Innovates volume entropy semi-norm, proving equivalence to simplicial volume.
problem Equivalence of volume entropy and simplicial volume in real homology.
method Introduces volume entropy semi-norm and proves its equivalence to simplicial volume.
result Equivalence of volume entropy semi-norm and simplicial volume in every dimension.
Positive simplicial volume implies virtually positive Seifert volume for 3-manifolds.
problem Understanding the relationship between simplicial and Seifert volumes in 3-manifolds.
method Utilizing recent results on virtual domination and virtual representation volumes of 3-manifolds.
result Closed orientable 3-manifolds with positive simplicial volume have virtually positive Seifert volume.
3-manifold volume simplified to one-fifth of traditional measure.
problem Calculating the volume of 3-manifolds efficiently.
method Cubical simplicial volume approach.
result Cubical simplicial volume equals one-fifth of ordinary volume.
Study shows integral foliated simplicial volume for certain manifolds equals classical simplicial volume.
problem Comparing integral foliated simplicial volume to classical simplicial volume for different types of manifolds.
method Used ergodic theory and weak containment of measure preserving actions to prove results.
result Integral foliated simplicial volume equals classical simplicial volume for certain manifolds.
Study on volumes of quasifuchsian manifolds, focusing on similarities and proximities.
problem Understanding the relationship between renormalized volume and dual volume of quasifuchsian manifolds.
method Analyzing similarities and proximities between renormalized volume and dual volume, using variational formulas and Weil-Petersson distance.
result Renormalized volume and dual volume are closely related, with bounded distance between related objects.
3-manifolds' volumes match stable integral values.
problem Determining 3-manifold volumes accurately.
method Integral foliated simplicial volume and ergodic theory.
result 3-manifolds' volumes equal stable integral simplicial volumes.
Defines ideal simplicial volume for manifolds with boundary, showing it's bounded by classical volume.
problem Measuring the complexity of manifolds with boundary.
method Introduces ideal simplicial volume, compares it to classical volume, and computes specific examples.
result Ideal simplicial volume is bounded by classical volume and can be strictly smaller for certain manifolds.
Polyhedron volume is a multiple root of its polynomial.
problem Understanding the volume of flexible polyhedra.
method Proof of volume polynomial properties.
result Volume of flexible polyhedra is a multiple root of its polynomial.
Infinite volumes of Bergman spaces on product manifolds.
problem Volume calculation of Bergman spaces on product manifolds.
method Calabi and Mabuchi volumes, inspired by Shiffman-Zelditch conjecture.
result Infinite volumes of Bergman spaces on product manifolds.
New finding links hyperbolic manifold systolic volume to triangulation complexity.
problem Understanding the relationship between systolic volume and triangulation complexity in hyperbolic manifolds.
method Proof based on Jørgensen and Thurston's theorem of hyperbolic volume.
result Systolic volume of hyperbolic manifolds is related to triangulation complexity.
We study hyperbolic bongles and find their volumes.
problem Characterizing and quantifying hyperbolic bongles.
method Provided necessary and sufficient conditions for hyperbolicity, calculated volumes, and established upper bounds.
result All balanced hyperbolic n-bongles have the same volume and this volume is an upper bound for any hyperbolic n-bongle. Study of p-adic simplicial volumes and their properties.
problem Understanding simplicial volumes over p-adic seminormed rings. method Definition and study of p-adic simplicial volumes, homology bounds, and computation of volumes for surfaces. result Established homology bounds and computed p-adic simplicial volumes for surfaces. Paper improves volume comparisons and entropy estimates for integral Ricci curvature.
problem Volume comparisons and entropy estimates for integral Ricci curvature.
method Several volume comparisons and an estimate for volume entropy.
result Improved estimates for volume entropy and algebraic entropy.
Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.
problem Bounding the volume spectrum of Riemannian manifolds.
method Proves upper bounds that depend on volume, dimension, and a conformal invariant.
result Upper bounds for the volume spectrum are established.
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.
Minimal surfaces found in finite volume manifolds.
problem Existence of minimal hypersurfaces in complete manifolds of finite volume.
method Independent result on volume sweeping by hypersurfaces; main tool is a volume constraint result.
result Proves existence of minimal hypersurfaces in complete non-compact manifolds of finite volume.
Study on price-volume correlation fractal features and market type effects.
problem Understanding the fractal features and market type effects of price-volume correlation.
method Applied MF-DXA method to analyze price, trading volume, and their coupling.
result Price, trading volume, and price-volume coupling exhibit power law and multifractal properties.
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…
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…
We show that complete uniform visibility manifolds of finite volume with sectional curvature −1≤K≤0 have positive simplicial volumes. This implies that their minimal volumes are non-zero.
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…
The paper proves bounded domains are biholomorphic to the unit ball.
problem Covering manifolds with finite volume.
method Analyzes C2 and C1,ε boundary conditions for bounded domains. result Bounded domains are biholomorphic to the unit ball under certain conditions.
The study counts cusps on finite volume Riemannian manifolds using volume and decay estimates.
problem Counting cusps on finite volume Riemannian manifolds.
method Volume and decay estimates, nonlinear theory of p-Laplacian, volume comparison theorems.
result An upper bound on the number of cusps based on the volume of the manifold.
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to (1+n)-dimensional Lorentz--Finsler manifolds. result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…
Estimates open sets for fibrations, leading to volume vanishing results.
problem Estimating open sets for fibrations.
method Straightforward estimate for open sets with fundamental group constraints.
result Vanishing results for simplicial volume and minimal volume entropy for certain mapping tori.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
Study of volume dynamics at market spread in Bitcoin/USD.
problem Understanding the statistical properties of order volumes in financial markets.
method Examined the dynamical properties of volume available at the spread, focusing on mean reversion, asymmetry, and clustering.
result Evidence of mean reverting volume changes and strong asymmetries in sell and buy orders.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is p-integrable for any 0<p<1. Hyperbolic volume correlates with chemical properties of fullerenes.
problem Understanding the relationship between fullerene structure and chemical properties.
method Calculated hyperbolic volumes of fullerenes and correlated them with topological indices.
result Hyperbolic volume correlates with Wiener index and other topological indices of fullerenes.
Defines half-volume spectrum for manifolds and proves Weyl law holds.
problem Understanding volume distribution in manifolds.
method Introduces half-volume spectrum and uses Weyl law and Allen-Cahn min-max theory.
result Weyl law holds for half-volume spectrum and half-volume constant achieved by specific surfaces.
Models predict equities' daily trading volume using Bayesian methods.
problem Predicting equities' daily trading volume accurately.
method Bayesian econometric methods combining historical and current inputs.
result Unified approach for predicting total, remaining, intra-day, close auction, and seasonal volumes.
Study pinches volume of CAT(1) spaces, proving sphere theorem and manifold recognition criterion.
problem Volume pinching problems in CAT(1) spaces.
method Characterization of compact geodesically complete CAT(1) spaces, sphere theorem proof, manifold recognition criterion formulation.
result Sphere theorem for compact CAT(1) homology manifolds of small volume, manifold recognition criterion under upper curvature bound.
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.
Formula for tube volume on special geometric manifolds.
problem Calculating volumes of geometric structures.
method Using Einstein constant and polarization volume.
result Formula for renormalized tube volume.
Study simplicial volume for fixed fundamental groups, finding gaps.
problem Understanding simplicial volume for manifolds with fixed fundamental group.
method Relate gap problem to rationality questions in bounded (co)homology.
result Show existence of gaps in simplicial volume spectrum at zero.
Finite volume ends found in quaternionic Kähler manifolds.
problem Existence of quaternionic Kähler manifolds with finite volume ends.
method Proof in all dimensions 4m≥4. result Existence of quaternionic Kähler manifolds with finite volume ends.
Study shows small transcendental simplicial volumes exist on manifolds.
problem Understanding the limits of simplicial volumes on manifolds.
method Examined closed manifolds and transcendental numbers.
result Found manifolds with arbitrarily small transcendental simplicial volumes.
Study links' simplicial volume using diagrams and decompositions.
problem Calculate the simplicial volume of link complements.
method Analyze link diagrams, use toroidal decompositions, and refine upper bounds.
result Proved a refined upper bound for the volume in terms of twists.
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.
In arxiv:1205.1274 Rieck and Yamashita defined the link volume of 3-manifolds and studied some of its basic properties. Many of these properties are similar to the corresponding properties of the hyperbolic volume. In this paper we calculate the link volume of an infinite family of prism manifolds. As a corollary, we s…
Study on volume growth, Liouville theorems, and f-harmonic functions in Ricci shrinkers.
problem Volume growth and Liouville theorems in Ricci shrinkers.
method Analysis of f-harmonic functions and volume comparison properties. result Integral properties and local gradient estimates of f-harmonic functions on Ricci shrinkers. New formula calculates volumes of ideal hyperbolic drums.
problem Computing volumes of ideal hyperbolic drums.
method Proved a volume formula for arbitrary ideal hyperbolic antiprisms (drums).
result Volume formula for ideal hyperbolic drums.
Fully augmented links have dense volume densities but discrete in certain ranges.
problem Characterizing the volume density spectrum of fully augmented links.
method Analyzing the ratio of volume to the number of augmentations.
result The set of FAL volume densities is dense in $[2\voct, 10\vtet)$ but discrete in $[\voct,2\voct)$.
Study shows volume and genus unrelated for hyperbolic fibred knots.
problem Volume and genus of hyperbolic fibred knots are unrelated.
method Analyzes hyperbolic fibred knots in three-sphere.
result Volume and genus are unrelated for hyperbolic fibred knots.