Multipliers of elementary operators and comparison of row and column space Schatten p norms

Sharp upper estimates for the norm of the weighted elementary operator of the form , acting from one symmetrically normed ideal of compact Hilbert space operators to another, are given. Particularly, we relate the norm of with norms of and on the appropriate domains and co-domains.     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

On structure of bivariate spline space of lower degree over arbitrary triangulation

The structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property. A new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces. The upper bound of the dimension of spline spaces over triangulation given by L.L. Schumaker is slightly improved via the new vertex coding method. The structure of multivariate spline spaces and over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by Luo and Wang. A kind of sufficient conditions on judging non-singularity of the and spaces over arbitrary triangulation is given, which only depends on the topological property of the triangulation. From the sufficient conditions, a triangulation strategy is presented at the end of the paper. The strategy ensures that the constructed triangulation is non-singular (or generic) for and .     

On the Brown–Shields conjecture for cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function to be cyclic, i.e. for to be dense in . This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in iff it is outer and its zero set (defined appropriately) is of capacity zero.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I

Let be the (2ν+1+l)-dimensional vector space over the finite field . In the paper we assume that is a finite field of characteristic 2, and the singular pseudo-symplectic groups of degree 2ν+1+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in and the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , and the characteristic polynomial of .     

A spectral equivalence for Jacobi matrices

We use the classical results of Baxter and Golinskii–Ibragimov to prove a new spectral equivalence for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in or for s1.     

Formal extension and quotients of the invariant holonomic system

Let be a semisimple Lie algebra and a Cartan subalgebra of . Fix . Let be the invariant holonomic system (see [R. Hotta, M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984) 327–358]). First we investigate its formal extension . In the sequel we calculate the characteristic variety of some simple quotients of and its Fourier transform .     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

-structure on the deformation complex of a morphism

-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is the extension of a -algebra by an associative algebra. Actions of -algebras on associative and -algebras are analyzed; extensions of -algebras by associative and -algebras that they act upon are constructed. The resulting -algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the -algebra on the deformation complex of the corresponding diagram algebra.     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

Propagating sharp group homology decompositions

A collection of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either the subgroups in or their centralizers or their normalizers. We give a short but systematic study of the relationship among such formulas for nine standard collections of p-subgroups, obtaining some new formulas in the process. To do this, we exhibit some sufficient conditions on the poset which imply comparison results.     

Generalized rough approximations in

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call -inner and -outer approximations respectively ( being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic by means of step functions from the hypercube [0,1]^k and taking value in an arbitrary (finite) subset of . Then, we will show that if a fuzzy set is definable as truth table of a formula of , then both its -inner and -outer approximation are definable as truth table of formulas of . Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set.     

On the sizes of Delaunay meshes

Let be a polyhedral domain occupying a convex volume. We prove that the size of a graded mesh of with bounded vertex degree is within a factor of the size of any Delaunay mesh of with bounded radius-edge ratio. The term depends on the geometry of and it is likely a small constant when the boundaries of are fine triangular meshes. There are several consequences. First, among all Delaunay meshes with bounded radius-edge ratio, those returned by Delaunay refinement algorithms have asymptotically optimal sizes. This is another advantage of meshing with Delaunay refinement algorithms. Second, if no input angle is acute, the minimum Delaunay mesh with bounded radius-edge ratio is not much smaller than any minimum mesh with aspect ratio bounded by a particular constant.     

Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball of and the radial derivative of h. In this paper we study the boundedness and compactness of the following integral operator:, from iterated logarithmic Bloch spaces to Zygmund-type spaces.     

Right coideal subalgebras of the quantum Borel algebra of type

In this paper we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum group , where is a simple Lie algebra of type G₂, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is finite, t>4, t≠6, then the same classification remains valid for homogeneous right coideal subalgebras of the positive part of the multiparameter version of the small Lusztig quantum group.