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.     

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.     

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 .     

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 .     

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.     

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 .     

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 .     

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 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 .     

Orthonormal polynomials with exponential-type weights

Let and let w_ρ(x)|x|^ρexp(-Q(x)), where and is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight , obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate A_n(x) and B_n(x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight .     

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 .     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

Classification of marginally trapped Lorentzian flat surfaces in and its application to biharmonic surfaces

A surface in a semi-Riemannian manifold is called marginally trapped if its mean curvature vector field is light-like at each point. In this article, we classify marginally trapped Lorentzian flat surfaces in the pseudo-Euclidean space . As an application, we obtain the complete classification of biharmonic Lorentzian surfaces in with light-like mean curvature vector.     

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.     

-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.     

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.     

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.     

The predicative Frege hierarchy

In this paper, we characterize the strength of the predicative Frege hierarchy, , introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that and are mutually interpretable. It follows that is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess’ PV is Robinson’s Q, The Journal of Symbolic Logic 72 (2) (2007) 619–624] using a different proof. Another consequence of the our main result is that is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, , Q3). The fact that interprets EA was proved earlier by Burgess. We provide a different proof.Each of the theories is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, , is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with is finitely axiomatizable.