Related papers: Maximal Orthoplectic Fusion Frames from Mutually U…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
We propose a new class of space-filling designs called rotated sphere packing designs for computer experiments. The approach starts from the asymptotically optimal positioning of identical balls that covers the unit cube. Properly scaled,…
The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their…
The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…
Recent years have witnessed the adoption of optical circuit switch (OCS) technology. How to design the physical topology, defined by the physical wiring between electrical switching equipments and the OCS, is fundamental to designing…
In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with…
Soft robots achieve functionality through tight coupling among geometry, material composition, and actuation. As a result, effective design optimization requires these three aspects to be considered jointly rather than in isolation. This…
Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in C^d\otimes…
A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in $\mathbb{R}^n$ was studied extensively for the last 70 years. In this paper, we…
We study the two-dimensional hierarchical rectangle packing problem, motivated by applications in analog integrated circuit layout, facility layout, and logistics. Unlike classical strip or bin packing, the dimensions of the container are…
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
Recently [Karimipour and Memarzadeh, Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space each of dimension $D$ with the following properties: (i) The family continuously…
The aim of this paper is to highlight recent progress in using conic optimization methods to study geometric packing problems. We will look at four geometric packing problems of different kinds: two on the unit sphere -- the kissing number…
In one of our earlier works, we proposed to approximate Pareto fronts to multiobjective optimization problems by two-sided approximations, one from inside and another from outside of the feasible objective set, called, respectively, lower…
We construct new maximally symmetric solutions for the metric. We then show that for a string moving in a background consisting of maximally symmetric gravity, dilaton field and second rank antisymmetric tensor field, the O(d) $\otimes$…
Near isometric orthogonal embeddings to lower dimensions are a fundamental tool in data science and machine learning. In this paper, we present the construction of such embeddings that minimizes the maximum distortion for a given set of…
A cutting-plane model for a nonsmooth function is the maximum of several first-order expansions centered at different points. Using such a model in a bundle method leads to linear convergence (of serious steps) to a minimum. In smooth…
We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra $U(1)^c \times U(1)^c$, or equivalently the linear programming bound for sphere packing in $2c$ dimensions. We give a more…
Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming…