Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.
problem Comparing solutions of Poisson equations on Riemannian manifolds with Robin boundary.
method Using Schwarz rearrangement and isoperimetric inequalities.
result Extends results on Poisson equations with Ric≥(n−1)κ. The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative…
The paper shows how to rearrange arcs to form closed curves.
problem Creating closed curves from planar arcs.
method Splitting a curve into arcs and rearranging them to form a closed curve.
result Closed curves can be formed by rearranging arcs under weak assumptions.
Simplified and extended a method for rearranging infinite configurations of cubes.
problem Constructing homotopies for isotopically rearranging cubes.
method Simplified and extended Eda and Kawamura's procedure.
result Simplified and extended the construction of homotopies.
This note proves a Gaussian version of a Pólya-Szegö conjecture using rearrangement techniques.
problem Finding the domain with the minimum Gaussian principal frequency when the Gaussian torsional rigidity is fixed.
method Adapted Kohler-Jobin rearrangement technique to the Gauss space, considering a modified torsional rigidity and rearranging layers to half-spaces.
result The Gaussian principal frequency is minimized for the half-space when the Gaussian torsional rigidity is fixed.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating wors…
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Schwarz lemma extended to equality cases and curvature on manifolds.
problem Extending Schwarz lemma to equality cases and studying curvature.
method Analyzing Schwarz lemma inequalities and equalities, studying holomorphic sectional curvature.
result Holomorphic maps are totally geodesic and have constant rank when Schwarz lemma equality holds.
Paper introduces capillary Schwarz symmetrization in half-space.
problem Capillary problems in half-space.
method Introduces a special anisotropic gauge to transform capillary symmetrization to convex symmetrization.
result Capillary Schwarz symmetrization in half-space established.
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Paper develops polynomial approximations for complex probability densities.
problem Approximating high-dimensional concentrated probability densities.
method Tensor-product spectral polynomials and KR rearrangements.
result Efficient approximation of complex densities using composite maps.
Higher dimensional generalizations of Schwarz's P-surface, Schwarz's D-surface and Scherk's second surface are constructed as complete embedded periodic minimal hy- persurfaces in Rn.
The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.
problem Proving a Moser-Trudinger inequality on metric measure spaces.
method Rearrangement of functions on CD(k,n)-spaces satisfying a Polya-Szegö type inequality.
result Characterization of manifolds with lower bounded Ricci curvature admitting a Moser-Trudinger inequality.
Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.
problem Improving Schwarz lemma for holomorphic maps between Hermitian manifolds.
method Introducing new curvature constraints on source and target manifolds, controlling by holomorphic sectional curvature.
result Significant improvements on the Wu--Yau theorem and Schwarz lemma for Gauduchon connections.
Method calibrates basket options using rearranged samples from constituent processes.
problem Calibrate basket options with non-linear dependency structure.
method Propose a method to extract dependency structure from market data through systematic sampling rearrangement, then calibrate a local volatility model.
result Efficiently calibrates basket options with near-perfect accuracy.
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.
problem Applying Milnor-Schwarz lemma to inverse monoids.
method Two proofs provided: elementary and using Vietoris-Rips complex.
result Generalization of Milnor-Schwarz lemma to inverse monoids.
Paper calculates eigenvalues of a specific triangle on a sphere.
problem Computing eigenvalues of a specific triangle on a sphere.
method Computed first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2) on the sphere.
result Computed the first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2).
The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
problem Analyzing holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
method Using Bochner formulas and comparison theorems.
result Established Schwarz type lemmas for holomorphic maps.
The paper creates symmetrical discrete minimal nets using Schwarz reflection.
problem Creating symmetrical discrete minimal nets.
method Extending Schwarz reflection principle to discrete minimal surfaces.
result Global examples of discrete minimal nets with high symmetry.
The paper generalizes the Cauchy-Schwarz-Bunyakovsky inequality and applies it to elasticity problems.
problem Applying the Cauchy-Schwarz-Bunyakovsky inequality to elasticity problems.
method Presentation of discrete and integral forms, n-dimensional generalizations, and strengthened CBS inequality.
result The strengthened CBS inequality is crucial for elasticity problems.
Curve shortening problem solved via Schwarz function.
problem Solving the curve shortening problem in the z-plane. method Using the Schwarz function to solve the differential equation StSz=Szz. result Explicit solutions for known curve shortening flow shapes can be recovered.
It has been shown that the Alvarez-Gaumeˊ-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…
Stochastic Schwarz lemma on Kähler manifolds via couplings.
problem Develop a new Schwarz lemma for Kähler manifolds.
method Probabilistic approach using Markovian couplings.
result Improved gradient estimates for harmonic functions.
Survey explores interactions between four conformal dynamics branches.
problem Understanding complex dynamics through different mathematical concepts.
method Examples and general results with technical tools.
result Dynamical relations between Schwarz reflection parameter spaces and anti-rational maps/ reflection groups.
Paper generalizes Schwarz lemma for polydisc mappings with specific metrics.
problem Schwarz lemma for holomorphic mappings with invariant metrics.
method Analyzes mappings between polydiscs with $\mbox{Aut}(P_m)$-invariant Kähler-Berwald metrics.
result Generalizes Schwarz lemma for polydisc mappings with specific metrics.
Describes the space of spherical triangles on a smooth 3-manifold.
problem Understanding the geometric structure of spherical triangles.
method Analyzes the homotopy and analytic properties of the space.
result The space is a smooth 3-manifold embedded in R^6.
Study fine Pólya-Szegő inequalities in metric spaces with applications.
problem Fine Pólya-Szegő rearrangement inequalities in metric spaces.
method Theory of Sobolev and BV functions, synthetic Ricci bounds, isoperimetric inequality.
result New geometric and functional inequalities under Ricci lower bounds.
The Schwarz--Pick lemma is a fundamental result in complex analysis. It is well-known that Yau generalized it to the higher dimensional manifolds by applying his maximum principle for complete Riemannian manifolds. Jeffres obtained Schwarz lemma for volume forms of conical Kähler metrics, based on a barrier function an…
Pixle attacks images by rearranging pixels, bypassing neural networks.
problem Vulnerability of neural networks to black-box adversarial attacks.
method A novel attack that rearranges a small number of pixels in images.
result Successfully attacks a high percentage of samples on various datasets and models.
Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.
problem Estimating distance functions and proving Schwarz lemma for weakly Kähler-Finsler manifolds.
method Establishing theorems about distance functions and applying them to prove the Schwarz lemma.
result Holomorphic mappings from weakly Kähler-Finsler manifolds to pseudoconvex Finsler manifolds are constant under certain conditions.
We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.
This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space X. In particular, we show that order closedness, σ(X,Xn∼)-closedness and σ(X,L∞)-closedness of a law-invariant convex set in $\mathc…
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in (−1,1], provided an additional condition on triangle weights. The paper proves a Schwarz lemma for mappings between specific types of manifolds.
problem Establishing a Schwarz lemma for mappings between Kähler and complex Finsler manifolds.
method Using properties of holomorphic sectional curvature and radial sectional curvature.
result A Schwarz lemma for holomorphic mappings between Kähler and complex Finsler manifolds.
In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with C2 boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. T…
We prove estimates interpolating the Schwarz Lemmata of Royden-Yau and the ones recently established by the author. These more flexible estimates provide additional information on (algebraic) geometric aspects of compact Kähler manifolds with nonnegative holomorphic sectional curvature, nonnegative $\Ric_\ell$ or posit…
DNA rearrangement processes recombine gene segments that are organized on the chromosome in a variety of ways. The segments can overlap, interleave or one may be a subsegment of another. We use directed graphs to represent segment organizations on a given locus where contigs containing rearranged segments represent ver…
Study characterizes cohomology and homotopy types for M-theory extensions.
problem Anomaly cancellation in M-theory extensions.
method Characterized integral cohomology and rational homotopy type of combined fibration.
result Subtle cohomology relations match Green-Schwarz mechanism.
In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on…
Introduces Cauchy-Schwarz divergence for domain adaptation.
problem Evaluating discrepancy between source and target domains in unsupervised domain adaptation.
method Introduces Cauchy-Schwarz divergence as a measure for evaluating discrepancy between marginal and conditional distributions.
result CS divergence offers a tighter generalization error bound than Kullback-Leibler divergence.
Through the Schwarz lemma, we provide a new point of view on three well-known results of the geometry of hyperbolic surfaces. The first result deal with the length of closed geodesics on hyperbolic surfaces with boundary (Thurston, Parlier, Papadopoulos-Théret). The two others give sharp lower bounds on two metric inva…