We give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
Trend · papers per month
48 results for “products of simplices”
Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.
problem Understanding cup product behavior in bounded cohomology.
method Analyzes map Ψ* associating closed forms to bounded cohomology classes via integration.
result Proves Ψ* preserves cup product in sufficiently high degrees.
New oriented matroids from simplex triangulations.
problem Constructing oriented matroids from triangulations.
method Using polyhedral matching fields and matroids over hyperfields.
result Generalized construction of matroids over hyperfields.
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
Study shows exact forms in bounded cohomology are in radical of cup product.
problem Understanding bounded cohomology of negatively curved manifolds.
method Integrating 2-forms over simplices to define bounded cocycles and studying cup products.
result Exact forms in bounded cohomology are in the radical of the cup product.
We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …
We describe discrete restricted Boltzmann machines: probabilistic graphical models with bipartite interactions between visible and hidden discrete variables. Examples are binary restricted Boltzmann machines and discrete naive Bayes models. We detail the inference functions and distributed representations arising in th…
The recent crisis and the following flight to simplicity put most derivative businesses around the world under considerable pressure. We argue that the traditional modeling techniques must be extended to include product design. We propose a quantitative framework for creating products which meet the challenge of being …
An analysis of Uniswap marketsq-fin.TR
Uniswap -- and other constant product markets -- appear to work well in practice despite their simplicity. In this paper, we give a simple formal analysis of constant product markets and their generalizations, showing that, under some common conditions, these markets must closely track the reference market price. We al…
New bounded cohomology classes found for exact forms on curved manifolds.
problem Finding non-trivial bounded cohomology classes for exact forms on negatively curved manifolds.
method Integration of forms over simplices to associate bounded cocycles.
result Exact non-zero 2-forms define non-trivial bounded cohomology classes.
Verma Howe duality connects tensor products of Verma modules to LKB representations.
problem Understanding the relationship between tensor products of Verma modules and LKB representations.
method Established a quantized version of Verma Howe duality and used it to prove the simplicity of LKB representations.
result LKB representations arise from the quantized Verma Howe duality and are shown to be simple modules.
The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of -products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams.
result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
We propose (WIPS) for neural network-based graph embedding. In addition to the parameters of neural networks, we optimize the weights of the inner product by allowing positive and negative values. Despite its simplicity, WIPS can approximate arbitrary general similarities in…
For a small cover Q^n and any principal (Z_2)^m-bundle M^n over Q^n, it was shown in a previous work of the author that the total sum of Z_2-Betti numbers of M^n is at least 2^m. In this paper, we prove that when M^n is connected, the total sum of Z_2-Betti numbers of such an M^n exactly equals 2^m if and only if M^n i…
We consider three different approaches to define natural Riemannian metrics on polytopes of stochastic matrices. First, we define a natural class of stochastic maps between these polytopes and give a metric characterization of Chentsov type in terms of invariance with respect to these maps. Second, we consider the Fish…
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to -trees and estimate their dimensions.
result Estimates the dimensions of maximal simplices for both rational and irrational trees.
Researchers found a quadratic estimate for embedding higher-dimensional simplices into sphere-connected sums.
problem Estimating the number of handles required for embedding higher-dimensional simplices into sphere-connected sums.
method Combining geometric topology, combinatorics, and linear algebra.
result Presented a quadratic estimate for embedding -faces of -simplex.
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
In the present paper we find a bijection between the set of small covers over an -cube and the set of acyclic digraphs with labeled nodes. Using this, we give a formula of the number of small covers over an -cube (generally, a product of simplices) up to Davis-Januszkiewicz equivalence classes and $\mathbf{Z}…
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in pseudo-hyperbolic space has finite volume.
In this paper we study non-negatively curved and rationally elliptic GKM manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…
Quantitative PL bordismmath.GT
Study PL bordism theories with quantitative bounds on filling simplices.
problem Understanding PL bordism theories with geometric constraints.
method Quantitative analysis of PL manifolds and exotic theories.
result Bounding the number of simplices in fillings of cycles.
Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embe…
Framework reduces simplicity bias in NNs, improving OOD generalization and robustness.
problem Simplicity bias in deep learning models leads to biased predictions and poor OOD generalization.
method Proposes a framework that regularizes conditional mutual information to encourage use of diverse features.
result Demonstrates effectiveness in various settings, enhancing OOD generalization and robustness.
New framework shows -simplicity for groups without certain subalgebras.
problem Characterizing -simplicity of groups.
method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is -simple if it has no non-trivial amenable confined subalgebras.
In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.
An incompressible surface on the boundary of a compact orientable 3-manifold is arc-extendible if there is an arc on Int such that is incompressible, where is a regular neighborhood of in . Suppose for simplicity that is irreducible, and has n…
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension , and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…
The paper studies actions on Bass-Serre trees and identifies new -simple groups.
problem Investigating actions of fundamental groups on Bass-Serre trees and their -algebraic properties.
method Analyzing boundary actions of fundamental groups of graphs of groups on their Bass-Serre trees.
result Identification of new families of -simple groups, including tubular groups and certain graphs of groups.
Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.
problem Simplicity bias in dynamical systems and its impact on data analysis and prediction.
method Examined the logistic map and random logistic map, focusing on simplicity bias and noise effects.
result Simplicity bias is observable in the random logistic map, persisting even with small noise levels.
The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.
problem Generalizing the Apollonius theorem for m-simplices.
method Direct generalization of the theorem to m-simplices in n-dimensional space.
result Applications in geometry and optimization, including minimal surface enclosures, simplex thickness, and root-finding methods.
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
Simplicial sets deformation retract onto transverse simplices.
problem Deformation retraction of simplicial sets.
method Showed deformation retraction of singular simplicial set onto transverse simplices.
result Singular simplicial set deformation retracts onto transverse simplices.
We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary (n-1)-simplices of each n-simplex are replaced by combinations of internal (n-1)- sim…
Efficiently predicts long-time dynamics of quantum spin models using MLP regression.
problem Challenges in calculating long-time expectation values for quantum spin models.
method Utilized a multi-layer perceptron (MLP) model for regression on matrix product states (MPS) expectation values.
result Significantly reduced computational cost for generating long-time dynamics while maintaining high accuracy.
New method uses patchworking to represent oriented matroids.
problem Representing oriented matroids from polyhedral structures.
method Variant of Viro's patchworking, controlled cell merging process.
result Achieved a combinatorial manifestation of patchworking.
The paper explores how simplicity leads to better out-of-distribution generalization in models.
problem Understanding the theoretical principles behind out-of-distribution (OOD) generalization in modern models.
method Examining diffusion models in image generation to analyze compositional generalization abilities and develop a theoretical framework for simplicity-based OOD generalization.
result The true, generalizable model corresponds to the simplest among consistent models, and this simplicity can be quantified and used to establish sample complexity guarantees.
The study reveals simplicity bias in neural networks leading to better compositional mappings.
problem Understanding when and how to encourage neural networks to learn compositional mappings.
method Examined compositional mappings through coding length and gradient descent dynamics.
result Neural networks tend to learn the simplest bijections, explaining their good generalization.
Study on simplicity of Lie skew braces, proving new results for compact cases.
problem Simplicity of Lie skew braces, focusing on compact connected cases.
method Reviewing correspondence, investigating ideals and rigidity, proving main result for compact Lie skew braces.
result Compact connected simple Lie skew braces are either trivial or have simple underlying Lie groups.
Triangulations of R^n have at least tensor rank of determinant simplices.
problem Understanding the minimum number of simplices in periodic triangulations of R^n.
method Proving lower bounds on the number of simplices in periodic triangulations of R^n.
result Lower bounds on the number of simplices in periodic triangulations of R^n.
Attention improves edge prediction in e-commerce graphs.
problem Predicting edges in graphs from node attributes in e-commerce.
method Used attention mechanism in simple feedforward networks, derived analytically tractable model AttEST.
result Attention network outperforms LSTM architectures by over 20% on F-1 score.
Two-layer networks favor simple features, especially in complex datasets.
problem Simplicity bias in neural networks over-reliing on simple features.
method Characterization of two-layer neural networks with small weights and gradient flow.
result Features learned in middle training stages are more useful for out-of-distribution transfer.
Adam avoids simplicity bias in neural networks, leading to better generalization.
problem Simplicity bias in neural networks trained with SGD.
method Comparison of Adam and GD on binary classification tasks with Gaussian data.
result Adam leads to richer and more diverse features, improving generalization.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.
Neural nets learn simple distributions first, then more complex ones.
problem Understanding how neural networks generalize from simple to complex functions.
method Stochastic gradient descent training, synthetic data, CIFAR10, ImageNet pre-training.
result Neural networks initially use lower-order statistics, then higher-order ones.
The paper proves eigenvalues are simple for specific operators on bundles.
problem Eigenvalue simplicity for connection Laplacian and -simplicity on bundles.
method Analyzes connections on vector bundles and principal bundles, proving eigenvalue simplicity for a residual set of connections.
result Eigenvalues of the connection Laplacian and Laplace-Beltrami operator are simple for specified conditions.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.