arXiv research

The study classifies all compact 5D polytopes with 9 facets.

The study classifies all compact hyperbolic polytopes with eight facets.

New cube complexes disprove Kalai's conjecture about sphere facets.

The study classifies 331 specific 4D polytopes with 7 facets.

We define a notion of facets-pairing structure and its seal space on a nice manifold with corners. We will study facets-pairing structures on any cube in detail and investigate when the seal space of a facets-pairing structure on a cube is a closed manifold. In particular, for any binary square matrix $A$ with zero dia…

New methods classify hyperbolic polytopes with up to 40 facets.

SessionPath improves category suggestions in type-ahead search.

Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed {\em facet component analysis} (FCA). The approach is based on facet iden…

Reconstructing polytopes with fixed facet directions from support function evaluations.

We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold $M$ is at least the weight of $π(M, \ast)$. This lower bound is sharp for the 3-m…

MFCVAE clusters data over multiple facets, improving disentanglement and generation.

The study sets limits on dihedral angles of large hyperbolic polyhedra.

Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polyt…

We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic $n$-spaces. There we give s…

This paper introduces Sigma, a domain-specific computational representation for collaboration in large-scale for the field of economics. A computational representation is not a programming language or a software platform. A computational representation is a domain-specific representation system based on three specific …

The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.

New algorithms for SSMF with weaker identifiability conditions than SSC.

The study broadens the concept of cyclic polytopes to Veronese polytopes.

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…

The aim of this article is to explain how to parameterize the equations of the facets of the Kirwan polyhedron using the notion of Ressayre's pairs.

The goal of this article is to invite the reader to get to know and to get involved into higher Teichmüller theory by describing some of its many facets.

The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d+1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a …

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The…

We prove that among four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.

New method finds 198,846 toric-colorable seeds of Picard number 5.

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study…

We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types …

For sales and marketing organizations within large enterprises, identifying and understanding new markets, customers and partners is a key challenge. Intel's Sales and Marketing Group (SMG) faces similar challenges while growing in new markets and domains and evolving its existing business. In today's complex technolog…

The paper constructs simplicial maps of any degree on spheres, solving a long-standing problem.

Maps discrete manifolds to partitions to define new manifolds.

Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with exactly one fa…

This paper provides a theoretical justification of the superior classification performance of deep rectifier networks over shallow rectifier networks from the geometrical perspective of piecewise linear (PWL) classifier boundaries. We show that, for a given threshold on the approximation error, the required number of b…

Minimal simplicial maps constructed for spheres and manifolds.

A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $ω$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a one-to-one correspondence between symplectic toric orbifolds and convex rational sim…

The paper proves a convex polytope conjecture with specific symmetry conditions.

New $t$-LC triangulated manifolds are exponentially many.

Consider a structured matrix factorization model where one factor is restricted to have its columns lying in the unit simplex. This simplex-structured matrix factorization (SSMF) model and the associated factorization techniques have spurred much interest in research topics over different areas, such as hyperspectral u…

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus acti…

Online reviews provided by consumers are a valuable asset for e-Commerce platforms, influencing potential consumers in making purchasing decisions. However, these reviews are of varying quality, with the useful ones buried deep within a heap of non-informative reviews. In this work, we attempt to automatically identify…

3-manifold triangulation can be reconstructed from its intersection matrix.

Machine Learning and Artificial Intelligence are considered an integral part of the Fourth Industrial Revolution. Their impact, and far-reaching consequences, while acknowledged, are yet to be comprehended. These technologies are very specialized, and few organizations and select highly trained professionals have the w…

Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.

FOSS is an acronym for Free and Open Source Software. The FOSS 2013 survey primarily targets FOSS contributors and relevant anonymized dataset is publicly available under CC by SA license. In this study, the dataset is analyzed from a critical perspective using statistical and clustering techniques (especially multiple…

New noncompact Coxeter polytopes found in various dimensions.

Unified analysis of tree-based methods for online reinforcement learning.

Systematizes blockchain decentralization taxonomy and metrics.

Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …

Current recommender systems exploit user and item similarities by collaborative filtering. Some advanced methods also consider the temporal evolution of item ratings as a global background process. However, all prior methods disregard the individual evolution of a user's experience level and how this is expressed in th…