The paper examines how stable solutions of elliptic problems affect the geometry of manifolds.
problem Characterizing manifolds with stable solutions of elliptic problems.
method Analyzing geometric rigidity under stability conditions and non-negative Ricci curvature.
result Characterization and splitting results for manifolds with stable solutions.
Studied stable solutions for symmetric systems with hypoelliptic operators.
problem Analyzing stable solutions for symmetric systems with hypoelliptic operators.
method Examined stable solutions for symmetric systems with hypoelliptic operators.
result Identified stable solutions for symmetric systems with hypoelliptic operators.
Stable solutions to Yang-Mills-Higgs equations on spheres and tori identified.
problem Stable solutions to abelian Yang-Mills-Higgs equations on S2 and T2. method Reduction to vortex equations and application of Bourguignon-Lawson's method for stable SU(2) Yang-Mills connections. result Stable solutions to abelian Yang-Mills-Higgs equations on S2 and T2 are identified as satisfying vortex equations. Stable solutions to a specific equation are one-dimensional.
problem Stability and dimensionality of solutions to the Allen-Cahn equation.
method Analysis of stable solutions with bounded energy density.
result Stable solutions to the Allen-Cahn equation are one-dimensional.
Paper proves convex domains have one maximum for semi-stable solutions.
problem Analyzing critical points of semi-stable solutions on convex domains.
method Relating critical points to an auxiliary function and using topological degree.
result Positive, semi-stable solutions have exactly one non-degenerate critical point.
Stable blowup solutions found for supercritical Yang-Mills equations.
problem Understanding blowup solutions for supercritical Yang-Mills equations.
method Investigated equivariant self-similar blowup solutions and their stability.
result Stability of blowup mechanism for odd dimensions greater than or equal to 5.
Stable solutions found for a specific physics model.
problem Stability of solutions to the U(1)-Yang-Mills-Higgs model. method Gluing method and detailed analysis of linearized operators.
result Found a family of stable critical points in higher dimensions.
New LP method recovers MAP solution from noisy stable instances.
problem MAP inference on noisy stable instances.
method Designing an algorithm to find nearby perturbation stable instances and using LP relaxation.
result LP approximately recovers the MAP solution from noisy stable instances.
Paper improves estimates for solutions to a specific equation.
problem Establishing uniform C2,θ estimates for stable solutions. method Combines infinite dimensional reduction and small regularity theorems.
result Uniform C2,θ estimates for stable solutions in dimensions ≤10. Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
Strictly stable Allen-Cahn hypersurfaces have multiplicity one.
problem Understanding the multiplicity of stable hypersurfaces in Allen-Cahn equations.
method Analyzing strictly stable components without variational assumptions.
result Strictly stable components occur with multiplicity one.
We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold M without boundary, \begin{equation*} -Δ_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when Δg stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb…
The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.
problem Symmetry phenomena in solutions of semilinear PDEs on Riemannian domains.
method General framework for formulating the symmetry problem; evidence from stable solutions; consideration of manifolds with density.
result Evidence that the framework is natural, with results for stable solutions.
New potential theory on minimal hypersurfaces shows stable growth of solutions near singularities.
problem Analyzing potential theory on minimal hypersurfaces.
method Introducing Hardy structures to study classical operators and showing stable growth of solutions.
result Minimal growth of positive solutions of Lw = 0 is stable and persists under perturbations or blow-ups.
Unique solution found for Demailly's equation on stable bundles.
problem Existence of a Griffiths positively curved metric on Hartshorne ample vector bundles.
method Proved an essentially unique solution to a Hermitian-Einstein-type equation for stable bundles.
result The proposed approach by Demailly must be modified to tackle the conjecture.
The paper tackles stable maxima optimization for expensive functions.
problem Finding stable maxima of expensive functions with input variations.
method Uses multiple gradient Gaussian Process models to estimate stability and guide optimization.
result Demonstrates effective finding of stable maxima on synthetic and real-world problems.
We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the eq…
This study presents a method for constructing a sequence of approximate solutions of increasing accuracy to general equilibrium models on nonlocal domains. The method is based on a technique originated from dynamical systems theory. The approximate solutions are constructed employing the Contraction Mapping Theorem and…
We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces Σn⊂Rn+1 that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We sho…
Following similar results in arXiv:1301.5934 for flat tori and round spheres, in this paper is presented a proof of the fact that, for "arbitrary" initial conditions f0, the solution ft at time t of the heat equation on real or complex projective spaces eventually becomes (and remains) a minimal Morse function.…
In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation R imposed on smooth maps of manifolds determines cohomology theories k∗ and h∗; the cohomology theory k∗ describes invariants of solutions of R, whil…
Self-similar solutions to geometric flows are stable under small perturbations.
problem Stability of self-similar solutions in geometric flows.
method Global analytic solutions, compactness arguments, spatial equi-decay properties, and estimates of linearized operator.
result Perturbed solutions are asymptotically self-similar as time tends to infinity.
Paper fills in technical details for Hitchin's self-duality equations proof.
problem Existence of solutions to Hitchin's self-duality equations.
method Reduction to minimizing energy functional, Coulomb gauge construction.
result Smooth solution constructed using unitary gauge transformation.
We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If (Mm,g) is a closed manifold of constant positive scalar curvature, which we normalize to be m(m−1), we consider the Riemannian product with the n-dimensional Euclidean space: $(M^m …
Study shows almost all Arnold stable solutions have no conjugate points.
problem Existence of conjugate points in Arnold stable solutions.
method Analysis of Misiołek curvature for Arnold stable solutions.
result Almost all Misiołek curvature is nonpositive for Arnold stable solutions.
Constructs a stable finite-time blowup solution for a specific harmonic map heat flow problem.
problem Energy-supercritical harmonic map heat flow with 1-corotational symmetry in 7 dimensions.
method Constructs a stable finite time blowup solution under corotational symmetry.
result Constructs a stable finite time blowup solution with concentration of the universal profile.
Researchers find stable solutions for heat map flow in higher dimensions.
problem Stability of shrinkers for harmonic map heat flow in higher dimensions.
method Construction of specific target manifolds allowing for stable shrinkers.
result Existence of corotational self-similar shrinkers representing stable blowup mechanisms.
Stable saddle solutions found for specific dimensions of the Allen-Cahn equation.
problem Stability of saddle solutions for the Allen-Cahn equation in specific dimensions.
method Analyzing the Simons cone and energy functional to confirm saddle solutions' stability.
result Stable saddle solutions found for m=4,5,6. New solutions found for G2 system using K3 orbifolds.
problem Finding smooth solutions to the G2 Hull-Strominger system. method Torus fibrations over K3 orbifolds, adapted Serre construction for singular settings.
result Constructed new smooth solutions to the G2 Hull-Strominger system. The paper establishes a correspondence between solutions of extended Bogomolny equations and Higgs bundles.
problem Solutions of the extended Bogomolny equations on $Σ imes \RP$ with specific singularities.
method Develops a Kobayashi-Hitchin type correspondence and partial correspondence for solutions with Nahm pole and knot singularities.
result Verifies a conjecture and proves existence and uniqueness of solutions with knot singularities.
Given an irreducible unitary representation of a cocompact lattice of SL(2,C), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.
Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.
problem Constructing Hermitian-Einstein metrics on stable holomorphic vector bundles
method Dynamical construction
result Provided a dynamical construction of Hermitian-Einstein metrics on stable holomorphic vector bundles
In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…
Study on non-negative solutions for stochastic Volterra equations with jumps.
problem Existence and uniqueness of non-negative solutions for stochastic Volterra equations with jumps and non-Lipschitz coefficients.
method Developed a nonnegative approximation approach and used Yamada--Watanabe approximation technique for convergence proof.
result Established conditions for strong existence and pathwise uniqueness of non-negative solutions.
Static spacetimes are stable attractors in a flow equation.
problem Stability of static spacetimes with negative cosmological constant.
method New expander entropy for Ricci-harmonic flow.
result Static metrics are stable if and only if a positive mass theorem holds.
Stable neural flows ensure robustness and efficiency in deep learning.
problem Ensuring robustness and stability in deep learning models.
method Introducing a stable variant of neural ODEs with a neural network parametrizing an energy functional, solving as an optimal control problem with adjoint sensitivity analysis.
result The proposed model provides robustness against input perturbations and low computational burden.
We prove the existence of a (spectrally) stable self-similar blow-up solution f0 to the heat flow for corotational harmonic maps from R3 to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear s…
In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture that all solvsolitons are line…
New stable minimal surfaces generalize classical Henneberg surface.
problem Finding new stable minimal surfaces in 3D.
method Generalized Henneberg surface with infinite families of complete, non-orientable surfaces.
result Infinite families of complete, finitely branched, non-orientable, stable minimal surfaces.
Study on stability of cylindrical singularities in MCF of finite codimensions.
problem Stability of cylindrical singularities in mean curvature flow.
method Construction of stable manifold, explicit solutions, asymptotic analysis.
result Asymptotic stability of cylindrical singularities under generic perturbations.
This paper applies Thompson Sampling to asymmetric α-stable bandits for financial and wireless data.
problem Optimizing exploration-exploitation in multi-armed bandits with asymmetric α-stable distributions. method Thompson Sampling applied to unknown asymmetric α-stable reward distributions. result Demonstrates effectiveness of Thompson Sampling for asymmetric α-stable bandits. Undecidability proved for DG algebras problems.
problem Stable isomorphism, quasi-isomorphism, and Morita equivalence problems for semifree DG algebras.
method Essentially autonomous solutions by Gemini Deep Think and Aletheia.
result Proved undecidability of problems for semifree DG algebras.
We introduce a new convex formulation for stable principal component pursuit (SPCP) to decompose noisy signals into low-rank and sparse representations. For numerical solutions of our SPCP formulation, we first develop a convex variational framework and then accelerate it with quasi-Newton methods. We show, via synthet…
Finite index solutions to Bernoulli problem are always axially symmetric.
problem Entire solutions to the Bernoulli free boundary problem with finite Morse index in 3D.
method Proof of axial symmetry for finite index solutions.
result Finite index solutions to the Bernoulli problem in 3D are axially symmetric.
I-SPEC learns stable models from data without full causal knowledge.
problem Learning models that generalize well across shifts in environment.
method End-to-end framework using partial ancestral graph to learn stable interventional distribution.
result I-SPEC can learn robust models without full causal knowledge.
We formalize the construction by Batalin and Vilkovisky of a solution of the classical master equation associated with a regular function on a nonsingular affine variety (the classical action). We introduce the notion of stable equivalence of solutions and prove that a solution exists and is unique up to stable equival…
Mean curvature flow with a conical end becomes stable.
problem Stability of mean curvature flow with a conical end.
method Analysis of asymptotic behavior and entropy.
result Expanding solution is asymptotically stable.
New method improves stability of collaborative filtering.
problem Stability issues in matrix approximation for recommender systems.
method Introduces new optimization objectives and solves the optimization problem for stable matrix approximation.
result Achieves better accuracy in rating prediction and top-N recommendation tasks.