Poles of Zeta Functions on Normal Surfaces

Let (S, 0) be a normal surface germ and Let f a non-constantregular function on Let (S, 0) with Let f(0) = 0. Using anyadditive invariant on complex algebraic varieties one can associatea zeta function to these data, where the topological and motiviczeta functions are the roughest and the finest zeta functions,respectively. In this paper we are interested in a geometricdetermination of the poles of these functions. The second authorhas already provided such a determination for the topologicalzeta function in the case of non-singular surfaces. Here wegive a complete answer for all normal surfaces, at least onthe motivic level. The topological zeta function however seemsto be too rough for this purpose, although for negative poles,which are the only ones in the non-singular case, we are ableto prove exactly the same result as for non-singular surfaces. We also give and verify a (natural) definition for when a rationalnumber is a pole of the motivic zeta function. 2000 MathematicsSubject Classification 14B05, 14E15, 14J17 (primary), 32S50(secondary).     

Mixing in the Absence of the Shrinking Target Property

Using reparametrizations of linear flows, we show that thereexist area-preserving real analytic maps of the three-dimensionaltorus that are ‘mixing of all orders’ and do notenjoy the monotone shrinking target property. Prior to that,we give a short proof of a result of Kurzweil from 1955: namely,that a translation T of the torus T^d has the monotone shrinkingtarget property if and only if the vector is badly approximable(that is, of constant type). 2000 Mathematics Subject Classification37E45, 37A25, 11J13. 2000 Mathematics Subject Classification37E45, 37A25, 11J13.     

On the Homeomorphism Groups of Cantor's Discontinuum and the Spaces of Rational and Irrational Numbers

The paper shows that the homeomorphism groups of, respectively,Cantor's discontinuum, the rationals and the irrationals haveuncountable cofinality. It is well known that the homeomorphismgroup of Cantor's discontinuum is isomorphic to the automorphismgroup Aut B of the countable, atomless boolean algebra B. Soalso Aut B has uncountable cofinality, which answers a questionposed earlier by the first author and H. D. Macpherson. Thecofinality of a group G is the cardinality of the length ofa shortest chain of proper subgroups terminating at G. 2000Mathematics Subject Classification 20B22, 20E15.     

Boundedness of Higher order Hankel Forms, Factorization in Potential Spaces and Derivations

We prove a bounded decomposition for higher order Hankel formsand characterize the first order Hochschild cohomology groupsof the disk algebra with coefficients in the space of boundedHankel forms of some fixed order. Although these groups arenon-trivial, we prove that every bounded derivation is innerand necessarily implemented by a Hankel form of order one higher.In terms of operators, this result extends the similarity resultof Aleksandrov and Peller. Both of the main structural theoremshere rely on estimates involving multilinear maps on the n-foldproduct of the disk algebra and we obtain several higher orderanalogues of the factorization results due to Aleksandrov andPeller. 2000 Mathematics Subject Classification: 47B35, 46E15,46E25.     

A rate of convergence for asymptotic contractions

In [P. Gerhardy, A quantitative version of Kirk's fixed point theorem for asymptotic contractions, J. Math. Anal. Appl. 316 (2006) 339-345], P. Gerhardy gives a quantitative version of Kirk's fixed point theorem for asymptotic contractions. This involves modifying the definition of an asymptotic contraction, subsuming the old definition under the new one, and giving a bound, expressed in the relevant moduli and a bound on the Picard iteration sequence, on how far one must go in the iteration sequence to at least once get close to the fixed point. However, since the convergence to the fixed point needs not be monotone, this theorem does not provide a full rate of convergence. We here give an explicit rate of convergence for the iteration sequence, expressed in the relevant moduli and a bound on the sequence. We furthermore give a characterization of asymptotic contractions on bounded, complete metric spaces, showing that they are exactly the mappings for which every Picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

Dual Pairs of Hopf *-Algebras

If A is a Hopf *-algebra, the dual space A' is again a *-algebra.There is a natural subalgebra A° of A' that is again a Hopf*-algebra. In many interesting examples, A° will be largeenough (to separate points of A). More generally, one can considera pair (A, B) of Hopf *-algebras and a bilinear form on A xB with conditions such that, if the pairing is non-degenerate,one algebra can be considered as a subalgebra of the dual ofthe other. In these notes, we study such pairs of Hopf *-algebras. We startfrom the notion of a Hopf *-algebra A and its reduced dual A°.We give examples of pairs of Hopf *-algebras, and discuss theproblem of non-degeneracy. The first example is an algebra pairedwith itself. The second example is the pairing of a Hopf *-algebra(due to Jimbo) and the twisted SU(n) of Woronowicz. We alsodiscuss the notion of the quantum double of Drinfeld in thisframework of dual pairs.     

Various Slicing Indices on Banach Spaces

We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(α) spaces, where α is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is ω (first infinite ordinal), then its weak*-dentability index is at most ω² and that this estimate is optimal. The first author was supported by the grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

The A-Complexity of a Space

Suppose that A is a pointed CW-complex. The paper looks at howdifficult it is to construct an A-cellular space B from copiesof A by repeatedly taking homotopy colimits; this is determinedby an ordinal number called the complexity of B. Studying thecomplexity leads to an iterative technique, based on resolutions,for constructing the A-cellular approximation CW_A(X) of an arbitraryspace X.     

First countable, countably compact spaces and the continuum hypothesis

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah's iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman's (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah's club-guessing sequences) that shows similar results do not hold for closed pre-images of .

     

A multigrid-type method for thin plate spline interpolation on a circle

We consider the problem of thin plate spline interpolation ton equally spaced points on a circle, where the number of datapoints is sufficiently large for work of O(n³ to be unacceptable.We develop an iterative multigrid-type method, each iterationcomprising ngrid stages, and n being an integer multiple of2^ngrid–1. We let the first grid, V₁ be the full set ofdata points, V say, and each subsequent (coarser) grid, V_k,k=2, 3,...,ngrid, contain exactly half of the data points ofthe preceding (finer) grid, these data points being equallyspaced. At each stage of the iteration, we correct our current approximationto the thin plate spline interpolant by an estimate of the interpolantto the current residuals on V_k, where the correction is constructedfrom Lagrange functions of interpolation on small local subsetsof p data points in V_k. When the coarsest grid is reached, however,then the interpolation problem is solved exactly on its q=n/2^ngrid–1points. The iterative process continues until the maximum residualdoes not exceed a specified tolerance. Each iteration has the effect of premultiplying the vector ofresiduals by an n x n matrix R, and thus convergence will dependupon the spectral radius, (R), of this matrix. We investigatethe dependence of the spectral radius on the values of n, p,and q. In all the cases we have considered, we find (R) <<1, and thus rapid convergence is assured.     

Bifurcation from Simple Eigenvalues and Eigenvalues of Geometric Multiplicity One

A counterexample has been constructed to show that a conjecturedglobal solution structure for bifurcation of non-trivial solutionsfrom a simple eigenvalue of the linearization at zero reallycan occur. In addition, new results and counterexamples havebeen obtained for bifurcation from an eigenvalue of geometricmultiplicity 1 and odd algebraic multiplicity. 2000 MathematicsSubject Classification 58E07, 47J15, 37G10.     

Composition Operators with Closed Image on Spaces of Real Analytic Functions

We characterize composition operators on spaces of real analyticfunctions which at the same time have closed image and are openonto their images. Under some mild assumptions, we also characterizecomposition operators with closed range and composition operatorsopen onto their images. 2000 Mathematics Subject Classification46E10, 47B33, 32C07 (primary), 26E05, 32E30, 32D15 (secondary).     

A stochastic gradient type algorithm for closed-loop problems

We focus on the numerical solution of closed-loop stochastic problems, and propose a perturbed gradient algorithm to achieve this goal. The main hurdle in such problems is the fact that the control variables are infinite-dimensional, due to, e.g., the information constraints. Alternatively said, control variables are feedbacks, i.e., functions. Such controls have hence to be represented in a finite way in order to solve the problem numerically. In the same way, the gradient of the criterion is itself an infinite-dimensional object. Our algorithm replaces this exact (and unknown) gradient by a perturbed one, which consists of the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point. Proceeding this way, we explore the whole space iteration after iteration through random points. Since each kernel function is perfectly known by a small number of parameters, say N, the control at iteration k is perfectly known as an infinite-dimensional object by at most N × k parameters. The main strength of this method is that it avoids any discretization of the underlying space, provided that we can sample as many points as needed in this space. Moreover, our algorithm can take into account the possible measurability constraints of the problem in a new way. Finally, the randomized strategy implemented by the algorithm causes the most probable parts of the space to be the most explored ones, which is a priori an interesting feature. In this paper, we first prove two convergence results of this algorithm in the strongly convex and convex cases, and then give some numerical examples showing the interest of this method for practical stochastic optimization problems. In Memoriam: Jean-Sébastien Roy passed away July 04, 2007. He was 33 years old.     

Analysis and numerical simulation for a class of time fractional diffusion equation via tension spline

In this work, we prove the weak and strong convergence of a sequence generated by a modified S-iteration process for finding a common fixed point of two G-nonexpansive mappings in a uniformly convex Banach space with a directed graph. We also give some numerical examples for supporting our main theorem and compare convergence rate between the studied iteration and the Ishikawa iteration.     

The Finite Dimensional Approximation Property and the AR-Property in Needle Point Spaces

We introduce the notion of the finite dimensional approximationproperty (the FDAP) and prove that if a subset X of a linearmetric space has the FDAP, then every non-empty convex subsetof X is an AR. As an application we show that every needle point space X containsa dense linear subspace E with the following properties: (i) E contains a non-empty compact convex set with no extremepoints; (ii) all non-empty convex subsets of E are AR.     

3×3阶上三角算子矩阵的点谱、剩余谱和连续谱

记■为Hilbert空间■上的上三角算子矩阵.我们借助对角元A,B和C的谱性质给出了σ_*(M_(D,E,F))=σ_*(A)∪σ_*(B)∪σ_*(C)对任意D∈B(H_2,H_1),E∈B(H_3,H_1),F∈B(H_3,H_2)均成立的充要条件,其中σ_*代表某类特定的谱,如点谱、剩余谱和连续谱等.此外,给出了一些例证.     

Constructions for chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphismgroup has two orbits on flags, with adjacent flags lying indifferent orbits. In this paper, we describe a method for constructingfinite chiral n-polytopes, by seeking particular normal subgroupsof the orientation-preserving subgroup of an n-generator Coxetergroup (having the property that the subgroup is not normalizedby any reflection and is therefore not normal in the full Coxetergroup). This technique is used to identify the smallest examplesof chiral 3- and 4-polytopes, in both the self-dual and non-self-dualcases, and then to give the first known examples of finite chiral5-polytopes, again in both the self-dual and non-self-dual cases.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

The Elementary Geometric Structure of Compact Lie Groups

We give geometric proofs of some of the basic structure theoremsfor compact Lie groups. The goal is to take a fresh look atthese theorems, prove some that are difficult to find in theliterature, and illustrate an approach to the theorems thatcan be imitated in the homotopy theoretic setting of p-compactgroups. 1991 Mathematics Subject Classification 22E15, 55P35.