mm-1038 Subject: Re: Intersection of Zariski dense sets with hypersurfaces > David, assume that $D$ is a dense subset of an afÞne irreducible > variety $V$, and that $D$ has the property that it does not intersect > any hypersurface $H$ of $V$ densely: > Then does this not imply that THERE ARE NO SUBSETS $S$ of $D$ having > the property that the closure of $S$ in $V$ is of co-dimension one in > $V$? > This is rather striking given that > $Dim(Cl(D))=Dim(V)$ > Where by $Cl(D)$ is meant the zariski closure of the set $D$. Yes, it would seem so: the dimension of the closure of a subset of D, for D a subset of the plane as was constucted, can be either 0 or 2 but not 1, strage as that may seem. -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Re: A continuous iteration of f(x,y) r>0 ; f(x,y)^[r] Originator: israel@math.ubc.ca (Robert Israel) > f:R*R->R .Let us start with a simple case: > g(x,g(x,y))= f(x,y) f given (1) > or f(x,y)^[1/2]=g(x,y) > Example: f(x,y)=y/(1+x*y) , g(x,y)=2*y/(2+x*y) veriÞes (1). > By the same way will have:f(x,y)^[1/p]=p*y/(p+x*y)=g(x,y) p integer > verifying g(x,g(x....)))n times =f(x,y)=y/(1+x*y). > We may generalize: > if f(x,y)=phi^-1(phi(y)+n(x)) or m^[n(x)](y) , > phi(m(y))=phi(y)+1 ; > then f(x,y)^[r]=phi^-1(phi(y)+r.n(x)) or m^[r.n(x)](y) , > here r is a positive real number. > Please your comments and ideas,alain. Alain, my tetration.org web site is dedicated to continuous iteration. See http://www.tetration.org/Dynamics/index.html for resources on continuous iteration and http://www.tetration.org/Combinatorics/index.html for an outline of my combinatorial approach to the subject. For ideas consider Stephen Wolfram¹s question: > How can one extend recursive function deÞnitions to continuous > numbers? What is the continuous analog of the Ackermann function? The > symbolic forms of the Ackermann function with a Þxed Þrst argument > seem to have obvious interpretations for arbitrary real or complex > values of the second argument. But is there a general way to extend > these kinds of recursive deÞnitions to continuous cases? Given a way > to do this, how does it apply to recursive deÞnitions like those on > page 130? What happens to all the irregularities when one is between > integer values? Or is it only possible to Þnd simple continuous > generalizations to functions that show fundamentally simple behavior? > Can this be used as a characterization of when the behavior is simple? For my non-peer-reviewed response to Wolfram¹s question on the NKS Forum see http://forum.wolframscience.com/showthread.php?s=&threadid=488 Here¹s why I¹m interested in the subject. A theory of continuously iterated smooth matrix functions would probably encompass all of dynamics in physics. A theory of continuously iterated smooth complex functions would be powerful enough to extend the Ackermann function to complex numbers. Daniel Geisler === Subject: Message from the moderator Originator: israel@math.ubc.ca (Robert Israel) As you may have noticed, sci.math.research, along with some other newsgroups, has been attacked by vandals again, posting messages === were not approved by the moderators. There seems to be no way to prevent this from happening. However, depending on your news try putting the following lines in your killÞle. Robert Israel moderator for sci.math.research === Subject: Re: Recurrence relation, part 3 Originator: israel@math.ubc.ca (Robert Israel) > I am sorry I wasn¹t clear enough in my previous message. The > polynomial p(x,y) is a general polynomial, namely, p(x,y) = a_{00} + > a_{10}x + a_{01}y + a_{11}xy + a_{20}x^2 + a_{02}y^2 + ... + > a_{04}y^4. > [ Moderator¹s note. > I was talking about the simplest expansion, namely, Taylor. > a_{00} may be taken equal 1 if this simpliÞes things. > -ri] > Alex You might write p as 1 + q(x,y), where q has no constant term. Then the inÞnite series 1 - q + q^2 - q^3 + ... automatically makes sense as a formal power series, where it is indeed the reciprocal of 1+q. It also converges absolutely for small enough values of x and y, so it is what you need. Of course, it¹s still a mess to sort out the terms, since ones of degree n may occur all the way from q^{n/4} to q^n. But this at least allows straightforward computation of as much of the expansion as you want. William C. Waterhouse Penn State === Subject: Re: Graph Theory: A minimal subgraph problem 3QLpj-NoP*NzsIC,boYU]bQ]H¹y<#4ga3$21: > This is what I need to do: Given a graph with vertices V and edges > given by the matrix W, I need a connected subgraph (V¹,W¹) with the > minimal number of nodes so that from V¹ there are edges in the > original graph (V,W) to all the nodes of the original graph (for > example this means that for a fully connected graph I need only one > node in V¹). (The costs of these removed original edges could also be > minimized). > I do not have experience with the graph theory and I do not know where > to search. This is the minimum connected dominating set problem, equivalent to the maximum leaf spanning tree problem. -- David Eppstein Computer Science Dept., Univ. of California, Irvine http://www.ics.uci.edu/~eppstein/ === Subject: Has anybody encountered the following strange object ... Epigone-thread: dijerdnem Originator: israel@math.ubc.ca (Robert Israel) Hi everybody, has anybody seen an object like this =c d(x-v) d(y-u) where c is a number, d stands for the Dirac-Deltafunction, and the <> is a functional average over the function f. Does anyone know a name for this? Has anyone done some work using/involving this? We are doing research in physics and do not know how to deal with this object and wonder if any of you mathematicians have ideas. Any help will be appreciated! Shoshi === Subject: Re: Has anybody encountered the following strange object ... > has anybody seen an object like this > =c d(x-v) d(y-u) > where c is a number, d stands for the Dirac-Deltafunction, and > the <> is a functional average over the function f. Does anyone know > a name for this? Has anyone done some work using/involving this? > We are doing research in physics and do not know how to deal with > this object and wonder if any of you mathematicians have ideas. > Any help will be appreciated! What is a functional average? Your formula looks like something from the theory of vertex algebras (or vertex Poisson algebras, as you say f is a function), you might look at Kac, Vertex algebras for beginners, or Frenkel/Ben-Zvi. -- Maarten Bergvelt === Subject: Re: modular elliptic curves over number Þelds? Originator: israel@math.ubc.ca (Robert Israel) >> Are there examples of >> elliptic curves over number Þelds which are known not to be modular? >It is strongly believed that there are elliptic curves even over real >quadratic Þelds that do not occur as a quotient of the Albanese of any >Shimura variety (see, for example, a recent paper of Blasius). On the other hand, for any elliptic curve over any totally real Þeld there should be a Hilbert modular form which gives rise to the curve, although as Milne points out one has to be a bit careful as to what one means here---the standard recipe over Q, cutting the curve out from a Jacobian, does not generalise well in the real quadratic case (although as it happens the technicalities have been dealt with in this case by Taylor). The problem with the question is that it is sufÞciently vague to admit several answers. If the questioner deÞnes precisely what he/she means by modular then it would be possible to give a better answer. The problem is that it¹s very hard to make the question rigorous in general! The heart of the matter is that given a classical cuspidal modular eigenform whose q-expansion has integral coefÞcients, a standard construction of Shimura et al gives you an elliptic curve over Q. The big conjecture was that all elliptic curves arose in this way, and this was proved recently by Breuil, Conrad, Diamond and Taylor via a combination of a technical generalisation of Wiles¹ ideas and a bit of good fortune with the prime 3. Now change Q to a number Þeld. What should one replace the notion of modular form by? One could try using geometry (Shimura varieties), or one could try using something more analytic (automorphic forms). But one quickly runs into technicalities---for example the analogue of Shimura¹s construction modular form-->elliptic curve is not known even for automorphic forms for GL_2 over an imaginary quadratic Þeld (although again there has been progress by Taylor and others). At the end of the day, one could dream, and the dream is probably that given any elliptic curve over any number Þeld K there exists a cuspidal automorphic representation of the GL_2 of the adeles of K such that the representations of the absolute Galois group of K associated to the curve and the automorphic representation are isomorphic. But this is more like a fantasy at the minute---the Galois representation associated to the automorphic form has not yet been constructed and, given that essentially the only construction of Galois representation that we know is via Shimura varieties and that these don¹t seem to give enough Galois representations, this dream is rather a long way from being realised. Kevin === 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-4-52.abs.html Title: CategoriÞcation of the Kauffman bracket skein module of $I$-bundles over surfaces Author(s): Marta M. Asaeda Jozef H. Przytycki Adam S. Sikora Abstract: Khovanov deÞned graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov¹s construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefÞcients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefÞcients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefÞcients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a `categoriÞcation¹ of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro¹s exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L. Secondary: 57M25, 57R56 Keywords: Khovanov homology, categoriÞcation, skein module, Kauffman bracket Author(s) address(es): Dept of Mathematics, 14 MacLean Hall University of Iowa, Iowa City, IA 52242, USA and Dept of Mathematics, Old Main Bldg, The George Washington University 1922 F St NW, Washington, DC 20052, USA and Dept of Mathematics, 244 Math Bldg, SUNY at Buffalo Buffalo, NY 14260, USA, and Inst for Adv Study, School of Math, Princeton, NJ 08540, USA Email: asaeda@math.uiowa.edu, przytyck@gwu.edu, asikora@buffalo.edu === Subject: Re: Banach space Of Analytic Functions Originator: israel@math.ubc.ca (Robert Israel) On 14 Dec 04 13:54:46 -0500 (EST), Ali Taghavi In Fact I Need To Such Structure For The Following Fredholm Index >Interpretation for the main Object In The Hilbert 16th Problem: >http://www.arxiv.org/abs/math.DS/0408037 >On The Other Hand,: >I Found The Following Thesis in > Math Project genealogy Banach space Of Analytic Maps... >http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=48919 >can one give me some references containing materials simillar to above >this (and Simillar topics) thesis >thank you >Ali Taghavi >PS:What other space you suggest for action of a polynomial Vector >Þeld X in order to obtaining a bounded fredholm operator which index >is equal to the number of limit cycles of X ************************ David C. Ullrich === Subject: Re: Banach space Of Analytic Functions Epigone-thread: gumÞkend 1)What Fredholm Operator Theory exist for (Complete) TVS(Topological Vector Space) which assert that there are only a Þnite Number Of Possible Index On a given connected Component of The Space Of Fredholm Operators (this would be an Attempt to prove a uniform Upper Bound For The Number Of Limit Cycles Of Polynomial Vector Field of degree n 2)The Argument Below can not work on Schwarts maps since exponential maps are not schwarts maps! Note that schwarts maps Are Invariant Under Action Of a pollynomial Vector Field. (Can we change the Norm of Schwarts Maps Such That Differentiation(and Action of Polynomial Vector Field Be A Bounded Operator)? >On 14 Dec 04 13:54:46 -0500 (EST), Ali Taghavi I Search for a Banach structure on C^inf(R^2) >There is no norm on C^inf(R^2) which induces the standard >topology. This is easy to see: In the standard topology on >this space partial differentiation operators are continuous. >But these operators cannot be bounded with respect to a norm, >because they have unbounded spectrum (exponential functions >are eigenvectors with large eigenvalues). === Subject: Re: Banach space Of Analytic Functions >1)What Fredholm Operator Theory exist for (Complete) TVS(Topological >Vector Space) which assert that there are only a Þnite Number Of >Possible Index On a given connected Component of The Space Of Fredholm >Operators (this would be an Attempt to prove a uniform Upper Bound For >The Number Of Limit Cycles Of Polynomial Vector Field of degree n >2)The Argument Below can not work on Schwarts maps since exponential >maps are not schwarts maps! If the Question was About Schwarz Maps then you should have Asked About Schwarz Maps! >Note that schwarts maps Are Invariant Under Action Of a pollynomial >Vector Field. (Can we change the Norm of Schwarts Maps Such That >Differentiation(and Action of Polynomial Vector Field Be A Bounded >Operator)? What Norm are you Referring To? Assuming that Schwarts Maps are what is commonly called Schwarz functions, ie rapidly decreasing functions: It¹s easy to see that there is no norm on the Schwarz space which induces the standard topology. Proof: Let S be the Schwarz space and S¹ the Dual, the space of Tempered Distributions. Suppose the Topology on S is given by a Norm. Then Partial Derivatives are bounded on S. Since they are Essentially Self-Adjoint, Partial Derivative Operators are Also Bounded on S¹. But Exponentials Are Tempered Distributions. QED! >>On 14 Dec 04 13:54:46 -0500 (EST), Ali Taghavi >There is no norm on C^inf(R^2) which induces the standard >>topology. This is easy to see: In the standard topology on >>this space partial differentiation operators are continuous. >>But these operators cannot be bounded with respect to a norm, >>because they have unbounded spectrum (exponential functions >>are eigenvectors with large eigenvalues). ************************ David C. Ullrich === Subject: Re: Banach space Of Analytic Functions Epigone-thread: gumÞkend Of Course I Search for a NON STANDARD topology for some Functional Space(For Example schwars maps) which is invariant under the action of a polynomial vector Þeld X (As a Bounded operator ) Such that the Number of Attractores Of X can be interprated in term of Fredholm Index of the operator(action) For Example I Search some reference Containing materials As Follow: http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=48919 which can help the idea in following: http://www.arxiv.org/abs/math.DS/0408037 Finally,let we remove the banachness of linear space in the deÞnition of a fredholm operator,what consequence would appear?does Dieudone¹s theorem on connected component of the fredholm operators valid? >If the Question was About Schwarz Maps then you should have >Asked About Schwarz Maps! >>Note that schwarts maps Are Invariant Under Action Of a pollynomial >>Vector Field. (Can we change the Norm of Schwarts Maps Such That >>Differentiation(and Action of Polynomial Vector Field Be A Bounded >>Operator)? >What Norm are you Referring To? Assuming that Schwarts Maps >are what is commonly called Schwarz functions, ie rapidly >decreasing functions: It¹s easy to see that there is no >norm on the Schwarz space which induces the standard >topology. >Proof: Let S be the Schwarz space and S¹ the Dual, the >space of Tempered Distributions. Suppose the Topology >on S is given by a Norm. Then Partial Derivatives >are bounded on S. Since they are Essentially Self-Adjoint, >Partial Derivative Operators are Also Bounded on S¹. >But Exponentials Are Tempered Distributions. QED! === Subject: Re: Uniqueness of implicit functions Epigone-thread: zhoabrarshel Originator: israel@math.ubc.ca (Robert Israel) >>Consider two implicit functions: >>f(x,y)=0, >>g(x,y)=0. >>What can be said about relation between them, if it is known that >>f(x,y)=0 iff g(x,y)=0. >Maybe I don¹t understand what an implicit function is. If you >mean, what can be said about two functions f,g : R^2 --> R knowing >that they vanish at the same points? the answer might be Not much; >I would say, for example, that f(x,y) = (sin(x) - cos(y) + xy)^2 + 1 >and g(x,y) = ( exp(xy) + x - y )^2 + 1 are very different, but >they do vanish at the same points... . >But that¹s assuming you meant x and y to be real. If you allow them to be >complex the situation is different. IN particular, when f and g are both >_polynomials_ and they vanish at the same points (x,y) in C^2, then by >Hilbert¹s Nullstellensatz, the ideals (f) and (g) have the same radical >in the ring C[x,y], so that f and g have the same irreducible divisors >(possibly to different powers, e.g. f = x^2y and g = x y^3.) >>For example, if they are differentiable, then >>f1(x,y)*g2(x,y) = f2(x,y)*g1(x,y), where f1,f2,g1,g2 are derivatives >>of corresponding function by the Þrst (second) variable. >Fails in my example, above. More generally, if the factorizations are >f = product( (p_i)^(n_i) ) and g = product( (p_i)^(m_i) ) , then >f_1 / f = sum( (n_i) (p_i)_1 / (p_i) ) and similarly for f_2 / f, >so that f_1 / f_2 = > sum( (n_i) (p_i)_1 / (p_i) )/ sum( (n_i) (p_i)_2 / (p_i) ) >Of course g_1 / g_2 is almost identical; one need only change the n_i >to m_i. So we don¹t have f_1 / f_2 = g_1 / g_2 unless f and g >are powers of the same irreducible. >I am told that the ring of analytic functions behaves in a very similar way; >certainly the previous example suggests ways that your conjecture could fail. >dave >PS -- I tried to reach you much earlier but you did not provide a way >to determine a valid email address. Sorry, in the previous post I, certainly, had to add some remarkable comments: There variables x,y are reals and both implicit functions f(x,y)=0, g(x,y)=0 deÞnes the same regular curve on the plane. May be we should start with easier case: Consider a regular (say, continuous) function y=f(x) of real variable x on some interval B. It is required to Þnd all functions g(x,y) such, that equation g(x,y)=0 has y=f(x) as the only solution, when x is from B. The current email is valid. === Subject: Re: Uniqueness of implicit functions >Consider two implicit functions: >f(x,y)=0, >g(x,y)=0. >What can be said about relation between them, if it is known that >f(x,y)=0 iff g(x,y)=0. >>Maybe I don¹t understand what an implicit function is. If you >>mean, what can be said about two functions f,g : R^2 --> R knowing >>that they vanish at the same points? the answer might be Not much; >>I would say, for example, that f(x,y) = (sin(x) - cos(y) + xy)^2 + 1 >>and g(x,y) = ( exp(xy) + x - y )^2 + 1 are very different, but >>they do vanish at the same points... . >Sorry, in the previous post I, certainly, had to add some remarkable >comments: There variables x,y are reals and both implicit functions >f(x,y)=0, g(x,y)=0 deÞnes the same regular curve on the plane. >May be we should start with easier case: >Consider a regular (say, continuous) function y=f(x) of real >variable x on some interval B. It is required to Þnd all functions >g(x,y) such, that equation g(x,y)=0 has y=f(x) as the only solution, >when x is from B. g(x,y) = h(x,y-f(x)) where h is any function (continuous if you¹re requiring g to be continuous) such that, for x in B, h(x,y) = 0 if and only if y = 0. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Uniqueness of implicit functions Epigone-thread: zhoabrarshel >>May be we should start with easier case: >>Consider a regular (say, continuous) function y=f(x) of real >>variable x on some interval B. It is required to Þnd all functions >>g(x,y) such, that equation g(x,y)=0 has y=f(x) as the only solution, >>when x is from B. >g(x,y) = h(x,y-f(x)) where h is any function (continuous if you¹re >requiring g to be continuous) such that, for x in B, h(x,y) = 0 if and >only if y = 0. Please, explain, how one can prove, that all such functions have the form g(x,y) = h(x,y-f(x)). Is there a way to generalized the result for the following case: Suppose that a function f(x1,...,xn)=0 (of real variables x1,....,xn on some connected set B^n) deÞnes a reqular curve in R^(n-1). It is required to Þnd all functions g(x1,...xn) such, that equation g(x1,...,xn)=0 has this curve as the only solution in B^n. === Subject: Re: Uniqueness of implicit functions >May be we should start with easier case: >Consider a regular (say, continuous) function y=f(x) of real >variable x on some interval B. It is required to Þnd all functions >g(x,y) such, that equation g(x,y)=0 has y=f(x) as the only solution, >when x is from B. >>g(x,y) = h(x,y-f(x)) where h is any function (continuous if you¹re >>requiring g to be continuous) such that, for x in B, h(x,y) = 0 if >and >>only if y = 0. >Please, explain, how one can prove, that all such functions have the >form g(x,y) = h(x,y-f(x)). Given g, simply deÞne h(x,y) = g(x,y+f(x)). >Is there a way to generalized the result for the following case: >Suppose that a function f(x1,...,xn)=0 (of real variables x1,....,xn >on some connected set B^n) deÞnes a reqular curve in R^(n-1). It is >required to Þnd all functions g(x1,...xn) such, that equation >g(x1,...,xn)=0 has this curve as the only solution in B^n. I¹m not sure what you mean by a regular curve in R^(n-1). It¹s certainly not a curve. Perhaps what you have in mind is something like this: Suppose there¹s a continuous function r(x_1,...,x_{n-1}) on a subset A of R^(n-1) such that for (x_1,...,x_n) in A and x_{n-1} in R, g(x_1,...,x_n) = 0 iff x_n = r(x_1,...,x_{n-1}). Then h(x_1,...,x_n) = g(x_1,...,x_{n-1},x_n+r(x_1,...,x_{n-1})) is a continuous function on A x R such that h(x_1,...,x_n) = 0 iff x_n = 0 and g(x_1,...,x_n) = h(x_1,...,x_{n-1},x_n-r(x_1,...,x_{n-1})). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Why exp(-st) in the Laplace Transform? Paul, I am not sure that I comprehend all of your commentary, however I do = note one of your statements If you take the Laplace transform, ... Shifting an exponential (exp(-st)) by a constant (t -> t + a) is equivalent to multiplying it by a real number (exp(-sa)). Your statement is not quite valid, in that, for the case of a Laplace = transform, the transform variable s has both real and imaginary = components, thus exp(-sa) has both real and imaginary components as = well. As in the Fourier transform, you would integrate over the = imaginary component. It is the real component, however, which = distinguishes the Laplace Transform from the Fourier Transform. That = is, the real component of our multiplier will tend to attenuate the = function. An example of the use of a Laplace transform would be to = transform the unit step function U(t). Note that it¹s Laplace Transform = is determined to be 1/s, under the restriction re(s) > 0. The Fourier = Transform of U(t) does not exist. Ed -- Edward Hyman EdwardH@email.uophx.edu Other EMail: e.hyman@worldnet.att.net > Does anyone have an explaination why the kernel function exp(-st) = was > used in the deÞnition of the Laplace transform? > Is there a physical meaning to the use of this function? Here is a group-theoretic answer. The existence (and utility) of both the Laplace and Fourier transforms come from the symmetry of the real line under translations (t -> t + a, where a is some constant shift). This is a one-dimensional Lie group, so it is Abelian (commutative), and its unitary irreducible representations are therefore one-dimensional. These are the Fourier transform kernels, exp(-i w t), for w any real number. Under the translation operation (t -> t + a), these get multiplied by the (unitary) number exp(-i w a). In other words, when you take the Fourier transform of a well-behaved complex function, you are splitting it into functions which transform in a simple way under translations! This splits the inÞnite- dimensional space of functions over the real line (t) into an inÞnite number of one-dimensional subspaces (the Fourier kernels) which are dense (in nice functions), and which transform trivially (multiplication by a constant) under translations. If you draw the Fourier kernels as their graph in the complex plane versus t, they are constant-pitch helices. Moving a constant-pitch helix by a constant is equivalent to rotating it! If you take the Laplace transform, you are splitting a well-behaved function into its *real* irreducible representations: exponentials! Shifting an exponential (exp(-st)) by a constant (t -> t + a) is equivalent to multiplying it by a real number (exp(-sa)). Taking the Laplace transform is splitting the inÞnite-dimensional space of well-behaved functions into an inÞnite number of one-dimensional subspaces which are dense in well-behaved functions, and which each behave nicely under translations. So, both the Laplace and Fourier transforms are sort of splitting functions into the normal modes of the group of translations of the real line. === Subject: This week in the mathematics arXiv (6 Dec - 10 Dec) Here are this week¹s 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 modiÞcation. Titles in the mathematics arXiv (6 Dec - 10 Dec) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0412194 Srikanth Iyengar, Tony J. Puthenpurakal: Hilbert-Samuel functions of modules over Cohen-Macaulay rings math.AC/0412140 Juergen Herzog, Yukihide Takayama, Naoki Terai: On the radical of a monomial ideal AG: Algebraic Geometry ---------------------- math.AG/0412189 Gunther Cornelissen, Ariane Mezard: Relevements des revetements de courbes faiblement ramiÞes (Lifts of weakly ramiÞed coverings of curves) math.AG/0412159 Johannes Huisman, Frederic Mangolte: Every connected sum of lens spaces is a real component of a uniruled algebraic variety math.AG/0412152 Matthieu Willems: K-theorie equivariante des varietes de Bott-Samelson. Application a la structure multiplicative de la K-theorie equivariante des varietes de drapeaux math.AG/0412150 Sergey Lysenko: Geometric Bessel models for GSp_4 and multiplicity one math.AG/0412148 Skip Garibaldi, Detlev Hoffmann: Totaro¹s question for G_2, F_4, and E_6 math.AG/0412142 Marcos Jardim: Instanton sheaves on complex projective spaces math.AG/0412134 Marian Aprodu: Remarks on syzygies of $d$-gonal curves math.AG/0412121 Roman M. Fedorov: Algebraic and hamiltonian approaches to isostokes deformations math.AG/0412117 GianMario Besana, Maria Lucia Fania: The dimension of the Hilbert scheme of special threefolds math.AG/0412103 H. Lange, S. Recillas: A family of Prym-Tyurin varieties of exponent 3 math.AG/0412102 Izzet Coskun: The arithmetic and the geometry of Kobayashi hyperbolicity math.AG/0412089 Jun Li, Kefeng Liu, Jian Zhou: Topological String Partition Functions as Equivariant Indices math.AG/0412075 Qing Liu: Stable reduction of Þnite covers of curves math.AG/0412073 A. S. Buch, R. Rimanyi: A formula for non-equioriented quiver orbits of type $A$ AP: Analysis of PDEs -------------------- math.AP/0412160 Pierre Germain: Existence globale de solutions d¹energie inÞnie de l¹equation de Navier-Stokes 2D math.AP/0412146 G. Barbatis, A. Tertikas: On a class of Rellich inequalities math.AP/0412125 Daniel Alayon-Solarz: On Some ModiÞcations of the Fueter Operator math.AP/0412106 Khalil El Mehdi: On Conformal Paneitz Curvature Equations in Higher Dimensional Spheres math.AP/0412105 Khalil El Mehdi: Single Blow up Solutions for a Slightly Subcritical Biharmonic Equation math.AP/0412088 Olga S. Rozanova: Hydrodynamic approach to constructing solutions of Hydrodynamic approach to constructing solutions of nonlinear Schrodinger equation in the critical case nlin.SI/0412010 A.S. Fokas: Linearizable Initial-Boundary Value Problems for the sine-Gordon Equation on the Half-Line nlin.SI/0412009 A.S. Fokas, J.T. Stuart: The Time Periodic Solution of the Burgers Equation on the Half-Line and an Application to Steady Streaming nlin.SI/0412008 A.S. Fokas, A.R. Its: The Nonlinear Schrodinger Equation on the Interval nlin.SI/0412005 A.S. Fokas: The Davey-Stewartson I Equation on the Quarter Plane with Homogeneous Dirichlet Boundary Conditions CA: Classical Analysis and ODEs ------------------------------- math.CA/0412199 Michael Aristidou, Mark Davidson, Gestur ŒOlafsson: Laguerre Functions on Symmetric Cones and recursion relations in the Real Case math.CA/0412174 Carlos Cabrelli, Michael Lacey, Ursula Molter, Jill C Pipher: Variations on the Theme of Journe¹s Lemma math.CA/0412115 V.Poberezhny: On Special monodromy groups and Riemann-Hilbert problem for Riemann equation math.CA/0412081 J. M. Aldaz: An example on the maximal function associated to a nondoubling measure math.CA/0412080 Michael Schlosser: Noncommutative extensions of Ramanujan¹s 1-psi-1 summation CO: Combinatorics ----------------- math.CO/0412153 Hasan Coskun, Robert A. Gustafson: Well--Poised Macdonald Functions W_lambda and Jackson CoefÞcients omega_lambda On BC_n math.CO/0412130 C. De Concini, C. Procesi: On the geometry of graph arrangements cs.CC/0412013 Jean-Christophe Dubacq, Veronique Terrier: Signals for Cellular Automata in dimension 2 or higher math.CO/0412124 Richard Ehrenborg, Margaret Readdy: The Tchebyshev transforms of the Þrst and second kind math.CO/0412118 Theresia Eisenkolbl: (-1)-enumeration of self-complementary plane partitions math.CO/0412114 Lars Engebretsen: Bipartite Multigraphs with Expander-Like Properties math.CO/0412091 Dan Bernstein: Euler-Mahonian polynomials for C_a wr S_n cond-mat/0411630 Otto Pulkkinen, Juha Merikoski: Phase transitions on Markovian bipartite graphs - an application of the zero-range process cond-mat/0407278 Dimitris Achlioptas, Cristopher Moore: The Chromatic Number of Random Regular Graphs CV: Complex Variables --------------------- math.CV/0412096 N.Kruzhilin, A.Sukhov: Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces math.CV/0412095 H.Gaussier, A.Sukhov: On the geometry of model almost complex manifolds with boundary DG: Differential Geometry ------------------------- math.DG/0412197 Ji-Ping Sha, Bruce Solomon: No skew branes on non-degenerate hyperquadrics math.DG/0412190 Isabel Fernandez, Francisco J. Lopez, Rabah Souam: The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space $l^3$ math.DG/0412185 D.H. Phong, Jacob Sturm: On stability and the convergence of the Kahler-Ricci þow math.DG/0412169 Emilio Musso, Lorenzo Nicolodi: Tableaux over Lie algebras, integrable systems, and classical surface theory math.DG/0412151 Anda Degeratu, Mark Stern: The Positive Mass Conjecture for Non-spin Manifolds nlin.SI/0411051 V. D. Gershun: Integrable string and hydrodynamical type models and nonlocal brackets math.DG/0412127 John Lott, Cedric Villani: Ricci curvature for metric-measure spaces via optimal transport gr-qc/0412020 Michael T. Anderson, Piotr T. Chrusciel: Asymptotically simple solutions of the vacuum Einstein equations in even dimensions math.DG/0412123 Bruce Kleiner, Bernhard Leeb: Rigidity of invariant convex sets in symmetric spaces math.DG/0412109 Ioan Bucataru: Metric nonlinear connections math.DG/0412097 Marius Crainic: Generalized complex structures and Lie brackets math.DG/0412084 Mohammed Abouzaid, Mitya Boyarchenko: Local structure of generalized complex manifolds math.DG/0412071 T. Hasanis, A. Savas-Halilaj, T. Vlachos: Complete minimal hypersurfaces of $S^4$ with zero Gauss-Kronecker curvature math.DG/0412068 Li Ma: Some properties of non-compact complete Riemannian manifolds DS: Dynamical Systems --------------------- math.DS/0412195 Abdelouahab Arouche, Mohamed Deffaf, Abdelghani Zeghib: On Lorentz dynamics math.DS/0412180 Juan Rivera-Letelier: Sur la structure des ensembles de Fatou p-adiques math.DS/0412177 Nikos Frantzikinakis Bryna Kra: Polynomial conÞgurations on integer subsets with positive density math.DS/0412175 Bassam Fayad: Rank one and mixing differentiable þows math.DS/0412172 Bassam Fayad: Smooth mixing þows with singular spectra math.DS/0412167 J.-R. Chazottes, P. Collet, B. Schmitt: Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems math.DS/0412166 J.-R. Chazottes, P. Collet, B. Schmitt: Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems math.DS/0412162 Romain Dujardin: Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms math.DS/0412158 Konstantin Igudesman: Dynamics of inÞnite-multivalued transformations math.DS/0412098 S. Aranson, E. Zhuzhoma: Proof of the Morse conjecture for analytic þows on orientable surfaces math.DS/0412072 Ali Tahzibi, Vanderlei Horita: Partial Hyperbolicity for Symplectic Diffeomorphisms math.DS/0412066 Julie Deserti, Dominique Cerveau: Feuilletages et actions de groupes sur les espaces projectifs FA: Functional Analysis ----------------------- math.FA/0412171 G. Androulakis, K. Beanland, S.J. Dilworth, F. Sanacory: Embedding $ell_{infty}$ into the space of all Operators on Certain Banach Spaces math.FA/0412165 Dmitry S. Kalyuzhnyu{i}-Verbovetzkiu{i}, Victor Vinnikov: Non-commutative positive kernels and their matrix evaluations math.FA/0412164 Dmitry S. Kalyuzhnyu{i}-Verbovetzkiu{i}: On the {Bessmertnyu{i}} Class of Homogeneous Positive Holomorphic Functions on a Product of Matrix Halfplanes math.FA/0412163 Dmitry S. Kalyuzhnyu{i}-Verbovetzkiu{i}: Multivariable $rho$-contractions math.FA/0412161 Dmitry S. Kalyuzhnyu{i}-Verbovetzkiu{i}: Carath¹{e}odory interpolation on the non-commutative polydisk math.FA/0412116 A.A.Shkalikov: On invariant subspaces of dissipative operators in a space with indeÞnite metric math.FA/0412108 L. Biliotti, R. Exel, P. Piccione, D. V. Tausk: On the Singularities of the Exponential Map in InÞnite Dimensional Riemannian Manifolds GM: General Mathematics ----------------------- math.GM/0412157 Frank Swenton: Limits and the system of near-numbers math.GM/0412133 Jean-Marie Didry, Pierre-Yves Gaillard: Around the Chinese Remainder Theorem math.GM/0412087 B.G.Sidharth: A New Integral Transform GR: Group Theory ---------------- math.GR/0412136 Danny Calegari, Nathan M. DunÞeld: An ascending HNN extension of a free group inside SL(2,C) math.GR/0412128 Ilya Kapovich: Currents on free groups math.GR/0412101 Kai-Uwe Bux, Kevin Wortman: A Geometric proof that SL(2,Z[t,1/t]) is not Þnitely presented GT: Geometric Topology ---------------------- math.GT/0412191 Benjamin Himpel: A Splitting Formula for the Spectral Flow of the Odd Signature Operator on 3-Manifolds Coupled to a Path of SU(2) Connections math.GT/0412187 Ekaterina Pervova, Carlo Petronio: Complexity and T-invariant of Abelian and Milnor groups, and complexity of 3-manifolds math.GT/0412184 Olga Plamenevskaya: Transverse knots and Khovanov homology math.GT/0412183 Olga Plamenevskaya: Transverse knots, branched double covers and Heegaard Floer contact invariants math.GT/0412147 Ben Klaff, Peter B. Shalen: The diameter of the set of boundary slopes of a knot math.GT/0412139 Ivan Izmestiev: On the hull numbers of torus links math.GT/0412126 Ronald Fintushel, Ronald J. Stern: Double node neighborhoods and families of simply connected 4-manifolds with b^+=1 math.GT/0412120 Denis Auroux: A stable classiÞcation of Lefschetz Þbrations math.GT/0412078 Kenneth J. Shackleton: Tightness and computing distances in the curve complex math.GT/0412074 Naoko Kamada: Span of the Jones polynomial of an alternating virtual link HO: History and Overview ------------------------ math.HO/0412154 Leonhard Euler: An analytical exercise KT: K-Theory and Homology ------------------------- math.KT/0412156 Wolfgang Lueck: K-and L-theory of the semi-direct product of the discrete three-dimensional Heisenberg group by Z/4 math.KT/0412131 Christian Voigt: A new description of equivariant cohomology for totally disconnected groups LO: Logic --------- math.LO/0412144 J. Michael Dunn, Tobias J. Hagge, Lawrence S. Moss, Zhenghan Wang: Quantum logic as motivated by quantum computing MG: Metric Geometry ------------------- math.MG/0412111 Thierry Barbot, Francois Beguin, Abdelghani Zeghib: Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on $AdS_3$ math.MG/0412093 Gunter M. Ziegler: Polyhedral surfaces of high genus cs.DS/0412008 Robert Krauthgamer, James R. Lee, Manor Mendel, Assaf Naor: Measured descent: A new embedding method for Þnite metrics MP: Mathematical Physics ------------------------ quant-ph/0412053 M. Keyl: Quantum state estimation and large deviations math-ph/0412029 Michel L. Lapidus, Erin P. J. Pearse: A tube formula for the Koch snowþake curve, with applications to complex dimensions math-ph/0412028 Christopher J. Fewster, Stefan Hollands: Quantum Energy Inequalities in two-dimensional conformal Þeld theory math-ph/0412027 Christopher J. Fewster, Izumi Ojima, Martin Porrmann: p-Nuclearity in a New Perspective math-ph/0412026 F. Hiroshima, K. R. Ito: Mass Renormalization in Non-relativistic Quantum Electrodynamics with Spin 1/2 math-ph/0412025 Mihai Stoiciu: The Statistical Distribution of the Zeros of Random Paraorthogonal Polynomials on the Unit Circle math-ph/0412024 Antonio Hern¹andez-Gardu~no, Ernesto A. Lacomba: Collisions and regularization for the 3-vortex problem hep-th/0412091 D. D. Dimitrijevic, G. S. Djordjevic, Lj. Nesic: Propagator for the Free math-ph/0412023 Elliott H. Lieb, Robert Seiringer, Jakob Yngvason: JustiÞcation of c-Number Substitutions in Bosonic Hamiltonians math-ph/0412022 Aarti Sawant, Amit Acharya: Model Reduction via Parametrized Invariant Manifolds: Some Examples math-ph/0412021 Zhi-Ming Gu: The Lie Group Structure of the $eta-xi$ Space-time and its Physical SigniÞcance math-ph/0412020 T. Sakaguchi: An asymptotic formula for models with caustics hep-th/0412061 Jasbir Nagi: On Extensions of Superconformal Algebras hep-th/0412058 Andrzej Herdegen: Asymptotic algebra of quantum electrodynamics quant-ph/0411197 Bernd Kuckert, Jens Mund: Spin & Statistics in Nonrelativistic Quantum Mechanics, II nlin.SI/0412018 M. Blaszak, A. Sergyeyev: Maximal superintegrability of Benenti systems nlin.SI/0412017 Fabio Musso, Matteo Petrera, Orlando Ragnisco, Giovanni Satta: Backlund transformations for the rational Lagrange chain math-ph/0412019 Roman Shvydkoy: The essential spectrum of advective equations math-ph/0412018 Christian Hainzl, Mathieu Lewin, Christof Sparber: Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation math-ph/0412017 Yan V. Fyodorov: Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond math-ph/0412016 Boyka Aneva, Todor Popov: Hopf Structure and Green Ansatz of Deformed Parastatistics Algebras math-ph/0412014 Giuseppe Ruzzi: Homotopy, net-cohomology and superselection sectors in globally hyperbolic spacetimes hep-th/0411094 Vladimir V. Bazhanov, Vladimir V. Mangazeev: Eight-vertex model and non-stationary Lame equation math-ph/0412015 Didier Robert: Non linear eigenvalue problems math-ph/0410002 P. Di Francesco, P. Zinn-Justin, J.-B. Zuber: Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops gr-qc/0412012 Jorma Louko, Robert B. Mann, Donald Marolf: Geons with spin and charge cond-mat/0412034 M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason: Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice quant-ph/0412016 B. Bagchi, A. Banerjee, C. Quesne, V. M. Tkachuk: Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass math-ph/0412013 Daniel Alayon-Solarz: We call attention to the unusual properties that the 4 dimensional solutions for a modiÞed Fueter-Dirac equations satisfy: In a coordinate-free, constant-free and strictly mathematical way it is possible to show that all the solutions for a modiÞed Fueter-Dirac Equation, which are radial symmetric when restricted to the 3-space, have a nice algebraic structure. Locally, these solutions behave naturally in a algebraic way determined by a certain commutative ring related to the quaternions with non-invertibles. This ring has a nontrivial localization. By using left and right versions of the operator we obtain quirality math-ph/0412012 Hatem Najar: About a result of S.M. Kozlov math-ph/0412011 Sebastian Formanski, Maciej Przanowski: $ast$-SDYM Fields and Heavenly Spaces. I. $ast$-SDYM equations as an integrable system math-ph/0412010 Takao Suzuki: Classical solutions of the degenerate Garnier system and their coalescence structures math-ph/0412009 Elliott H. Lieb, Robert Seiringer: A Stronger Subadditivity of Entropy math-ph/0412008 Andrei Okounkov: Random surfaces enumerating algebraic curves NA: Numerical Analysis ---------------------- physics/0412030 A.S. Fokas, A. Iserles, V. Marinakis: Reconstruction Algorithms for Positron Emission Tomography and Single Photon Emission Computed Tomography and their Numerical Implementation physics/0412028 A.S. Fokas, Y. Kurylev, V. Marinakis: The Unique Determination of Neuronal Currents in the Brain via Magnetoencephalography math.NA/0412112 Massimo Fornasier: Nonlinear projection digital image inpainting and restoration methods NT: Number Theory ----------------- math.NT/0412181 Michael O. Rubinstein: Computational methods and experiments in analytic number theory math.NT/0412178 Luis Dieulefait: Appendix to: The level 1 weight 2 case of Serre¹s conjecture - a strategy for a proof math.NT/0412176 Hershy Kisilevsky, Jack Sonn: Abelian extensions of global Þelds with constant local degrees math.NT/0412173 Matilde N. Lalin: Mahler measure of some n-variable polynomial families math.NT/0412145 Francesca Aicardi, Vladlen Timorin: On perfect binary quadratic forms math.NT/0412141 Victor Beresnevich, Sanju Velani: A Mass Transference Principle and the DufÞn-Schaeffer conjecture for Hausdorff measures math.NT/0412135 A. Granville, P. Kurlberg: Poisson statistics via the Chinese remainder theorem math.NT/0412104 Ariel Pacetti, Gonzalo Tornaria: Examples of Shimura correspondence for level p^2 and real quadratic twists math.NT/0412099 Luis Dieulefait: The level 1 weight 2 case of Serre¹s conjecture - a strategy for a proof math.NT/0412090 Shinji Fukuhara: Hecke operators on weighted Dedekind symbols math.NT/0412086 R. de la Breteche, T.D. Browning: On Manin¹s conjecture for singular del Pezzo surfaces of degree four, I math.NT/0412083 Brian Conrey, Jon Keating, Michael Rubinstein, Nina Snaith: Random Matrix Theory and the Fourier CoefÞcients of Half-Integral Weight Forms math.NT/0412079 G. Everest, S. Stevens, D. Tamsett, T. Ward: Primitive Divisors of Quadratic Polynomial Sequences math.NT/0412076 Chandrashekhar Khare, Jean-Pierre Wintenberger: On Serre¹s reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q math.NT/0412067 Yoshinori Yamasaki: q-Analogues of the Barnes multiple zeta functions math.NT/0412065 Lei Yang: Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations math.NT/0412063 Eduardo Duenez, Steven J. Miller, Howard Straubing, Amitabha Roy: Incomplete Quadratic Exponential Sums in Several Variables OA: Operator Algebras --------------------- math.OA/0412170 Ilwoo Cho: Diagonal Tracial Graph W*-Probability Theory math.OA/0412138 George Eleftherakis: Decompositions of Reþexive Bimodules over Maximal Abelian Selfadjoint Algebras math.OA/0412137 Bojan Magajna: On tensor products of operator modules math.OA/0412129 Huaxin Lin, Hiroki Matui: Minimal dynamical systems on the product of the Cantor set and the circle II math.OA/0412107 Rolf Gohm: Decompositions of Beurling Type for E_0-Semigroups OC: Optimization and Control ---------------------------- math.OC/0412070 Lorenzo Finesso, Peter Spreij: Nonnegative Matrix Factorization and I-Divergence Alternating Minimization PR: Probability --------------- math.PR/0412198 Pablo A. Ferrari, James B. Martin, Leandro P. R. Pimentel: Roughening and inclination of competition interfaces math.PR/0412196 Jan Obloj, Marc Yor: On local martingale and its supremum: harmonic functions and beyond math.PR/0412193 Djalil Chafai, Didier Concordet: A continuous stochastic maturation model math.PR/0412188 Hanene Mohamed, Philippe Robert: A Probabilistic Analysis of Some Tree Algorithms math.PR/0412182 Misja Nuyens: The Foregound-Background Processor Sharing Queue: an overview math.PR/0412155 James Allen Fill, Nevin Kapur, Alois Panholzer: Destruction of very simple trees cond-mat/0412166 Thierry Huillet: Random Partitioning Problems Involving Poisson Point Processes On The Interval math.PR/0412092 Oliver Delzeith: On Skorohod spaces as universal sample path spaces QA: Quantum Algebra ------------------- math.QA/0412192 D.I. Gurevich, P.N. Pyatov, P.A. Saponov: Cayley-Hamilton theorem for quantum matrix algebras of GL(m|n) type math.QA/0412149 Kevin Costello: Topological conformal Þeld theories and Calabi-Yau categories math.QA/0412143 Pavel Etingof, Shlomo Gelaki: Liftings of graded quasi-Hopf algebras with radical of prime codimension math.QA/0412122 I. Balint, K. Szlachanyi: Finitary Galois extensions over noncommutative bases math.QA/0412113 Alice Fialowski, Martin Schlichenmaier: Global geometric deformations of current algebras as Krichever-Novikov type algebras math.QA/0412100 Shao-You Zhao, Kang-Jie Shi, Rui-Hong Yue: The Center for the Elliptic Quantum Group $E_{tau,eta}(sl_n)$ math.QA/0412094 Ferenc Gerlits: The Euler characteristic of graph complexes via Feynman diagrams math.QA/0412069 Anatol. N. Kirillov, Toshiaki Maeno: A note on quantization operators on Nichols algebra model for Schubert calculus on Weyl groups RT: Representation Theory ------------------------- math.RT/0412186 Issai Kantor, Gregory Shpiz: Graded Lie algebras deÞned by Jordan algebras and their representations math.RT/0412179 Meighan I. Dillon: Constructing Graded Lie Algebras math.RT/0412168 G. Lusztig: Character sheaves on disconnected groups, VII math.RT/0412119 Yuly Billig: Jet modules math.RT/0412085 Henning Krause, Jue Le: The Auslander-Reiten formula for complexes of modules math.RT/0412077 Aslak Bakke Buan, Robert J. Marsh, Idun Reiten: Cluster mutation via quiver representations SG: Symplectic Geometry ----------------------- math.SG/0412110 Paul Biran: Lagrangian non-Intersections math.SG/0412082 Timothy J. Hodges, Milen Yakimov: Triangular Poisson structures on Lie groups and symplectic reduction SP: Spectral Theory ------------------- math.SP/0412132 B. Chenaud, P. Duclos, P. Freitas, D. Krejcirik: Geometrically induced discrete spectrum in curved tubes math.SP/0412064 Colin Guillarmou: Resonances on some geometrically Þnite hyperbolic manifolds ST: Statistics -------------- cs.NI/0412037 David B. Chua, Eric D. Kolaczyk, Mark Crovella: A Statistical Framework for EfÞcient Monitoring of End-to-End Network Properties -- / 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 that¹s Þt to e-print * === Subject: Schatten Class Operators Epigone-thread: jongþeldbrend Hi all, Could someone please give me a good reference on the Schatten Class Operators? I read about this kind of Operators in the book Operator theory in function spaces by K. Zhu but I can¹t Þgure out how to show that S_p (the p-Schatten Class Operators) is a complete normed space for 1<=p=N (2) sum(ak-bk)=0 If (2) then the respective convergents pn/qn and cn/dn lie on the same horizontal line in the Stern-Brocot tree (for n>=N). If (1) then Lagrange showed that x2=f(x1) where f(x)=(ax+b)/(cx+d) with integer coefÞcients (if I remember correctly). Question 1: is there en explicit form for f (for instance knowing ak,bk for k[a0+b0;a1+b1,a2+b2,...] in terms of x1 and x2? Loic