Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to…
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48 results for “secondary polytope”
Constructs combinatorial 2D topological field theories from cyclic A-infinity algebras.
problem Developing a combinatorial framework for 2D topological field theories.
method Using triangulations and polygonal decompositions, constructing cochains on a CW complex.
result Existence of combinatorial 2D topological field theories based on cyclic A-infinity algebras.
A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of . That fan arises as the normal fan of a convex polytope. In a complete…
We study algebraic structures ( and -algebras) introduced by Gaiotto, Moore and Witten in their recent work devoted to certain supersymmetric 2-dimensional massive field theories. We show that such structures can be systematically produced in any number of dimensions by using the geometry of seconda…
We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometr…
Machine learning improves RNA secondary structure prediction.
problem Stagnant performance of RNA secondary structure prediction methods.
method Machine learning, especially deep learning, is used to predict RNA secondary structures.
result Machine learning methods have improved the prediction of RNA secondary structures.
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.
The paper studies deformation spaces of Coxeter truncation polytopes.
problem Understanding the geometric properties and deformations of Coxeter truncation polytopes.
method Analyzing Coxeter truncation polytopes and their deformation spaces.
result Description of deformation spaces for Coxeter truncation polytopes of dimension .
Neural Polytopescs.LG
Neural networks approximate unit spheres as polytopes.
problem Approximating unit spheres with neural networks.
method Using ReLU activation in neural networks to generate polytopes.
result Neural networks can approximate unit spheres as polytopes.
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.
Study of spectral flow in symmetric Toeplitz operator families.
problem Understanding spectral flow in families of symmetric Toeplitz operators.
method Analog of Atiyah-Singer-Robbin-Salamon theorem for -valued spectral flow.
result Graded secondary spectral flow equals secondary index of a Callias-type operator.
The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…
The paper tackles online learning with two types of losses and shows it's impossible without certain assumptions.
problem Online learning with primary and secondary losses where the secondary loss is bounded by a linear threshold.
method Analyzes the feasibility of achieving low regret with respect to the primary loss while keeping the secondary loss within a linear threshold.
result Achieving the goal is impossible without bounded variance assumption on the secondary loss.
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
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 problem of distributed learning and channel access is considered in a cognitive network with multiple secondary users. The availability statistics of the channels are initially unknown to the secondary users and are estimated using sensing decisions. There is no explicit information exchange or prior agreement amon…
Fuses ITRs for primary and secondary outcomes to minimize harm.
problem Learn an ITR maximizing primary outcome while minimizing harm to secondary outcomes.
method Introduces fusion penalty to encourage similar recommendations for different outcomes. Two algorithms estimate the ITR using surrogate loss functions.
result Agreement rate between primary and secondary optimal ITRs converges faster than ignoring secondary outcomes.
This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…
We introduce two invariants called the secondary cuspidal curvature and the bias on -cuspidal edges, and investigate their basic properties. While the secondary cuspidal curvature is an analog of the cuspidal curvature of (ordinary) cuspidal edges, there are no invariants corresponding to the bias. We prove that t…
In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…
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.
Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.
problem Proving smoothness of deformation space for Coxeter polytopes.
method Analyzing natural map into realization space.
result Deformation space of Coxeter 3-polytopes is smooth.
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
problem Formalizing and simplifying the study of partial differential equations (PDEs).
method Using cohomology of diffieties to formalize PDEs and their properties.
result Differential calculus on PDE solution spaces is homotopy calculus on horizontal De Rham algebras of diffieties.
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.
Contact manifolds' momentum polytopes are convex.
problem Understanding the structure of contact manifolds.
method Using isomorphism to toric varieties.
result Momentum polytopes of contact manifolds are convex.
Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Polytopic Matrix Factorization models data as latent vectors from a polytope, maximizing determinant for identifiability.
problem Data decomposition with semi-structured latent vectors and polytope constraints.
method Model input data as latent vectors from a polytope, using determinant maximization for identifiability.
result Identifiability condition for polytopes with specific symmetry restrictions.
The article studies factorization structures in geometry and their applications to cones and polytopes.
problem Understanding and characterizing factorization structures in geometry.
method Comprehensive study of factorization structures, including structure theory, construction of compatible polytopes and cones, and derivation of generalised Gale's evenness condition.
result Established generalised Vandermonde identities and found examples of Delzant and rational Delzant compatible polytopes.
Examines nonrational polytopes and fans in toric geometry.
problem Understanding nonrational convex polytopes and fans in toric geometry.
method Discussion and interrelation of recent developments.
result Exploration of nonrational polytopes and fans in toric geometry.
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.
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.
We introduce two operations named biflip and puzzle-move on simple polytopes producing polytopes with diffeomorphic moment-angle manifolds.
Proves necessity of at least log2(n) layers to compute maximum of n numbers.
problem Computing the maximum of n numbers with ReLU neural networks.
method Uses lattice polytopes and duality with Newton polytopes to prove depth lower bounds.
result Proves that log2(n) hidden layers are necessary and sufficient.
Unified cosmological and Einstein polytope theories.
problem Unified understanding of cosmological and Einstein polytope theories.
method Unified combinatorial perspective of cosmological and Einstein polytope theories.
result Unified construction of cosmological and Einstein polytope theories.
The study of symmetries in manifolds derived from colored polytopes.
problem Existence and types of symmetries in rational homology 3-spheres.
method Analysis of hyperbolic manifolds and right-angled polytopes.
result Described how to create colorings with specific symmetries.
We verify that the formula of X. Ma for the analytic torsion form of an iterated fibration implies that Lott's secondary analytic index is functorial.
Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.
problem Finding higher-dimensional analogs of the Calabi invariant and its transgression to the Euler class.
method Constructing a cohomology class of volume-preserving diffeomorphisms and proving transgression to the Euler class of foliated sphere bundles.
result The cohomology class transgresses to the Euler class of foliated sphere bundles.
Diffeomorphisms of convex polytopes form a Lie group.
problem Understanding transformations of convex polytopes.
method Forming a Lie group from diffeomorphisms of convex polytopes.
result The group of diffeomorphisms of a convex polytope is a regular Lie group.
The study improves credit evaluation in peer-to-peer lending using machine learning.
problem Traditional credit histories are insufficient for distinguishing good from bad borrowers.
method Used machine learning classification and clustering algorithms to predict creditworthiness.
result Achieved 65% F1 and 73% AUC on LendingClub data, identifying key secondary attributes.
Efficiently projects points onto polytopes, especially useful in web-scale applications.
problem Efficiently projecting points onto polytopes in large-scale applications.
method Developed a vertex-oriented incremental algorithm for polytope projection, tailored for simplex and unit-box cut polytopes.
result Majority of projections lie on vertices of polytopes, leading to significant performance improvements.
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative …
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
New noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
Proves weight polytope matches with energy vectors in toric varieties.
problem Understanding the relationship between weight polytopes and energy functionals in toric varieties.
method Combines two slope formulas of K-energy in the toric setting.
result Weight polytope of Hurwitz form matches with convex hull of characteristic vectors.
Unique floating and buoyancy surfaces identify convex polytopes.
problem Identifying convex polytopes from their flotation and buoyancy surfaces.
method Proving uniqueness of surfaces for polytopes with uniform or prescribed density.
result Floating and buoyancy surfaces uniquely determine convex polytopes.
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.