Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

On closures of cycle spaces of flag domains

Open orbits D of noncompact real forms G ₀ acting on flag manifolds Z = G/Q of their semisimple complexifications G are considered. Given D and a maximal compact subgroup K ₀ of G ₀, there is a unique complex K ₀–orbit in D which is regarded as a point in the space of q-dimensional cycles in D. The group theoretical cycle space is defined to be the connected component containing C ₀ of the intersection of the G–orbit G(C ₀) with . The main result of the present article is that is closed in . This follows from an analysis of the closure of the universal domain in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K ₀ in G.     

The amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable manifold, and F be an essential closed surface which cuts M into two 3-manifolds M ₁ and M ₂. Let be a Heegaard splitting for i = 1, 2. We denote by d(S _i ) the distance of . If d(S ₁), d(S ₂) ≥ 2(g(M ₁) + g(M ₂) − g(F)), then M has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of and . Ruifeng Qiu is supported by NSFC(10625102).     

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.      

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces

By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space on the unit disk can be represented as where Φ is an inner multiplier of . This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert space will be called Beurling decomposable if there exist (operator-valued) multipliers such that and . We characterize the finite-codimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules , every finite-codimensional submodule M can be written as with suitable polynomials p _i .      

On Similarity of Multiplication Operator on Weighted Bergman Space

Let D be the unit disk and be the weighted Bergman space. In this paper, we prove that the multiplication operator is similar to M _z on . The author was supported in part by NSF Grant (10571041, L2007B05).     

Finite Blaschke product and the multiplication operators on Sobolev disk algebra

Let R(D) be the algebra generated in Sobolev space W22(D) by the rational functions with poles outside the unit disk D. In this paper the multiplication operators Mg on R(D) is studied and it is proved that Mg ～ Mzn if and only if g is an n-Blaschke product. Furthermore, if g is an n-Blaschke product, then Mg has uncountably many Banach reducing subspaces if and only if n > 1.     

Convex Sets in Acyclic Digraphs

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by and its size by co(D) (cc(D)). Gutin et al. (2008) conjectured that the sum of the sizes of all convex sets (connected convex sets) in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with and . We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n − k + 1. This is a strengthening of a theorem of Gutin and Yeo.     

The generalized Verschiebung map for curves of genus 2

Let C be a smooth curve, and M _r (C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map induced by pulling back under Frobenius. Our main result is a computation of the degree of V ₂ for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.This paper was partially supported by fellowships from the National Science Foundation and Japan Society for the Promotion of Sciences.     

Weak Riemannian manifolds from finite index subfactors

Let N ⊂ M be a finite Jones’ index inclusion of II₁ factors and denote by U _N ⊂ U _M their unitary groups. In this article, we study the homogeneous space U _M /U _N , which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit of the Jones projection of the inclusion. We endow with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete); therefore, is a weak Riemannian manifold. We show that enjoys certain properties similar to classic Hilbert–Riemann manifolds. Among them are metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point p ₁ of , there is a ball (of uniform radius r) of the usual norm of M, such that any point p ₂ in the ball is joined to p ₁ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and .      

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

Symmetric groups and expander graphs

We construct explicit generating sets S _n and of the alternating and the symmetric groups, which turn the Cayley graphs and into a family of bounded degree expanders for all n.     

Region-Fault Tolerant Geometric Spanners

We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph on a point set P and a region F, we define to be what remains of after the vertices and edges of intersecting F have been removed. A  -fault tolerant t-spanner is a geometric graph  on P such that for any convex region F, the graph is a t-spanner for , where is the complete geometric graph on P. We prove that any set P of n points admits a -fault tolerant (1+ε)-spanner of size for any constant ε>0; if adding Steiner points is allowed, then the size of the spanner reduces to  , and for several special cases, we show how to obtain region-fault tolerant spanners of size without using Steiner points. We also consider fault-tolerant geodesic t -spanners: this is a variant where, for any disk D, the distance in between any two points u,v∈P∖D is at most t times the geodesic distance between u and v in ℝ²∖D. We prove that for any P, we can add Steiner points to obtain a fault-tolerant geodesic (1+ε)-spanner of size  . M.A. Abam was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 612.065.307 and by the MADALGO Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation. M. de Berg was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301. M. Farshi was supported by Ministry of Science, Research and Technology of I.R. Iran. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.     

Sparse quasi-Newton updates with positive definite matrix completion

Quasi-Newton methods are powerful techniques for solving unconstrained minimization problems. Variable metric methods, which include the BFGS and DFP methods, generate dense positive definite approximations and, therefore, are not applicable to large-scale problems. To overcome this difficulty, a sparse quasi-Newton update with positive definite matrix completion that exploits the sparsity pattern of the Hessian is proposed. The proposed method first calculates a partial approximate Hessian , where , using an existing quasi-Newton update formula such as the BFGS or DFP methods. Next, a full matrix H _k+1, which is a maximum-determinant positive definite matrix completion of , is obtained. If the sparsity pattern E (or its extension F) has a property related to a chordal graph, then the matrix H _k+1 can be expressed as products of some sparse matrices. The time and space requirements of the proposed method are lower than those of the BFGS or the DFP methods. In particular, when the Hessian matrix is tridiagonal, the complexities become O(n). The proposed method is shown to have superlinear convergence under the usual assumptions.      

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M ⁿ is the hypersurface .      

Inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits

Let p be a prime and let be an elliptic curve defined over the finite field of p elements. For a given point the linear congruential genarator on elliptic curves (EC-LCG) is a sequence (U _n ) of pseudorandom numbers defined by the relation: where denote the group operation in and is the initial value or seed. We show that if G and sufficiently many of the most significants bits of two consecutive values U _n , U _n+1 of the EC-LCG are given, one can recover the seed U ₀ (even in the case where the elliptic curve is private) provided that the former value U _n does not lie in a certain small subset of exceptional values. We also estimate limits of a heuristic approach for the case where G is also unknown. This suggests that for cryptographic applications EC-LCG should be used with great care. Our results are somewhat similar to those known for the linear and non-linear pseudorandom number congruential generator.      

Equivariant vector bundles on Drinfeld’s upper half space

Let be Drinfeld’s upper half space over a finite extension K of ℚ _p . We construct for every GL _d+1-equivariant vector bundle on ℙ ^d _K , a GL _d+1(K)-equivariant filtration by closed subspaces on the K-Fréchet . This gives rise by duality to a filtration by locally analytic GL _d+1(K)-representations on the strong dual . The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1] and that of the structure sheaf by Pohlkamp [P].