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 .     

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 .     

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 .     

Oscillation criteria for a class of second-order neutral delay differential equations

In this article, we investigate oscillation and asymptotic behaviour of all solutions of a class of neutral delay differential equations of second-order with several positive and negative coefficients having the form

where R,P,Q are bounded beginning segments of positive integers, , , are delay functions and f is a continuous function. Our results improve and extend the recent results given in the papers [J. Manojlović, Y. Shoukaku, T. Tanigawa, N. Yoshida, Oscillation criteria for second-order differential equations with positive and negative coefficients, Appl. Math. Comput. 181 (2006) 853–863] and [A. Weng, J. Sun, Oscillation of second order delay differential equations, Appl. Math. Comput. 198 (2) (2008) 930–935].     

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.     

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.     

The simplicity of certain groups of subanalytic diffeomorphisms

Let M be a connected real analytic manifold. We denote by , 1r<∞, the group of subanalytic C^r diffeomorphisms of M which are isotopic to the identity via a compactly supported subanalytic C^r isotopy. We show that satisfies Epstein's axioms. This implies that the commutator subgroup of is simple. Moreover, we show that the commutator subgroup of is dense in . As a corollary we obtain that is topologically simple.     

Poincaré's theorem for the modular group of real Riemann surfaces

Let Mod_g denote the modular group of (closed and orientable) surfaces S of genus g. Each element [h]Mod_g induces a symplectic automorphism H([h]) of . Poincaré showed that is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution σ. Let (S,σ) be a real Riemann surface, be the group of orientation preserving homeomorphisms of S such that h○σ=σ○h and be the subgroup of consisting of those isotopic to the identity by an isotopy in . The group plays the role of the modular group in the theory of real Riemann surfaces. In this work we describe the image by H of . Such image depends on the topological type of the involution σ.     

(Semi-)Riemannian geometry of (para-)octonionic projective planes

We use reduced homogeneous coordinates to construct and study the (semi-)Riemannian geometry of the octonionic (or Cayley) projective plane , the octonionic projective plane of indefinite signature , the para-octonionic (or split octonionic) projective plane and the hyperbolic dual of the octonionic projective plane . We also show that our manifolds are isometric to the (para-)octonionic projective planes defined classically by quotients of Lie groups.     

Irreducible complex skew-Berger algebras

Irreducible skew-Berger algebras , i.e. algebras spanned by the images of the linear maps satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in .     

3-structures with torsion

We find conditions which ensure the integrability of the canonical 3-dimensional distribution spanned by the Reeb vector fields of an almost 3-contact manifold, showing by an explicit counterexample that the normality of the structures does not necessarily imply the integrability of . Then we focus on those almost 3-contact metric manifolds for which is integrable and we define an appropriate notion of almost 3-contact metric connection with torsion. The geometry of an almost 3-contact manifold with torsion is then studied and put in relation with the well-known HKT-geometry.     

About the geometry of almost para-quaternionic manifolds

We provide a general criterion for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold of dimension 4m8 in terms of the integrability of two or three sections of the defining rank three vector bundle . We relate it with the integrability of the canonical almost complex structure of the twistor space and with the integrability of the canonical almost para-complex structure of the reflector space of . We deduce that has plenty of locally defined, compatible, complex and para-complex structures, provided that is integrable.     

Projective change between two classes of -metrics

In this paper, we find equations to characterize projective change between (α,β)-metric and Randers metric on a manifold with dimension n3, where α and are two Riemannian metrics, β and are two nonzero one forms. Moreover, we consider this projective change when F has some special curvature properties.     

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.     

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 .     

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 .     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,