Clique matchings in the <Emphasis Type="Italic">k</Emphasis>-ary <Emphasis Type="Italic">n</Emphasis>-dimensional cube

A clique matching in the k-ary n-dimensional cube (hypercube) is a collection of disjoint one-dimensional faces. A clique matching is called perfect if it covers all vertices of the hypercube. We show that the number of perfect clique matchings in the k-ary n-dimensional cube can be expressed as the k-dimensional permanent of the adjacency array of some hypergraph. We calculate the order of the logarithm of the number of perfect clique matchings in the k-ary n-dimensional cube for an arbitrary positive integer k as n→∞.     

Inverse Spectra with Two and Three Maps

It is shown that, for any 1 n < , there exist four maps of the n-dimensional cube to itself such that the limit of any inverse sequence of n-cubes is the limit of some sequence with only these four bonding maps. A universal continuum in the class of all limits of sequences of n-cubes is constructed as the limit of an inverse sequence of n-cubes with one bonding map. All compact sets of trivial shape are represented by using only three maps of the Hilbert cube to itself. Two maps of the closed interval to itself such that any Knaster continuum can be obtained as the limit of an inverse sequence with only these two bonding maps are constructed.     

The cardinality of the separated vertex set of a multidimensional cube

An n-dimensional cube and the sphere inscribed into it are considered. The conjecture of A. Ben-Tal, A. Nemirovski, and C. Roos states that each tangent hyperplane to the sphere strictly separates not more than 2 ⁿ⁻² cube vertices. In this paper this conjecture is proved for n ≤ 6. New examples of hyperplanes separating exactly 2 ⁿ⁻² cube vertices are constructed for any n. It is proved that hyperplanes orthogonal to radius vectors of cube vertices separate less than 2 ⁿ⁻² cube vertices for n ≥ 3.     

A reduction of Keller's conjecture

A family of translates of a closedn-dimensional cube is called a cube tiling if the union of the cubes is the wholen-space and their interiors are disjoint. According to a famous unsolved conjecture of O. H. Keller, two of the cubes in ann-dimensional cube tiling must share a complete (n – 1)-dimensional face. In this paper we shall prove that to solve Keller's conjecture it is sufficient to examine certain factorizations of direct sum of finitely many cyclic group of order four.     

Edge Domination in Graphs of Cubes

The signed edge domination number and the signed total edge domination number of a graph are considered; they are variants of the domination number and the total domination number. Some upper bounds for them are found in the case of the n-dimensional cube Q _n.     

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Tucker’s well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1,±2,…,±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1,±2,…,±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0,1} ⁿ +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.     

Menger (N?beling) Manifolds versus Hilbert Cube (Space) Manifolds – A Categorical Comparison

 We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each .     

Demushkin's Theorem in codimension one

Demushkin's Theorem says that any two toric structures on an affine variety X are conjugate in the automorphism group of X. We provide the following extension: Let an (n–1)-dimensional torus T act effectively on an n-dimensional affine toric variety X. Then T is conjugate in the automorphism group of X to a subtorus of the big torus of X. Mathematics Subject Classification: 13A50, 14L30, 14M25, 14R20.     

Ceva, Menelaus, and Selftransversality

The purpose of this paper is to state and prove a theorem (the CMS Theorem) which generalizes the familiar Ceva's Theorem and Menelaus' Theorem of elementary Euclidean geometry. The theorem concernsn -acrons (generalizations of n-gons) in affine space of any number of dimensions and makes assertions about circular products of ratios of lengths, areas, volumes, etc. In particular it contains, as special cases, many results in this area proved by earlier authors.     

Spectra of graphs attached to the space of melodies

For any integer n greater than or equal to two, two intimately related graphs on the vertices of the n-dimensional cube are introduced. All of their eigenvalues are found to be integers, and the largest and the smallest ones are also determined. As a byproduct, certain kind of generating function for their spectra is introduced and shown to be quite effective to compute the eigenvalues of some broader class of adjacency matrices of graphs.     

Menger (Nöbeling) Manifolds versus Hilbert Cube (Space) Manifolds – A Categorical Comparison

 We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each . (Received 30 August 1999; in revised form 7 December 1999)     

The View-Obstruction Problem for 5-Dimensional Cubes

 The view-obstruction problem for the n-dimensional cube is equivalent to the conjecture that for any n positive integers there is a real number x such that each (here denotes the distance from y to the nearest integer). This conjecture has been previously solved for . In this paper we prove that when we can find x which gives each ; this is the first improvement over the easy result . Received 5 August 1997; in revised form 19 January 1998     

Configurations Maximizing the Number of Pairs of Hamming-Adjacent Lattice Points

The following question is considered: Which sets of k lattice points among the n^d points in a d-dimensional cube of length n maximize the number of pairs of points differing in only one coordinate? It is shown that maximal configurations for any (d, n, k) are obtained by choosing the first k points in a lexicographic ordering of the points by coordinates. Some possible generalizations of the problem are discussed.     

An <Emphasis Type="Italic">n</Emphasis>-dimensional Contou–Carrère symbol over an artinian local ring

The aim of this work is to give a definition of an n-dimensional Contou–Carrère symbol. This symbol allows us to define an n-dimensional Witt residue. Moreover, we offer a reciprocity law for the symbol related to a flag of irreducible varieties, in order to obtain an n-dimensional Witt residue Theorem as a particular case of this new reciprocity law. This work is partially supported by the DGI research contract no. MTM2006-07618 and Castilla y León regional government contract SA071/04.     

An Elementary Deduction of the Topological Radon Theorem from Borsuk–Ulam

The Topological Radon Theorem states that, for every continuous function from the boundary of a (d+1)-dimensional simplex into ℝ ⁿ , there exists a pair of disjoint faces in the domain whose images intersect in ℝ ⁿ . The similarity between that result and the classical Borsuk–Ulam Theorem is unmistakable, but a proof that the Topological Radon Theorem follows from Borsuk–Ulam is not immediate. In this note we provide an elementary argument verifying that implication.     

Cube term blockers without finiteness

We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety \({\mathcal{V}}\) of finite signature whose fundamental operations have arities n ₁, . . . , n _k, has a d-dimensional cube term for some d if and only if it has one of dimension \({1 + \sum_{i=1}^{k} (n_{i} - 1)}\). This upper bound on the dimension of a minimal-dimension cube term for \({\mathcal{V}}\) is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions.     

The dimension of leaf closures of K-contact flows

For K-contact flows on (2n+1)-dimensional compact manifolds, we show that the dimension of any leaf closure is at most the smaller of (n+1) and 2n+1) minus the rank of the vector space of harmonic vector fields.     

Sets of disjoint snakes based on a Reed-Muller code and covering the hypercube

A snake-in-the-box code (or snake) of word length n is a simple circuit in an n-dimensional cube Q _n , with the additional property that any two non-neighboring words in the circuit differ in at least two positions. To construct such snakes a straightforward, non-recursive method is developed based on special linear codes with minimum distance 4. An extension of this method is used for the construction of covers of Q _n consisting of 2 ^m-1 vertex-disjoint snakes, for 2 ^m-1 < n ≤ 2 ^m . These covers turn out to have a symmetry group of order 2 ^m .      

Projecting <Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> onto Classical Spaces

We describe an explicit construction of a linear projection of a symmetric conical section of the n-dimensional cube onto a (1+ε)-isomorphic version of the Euclidean ball of proportional dimension, or more generally onto a (1+ε)-isomorphic image of an l _p ^m -ball. Allowing non-linear projections (of logarithmic polynomial nonlinearity) we may even project the full n-dimensional cube onto the same images. This is done by gluing together explicit projections onto two-dimensional spaces, interpreting and modifying a paper of Ben-Tal and Nemirowski on polynomial reductions of conic quadratic programming problems to linear programming problems in terms of Banach spaces.