New singularity concept in GR: volume singularities.
problem Understanding spacetime singularities in General Relativity.
method Introducing a new type of singularity: volume singularities.
result Volume singularities are hidden by event horizons.
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48 results for “Volume generation”
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, N…
The general volume of a star body, a notion that includes the usual volume, the th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the -dual curvature measures introduced recently by Lutwak, Ya…
Study of -adic simplicial volumes and their properties.
problem Understanding simplicial volumes over -adic seminormed rings.
method Definition and study of -adic simplicial volumes, homology bounds, and computation of volumes for surfaces.
result Established homology bounds and computed -adic simplicial volumes for surfaces.
For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannia…
DiffVolume generates realistic volume snapshots for LOBs.
problem Generating high-dimensional volume snapshots in LOBs is challenging.
method Conditional Diffusion model for volume generation.
result DiffVolume outperforms in realism, counterfactual generation, and downstream prediction.
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, the…
The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…
The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, no…
Upper bounds for volumes of hyperbolic polyhedra and links are derived.
problem Finding upper limits for volumes of generalized hyperbolic polyhedra and links.
method Application of Belletti's theorem and analysis of polyhedra with triangular faces and trivalent vertices.
result Improved upper bounds for volumes of hyperbolic polyhedra and links are derived.
Generalizes inequality for complete manifolds involving homology classes.
problem Volume and simplicial volume inequality for closed manifolds.
method Extends inequality to -norm of homology classes on complete manifolds.
result Inequality involving critical exponent and mass of homology classes.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Upper bounds for the second nonzero eigenvalue of the Laplacian via folding and conformal volumemath.SP
Upper bound for Laplacian eigenvalue via conformal volume.
problem Finding upper bounds for Laplacian eigenvalues.
method Using conformal volume to derive an upper bound.
result Effective upper bound for Laplacian eigenvalues on manifolds.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Proves generic nondegeneracy for solutions under volume constraint in closed manifolds.
problem Proving nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation.
method Adapting techniques from previous research to prove nondegeneracy.
result Generic nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint in closed manifolds.
The volume of the quantum mechanical state space over -dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endo…
In recent years, several families of hyperbolic knots have been shown to have both volume and (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist nu…
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …
Proof confirms volume conjecture for a specific knot.
problem Verifying the volume conjecture for a specific knot.
method Using a generalized topological quantum field theory and a tetrahedral decomposition.
result Volume conjecture holds for the knot in .
We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational. Dunfield's proof involves the rigidity of maximal volume. The volume is still an important ingredient in this paper. Unfortunately at this poi…
Estimates volume of convex Alexandrov spaces with boundary.
problem Estimating volume of Alexandrov spaces with convex boundaries.
method Gradient flow of semi-concave functions.
result Volume upper bound achieved implies Boundary Conjecture.
The Weyl principle holds in some Finsler settings despite general failure.
problem Applying the Weyl principle to Finsler manifolds.
method Investigation of the Weyl principle in Finsler geometry.
result A weak form of the Weyl principle persists in certain Finsler settings.
New recursion found for hyperbolic sphere volumes.
problem Volume calculation of hyperbolic sphere moduli spaces.
method Proved a non-linear recursive relation.
result Generalized Zograf's result for conical points and geodesic boundaries.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in t…
Small volume link orbifoldsmath.GT
This paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose singular locus is a link. We identify the unique smallest volume orbifold whose singular locus is a knot or link in the 3-sphere, or more generally in a Z_6 homology sphere. We also prove more general lower bounds under mild homological hypot…
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
Derives Weyl law for volume spectrum using parametric inequalities.
problem Deriving the Weyl law for the volume spectrum in compact Riemannian manifolds.
method Proves parametric generalizations of isoperimetric and coarea inequalities to derive the Weyl law.
result Derives the Weyl law for 1-cycles in 3-manifolds.
Study of -cylinder surfaces to calculate Masur-Veech volumes.
problem Calculating Masur-Veech volumes for hyperbolic surfaces.
method Combinatorial approach using metric ribbon graphs and plane trees.
result Found generating function for -cylinder contributions.
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.
A dynamic herding model with interactions of trading volumes is introduced. At time , an agent trades with a probability, which depends on the ratio of the total trading volume at time to its own trading volume at its last trade. The price return is determined by the volume imbalance and number of trades. The …
Ricci curvature links volume convexity and minimal submanifolds.
problem Volume functional convexity and minimal submanifolds in Kaehler geometry.
method Toric Kaehler geometry and quasi-homogeneous manifolds.
result Sign of Ricci curvature correlates with volume functional convexity.
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 …
We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first var…
Extends cohomology theory for infinite volume transformation groups.
problem Generalizing cohomology for infinite volume transformation groups.
method Introduces norm-controlled cohomology as a generalization of bounded cohomology.
result Establishes norm-controlled cohomology for infinite volume transformation groups.
This work generalizes a geometric Laplacian determinant description to higher dimensions.
problem Defining and understanding the Laplacian determinant in higher dimensions with non-Delaunay triangulations.
method Geometric description of the Laplacian determinant in higher dimensions, relating it to volume quantities derived from simplex geometry.
result Generalizes geometric Laplacian determinant description to higher dimensions, showing negative semidefiniteness and kernel of constants.
New inequality linking geodesic length and volume in complex projective plane.
problem Understanding geometric properties of complex projective plane.
method Combining recent results on area minimizers and geodesics with Kronheimer-Mrowka's proof.
result Proved a new inequality relating volume and length of geodesics.
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
Study volume conjecture for links with multiple hyperbolic pieces.
problem Volume conjecture for links with more than one hyperbolic piece.
method Constructing infinite families of prime links, analyzing their complements, and using colored Jones polynomials and simplicial volume.
result Exponential growth rates of colored Jones polynomials capture the simplicial volume of link complements.
We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.
This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the parameter of the colored Jones polynomial we also conjecture that it would also g…
Develops new methods for Epstein surfaces and W-volume.
problem Constructing and understanding Epstein surfaces and W-volume.
method Alternate construction using Osgood-Stowe differential; variational formulas.
result Generalizations of Epstein's univalence criterion and variational formulas for W-volume.
We examine the relationship between trading volumes, number of transactions, and volatility using daily stock data of the Tokyo Stock Exchange. Following the mixture of distributions hypothesis, we use trading volumes and the number of transactions as proxy for the rate of information arrivals affecting stock volatilit…
Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
problem Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
method Define the -illumination body and prove a generalization of Werner's formula.
result Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link complements by analyzing the corresponding toroidal decompositions. We then use it to prov…
Volume vs. rank of latticesmath.GR
We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume.
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.