arXiv research

Topological obstructions to admissibility in $σ_k$-Loewner--Nirenberg problem

Flow approach solves Ricci equation boundary problem.

Solves Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.

Proves existence of smooth metrics with specific curvature properties.

Two flow methods solve a problem on Riemannian manifolds, proving convergence to the Loewner-Nirenberg solution.

The paper proves existence of solutions to a Loewner-Nirenberg problem on Riemannian manifolds.

Solves geometric problems using fully nonlinear equations and Morse theory.

Solves nonlinear problems on metric structures through eigenvalue counting.

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Further…

Maximal solution of a PDE shows boundary smoothness for certain domains.

The paper examines the smoothness of solutions to a specific partial differential equation on smooth domains.

The paper classifies solutions to a specific equation and finds counterexamples to boundary estimates.

We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence a…

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

Study on metrics on manifolds with specific curvature properties.

Study properties of solutions with singularities in the negative cone.

This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a confor…

The negative case of the Singular Yamabe Problem concerns the existence and behavior of complete metrics with constant negative scalar curvature on the complement of a closed set in a compact Riemannian manifold which are conformally equivalent to a smooth metric on this compact manifold. When the closed set is a smoot…

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the…

The paper proves Ricci flow convergence on compact manifolds with boundary.

The paper extends a Ricci flow result for compact manifolds with boundary.

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…

Optimal transport reformulates multiple quantile hedging problem.

Solves four problems related to circle families in the plane.

Solves four problems related to sphere families in 3D space.

The paper solves optimal control problems for various convex sets using convex trigonometry.

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we anal…

Study proves only origin-centered spheres solve certain curvature problems.

MathChat uses LLM agents to solve challenging math problems through conversational problem-solving.

The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.

Paper solves Gromov-Wasserstein for point clouds efficiently.

The paper explains how microlocal analysis solves geometric inverse problems.

Proves NP and co-NP status for knot core recognition in solid torus.

The min-max problem, also known as the saddle point problem, is a class of optimization problems which minimizes and maximizes two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal processing and communication (SPCOM) problems. Despite its popularity, most exist…

A new method solves complex control problems with random coefficients.

This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For ea…

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…

New method solves generalized Minkowski problem for torsional rigidity.

We present 27 problems encountered in automating the translation of movie/TV show subtitles. We categorize each problem in one of the three categories viz. problems directly related to textual translation, problems related to subtitle creation guidelines, and problems due to adaptability of machine translation (MT) eng…

Solves Brezis' first open problem on ball solutions.

Classical knot recognition problem solved in NP with exponential time algorithm.

Ranking problems, also known as preference learning problems, define a widely spread class of statistical learning problems with many applications, including fraud detection, document ranking, medicine, credit risk screening, image ranking or media memorability. In this article, we systematically review different types…

Paper solves four problems of pseudo-circle envelopes in Minkowski plane.

In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intra…

Conference compiles problems on foliations and diffeomorphisms.

Solves double coset problem for braid group H_n.