Related papers: A lower bound for essential covers of the cube

We establish a connection between linear complementary dual (LCD) codes and caps in projective space. Using this framework and the structure theory of maximal caps, we derive nonexistence theorems for LCD codes with minimum distance at…

We improve the previously best known upper bounds on the sizes of $\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing…

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the…

This paper presents new lower and upper bounds for the optimal compression of binary prefix codes in terms of the most probable input symbol, where compression efficiency is determined by the nonlinear codeword length objective of…

Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in…

We argue that if, in order to reverse an object's dynamics, we need to be able to keep track of it with enough precision, then there is an upper bound on the size of the object whose dynamics we can reverse - even using all the available…

We give various applications of essential circles (introduced in an earlier paper by the authors) in a compact geodesic space X. Essential circles completely determine the homotopy critical spectrum of X, which we show is precisely 2/3 the…

The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{ess-Minvol}(M)$,…

In this paper, we consider the essential spectrum of submanifolds in Euclidean spaces under various geometric hypotheses. Our results involve extrinsic conditions such as finite total mean curvature, the convergence of the gradient of the…

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm gave an upper bound for modular forms of a given…

We study the $(n-1)$-dimensional volume of central hyperplane sections of the $n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular…

In 1974, Witsenhausen asked for the maximum possible density $\alpha_n$ of a measurable subset $A$ of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known…

Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the $n$-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the…

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that $M$…

The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the…

Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…

Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18.…

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…

We obtain a new lower bound for the eternal vertex cover number of an arbitrary graph $G$, in terms of the cardinality of a vertex cover of minimum size in $G$ containing all its cut vertices. The consequences of the lower bound includes a…

The fundamental lower size limit in wavelength selecting structures is explored with the aid of the Heisenberg uncertainty principle. The analysis shows that for a given wavelength selectivity resonating structures with optical feedback…