A sampling theorem for multivariate stationary processes

The notion of sampling for second-order q-variate processes is defined. It is shown that if the components of a q-variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q-variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q-variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.     

A Full and Faithful Nerve for 2-Categories

The notion of geometric nerve of a 2-category (Street, J. Pure Appl. Algebra 49 (1987), 283–335) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding simplicial maps. These facts allow us to prove a representation theorem of the general non-abelian cohomology of groupoids (classifying non-abelian extensions of groupoids) by means of homotopy classes of simplicial maps.Mathematics Subject Classifications (2000) 18D05, 18G30, 55P15.     

Tverberg's theorem with constraints

The topological Tverberg theorem claims that for any continuous map of the (q−1)(d+1)-simplex σ(d+1)(q−1) to Rd there are q disjoint faces of σ(d+1)(q−1) such that their images have a non-empty intersection. This has been proved for affine maps, and if q is a prime power, but not in general.We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem.The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for d=2 and q=3.     

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

We prove that (q ², 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q ² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q ² + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q ³, 6, q ³ ?q ² ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear [q ³ + q, 6,q ³ ?q ² ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 ^r . Furthermore, we construct 2-arcs of size (q + 1)²/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p ².     

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq ² and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq ²(q?1) then it is shown thatP is either a semifield plane or is a Lüneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.     

Structure of A*-Fibrations over One-Dimensional Seminormal Semilocal Domains

In this paper we give a structure theorem for an A*-fibration over a one-dimensional noetherian seminormal semilocal domain and show that, in this situation, any A*-fibration whose spectrum occurs as an affine open subscheme of the spectrum of an A¹-fibration (equivalently, an affine line A¹) is actually A*. The structure theorem provides examples of A*-fibrations over one-dimensional noetherian seminormal semilocal domains whose spectra are not affine open subschemes of any affine line A¹ over the base ring. We also construct examples of nontrivial A*-fibrations over one-dimensional noetherian non-seminormal local domains whose spectra are open subschemes of A¹-fibrations over the base ring.     

A general limit lifting theorem for 2-dimensional monad theory

We give a definition of weak morphism of T-algebras, for a 2-monad T, with respect to an arbitrary family Ω of 2-cells of the base 2-category. By considering particular choices of Ω, we recover the concepts of lax, pseudo and strict morphisms of T-algebras. We give a general notion of weak limit, and define what it means for such a limit to be compatible with another family of 2-cells. These concepts allow us to prove a limit lifting theorem which unifies and generalizes three different previously known results of 2-dimensional monad theory. Explicitly, by considering the three choices of Ω above our theorem has as corollaries the lifting of oplax (resp. σ, which generalizes lax and pseudo, resp. strict) limits to the 2-categories of lax (resp. pseudo, resp. strict) morphisms of T-algebras.     

Symmetry codes and their invariant subcodes

We define and study the invariant subcodes of the symmetry codes in order to be able to determine the algebraic properties of these codes. An infinite family of self-orthogonal rate $\text{1}\text{2}$ codes over GF(3), called symmetry codes, were constructed in [3]. A (2q + 2, q + 1) symmetry code, denoted by C(q), exists whenever q is an odd prime power ≡ ?1, (mod 3). The group of monomial transformations leaving a symmetry code invariant is denoted by G(q). In this paper we construct two subcodes of C(q) denoted by R_σ(q) and R_μ(q). Every vector in R_σ(q) is invariant under a monomial transformation τ in G(q) of odd order s where s divides (q + 1). Also R_μ(q) is invariant under τ but not vector-wise. The dimensions of R_σ(q) and R_μ(q) are determined and relations between these subcodes are given. An isomorphism is constructed between R_σ(q) and a subspace of $\text{W = V}{}_3{}^{\mathrm{<mtext>(2q+2)</mtext><mtext>s</mtext>}}$. It is shown that the image of R_σ(q) is a self-orthogonal subspace of W. The isomorphic images of R_σ(17) (under an order 3 monomial) and R_σ(29) (under an order 5 monomial) are both demonstrated to be equivalent to the (12, 6) Golay code.     

On the categorical meaning of Hausdorff and Gromov distances, I

Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov “distance” between V-categories X and Y we use V-modules between X and Y, rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K, yields a lax extension to the category V-Mod of V-categories, with V-modules as morphisms.     

The Existence of Complete Mappings of SL(2, q), q≡1 Modulo 4

In 1955, Hall and Paige conjectured that any finite group with a noncyclic Sylow 2-subgroup admits complete mappings. For the groups GL(2, q), SL(2, q), PSL(2, q), and PGL(2, q) this conjecture has been proved except for SL(2, q), q odd. We prove that SL(2, q), q≡1 modulo 4 admits complete mappings.     

Arc-regular cubic graphs of order four times an odd integer

A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4m where m is an odd integer. Using the Gorenstein?CWalter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of automorphisms that is isomorphic to a subgroup of Aut(PSL(2,q)) containing PSL(2,q) for some odd prime-power?q. Also a construction is given for infinitely many such graphs??namely a family of Cayley graphs for the groups PSL(2,p ³) where p is an odd prime; the smallest of these has order?9828.     

Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zⁿ of the n-dimensional Euclidean space Rⁿ. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zⁿ and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.     

2-nerves for bicategories

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. This becomes the object part of a 2-functor N : NHom → [Δ^op,Cat], where NHom is a 2-category whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories. The 2-functor N is fully faithful and has a left biadjoint, and we characterize its image. The 2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2-category whose objects are Tamsamani weak 2-categories. The hospitality of Macquarie University and the support of the Australian Research Council are gratefully acknowledged by S. Lack. The support of the Australian Research Council is gratefully acknowledged by S. Paoli.     

Tchebyshev triangulations of stable simplicial complexes

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F-polynomials j_∑fj−1(j(x−1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev transform of the second kind. Using the reverse implication of a theorem by Schelin we observe that the Tchebyshev transforms of Schur stable polynomials with real coefficients have interlaced real roots in the interval (−1,1), and present ways to construct simplicial complexes with Schur stable F-polynomials. We show that the order complex of a Boolean algebra is Schur stable. Using and expanding the recently discovered relation between the derivative polynomials for tangent and secant and the Tchebyshev polynomials we prove that the roots of the corresponding pairs of derivative polynomials are all pure imaginary, of modulus at most one, and interlaced.     

n-connectedness in pure 2-complexes

A graph is a 1-dimensional simplicial complex. In this work we study an interpretation of “n-connectedness” for 2-dimensional simplicial complexes. We prove a 2-dimensional analogue of a theorem by Whitney for graphs: Theorem (A Whitney type theorem for pure 2-complexes).Let G be a pure 2-complex with no end-triangles. Then G is n-connected if and only if the valence of e is at least n for every interior edge e of G, and there does not exist a juncture set J of less than n edges of G. Examples ofn-connected pure 2-complexes are then given, and some consequences are proved.     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

Matroid optimization with the interleaving of two ordered sets

We study the structure of the minimum weight base of a matroid M = (E, I) the order of whose element set E is determined by the interleaving of two ordered subsets of E, R and W. The results imply an interesting application in economics, and are useful for the rapid recomputation of the minimum weight base when the order of E is successively modified by changing the interleaving of R and W. As a special case of the main result, the following parametric problem is efficiently solved: For M = (E, I) a matroid with weighted element set E, and R a subset of E, find for all feasible values of q, the minimum weight base of M containing exactly q elements of R. This parametric problem is a weighted matroid intersection problem and hence can be solved by known matroid intersection algorithms. The approach in this paper is different, and vastly improves the efficiency of the solution, as well as determining structural information about the bases.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.