Classifies reversible Finsler metrics with positive curvature.
problem Classifying homogeneous reversible Finsler metrics with positive flag curvature.
method Classification based on G-invariant metrics and curvature properties.
result Exceptions exist where homogeneous Finsler metrics with positive flag curvature are not known.
Paper proves rigidity theorems for geodesically reversible Finsler metrics.
problem Understanding geodesically reversible Finsler metrics in closed manifolds.
method Applied theory of volumes and areas on Finsler spaces to establish rigidity theorems.
result Partial explanation of the scarcity of geodesically reversible Finsler metrics in closed manifolds.
Boundary rigidity proven for non-reversible Finsler metrics.
problem Recovering non-reversible Finsler metrics from boundary distance data.
method Sum of reversible Finsler norm and closed 1-form, boundary rigidity results.
result 1-form can be uniquely recovered from boundary distance data.
Study on Finsler spaces with specific metric changes and reversible geodesics.
problem Characterizing Finsler spaces with reversible geodesics.
method Analyzing a Finsler space with a Randers change of Quartic metric and deriving conditions for reversible geodesics.
result The Finsler metric F induces a generalized weighted quasi-distance on the space.
In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of Finsler metrics of Einstein-reversibility are always Einstein metrics. In parti…
A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…
In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective n-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…
We study the necessary and sufficient conditions for a Finsler surface with (α,β)-metrics to be with reversible geodesics.
The paper introduces new Finsler metrics and associated weighted quasi-metrics.
problem Geometric properties of Finsler metrics and quasi-metric spaces.
method Construction of weighted quasi-metrics associated with Finsler metrics.
result Investigation of geometric properties of weighted quasi-metric spaces.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
Characterizes isometries between non-reversible Finsler manifolds.
problem Understanding isometries in non-reversible Finsler manifolds.
method Generalization of Myers-Nakai Theorem for Riemannian manifolds, modification of function spaces to accommodate asymmetric structure.
result Functional characterization of isometries between non-reversible Finsler manifolds.
Paper classifies Randers metrics based on Ricci curvature properties.
problem Investigating isotropic projective Ricci curvature in Randers metrics.
method Classification of Randers metrics based on isotropic projective Ricci curvature properties.
result Randers metric of isotropic projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.
Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.
problem Establishing reverse Hölder inequalities on Kähler metrics of Fano varieties.
method Using log-concavity and properties of Ricci potentials, the inequality is proven for Fano manifolds with log terminal singularities.
result The inequality holds for Fano varieties with log terminal singularities and the constant depends only on p and the dimension of X.
The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.
problem Approximating Riemannian metrics and proving geometric conjectures.
method Discretization of metrics using walls and triangulations.
result The discrete filling area conjecture is equivalent to Gromov's original conjecture.
Paper studies metric properties on geodesic spaces to characterize curvature negativity.
problem Characterizing curvature negativity in geodesic spaces.
method Introduces metric properties via metric projection and applies to Alexandrov and Busemann NPC spaces.
result Proves both properties characterize non-positivity of sectional curvature on Riemannian manifolds.
Study of parabolas in Funk metric on unit disk.
problem Understanding parabolas in Funk metric on a disk.
method Analyzing four types of parabolas due to Funk metric's non-reversibility.
result Two known conics and two irreducible quartics found.
Study functional inequalities on non-reversible Finsler manifolds.
problem Functional inequalities on non-reversible Finsler manifolds.
method Application of Bochner inequality and Γ-calculus.
result Dimensional versions of Poincare--Lichnerowicz, logarithmic Sobolev, and Sobolev inequalities hold for non-reversible metrics.
A new trading strategy using reinforcement learning for statistical arbitrage.
problem Traditional statistical arbitrage models rely on model assumptions and price deviations from a long-term mean.
method Empirical reversion time metric, reinforcement learning framework, and state space optimization.
result Optimal mean reversion strategy identified through reinforcement learning.
Complete Finsler spaces with negative Ricci curvature are reversible.
problem Characterizing Finsler spaces with constant negative Ricci curvature.
method Utilizing projectively invariant pseudo-distance and Schwarzian derivative.
result Every connected complete Finsler space with constant negative Ricci scalar is reversible.
In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional p…
We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.
Study shows reversible Finsler metrics on symmetric spaces are symmetric.
problem Characterizing Finsler metrics on symmetric spaces.
method Analyzes C2 Finsler metrics on compact spaces, proving reversibility implies symmetry. result Reversible Finsler metrics on symmetric spaces are symmetric.
We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endo…
Study homology products on loop spaces with orientation reversal.
problem Computing homology products on loop spaces with orientation reversal.
method Defined transfer product on loop space quotients using transfer maps and Chas-Sullivan loop product.
result Computed homology of loop spaces with orientation reversal and defined product.
Study geodesics on spheres with constant curvature, showing integrability and invariant properties.
problem Characterizing geodesics on spheres with constant flag curvature.
method Analyzing non-reversible Finsler metrics on S2 with constant flag curvature 1. result Geodesic flow is conjugate to Katok's examples and length of shortest closed geodesic is invariant.
New findings show functional inequalities fail on Finsler manifolds with positive S-curvature.
problem Failure of functional inequalities on Finsler manifolds with positive S-curvature.
method Analysis of Finsler metric measure manifolds with reversibility, flag curvature, and S-curvature.
result Functional inequalities fail on Finsler manifolds with positive S-curvature.
We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented complex projective planes admits such metrics.
Develops semigroup approach for Finsler geometry, proving isoperimetric inequality.
problem Proving isoperimetric inequality for Finsler manifolds.
method Semigroup approach, Bochner-Weitzenböck formula, gradient estimate, lower weighted Ricci curvature bound.
result Proves Bakry-Ledoux's Gaussian isoperimetric inequality for non-reversible metrics.
We construct all Finsler metrics on the two-sphere for which geodesics are circles and show that any (reversible) path geometry on a two-dimensional manifold is locally the system of geodesics of a Finsler metric.
New inequality for eigenfunctions on curved spaces.
problem Eigenfunctions on non-smooth spaces with Ricci curvature.
method Sharp reverse-Hölder inequality for Dirichlet Laplacian eigenfunctions.
result Generalizes classical comparison theorem to curved spaces.
This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.
problem Characterize all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces.
method Unified construction using axi-symmetric harmonic functions and methods from scalar-flat Kähler metrics.
result All such metrics are ALF and belong to known families.
Study on how non-reversible diffusion processes affect homology on manifolds.
problem Understanding the asymptotic behavior of random homology in diffusion processes.
method Investigation of asymptotic properties of random homology associated with stochastic diffusion processes on compact Riemannian manifolds.
result For quadratic rate, manifold is a locally trivial fiber bundle over a flat torus with minimal fibers.
Paper proposes a new method to solve Schrödinger Bridge Problem using kernel regression.
problem Schrödinger Bridge Problem in the context of entropic optimal transport.
method Forward-reverse iterative Monte Carlo procedure using kernel regression.
result Developed a provably convergent algorithm for approximating Schrödinger potentials.
The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.
problem Existence and multiplicity of orthogonal Finsler geodesic chords in a disk-like manifold.
method Study of Finsler geodesic chords under reversibility assumption.
result At least N orthogonal Finsler geodesic chords found in a disk-like manifold.
Eigenvalue bounds for Finsler manifolds controlled by metric properties.
problem Bounding p-Laplacian eigenvalues on Finsler manifolds. method Using volume, reversibility, and metric properties to bound eigenvalues.
result Eigenvalue bounds for Finsler manifolds controlled by metric properties.
Post-hoc transforms can reverse model performance trends, especially in noisy settings.
problem Post-hoc transforms can reverse model performance trends, especially in noisy settings.
method Empirical study and analysis of post-hoc transforms like temperature scaling, ensembling, and SWA.
result Post-hoc reversal can prevent double descent and mitigate mismatches between test loss and test error.
Sharp bounds found on shortest geodesic on punctured spheres.
problem Finding the shortest closed geodesic on punctured spheres.
method Sharp curvature-free upper bounds expressed in terms of area, extremal metrics described.
result Optimal bounds for spheres with up to four ends, extended to larger numbers of punctures.
We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n>2.
For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.
Two distinct geodesics on spheres with a bumpy metric were shown.
problem Existence of two distinct closed geodesics on spheres with a non-reversible and bumpy Finsler metric.
method Simplified proof using covering geodesics and minimal index growth.
result Contradiction for infinite index, leading to the existence of two distinct geodesics.
Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by opti…
We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex F…
A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit met…
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
New near-metrics capture similarity for diverse data types.
problem Measuring similarity across different data types and forms.
method Local graph diffusion to create quasi-metrics without metric axioms.
result Near-metrics perform well for various data types, especially for categorical and continuous data.
For odd-dimensional spheres, there's always a second short geodesic.
problem Finding the second shortest closed geodesic on odd-dimensional spheres.
method Analyzing non-reversible Finsler metrics on spheres of odd dimension.
result There is a second closed geodesic with Morse index ≤ 4(m+2)(m-1)+2.
New loops found in universe's timeline, challenging traditional time direction.
problem Signature changing spacetimes and time origins.
method Developed framework for signature changing manifolds, adapted Lorentzian tools.
result Pseudo-timelike loops exist in every point on the time origin hypersurface.