A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients

In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). These numerical simulations indicate that DNNs seem to possess the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the dimension $d \in \mathbb{N}$ of the function which the DNN aims to approximate in such computational problems. There is also a large number of rigorous mathematical approximation results for artificial neural networks in the scientific literature but there are only a few special situations where results in the literature can rigorously justify the success of DNNs in high-dimensional function approximation. The key contribution of this paper is to reveal that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. We prove that the number of parameters used to describe the employed DNN grows at most polynomially in both the PDE dimension $d \in \mathbb{N}$ and the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$. A crucial ingredient in our proof is the fact that the artificial neural network used to approximate the solution of the PDE is indeed a deep artificial neural network with a large number of hidden layers.Comment: 48 page

Akt-dependent Pp2a activity is required for epidermal barrier formation during late embryonic development

Acquisition of epidermal barrier function occurs late in mouse gestation. Several days before birth a wave of barrier acquisition sweeps across murine fetal skin, converging on dorsal and ventral midlines. We investigated the molecular pathways active during epidermal barrier formation. Akt signaling increased as the barrier wave crossed epidermis and Jun was transiently dephosphorylated. Inhibitor experiments on embryonic explants showed that the dephosphorylation of Jun was dependent on both Akt and protein phosphatase 2A (Pp2a). Inhibition of Pp2a and Akt signaling also caused defects in epidermal barrier formation. These data are compatible with a model for developmental barrier acquisition mediated by Pp2a regulation of Jun dephosphorylation, downstream of Akt signaling. Support for this model was provided by siRNA-mediated knockdown of Ppp2r2a (Pr55α or B55α), a regulatory subunit of Pp2a expressed in an Akt-dependent manner in epidermis during barrier formation. Ppp2r2a reduction caused significant increase in Jun phosphorylation and interfered with the acquisition of barrier function, with barrier acquisition being restored by inhibition of Jun phosphorylation. Our data provide strong evidence that Ppp2r2a is a regulatory subunit of Pp2a that targets this phosphatase to Jun, and that Pp2a action is necessary for barrier formation. We therefore describe a novel Akt-dependent Pp2a activity that acts at least partly through Jun to affect initial barrier formation during late embryonic epidermal development

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are $n$-times continuously Fr\'{e}chet differentiable, then the solutions of the considered SEEs are also $n$-times continuously Fr\'{e}chet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes

Concepciones de estudiantes y profesores acerca de la energía de las ondas

El propósito de este trabajo es examinar algunas dificultades que tienen estudiantes y profesores para interpretar y describir los mecanismos físicos asociados con la generación y propagación de una onda y la energía involucrada en estos procesos. Se analizan resultados de encuestas y entrevistas realizadas a docentes y alumnos de un curso introductorio de física de nivel universitario y se los compara con enunciados de libros de texto que son, o bien claramente incorrectos, o bien ambiguos y más o menos compatibles con los pensamientos de los estudiantes. Se consideran posibles causas de la difusión de estas ideas entre los que participan en el proceso de enseñanza-aprendizaje de las ciencias, y se esbozan algunas pautas para lograr un aprendizaje significativo de la temática analizada.The purpose of this paper is to examine some difficulties students and professors have to interpret and describe the physical mechanisms associated with the generation and propagation of a wave and the energy involved in these processes. Results of surveys and interviews carried out to teachers and students of an introductory Physics course of university level are analyzed and compared with text-book statements that are, either clearly incorrect, or ambiguous and more or less compatible with the thoughts of the students. They are also considered the possible causes of the diffusion of these ideas among the actors participating in the process of teaching-learning of the sciences and rules are given to reach a significant learning of the analyzed subject matter

Information Rates and post-FEC BER Prediction in Optical Fiber Communications

Information-theoretic metrics to analyze optical fiber communications systems with binary and nonbinary soft-decision FEC are reviewed. The numerical evaluation of these metrics in both simulations and experiments is also discussed. Ready-to-use closed-form approximations are presented.Comment: Invited paper, OFC 201

Regulation of dolichol-linked glycosylation

In the majority of congenital disorders of glycosylation, the assembly of the glycan precursor GlcNAc2Man9Glc3 on the polyprenol carrier dolichyl-pyrophosphate is compromised. Because N-linked glycosylation is essential to life, most types of congenital disorders of glycosylation represent partial losses of enzymatic activity. Consequently, increased availability of substrates along the glycosylation pathway can be beneficial to increase product formation by the compromised enzymes. Recently, we showed that increased dolichol availability and improved N-linked glycosylation can be achieved by inhibition of squalene biosynthesis. This review summarizes the current knowledge on the biosynthesis of dolichol-linked glycans with respect to deficiencies in N-linked glycosylation. Additionally, perspectives on therapeutic treatments targeting dolichol and dolichol-linked glycan biosynthesis are examine