Probabilistic n-normed spaces, -n-compact sets and -n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and -n-compactness, also we prove some theorems and inequalities. In the next section we define -n-boundedness and prove some results in relation between -n-compact and -n-bounded sets in these spaces.     

Common hermitian and positive solutions to the adjointable operator equations ,

Let be a C^*-algebra. For any Hilbert -modules H and K, let be the set of adjointable operators from H to K. Let H,K,L be Hilbert -modules, and . In this paper, we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions to the equations , and obtain the formulae for the general forms of these solutions. Some results, known for finite matrices and Hilbert space operators, are extended to the adjointable operators acting on Hilbert C^*-modules.     

Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball

Let H(X) be the class of all holomorphic functions on the set and uH(X). We calculate operator norms of the multiplication operators M_u(f)=uf, on the weighted Bergman space , as well as on the Hardy space H^p(X), where X is the unit polydisk or the unit ball in . We also calculate the norm of the weighted composition operator from the weighted Bergman space , and the Hardy space , to a weighted-type space on the unit polydisk.     

Erdős–Ko–Rado theorems for permutations and set partitions

Let Sym([n]) denote the collection of all permutations of [n]={1,…,n}. Suppose is a family of permutations such that any two of its elements (when written in its cycle decomposition) have at least t cycles in common. We prove that for sufficiently large n, with equality if and only if is the stabilizer of t fixed points. Similarly, let denote the collection of all set partitions of [n] and suppose is a family of set partitions such that any two of its elements have at least t blocks in common. It is proved that, for sufficiently large n, with equality if and only if consists of all set partitions with t fixed singletons, where B_n is the nth Bell number.     

Levels of multi-continued fraction expansion of multi-formal Laurent series

The multi-continued fraction expansion of a multi-formal Laurent series is a sequence pair consisting of an index sequence and a multi-polynomial sequence . We denote the set of the different indices appearing infinitely many times in by H_∞, the set of the different indices appearing in by H₊, and call |H_∞| and |H₊| the first and second levels of , respectively. In this paper, it is shown how the dimension and basis of the linear space over F(z) (F) spanned by the components of are determined by H_∞ (H₊), and how the components are linearly dependent on the mentioned basis.     

Sur le groupe d'automorphismes des géométries paraboliques de rang 1

The aim of this article is to prove the following result, which generalizes the Ferrand–Obata theorem, concerning the conformal group of a Riemannian manifold, and the Schoen–Webster theorem about the automorphism group of a strictly pseudo-convex CR structure: let M be a connected manifold endowed with a regular Cartan geometry, modelled on the boundary of the hyperbolic space of dimension d2 over , being , , or the octonions . If the automorphism group of M does not act properly on M, then M is isomorphic, as a Cartan geometry, to X, or X minus a point.     

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 .     

There's something about the diameter

We study diameter preserving linear bijections from onto where X, Y are compact Hausdorff spaces and V, Z are Banach spaces. For instance, we obtain that if X has at least four points, Z is linearly isometric to V and either Z is a space or Z^* is strictly convex or smooth, then there is a diameter preserving linear bijection from onto if and only if X is homeomorphic to Y. We also consider the case when X and Y are not compact but locally compact spaces.     

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 .     

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 ).     

Additive maps preserving the ascent and descent of operators

Let and be the algebras of all bounded linear operators on infinite dimensional complex Banach spaces X and Y, respectively. We characterize additive maps from onto preserving different quantities such as the nullity, the defect, the ascent, and the descent of operators.     

Zeros of Bernoulli-type functions and best approximations

The zero sets of (D+a)ⁿg(t) with in the (t,a)-plane are investigated for and .The results are used to determine entire interpolations to functions , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η>0 to .     

Interpolation and approximation in

Assume a standard Brownian motion W=(W_t)_t[0,1], a Borel function such that f(W₁)L₂, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space , obtained via the real interpolation method, by the behavior of , where is a deterministic time net and the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a_X(f(X₁);τ) can be used to describe the L₂-error in discrete time simulations of the martingale generated by f(W₁) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.     

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 .     

Strong uniform continuity

Let be an ideal of subsets of a metric space X,d. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of which reduces to ordinary continuity when consists of the finite subsets of X and agrees with uniform continuity on members of when is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions.     

Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σ_k has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δ_k∩Δ_k+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions r_k, k=1,…,m, whose zeros and poles lie in .     

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 .     

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.     

On Jordan ideals and matrix rings

For A, a commutative ring, and results by Costa and Keller characterize certain -normalized subgroups of the symplectic group, via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for -graded rings, A₀A₁, and a -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.