Proves local strong solutions for General Landau-Lifshitz-Bloch equation.
problem Proving existence of solutions for complex equations.
method Analytical proof of local strong solutions.
result Local strong solutions exist for General Landau-Lifshitz-Bloch equation.
Local solutions found for nonautonomous Schrödinger flows on Kähler manifolds.
problem Existence and uniqueness of solutions for nonautonomous Schrödinger flows.
method Proved the existence of local solutions under certain conditions.
result Existence and uniqueness of solutions with higher regularity.
Study bifurcations and local rigidity on flag manifolds for Yamabe solutions.
problem Yamabe problem on maximal flag manifolds
method Determine bifurcation and local rigidity points for 1-parameter families of solutions
result Identify specific points of bifurcation and local rigidity
Study on Laplacian flow and coflow of Locally Conformal Parallel G2-structures.
problem Analyzing the Laplacian flow and coflow of Locally Conformal Parallel G2-structures.
method Examined the Laplacian flow and coflow of Locally Conformal Parallel G2-structures on specific Lie groups.
result Found long time solutions of the Laplacian flow and coflow, including ancient Laplacian solitons and Einstein metrics.
Study local perturbations of vector bundles with polynomial curvature solutions.
problem Existence and stability of solutions to geometric PDEs under deformations.
method Geometric invariant theory, moment map framework, polystability conditions.
result Existence and uniqueness of solutions under local polystability conditions.
Unique solutions found for diffusive martingale problems.
problem Finding unique solutions to Cauchy problems for diffusive real-valued strict local martingales.
method Provided sets of smooth functions under local Hölder and Engelbert-Schmidt conditions for unique classical and weak solutions.
result Unique solutions found for specific martingale models.
The paper constructs local solutions concentrating near singular points of spinors.
problem Constructing solutions near singular points of spinors.
method Constructs local solutions parameterized by ε, concentrating near singular points.
result Local solutions concentrate in tubular neighborhoods of singular points, converging to original spinors after renormalization.
Solutions to a specific problem are shown to be locally Lipschitz but not differentiable.
problem Locally Lipschitz viscosity solutions to the σk-Loewner-Nirenberg problem on annuli. method Analytical proof of regularity and non-differentiability.
result Solutions are $C^{1, rac{1}{k}}_{
m loc}$ in each of the annulus regions and have a jump in radial derivative.
Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.
problem Yamabe problem on Aloff-Wallach Spaces
method Constructing 1-parameter families of solutions and examining changes in the Morse index as the parameter varies.
result Identifies bifurcation and local rigidity points for homogeneous solutions to the Yamabe problem.
For any locally defined G2−structure and Hermitian-Yang-Mills connection on S6, the G2−monopole equation always admits a local solution that is asymptotic to the HYM-connection.
This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.
In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point p in a spacetime N, we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving …
The paper tackles hierarchical reinforcement learning by approximating optimal solutions for the Traveling Salesman Problem.
problem Approximating optimal solutions for the Traveling Salesman Problem using hierarchical reinforcement learning.
method Mapping the problem into a Reward Discounted Traveling Salesman Problem and deriving approximate solutions using local policies.
result Three stochastic policies are proposed that guarantee better performance than any deterministic policy.
Modified K-means ensures local optimality with same complexity.
problem Lack of rigorous analysis on local optimality guarantees of K-means.
method Proposed modifications to K-means ensuring local optimality.
result Proposed methods provide improved locally optimal solutions.
New method uses Gaussian processes to find local minima efficiently.
problem Finding all local minima of a black-box function with unknown derivatives.
method Sequentially selects input points to update GP derivatives' confidence intervals.
result Theoretical analysis and numerical experiments show the method's effectiveness.
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
Proves local noncollapsing estimate for mean curvature flow.
problem Ensuring noncollapsing in mean curvature flow.
method Combining local estimate with earlier work on ancient solutions.
result Ancient convex solutions that sweep out entire space are noncollapsed.
In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting …
Fractional porous media equations yield q-Gaussian solutions for stock price returns.
problem Modeling stock price returns using fractional porous media equations.
method Analyzed three types of fractional extensions of the porous media equation.
result Local and non-local fractional extensions fit S&P 500 data better than classical models.
We present local estimates for solutions to the Ricci flow, without the assumption that the solution has bounded curvature. These estimates lead to a generalisation of one of the pseudolocality results of G.Perelman in dimension two.
The study finds static solutions in symplectic curvature flow in 4D.
problem Finding static solutions in symplectic curvature flow in 4D.
method Derived a local normal form for static solutions and used Cartan-Kahler theorem for solitons.
result Every complete static solution to symplectic curvature flow in 4D is Kahler-Einstein.
Study bounds derivatives of solutions to a specific equation on domains.
problem Bounding second derivatives of solutions to the σk-Yamabe equation. method Proves local pointwise second derivative estimates for positive W2,p solutions. result Establishes bounds for derivatives of solutions to the σk-Yamabe equation. Ancient solutions of Ricci flow with Type I growth are classified.
problem Understanding ancient solutions of Ricci flow with specific curvature growth.
method Analyzing ancient solutions with Type I curvature growth in arbitrary dimensions.
result Ancient solutions with Type I growth are classified into specific types.
Study finds non-uniqueness in sphere metrics with constant fractional curvature.
problem Non-uniqueness of metrics with constant positive fractional curvature on spheres.
method Bifurcation techniques applied to non-local equations with critical non-linearity.
result Non-uniqueness results for complete metrics on Sn∖Sk. Geometric perspective on unique solution in matrix completion with a deterministic pattern.
problem Identifying unique solutions in matrix completion with a specific pattern of observed entries.
method Geometric and algebraic analysis, focusing on the well-posedness condition and local stability.
result A sufficient condition for local uniqueness of matrix completion solutions, called the well-posedness condition.
We introduce the localized Lasso, which is suited for learning models that are both interpretable and have a high predictive power in problems with high dimensionality d and small sample size n. More specifically, we consider a function defined by local sparse models, one at each data point. We introduce sample-wis…
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold …
Important models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G-invariant solutions on principal bundles, where G is a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain R^{N}…
Proves existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.
problem Proving existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.
method Analyzes the equation on Hilbert manifold, proving existence and uniqueness of solutions.
result Demonstrates that solutions are in the Hilbert manifold and are gradient flows.
New boundary condition for Black-Scholes equations in strict local martingale models.
problem Computing prices of European options with underlying asset as a strict local martingale.
method Numerical procedure using finite difference methods with a new boundary condition at infinity.
result The minimal solution, satisfying a discrete maximum principle, is the correct derivative price.
Graph Neural Networks and Guided Local Search improve TSP solutions.
problem Finding optimal solutions to the Traveling Salesperson Problem quickly.
method Hybrid approach combining Graph Neural Networks and Guided Local Search.
result Significant reduction in optimality gap for TSP solutions.
Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a st…
In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower boun…
We compute approximate solutions to L0 regularized linear regression using L1 regularization, also known as the Lasso, as an initialization step. Our algorithm, the Lass-0 ("Lass-zero"), uses a computationally efficient stepwise search to determine a locally optimal L0 solution given any L1 regularization solution. We …
Algorithm learns CNF formulas from random solutions under specific conditions.
problem Learning a CNF formula from uniform random solutions.
method Revisits Valiant's algorithm and applies Lovász local lemma conditions.
result Significantly reduces sample complexity for learning CNFs.
We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.
Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifo…
Assuming local uniform bounds on the metric for a solution of the Chern-Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.
We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non homothetic solutions of the Yamabe problem. Using local rigi…
We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to efficiently sample solutions of random Constraint Satisfaction Problems (CSPs). First, we extend a recent result that, using a large-deviation analysis, shows that the geometry of the space of solutions of the Binary Perceptron Learning Problem (a proto…
Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.
problem Analyzing the behavior of Ricci flow on finite graphs.
method Local existence and uniqueness proof for solutions of the Bakry-Émery Ricci flow.
result Local existence and uniqueness of solutions to the Ricci flow on finite graphs.
In this paper, we study global existence and blow up properties to Lp norm preserving non-local heat flows. We first study two kinds of Lp norm preserving non-local flows and prove that these flows have the global solutions. Finally, we give a example to show that one kind of this heat flow may blow up in $L^{\in…
New solutions found with negative mass in general relativity.
problem Finding metrics with negative mass in general relativity.
method Constructing families of metrics with specific properties.
result Obtained new classes of solutions with negative mass.
Local search improves GFlowNets' ability to generate high-reward samples.
problem GFlowNets struggle with over-exploration in high-reward space.
method Local search focusing on high-reward samples via backtracking and reconstruction.
result Significant performance improvement in biochemical tasks.
This paper improves GANs by ensuring local equilibria are also Nash equilibria, leading to better solutions.
problem Current GAN training methods often converge to local Nash equilibria, which may not be optimal.
method Formalizes GANs as finite games in mixed strategies, ensuring every local equilibrium is a Nash equilibrium.
result The proposed method converges to a resource-bounded Nash equilibrium, producing solutions closer to theoretical predictions.
We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.
New insights into spurious local minima in k-means clustering.
problem Understanding and mitigating spurious local minima in k-means clustering.
method Investigating spurious local minima under a probabilistic generative model.
result Proven structures of spurious local minima for k-means clustering.
New algorithm reduces communication in distributed eigenspace estimation.
problem Efficiently estimating eigenspaces in distributed settings without excessive communication.
method Communication-efficient distributed algorithm using Procrustean alignment.
result Achieves similar error rate to centralized estimator for PCA.