Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
problem Solving Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
method Uses interior C2 estimate. result Result is sharp, showing existence of singular solutions in subcritical phase.
Characterizes critical points in convex double and triple bubbles.
problem Critical points of double and triple bubbles in convex shapes.
method Characterization through stationary varifolds in Rn and R3. result Characterization of critical points in convex shapes.
We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.
problem Foliation of an asymptotically flat end by critical hypersurfaces.
method Constructing hypersurfaces as critical points of a functional, solving an over-determined boundary value problem.
result Solutions to the Laplace-Beltrami operator over a foliation of critical spheres.
The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.
problem Eigenvalue problem on the sphere with boundary conditions.
method Classifying positive solutions as rotationally symmetric and analyzing boundary conditions.
result Characterization of the critical catenoid as the only embedded free boundary minimal annulus.
Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.
problem Existence and rigidity of critical Z2 eigenvalues on sphere configurations.
method Algebraic identities and finite group representation theory.
result Construction of infinitely many configurations admitting critical eigensections and proof of deformation rigidity of Taubes-Wu tetrahedral eigensections.
In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy …
New actor-critic method reduces sample complexity for reinforcement learning.
problem Improving sample complexity for actor-critic algorithms in reinforcement learning.
method Integrates Monte Carlo rollouts into policy search steps for better control over bias.
result Established sample complexity for actor-critic algorithms with policy gradient.
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
Learning the distribution of natural images is one of the hardest and most important problems in machine learning. The problem remains open, because the enormous complexity of the structures in natural images spans all length scales. We break down the complexity of the problem and show that the hierarchy of structures …
A classical result due to Blaschke states that for every analytic self-map f of the open unit disk of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map f of…
Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.
problem Proving mass-capacity inequalities for critical area-normalized capacitors.
method Analyzes asymptotically flat manifolds with boundary capacity potential satisfying an overdetermined problem.
result Improves Schwarzschild metric uniqueness and results for spin asymptotically flat spacetimes.
Paper finds infinitely many solutions changing sign for critical fractional equations.
problem Critical fractional equations with sign-changing solutions.
method Reduction to equivalent problem on sphere, blow-up arguments, Pohozaev's identity, regularity results, symmetries of sphere.
result Unbounded sequence of sign-changing solutions for critical problems.
We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem of arXiv:math.DG/0402212.
Paper proves gradient estimates for Lagrangian mean curvature equation.
problem Proving gradient estimates for Lagrangian mean curvature equation.
method Interior gradient estimates for critical and supercritical Lagrangian mean curvature equation.
result Solves Dirichlet boundary value problem for critical and supercritical Lagrangian mean curvature equation.
Research characterizes critical points of scalar curvature functionals.
problem Characterizing critical points of scalar curvature functionals.
method Translation and analysis of a previous Russian paper.
result Provides insights into critical points of scalar curvature functionals.
Study on CR curves in 3-sphere, focusing on critical curves integration and existence.
problem Addressing the integration and existence of critical curves in the CR 3-sphere.
method Provided a procedure for the explicit integration of general critical curves and characterized closed curves.
result Existence of infinite countably many closed critical curves.
The paper connects reflection groups to maps with specific dynamical properties.
problem Understanding the relationship between reflection groups and anti-rational maps.
method Established a correspondence using planar graphs.
result Complete answers to geometric mating problems for anti-rational maps.
The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…
Numerically locating the critical points of non-convex surfaces is a long-standing problem central to many fields. Recently, the loss surfaces of deep neural networks have been explored to gain insight into outstanding questions in optimization, generalization, and network architecture design. However, the degree to wh…
Solves critical LYZ equation in Kähler geometry.
problem Solvability of LYZ equation at critical phase.
method Establishes existence of smooth solutions.
result Solves critical case of LYZ equation.
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the topological classificat…
New algorithm solves mean-field control problems using actor-critic learning with moment neural networks.
problem Solving mean-field control problems in continuous time reinforcement learning.
method Gradient-based policy and value function learning with moment neural networks on the Wasserstein space.
result Effective solution for diverse mean-field control problems, including multi-dimensional and nonlinear settings.
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.
CoRMF uses RNNs to solve Ising models efficiently by ordering critical edges.
problem Solving Ising models efficiently and accurately.
method Criticality-ordered spin sequence and RNNs for mean-field factorization.
result Proves tighter error bounds than naive mean-field.
Actor-critic methods can achieve incredible performance on difficult reinforcement learning problems, but they are also prone to instability. This is partly due to the interaction between the actor and critic during learning, e.g., an inaccurate step taken by one of them might adversely affect the other and destabilize…
We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…
Actor-critic methods solve reinforcement learning problems by updating a parameterized policy known as an actor in a direction that increases an estimate of the expected return known as a critic. However, existing actor-critic methods only use values or gradients of the critic to update the policy parameter. In this pa…
The critical catenoid is uniquely determined by certain symmetries of its boundary.
problem Uniqueness of free boundary minimal annuli in a half-ball.
method Symmetry analysis and boundary conditions.
result An embedded free boundary minimal annulus with specific symmetries is congruent to the critical catenoid.
Solves local minima problems on smooth manifolds.
problem Local minima issues on smooth manifolds.
method Introducing valley functions and applying Morse's lemma.
result Eliminates critical points and reduces to 1D.
New method improves off-policy critic evaluation in reinforcement learning.
problem High variance and instability in off-policy policy evaluation.
method Doubly robust estimators applied to actor-critic algorithms.
result Doubly robust estimation significantly improves performance in continuous control tasks.
Paper resolves ambiguity in non-convex bilevel optimization problems.
problem Ambiguity in bilevel optimization with non-convex lower-level objectives.
method Introduces selection maps to define critical points and resolves ambiguity.
result Validates new analytical tools in Morse theory for implicit differentiation.
The paper studies the smoothness of critical points of variational integrals on Hessian spaces.
problem The study focuses on the regularity of critical points of variational integrals defined on Hessian spaces.
method The approach involves solving a fourth order nonlinear equation and analyzing the Hessian of the critical points.
result Smooth critical points with bounded Hessian are shown to be smooth provided their Hessian has small BMO.
Proves the index of a Möbius band in 4D ball equals 5.
problem Determining the Morse index of critical non-orientable surfaces.
method Comparison theorem between Steklov spectral index and energy index.
result Proves the index of critical Möbius band in B4 equals 5. Neural networks solve high-dimensional HJB PDEs with asymptotic guarantees.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs in stochastic control theory.
method Actor-critic machine learning algorithm with a structured critic and biased gradient actor.
result The training dynamics converge to an ODE, ensuring solutions to the original problem.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Critical points of scale-invariant curvature energies in 4D are analytic.
problem Analyzing critical points of curvature energies in 4D manifolds.
method Applying Noether's theorem to identify conservation laws and lower order elliptic system of PDEs, then using integrability by compensation and interpolation theory.
result Critical points of scale-invariant curvature energies in 4D are analytic.
For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvat…
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman-Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solut…
We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of suc…
The paper discusses quantifying realism in generated images.
problem Designing functions to reliably distinguish realistic data from unrealistic data.
method Drawing on insights from algorithmic information theory, the paper introduces the notion of a universal critic.
result A good generative model alone is insufficient to solve the problem of realism quantification.
New actor-critic algorithm achieves optimal sample efficiency in RL.
problem Achieving ε-optimal policies with minimal samples in RL. method Integrates optimism, off-policy critic estimation, and rare-switching policy resets.
result Sample complexity of O(dH5log∣A∣/ε2+dH4log∣F∣/ε2) trajectories. The paper finds sign-changing solutions for a specific type of elliptic equation.
problem Existence of sign-changing solutions for a Yamabe type equation.
method Investigates a critical elliptic equation with a Yamabe type operator on a compact manifold with boundary.
result Existence of sign-changing solutions assured under certain geometric conditions.
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. The paper tackles learning energies from time-evolving critical points.
problem Learnability of energies from data of critical points.
method Formulates a variational problem and uses Gamma-convergence arguments.
result Minimal solutions from finite observations converge to the exact energy.
Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
Generative adversarial networks (GANs) are an exciting alternative to algorithms for solving density estimation problems---using data to assess how likely samples are to be drawn from the same distribution. Instead of explicitly computing these probabilities, GANs learn a generator that can match the given probabilisti…
We consider a variational problem for submanifolds Q ⊂ M with nonempty boundary ∂Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutio…