Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
problem Existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
method Analyzes existence and uniqueness of solutions for different values of p.
result Existence and uniqueness of smooth solutions for p > n.
Proves existence of longest paths in sub-Lorentzian problems.
problem Existence of longest paths in sub-Lorentzian problems.
method Generalizes classical theorem for Lorentzian manifolds to sub-Lorentzian problems.
result Longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups.
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including Lp versions. Study on geometric variational problems for existence, regularity, and uniqueness of solutions.
problem Geometric variational problems, focusing on existence, regularity, and uniqueness of solutions.
method Formulated in Federer and Fleming's theory of currents, discussed the existence theory, and presented core ideas of the (interior) regularity theory for area-minimizing currents and optimal transport paths. Two original results on generic uniqueness of solutions were presented.
result Generic uniqueness of solutions for both Plateau's problem and optimal branched transport problem.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. Problems on region choices for knot and link diagrams solved using Alexander numbering.
problem Existence of solutions for region choice problems on knot and link diagrams.
method Alexander numbering for regions, alternative proofs, necessary and sufficient conditions.
result Existence of solutions for region choice problems on link diagrams.
Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic p-Laplacian equation, presenting sufficient and necessary conditions for existence. result Presented sufficient and necessary conditions for the existence of a solution.
Paper investigates curvature problems and existence of solutions.
problem Existence of admissible solutions to curvature problems.
method Investigates curvature problems with prescribed Lp quotient type, proving existence under specific conditions. result Proves existence of admissible solutions without additional conditions.
Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.
problem Orlicz-Aleksandrov problem
method Gauss curvature flow
result Existence of smooth solutions
Paper finds smooth convex solutions to curvature problem.
problem Finding smooth, convex solutions to curvature problems.
method Established existence of solutions through mathematical analysis.
result Smooth, origin-symmetric, strictly convex solutions found.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
Study an anisotropic capillary flow to solve capillary Orlicz-Minkowski problem.
problem Solve capillary Orlicz-Minkowski problem without evenness assumption.
method Analyze an anisotropic capillary Gauss curvature flow to prove convergence and establish existence.
result Establish existence result for capillary Orlicz-Minkowski problem without evenness assumption.
We study a bad arm existing checking problem in which a player's task is to judge whether a positive arm exists or not among given K arms by drawing as small number of arms as possible. Here, an arm is positive if its expected loss suffered by drawing the arm is at least a given threshold. This problem is a formalizati…
New framework studies policy learning problems under data scarcity.
problem Learning improving policies when data is insufficient.
method Developed a mathematical framework for policy learning problems.
result Reduced policy learning problems to simpler ones in sample complexity.
Solves a specific Dirichlet problem on Riemannian manifolds.
problem Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds.
method Derives existence of C1,1-solutions under appropriate assumptions. result Existence of C1,1-solutions. New existence results for curvature problem on balls with specific conditions.
problem Existence of solutions for a prescribed mean curvature problem on a ball.
method Combining critical points at infinity approach with Morse theory.
result New existence results for higher dimensional case n≥5 under pinching conditions. Verified numerics prove existence of a curvature solution with known symmetries.
problem Existence of a curvature solution for the Nirenberg problem.
method Verified numerics and computer assistance.
result Existence of a genuine solution with known symmetry groups.
We present a method for proving the existence of solutions to a class of one dimensional variational problems. The method is demonstrated by two examples of optimal interpolation problems which are motivated by engineering applications. In each case we prove that the variational problem satisfies the Palais-Smale condi…
Solves a generalized dual Minkowski problem for specific values of q.
problem Finding solutions to the generalized dual Minkowski problem for given q and star bodies.
method Variational methods
result Existence of solutions for q<0 and 0≤q≤1, sufficient condition for q>1.
Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.
problem Existence and uniqueness of solutions for the Yamabe problem on non-compact manifolds of negative curvature type.
method Used partial C2 decay of the metric and local volume ratio condition to establish existence and uniqueness results. result Established existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.
The paper addresses how to add points to existing configurations on surfaces without disrupting continuity.
problem Adding points to existing configurations on orientable surfaces without disrupting continuity.
method Algebraic approach for g≥1 and m≥2, geometric approach for g≥1 and m=1. result A necessary condition for the existence of a section is that n must be a multiple of m+(2g−2) for g≥1 and m≥2. This paper concerns a fully nonlinear version of the Yamabe problem on manifolds with boundary. We establish some existence results and estimates of solutions.
We study the existence problem for tilted unduloids in H2×R. These are singly periodic annuli with constant mean curvature H>1/2 in H2×R, and the periodicity of these surfaces is with respect to a discrete group of translations along a geodesic that is neither verti…
In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.
Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.
problem Solving capillary curvature problems in Euclidean half-spaces.
method Applying a capillary John ellipsoid theorem to derive non-collapsing estimates and gradient estimates.
result Established existence of solutions to capillary curvature problems in certain ranges of p and q. New framework tackles geometric structure existence and classification.
problem Existence and classification of geometric structures.
method Developed a new framework of relative algebroids.
result New framework addresses geometric structure problems.
Study on new Monge-Ampère functionals and their variational problems.
problem Existence and uniqueness of solutions for nonlinear eigenvalue problems.
method Introduction of a family of real Monge-Ampère functionals and proving Sobolev type inequalities.
result Existence of solutions for a nonlinear eigenvalue problem.
The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classificatio…
Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.
problem Existence of sign-changing solutions to the Yamabe problem on manifolds with boundary.
method Variational approach, analysis of conformal invariants, and sharp energy estimates.
result Existence of least-energy nodal solutions when the manifold is positive and the boundary has non-negative constant mean curvature.
Math proves deep learning unstable, despite stable neural networks existing.
problem Unstable neural networks in deep learning despite stable ones existing.
method Mathematical proof showing instability of current training procedures.
result Proven existence of stable and accurate neural networks with variable dimensions, but current algorithms cannot compute them.
Proves Chen-Lin conjecture for sphere scalar curvature problem.
problem Existence of solutions for prescribed scalar curvature on spheres.
method Criterion proving existence based on Chen-Lin conjecture.
result Proves conjecture for standard n-dimensional sphere.
We consider the existence problem for `Steiner networks' (trivalent graphs with 120 degree angles at each junction) in strictly convex domains, with `Neumann' boundary conditions (orthogonal intersection with the domain boundary.) For each of the three possible combinatorial possibilities, sufficient conditions on the …
The paper solves problems related to curvature on a 3-sphere.
problem Prescribing positive cross curvature on the three-dimensional sphere.
method Existence results and a non-uniqueness example.
result Disproved a conjecture of Hamilton's about uniqueness.
There is an obstruction to the existence of Kähler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of Kähler -Ricci solitons, Sasaki-Einstein metrics and (con…
Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hija…
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
Study shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.
problem Existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems of Novikov equation.
method Kato's semi-group approach to establish well-posedness and C1 regularity. result Existence of exactly two families of periodic one-forms defining metrics and Levi-Civita connections.
In this paper we study the non-existence of solutions to the Dirichlet problem for minimal graphs of codimension ≥2, including certain situations over domain Ω even with non-C1 boundary ∂Ω.
In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying mani…
Study finds solutions for complex problems on non-compact manifolds.
problem Solving fully nonlinear Yamabe-type problems on non-compact manifolds.
method Existence results for a class of problems, considering both positive and negative cases.
result Explicit examples of manifolds satisfying the hypotheses of the theorems.
Open problem seeks an online learning algorithm for binary classification.
problem Existence of an online learning algorithm for binary classification with sublinear mistakes.
method Assumption of sequence allowing learning algorithm's existence.
result Specific condition determines sequence's learnability.
Proves existence of smooth convex solutions to capillary curvature equations.
problem Proving existence of smooth convex solutions to capillary curvature equations.
method Gradient estimate for capillary curvature equations in half-space.
result Existence of even, smooth, strictly convex solutions for all 1<p<k+1 and θ∈(0,π/2). We consider the fractional Nirenberg problem on the standard sphere Sn with n≥4. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.
Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k=1, it reduces to the well-…
Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.
problem Existence of least energy solutions for nonlinear p-Laplacian problems with critical exponent.
method Proving existence of solutions through critical point theory and variational methods.
result Significant difference in existence results between p-Laplacian and Laplacian cases.
Let ψ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface M for which ψ, when evaluated on M, coincides with the m−th elementary symmetric function of principal curvatures of M for a given m? The corresponding existence and uniqueness…