Lipschitz regularity for minima without strict convexity of the Lagrangian

We give, in a non-smooth setting, some conditions under which (some of) the minimizers of among the functions in W^1,1(Ω) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function are bounded and the boundary datum u₀ satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of on , whenever they exist, are Lipschitz. A relaxation result follows.     

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 .     

-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.     

On realistic terrains

We study worst-case complexities of visibility and distance structures on terrains under realistic assumptions on edge length ratios and the angles of the triangles, and a more general low-density assumption. We show that the visibility map of a point for a realistic terrain with n triangles has complexity . We also prove that the shortest path between two points p and q on a realistic terrain passes through triangles, and that the bisector of p and q has complexity . We use these results to show that the shortest path map for any point on a realistic terrain has complexity , and that the Voronoi diagram for any set of m points on a realistic terrain has complexity and . Our results immediately imply more efficient algorithms for computing the various structures on realistic terrains.     

Hypercyclic sequences of PDE-preserving operators

A sequence of continuous linear operators is said to be hypercyclic if there exists a vector , called hypercyclic for , such that {T_nx:n0} is dense. A continuous linear operator, acting on some suitable function space, is PDE-preserving for a given set of convolution operators, when it map every kernel set for these operators invariantly. We establish hypercyclic sequences of PDE-preserving operators on , and study closed infinite-dimensional subspaces of, except for zero, hypercyclic vectors for these sequences.     

Analyticity of functions analytic on circles

Let Δ be the open unit disc in, let pbΔ, and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p. Then f is holomorphic on Δ.     

On the dual of a Coulter–Matthews bent function

Coulter–Matthews (CM) bent functions are from to defined by , where and (α,2n)=1. It is not known if these bent functions are weakly regular in general. In this paper, we show that when n is even and α=n+1 (or n−1), the CM bent function is weakly regular. Moreover, we explicitly determine the dual of the CM bent function in this case. The dual is a bent function not reported previously.     

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 ).     

Construction of highly stable parallel two-step Runge–Kutta methods for delay differential equations

It is shown that any A-stable two-step Runge–Kutta method of order and stage order for ordinary differential equations can be extended to the P-stable method of uniform order for delay differential equations.     

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 .     

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 .     

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.     

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 .     

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.     

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 .     

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.     

Classical and non-classical solutions of a prescribed curvature equation

We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem

depending on the behaviour at the origin and at infinity of the potential . Besides solutions in W^2,1(0,1), we also consider solutions in which are possibly discontinuous at the endpoints of [0,1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.     

Binary Hermitian forms over a cyclotomic field

Let ζ be a primitive fifth root of unity and let F be the cyclotomic field . Let be the ring of integers. We compute the Voronoï polyhedron of binary Hermitian forms over F and classify -conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group and Hecke eigenforms.