Cube Polynomial of Fibonacci and Lucas Cubes

The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k≥0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.     

A Cube Tiling of Dimension Eight with No Facesharing

Abstract. A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.     

A Cube Tiling of Dimension Eight with No Facesharing

   Abstract. A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.     

Integration over a simplex,truncated cubes,and Eulerian numbers

Summary A method of integrating a function over a simplex is described in which (i) the simplex is first transformed into a right-angled isosceles simplex; (ii) this simplex is dissected into small cubes and truncated cubes; (iii) the integration over the truncated cubes is performed by the centroid method or by Stroud's method, and this requires the use of formulae for the moments of a truncated cube. These formulae are developed and are expressed in terms of Eulerian numbers. In the special case when the truncated cube is itself a right-angled isoceles simplex a new algorithm is given, depending on the discrete Fourier transform, for calculating the moments as polynomials inn wheren is the dimensionality.     

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.     

On-line covering the unit cube by cubes

We consider two on-line methods of covering the unit cube of Euclideand-space by sequences of cubes. The on-line restriction means that we are given the next cube from the sequence only after the preceding cube has been put in place without the possibility of changing the placement. The first method enables on-line covering of the unit cube by an arbitrary sequence of cubes whose total volume is at least 3...2 ^d −4. The second method is more complicated, but, asymptotically, asd tends to infinity, it yields an efficiency of the order of magnitude 2 ^d with factor 1. So, asymptotically, it is as good as the best possible non-on-line method of covering the unit cube by cubes. This research was supported in part by Komitet Badań Naukowych (Committee of Scientific Research), Grant Number 2 2005 92 03.     

Crumpled Cube and Solid Horned Sphere Space Fillers

We construct two classes of wildly embedded space fillers of R³. First, every crumpled cube is shown to have an embedding in R³ that admits a monohedral tiling of R³. Second, a solid Alexander horned sphere with a topologically trivial interior is shown to admit a monohedral tiling of a cube and hence R³. By joining a solid horned sphere with compact polyhedral 3-submanifolds of R³ with one boundary component, we construct space fillers homeomorphic to the polyhedral submanifolds but of different embedding types. Using the suitably embedded crumpled cubes instead of a solid horned sphere, space fillers of even more different topological types can be produced.     

Rates for the probability of large cubes being non-internally spanned in modified bootstrap percolation

Summary We find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation. It is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line.Research partially supported by NSF DMS 9157461 and a grant from the Sloan Foundation     

Special John-Nirenberg-Campanato spaces via congruent cubes

Let p ∈ [1, ∞), q ∈ [1, ∞), α∈ R, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg(1961) and the space B introduced by Bourgain et al.(2015), we introduce a special John-Nirenberg-Campanato space JN^con_(p,q,s) over Rⁿ or a given cube of R;with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p →∞ is just the Campanato space which coincides with the space BMO(the space of functions with bounded mean oscillations)when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO(the space of functions with vanishing mean oscillations) over Rⁿ or a given cube of Rⁿ with finite side length.Furthermore, some VMO-H¹-BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.     

Stochastic algorithm for solving transient diffusion equations with a precise accounting of reflection boundary conditions on a substrate surface

A new random walk based stochastic algorithm for solving transient diffusion equations in domains where a reflection boundary condition is imposed on a plane part of the boundary is suggested. The motivation comes from the field of exciton transport and recombination in semiconductors where the reflecting boundary is the substrate plane surface while on the defects and dislocations an absorption boundary condition is prescribed. The idea of the method is based on the exact representations of the first passage time and position distributions on a parallelepiped (or a cube) with a reflection condition on its bed face lying on the substrate. The algorithm is meshfree both in space and time, the particle trajectories are moving inside the domain in accordance with the Random Walk on Spheres (RWS) process but when approaching the reflecting surface they switch to move on parallelepipeds (or cubes). The efficiency of the method is drastically increased compared with the standard RWS method. For illustration, we present an example of exciton flux calculations in the cathodoluminescence imaging method in semiconductors with a set of threading dislocations.     

Embeddability of open-ended carbon nanotubes in hypercubes

A graph that can be isometrically embedded into a hypercube is called a partial cube. An open-ended carbon nanotube is a part of hexagonal tessellation of a cylinder. In this article we determine all open-ended carbon nanotubes which are partial cubes.     

Recursive fault-tolerance of Fibonacci cube in hypercubes

Fibonacci cube is a subgraph of hypercube induced on vertices without two consecutive 1's. If we remove from Fibonacci cube the vertices with 1 both in the first and the last position, we obtain Lucas cube. We consider the problem of determining the minimum number of vertices in n-dimensional hypercube whose removal leaves no subgraph isomorphic to m-dimensional Fibonacci cube. The exact values for small m are given and several recursive bounds are established using the symmetry property of Lucas cubes and the technique of labeling. The relation to the problem of subcube fault-tolerance in hypercube is also shown.     

An Analogue of Gromov’s Waist Theorem for Coloring the Cube

It is proved that if we partition a d-dimensional cube into \(n^d\) small cubes and color the small cubes in \(m+1\) colors then there exists a monochromatic connected component consisting of at least \(f(d, m) n^{d-m}\) small cubes. Another proof of this result is given in Matdinov’s preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011)     

Cubes and orientability

We define and study a new class of matroids: cubic matroids. Cubic matroids include, as a particular case, all affine cubes over an arbitrary field. There is only one known orientable cubic matroid: the real affine cube. The main results establish as an invariant of orientable cubic matroids the structure of the subset of acyclic orientations with LV-face lattice isomorphic to the face lattice of the real cube or, equivalently, with the same signed circuits of length 4 as the real cube.     

Netlike partial cubes, V: Completion and netlike classes

We define a completion of a netlike partial cube G by replacing each convex 2n-cycle C of G with n≥3 by an n-cube admitting C as an isometric cycle. We prove that a completion of G is a median graph if and only if G has the Median Cycle Property (MCP) (see N. Polat, Netlike partial cubes III. The Median Cycle Property, Discrete Math.). In fact any completion of a netlike partial cube having the MCP is defined by a universal property and turns out to be a minimal median graph containing G as an isometric subgraph. We show that the completions of the netlike partial cubes having the MCP preserves the principal constructions of these graphs, such as: netlike subgraphs, gated amalgams and expansions. Conversely any netlike partial cube having the MCP can be obtained from a median graph by deleting some particular maximal finite hypercubes. We also show that, given a netlike partial cube G having the MCP, the class of all netlike partial cubes having the MCP whose completions are isomorphic to those of G share different properties, such as: depth, lattice dimension, semicube graph and crossing graph.     

Writing Commutators of Commutators as Products of Cubes in Groups

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(γ) of an element γ of a group G, we mean the least natural number k such that γ is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = F?C _∞ is the wreath product of a free group F by the infinite cyclic group, we show that every element of W″ is a product of at most three cubes in W.     

Equivariant cohomology distinguishes toric manifolds

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are weakly isomorphic. We also prove that quasitoric manifolds, which can be thought of as a topological counterpart to toric manifolds, are equivariantly homeomorphic if and only if their equivariant cohomology algebras are isomorphic.     

Pell graphs

In this paper, we introduce the Pell graphs, a new family of graphs similar to the Fibonacci cubes. They are defined on certain ternary strings (Pell strings) and turn out to be subgraphs of Fibonacci cubes of odd index. Moreover, as well as ordinary hypercubes and Fibonacci cubes, Pell graphs have several interesting structural and enumerative properties. Here, we determine some of them. Specifically, we obtain a canonical decomposition giving a recursive structure, some basic properties (bipartiteness and existence of maximal matchings), some metric properties (radius, diameter, center, periphery, medianicity), some properties on subhypercubes (cube coefficients and polynomials, cube indices, decomposition in subhypercubes), and, finally, the distribution of the degrees.     

A Second Look at the Toric h-Polynomial of a Cubical Complex

We provide an explicit formula for the toric h-contribution of each cubical shelling component, and a new combinatorial model to prove Chan??s result on the non-negativity of these contributions. Our model allows for a variant of the Gessel-Shapiro result on the g-polynomial of the cubical lattice, this variant may be shown by simple inclusion-exclusion. We establish an isomorphism between our model and Chan??s model and provide a reinterpretation in terms of noncrossing partitions. By discovering another variant of the Gessel-Shapiro result in the work of Denise and Simion, we find evidence that the toric h-polynomials of cubes are related to the Morgan-Voyce polynomials via Viennot??s combinatorial theory of orthogonal polynomials.     

Aristotle’s Cubes and Consequential Implication

It is shown that the properties of so-called consequential implication allow to construct more than one aristotelian square relating implicative sentences of the consequential kind. As a result, if an aristotelian cube is an object consisting of two distinct aristotelian squares and four distinct “semiaristotelian” squares sharing corner edges, it is shown that there is a plurality of such cubes, which may also result from the composition of cubes of lower complexity.