Analysis and Applications

In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in $C\left(\right.0,T;L^q(\mathrm{\Omega };L^q({ℝ}^N))\left)\right.$ under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms.

We consider the following fractional $p$-Laplacian logarithmic Schrödinger equation:

Various powerful deep neural network architectures have made great contributions to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets are the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at studying the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix–vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notion of activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. We also conduct experiments on benchmark machine learning data to illustrate the potential usefulness of the results.

We develop Taylor type series expansions for entire functions of order zero using general interpolation sequences that diverge to infinity sufficiently fast. We also derive two-point and multi-point Lidstone type series expansions for entire functions of order zero. The coefficients of these Taylor and Lidstone type series are expressed in terms of values of a certain integral operator. Our results imply as special cases several recent results on $q$-Taylor and $q$-Lidstone expansions of order zero entire functions, and provide a general technique for deriving such series.

In this paper, we investigate the various aspects of normal hyperbolic mean curvature flow by LeFloch and Smoczyk. It is remarkable that the equation admits the null condition in three-dimensional case and only satisfies the first null condition when $n=2$. Based on the interesting findings, we can obtain the results of global existence of smooth solutions, as well as the stability of hyperplanes under this flow when $2\le n\le 7$, which relates to the famous Bernstein theorem. Some explicit solutions for this flow have been also derived. It should be emphasized that the null structures of this hyperbolic mean curvature flow have not been found before.