Intersections of maximal staircase sets

A compact set is staircase connected if every two points a, b ∈ S can be connected by an x-monotone and y-monotone polygonal path with sides parallel to the coordinate axes. In [5] we have introduced the concepts of staircase k-stars and kernels. In this paper we prove that if the staircase k-kernel is not empty, then it can be expressed as the intersection of a covering family of maximal subsets of staircase diameter k of S.      

( m,n)-Equidista nt Sets i n $\ mathbb{R}^{k},\ mathbb{S}^{k}$ , a nd ℙk

We call a metric space X (m,n)-equidistant if, when A⊆X has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝ ^k contains an (m,1)-equidistant set if and only if k≥m. Although the sphere is (3,2)-equidistant, and ℝ⁴ contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about analogous to the Double Midset Conjecture.     

The dimension of the kernel in an intersection of starshaped sets

We establish the following Helly-type theorem: Let ${\cal K}$ be a family of compact sets in $\mathbb{R}^d$. If every d + 1 (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel contains a translate of set A, then $\cap \{ K : K\; \hbox{in}\; {\cal K} \}$ also is a starshaped set whose kernel contains a translate of A. An analogous result holds when ${\cal K}$ is a finite family of closed sets in $\mathbb{R}^d$. Moreover, we have the following planar result: Define function f on $\{0, 1, 2\}$ by f(0) = f(2) = 3, f(1) = 4. Let ${\cal K}$ be a finite family of closed sets in the plane. For k = 0, 1, 2, if every f(k) (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel has dimension at least k, then $\cap \{K : K\; \hbox{in}\; {\cal K}\}$ also is a starshaped set whose kernel has dimension at least k. The number f(k) is best in each case.Received: 4 June 2002     

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine H^k([`(S)]<sub>g</sub>, \mathbb Q){H^k\left({\bar S}_{g}, {\mathbb Q}\right)} for g ≥ 2 and k ≤ 3, where [`(S)]_g{{\bar S}_{g}} denotes the moduli space of spin curves of genus g.     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

Vertex‐pancyclicity of hypertournaments

A hypertournament or a k‐tournament, on n vertices, 2≤k≤n, is a pair T=(V, E), where the vertex set V is a set of size n and the edge set E is the collection of all possible subsets of size k of V, called the edges, each taken in one of its k! possible permutations. A k‐tournament is pancyclic if there exists (directed) cycles of all possible lengths; it is vertex‐pancyclic if moreover the cycles can be found through any vertex. A k‐tournament is strong if there is a path from u to v for each pair of distinct vertices u and v. A question posed by Gutin and Yeo about the characterization of pancyclic and vertex‐pancyclic hypertournaments is examined in this article. We extend Moon's Theorem for tournaments to hypertournaments. We prove that if k≥8 and n≥k + 3, then a k‐tournament on n vertices is vertex‐pancyclic if and only if it is strong. Similar results hold for other values of k. We also show that when n≥7, k≥4, and n≥k + 2, a strong k‐tournament on n vertices is pancyclic if and only if it is strong. The bound n≥k+ 2 is tight. We also find bounds for the generalized problem when we extend vertex‐pancyclicity to require d edge‐disjoint cycles of each possible length and extend strong connectivity to require d edge‐disjoint paths between each pair of vertices. Our results include and extend those of Petrovic and Thomassen. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 338–348, 2010     

Indecomposable Coverings with Concave Polygons

We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two coverings.     

Fields of moduli and definition of hyperelliptic curves of odd genus

Mestre has shown that if a hyperelliptic curve C of even genus is defined over a subfield k ì \mathbbC{k \subset \mathbb{C}} then C can be hyperelliptically defined over the same field k. In this paper, for all genera g > 1, g o 1{g\equiv1} mod 4, hence odd, we construct an explicit hyperelliptic curve defined over \mathbbQ{\mathbb{Q}} which can not be hyperelliptically defined over \mathbbQ{\mathbb{Q}}.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

The Erdős–Pósa Property for Odd Cycles in Graphs of Large Connectivity

Dedicated to the memory of Paul Erdős A graph G is k-linked if G has at least 2k vertices, and, for any vertices , , ..., , , , ..., , G contains k pairwise disjoint paths such that joins for i = 1, 2, ..., k. We say that G is k-parity-linked if G is k-linked and, in addition, the paths can be chosen such that the parities of their lengths are prescribed. We prove the existence of a function g(k) such that every g(k)-connected graph is k-parity-linked if the deletion of any set of less than 4k-3 vertices leaves a nonbipartite graph. As a consequence, we obtain a result of Erdős–Pósa type for odd cycles in graphs of large connectivity. Also, every -connected graph contains a totally odd -subdivision, that is, a subdivision of in which each edge of corresponds to an odd path, if and only if the deletion of any vertex leaves a nonbipartite graph. Received May 13, 1999/Revised June 19, 2000     

Maximal Volume Representations are Fuchsian

We prove a volume-rigidity theorem for Fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom . Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of π₁(M) into isom , 3 ≤ k ≤ n, is less than the volume of M, and the volume is maximal if and only if the representation is discrete, faithful and ‘k-Fuchsian’ Stefano Francaviglia: Supported by an INdAM and a Marie Curie Intra European fellowship Ben Klaff: Supported by a CIRGET fellowship and by the Chaire de Recherche du Canada en algèbre, combinatoire et informatique mathématique de l’UQAM.     

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

Mass equidistribution of Hilbert modular eigenforms

Let \mathbbF\mathbb{F} be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on \operatornameGL₂/\mathbbF\operatorname{GL}_{2}/\mathbb{F} of weight (k₁,?,k_{[\mathbbF:\mathbbQ]})(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k₁,?,k_{[\mathbbF:\mathbbQ]}) ? ￥\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty.     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

Gabor frame sets for subspaces

This paper investigates Gabor frame sets in a periodic subset \mathbb S\mathbb S of \mathbb R\mathbb R. We characterize tight Gabor sets in \mathbb S\mathbb S, and obtain some necessary/sufficient conditions for a measurable subset of \mathbb S\mathbb S to be a Gabor frame set in \mathbb S\mathbb S. We also characterize those sets \mathbb S\mathbb S admitting tight Gabor sets, and obtain an explicit construction of a class of tight Gabor sets in such \mathbb S\mathbb S for the case that the product of time-frequency shift parameters is a rational number. Our results are new even if \mathbb S=\mathbb R\mathbb S=\mathbb R.     

Algebraic Torsion in Contact Manifolds

We extract an invariant taking values in \mathbbNè{￥}{\mathbb{N}\cup\{\infty\}} , which we call the order of algebraic torsion, from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order 0 if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order 1 (though the converse is not true). We also construct examples for each k ? \mathbbN{k \in \mathbb{N}} of contact 3-manifolds that have algebraic torsion of order k but not k − 1, and derive consequences for contact surgeries on such manifolds.     

A Semigroup Approach to Automaticity

We present an abstract, yet rather natural concept of automaticity which is based on semigroup actions. The standard notion of k-automaticity as well as many, if not all of its generalizations are recovered by specifying appropriate semigroup actions on $ \mathbb{N}_0 $ or $ \mathbb{N}^N_0 $ or other similar objects—e.g., the standard notion of k-automaticity is recovered by considering a certain "natural" action of the free semigroup with k generators on $ \mathbb{N}_0 $. We show that the main results characterizing automatic sequences (Cobham, 1972, Math. Systems Theory 6) hold almost verbatim in the general context, too.AMS Subject Classification: 11B85, 68Q45, 16W22, 18DXX.     

Deep Matrix Modules

A deep matrix algebra, DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), is a unital associative algebra over a field \mathbbK\mathbb{K} with basis all deep matrix units, \mathfrake(h,k)\mathfrak{e}(h,k), indexed by pairs of elements h and k taken from a free monoid generated by a set X. After briefly describing the construction of DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), we determine necessary and sufficient conditions for constructing representations for DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}). With these conditions in place, we define null modules and give three canonical examples of such. A classification of general null modules is then given in terms of the canonical examples along with their submodules and quotients. In the final section, additional examples of natural actions for DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}) are given and their submodules determined depending on the cardinality of the set X.