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.
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.
169,051 papers · 148 categories
Trend · papers per month
48 results for “simplicity volume”
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.
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.
New proof for Ending Lamination Theorem using hyperbolic 3-simplices.
problem Proving the Ending Lamination Theorem for hyperbolic 3-manifolds of infinite volume.
method Relies on Maximum Volume Law for hyperbolic 3-simplices.
result New proof of Ending Lamination Theorem.
Study simplicial volume via foliated simplices and duality.
problem Calculate simplicial volume using foliated simplices and duality.
method Defined real singular foliated homology, constructed foliated fundamental class, and established isometric isomorphism with measurable bounded cohomology.
result Norm of foliated fundamental class equals simplicial volume of .
The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…
Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.
problem Understanding the finite domination of manifolds by simplicial complexes.
method Proving that a manifold can be dominated by the -skeleton of a finite simplicial complex with a bounded number of simplices.
result The total number of simplices in the -skeleton is bounded above by a constant depending only on and the embolic volume of the manifold.
We prove the Proportionality Principle for the Lipschitz simplicial volume without any restriction on curvature. Our argument is based on the construction of a suitable \emph{pseudostraightening} for simplices.
Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …
Researchers calculate the minimal entropy of 3-manifolds, proving it's additive.
problem Calculating the minimal entropy of 3-manifolds.
method Using the JSJ decomposition and properties of hyperbolic components.
result The cube of the minimal entropy is proportional to the simplicity volume.
Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXi…
Proves simplicity of Lyapunov exponents for specific Anosov flows.
problem Proving all Lyapunov exponents have multiplicity 1 for certain Anosov flows.
method Perturbative results for flows, modification of eigenvalues, Markov partition, and simplicity criterion.
result In a -open and -dense set of Anosov flows, all Lyapunov exponents have multiplicity 1.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412…
We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold measures the minimal size of possibly ideal triangulations of "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…
We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G,…
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
The systole length of hyperbolic n-manifolds is bounded by a function of n and t.
problem Bounding the systole length of hyperbolic n-manifolds.
method Relating the number of simplices to the diameter, and using bounds on Margulis tubes.
result The systole length is bounded by a function of n and t.
The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in 's that every CPE metric must be Einstein. We prove that a -dimensional CPE…
Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonah…
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…
Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyag…
New theorem bounds group quotient size to subgroups index.
problem Understanding subgroup structure in hyperbolic groups.
method Proved quotient size bounds on Eilenberg-MacLane spaces.
result One-ended hyperbolic groups cannot have isomorphic finite-index subgroups.
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles (``The continuity conjecture''), and furthermore, the limit at a bou…
We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…
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.
A new method defines bounded cohomology classes from differential forms.
problem Defining bounded cohomology classes from differential forms.
method Integration over geodesic simplices and Fourier analysis on the hyperbolic plane.
result The method defines an injective embedding of differential forms into bounded cohomology classes.
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.
Arithmetic spaces simplified to simplicial complexes.
problem Understanding the complexity of arithmetic locally symmetric spaces.
method Homotopy equivalence to a simplicial complex with linearly bounded simplices, using a strengthened Margulis collar lemma.
result Arithmetic locally symmetric spaces are homotopy equivalent to simplicial complexes with linearly bounded simplices.
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.
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.
Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We pr…
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.
Unimodular classification of symmetric matrix map-germs.
problem Classifying symmetric matrix map-germs under volume-preserving equivalence.
method Introducing symmetrical quasi-homogeneity and volume-preserving equivalence.
result All simple -equivalence classes coincide with volume-preserving equivalence classes.
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…
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.
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 …
Improved financial predictions with OHLC data and timestamps.
problem Improving VWAP predictions in financial markets.
method Investigated the impact of timing features on machine learning models for VWAP prediction.
result Incorporating timing features consistently improves predictive performance across multiple ML architectures.
We consider polyhedra and 4-polytopes in Minkowski spacetime - in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, "tetrahedral diamonds" and 4-parallelotopes. We propose …
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the -skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…
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…
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.