Local minimizers and Gamma-convergence for nonlocal perimeters in Carnot groups

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $\Gamma$-liminf of the rescaled energy in terms of the horizontal perimeter.

Necessary and sufficient conditions for the strong local minimality of C1 extremals on a class of non-smooth domains

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for C1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky and Mengesha (2009), where smoothness assumptions are made on the boundary.

Boosting sparsity-induced autoencoder: A novel sparse feature ensemble learning for image classification

As a kind of unsupervised learning model, the autoencoder is usually adopted to perform the pretraining to obtain the optimal initial value of parameter space, so as to avoid the local minimality that the nonconvex problem may fall into and gradient vanishment of the process of back propagation. However, the autoencoder and its variants have not taken the statistical characteristics and domain knowledge of the train set and also lost plenty of essential representaions learned from different levels when it comes to image processing and computer vision. In this article, we firstly add a sparsity-induced layer into the autoencoder to exploit and extract more representative and essential features exist in the input and then combining the ensemble learning mechanism, we propose a novel sparse feature ensemble learning method, named Boosting sparsity-induced autoencoder, which could make full use of hierarchical and diverse features, increase the accuracy and the stability of a single model. The classification results on different data sets illustrated the effectiveness of our proposed method.

Local minimality of the ball for the Gaussian perimeter

We prove that balls centered at the origin and with small radius are stable local minimizers of the Gaussian perimeter among all symmetric sets. Precisely, using the second variation of the Gaussian perimeter, we show that if the radius is smaller than{\sqrt{n+1}}, then the ball is a local minimizer, while if it is larger, the ball is not a local minimizer.

Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity

The main focus of this paper is to develop a sufficiency criterion for optimality in nonlinear optimal control problems defined on time scales. In particular, it is shown that the coercivity of the second variation together with the controllability of the linearized dynamic system are sufficient for the weak local minimality. The method employed is based on a direct approach using the structure of this optimal control problem. The second aim pertains to the sensitivity analysis for parametric control problems defined on time scales with separately varying state endpoints. Assuming a slight strengthening of the sufficiency criterion at a base value of the parameter, the perturbed problem is shown to have a weak local minimum and the corresponding multipliers are shown to be continuously differentiable with respect to the parameter. A link is established between (i) a modification of the shooting method for solving the associated boundary value problem, and (ii) the sufficient conditions involving the coercivity of the accessory problem, as opposed to the Riccati equation, which is also used for this task. This link is new even for the continuous time setting.

Characterizing Lie groups by controlling their zero-dimensional subgroups

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows:(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.(ii) An abelian topological groupGis a Lie group if and only ifGis locally minimal, locally precompact and all closed metric zero-dimensional subgroups ofGare discrete.(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional

This paper is the first part of an ongoing project aimed at providing a local minimality criterion, based on a second variation approach, for the triple point configurations of the Mumford-Shah functional.