbitrary metric space, such that all orbits of $G$ are finite. We suppose that there exists $pin N$ such that for all $gin G$ we have $g^p = 1$. The group $G$ is it finite? === Subject: Re: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an >arbitrary metric space, such that all orbits of $G$ are finite. >We suppose that there exists $pin N$ such that for all $gin G$ we >have $g^p = 1$. The group $G$ is it finite? The answer is no. This was essentially known as the Burnside problem, and solved in the negative by Novikov and Adjan in 1968: there exist infinite two-generator groups identically satisfying x^n=1 with n any sufficiently large odd integer. (The additional condition stipulating that G should be a subgroup of Homeo(E) adds nothing to the problem, for every group can be so represented: choose E = G, with the discrete metric where distinct points always have distance 1, and let G operate on itself by left translations; these, of course, are homeomorphisms in the given metric.) === Subject: Re: groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is >>an arbitrary metric space, such that all orbits of $G$ are finite. >>We suppose that there exists $pin N$ such that for all $gin G$ we >>have $g^p = 1$. The group $G$ is it finite? >The answer is no. This was essentially known as the Burnside >problem, and solved in the negative by Novikov and Adjan in 1968: >there exist infinite two-generator groups identically satisfying >x^n=1 with n any sufficiently large odd integer. (The additional >condition stipulating that G should be a subgroup of Homeo(E) adds >nothing to the problem, for every group can be so represented: choose >E = G, with the discrete metric where distinct points always have >distance 1, and let G operate on itself by left translations; these, >of course, are homeomorphisms in the given metric.) But among the hypotheses you have that the orbits of G are finite. This implies that G is residually finite and thus, by Zelmanovs positive solution to the restricted Burnside problem, G is indeed finite. Andreas === Subject: Re: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Sorry for my oversight! Of course, Andreas is right; I overlooked the requirement that the orbits should be finite. Regretfully, Peter === Subject: Re: groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an >>arbitrary metric space, such that all orbits of $G$ are finite. >>We suppose that there exists $pin N$ such that for all $gin G$ we >>have $g^p = 1$. The group $G$ is it finite? >The answer is no. This was essentially known as the Burnside >problem, and solved in the negative by Novikov and Adjan in 1968: >there exist infinite two-generator groups identically satisfying x^n=1 >with n any sufficiently large odd integer. (The additional condition >stipulating that G should be a subgroup of Homeo(E) adds nothing to >the problem, for every group can be so represented: choose E = G, with >the discrete metric where distinct points always have distance 1, and >let G operate on itself by left translations; these, of course, are >homeomorphisms in the given metric.) But the orbits of G will not be finite in this example. G having a faithful representation as mappings of a set with all orbits finite means that G has a collection of normal subgroups of finite index whose intersection is the identity. I dont know if this is true for any of the known examples in the Burnside problem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Fall Pacific NW Geometry Seminar at U of Oregon Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Final announcement: PACIFIC NORTHWEST GEOMETRY SEMINAR University of Oregon Eugene, OR SCHEDULE Saturday, November 6 10:30 - 11:00 Morning Reception 11:00 - 12:15 Charles Doran (University of Washington) Mirror Symmetry, K-Theory, and Toric Geometry 12:15 - 2:00 Lunch 2:00 - 3:15 Mutao Wang (Columbia) Mean curvature ßows of Lagrangian submanifolds 3:15 - 4:00 Break 4:00 - 5:15 Lei Ni (UCSD) Ancient Solutions of the K.8ahler-Ricci Flow 7:00 Party at Botvinniks Sunday, November 7 8:30 - 9:00 Morning Reception 9:00 - 10:15 John Lott (University of Michigan) Ricci curvature for metric-measure spaces 10:15 - 10:45 Break 10:45 - 12:00 David Auckly (Kansas State University) The structure of maps into homogenous spaces and the Faddeev and Skyrme models Note: Each speakers time allotment includes 15 minutes for a discussion of Open Problems related to his topic. The talks will be in 110 Fenton Hall (D-7 on the campus map). The receptions and breaks will be right outside 110 Fenton. ----------------------------------------------------------- For general information about the PNGS, visit the PNGS web site: http://www.math.washington.edu/~lee/PNGS It contains up-to-date information about this meeting, travel and lodging information, general information about the PNGS, and a historical record of all PNGS meetings and speakers. ----------------------------------------------------------- For more information about this meeting, contact the organizers: Boris Botvinnik (botvinn@math.uoregon.edu) Jim Isenberg (jim@newton.uoregon.edu) === Subject: Non linear hyperbolic PDEs : ill posedness ? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi everybody I am studying the well-posedness of the Cauchy problem for systems of PDE of the form : d U/dt = A d U/dx, with unknown vector U(x,t) In the linear case (A is a matrix function of (x,t)), it is well known that the problem is well-posed (existence of a unique solution depending continuously on the initial data) iff A is diagonalisable with real eigenvalues for all x and t. Here is my question : In the non linear case (A is a function of U, x and t) does one know such a system where the matrix A is not always diagonalisable but wich is still well-posed ? Michael === Subject: Partitioning 4 space with ultraskew lines, and the three body problem. Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Suppose we are in 4 space. Two lines are skew if they do not lie in the same plane(2 space). Skewness is a binary relation on lines. Now two skew lines will lie in the same 3 space. But it is possible for three lines to not lie in the same 3 space. This is a ternary relation on lines. I dont know if theres a name for this relation, so lets call it ultraskewness until we find out. 1) Can one foliate, or at least partition, 4 space with lines that are tripletwise ultraskew? I mean EACH 3 line subset of the partition must not lie in the same 3 space. 2) In the three-body problem, one could approximate the trajectory of each of the 3 masses as a straight lines until they became close enough to each other for their gravitation to have an appreciable effect. I think much of the work on this problem assumes all three trajectories lie in the same plane.(The restricted 3-body problem.) But some work has been where the trajectories are not coplanar. Wouldnt it be fun to explore the three-body problem when the trajectories are not cospatial? Richard Peterson, CSU Sacramento === Subject: Re: Partitioning 4 space with ultraskew lines, and the three body problem. ath: nntpswitch.com Originator: israel@math.ubc.ca (Robert Israel) > Suppose we are in 4 space. Two lines are skew if they do not lie in > the same plane(2 space). Skewness is a binary relation on lines. Now > two skew lines will lie in the same 3 space. But it is possible for > three lines to not lie in the same 3 space. This is a ternary relation > on lines. I dont know if theres a name for this relation, so lets > call it ultraskewness until we find out. > 1) Can one foliate, or at least partition, 4 space with lines that > are tripletwise ultraskew? I mean EACH 3 line subset of the partition > must not lie in the same 3 space. > 2) In the three-body problem, one could approximate the trajectory > of each of the 3 masses as a straight lines until they became close > enough to each other for their gravitation to have an appreciable > effect. I think much of the work on this problem assumes all three > trajectories lie in the same plane.(The restricted 3-body problem.) > But some work has been where the trajectories are not coplanar. > Wouldnt it be fun to explore the three-body problem when the > trajectories are not cospatial? > Richard Peterson, CSU Sacramento I dont know about the three-body problem, or about foliations, but I think one can partition 4-space into ultraskew lines without much trouble. The proof is via a transfinite induction of c (the cardinality of the continuum) many steps; at step k one considers the k-th point p in a fixed enumeration of 4-space, and one has already constructed a collection L of |k|-many (fewer than c) lines. If p is in the union of L there is nothing to do. Otherwise one need only find a unit tangent vector u at p so that the line through p in direction u is (a) disjoint from each line in L and (b) ultraskew to every pair of lines in L. Since p lies on no line in L (a) is satisfied so long as u does not lie in any plane containing both p and a line in L. Also, (b) is satisfied so long as u does not lie in any translate containing p of an (affine) 3-space generated by a pair of lines in L. So we need a point on the unit 3-sphere in R^4 not lying in any of a collection of fewer than c many subspaces of R^4 of dimension at most 3. Without loss of generality we may assume all the subspaces have dimension 3, so each has a perp that meets the 3-sphere in at most 2 points. As we have fewer than c subspaces, there is a point w on the 3-sphere not in the perp of any of them. Then the perp P of the line spanned by w is a 3-dimensional space meeting each of the spaces we want to avoid in a subspace of dimension at most 2. Thus it suffices to find a point on the intersection of the unit 3-sphere with P (i.e., a unit 2-sphere) not in any of a collection of fewer than c subspaces of P of dimension at most 2. By a similar argument we can drop the dimension once again, and then we need to find a point on the unit circle avoiding fewer than c lines through the origin. As the circle has c points, and each line meets it in two point, that is easy. Well, its late and Im hurrying, but I think this argument holds water. Bob Beaudoin === Subject: Integral recurrence relation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I have encountered the following integral in some research in the physical sciences int |u-A|^(2a) |u-B|^(2b) Exp[-|u|^2] du where u, A and B are cartesian vectors in 3 dimensions and the integral is to performed over all space. This seems like quite a straightforward integral but the best I can do is to write it as a triple infinite series in A^2, B^2 and |A-B|^2 (which quickly truncates, depending on the values of a and b). I was wondering if anyone has any suggestions as how I might produce a more useful formulation. Even more useful would be a suggestion as to how I might derive a recurrence relation to generate integrals of higher values of a and b or if it is possible to prove or disprove the existence of such a relation. Darragh === Subject: Re: Integral recurrence relation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >I have encountered the following integral in some research in the >physical sciences > int |u-A|^(2a) |u-B|^(2b) Exp[-|u|^2] du >where u, A and B are cartesian vectors in 3 dimensions and the >integral is to performed over all space. This seems like quite a >straightforward integral but the best I can do is to write it as a >triple infinite series in A^2, B^2 and |A-B|^2 (which quickly >truncates, depending on the values of a and b). I was wondering if >anyone has any suggestions as how I might produce a more useful >formulation. Let your integral be F(a,b) (for nonnegative integers a,b). Consider the exponential generating function f(s,t) = sum_{a=0}^infinity sum_{b=0}^infinity F(a,b) s^a t^b/(a! b!) = int_{R^3} exp(s |u-A|^2) exp(t |u-B|^2) exp(-|u|^2) du = int_{R^3} exp(-(1-s-t) |u|^2 - 2 u.(sA+tB) + s|A|^2 + t|B|^2) du = exp(s|A|^2 + t|B|^2 + |sA+tB|^2/(1-s-t)) int_{R^3} exp(-(1-s-t) |u-(sA+tB)/sqrt(1-s-t)|^2) du = exp(s|A|^2 + t|B|^2 + |sA+tB|^2/(1-s-t)) (pi/(1-s-t))^(3/2) for |s|+|t| < 1. Then F(a,b) can be obtained from the coefficients of the bivariate Taylor series for f(s,t) around (0,0). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Paper published by Geometry and Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper38.abs.html Title: Noncommutative localisation in algebraic K-theory I Author(s): Amnon Neeman, Andrew Ranicki Abstract: associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A-->B be the localisation with respect to a set sigma of maps between finitely generated projective A-modules. Suppose that Tor_n^A(B,B) vanishes for all n>0. View each map in sigma as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D^perf(A). Denote by the thick subcategory generated by these complexes. Then the canonical functor D^perf(A)-->D^perf(B) induces (up to direct factors) an equivalence D^perf(A)/--> D^perf(B). As a consequence, one obtains a homotopy fibre sequence K(A,sigma)-->K(A)-->K(B) (up to surjectivity of K_0(A)-->K_0(B)) of Waldhausen K-theory spectra. consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor_n^A(B,B), we also assume that every map in sigma is a monomorphism, then there is a description of the homotopy fiber of the map K(A)-->K(B) as the Quillen K-theory of a suitable exact category of torsion modules. Secondary: 19D10, 55P60 Keywords: Noncommutative localisation, $K$--theory, triangulated category Proposed: Bill Dwyer Seconded: Thomas Goodwillie, Gunnar Carlsson Author(s) address(es): Centre for Mathematics and its Applications The Australian National University Canberra, ACT 0200, Australia and School of Mathematics, University of Edinburgh Edinburgh EH9 3JZ, Scotland, UK Email: Amnon.Neeman@anu.edu.au, a.ranicki@ed.ac.uk === Subject: monoidal enriched natural transformations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi. I have a problem which, it seems to me, requires the notion of enriched natural transformation between enriched monoidal functors, but I havent been able to find a good reference for it. Ive taken a look at the books of Borceux (Handbook of categorical algebra), Kelly (Basic notions of enriched category) and the appendix in Levines Mixed motives, but they all end just where I need them, or even long before that point. More precisely, the situation Ive encountered seems to be the following. I have: - Two monoidal (symmetric) categories: V and W . - A couple of monoidal (symmetric) functors between them: S, T : V ---> W - A monoidal natural transformation between S and T : And here is where my problems begin. There is a well-known notion of what is a V-functor between V-categories and what a V-natural transformation is. My first need is to understand what an S-functor between a V-category C and a W-category D should be: F : C ---> D I didnt find this thing in the literature, but I expect it ought to be something like a V-functor, but with a family of morphisms in W lambda_{XY} : S[X,Y] ---> [FX,FY] for every pair of objects X, Y in C . (Here the square brackets [,] stand for the objects in V and W of morphisms of C and D , and Im leaving aside units and commutative isomorphisms for the moment.) This seems reasonable to me, since (a) is the situation I have in the real world and (b) if I put S = id_V , I find the definition of a V-functor. Next, I would need the notion of an omega - natural transformation and I think this should be something like a V-natural transformation between an S-functor F : C ---> D and a T-functor G: C ---> D, but placing at the beginning of the commutative diagram which defines a V-natural transformation an arrow like omega_{[X,Y]} : S[X,Y] ---> T[X,Y] . Assuming that this is ok, I should also need to understand what might be the definition of a monoidal omega - natural transformation. That is to say, C is a monoidal V-category, D is a monoidal W-category, F is a monoidal S-functor and G a monoidal T-functor: what is a monoidal natural transformation between F and G , over Ive drawn a couple of commutative diagrams that should appear in the definition of such a construct, but I feel I could be forgetting a dozen more. Any references for it? Unfortunately for me, Kellys book ends before this point: it explicitely says: is our decision not to discuss the Ôchange of base-category given by a symmetric monoidal functor V ---> W. Has someone else done the job after Kellys book? Agust.92 Roig === Subject: Re: monoidal enriched natural transformations Epigone-thread: tixslongcax Content-Length: 6836 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Hi. >I have a problem which, it seems to me, requires the notion of >enriched natural transformation between enriched monoidal functors, >but I havent been able to find a good reference for it. >Ive taken a look at the books of Borceux (Handbook of categorical >algebra), Kelly (Basic notions of enriched category) and the appendix >in Levines Mixed motives, but they all end just where I need them, >or even long before that point. >More precisely, the situation Ive encountered seems to be the >following. I have: >- Two monoidal (symmetric) categories: > V and W . >- A couple of monoidal (symmetric) functors between them: > S, T : V ---> W >- A monoidal natural transformation between S and T : >And here is where my problems begin. >There is a well-known notion of what is a V-functor between >V-categories and what a V-natural transformation is. >My first need is to understand what an S-functor between a >V-category C and a W-category D should be: > F : C ---> D >I didnt find this thing in the literature, but I expect it ought to >be something like a V-functor, but with a family of morphisms in W > lambda_{XY} : S[X,Y] ---> [FX,FY] >for every pair of objects X, Y in C . (Here the square brackets >[,] stand for the objects in V and W of morphisms of C and D >, and Im leaving aside units and commutative isomorphisms for the >moment.) >This seems reasonable to me, since (a) is the situation I have in the >real world and (b) if I put S = id_V , I find the definition of a >V-functor. >Next, I would need the notion of an omega - natural transformation >and I think this should be something like a V-natural transformation >between an S-functor F : C ---> D and a T-functor G: C ---> D, >but placing at the beginning of the commutative diagram which defines >a V-natural transformation an arrow like > omega_{[X,Y]} : S[X,Y] ---> T[X,Y] . >Assuming that this is ok, I should also need to understand what might >be the definition of a monoidal omega - natural transformation. >That is to say, C is a monoidal V-category, D is a monoidal >W-category, F is a monoidal S-functor and G a monoidal T-functor: >what is a monoidal natural transformation between F and G , over >Ive drawn a couple of commutative diagrams that should appear in the >definition of such a construct, but I feel I could be forgetting a >dozen more. Any references for it? >Unfortunately for me, Kellys book ends before this point: it >explicitely says: is our decision not to discuss the Ôchange of >base-category given by a symmetric monoidal functor V ---> W. Has >someone else done the job after Kellys book? If you havent done so already, I recommend that you get on the categories mailing list: categories@mta.ca where you would surely get a reply and advice about the literature. Alternatively, you might write Max Kelly (at the University of Sydney) or Ross Street (Macquarie University) directly. I wish I had suitable references at hand, but here are some remarks on your query. First, a monoidal functor S: V --> W induces a 2-functor S_{*}: V-Cat --> W-Cat, making straightforward use of the monoidal structure on S. If C is a V-category and D is a W-category, then what you call an S-functor is undoubtably the same as a W-functor of the form F: S_{*}C --> D. Next, a monoidal natural transformation omega: S --> T induces a 2-natural transformation between 2-functors omega_{*}: S_{*} --> T_{*} and in particular provides, for each V-category C, a W-functor of the form omega_{*}(C): S_{*}C --> T_{*}C This too is straightforward, using just the data and equations for an m.n.t. Then what you call an omega-natural transformation from F to G is undoubtably the same as a W-transformation of the form F --> G(omega_{*}(C)) where F: S_{*}C --> D and G: T_{*}C --> D are W-functors, and the target on the right is a composite of W-functors. Now to define a monoidal omega-natural transformation, you want to do a jazzed-up version of the above definitions. Heres what youll need (minimally): -- V, W braided monoidal categories -- S, T braided monoidal functors of the form V --> W -- omega a m.n.t. of the form S --> T (You can of course replace braided by symmetric, but you lose some generality in doing so.) Since V is braided monoidal, V-Cat is a monoidal 2-category, and monoidal V-categories are the same as (pseudo-)monoids in V-Cat as a monoidal 2-category. Indeed, the 2-category Mon(V-Cat) whose objects are monoidal V-categories, whose objects are monoidal V-functors, and whose 2-cells are monoidal V-transformations, is definable purely in terms of the monoidal 2-category structure on V-Cat, and therefore, the desired change of base induced by S: V --> W, Mon(V-Cat) --> Mon(W-Cat), requires only a monoidal 2-functor S_{*}: V-Cat --> W-Cat to get off the ground. The point of demanding that S be braided monoidal is so that the 2-functor S_{*} is in fact monoidal. So: under these hypotheses, a monoidal S-functor (to use your terminology) should be the same as a monoidal W-functor of the form F: S_{*}C --> D where C is a monoidal V-category and D is a monoidal W-category. Finally, if omega: S --> T is an m.n.t., there is a monoidal W-functor omega_{*}(C): S_{*}C --> T_{*}D and what you call a monoidal omega-natural transformation should just be a monoidal W-transformation of the form F --> G(omega_{*}(C)). Notice that it is unnecessary to write down a whole bunch of commutative diagrams to define these constructs: the data and axioms inherent in ordinary enriched notions and in braided monoidal notions do the work for you. However, this approach does involve some machinery of higher-dimensional categories (monoidal 2-categories, monoidal 2-functors), which is probably why Kelly didnt touch this in his book -- the relevant notions hadnt yet been formulated properly. If you want to follow up on this machinery, you might want to look at Coherence for Tricategories by Gordon, Power & Street. The full-ßedged definition of monoidal 2-category can be found there, and is shown to be equivalent in an appropriate sense to so-called Gray-monoids, which are much simpler (indeed, for V braided monoidal, V-Cat *is* a Gray-monoid). Also look at the havent already done so. Todd Trimble === Subject: Re: monoidal enriched natural transformations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) vas dir: >>Assuming that this is ok, I should also need to understand what might >>be the definition of a monoidal omega - natural transformation. [...] >>Unfortunately for me, Kellys book ends before this point: it >>explicitely says: is our decision not to discuss the Ôchange of >>base-category given by a symmetric monoidal functor V ---> W. Has >>someone else done the job after Kellys book? >If you havent done so already, I recommend that you get on the >categories mailing list: > categories@mta.ca >where you would surely get a reply and advice about the literature. >Alternatively, you might write Max Kelly (at the University of >Sydney) or Ross Street (Macquarie University) directly. >I wish I had suitable references at hand, but here are some remarks >on your query. [...] >Todd Trimble Agust.92 Roig === Subject: weighted tree generation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For a given undirected graph G(V,E) I would like to generate all possible rooted trees. How many will there be? If there are a lot, at least I would like to generate a set of different ones or make a single change to a given tree. Secondly, is there any standard method to assign weights to an undirected graph such that for a given root, the all shortest paths algorithm (e.g. dijkstra) will yield a given tree? (I guess that you could assign low values to the links that are elements of the tree and high values for all others, but can it be proven that it will always yield the given tree?) Diego diego at aulignac dot com www.aulignac.com === Subject: fundamental bounds on the elements of covariance matrices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let x and y be n-tuplets of normally distributed random variables, and let C = cov(x,y) be the nxn covariance matrix of x and y. Let z be some fixed n-tuplet whose elements are all strictly positive. 1) Are all the elements of the n-vector, Cz, non-negative? 2) Is zCz > 0? Where can I read more about bounds on the elements of covariance matrices? === Subject: Re: fundamental bounds on the elements of covariance matrices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) See Morris L. Eatons book with multivariate in the title. I was about to answer that any matrix whatsoever could be in the role of C, but I find Im a bit rusty. -- Mike Hardy > Let x and y be n-tuplets of normally distributed random variables, and let > C = cov(x,y) be the nxn covariance matrix of x and y. > Let z be some fixed n-tuplet whose elements are all strictly positive. > 1) Are all the elements of the n-vector, Cz, non-negative? > 2) Is zCz > 0? > Where can I read more about bounds on the elements of covariance matrices? === Subject: Re: binary vector packings Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Could someone please help me identify the following problem? > Consider binary arrays {u_i} of length n with k 1s (and n-k 0s). How > many can we choose such that all pairwise inner products > sum_i u_i v_i < t ? > Equivalently, what is the maximum number of k-subsets of the n-set with > pairwise intersections less than t elements? > Does this problem, or some equivalent, have a name? Any references? Looks like your problem is more or less equivalent to finding the maximum size of a certain constant weight code (aka fixed weight code). Searching with those buzzwords should lead you to sources of known results, tables of upper bounds etc. Jyrki Lahtonen, Turku, Finland === Subject: two fibrations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hallo, consider an isometric action of a compact Lie group G on a connected Riemannian manifold M with a fixed point p. I am interested in the topology (especially the homology) of the space of paths in M, starting at p and ending in some fixed orbit N=Gq, denoted by P(M,ptimes N). Even for arbitrary p and submanifolds N, the end point map P(M,ptimes N)to N; c mapsto c(1) is a fibration with fibre P(M,ptimes q), the space of paths from p to q (which ist homotopy equivalent to the space of loops on M). So I can deduce some information on the homology from this fibration, e.g. by using the Leray-Serre spectral sequence. Furthermore, now restricting to the case of N being some orbit, we have P(M,ptimes Gq)=P(M,ptimes q)times_{G_q} G (twisted product - G_q is the isotropy group at q), which can be easily seen by regarding the mapping P(M,ptimes q)times Gto P(M,ptimes Gq); (c,g)mapsto gc (well-defined since p is fixed). Summarizing, we have two fibrations: i) P(M,ptimes q)to P(M,ptimes Gq)to Gq and ii) G_qto P(M,ptimes q)times G to P(M,ptimes Gq). Now my question: Does the existence of such two fibrations give any new relation between the topology of these spaces? Is there some method of extracting information from such two similar-looking fibrations? Oliver Goertsches === Subject: RA Positions at UNR Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The Computer Vision Laboratory (CVL) at the University of Nevada, Reno (UNR) invites applications for research assistant positions starting in Spring2005/Fall 2005. Preference will be given to students who want to pursue a PhD degree in Computer Vision. Active research areas within CVL include object recognition, visual motion analysis, face detection and recognition, biometrics, tracking and pose estimation of human body/head/hand/eye-gaze, surveillance and activity recognition. CVL is currently funded by NSF, NASA, ONR, and Ford Motor Company. We are also collaborating with several government and industry laboratories. For more information, please visit http://www.cs.unr.edu/CVL Requirements: You must have a first degree in either an Engineering subject, in Mathematics, in Physics, or in Computer Science. Good Mathematical background, programming skills in C or C++, and familiarity with Unix/Linux/Windows are necessary. Prior familiarity with Image Processing, Computer Vision, Pattern Recognition, and Machine Learning is desirable. Good communication and writing skills in English are essential. Interested students should send their CV by regular mail, e-mail, or fax to Dr. George Bebis (bebis@cs.unr.edu) or Dr. Mircea Nicolescu (mircea@cs.unr.edu) Dr. George Bebis Department of Computer Science & Engineering University of Nevada Reno, NV 89557, USA phone: (775) 784-6463 email: bebis@cs.unr.edu http://www.cs.unr.edu/~bebis Dr. Mircea Nicolescu Department of Computer Science & Engineering University of Nevada Reno, NV 89557, USA phone: (775) 784-4356 email: mircea@cs.unr .edu http://www.cs.unr.edu/~mircea === Subject: A type of regular graph Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For a graph with verticies the integers mod n, we may draw an edge between a and b iff b=a+i for some i in a set of residues mod n. Clearly such a graph is a regular graph, is it possible to characterize it further with known graph-theoretic properties? === Subject: Re: A type of regular graph 3QLpj-NoP*NzsIC,boYU]bQ]Hy<#4ga3$21: Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > For a graph with verticies the integers mod n, we may draw an edge > between a and b iff b=a+i for some i in a set of residues mod n. > Clearly such a graph is a regular graph, is it possible to > characterize it further with known graph-theoretic properties? These graphs are known as circulants, e.g. see . That doesnt answer your question, but it should at least help in searching for an answer. -- David Eppstein Computer Science Dept., Univ. of California, Irvine http://www.ics.uci.edu/~eppstein/ === Subject: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) hi world, im looking for toolkits of segments / magnets / plastic caps to build 3d shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. i want to play around with the cell decomposition of the lattice a_3, so ill need quite a bunch of these. i quickly searched on the net for some way of buying these online, with no success. does anybody know where i can find them? tia, laurent -- Laurent Bartholdi laurent.bartholdiepßch EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 === Subject: Re: building 3d shapes Originator: israel@math.ubc.ca (Robert Israel) > hi world, > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? One possibility: http://www.zometool.com/ -- http://hertzlinger.blogspot.com === Subject: Re: building 3d shapes Originator: israel@math.ubc.ca (Robert Israel) > hi world, > im looking for toolkits of segments / magnets / plastic caps to > build 3d shapes like the 1- or 2-skeleta of tetrahedra, octahedra, > etc. > i want to play around with the cell decomposition of the lattice a_3, > so ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, > with no success. does anybody know where i can find them? > tia, laurent Or try Geomag and Supermag (dont know if these are the same), e.g. at http://www.toymagnets.com/geomag/index.cfm or http://www.geomags.com/. Ive seen them sold in Swiss toyshops as well. Christian Graf === Subject: Re: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > hi world, > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? > tia, laurent > -- > Laurent Bartholdi laurent.bartholdiepßch > EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 > CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 For magnets look at http://www.supermagnete.ch/magnets.php?at=Z Hugo Pfoertner === Subject: Re: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > .... > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? .... Is http://www.korthalsaltes.com/ any use to you? Ken Pledger. === Subject: natural numbers as coequalizer (Re: Those Naughty Category Theorists) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) begin{quote} Incidentally for those who think that by defining numbers as lengths of strings Im making some sort of obscure automata theoretic point with my computer science hat on, let me just point out that not only can this representation be made entirely respectable mathematically, but via a slicker mathematical name than any other Ive seen proposed in this thread so far: the free monoid (N,+,0) on one generator. Just as punchy, and as clear to a category theorist as free monoid is to an algebraist: the coequalizer of the elements of (the ordinal) *2. The set-theoretic explanation of this is a bit clumsy, but here goes. The scene is Cat, the category of all small categories. The diagram to be coequalized is the left half of 0 ----> F *1 *2 ----> Coeq(0,1) ----> 1 with the coequalizer Coeq(0,1) and its coequalizing arrow F shown on the right. The ordinal *2 is the one-nonidentity-arrow category {0->1}, the ordinal *1 is the evident {0}, the elements 0,1 form the set Hom(*1,*2). The coequalizer of 0 and 1 creates Coeq(0,1), a copy of *2 which identifies 0 and 1. This has the side effect of looping the nonidentity arrow back on itself. Since we are in Cat, we now have to specify a composition law for this arrow with itself in the least constraining way, i.e. Coeq(0,1) has to be universal. Clearly we need all composites f, ff, fff, etc. Identifying any two of these is an unwanted constraint, so we leave them all unidentified. We now have a monoid whose arrows 1,f,ff,fff,... represent the natural numbers 0,1,2,3,..., composition represents addition, and the identity arrow represents 0. (Represent is meaningful only in this set-theoretic view.) The functor F takes both objects of *2 to the object of Coeq(0,1), and takes the nonidentity arrow of *2 to f or 1, the generator of (N,+,0). This construction may seem a bit contrived until you look at how category theoretic foundations are typically organized. (Good reading: McLarty, Axiomatizing a Category of Categories, J.Symbolic Logic, 56:4(Dec91).) Ordinal constructions involving the four ordinals up to *3, along with the product *2 x *2, are at the heart of this organization, and the above construction of the natural numbers as a monoid in Cat is not only slick but very natural and in that setting. end{quote} Can we construct in a similar manner integers, rational, reals and complex numbers? David, === Subject: Re: Schlomilchs series Epigone-thread: yendwholgrimp Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi--Im a newbie to special functions, I hope somebody can help me out. Im doing a problem in scattering, and it would be really helpful if I could sum the following series: sum_{n=1}^inf J_0(n x + b) where b is a real number. If b=0 then this is a bread & butter Schlomilch series (see e.g. Gradshteyn and Ryzhik 5th ed. 8.521). I tried expanding J_0(n x + b) via the addition theorem. But I didnt get too far. I was wondering if this was a known series...hoping the experts have some help! :) === Subject: Re: Schlomilchs series Epigone-thread: yendwholgrimp Content-Length: 782 Originator: rusin@vesuvius There is a sort of obvious way to do it if b is an integer...take your original sum, replace x by 2x and then subtract the two infinite sums, etc. For arbitrary b Im really not sure. >Hi--Im a newbie to special functions, I hope somebody can help me >out. Im doing a problem in scattering, and it would be really >helpful if I could sum the following series: > sum_{n=1}^inf J_0(n x + b) >where b is a real number. >If b=0 then this is a bread & butter Schlomilch series (see e.g. >Gradshteyn and Ryzhik 5th ed. 8.521). >I tried expanding J_0(n x + b) via the addition theorem. But I didnt >get too far. >I was wondering if this was a known series...hoping the experts have >some help! :) === Subject: Re: Schlomilchs series Content-Length: 1302 Originator: rusin@vesuvius Please dont top-post. Im putting the original question here where it belongs: >>Hi--Im a newbie to special functions, I hope somebody can help me >>out. Im doing a problem in scattering, and it would be really >>helpful if I could sum the following series: >> sum_{n=1}^inf J_0(n x + b) >There is a sort of obvious way to do it if b is an integer...take >your original sum, replace x by 2x and then subtract the two infinite >sums, etc. Sorry, I dont understand this. You seem to be saying take the original sum, J_0(x+b) + J_0(2x+b) + J_0(3x+b) + ... and subtract J_0(2x+b) + J_0(4x+b) + J_0(6x+b) + .... But that will just give you J_0(x+b) + J_0(3x+b) + J_0(5x+b) + ... and I dont see how that helps, or what difference b being an integer makes. On the other hand, if b is an integer multiple of x, you can say something: if F(x,b) is the sum, F(x,b+x) = F(x,b) - J_0(x+b) so F(x, kx) = F(x,0) - sum_{j=1}^k J_0(jx) for positive integers k Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: On a property of Bernoulli numbers Epigone-thread: clephoystald Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Let A(n,k)=(n^(2k)-1)*B(2k) where B(2k) denotes the 2k-th Bernoulli >number. Then I suspect the existence of a minimal positive rational >value, depending on n, r(n)=P(n)/Q(n) with (P(n),Q(n))=1 and such >that : >for any k>0 r(n)*A(n,k) is an integer value. >r(2)=2, r(3)=3/4 .... >P(n) appears to be the largest square-free divisor of n but I didnt >observation when n is a power of 2 : >for p prime, if 2^p-1 and (2^p+1)/3 are both primes then >Q(2^p)=(4^p-1)/3 (converse doesnt hold). >Can anyone confirm theorically the existence of r(n) and the formula >for P(n)? If so, what is the formula for Q(n)? Update : I found that r(n)=rad(n^3-n)/(n^2-1) where rad(n) is the square-free kernel of n, the largest square-free divisor of n. This explains why P(n)=rad(n) and we have Q(n)=(n^2-1)/rad(n^2-1). I cant say if this property of Bernoullis numbers is known. Studying r(n) I came across something looking as an integer formulation of Agohs conjecture : p is prime iff p divides (p^p-p)*B(p-1)-1 and I unearthed this amusing connection with 3-smooth numbers (numbers of form 2^i*3^j i,j>=0) : fractional part of ((n^(2k)-1)*B(2k)) is constant for any k>0 iff n is a 3-smooth number. Benoit Cloitre === Subject: Paper published by Algebraic and Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-42.abs.html Title: A class of tight contact structures on Sigma_2 x I Author(s): Tanya Cofer Abstract: We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case. Secondary: 53C15 Keywords: Tight, contact structure, genus-2 surface Author(s) address(es): Department of Mathematics, Northeastern Illinois University 5500 North St Louis Avenue, Chicago, IL 60625-4699, USA Email: T-Cofer@neiu.edu URL: http://www.neiu.edu/~tcofer/ === Subject: This week in the mathematics arXiv (18 Oct - 22 Oct) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Here are this weeks 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 (18 Oct - 22 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410375 Kiran S. Kedlaya: Finite automata and algebraic extensions of function fields math.AC/0410340 Claudia Polini, Bernd Ulrich: A formula for the core of an ideal AG: Algebraic Geometry ---------------------- math.AG/0410469 Frederic Campana: Fibres multiples des surfaces math.AG/0410458 Samuel Boissiere: Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane math.AG/0410444 S. Kaplan, E. Liberman, M. Teicher: Braid Monodromy Computation of Real Singular Curves math.AG/0410442 Nickolas Michelacakis, Apostolos Thoma: On the geometry of complete intersection toric varieties math.AG/0410432 Qi Zhang: On projective varieties with nef anticanonical divisors hep-th/0410055 Volker Braun, Burt A. Ovrut, Tony Pantev, Rene Reinbacher: Elliptic Calabi-Yau Threefolds with Z_3 x Z_3 Wilson Lines math.AG/0410408 Ivan Cheltsov: Double cubics and double quartics hep-th/0410018 A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger: Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections math.AG/0410401 Gabor Szekelyhidi: Extremal metrics and K-stability math.AG/0410394 Ana Cristina Lopez: Relative Jacobians of elliptic fibrations with reducible fibers math.AG/0410393 Ana Cristina Lopez: Simpson Jacobians of reducible curves math.AG/0410392 variety math.AG/0410388 M. E. Kazaryan, S. K. Lando: Towards the Intersection Theory on Hurwitz Spaces math.AG/0410383 Philibert Nang, Kiyoshi Takeuchi: Addendum to the paper Characteristic Cycles of Perverse Sheaves and Milnor Fibers math.AG/0410379 Seongchun Kwon: Transversality properties on the moduli space of genus 0 stable maps to a smooth rational projective surface and their real enumerative implications math.AG/0410378 Silvano Baggio: Equivariant K-Theory of Smooth Toric Varieties math.AG/0410360 Tyler J. Jarvis, William E. Lang, Nansen Petrosyan, Gretchen Rimmasch, Julie Rogers, Erin D. Summers: Classification of Singular Fibres on Rational Elliptic Surfaces in Characteristic Three hep-th/0410170 Bjorn Andreas, Daniel Hernandez Ruiperez: U(n) Vector Bundles on Calabi-Yau Threefolds for String Theory Compactifications math.AG/0410349 Igor Burban, Bernd Kreussler: On a relative Fourier-Mukai transform on genus one fibrations math.AG/0410346 Oliver Lorscheid: Completeness and compactness for varieties over local fields AP: Analysis of PDEs -------------------- math.AP/0410475 Zhongwei Shen: Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators math.AP/0410462 Fernando Cardoso, Georgi Vodev: Weighted L^p decay estimates of solutions to the wave equation with a potential math.AP/0410452 A.G.Ramm: Existence of a solution to a nonlinear equation math.AP/0410451 A.G.Ramm: A singular perturbation problem math.AP/0410443 Arnaud Debussche, Cyril Odasso: Ergodicity for the weakly damped stochastic non-linear Schrodinger equations math.AP/0410441 Giuseppe Da Prato, Arnaud Debussche, Luciano Tubaro: Coupling for some partial differential equations driven by white noise math.AP/0410431 Burak Erdogan, Wilhelm Schlag: Dispersive estimates for Schr{o}dinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I math.AP/0410416 Dian Palagachev, Lubomira Softova: Fine regularity for elliptic systems with discontinuous ingredients math.AP/0410415 Dian K. Palagachev, Lubomira G. Softova: Apriori estimates and precise regularity for parabolic systems with discontinuous data math.AP/0410380 Fabian Waleffe: On some dyadic models of the Euler equations math.AP/0410344 Isabelle Gallagher, Thierry Gallay, Pierre-Louis Lions: On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity AT: Algebraic Topology ---------------------- math.AT/0410405 Scott O. Wilson: Partial Algebras Over Operads of Complexes and Applications math.AT/0410398 R. Brown, H.K. Kamps, T. Porter: A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem math.AT/0410374 Norio Iwase, Donald Stanley, Jeffrey Strom: Implications of the Ganea Condition math.AT/0410367 Z. Fiedorowicz, R. M. Vogt: Topological Hochschild Homology of $E_n$-Ring Spectra math.AT/0410363 A.D.R. Choudary, A. Dimca, S. Papadima: Some Analogs of Zariski Theorem on Nodal Line Arrangements math.AT/0410342 Nicholas J. Kuhn: Goodwillie towers and chromatic homotopy: an overview CA: Classical Analysis and ODEs ------------------------------- math.CA/0410439 Jose L. Lopez, Nico M. Temme: Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials math.CA/0410436 Jose L. Lopez, Nico M. Temme: Multi-point Taylor Expansions of Analytic Functions math.CA/0410395 Sever Silvestru Dragomir: Some Inequalities for Functions of Bounded Variation with Applications to Landau Type Results CO: Combinatorics ----------------- math.CO/0410471 Michiel Hazewinkel: Word Hopf algebras math.CO/0410466 Charles F. Dunkl: Hook-lengths and Pairs of Compositions math.CO/0410455 David E Speyer: Tropical Linear Spaces math.CO/0410429 Jens Christian Claussen: Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration math.CO/0410425 Joseph E. Bonin, Omer Gimenez: Multi-Path Matroids math.CO/0410410 Annalies Vuong, M. Ian Wyckoff: Conditions for Weighted Cover Pebbling of Graphs math.CO/0410404 F. Bonetto, H. Matzinger: Fluctuations of the Longest Common Subsequence in the Asymmetric Case of 2- and 3-Letter Alphabets math.CO/0410382 B.M. Kim, Y. Rho: Van der Waerdens Theorem on Homothetic copies of {1,1+s, 1+s+t} math.CO/0410373 Ira M. Gessel, Louis H. Kalikow: Hypergraphs and a functional equation of Bouwkamp and de Bruijn math.CO/0410366 Michiel Hazewinkel: Explicit polynomial generators for the ring of quasi-symmetric functions over the integers math.CO/0410361 Howard Kleiman: The Floyd-WarshallAlgorithm and the Asymmetric TSP nlin.AO/0407024 Fatihcan M. Atay, Tuerker Biyikoglu, Juergen Jost: On the synchronization of networks with prescribed degree distributions math.CO/0410347 Svante Linusson, Johan Waestlund: Completing a k-1 assignment math.CO/0410345 Svante Linusson, John Shareshian, Volkmar Welker: Complexes of graphs with bounded matching size CT: Category Theory ------------------- math.CT/0410412 Dominic Verity: Complicial Sets CV: Complex Variables --------------------- math.CV/0410445 P. Ebenfelt, L. P. Rothschild: Transversality of CR mappings math.CV/0410420 Rostyslav O. Hryniv, Yaroslav V. Mykytyuk: Asymptotics of zeros for some entire functions math.CV/0410399 Vladimir V. Kisil, Debapriya Biswas: Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains math.CV/0410390 Franc Forstneric, Joerg Winkelmann: Holomorphic discs with dense images math.CV/0410386 Franc Forstneric, Christine Laurent-Thiebaut: Stein compacts in Levi-ßat hypersurfaces math.CV/0410376 Bertrand Deroin: Laminations dans les esapces projectifs complexes math.CV/0410362 Young-Heon Kim: Holomorphic extensions of determinants of Laplacians math.CV/0410353 J. J. Kohn: Superlogarithmic estimates on pseudoconvex domains and CR manifolds math.CV/0410343 Mikhail Sodin: Zeroes of Gaussian analytic functions math.CV/0410341 Fedor Nazarov, Mikhail Sodin: Coarse equidistribution of the argument of entire functions of finite order DG: Differential Geometry ------------------------- math.DG/0410461 Josef Janyv{s}ka: Natural connections given by general linear and classical connections math.DG/0410460 Tom Mestdag, Bavo Langerock: A Lie algebroid framework for non-holonomic systems math.DG/0410456 Mikhail G. Katz, Yuli B. Rudyak: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds nlin.SI/0407057 Paolo Lorenzoni, Marco Pedroni: On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations math.DG/0410435 Isabel Fernandez, Francisco J. Lopez: Relative parabolicity of zero mean curvature surfaces in $R^3$ and $R_1^3$ math.DG/0410434 Michael Schulze: On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry math.DG/0410418 Jian Song, Ben Weinkove: On the convergence and singularities of the J-ßow with applications to the Mabuchi energy math.DG/0410413 2-surfaces of prescribed mean curvature DS: Dynamical Systems --------------------- math.DS/0410464 I.Dynnikov, S.Novikov: Topology of quasiperiodic functions on the plane physics/0410160 R. Ball: The case of the trapped singularities math.DS/0410417 Charles Favre, Mattias Jonsson: Eigenvaluations math.DS/0410384 N. Haydn, Y. Lacroix & S. Vaienti: Hitting and return times in ergodic dynamical systems math.DS/0410355 Marco Lenci: Typicality of recurrence for Lorentz gases nlin.CD/0410019 Sylvie Oliffson Kamphorst, Sonia Pinto de Carvalho: The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on Billiards FA: Functional Analysis ----------------------- math.FA/0410427 W. B. Johnson, N. L. Randrianarivony: $ell_p$ (p>2) does not coarsely embed into a Hilbert space math.FA/0410422 Assaf Naor, Yuval Peres, Oded Schramm, Scott Sheffield: Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces math.FA/0410403 Stefan Bildea, Dorin Ervin Dutkay, Gabriel Picioroaga: MRA Super-wavelets math.FA/0410391 Roberto Giambo, Fabio Giannoni, Paolo Piccione: Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds math.FA/0410351 Frederic Bayart, Catherine Finet, Daniel Li, Herve Queffelec: Composition operators on the Wiener-Dirichlet algebra math.FA/0410348 Wojciech Czaja: Remarks on Naimarks duality GM: General Mathematics ----------------------- math.GM/0410377 Jacky Cresson: Non-differentiable variational principles GT: Geometric Topology ---------------------- math.GT/0410476 J.-F. Lafont: Strong Jordan separation and applications to rigidity math.GT/0410474 Brent Everitt, John Ratcliffe, Steven Tschantz: The smallest hyperbolic 6-manifolds math.GT/0410433 Carlo Petronio: Complexity of 3-orbifolds math.GT/0410381 Suhyoung Choi: Drilling cores of hyperbolic 3-manifolds to prove tameness math.GT/0410370 Jerzy Dydak, Michael Levin: Extension of maps to the projective plane math.GT/0410369 Michael Levin: Rational acyclic resolutions math.GT/0410368 Michael Levin: Universal acyclic resolutions for arbitrary coefficient groups math.GT/0410358 Robion Kirby, Paul Melvin: Local surgery formulas for quantum invariants and the Arf invariant math.GT/0410356 Eaman Eftekhary: Filtration of Heegaard Floer homology and gluing formulas HO: History and Overview ------------------------ math.HO/0410411 Tommaso Toffoli: Maxwells daemon, the Turing machine, and Jaynes robot math.HO/0410397 V.G.Gurzadyan: Kolmogorov and Aleksandrov in Sevan Monastery, Armenia, 1929 MG: Metric Geometry ------------------- math.MG/0410440 Andreas Balser, Alexander Lytchak: Centers of convex subsets of buildings math.MG/0410437 Andreas Balser, Alexander Lytchak: Building-like spaces math.MG/0410421 Alexander Lytschak, Viktor Schroeder: Affine functions on CAT(kappa) spaces MP: Mathematical Physics ------------------------ quant-ph/0410131 Xiong-Jun Liu, Hui Jing, Xin Liu, Mo-Lin Ge: Dynamical Symmetry and Its Applications In Electromagnetically Induced Transparency math-ph/0410046 Michiel Hazewinkel, Hugo H Torriani: Coherence and uniqueness theorems for averaging processes in statistical mechanics hep-th/0410199 A.P. Balachandran, A. Pinzul: On Time-Space Noncommutativity for Transition Processes and Noncommutative Symmetries hep-th/0008117 M. Hssaini, M. Kessabi, B. Maroufi, M.B.Sedra: Central extended D=2 N=4 SU(2) Liouville self interacting model and explicit hyperkahler metric math-ph/0410045 Alexei F. Cheviakov: Plasma equilibrium equations in coordinates connected with magnetic surfaces. Exact equilibrium solutions gr-qc/0410069 Antonio Lopez-Pinto: Nonstandard spin 2 field theory physics/0410127 A. Figotin, J. H. Schenker: Hamiltonian treatment of time dispersive and dissipative media within the linear response theory math-ph/0410044 Daniel Peralta-Salas: A geometric approach to the equilibrium shapes of self-gravitating ßuids hep-th/0410013 Patrick Dorey, Adam Millican-Slater, Roberto Tateo: Beyond the WKB approximation in PT-symmetric quantum mechanics cond-mat/0410435 F. Guerra: Mathematical aspects of mean field spin glass theory math-ph/0410043 Volodymyr Sushch: On some discrete model of the magnetic Laplacian math-ph/0410042 Jochen Bruening, Vladimir Geyler, Konstantin Pankrashkin: Continuity of integral kernels related to Schrodinger operators on manifolds math-ph/0410041 O.M. Kiselev, S.G. Glebov, V.A. Lazarev: Resonant pumping in nonlinear Klein-Gordon equation and solitary packets of waves math-ph/0410040 G.Giachetta, L.Mangiarotti, G.Sardanashvily: Geometric and Algebraic Topological Methods in Quantum Mechanics hep-th/0408241 S. Meljanac, A. Samsarov: Matrix oscillator and Calogero-type models math-ph/0410039 Nasser Saad, Richard L. Hall, Qutaibeh D. Katatbeh: Study of anharmonic singular potentials NT: Number Theory ----------------- math.NT/0410428 L.A.Gutnik: On the difference equation of the Poincare type math.NT/0410409 A. Agboola: Galois modules and p-adic representations math.NT/0410387 C. S. Rajan: Recovering modular forms and representations from tensor and symmetric powers math.NT/0410372 Mark van Hoeij: Solving conics over Q(t1,..,tk) OA: Operator Algebras --------------------- math.OA/0410449 Kenneth Davidson, Elias Katsoulis: Nest representations of directed graph algebras math.OA/0410426 Benjam{i}n Itza-Ortiz: Eigenvalues, K-theory and Minimal Flows math.OA/0410400 Marius Dadarlat: On the topology of the Kasparov groups and its applications OC: Optimization and Control ---------------------------- math.OC/0410467 Antonios Armaou, Ioannis G. Kevrekidis: Equation-free optimal switching policies for bistable reacting systems using coarse time-steppers PR: Probability --------------- math.PR/0410465 Federico Camia: Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation math.PR/0410459 Florent Benaych-Georges: Taylor expansions of R-transforms, application to supports and moments math.PR/0410457 Catherine Donati-Martin: Large deviations for Wishart processes math.PR/0410453 Patrick Cheridito, Freddy Delbaen, Michael Kupper: Dynamic monetary risk measures for bounded discrete-time processes math.PR/0410447 Michail Loulakis: On the Symmetry of the Diffusion Coefficient in Asymmetric Simple Exclusion math.PR/0410430 Yuval Peres, David Revelle: Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs math.PR/0410414 Robert C. Dalang, Carl Mueller, Lorenzo Zambotti: Hitting properties of s.p.d.e.s with reßection math.PR/0410402 David J. Aldous, Lea Popovic: A critical branching process model for biodiversity math.PR/0410371 Harry Kesten, Vladas Sidoravicius: A phase transition in a model for the spread of an infection math.PR/0410359 Bela Bollobas, Oliver Riordan: A short proof of the Harris-Kesten Theorem cond-mat/0410309 V. Sood, S. Redner, D. ben-Avraham: First Passage Properties of the Erdos-Renyi Random Graph QA: Quantum Algebra ------------------- math.QA/0410470 Michiel Hazewinkel: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions II math.QA/0410468 Michiel Hazewinkel: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions math.QA/0410463 S. Sinelshchikov, A. Stolin, L. Vaksman: A Quantum Analogue of the Bernstein Functor math.QA/0410450 J.E. McClure: On the chain-level intersection pairing for PL manifolds math.QA/0410448 Christian Blohmann: Reconstruction of universal Drinfeld twists from representations math.QA/0410446 Haisheng Li, Gaywalee Yamskulna: On certain vertex algebras and their modules associated with vertex algebroids math.QA/0410407 K. Szlachanyi: Monoidal Morita equivalence math.QA/0410396 S. Sinelshchikov, L. Vaksman: Quantum groups and non-commutative complex analysis math.QA/0410389 Harald Grosse, Stefan Schraml: The Eigenfunctions of the q-Harmonic Oscillator on the Quantum Line math.QA/0410365 Michiel Hazewinkel: The primitives of the Hopf algebra of noncommutative symmetric functions math.QA/0410364 Michiel Hazewinkel: Hopf algebras of endomorphisms of Hopf algebras math.QA/0410350 Henrique Bursztyn, Stefan Waldmann: Hermitian star products are completely positive deformations RA: Rings and Algebras ---------------------- math.RA/0410473 Gizem Karaali: A New Lie Bialgebra Structure on sl(2,1) math.RA/0410406 Michael Pinsker: The number of unary clones containing the permutations on an infinite set RT: Representation Theory ------------------------- math.RT/0410472 Paolo Bravi, Guido Pezzini: Wonderful varieties of type D math.RT/0410454 Francois Digne, Jean Michel, Raphael Rouquier: Cohomologie des varietes de Deligne-Lusztig math.RT/0410423 Calin Chindris: Quivers, long exact sequences and Horn type inequalities math.RT/0410357 Helmer Aslaksen, Mong Lung Lang: Extending $pi$-systems to bases of root systems SG: Symplectic Geometry ----------------------- math.SG/0410352 Eaman Eftekhary: Embedded curves and Gromov-Witten invariants of three-folds SP: Spectral Theory ------------------- math.SP/0410438 M.A. Kaashoek, A.L. Sakhnovich: Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model ST: Statistics -------------- math.ST/0410424 George Kahrimanis, Daniel Berleant: Direct pivotal predictive inference math.ST/0410419 Grace Wahba: An introduction to (smoothing spline) ANOVA models in RKHS with examples in geographical data, medicine, atmospheric science and machine learning math.ST/0410385 Mario Ruetti, Matthias Troyer, Wesley P. Petersen: A Generic Random Number Generator Test Suite math.ST/0410354 Stephane Gaiffas: Rates of convergence for pointwise curve estimation with a degenerate design -- / 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 thats fit to e-print * === Subject: Question about Gomory-Cut Originator: israel@math.ubc.ca (Robert Israel) for an application I have to find an integer solution. I use the simplex-algorithm and the Gomory-cut. It works fine, but it is to slowly. There are multiply opportunities to make a Gomory-cut. Which Gomory-cuts should I take, to get a solution in shortest time ? Where can I get information about that (in the internet) ? Ulrich === Subject: Combinatorial inequality Epigone-thread: wunloifo Originator: israel@math.ubc.ca (Robert Israel) Can anyone help me in deriving this inequality which appears to be true? {C(n,k)/C(a,k)}*(m/n)^k <={((n-m)/(n-a))^(n-a)}*{(n/a)^a} where k=(a-m)/(1-m/n); and 0Can anyone help me in deriving this inequality which appears to be >true? > {C(n,k)/C(a,k)}*(m/n)^k <={((n-m)/(n-a))^(n-a)}*{(n/a)^a} > where k=(a-m)/(1-m/n); and 1 Here C(n,k) means the binomial coefficient and ^ means >exponentiation,* denotes ordinary multiplication. === Subject: Conformal Mapping Question Originator: israel@math.ubc.ca (Robert Israel) Id like to find the explicit formula of a bijective conformal mapping from the following region, say K, K={ x+iy in C mid x>0, y > arccos(e^{-x}) } to the interior of the unit disk (K is just the subregion in the first quadrant of the complex plane bounded below by the graph of e^{x}cos(y)=1). Ive tried basic ones and looked into Dictionary of Conformal Representations by H. Kober, but so far nothing works for me. Any suggestions are greatly appreciated. -- So Okada Ph.D. Student of Math at UMass Amherst okada@math.umass.edu === Subject: This week in the mathematics arXiv (25 Oct - 29 Oct) Originator: israel@math.ubc.ca (Robert Israel) Here are this weeks 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 Oct - 29 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410598 Kamran Divaani-Aazar, Amir Mafi: Associated primes of local cohomology module math.AC/0410585 Nicholas Baeth: A Krull-Schmidt Theorem for One-dimensional Rings of Finite Cohen-Macaulay Type math.AC/0410535 Anurag K. Singh, Uli Walther: On the arithmetic rank of certain Segre products math.AC/0410497 Juan C. Migliore, Uwe Nagel, Tim Romer: The Multiplicity Conjecture in low codimensions math.AC/0410478 Carlos DAndrea, Laurent Buse: Properness and inversion problems by means of matrices AG: Algebraic Geometry ---------------------- math.AG/0410604 Elizabeth S. Allman, John A. Rhodes: Phylogenetic ideals and varieties for the general Markov model math.AG/0410602 E. Carlini: Codimension one decompositions and Chow varieties math.AG/0410600 J. C. Sierra, L. Ugaglia: On double Veronese embeddings in the Grassmannian G(1,N) math.AG/0410584 Carolina Araujo: Rational curves of minimal degree and characterizations of ${mathbb P}^n$ math.AG/0410572 Israel Moreno Mej{i}a: The trace of an automorphism on H^0(J,O(nTheta)) math.AG/0410558 Ivan Cheltsov: Birationally superrigid cyclic triple spaces math.AG/0410554 Meirav Amram, David Goldberg: Higher degree Galois covers of CP^1 x T math.AG/0410547 D. A. Stepanov: Non-rational divisors over non-Gorenstein terminal singularities math.AG/0410540 Pan Peng: A simple proof of Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds math.AG/0410537 Eduardo Esteves, Steven Kleiman: The compactified Picard scheme of the compactified Jacobian math.AG/0410527 Cristiano Bocci: Special effect varieties and (-1)-curves math.AG/0410526 Shihoko Ishii: Arcs, valuations and the Nash map math.AG/0410524 Boris E. Kunyavskii, Louis H. Rowen, Sergey V. Tikhonov, Vyacheslav I. Yanchevskii: Division algebras that ramify only on a plane quartic curve math.AG/0410520 Laurent Manivel, Emilia Mezzetti: On linear spaces of skew-symmetric matrices of constant rank math.AG/0410518 Elena Drozd: Curves on a nonsingular Del Pezzo Surface in $P^4_k$ math.AG/0410513 Kalle Karu: The cd-index of fans and lattices AP: Analysis of PDEs -------------------- math.AP/0410581 G. Olafsson, A. Pasquale: Support properties and Holmgrens uniqueness theorem for differential operators with hyperplane singularities math.AP/0410564 James Nolen, Jack Xin: A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears math.AP/0410546 Plamen Stefanov, Gunther Uhlmann: Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map math.AP/0410538 J. Colliander, W. Staubach: $L^2$ blowup solutions of cubic NLS on $R^2$ concentrate a fixed amount of mass math.AP/0410525 Gianni Dal Maso, Rodica Toader: On a notion of unilateral slope for the Mumford-Shah functional math.AP/0410499 Hans Lindblad, Jacob Sterbenz: Global Stability for Charge Scalar Fields on Minkowski Space AT: Algebraic Topology ---------------------- math.AT/0410589 Nora Ganter: Smash products of E(1)-local spectra at an odd prime math.AT/0410552 Javier Turiel: Polynomials Maps and Even Dimensional Spheres math.AT/0410503 Kathryn Hess, Ran Levi: An algebraic model for the loop homology of a homotopy fiber CA: Classical Analysis and ODEs ------------------------------- math.CA/0410548 Marc Artzrouni: A new family of periodic functions as explicit roots of a class of polynomial equations math.CA/0410542 Projections And Universal Encoding Strategies math.CA/0410508 Stephen Semmes: Potpourri, 8 math.CA/0410490 Stephen Semmes: Potpourri, 7 math.CA/0410489 Stephen Semmes: Potpourri, 6 math.CA/0410483 A. A. Bolibruch, S. Malek, C. Mitschi: On the generalized Riemann-Hilbert problem with irregular singularities CO: Combinatorics ----------------- math.CO/0410592 S. Ole Warnaar: Hall--Littlewood functions and the A_2 Rogers--Ramanujan identities quant-ph/0410226 P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon: Deformed Bosons: Combinatorics of Normal Ordering math.CO/0410550 Ewa Krot: Further develpoements in finite fibonomial calculus math.CO/0410529 math.CO/0410482 Michael Anshelevich: Orthogonal polynomials with a resolvent-type generating function CT: Category Theory ------------------- math.CT/0410555 Alan Robinson: Partition complexes, duality and integral tree representations gr-qc/0410104 J. Daniel Christensen, Louis Crane: Causal sites as quantum geometry CV: Complex Variables --------------------- math.CV/0410599 Laurent Gendre: Inegalites de Markov tangentielles locales sur les courbes algebriques singulieres de R^n math.CV/0410578 Dmitri Prokhorov, Alexander Vasilev: Optimal control in Bombieris and Tammis conjectures math.CV/0410509 David E. Barrett: A ßoating body approach to Feffermans hypersurface measure DG: Differential Geometry ------------------------- math.DG/0410610 Francisco Martin Cabrera: SU(3)-structures on hypersurfaces of manifolds with $G_2$-structure hep-th/0410183 Anton Alekseev, Thomas Strobl: Current Algebras and Differential Geometry math.DG/0410579 Jorge Lauret: A canonical compatible metric for geometric structures on nilmanifolds math.DG/0410575 Joseph H.G. Fu: Structure of the unitary valuation algebra math.DG/0410561 math.DG/0410559 Tomasz S. Mrowka, Yann Rollin: Legendrian knots and monopoles math.DG/0410557 A. V. Kiselev, G. Manno: On the symmetry structure of the minimal surface equation math.DG/0410553 Anton Deitmar: A prime geodesic theorem for higher rank II: singular geodesics math.DG/0410551 Eduardo Martinez: Classical field theory on Lie algebroids: Variational aspects math.DG/0410512 Maks A. Akivis, Vladislav V. Goldberg, Arto V. Chakmazyan: Induced connections on submanifolds in spaces with fundamental groups math.DG/0410511 Maks A. Akivis, Vladislav V. Goldberg: Dually degenerate varieties and the generalization of a theorem of Griffiths--Harris math.DG/0410498 Boris S. Kruglikov, Vladimir S. Matveev: Strictly non-proportional geodesically equivalent metrics have $h_text{top}(g)=0$ math.DG/0410494 George Papadopoulos: Spin Cohomology math.DG/0410493 Abdenago Barros G. Pacelli Bessa: Estimates of the first eigenvalue of minimal hypersurfaces of $mathbb{S}^{n+1} math.DG/0410487 Hiroshi Iritani: Quantum D-modules and equivariant Floer theory for free loop spaces math.DG/0410484 Aleksis Raza: An application of Guillemin-Abreu theory to a non-abelian group action DS: Dynamical Systems --------------------- math.DS/0410580 I. Binder, M. Braverman, M. Yampolsky: Filled Julia sets with empty interior are computable math.DS/0410517 Le Van Hien: Stability of Solutions of Fuzzy Differential Equations math.DS/0410507 Topologies on the group of homeomorphisms of a Cantor set math.DS/0410506 Topologies on the group of Borel automorphisms of a standard Borel space math.DS/0410505 Sergey Bezuglyi, Anthoni H. Dooley, Konstantin Medynets: The Rokhlin lemma for homeomorphisms of a Cantor set math.DS/0410504 Sergey Bezuglyi, Konstantin Medynets: Smooth automorphisms and path-connectedness in Borel dynamics math.DS/0410500 Ara Basmajian, Mahmoud Zeinalian: Maximal Convergence Groups and Rank One Symmetric Spaces math.DS/0410481 Pavlos B. Konstadinidis: The Real 3x+1 Problem FA: Functional Analysis ----------------------- math.FA/0410596 Ralf Meyer: Embeddings of derived categories of bornological modules math.FA/0410573 Jorge Antezana, Gustavo Corach, Demetrio Stojanoff: Spectral shorted operators math.FA/0410571 Massimo Fornasier, Holger Rauhut: Continuous Frames, Function Spaces, and the Discretization Problem math.FA/0410567 A. Brudnyi: Contractibility of Maximal Ideal Spaces of Certain Algebras of Almost Periodic Functions math.FA/0410549 Massimo Fornasier: Banach frames for alpha-modulation spaces math.FA/0410501 V.Yaskin: The Busemann-Petty problem in hyperbolic and spherical spaces math.FA/0410496 A.Koldobsky, V.Yaskin, M.Yaskina: Modified Busemann-Petty problem on sections of convex bodies math.FA/0410491 T. Banks, T. Constantinescu, Nermine El-Sissi: Tensor algebras and displacement structure. IV. Invariant kernels math.FA/0410479 A.G.Ramm: Dynamical systems method (DSM) for nonlinear equations in Banach spaces GM: General Mathematics ----------------------- math.GM/0410556 Joao R. Cardoso: An Explicit Formula for the Matrix Logarithm GR: Group Theory ---------------- math.GR/0410593 Henrik Baarnhielm: The Schreier-Sims algorithm for matrix groups math.GR/0410590 Edith Adan-Bante: Products of characters with few irreducible constituents math.GR/0410583 Edith Adan-Bante: Products of characters and derived length II math.GR/0410582 Edith Adan-Bante: Squares of characters and groups of odd order math.GR/0410539 Daniel Farley, Lucas Sabalka: Discrete Morse theory and graph braid groups math.GR/0410533 Stephen DeBacker: Parametrizing nilpotent orbits via Bruhat-Tits theory math.GR/0410516 J.Mostovoy, J.M. Perez-Izquierdo: Dimension filtration on loops math.GR/0410515 Jacob Mostovoy: On the notion of lower central series for loops GT: Geometric Topology ---------------------- math.GT/0410606 Greg Friedman: Knot spinning math.GT/0410603 R. C. Penner, Dennis Sullivan: The Structure and Singularities of Arc Complexes math.GT/0410595 Pascal Hubert, Samuel Lelievre: Noncongruence subgroups in H(2) math.GT/0410570 Andras Nemethi: On the Heegaard Floer homology of S^3_{-p/q}(K) math.GT/0410565 Brooke Brennan, Thomas W. Mattman, Roberto Raya, Dan Tating: Ribbonlength of torus knots math.GT/0410541 Ensil Kang, J. Hyam Rubinstein: Ideal triangulations of 3--manifolds I: spun normal surface theory math.GT/0410495 Dror Bar-Natan: Khovanovs Homology for Tangles and Cobordisms KT: K-Theory and Homology ------------------------- math.KT/0410597 Ralf Meyer: Combable groups have group cohomology of polynomial growth LO: Logic --------- math.LO/0410523 Fredrik Engstrom: Expansions, omitting types, and standard systems MG: Metric Geometry ------------------- math.MG/0410566 Piotr W. Nowak: On coarse embeddability into $ell_p$-spaces and a conjecture of Dranishnikov MP: Mathematical Physics ------------------------ quant-ph/0410201 Kazuyuki Fujii: Jaynes-Cummings Model and a Non-Commutative Geometry : A Few Problems Noted math-ph/0410062 David Damanik, Daniel Lenz, Gunter Stolz: Lower Transport Bounds for One-Dimensional Continuum Schrodinger Operators math-ph/0410061 P. Di Francesco, P. Zinn-Justin: Razumov-Stroganov sum rule: a proof based on multi-parameter generalizations math-ph/0410060 A.W.Beckwith: How false vacuum synthesis of a universe sets initial conditions which permit the onset of variations of a nucleation rate per Hubble volume per Hubble time math-ph/0410059 Manfred Requardt: Supersymmetry on Graphs and Networks math-ph/0410058 Vitaly V. Bulatov, Yuriy V. Vladimirov, Vasily A. Vakorin: Weak Singularity for Two-Dimensional Nonlinear Equations of Hydrodynamics and Propagation of Shock Waves math-ph/0410057 Joseph V. Pule, Andre F. Verbeure, Valentin A. Zagrebnov: Models with Recoil for Bose-Einstein Condensation and Superradiance math-ph/0410056 Petko Nikolov, Tihomir Valchev: Description of all conformally invariant differential operators, acting on scalar functions astro-ph/0404408 Ing-Guey Jiang, Li-Chin Yeh: On the Chaotic Orbits of Disc-Star-Planet Systems math-ph/0410055 A. A. Hujeirat: Problem-orientable numerical algorithm for modelling multi-dimensional radiative MHD ßows in astrophysics -- the hierarchical solution scenario math-ph/0410054 A.A. Hujeirat: A method for enhancing the stability and robustness of explicit schemes in astrophysical ßuid dynamics hep-th/0410172 Beatriz Gato-Rivera: The Adapted Ordering Method in Representation Theory math-ph/0410053 Alexander Rybko, Senya Shlosman: Poisson Hypothesis for Information Networks II. Cases of Violations and Phase Transitions math-ph/0410052 A.C.D.van Enter, E.A.Verbitskiy: On the Variational Principle for Generalized Gibbs Measures math-ph/0410051 Xavier Gracia, Ruben Martin: Time-dependent singular differential equations math-ph/0410050 Nicolae Cotfas: Systems of orthogonal polynomials defined by hypergeometric type equations math-ph/0410049 Joachim Kupsch, Subhashish Banerjee: Ultracoherence and Canonical Transformations math-ph/0410048 Sebastian Bauer: Post-Newtonian approximation of the Vlasov-Nordstrom system hep-th/0410212 C. Chryssomalakos, E. Okon: Generalized Quantum Relativistic Kinematics: a Stability Point of View cond-mat/0410424 Sandeep Tyagi: New series representation for Madelung constant quant-ph/0410151 S. Twareque Ali, F. Bagarello: Some Physical Appearances of Vector Coherent States and CS Related to Degenerate Hamiltonians math-ph/0410047 Volodymyr Sushch: Discrete model of Yang-Mills equations in Minkowski space hep-th/0410109 S. Odake, R. Sasaki: Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials hep-th/0410102 S. Odake, R. Sasaki: Shape Invariant Potentials in Discrete Quantum Mechanics NA: Numerical Analysis ---------------------- math.NA/0410488 Paul Sablonniere: Recent Results on Near-Best Spline Quasi-Interpolants NT: Number Theory ----------------- math.NT/0410563 Dragos Ghioca: The Mordell-Lang Theorem for Drinfeld modules math.NT/0410536 construction of some Galois modules math.NT/0410531 Takashi Taniguchi: A mean value theorem for the square of class numbers of quadratic fields math.NT/0410522 A. Ivic, E. Kratzel, M. Kuhleitner, W.G. Nowak: Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic math.NT/0410519 Howard Kleiman: Bounds for the Solutions of Cubic Diophantine Equations math.NT/0410502 Avner Ash, David Pollack, Warren Sinnott: A_6-extensions of Q and the mod p cohomology of GL(3,Z) math.NT/0410477 Bernd C. Kellner: Some remarks on Kurepas left factorial OA: Operator Algebras --------------------- math.OA/0410607 P. S. Muhly, M. Skeide, B. Solel: Representations of B^a(E) math.OA/0410601 Marek Bozejko, Wlodzimierz Bryc: On a class of free Levy laws related to a regression problem math.OA/0410594 Kenley Jung: Some free entropy dimension inequalities for subfactors math.OA/0410587 Wei Wu: Non-commutative metric topology on matrix state space math.OA/0410534 Todd Kemp: Strong hypercontractivity in non-commutative holomorphic spaces math.OA/0410492 Gelu Popescu: Unitary invariants in multivariable operator theory math.OA/0410480 Marius Ionescu, Yasuo Watatani: $C^{ast}$-Algebras associated with Mauldin-Williams Graphs PR: Probability --------------- math.PR/0410569 Aaron Abrams, Henry Landau, Zeph Landau, James Pommersheim, Eric Zaslow: Random Multiplication Approaches Uniform Measure in Finite Groups math.PR/0410560 Elchanan Mossel, Ryan ODonnell, Oded Regev, Jeffrey Steif, Benjamin Sudakov: Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality math.PR/0410545 Ravi Montenegro: Vertex and edge expansion properties for rapid mixing math.PR/0410544 Frederik Herzberg: The fairest price of an asset in an environment of temporary arbitrage math.PR/0410543 Frederik Herzberg: On measures of unfairness and an optimal currency transaction tax math.PR/0410532 Anna Rudas: Random tree growth with general weight function math.PR/0410514 Jianjun Tian, Xiao-Song Lin: Colored Genealogical Trees and Coalescent Theory math.PR/0410510 Anna Karczewska: Properties of convolutions arising in stochastic Volterra equations math.PR/0410485 Jacques Franchi, Yves Le Jan: Relativistic Diffusions and Schwarzschild Space cond-mat/0410543 Federico Camia: A Note on Edwards Hypothesis for Zero-Temperature Ising Dynamics QA: Quantum Algebra ------------------- math.QA/0410605 D. Shklyarov, S. Sinelshchikov, L. Vaksman: Fock Representations and Quantum Matrices math.QA/0410562 Vasiliy Dolgushev, Pavel Etingof: Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology math.QA/0410530 S. Sinelshchikov, L. Vaksman: Quantum Groups and Bounded Symmetric Domains math.QA/0410528 Michel Van den Bergh: Double Poisson algebras math.QA/0410486 Vladimir D. Lyakhovsky: On a class of skew classical r-matrices with large carrier RA: Rings and Algebras ---------------------- math.RA/0410591 Aaron Lauve: NSym into Q_{infty} is not a Hopf Map math.RA/0410576 Friedrich Wehrung, Jiri Tuma: Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties math.RA/0410521 Jerzy Matczuk: Ore Extensions over Duo Rings RT: Representation Theory ------------------------- math.RT/0410588 Konstantin Styrkas: Regular representation on the big cell and big projective modules in the category O SG: Symplectic Geometry ----------------------- math.SG/0410609 Joa Weber: Noncontractible periodic orbits in cotangent bundles and Floer homology math.SG/0410608 Weimin Chen: Pseudoholomorphic curves in four-orbifolds and some applications math.SG/0410568 Eugene Lerman: Gradient ßow of the norm squared of a moment map SP: Spectral Theory ------------------- math.SP/0410577 Sergio Albeverio, Alexander K. Motovilov: Operator integrals with respect to a spectral measure and solutions to some operator equations ST: Statistics -------------- math.ST/0410586 Rasa Karapandza, Milos Bozovic: You Can Fool Some People Sometimes math.ST/0410574 Igor Podlubny: A note on comparison of scientific impact expressed by number of citations in different fields of science -- / 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 thats fit to e-print * === Subject: 3rd cohomology of loop group Originator: israel@math.ubc.ca (Robert Israel) Id like to know the 3rd cohomology group of the loop group OmegaE_8 over E_8, H^3(OmegaE_8,H) , with H some abelian group. Does anyone know where I could find respective information? === Subject: Re: Open questions related to periodic continued fractions Originator: israel@math.ubc.ca (Robert Israel) Diana Mecum asked: > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? If n is not a perfect square, let p(n) be the length of the period of the s.c.f. of sqrt(n). In terms of n, how large can p(n) be? Its been shown that p(n) = O(sqrt(n) log(n)). It follows from some generalization of the Riemann hypothesis that p(n) = O(sqrt(n) log(log(n))). It seems likely that p(n)/sqrt(n) is unbounded, but I dont think its even been shown that it doesnt tend to 0. The first 23 record-setting values of p(n)/sqrt(n) are shown below: n p(n) p(n)/sqrt(n) 2 1 0.70711 3 2 1.15470 7 4 1.51186 43 10 1.52499 46 12 1.76930 211 26 1.78991 331 34 1.86881 631 48 1.91085 919 60 1.97922 1726 88 2.11818 4846 152 2.18349 7606 194 2.22445 10399 228 2.23583 10651 234 2.26736 10774 238 2.29292 18379 322 2.37517 19231 332 2.39407 32971 438 2.41217 48799 544 2.46260 61051 614 2.48497 78439 696 2.48510 82471 716 2.49323 111094 834 2.50219 See http://www.research.att.com/projects/OEIS?Anum=A003285 for some references. Dean Hickerson dean@math.ucdavis.edu === Subject: Re: Open questions related to periodic continued fractions > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? As you may know, the outstanding problem in this area is to improve on known conditions for the length of the period of the continued fraction expansion of sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having solutions). The following might be a current summary of known conditions: B. D. Beach and H. C. Williams, A Numerical Investigation of the Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Mathematica Publishing Inc., Winnipeg, Canada, 1972, pages 37 to 52. A less well known problem is as follows. As far as I know, this is an open problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo 4, let L1(D) and L4(D) denote the lengths of the periods of the continued fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). As far as I can tell, this is not prohibited by results in the literature. I have empirical evidence that it is not possible based on testing those D up to 30 billion that have L4(D) <= 255. It is not hard to show this for one particular case, namely if L4(D) = 3 then L1(D) cannot be 7. See discussion under the heading ``Periods of Continued Fractions in April and May of 2000 in the archives of the Number Theory Listserver at http://listserv.nodak.edu/archives/nmbrthry.html for related comments and some references that might be of interest. John Robertson === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox4.ucsd.edu: domain of news@newsread1.news.pas.earthlink.net does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? > As you may know, the outstanding problem in this area is to improve on known > conditions for the length of the period of the continued fraction expansion of > sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having > solutions). The following might be a current summary of known conditions: > B. D. Beach and H. C. Williams, A Numerical Investigation of the > Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, > Proceedings of the Third Southeastern Conference on Combinatorics, > Graph Theory and Computing, Utilitas Mathematica Publishing Inc., > Winnipeg, Canada, 1972, pages 37 to 52. > A less well known problem is as follows. As far as I know, this is an open > problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo > 4, let L1(D) and L4(D) denote the lengths of the periods of the continued > fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question > is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). > As far as I can tell, this is not prohibited by results in the literature. I > have empirical evidence that it is not possible based on testing those D up to > 30 billion that have L4(D) <= 255. It is not hard to show this for one > particular case, namely if L4(D) = 3 then L1(D) cannot be 7. > See discussion under the heading ``Periods of Continued Fractions in April > and May of 2000 in the archives of the Number Theory Listserver at > http://listserv.nodak.edu/archives/nmbrthry.html > for related comments and some references that might be of interest. > John Robertson === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Regarding the maximal element in the CF of sqrt(n) (say M(n)) . Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this behaviour : sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 and = 0.8... I dont know references on this subject. Should exist some. B. Cloitre === Subject: Re: Open questions related to periodic continued fractions Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Regarding the maximal element in the CF of sqrt(n) (say M(n)) . > Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this > behaviour : > sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 > and = 0.8... > I dont know references on this subject. Should exist some. Im not sure what ``sqrtint(n) is. The maximum partial quotient in the continued fraction expansion of sqrt(n) (for n not a square) is 2 times the integer part of the square root of n. This is proved in most references that consider the continued fraction expansion of sqrt(n), e.g., Mollins Fundamental Number Theory with Applications, or Niven, Zuckerman, and Montgomery. Also, (1/n) Sum_{i=1}^{n} [sqrt(i) - int(sqrt(i))] tends to 1/2. John Robertson === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Regarding the maximal element in the CF of sqrt(n) (say M(n)) . >Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this >behaviour : >sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 and = >0.8... >I dont know references on this subject. Should exist some. >B. Cloitre May be the following remarks are all well known: observe for instance sqrt(41) giving sequence (6;1/2,1/2,1/12,...) the period (1/2,1/2,1/12 ..)is linked to function (1/2,1/2,1/12,x) or (5x+62)/(2x+25) with two fixed points ,the positive -6+sqrt(41) is related to our continuous fraction. f(x)=(5x+62)/(2x+25) is easily iterated (sci.math 20 oct), Friendly yours,Alain. === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Content-Length: 2687 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Diana, in case it can help. Is anything known about the continued fraction of log_2(3)? (that is, of the logarithm of 3, base 2) Is it bounded,...!?? Maybe there are some results on the continued fractions of logarithms.. I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, for that number. (One more thing, is log_2(3) algebraic or trascendent?..!) Jose >> I am starting research for a thesis on continued fractions, and want >> to look at open questions related to periodic continued fractions. Is >> anyone aware of current open questions of interest? > As you may know, the outstanding problem in this area is to improve on >known >> conditions for the length of the period of the continued fraction >expansion of >> sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 >having >> solutions). The following might be a current summary of known conditions: >> B. D. Beach and H. C. Williams, A Numerical Investigation of the >> Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, >> Proceedings of the Third Southeastern Conference on Combinatorics, >> Graph Theory and Computing, Utilitas Mathematica Publishing Inc., >> Winnipeg, Canada, 1972, pages 37 to 52. >> A less well known problem is as follows. As far as I know, this is an >open >> problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 >modulo >> 4, let L1(D) and L4(D) denote the lengths of the periods of the continued >> fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The >question >> is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod >6). >> As far as I can tell, this is not prohibited by results in the literature. >> have empirical evidence that it is not possible based on testing those D >up to >> 30 billion that have L4(D) <= 255. It is not hard to show this for one >> particular case, namely if L4(D) = 3 then L1(D) cannot be 7. >> See discussion under the heading ``Periods of Continued Fractions in >April >> and May of 2000 in the archives of the Number Theory Listserver at >> http:// listserv.no dak.edu/archives/nmbrthry.html> for related comments and some references that might be of interest. >> John Robertson === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox3.ucsd.edu: domain of news@nntp.itservices.ubc.ca does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Diana, > in case it can help. > Is anything known about the continued fraction of log_2(3)? (that >is, of the logarithm of 3, base 2) > Is it bounded,...!?? > Maybe there are some results on the continued fractions of >logarithms.. I doubt that this will help. Very little is known about the continued fractions of closed-form numbers apart from rationals and quadratic irrationals, and AFAIK there is no prospect, with currently available mathematical techniques, of being able to prove whether the continued fraction of a number such as this has bounded elements. IMHO this is not a problem to give to a student starting her thesis. > I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, >for that number. > (One more thing, is log_2(3) algebraic or trascendent?..!) Transcendental, by the Gelfond-Schneider Theorem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Open questions related to periodic continued fractions Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Diana, > in case it can help. > Is anything known about the continued fraction of log_2(3)? (that > is, of the logarithm of 3, base 2) > Is it bounded,...!?? No, its unbounded. But, so far, nobody knows how to prove that. > Maybe there are some results on the continued fractions of > logarithms.. > I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, > for that number. How does that compare to almost all continued fractions? > (One more thing, is log_2(3) algebraic or trascendent?..!) It is transcendental. -- G. A. Edgar edgar at math.ohio-state.edu === Subject: Re: Galois group of a given quartic equation Epigone-thread: twomprendlex Originator: israel@math.ubc.ca (Robert Israel) The pending issue is whether the quartic Q(x) = 0 may have the Galois group isomorphic to Z4 for a and b different. No progress has been posted. One way to rule out D4 in the Z4/D4 case is given below. (If Q(x) is irreducible over Z, the discriminant is not a square and the cubic resolvent R(t) = 0 of the quartic has one and only one integral root t0, then the Galois group G is either Z4 or D4.) Let R(t) = (t - t0) r(t) where r(t) is a monic irreducible quadratic with integral coefficients. Further, let the discriminant of r(t) be D. Let d be the squarefree part of D and E the splitting field of R(t) = 0. Then E = Q[Sqrt[d]] and therefore easy to determine. Then put E to use as follows: If Q(x) is reducible over E then G is Z4; otherwise G is D4. May be this can be used to settle the case whether G may be Z4 or not. Kent Holing === Subject: matrix derivative Epigone-thread: zoypryrwhou Originator: israel@math.ubc.ca (Robert Israel) I am currently having a matrix derivative problem What is derivative of trace{(A+F*B*F)^(-1)} with respect to matrix F where () is the transpose of a matrix. A and B is diagonal matrices. I searched online and was only able to find the derivative of d trace{(F*B*F)^(-1)}/d F =-2*B*F*(F*B*F)^(-2) without knowing how they get it. Moreover, all these matrix derivative problems seem to be difficult forme. Could anyone be kind enough to give me some good references on this topic. === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) > I am currently having a matrix derivative problem > What is derivative of > trace{(A+F*B*F)^(-1)} > with respect to matrix F > where () is the transpose of a matrix. > A and B is diagonal matrices. > I searched online and was only able to find the derivative of > d trace{(F*B*F)^(-1)}/d F > =-2*B*F*(F*B*F)^(-2) > without knowing how they get it. > Moreover, all these matrix derivative problems seem to be > difficult forme. Could anyone be kind enough to give me > some good references on this topic. d A denotes the matrix differential for the matrix A. x = trace{ (A+F*B*F)^(-1) } d x = trace{ d ( (A+F*B*F)^(-1) ) } = trace{ - (A+F*B*F)^(-1) ( d (A+F*B*F) ) (A+F*B*F)^(-1) } = trace{ - (A+F*B*F)^(-2) ( d (F*B*F) ) } = trace{ - (A+F*B*F)^(-2) ( d F *B*F + F*B* d F ) } = - ( trace{ (A+F*B*F)^(-2) d F *B*F } + trace{ (A+F*B*F)^(-2) F*B* d F } ) = - trace{ B * F * (A+F*B*F)^(-2) d F } - trace{ (A+F*B*F)^(-2) F*B* d F } = - trace{ (A+F*B*F)^(-2) * F * B * d F} - trace{ (A+F*B*F)^(-2) F*B* d F } (Note that B is symmetric, and B=B; also, (A+F*B*F) is also symmetric) = -2 trace{ (A+F*B*F)^(-2) * F * B * d F} Therefore, by the first identification theorem, the differential is as given above. Recall that trace( (d x / d F) * d F ) = d x I find working with matrix differential much easier than other alternatives. The book recommended by Peter has a great account on how to manipulate matrix differential. It also provides the theorectical justification. Hope that helps. === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) >I am currently having a matrix derivative problem >What is derivative of >trace{(A+F*B*F)^(-1)} >with respect to matrix F >where () is the transpose of a matrix. >A and B is diagonal matrices. >I searched online and was only able to find the derivative of >d trace{(F*B*F)^(-1)}/d F >=-2*B*F*(F*B*F)^(-2) >without knowing how they get it. >Moreover, all these matrix derivative problems seem to be >difficult forme. Could anyone be kind enough to give me >some good references on this topic. Derivatives are not appropriate for functions of more than one variable, and you are asking for derivatives with respect to a matrix. Differentials satisfy the usual properties, but the failure of commutativity is quite important. So if q is a differential function of an argument in a locally ßat space, the Frechet derivative is dq(x, m) = lim ((q(x+em) - q(x))/e) as e -> 0. Using this, and the result that d(X^(-1)) = - X^(-1) dX X^(-1), we get that the differential you seek is trace (-2*(A+F*B*F)^(-1)*F*B*dF*(A+F[CapitalOTilde ]*B*F)^(-1)); this uses the results about trace being invariant under permutation and transposition. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) >I am currently having a matrix derivative problem >What is derivative of >trace{(A+F*B*F)^(-1)} >with respect to matrix F >where () is the transpose of a matrix. >A and B is diagonal matrices. >I searched online and was only able to find the derivative of >d trace{(F*B*F)^(-1)}/d F >=-2*B*F*(F*B*F)^(-2) >without knowing how they get it. >Moreover, all these matrix derivative problems seem to be >difficult forme. Could anyone be kind enough to give me >some good references on this topic. some reference which might help: Magnus, J.R., Neudecker, H. Matrix Differential Calculus with Applications in Statistics and Econometrics. paul fackler has some notes at http://www4.ncsu.edu/~pfackler/MATCALC.ps hth peter === Subject: standard probability spaces Originator: israel@math.ubc.ca (Robert Israel) let $(Omega,F)$ be a standard probability space and $X:[0,t]times Omegato E$ a stochastic process with values in a Polish space $E$ and RCLL trajectories. Is it true that $(Omega,F^X)$ is standard where $F^X$ denotes the $sigma$-algebra generated by $X$? J. p.s.: Please reply to email as well. Thx. === Subject: Average number of vectors on a plane Originator: israel@math.ubc.ca (Robert Israel) Hi there, I try to solve the following problem: Given N points random (lets say with mean m) distributed on a plane(2D). Each of these points can be the beginning or the end of a vector of length less than R. Under the condition that the beginning of each vector must have distance greater than R from the end of all the other vectors, what is the average number of vectors I can have on the given plane? (e.g. if (a_i) is the beginning point and (b_i) is the end point of the i-th vector, what is the mean{i} under the constrains ||a_i-b_i||<=R and ||a_i-b_j||>R [for i != j] , where ||x-y|| denotes the Euclidean diatance) Any ideas on which direction I have to look or about any related work it would be very grateful. Thanos === Subject: Two papers published by AGT Originator: israel@math.ubc.ca (Robert Israel) The following two papers have been published: (1) The conjugacy problem for relatively hyperbolic groups by Inna Bumagin URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-43.abs.html (2) Mp-small summands increase knot width by Jacob Hendricks URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-44.abs.html Full details follow: (1) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-43.abs.html Title: The conjugacy problem for relatively hyperbolic groups Author(s): Inna Bumagin Abstract: Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups. Secondary: 20F10 Keywords: Negatively curved groups, algorithmic problems Received: 5 May 2002 Author(s) address(es): Department of Mathematics and Statistics, Carleton University 1125 Colonel By Drive, Herzberg Building Ottawa, Ontario, Canada K1S 5B6 Email: bumagin@math.carleton.ca (2) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-44.abs.html Title: Mp-small summands increase knot width Author(s): Jacob Hendricks Abstract: Scharlemann and Schultens have shown that for any pair of knots K_1 and K_2, w(K_1 # K_2) >= max{w(K_1),w(K_2)}. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann-Schultens, Rieck-Sedgwick and Thompson, it is shown that for K= #_{i=1}^n K_i a connected sum of mp-small knots and K any non-trivial knot, w(K # K)>w(K). Secondary: 57M27 Keywords: Thin position, knot width Author(s) address(es): Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA Email: jghendr@uark.edu === Subject: An Algebraic Structure Originator: israel@math.ubc.ca (Robert Israel) I have found a concrete and non trivial example of the following algebraic structure: Let V an non empty set where are defined two binary operations +,*: V->VxV and there exists a non empty subset D such that: (1) +,* are associative and commutative. (2) * is distributive with respect to +. (3) * has a neutral element: 1. (4) each element in (V-D) has an multiplicative inverse in (V-D) . (5) forall v in (V-D) exists w in (V-D) such that v+w is in D ( existence of additive pseudo-inverse). (6) forall v,v in V : v*v in D->v is in D or v is in D. It is something which look like as a field and it is a field when D contains only one element. If this stucture doesnt have a name I would call it pseudo-field (anyway do not stick on this detail). Can please anybody provide me reference sources for this algebraic structure, if any? Roberto Volpe PS I hope now is more readable of the previous version made with cut and paste by my LaTeX note. === Subject: Re: An Algebraic Structure Originator: israel@math.ubc.ca (Robert Israel) Roberto Volpe in litteris scripsit: > Let V an non empty set where are defined two binary operations +,*: V->VxV > and there > exists a non empty subset D such that: > (1) +,* are associative and commutative. > (2) * is distributive with respect to +. > (3) * has a neutral element: 1. > (4) each element in (V-D) has an multiplicative inverse in (V-D) . > (5) forall v in (V-D) exists w in (V-D) such that v+w is in D ( existence > of additive pseudo-inverse). > (6) forall v,v in V : v*v in D->v is in D or v is in D. Any (commutative) local ring satisfies these axioms and more (essentially, the converse of (6) and the existence of additive inverses not just pseudo-inverses wrt D) if D is the maximal ideal. I believe your axioms are, as they stand, too weak to make much of (for example, V = the set of natural numbers, and D = all of V, seems to work; and you can always replace a pair (V,D) which works by taking D to be all of V). -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Re: An Algebraic Structure Originator: israel@math.ubc.ca (Robert Israel) David Madore ha scritto nel messaggio > I believe your axioms are, as they stand, too weak to make much of > (for example, V = the set of natural numbers, and D = all of V, seems > to work; and you can always replace a pair (V,D) which works by taking > D to be all of V). You are right I was vague when I mentioned in my original post that I found a non trivial example. For me a non trivial example means that D is a proper subset of V. Most likely I have found a local ring . Roberto Volpe === Subject: Moment-ratios Originator: israel@math.ubc.ca (Robert Israel) I am looking for some good approximations for the moment-ratios (kurtosis as function of skewness) of the following distributions: Weibull (3 parameter) lognormal(2 parameter) logpearsonIII Gumbel would anyone know where to find them ? JH === Subject: Multidimensional Abels/Schroders functional equations Epigone-thread: snahkheeleld Originator: israel@math.ubc.ca (Robert Israel) I have enough references concerning Abel f(g(x)) = f(x) + a and Schroder f(g(x)) = b*f(x) functional equations, where x is one-dimensional variable. Has anybody seen any results (on existence/uniqueness of the solution) for multidimensional versions of them: f(g(x1),...,g(xn)) = f(x1,...,xn) + a f(g(x1),...,g(xn)) = b*f(x1,...,xn) ? Any reference will be appreciated. === Subject: Re: Multidimensional Abels/Schroders functional equations Epigone-thread: snahkheeleld Originator: israel@math.ubc.ca (Robert Israel) >I have enough references concerning Abel >f(g(x)) = f(x) + a >and Schroder >f(g(x)) = b*f(x) >functional equations, where x is one-dimensional variable. >Has anybody seen any results (on existence/uniqueness of the solution) >for multidimensional versions of them: >f(g(x1),...,g(xn)) = f(x1,...,xn) + a >f(g(x1),...,g(xn)) = b*f(x1,...,xn) ? >Any reference will be appreciated. ÔAbel is certainly a useful tool to solve functional equations. {phi ,f} phi(f(x))= phi(x)+1 and (1) phi(f^[r](x))= phi(x)+r r real (2) and also f^[r](x)=phi^[-1](phi(x)+r). (3) Power iterated forms are also very interesting ,for instance g(h(x))=h(g(x)) h unknown ,got a not exotic solution : h(x)=g^[phi(x)](L) phi Abel of g(L). []iterations. (4) both your equations are strictly equivalent : f(g(x1),...,g(xn)) = f(x1,...,xn) + a (5) >f(g(x1),...,g(xn)) = b*f(x1,...,xn) form => f(g(x1),...,g(xn))/a = f(x1,...,xn)/a + 1 Combining ideas (1) and (4) will give you a power iterated solution for (5). We can come back whith a Ôclassic writing ,see (3). Notice:we deal with f functions having Abel phi !. === Subject: Re: Multidimensional Abels/Schroders functional equations Epigone-thread: snahkheeleld Originator: israel@math.ubc.ca (Robert Israel) If it is possible, please, give some reference to literature with multidimensional case. Unfortunately, in the literature, I have, the only univariate version is considered. For example: Kuczma M. Functional Equations in a Single Variable. Warszawa: PWN-Polish Scientific Publishers, 1968. 383 p. Cermak J. Note on Simultaneous Solutions of a System of Schroders Equations // Mathematica Bohemica. Vol. 120. 3. 1995. P. 225236. Dubuc S. Problemes relatifs a literation de fonctions suggeres par les processus en cascade // Annales de linstitut Fourier. Vol. 21. 1. 1971. P. 171251. Szekeres G. Abels Equation and Regular Growth: Variations on a Theme by Abel // Experimental Mathematics, Vol. 7 (1998), No. 2. P. 85-100. Szekeres G. Regular iterations of real and complex function // Acta Math. 100. 1958. P. 202-258. I have no book: Kuczma M., Choczewski B., Ger R. Iterative Functional Equations. Cambridge: Cambridge University Press, 1990. 571 p. May be you know, if it contains multidimensional case. === Subject: Re: Multidimensional Abels/Schroders functional equations Epigone-thread: snahkheeleld Content-Length: 730 Originator: rusin@vesuvius >I have no book: >Kuczma M., Choczewski B., Ger R. Iterative Functional Equations. >Cambridge: Cambridge University Press, 1990. 571 p. >May be you know, if it contains multidimensional case. This book indeed contains some results on the multidimensional Schr.9ader equation; specifically, section 8.2 treats such topics. There are a few existence andd uniqueness theorems but only for local solutions, as far as I can find at a quick glance. References are, among others, to a paper by Kuczma in Ann.Polon.Math. 29 (1974), 75-81, and to a paper by Smajdor, also in Ann.Polon.Math. 19(1967), 169-176. Best wishes, Peter Flor in Graz. === Subject: Re: Multidimensional Abels/Schroders functional equations Content-Length: 1706 Originator: rusin@vesuvius > If it is possible, please, give some reference to literature with > multidimensional case. > Unfortunately, in the literature, I have, the only univariate version > is considered. For example: > Kuczma M. Functional Equations in a Single Variable. Warszawa: > PWN-Polish Scientific Publishers, 1968. 383 p. > Cermak J. Note on Simultaneous Solutions of a System of Schroder?s > Equations // Mathematica Bohemica. Vol. 120. 3. 1995. P. 225?236. > Dubuc S. Problemes relatifs a literation de fonctions suggeres par les > processus en cascade // Annales de linstitut Fourier. Vol. 21. 1. > 1971. P. 171?251. > Szekeres G. Abels Equation and Regular Growth: Variations on a Theme > by Abel // Experimental Mathematics, Vol. 7 (1998), No. 2. P. 85-100. > Szekeres G. Regular iterations of real and complex function // Acta > Math. 100. 1958. P. 202-258. > I have no book: > Kuczma M., Choczewski B., Ger R. Iterative Functional Equations. > Cambridge: Cambridge University Press, 1990. 571 p. > May be you know, if it contains multidimensional case. Hi Mikhail, A particular case : when g are polynomials is related with Mahlers functionnal equation . These are importants for transcendental number theory. In fact I am interested to know solution in other extension of these equations such that f(x^4)= r(x)f(x^2)+s(x)f(x) For your problem hope you may find something in polynomials cases in Kumiko NISHIOKA Lectures Notes N1631 Mahler functions and transcendence (may be something in the references included in this book). Laurent === Subject: generic point and sheaf Originator: israel@math.ubc.ca (Robert Israel) Is this true? X : noetherian scheme F : coherent sheaf of O_X-module U in X : affine open subset s in F(U) : a section such that s(U) != 0 s_y != o for some y in U z in U : a generic point of U (*) Then, s_z != 0 I made a proof by myself, but Im not sure of it. Since if that is true, a statement of Mumfords redbook is wrong. So, please point out the wrong point of my proof. My Proof) FACT) Closed subsets are stable under specialization. By the FACT, y and z are in the same irreducible component of X. So, let Y be the same irreducible component of X containing y and z. Suppose s_z = 0. Then, there exists an open set V in U containing z such that s(V) = 0. But, since s_y != 0 and y is in V, this is a contradiction. Thus, s_z != 0. (END) I dont know where I made a mistake. And, heres an another question. If above statement is true, how about we replace the statement (*) by z is a generilization of y.? Is that still hold? Ill be happy if you give me any comments. === Subject: Re: generic point and sheaf Originator: israel@math.ubc.ca (Robert Israel) > Is this true? > X : noetherian scheme > F : coherent sheaf of O_X-module > U in X : affine open subset > s in F(U) : a section such that s(U) != 0 > s_y != o for some y in U > z in U : a generic point of U (*) > Then, s_z != 0 No, it is false. Consider, for instance, the affine line X = U = Spec(F[t]) over a field F. You get open subsets as Spec(F[t][1/h(t)] for various nonconstant polynomials. The generic point is given by the ideal {0}. Let M be a finitely generated F[t]-module; this gives us a coherent sheaf on X whose sections on the open set Spec(F[t][1/h(t)]) are the tensor product of M and F[t][1/h] over F[t]. In particular, now, take M = F[t]/(t-1)F[t]. This is nonzero, and it has nonzero stalk at the point y given by the ideal (t-1)F[t]. But it becomes trivial once we restrict to the open subsset where t-1 is invertible. > I made a proof by myself, but Im not sure of it. Since if that is > true, a statement of Mumfords redbook is wrong. So, please point out > the wrong point of my proof. The problem in the proof is the assumption that the point y has to be in each open subset V of U. William C. Waterhouse Penn State > My Proof) > FACT) Closed subsets are stable under specialization. > By the FACT, y and z are in the same irreducible component of > X. > So, let Y be the same irreducible component of X containing y > and z. > Suppose s_z = 0. > Then, there exists an open set V in U containing z such that > s(V) = 0. > But, since s_y != 0 and y is in V, this is a contradiction. > Thus, s_z != 0. (END) > I dont know where I made a mistake. > And, heres an another question. > If above statement is true, how about we replace the statement (*) > by > z is a generilization of y.? Is that still hold? > Ill be happy if you give me any comments. === Subject: How to find maximal orders in crossed product algebras over number fields? Originator: israel@math.ubc.ca (Robert Israel) I would like to know more about the problem of finding maximal orders in crossed product algebras over number fields in general and cyclic division algebras over Q(i) and Q(sqrt{-3}) in particular, because these are most useful in the application we have in mind. What do I know? Ok: Let L/K be a finite cyclic Galois extension of number fields. Let s be the generating automorphism in Gal(L/K), and consider the cyclic algebra A=(L,s,gamma), where gamma is an algebraic integer in K. So A=L{u}, with the relations u^n=gamma, and u*rho=s(rho)*u for all the numbers rho in L. In this case an obvious starting point is the order Lambda= O_L{u}, where O_L is the ring of integers of the field L. However, this need not be a maximal O_K- order. The easiest example of that phenomenon is the case K=Q, L=Q(i), gamma=-1, u=j yielding the Lipschitz order Z[i,j,k] that is a proper subring of the Hurwitz ring Z[i,j,(1+i+j+k)/2]. Of course, computing the discriminant of the order O_L{u} will give severe limitations to its non-maximality, but I wouldnt expect to get anywhere near to a unit discriminant too often. Apropos Lipz vs. Hurwitz orders: Inside the skewfield of rational quaternions we have many splitting fields L_1,L_2,... that yield different O_K-orders O_{L_i}{u_i}. E.g. as above the choice L_1=Q(i) gives the Lipschitz ring, but the choice L_2= Q(sqrt(-3)), where we use i+j+k as the square root of -3, and let u=i-j, gives a different Z-order. This order has a large intersection with the Lipschitz ring, and also happens to contain the element (1+i+j+k)/2 because (1+sqrt(-3))/2 happens to be an integer of L_2! Can one always find a maximal O_k-order by combining suitable orders of the type O_{L_i}{u_i} in this manner? Of course one cannot use all such orders, because there are many conjugates of the splitting fields around:) As I said our main interest is in the cases K=Q(i), and K=Q(sqrt(-3)), because O_K is then a particularly nice lattice. We started looking at the order two extension K=Q(sqrt(2)), L=the 8th cyclotomic field, gamma=-1, and we wanted to find a maximal Z[i]-order (so its not always the O_K-order we want, but whatever). In this case the cyclic algebra is a subring of the usual quaternions, so it was no surprise that the Hurwitz unit rho=(1+i+j+k)/2 could be used to extend the Z[i]-order O_L{j}. Any help is appreciated, and all the thoughts and suggestions are most welcome. Earlier this week I leafed thru I.Reiners book Maximal Orders. It contains a lot of valuable theory for us, but my cursory study didnt immediately give answers to these questions. E.g. the main existence result was by a Zorns lemma argument, which is ok. Also the exercises gave another existence proof by a reduce the discriminant argument. May be more examples are buried in there somewhere? Jyrki Lahtonen, Turku, Finland === Subject: the number of Conjugacy Classes of PSO_{2n+1}(q) and PSp_{2n}(q) Originator: israel@math.ubc.ca (Robert Israel) 1) Is there an explicite formulae for the number of conjugacy classes of PSO_{2n+1}(q) and PSp_{2n}(q), where $n>2$ and $q$ is an odd prime power? 2) Is it true that The number of conjugacy classes of the groups PSO_{2n+1}(q) and PSp_{2n}(q), (n>2 and q is odd) are never equal. Any help and comment is appriciated. All the best Alireza Abdollahi === Subject: which semigroups are rings? Epigone-thread: skulthoosteh Originator: israel@math.ubc.ca (Robert Israel) Is any semigroup can be endowed with a structure of an abelian group such that the distributive laws are satisfied? Are there any necessary or sufficient conditions known for a semigroup to allow a structure of an associative ring? [ Moderators note: I assume Magpie wants the semigroup operation to be the multiplication and the new abelian group operation to be the addition. -RI ] === Subject: Re: which semigroups are rings? Originator: israel@math.ubc.ca (Robert Israel) > Is any semigroup can be endowed with a structure of an abelian group > such that the distributive laws are satisfied? > Are there any necessary or sufficient conditions known for a semigroup > to allow a structure of an associative ring? > [ Moderators note: > I assume Magpie wants the semigroup operation to be the multiplication > and the new abelian group operation to be the addition. > -RI I dont know how helpful it is but quite a bit is known about the complementary problem : Given an Abelian group, what rings have that group as its additive structure. See L. Fuchs; Infinite Abelian Groups, Vol II. -- Paul Sperry Columbia, SC (USA) === Subject: Re: which semigroups are rings? Originator: israel@math.ubc.ca (Robert Israel) > Is any semigroup can be endowed with a structure of an abelian group > such that the distributive laws are satisfied? No, certainly not. For example, a ring has a zero, so your semigroup must as well: that is, an element 0 with 0x=x for all x. Even with a zero, it is unlikely that a random seimgroup is the multiplication of a ring. For example, take any finite nonabelian group, add a zero. It isnt a ring, by the nontrivial theorem that any finite division ring is a field. -- G. A. Edgar edgar at math.ohio-state.edu === Subject: Hurwitzian numbers Originator: israel@math.ubc.ca (Robert Israel) I am wondering if something is known about arithmetic with Hurwitzian numbers (adding two of them, multiplicating, etc.). Besides, I found very little about these numbers online. Do you know if a good document speaks of them ? -- Thomas Baruchel To contact me, please, see at: http://cerbermail.com/?OEyHGNQFEa Home Page: http://tbaruchel.free.fr/ [ Moderators note: for the definition of Hurwitzian numbers, see e.g. -RI ] === Subject: Re: Hurwitzian numbers 2nd reply to question by Thomas Baruchel - (1st reply is not complete and contains an error.) ------------------------------------------------------------- --------------- --- In 1896 Adolf Hurwitz set up an integer arithmetic of quaternions, very much like the arithmetics of the real integers (Z) and the complex integers (Z + iZ), aka the Gaussian integers. See Adolf Hurwitz: Ueber die Zahlentheorie der Quaternionen. Nachrichten der koenigliche Gesellschaft der Wissenschaften zu Goettingen, mathematisch-physikalische Klasse, 1896, Seiten 313-340; reprinted in Adolf Hurwitz - Mathematische Werke, 2. Band, LXIV, 303-330. Birkhaeuser 1963 As far as I know no translation into English exists. Speaking algebraically, the essential thing is that one deals with a ring with identity and therefore with a multiplicative group of units. To obtain a full-ßedged integer arithmetic one needs a maximal subgroup of rational units without impairing the discrete character of the whole thing, i.e. a maximal =finite= subgroup of rational units. Hurwitz found out that in quaternions the group of units (1, -1, i, -i, j, -j, k, -k) is not maximal, and that one can add all quaternion numbers a + bi + cj + dk with a, b, c, d = 1/2 or -1/2 independently from each other. In this way one obtains the largest finite group of rational quaternion units. BTW, altogether one has the vertices of a 24-cell, one of the six regular polytopes in 4D space. In the end the Hurwitzian numbers are quaternion numbers in which all components are either integers or half-integers. There exists a German book in the Springer series Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen that deals with these Hurwitzian integers. (about 1920) I do not know by heart the author and title. URLs: http://www.gsu.edu/~oprdeb/qtrn/factor.html is a site on quaternion arithmetic; the editors seem not to be aware of the half-integer-component quaternions. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/ Hurwitz.html is a short biography of Hurwitz. I wish you success and pleasure. Johan E. Mebius > I am wondering if something is known about arithmetic with Hurwitzian > numbers > (adding two of them, multiplicating, etc.). > Besides, I found very little about these numbers online. Do you know > if a good > document speaks of them ? === Subject: Paper published by Algebraic and Geometric Topology Originator: israel@math.ubc.ca (Robert Israel) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-45.abs.html Title: sl(3) link homology Author(s): Mikhail Khovanov Abstract: We define a bigraded homology theory whose Euler characteristic is the quantum sl(3) link invariant. Secondary: 18G60 Keywords: Knot, link, homology, quantum invariant, sl(3) Author(s) address(es): Department of Mathematics, University of California Davis CA 95616, USA Email: mikhail@math.ucdavis.edu === Subject: This Weeks Finds in Mathematical Physics (Week 208) Originator: baez@math-cl-n01.math.ucr.edu (John Baez) Originator: israel@math.ubc.ca (Robert Israel) Also available at http://math.ucr.edu/home/baez/week208.html This Weeks Finds in Mathematical Physics Week 208 Last weekend I went to a conference at the Perimeter Institute: 1) Workshop on Quantum Gravity in the Americas, http://www.perimeterinstitute.ca/activities/scientific/PI-WORK -2/ It was great to see the new building. Id visited this institute before in its temporary location, which was a funky old hotel building complete with pool tables and a bar. The new building is very different: four stories of intensely modern architecture overlooking a lake, consisting mainly of an enormous atrium lined with walkways and glass-walled offices. Theres also a big lecture theater, a couple of smaller seminar rooms, a library, a restaurant whose walls are all blackboards, a reßecting pool, and lots of little places to sit and talk, complete with espresso machines. In short, a theoretical physicists idea of heaven! But perhaps the design of heaven shouldnt be left to theoretical physicists. Some aspects of the setup dont seem very comfortable. Like most modern architecture, the place is short on coziness - theres too much glass, metal and concrete for my taste. You also find yourself spending a lot of time climbing up and down uncomfortably narrow staircases. The last, at least, is no accident: they made the stairs skinny on purpose, so you have to say hello to anyone you meet going the other way. Itll be interesting to see how many collaborative papers come out of this. Abhay Ashtekar was supposed to give the first talk, but he got lost walking to the new building, so suddenly I had to give the first talk. Yikes! Jet-lagged and not fully awake, I sketched the problem of dynamics in quantum gravity: the problem of describing motion in a world where the geometry of spacetime is quantum-mechanical and interacts with matter. I gave a generally downbeat assessment of the progress so far in all known approaches: 2) John Baez, The problem of dynamics in quantum gravity, http://math.ucr.edu/home/baez/dynamics/ Even though the last few Weeks have been on quantum gravity conference, Ive been mainly working on n-categories lately, because Ive been sort of fed up with quantum gravity. I did, however, sketch some avenues for progress - and later in this conference I saw some work that really cheered me up! For example, Ive always been fascinated by John Wheelers old dream of matter without matter. In its original version, the idea was to an electric field going in one end and out the other, the ends will pair. But there were big problems with this idea: for example, More recently this idea was reincarnated in the spin network formalism by Lee Smolin, with spin network edges replacing wormholes: 3) Lee Smolin, Fermions and topology, available as gr-qc/9404010. A spin network is a gadget with vertices and edges, where the edges represent field lines - lines of the electric field or the analogous thing for other forces, including gravity. If a spin network edge goes between vertices that would otherwise be far apart, it acts a bit like a wormhole. It will be hidden from observers in the rest of they dont call them spin networks for nothing! A variant on this idea is to have spin networks with loose ends: edges that just fizzle out. This is more ad hoc, but easier to study in some ways. A decade ago, Kirill Krasnov and I showed how to describe 4) John Baez and Kirill Krasnov, Quantization of diffeomorphism- invariant theories with fermions, hep-th/9703112. However, the hard problem in quantum gravity is always dynamics. Does the dynamics of spin networks with loose ends actually mimic that this question in a toy model, 3-dimensional Lorentzian gravity: 5) Kirill Krasnov, Lambda<0 Quantum Gravity in 2+1 Dimensions I: Quantum States and Stringy S-Matrix, Class. Quant. Grav. 19 (2002) 3977-3998, also available as hep-th/0112164. Kirill Krasnov, Lambda<0 Quantum Gravity in 2+1 Dimensions II: 3999-4028, also available as hep-th/0202117. network ends - though you dont need to emphasize that viewpoint, since there are also other nice ways to think about whats going on, using hyperbolic geometry and complex analysis. It all fits together in a beautiful picture. In principle you can even calculate In this conference, Laurent Freidel explained how this idea works in 3-dimensional Riemannian gravity - a less physical but mathematically more tractable spin foam model. Some but not all of his work can be found here: 6) Laurent Freidel and David Louapre, Ponzano-Regge model revisited I: as hep-th/0401076. Laurent Freidel and David Louapre, Ponzano-Regge model revisited II: Equivalence with Chern-Simons, available as gr-qc/0410141. Freidel showed that if you take this theory and allow spin networks with automatically quantized. More surprisingly, so is their mass - and theres an upper bound on the mass! Thats because when we quantize this theory, its gauge group automatically gets replaced by a quantum group. Physically, this means that spacetime becomes quantum-mechanical in such a way that it no longer makes sense to talk about times shorter to the rate at which its wavefunction oscillates, this puts an upper Mathematically, part of the point is that we can describe 3d Riemannian gravity as a gauge theory where the gauge group is the double cover of the 3d Euclidean group - the analogue of the Poincare group in this context. But when we quantize the theory, this gets replaced by a quantum group: the quantum double of SU(2). As with the 3d Euclidean group, unitary representations of this quantum group are classified by mass and spin... but now both mass and spin are discrete, and both are bounded above. Anyway, whats great about Freidel and Louapres work is that it gives a simplified but mathematically rigorous testbed in which loose ends networks with hidden edges in this setup. So, we should be able to do calculations and see if a spin network with a hidden edge acts like pair. Unfortunately, all this work on gravity in 3d spacetime doesnt easily generalize to 4d spacetime. The reason is that gravitational waves are only possible when spacetime has dimension 4 or more... so 3d gravity all the fun comes from global topology, like wormholes. Thats why 3d theories are easy to calculate with - we can use ideas from topological quantum field theory. The danger, though, is that these calculations are misleading it comes to real-world physics. Indeed, thats precisely the sort of thing I was worrying about in my talk. So, it really cheered me up when a young guy named Artem Starodubtsev spoke about a promising new spin foam model of quantum gravity in 4 dimensions! Hes working on it now with Laurent Freidel. He has a couple of papers out that *hint* at the main ideas, but youll have to wait to see what theyre up to now: 7) Artem Starodubtsev, Topological excitations around the vacuum of quantum gravity I: The symmetries of the vacuum, available as hep-th/0306135. Artem Starodubtsev and Lee Smolin, General relativity with a topological phase: an action principle, available as hep-th/0311163. The basic idea is to treat 4d general relativity with positive cosmological constant as a perturbation of a topological quantum field theory. The topological theory has a single state, which corresponds to a quantum version of deSitter space: an exponentially expanding universe similar to the one we see today, but with no matter. To calculate in full-ßedged gravity, we then use perturbation theory, getting answers as power series in a coupling constant. But the cool part is that unlike ordinary perturbative quantum gravity this perturbation theory is manifestly diffeomorphism invariant term by term. And each term is a sum over spin foams! Even better, the coupling constant in this theory is the cosmological constant in Planck units! Thats an incredibly small dimensionless number: about 10^{-123}. Physicists like perturbation theory when the coupling constant is small, since then the first few terms tend to give reasonably accurate answers - even if the whole series diverges. For example, quantum electrodynamics gives high-precision answers because the fine structure constant is about 1/137, which is pretty small. But 10^{-123} is *really* small. Id seen Starodubtsev talk about this in Marseille (see week206) but now he and Freidel have done calculations recovering Newtons law of gravity in an appropriate approximation from this theory. That may not seem like a big deal, but its actually very cool to see Newtons law reemerge from a manifestly diffeomorphism-invariant theory of quantum gravity: no model had never managed this feat before. For those of you hungering for technical details, Ill just say that the topological theory in question is BF theory with the symmetry group of deSitter spacetime, namely SO(4,1), as the gauge group. General relativity can be regarded as a perturbation of this BF theory by borrowing some ideas from the MacDowell-Mansouri formulation of general relativity. If you havent heard of that, well, neither had I. Its a sort of old idea: 8) S. W. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38, 739742 (1977). ... but here we arent using anything anything about supergravity, just the fact that ordinary general relativity can be treated as a theory with gauge group SO(4,1) and a Lagrangian that breaks this symmetry down to the Lorentz group SO(3,1). The paper by Smolin and Starodubtsev listed above describes the details, but in the case of going from SO(5) down to SO(4). When we quantize BF theory in 4 dimensions we get a spin foam model called the Crane-Yetter model, where the spin foams are defined using the representation theory of a quantum group: a q-deformed version of the original gauge group. So, the real engine behind Freidel and Starodubtsevs calculations are spin foams involving the q-deformed version of SO(4,1), called SO_q(4,1). This is technically tricky because SO(4,1) is noncompact, and noncompact quantum groups are just beginning to be understood. So, theres probably still tons of mathematical work left to be done. But, the upshot is that Freidel and Starodubtsev calculate stuff as power series in the cosmological constant where each term is computed using SO_q(4,1) spin foams. Its sort of like a souped-up Feynman diagram expansion, but with spin foams replacing Feynman diagrams. Now that Ive thrown around enough buzzwords to scare off the kids, I can tell you about Lee Smolins talk, which was definitely for adults only, people who can listen to speculations with just the right mixture of disbelief and open-mindedness. It was the last talk in the conference. And it was about the possible physical effects of spin networks with hidden edges! ahead and suggested that hidden edges can cause nonlocal effects in physics, like the force of gravity decaying more slowly than 1/r^2 - just as it does in MOND, the wacky but strangely accurate explanation of galactic rotation curves that uses a modification of Newtonian gravity instead of positing dark matter! (See week206 for more on MOND.) Its hard to make up sensible theories of forces that decay more slowly than 1/r^2, but nonlocal interactions would be one way to do it... and hidden spin network edges might cause those. There are a million things that could go wrong with this idea, but I like it, because it suggests a way quantum gravity might try to explain one of the big mysteries of physics - dark matter. And until we get our theories to make contact with experiment, itll be very hard for us to tell if theyre on the right track. Anyway, Smolin hasnt come out with a paper on this stuff yet, so well have to wait for more details. By the way: In what Ive written this week, Ive had to seriously downplay the cool math involved, to give (I hope) some inkling of the cool physics. Krasnov work on 2+1-dimensional Lorentzian gravity with positive cosmological constant uses the fact that the phase space of this theory is closely related to Teichmueller space - the space of complex structures mod diffeomorphisms that are connected to the identity. I talked about this space in week28, but I forgot to say that we can think of it as a space of ßat SO(2,1) connections mod gauge transformations. Here SO(2,1) is just the Lorentz group in 3 dimensions. So, if we quantize 2+1 Lorentzian gravity with positive cosmological constant, we get a theory where states are described by SO_q(2,1) spin networks... but this is also a theory of quantum Teichmueller space. Again this is tricky because SO(2,1) is noncompact, of work started by Kashaev: 9) R. M. Kashaev, Quantization of Teichmueller spaces and the quantum dilogarithm, available as q-alg/9705021. 10) L. Chekhov and V. V. Fock, Quantum Teichmueller space, Theor. Math. Phys. 120 (1999) 1245-1259, also available as math.QA/9908165. You can get a sense of whos working on this stuff and what theyre doing by looking at the references for this recent conference on 3d quantum gravity in Edinburgh, which unfortunately took place when I was in Hong Kong: 11) Workshop on physics and geometry of 3-dimensional quantum gravity, http://www.ma.hw.ac.uk/~bernd/references.html I should also add that people dont usually dont talk about the 3d Lorentz group SO(2,1) here; they talk about its double cover SL(2,R). Anyway, Ill quit here. The next conference on loops and spin foams will probably happen in Berlin at the Albert Einstein Institute in 2005, which happens to be the hundredth birthday of special relativity. I hope we can make a lot of progress before then and make Al proud. ------------------------------------------------------------- ---------- 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 Weeks 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: Equivalence of Fundamental theorems of Analysis Originator: israel@math.ubc.ca (Robert Israel) In the book The implicit function theorem by S. Krantz and H. Parks, several equivalent formulations of the implicit function theorem are discussed. There is of course the well-known equivalence between the implicit and the inverse function theorem. Moreover, in section 4.1 of that book the equivalence of the implicit function theorem and the existence theorem for ordinary differential equations is discussed. Unfortunately, the proof that the existence theorem for ODEs implies the implicit function theorem is only sketched there and no literature is cited where one could find this result. More generally, I would suppose that also the Frobenius theorem and the straightening out theorem for vector fields on manifolds ought to be equivalent with the implicit function theorem. My question is: could anyone tell me about literature where these equivalences (most importantly: ODE => implicit) are proved? Any hints would be appreciated! === Subject: Int { f(x)* g(x , y) , x = x to h(x) } = n(y) ; via ABEL ! Epigone-thread: vendwhingzhald Originator: israel@math.ubc.ca (Robert Israel) Ive thought it interesting ,so I tell you. f,h,n continous functions R->R ;g(x,y) R*R->R unknown. We have here an integral equation. Let us put p(x ,y)= Int(f(x)*g(x ,y) dx) then re-writing the initial equation gives p(h(x),y)- p(x ,y) = n(y) (1) I suppose there is a Abel phi | phi(h(x))= phi(x) +1 a direct solution to (1) is :p(x ,y) = n(y)*phi(x) (2) thence f(x)*g(x ,y) =d/dx (phi(x)*n(y)) g(x ,y) =n(y)/f(x).d/dx phi(x) What about finding other likewise equations ! Notice:inside (2)we might have added invariant (x->h(x))additive terms. Hear you soon,Alain. === Subject: Functional analysis: space of bounded varation separable? Originator: israel@math.ubc.ca (Robert Israel) my course in functional analysis is so long ago. So, here my question: Is it true that the space of real valued functions on R of bounded variation is separable? /J. === Subject: Re: Functional analysis: space of bounded varation separable? Content-Length: 546 Originator: rusin@vesuvius >Is it true that the space of real valued functions on R of bounded >variation is separable? No, it isnt. For example, the indicator functions of intervals [a,b] form an uncountable family, and the total variation of the difference of any two of them is at least 2. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Functional analysis: space of bounded varation separable? Content-Length: 780 Originator: rusin@vesuvius >>Is it true that the space of real valued functions on R of bounded >>variation is separable? > No, it isnt. For example, the indicator functions of intervals > [a,b] form an uncountable family, and the total variation of the > difference of any two of them is at least 2. I need to think about your argument, because, meanwhile, I found out: BV[0,1] is the dual of C[0,1] (via the pairing induced by the Riemann-Stieltjes-integral). Since C[0,1] is a separable Banach space (Stone-Weierstrass), so is its dual BV[0,1]. Now, how do these two statements fit together? Jannick === Subject: Re: Functional analysis: space of bounded varation separable? Content-Length: 1205 Originator: rusin@vesuvius >Is it true that the space of real valued functions on R of bounded >variation is separable? >> No, it isnt. For example, the indicator functions of intervals >> [a,b] form an uncountable family, and the total variation of the >> difference of any two of them is at least 2. >I need to think about your argument, because, meanwhile, I found out: >BV[0,1] is the dual of C[0,1] (via the pairing induced by the >Riemann-Stieltjes-integral). Since C[0,1] is a separable Banach space >(Stone-Weierstrass), so is its dual BV[0,1]. >Now, how do these two statements fit together? The unit sphere of BV[0,1] is separable in the weak-* topology as the dual of C[0,1]. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Functional analysis: space of bounded varation separable? Content-Length: 1145 Originator: rusin@vesuvius >Is it true that the space of real valued functions on R of bounded >variation is separable? >> No, it isnt. For example, the indicator functions of intervals >> [a,b] form an uncountable family, and the total variation of the >> difference of any two of them is at least 2. >I need to think about your argument, because, meanwhile, I found out: >BV[0,1] is the dual of C[0,1] (via the pairing induced by the >Riemann-Stieltjes-integral). Since C[0,1] is a separable Banach space >(Stone-Weierstrass), so is its dual BV[0,1]. No, the dual of a separable Banach space is not necessarily separable. C[0,1] is one good example of this, l^1 (the space of summable sequences, whose dual is identified with l^infinity) is another. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: disk in a tetrahedron Originator: israel@math.ubc.ca (Robert Israel) Somebody recently told me -- oh, all right, it was a referee for a paper -- that he (or she) thought that Victor Klee has shown somewhere that the largest disk that fits in a tetrahedron necessarily lies in a face (facet) of that tetrahedron. I havent been able to turn up the reference. Can somebody tell me where this fact can be found in the literature? --J. Wetzel === Subject: weak compactnes vs. norm discreteness in uniformly convex Banachs spaces Content-Length: 399 Originator: rusin@vesuvius Let $E$ be a uniformly convex Banach space, and let $C subset E$ be weakly compact such that there is $epsilon > 0$ with $| x | geq epsilon$ for $x in C$ and $| x - y | geq epsilon$ for $x, y in C$ with $x neq y$, i.e. $C$ is bounded away from zero and uniformly discrete in the norm topology. Question: Has $C$ to be finite? Any pertinent hints will be appreciated! Volker Runde. === Subject: Re: weak compactnes vs. norm discreteness in uniformly convex Banachs spaces Content-Length: 812 Originator: rusin@vesuvius >Let $E$ be a uniformly convex Banach space, and let $C subset E$ be >weakly compact such that there is $epsilon > 0$ >with $| x | geq epsilon$ for $x in C$ and $| x - y | geq >epsilon$ for $x, y in C$ with $x neq y$, i.e. $C$ is bounded away >from zero and uniformly discrete in the norm topology. >Question: Has $C$ to be finite? >Any pertinent hints will be appreciated! Hint: In l^2, the standard unit vectors e_n converge weakly to 0. Add something to take care of the bounded away from 0 requirement. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: weak compactnes vs. norm discreteness in uniformly convex Banachs spaces Content-Length: 1172 Originator: rusin@vesuvius >>Let $E$ be a uniformly convex Banach space, and let $C subset E$ be >>weakly compact such that there is $epsilon > 0$ >>with $| x | geq epsilon$ for $x in C$ and $| x - y | geq >>epsilon$ for $x, y in C$ with $x neq y$, i.e. $C$ is bounded away >>from zero and uniformly discrete in the norm topology. >>Question: Has $C$ to be finite? >>Any pertinent hints will be appreciated! >Hint: In l^2, the standard unit vectors e_n converge weakly to 0. >Add something to take care of the bounded away from 0 requirement. Just in case hes as dense as me here: I read your Add something as Add some condition, and I couldnt figure out what you were getting at for a minute... Of course you meant Add some vector to give a counterexample. >Robert Israel israel@math.ubc.ca >Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia Vancouver, BC, Canada ************************ David C. Ullrich === Subject: CFP: CALL FOR PAPERS. ORP3 2005. EURO Operational Research Peripatetic Post-graduate Programme Content-Length: 5603 Originator: rusin@vesuvius Your help with circulating this announcement is very much appreciated. Apologies for multiple postings. ******************************************************** *********************** ORP3 2005 ********************** ******************************************************** Call for Papers: ORP3 2005 Operational Research Peripatetic Post-graduate Programme September 6-10, 2005 Valencia, Spain Home Page: http://www.cfp.upv.es/orp3/index.jsp http://www.orp3.com ******************************************************** ORP3 or =93the Operational Research Peripatetic Post-Graduate Programme=94 is a new instrument of EURO - The Association of European Operational=20 Research Societies =96 (http://www.euro-online.org) designed for young=20 OR researchers and practitioners. The objective is promoting scientific=20 and social exchanges between the members of the future generation of=20 Operational Researchers ORP3 is organized every two years and the 2005 edition in Valencia is=20 the third. The first edition of ORP3 took place in Paris, France, in=20 September 26-29, 2001 and the second edition took place in Lambrecht,=20 The main features of this edition of the ORP3 are: =95 ORP3 is organized by young OR researchers and is dedicated to young=20 doctoral researchers and/or OR analysts =95 ORP3 is mainly sponsored by EURO and offers really low registration=20 fees with meals and accommodation included =95 ORP3 covers all fields of OR =95 There are no parallel sessions in ORP3 and about 30-40 participants=20 will be giving talks in plenary sessions =95 Participants also get to chair sessions and to discuss another=20 participant=92s paper =95 Active social events and coffee breaks =95 Tutorial sessions given by 3 renowned senior speakers =95 Special issue in the European Journal of Operational Research (EJOR) planned for the best papers presented at the conference Topics for the conference ------------------------- All Operational Research/Management science related fields are welcome! However, special emphasis is put on real world applications of OR. Location -------- ORP3 will be held at the Valencia University of Technology (UPV)=20 http://www.upv.es, located in Valencia, on the east coast of Spain. Submission ---------- In order to attend ORP3 as a participant you must be one of the=20 following: =95 A Young PhD Student =95 A young post doctoral OR researcher (maximum two years after completing=20 your doctoral thesis at the deadline for submission) =95 A young OR analyst (maximum two years of professional experience at the=20 deadline for submission) The selection of participants is made by the scientific committee on the=20 basis of full paper submissions of publishable quality in an international=20 OR journal Co-authored papers are acceptable as long as they are presented by a=20 participant satisfying the aforementioned requirements. When preparing the paper any format is welcomed although LaTeX is strongly=20 encouraged since all papers will be edited in the conference proceedings.=20 Templates and more directions are given in the ORP3 web page You have to submit an electronic version of your paper (TEX, PS, PDF, DOC=20 or RTF) and a short Curriculum Vitae to: Concepcion Maroto Alvarez Operations Research Group Department of Applied Statistics, Operations Research and Quality Valencia University of Technology Camino de Vera S/N, I-3 Building 46022 Valencia, Spain Tel. +34 96 387 70 07 Ext: 74921 Fax: +34 96 387 74 99 Email: cmaroto@eio.upv.es Registration ------------ Registration fee is only 200 =80. This includes all expenses (conference fee, meals, coffee breaks, accommodation, social program, gala dinner, proceedings book,...). Registration details are given in the ORP3 web page. Important dates --------------- February 11th 2005 deadline for paper submission June 6th 2005 notification of acceptance July 15th 2005 deadline for registration Organization and committees --------------------------- Organized by: The Operations Research group (http://www.upv.es/gio) and=20 the Department of Applied Statistics, Operations Research and Quality=20 of the Valencia University of Technology. Scientific Committee: =95 David Alcaide (Spain) =95 Javier Alcaraz (Spain) =95 Denis Bouyssou (France) =95 Laureano Escudero (Spain) =95 Horst Hamacher (Germany) =95 Concepcion Maroto (Chair) (Spain) =95 Ethel Mokotoff (Spain) =95 Luis Paquete (Portugal) =95 Jesus Pastor (Spain) =95 Marie Claude Portmann (France) =95 Joaquin Sicilia (Spain) =95 Thomas St=FCtzle (Germany) =95 Ruben Ruiz (Spain) =95 Enriqueta Vercher (Spain) =09 Organization Committee: =95 Andres Carrion =95 Fortunato Crespo =95 Jose Miguel Carot =95 Juan Carlos Garcia-Diaz =95 Jose Jabaloyes =95 Ruben Ruiz (Chair) =95 Eva Vallada (Co-chair) =95 Elena Vazquez =20 For further information please contact: Ruben Ruiz Garcia Operations Research Group Department of Applied Statistics, Operations Research and Quality Valencia University of Technology Camino de Vera S/N, I-3 Building 46022 Valencia, Spain Tel. +34 96 387 70 07 Ext: 74946 Fax: +34 96 387 74 99 Email: rruiz @ eio.upv.es Concepcion Maroto Alvarez Operations Research Group Department of Applied Statistics, Operations Research and Quality Valencia University of Technology Camino de Vera S/N, I-3 Building 46022 Valencia, Spain Tel. +34 96 387 70 07 Ext: 74921 Fax: +34 96 387 74 99 Email: cmaroto @ eio.upv.es === Subject: a question about coding and threshold schemes Content-Length: 505 Originator: rusin@vesuvius i think my question is suitable here, it is about coding theory and threshold schemes. i want to find a way to construct a matrix with n lines and m columns, take the lines as vectors and add arbitary k lines of the all n lines, one get a vector with which the hamming weight is at least h (h is between 1 and m). i have got some examples: for a 3*3 matrix, ((1,0,0),(1,0,1),(0,1,0))the hamming weight of the sum of any two of the 3 lines is at least 1. so what is the general (k,n)-threshold scheme? === Subject: Resolvable Set References Content-Length: 438 Originator: rusin@vesuvius I recently became aware that the concept of the topological residue of a set Q, (cl(Q)Q) intersect Q, and its relation to resolvable sets are essentially related to some work Im doing on Borel sets in Standard Borel Spaces. The only reference Ive found is K. Kuratowskis Topology, Vol. I. Does anyone know of any other references? FYI, Im working on proving that if the closure of a set Q is perfect, then Q is Borel. Ôcid Ôooh === Subject: Re: Resolvable Set References Content-Length: 484 Originator: rusin@vesuvius > FYI, Im working on proving that if the closure of a set Q is perfect, > then Q is Borel. It seems to me that your conjecture is false: Since the Cantor set C is compact (and metric), it contains a countable dense subset A. Hence, the union of A with an arbitrary subset of C has a perfect closure. However, the cardinality of the family of sets obtained in this way is 2^{2^N} while in contrast there are at most 2^N Borel subsets of C. === Subject: Re: Resolvable Set References Content-Length: 543 Originator: rusin@vesuvius >FYI, Im working on proving that if the closure of a set Q is perfect, >then Q is Borel. Thats not true. For example, its easy to modify the standard construction of a nonmeasurable (and therefore non-Borel) set to make it dense in [0,1]. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Infinit Number Of Closed Geodesics On S^2 Epigone-thread: dyrshilploy Content-Length: 320 Originator: rusin@vesuvius Due To Finiteness theorem of limit cycles a celeberated theorem by Yu. Ilyashenko,the following question sound interesting(if you have any Idea please reply me): Question:(Can sphere (S^2) be equiped to an ANALYTIC riemanian metrics,with infinit number of disjoint simple closed Geodesics? Ali Taghavi === Subject: FFT result evaluation Content-Length: 657 Originator: rusin@vesuvius I have a signal with harmonics that are not integer multiples of the fundamental frequency. So it is not possible to guarantee that the FFT - Window get a full period of each harmonic of the analysed signal. The result of the FFT is a distributed spectrum for each harmonic and it is not possible to find out the correct magnitude of the harmonics. Is it possible to calculate the correct signal magnitude from the distributed spectrum for each harmonic ? I found a way to calculate the magnitude for each distributed spectrum, but I dont find the correct mathematical description for this problem. Has anybody an idea how to define this mathematically. === Subject: Re: FFT result evaluation Content-Length: 1725 Originator: rusin@vesuvius >I have a signal with harmonics that are not integer multiples of the >fundamental frequency. So it is not possible to guarantee that the FFT - >Window get a full period of each harmonic of the analysed signal. The result >of the FFT is a distributed spectrum for each harmonic and it is not >possible to find out the correct magnitude of the harmonics. Is it possible >to calculate the correct signal magnitude from the distributed spectrum for >each harmonic ? I found a way to calculate the magnitude for each >distributed spectrum, but I dont find the correct mathematical description >for this problem. Has anybody an idea how to define this mathematically. The FFT (or any other DFT) is covering a finite range in the time domain and equivalent to multiplication of the time series by a boxcar function. The standard convolution theorem of Fourier analysis describes this in the frequency domain as a convolution with the DFT of the boxcar, known as the sinc-function. To avoid that the convolution with the negative pieces of the oscillatory sinc-funcion in the frequency domain correct peak in the frequency domain, one usually modifies the boxcar function to round up the sharp edges in the time domain and avoid the Gibbs osciallations of the sinc-function; the keywords to search for these techniques are apodisation, Norton-Beer ... This way one can to a good extend ensure that the area under the dispersed peaks in the frequency domain represents the originals series amplitudes. http://www.strw.leidenuniv.nl/~mathar === Subject: differential equation in finite characteristic Content-Length: 289 Originator: rusin@vesuvius