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.     

Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -minor. Let be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of if either one of two things hold:

(i) For any two disjoint cocircuits and in it holds that .

(ii) For any two disjoint cocircuits and in it holds that .

Part (ii) implies Ore's Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c^*2k, then there is a circuit which intersects each cocircuit of size c^*−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.     

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 .     

The Burnside ring of fusion systems

Given a saturated fusion system on a finite p-group S we define a ring modeled on the Burnside ring of finite groups. We show that these rings have several properties in common. When is the fusion system of G we describe the relationship between these rings.     

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 .     

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.     

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 .     

Approximation of Sobolev classes by polynomials and ridge functions

Let be the usual Sobolev class of functions on the unit ball in , and be the subclass of all radial functions in . We show that for the classes and , the orders of best approximation by polynomials in coincide. We also obtain exact orders of best approximation in of the classes by ridge functions and, as an immediate consequence, we obtain the same orders in for the usual Sobolev classes .     

All-derivable points in continuous nest algebras

Let be an operator algebra on a Hilbert space. We say that an element is an all-derivable point of for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ(ST)=φ(S)T+Sφ(T) for any with ST=G) is a derivation. Let be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P(M) () is an all-derivable point of for the strong operator topology.     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

A unified class of k-uniformly convex functions defined by the Dziok–Srivastava linear operator

The main purpose of this paper is to introduce a new class of functions which are analytic in the open unit disk . We obtain various results including characterization, coefficient estimates, and distortion and covering theorems, for functions belonging to the class .     

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 ).     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

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 .     

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.     

Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

-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.     

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 .     

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.