Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.
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48 results for “generalized solutions”
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
The paper solves a specific type of curvature problem on curved surfaces.
problem Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds.
method Defined a new conformal area functional and generalized solution theory to prove existence.
result Existence of generalized solutions under certain conditions, including smooth solutions for mean convex domains.
Second-order generalized Monge--Ampère equations on a plane and its geometric singular solutionsmath.DG
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
We generalize here our general procedure for constructing constant curvature maps of 2-spheres into Grassmannian manifolds G(m,n) this time concentrating our attention on maps which are non-holomorphic. We present some expressions describing these solutions in the general case and discuss how to use these results to co…
A generalized geometric method is developed for constructing exact solutions of gravitational field equations in Einstein theory and generalizations. First, we apply the formalism of nonholonomic frame deformations (formally considered for nonholonomic manifolds and Finsler spaces) when the gravitational field equation…
Proves existence and uniqueness of weak solutions for specific equations.
problem Existence and uniqueness of solutions for generalized Monge-Ampère and deformed Hermitian-Yang-Mills equations.
method Combines viscosity-theoretic and pluripotential-theoretic techniques.
result Existence and uniqueness of weak solutions in boundary cases.
Fundamental solutions found for p-Laplace equations in Heisenberg and Grushin spaces.
problem Finding solutions to p-Laplace equations with drift terms in specific geometric spaces.
method Analyzing fundamental solutions in the Heisenberg group and Grushin-type planes.
result Natural generalizations of Beals, Gaveau, and Greiner's solutions for the Laplace equation with drift term.
New proof shows solutions to Lichnerowicz equation exist.
problem Existence of solutions to the Lichnerowicz equation in general relativity.
method A new proof approach.
result Existence and uniqueness of solutions proven.
Flat solutions don't guarantee generalization for logistic loss in neural networks.
problem Proving flat solutions imply generalization for logistic loss in neural networks.
method Analyzing overparameterized two-layer ReLU networks with univariate input under logistic loss.
result Flat solutions enjoy near-optimal generalization bounds within uncertain sets but can still overfit at infinity.
The Generalization Error of the Minimum-norm Solutions for Over-parameterized Neural Networksstat.ML
Minimum-norm solutions generalize well in over-parametrized neural networks.
problem Generalization error in over-parametrized neural networks.
method Analyzing three models: random feature model, two-layer neural network, and residual network.
result Generalization error for minimum-norm solutions is comparable to Monte Carlo rate, up to logarithmic terms.
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…
The paper proves uniqueness of a solution in general relativity.
problem Uniqueness of solutions in the conformal method for Einstein's constraint equations.
method Analyzes solutions with arbitrary mean curvature and volume constraint.
result The Holst-Nagy-Tsogtgerel--Maxwell solution is unique for volumes below a certain threshold.
Extends rough Heston model solution to general λ.
problem Improving the rough Heston model for various λ values.
method Generalized rational approximation for Mittag-Leffler kernel.
result Convergence of the solution for general λ.
The paper constructs solutions to a critical Dirac equation on spheres.
problem Solving the critical Dirac equation on spheres with singularities.
method Constructing Delaunay-type solutions and another kind of singular solutions.
result The constructed solutions are building blocks for singular solutions on Spin manifolds.
Develops methods for constructing exact, non-stationary solutions to Euler equations.
problem Constructing exact, non-stationary solutions to the incompressible Euler equations.
method Arnold's geometric framework with a generalized Coriolis force.
result Explicit, smooth, global-in-time solutions on curved surfaces and three-dimensional manifolds.
Continuous Parallel Relaxation for Finding Diverse Solutions in Combinatorial Optimization Problemsstat.ML
CPRA efficiently finds diverse solutions in CO problems using UL and parallelization.
problem Finding optimal solutions often requires diverse outcomes in real-world applications.
method CPRA, an UL-based framework, discovers shared representations to generate diverse solutions.
result CPRA outperforms existing UL-based solvers in generating diverse solutions.
We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions whi…
Study on generalization in gradient-based meta-learning, showing flatter solutions and coherence between adaptation trajectories.
problem Understanding generalization in gradient-based meta-learning.
method Analysis of objective landscapes, experimental demonstration of solution properties, and empirical evidence on coherence between adaptation trajectories.
result Meta-test solutions become flatter, lower in loss, and further away from the meta-train solution as meta-training progresses, even as generalization starts to degrade.
Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
In this paper, we study existence, regularity, classification, and asymptotical behaviors of solutions of some Monge-Ampère equations with isolated and line singularities. We classify all solutions of in with one puncture point. This can be applied to characterize ellipsoids, in the same spir…
New dual formulation reduces generalization error for ERM-fDR.
problem Generalization error in constrained optimization problems.
method Introduces a dual formulation of ERM-fDR using Legendre-Fenchel transform and implicit function theorem.
result Explicit characterizations of generalization error for algorithms under mild conditions.
ICON learns differential equation operators from examples, revealing probabilistic inference.
problem Learning operators for differential equations from limited examples.
method Probabilistic operator learning using ICON architectures trained on diverse datasets.
result ICON implicitly performs Bayesian inference on solution operators.
In this paper, we bring in General Landau-Lifshitz-Bloch equation and prove that it admits a local strong solution.
Maps dBKP solutions to MS system solutions, defining Einstein-Weyl structures.
problem Constructing solutions and structures for dBKP and MS systems.
method Map construction and spectral characterisation of reductions.
result Defines Einstein-Weyl structures for dBKP and BMS systems.
Ancient solutions of heat equation with exponential growth are analytic in time.
problem Analyticity of solutions to the heat equation in time.
method Proving analyticity for ancient solutions with exponential growth.
result Ancient solutions with exponential growth are analytic in time.
Study on modified Ricci flow equations for bundles, leading to rigidity and classification.
problem Understanding structure of solutions to generalized Ricci flow equations.
method Derived modified Perelman-type monotonicity formulas for solutions on principal bundles.
result Rigidity and classification results for nonsingular solutions.
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.
In a number of physically important cases, the nonholonomically (nonintegrable) constrained Ricci flows can be modelled by exact solutions of Einstein equations with nonhomogeneous (anisotropic) cosmological constants. We develop two geometric methods for constructing such solutions: The first approach applies the form…
This paper presents a new method for solving systems with polynomial stiffness.
problem Finding analytical solutions to nonlinear differential equations with polynomial stiffness is challenging.
method The paper introduces a geometric/algebraic method using generating series and shuffle product.
result The method provides a recursive schematic that can be automated and applied to systems with polynomial stiffness.
We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…
We present a 1-parameter family of finite action solutions to the Hitchin's equations and explore some of its basic properties. For a fixed value of the parameter, the solution is smooth. We conclude by showing a multi-particle generalization of our basic solutions.
Neural networks learn modular arithmetic but not all, extending known solutions to generalize.
problem Neural networks struggle with modular arithmetic, especially for polynomials.
method Developed analytical solutions for MLP networks to learn modular addition and multiplication, then combined these solutions to generalize on arbitrary modular polynomials.
result Neural networks can learn and generalize solutions to modular polynomials, supporting the hypothesis that some polynomials are learnable.
We analyze a generalized version of the Black-Scholes equation depending on a parameter . It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case . We show that the generalized equation is exactly solvable in terms of Hermite polynomials a…
Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We give a general criterion to construct non-holomorphic…
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.
We find the fundamental solution to the p-Laplace equation in a class of Hörmander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space…
Two new ridge solutions improve BLS on added nodes, achieving better accuracy.
problem Improving the Broad Learning System (BLS) for new nodes.
method Proposed two ridge solutions for BLS output weights, updating efficiently.
result Proposed ridge solutions achieve better testing accuracy than original BLS.
The paper provides estimates for positive solutions to a nonlinear equation under geometric flow.
problem Analyzing positive solutions to a nonlinear equation under geometric flow.
method Gradient estimates for positive solutions under geometric flow on manifolds.
result Gradient estimates for positive solutions to a nonlinear equation under geometric flow.
Ancient pancake solutions found for curvature flows.
problem Finding unique ancient solutions to curvature flows.
method Constructing and analyzing -invariant ancient solutions.
result Unique -invariant ancient solutions found.
Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of to construct solutions of .
result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.
Paper addresses statistical inference for GANs and minimax problems.
problem Statistical properties of GANs and minimax problems.
method Consistent estimation and confidence sets for GAN parameters.
result Confidence sets for GAN parameters contain the population solutions with desired coverage probability.
This study proposes an approach based on a perturbation technique to construct global solutions to dynamic stochastic general equilibrium models (DSGE). The main idea is to expand a solution in a series of powers of a small parameter scaling the uncertainty in the economy around a solution to the deterministic model, i…
Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
Study estimates for -Hessian equations on closed manifolds.
problem Estimating solutions to -Hessian equations on closed Riemannian manifolds.
method Introducing pseudo-solutions to generalize -subsolution and proving and estimates.
result Proves estimates for general -Hessian equations on closed manifolds under sharp conditions.
Given a collection of N solutions of the (3+1) vacuum Einstein constraint equations which are asymptotically Euclidean, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier…