相关论文: A variational theory for point defects in patterns

Large language models with a huge number of parameters, when trained on near internet-sized number of tokens, have been empirically shown to obey neural scaling laws: specifically, their performance behaves predictably as a power law in…

We study a variational model in nonlinear elasticity allowing for cavitation which penalizes both the volume and the perimeter of the cavities. Specifically, we investigate the approximation (in the sense of {\Gamma}-convergence) of the…

A new energy functional for pure traction problems in elasticity has been deduced in [23] as the variational limit of nonlinear elastic energy functional for a material body subject to an equilibrated force field: a sort of Gamma limit with…

We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of…

We present a scaling technique which transforms the evolution problem for a nonlinear wave equation with small initial data to a linear wave equation with a distributional source. The exact solution of the latter uniformly approximates the…

Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator…

We provide theoretical convergence bounds for the variational Monte Carlo (VMC) method as applied to optimize neural network wave functions for the electronic structure problem. We study both the energy minimization phase and the supervised…

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands…

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising--like) models in equilibrium statistical mechanics, improving on the mean--field approximation and the Bethe--Peierls approximation,…

This study seeks a better comprehension of anomalies by exploring (n+1)-point perturbative amplitudes in a 2n-dimensional framework. The involved structures combine axial and vector vertices into odd tensors. This configuration enables…

A nonparametric variant of the Kiefer--Weiss problem is proposed and investigated. In analogy to the classical Kiefer--Weiss problem, the objective is to minimize the maximum expected sample size of a sequential test. However, instead of…

Deep neural networks exhibit empirical neural scaling laws, with error decreasing as a power law with increasing model or data size, across a wide variety of architectures, tasks, and datasets. This universality suggests that scaling laws…

The fluctuation scaling law has universally been observed in a wide variety of phenomena. For counting processes describing the number of events occurred during time intervals, it is expressed as a power function relationship between the…

In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element…

We reconsider the geometrically nonlinear Cosserat model for a uniformly convex elastic energy and write the equilibrium problem as a minimization problem. Applying the direct methods of the calculus of variations we show the existence of…

In this paper, we establish an $\varepsilon$-regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence…

We discuss the universal scaling laws of order parameter fluctuations in any system in which the second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with the…

The recent work, Nemoto and Sasa [Phys. Rev. E, 83: 030105(R) (2011)], has shown that large deviations of the current characterizing a nonequilibrium system are obtained by observing the typical current for a modified system specified by a…

We develop a variationally consistent mesoscopic extension of Cosserat elasticity motivated by the breakdown of compatibility in classical formulations. By admitting compatibility-breaking perturbations, the classical theory ceases to…

We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $\Delta u=\nabla W(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, with $W:\mathbb{R}^m\to \mathbb{R}$,…