New method finds 198,846 toric-colorable seeds of Picard number 5.
problem Enumerating toric-colorable seeds of Picard number 5.
method Binary matroid approach and dynamic programming algorithm.
result 198,846 mod 2 toric-colorable seeds of dimension four and Picard number five.
Minimal simplicial maps constructed for spheres and manifolds.
problem Constructing minimal simplicial maps of specific degrees.
method Triangulations and degree constructions for manifolds and spheres.
result Minimal triangulations for degree d self-maps of Sn−1imesS1. We give a purely combinatorial construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel f…
New method to visualize hyperbolic 4-manifolds and study mutations.
problem Understanding and visualizing hyperbolic 4-manifolds and mutations.
method Orbifold covers of Coxeter polytopes with facet colorings and studying totally geodesic sub-manifolds.
result Construction of hyperbolic 4-manifolds with unique properties.
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
This paper investigates the impact of unconventional preprocessors on deep convolutional neural networks for face identification.
problem The impact of unconventional preprocessors on deep convolutional neural networks for face identification.
method Varying the preprocessing module across different approaches while keeping other network architecture facets constant, using transfer learning from Inception-V3 network features.
result Improvement in discriminative capability of deep networks through preprocessing with HE, quantization, and other methods.
The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
New cube complexes disprove Kalai's conjecture about sphere facets.
problem Disproving Kalai's conjecture about the number of facets of cubical spheres.
method Constructing cube complexes homeomorphic to the d-sphere with n vertices and Ω(n^(5/4)) facets.
result The conjecture is disproved for all d≥3 and n sufficiently large.
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
Explains parameterizing facets using Ressayre's pairs in Kähler geometry.
problem Parameterizing facets of Kirwan polyhedron.
method Using Ressayre's pairs in Kähler setting.
result Equations of facets parameterized successfully.
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.
problem Classifying compact hyperbolic Coxeter polytopes with specific facet counts.
method New combinatorial method via point set order types.
result Proves existence of a compact hyperbolic Coxeter 29-polytope with at least 40 facets.
SessionPath improves category suggestions in type-ahead search.
problem Improving precision and recall in eCommerce type-ahead suggestions.
method SessionPath uses session embeddings and a probability distribution model to predict facets.
result SessionPath outperforms count-based and neural models in eCommerce shops.
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…
Algorithm identifies spheres with maximal Buchstaber number.
problem Characterizing (n−1)-dimensional PL spheres with specific vertex counts. method Computational algorithm for weak pseudo-manifolds, toric colorable seeds enumeration.
result Comprehensive characterization of (n−1)-spheres with maximal Buchstaber number. Reconstructing polytopes with fixed facet directions from support function evaluations.
problem Reconstructing polytopes with known facet directions from limited data.
method Least-squares estimate via convex quadratic program, combinatorial characterization for uniqueness, algorithm convergence.
result The least-squares estimate for a fixed simplicial normal fan is a convex quadratic program, and the solution is unique under certain conditions.
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,∗). This lower bound is sharp for the 3-m…
MFCVAE clusters data over multiple facets, improving disentanglement and generation.
problem Clustering high-dimensional data like images over multiple characteristics.
method Variational autoencoder with hierarchical latent variables and Mixture-of-Gaussians priors.
result MFCVAE learns and clusters over multiple aspects of data in a disentangled manner.
The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
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 Rn, where n≥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…
The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.
problem Understanding the moment polytopes in real symplectic geometry.
method Parameterizing equations of facets of Delta(Z) in terms of real Ressayre's pairs of Z.
result Parameterization of facets of moment polytopes explained.
New method finds hyperelliptic 4-manifolds from polytope vector-colorings.
problem Finding hyperelliptic 4-manifolds from polytope vector-colorings.
method Introducing Hamiltonian subcomplexes and their corresponding subgroups.
result For dimensions ≤ 4, there is a bijection between Hamiltonian subcomplexes and hyperelliptic involutions.
New algorithms for SSMF with weaker identifiability conditions than SSC.
problem Identifying unique decompositions in simplex-structured matrix factorization.
method Extracting facets containing the largest number of points to ensure identifiability.
result Our algorithms recover unique decompositions under weaker conditions than SSC.
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
Sigma simplifies collaboration in economics with a streamlined computational representation.
problem Lack of effective collaboration tools in economics for large-scale projects.
method Introduces Sigma, a domain-specific computational representation for economics based on facets, contributions, and constraints of data.
result Sigma enables sharing and formalizing domain-specific concepts in economics for crowd-based scientific investigations.
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…
Minimal coloring number found for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings of Z-colorable links.
method Defined Z-coloring as a generalization of Fox coloring for links with zero determinants. Provided sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
result Sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
Intel's system identifies and categorizes businesses for sales opportunities.
problem Identifying relevant new markets and customers for large enterprises.
method Mining public business web pages, enriching with external data, and using deep learning.
result Significantly boosts sales personnel's ability to discover new customers and partnerships.
The paper explores minimal coloring numbers for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings on minimal diagrams of Z-colorable links. method Investigates minimal diagrams and Z-colorings for Z-colorable links. result For any positive integer N, there exists a minimal diagram of a Z-colorable link with at least N colors in any Z-coloring. The paper shows links can be colored with fewer colors than previously thought.
problem Coloring links using the symmetric group of degree three.
method Analyzing the number of colors for link colorings by S3. result 2-bridge links with 5 colors can be colored with only 4 colors.
Study finds minimal coloring numbers for torus links using rack colorings.
problem Determining the minimal number of colors for torus link diagrams.
method Using rack colorings on link diagrams to classify minimal colorings.
result Complete classifications of Z-colorings by four colors. Study on knots using 17 colors, finding specific color assignments.
problem Understanding the minimum number of colors needed for Fox colorings of knots.
method Investigated 17-colorable knots and their diagrams.
result Found that exactly 6 out of 17 colors are used in diagrams of 17-colorable knots.
Proves minimum colors needed for 11-colorable knots and ribbons.
problem Determining the minimum number of colors needed for 11-colorable knots and ribbons.
method Analyzes 11-colored diagrams to find the minimum number of colors required.
result Minimum number of colors needed is five for all non-trivially 11-colored diagrams.
Aicardi's invariant F(L) is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends F(L) to colored singular links, showing it's stronger than HOMFLY-PT polynomial. This paper proves the minimal coloring number for a specific type of link is exactly 4.
problem Determining the minimal coloring number for a specific type of link.
method Investigated Z-colorable links and used their properties to prove the minimal coloring number is 4. result The minimal coloring number of any non-splittable Z-colorable link is exactly 4. The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.
problem Finding the minimum number of colors for Dehn colorings of knots.
method Analyzes Dehn colorings for knots and defines R-palette graphs. result For Dehn p-colorable knots, the minimum number of colors is at least ⌊log2pfloor+2. Algorithm finds minimal colorings of tree structures.
problem Finding minimal unbounded factor complexity colorings of trees.
method Induction algorithm using colored balls.
result Characterization of Sturmian colorings.
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 …
The paper offers new ways to solve map coloring problems.
problem Coloring plane maps with four colors.
method Alternate formulations including a tautological expansion and an extended Penrose Bracket.
result Extended Penrose Bracket counts colorings of arbitrary cubic graphs.
This paper shows the minimal coloring number for certain Z-colorable links is four.
problem Determining the minimal number of colors for Z-colorings of links. method Analyzing diagrams of Z-colorable links and constructing specific diagrams to find the minimal coloring number. result The minimal coloring number for non-splittable Z-colorable links is four. 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…
Minimum 5 Fox colors needed for knot diagrams modulo 13.
problem Finding the minimum number of Fox colors for knot diagrams modulo 13.
method Using known results and equivalent diagram transformations.
result Minimum number of Fox colors modulo 13 is 5.
This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. The choice of n depends upon the knot. Each typ…
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
The paper discusses colorings and doubled colorings of virtual doodles.
problem Coloring virtual doodles using a new algebraic structure.
method Introduced a new algebra called a doodle switch and defined an invariant for virtual doodles.
result Introduced doubled colorings and defined an invariant for virtual doodles.
For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove …
The paper defines a new homotopy type for colored links and proves stabilization behavior.
problem Understanding the behavior of Khovanov homotopy types for colored links.
method Definition of a Khovanov homotopy type for colored links and quantum spin networks, and derivation of its properties.
result Stabilization of the homotopy types for n-colored B-adequate links as nightarrow∞.