Monotonicity results for the fractional p-Laplacian in unbounded domains

In this paper, we develop a direct method of moving planes in unbounded domains for the fractional p-Laplacians, and illustrate how this new method to work for the fractional p-Laplacians. We first proved a monotonicity result for nonlinear equations involving the fractional p-Laplacian in [Formula: see text] without any decay conditions at infinity. Second, we prove De Giorgi conjecture corresponding to the fractional p-Laplacian under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional p-Laplacian along a sequence of approximate maxima; (ii) estimating the lower bound of the solutions by constructing sub-solutions.

A monotonicity result under symmetry and Morse index constraints in the plane

This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

Equality case for an elliptic area condenser inequality and a related Schwarz type lemma

We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.

On gaps in the closures of images of divisor functions

Given a complex number [Formula: see text], define the divisor function [Formula: see text] by [Formula: see text]. In this paper, we look at [Formula: see text], the topological closure of the image of [Formula: see text], when [Formula: see text]. We exhibit new lower bounds on the number of connected components of [Formula: see text], bringing this bound from linear in [Formula: see text] to exponential. We also discuss the general structure of gaps of [Formula: see text] in order to work toward a possible monotonicity result.

Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning

The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses, we introduce variational characterizations that naturally suggest a two-step scheme for their optimization, based on the iterative shift of a probability density and the calculation of a best Gaussian approximation in Kullback–Leibler divergence. Disregarding approximation error in these two steps, the variational characterizations allow us to show a simple monotonicity result for training error along optimization iterates. The two-step optimization schemes for local entropy and heat regularized loss differ only over which argument of the Kullback–Leibler divergence is used to find the best Gaussian approximation. Local entropy corresponds to minimizing over the second argument, and the solution is given by moment matching. This allows replacing traditional backpropagation calculation of gradients by sampling algorithms, opening an avenue for gradient-free, parallelizable training of neural networks. However, our presentation also acknowledges the potential increase in computational cost of naive optimization of regularized costs, thus giving a less optimistic view than existing works of the gains facilitated by loss regularization.

Monotonicity of Avoidance Coupling on KN

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler [1], we show that the maximal number of non-colliding coupled simple random walks on the complete graph KN, which take turns, moving one at a time, is monotone in N. We use this fact to couple [N/4] such walks on KN, improving the previous Ω(N/log N) lower bound of Angel et al. We also introduce a new generalization of simple avoidance coupling which we call partially ordered simple avoidance coupling, and provide a monotonicity result for this extension as well.

An eigenvalue optimization problem for the p-Laplacian

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.