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Hyperuniform structures possess the ability to confine and drive light, although their fabrication is extremely challenging. Here we demonstrate that speckle patters obtained by a superposition of randomly arranged sources of Bessel beams…
Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective…
Clustering of high-dimensional data sets is a growing need in artificial intelligence, machine learning and pattern recognition. In this paper, we propose a new clustering method based on a combinatorial-topological approach applied to…
Most of the optimal guidance problems can be formulated as nonconvex optimization problems, which can be solved indirectly by relaxation, convexification, or linearization. Although these methods are guaranteed to converge to the global…
We study the $K$-ring of the wonderful variety of a hyperplane arrangement and give a combinatorial presentation that depends only on the underlying matroid. We use this combinatorial presentation to define the $K$-ring of an arbitrary…
Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct…
We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests,…
From condensed matter to quantum chromodynamics, multidimensional spins are a fundamental paradigm, with a pivotal role in combinatorial optimization and machine learning. Machines formed by coupled parametric oscillators can simulate spin…
Two methods are proposed for high-dimensional shape-constrained regression and classification. These methods reshape pre-trained prediction rules to satisfy shape constraints like monotonicity and convexity. The first method can be applied…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…
The main purpose of this paper is to provide combinatorial constraints on the constructability of free and nearly free arrangements of smooth plane conics admitting certain ${\rm ADE}$ singularites.
In this paper we describe the method which we applied to successfully compute the primary decomposition of a certain ideal coming from applications in combinatorial algebra and algebraic statistics regarding conditional independence…
The goal of this paper is to show that generalizing the notion of frequent patterns can be useful in extending association analysis to more complex higher order patterns. To that end, we describe a general framework for modeling a complex…
We show a method in constructing algebraic cycles via intersection theory. It leads to a proof of the Lefschetz standard conjecture.
We present a convex mixed-integer programming formulation for non-rigid shape matching. To this end, we propose a novel shape deformation model based on an efficient low-dimensional discrete model, so that finding a globally optimal…
A functional for joint variational object segmentation and shape matching is developed. The formulation is based on optimal transport w.r.t. geometric distance and local feature similarity. Geometric invariance and modelling of…
We can associate with any irreducible curve singularity (ics) a numerical semigroup. Two ics are said to be equisingular if they have the same semigroup. Two equisingular ics have the same Milnor number. Conversely, The set of ics with a…
Network representations often cannot fully account for the structural richness of complex systems spanning multiple levels of organisation. Recently proposed high-order information-theoretic signals are well-suited to capture synergistic…
This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the {\it generalized corner-vector method}. First we establish a uniform upper bound for the degree…
The monography considers the problem of constructing a Hamiltonian cycle in a complete graph. A rule for constructing a Hamiltonian cycle based on isometric cycles of a graph is established. An algorithm for constructing a Hamiltonian cycle…