Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
Weil-Petersson volumes vary continuously with weighted points on a projective line.
problem Continuity of Weil-Petersson volumes in moduli space with weighted points.
method Localization and geometric computation methods.
result CM volume converges to geometric volume as weights approach Calabi-Yau geometry.
The paper classifies hypersurfaces with constant weighted mean curvature.
problem Characterizing and classifying hypersurfaces with specific curvature properties.
method Using intrinsic properties of the second fundamental form and analyzing weighted volume and growth.
result Characterization of hyperplanes and generalized round cylinders.
Proposes volumization for neural networks to control bias-variance tradeoff.
problem Improving generalization and preventing memorization in neural networks.
method Defines a physical volume for weights, interpolating between L2 and L∞ regularization.
result Volumization interpolates between weight decay and clipping, improving generalization.
In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume…
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…
Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.
problem Characterizing weakly weighted Einstein-Finsler metrics.
method Showed isotropic S-curvature under certain conditions. Characterized via navigation expressions and α and β. result Weakly weighted Einstein-Kropina metrics have isotropic S-curvature and can be completely characterized.
Uniqueness found for elliptic equations with drift on manifolds.
problem Finding unique solutions to elliptic equations with drift on manifolds.
method Investigation in weighted Lebesgue spaces, focusing on conditions for uniqueness.
result Sharp conditions on drift term for uniqueness in polynomial volume growth manifolds.
In this paper, we introduce a definition of λ-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that λ-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete λ-hypersurfaces with …
New method calculates Ricci curvature from distances between weighted volumes.
problem Calculating Ricci curvature for weighted Riemannian manifolds.
method Asymptotic retrieval of generalized Ricci tensor from scaled metric derivatives of Wasserstein 1-distances.
result Limiting coarse curvature of random graphs converges to generalized Ricci tensor.
In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for f-stability index of a constant weighted mean curvature hypersurface…
Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.
problem Investigating inequalities on Finsler manifolds with weighted Ricci curvature.
method Volume comparison, Bonnet-Myers theorem, Poincaré-Lichnerowicz inequality.
result Sharp lower bound for the first eigenvalue on Finsler manifolds.
In this paper, we study the complete bounded λ-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with ∣A∣≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
The paper proves an infinite double bubble theorem in higher dimensions.
problem Characterizing minimizing partitions of infinite and finite volumes in Rn. method Proves a variant of the double bubble theorem for configurations with infinite and finite chambers.
result Locally minimizing (1,2)-clusters are unique in Rn for n≤7 and n≥8 under certain conditions. The study examines stable regions in weighted manifolds with boundary properties.
problem Studying stable regions in weighted manifolds with boundary properties.
method Using deformations constructed from parallel vector fields tangent to the boundary, the study deduces rigidity properties for stable sets.
result The classification of stable sets in some Riemannian cylinders and uniqueness results for minimizers.
We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…
Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
The paper studies weighted Ricci curvatures and characterizes Randers metrics.
problem Characterizing Randers metrics with weighted Ricci curvatures.
method General weighted Ricci curvatures and characterization of Randers metrics.
result Characterization of Randers metrics with almost isotropic weighted Ricci curvatures.
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 …
In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, λ1-weighted surface area, and λ2-weighted volume, for surfaces immersed in R3. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all …
Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.
problem Defining surface area in non-Euclidean geometries.
method Generalizes illumination bodies to Riemannian spaces of constant curvature and projective Finsler geometries, proving their volume derivative defines surface area.
result Derivative of volume of illumination bodies defines surface area in non-Euclidean geometries.
We study the problem of optimal execution of a trading order under Volume Weighted Average Price (VWAP) benchmark, from the point of view of a risk-averse broker. The problem consists in minimizing mean-variance of the slippage, with quadratic transaction costs. We devise multiple ways to solve it, in particular we stu…
A rigidity theorem for smooth Legendrian self-shrinkers is proven.
problem Understanding the structure of Legendrian self-shrinkers.
method Estimating weighted volume to prove optimal volume growth.
result Rigidity theorem for entire smooth Legendrian self-shrinkers.
In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…
Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.
problem Volume comparison and monotonicity for Bakry-Émery Ricci curvature.
method Relative volume comparison theorem for LP-bound of Bakry-Émery Ricci curvature and gradient of potential function. result Modified proof for volume comparison and monotonicity of Kähler-Ricci flow.
We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…
Synthetic Ricci flows defined for metric measure spaces.
problem Characterizing Ricci flows for non-smooth spaces.
method Heat flow, optimal transport, volume asymptotics.
result Equivalent characterizations of weighted Ricci flow.
Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.
problem Existence and classification of conical Calabi-Yau structures on horospherical cones.
method Variational approach to establish equivalence between volume minimization and existence of conical Calabi-Yau structures.
result Existence of many irregular horospherical cones with mild singularities.
We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a con…
Paper generalizes CR Obata theorem to weighted Sasakian manifolds.
problem Deriving eigenvalue estimates for weighted Kohn Laplacian.
method Derived weighted CR Reilly's formula and applied to Sasakian manifolds.
result CR Obata theorem proven for weighted Sasakian manifolds.
This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.
Study predicts intraday stock trading volume using ML models.
problem Predicting intraday trading volumes in equity markets.
method Used machine learning models with HF predictors.
result Intraday stock trading volume is highly predictable.
We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of Uq(sl2). We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of t…
Paper proves optimal systolic inequality for manifolds with positive triRic curvature.
problem Optimal systolic inequality for manifolds with positive triRic curvature.
method Stable weighted k-slicing, volume comparison theorem, and metric deformation. result Proves an optimal systolic inequality and characterizes the equality case.
This paper sets out to provide a general framework for the pricing of average-type options via lower and upper bounds. This class of options includes Asian, basket and options on the volume-weighted average price. We demonstrate that in cases under discussion lower bounds allow for the dimensionality of the problem to …
In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obt…
Study compares spectral properties of a specific tensor in geometry.
problem Comparing spectral properties of a specific tensor in geometry.
method Diameter and global weighted volume comparison with a positive lower bound on the N-Bakry-Emery Ricci tensor. result Established diameter and volume comparisons for tensors with positive lower bounds.
Study of weighted nonlinear flags in symplectic geometry.
problem Understanding the geometry of weighted nonlinear flags.
method Generalizing weighted nonlinear Grassmannians to Frechet manifolds and using them to describe coadjoint orbits.
result Description of coadjoint orbits of Hamiltonian diffeomorphisms using weighted isotropic nonlinear flags.
A new distillation framework predicts stock trading volumes more accurately with less model size.
problem Predicting stock trading volumes using regression models without class correlations.
method Transformed regression model into a probabilistic forecasting model, matching distributions and correlational relationships.
result Framework achieves superior prediction accuracy with significantly smaller model size.
Smoothly conjugate Anosov flows on 3D manifolds are actually smoothly conjugate.
problem Smoothly conjugate 3D Anosov flows are not always smoothly conjugate.
method Proved smooth rigidity for volume preserving Anosov flows on 3-manifolds.
result Smooth conjugacy implies smooth conjugacy for volume preserving Anosov flows.