Related papers: Hypercubes are determined by their distance spectr…
We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.
Let $D(G)$ denote the distance matrix of a connected graph $G$ with $n$ vertices. The distance spectral gap of a graph $G$ is defined as $\delta_{D^G} = \rho_1 - \rho_2$, where $\rho_1$ and $\rho_2$ represent the largest and second largest…
The spectrum of a density matrix $\rho(t)$ is conserved by a Lie-Nambu dynamics if $\rho(t)$ is a self-adjoint and Hilbert-Schmidt solution of a nonlinear triple-bracket equation. This generalizes to arbitrary separable (positive- and…
For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely…
We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…
Some issues concerning the question if the two-dimensional Hubbard model really show d-wave superconductivity are briefly discussed.
We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the…
Two points are randomly selected inside a three-dimensional euclidian cube. The value l of their separation lies somewhere between zero and the length of a diagonal of the cube. The probability density P(l) of the separation is obtained…
The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this…
It is verified that the number of vertices in a $d$-dimensional cubical pseudomanifold is at least $2^{d+1}$. Using Adin's cubical $h$-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for…
For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…
The cognitive framework of conceptual spaces [3] provides geometric means for representing knowledge. A conceptual space is a high-dimensional space whose dimensions are partitioned into so-called domains. Within each domain, the Euclidean…
It is shown that $N$ points on a real algebraic curve of degree $n$ in $\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the…
We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming…
Angle-resolved photoemission spectra are calculated microscopically for the two-dimensional attractive Hubbard model. A system of self-consistent T-matrix equations are solved numerically in the real-time domain. The single-particle…
The multifractal dimensions D2^mu and D2^psi of the energy spectrum and eigenfunctions, resp., are shown to determine the asymptotic scaling of the width of a spreading wave packet. For systems where the shape of the wave packet is…
Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side…
The $d$-dimensional Goldfarb cube is a polytope with the property that all its $2^d$ vertices appear on some \emph{shadow} of it (projection onto a 2-dimensional plane). The Goldfarb cube is the solution set of a system of 2d linear…
We study the spectrum of a random matrix, whose elements depend on the Euclidean distance between points randomly distributed in space. This problem is widely studied in the context of the Instantaneous Normal Modes of fluids and is…