On an interior Calder\'{o}n operator and a related Steklov eigenproblem for Maxwell's equations

We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calder\'{o}n operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. With an approach similar to that developed by Auchmuty for the Laplace operator, we provide natural spectral representations for the appropriate trace spaces, for the Calder\'{o}n operator itself and for the solutions of the corresponding boundary value problems subject to electric or magnetic boundary conditions on a cavity.Comment: Submitted for publication to Siam Journal on Mathematical Analysis on 21 March 2019, revised on 12 May 2020, accepted for publication on 16 July 202

Electromagnetic fields in linear and nonlinear chiral media: a time-domain analysis

We present several recent and novel results on the formulation and the analysis of the equations governing the evolution of electromagnetic fields in chiral media in the time domain. In particular, we present results concerning the well-posedness and the solvability of the problem for linear, time-dependent, and nonlocal media, andresults concerning the validity of the local approximation of the nonlocal medium (optical response approximation). The paper concludes with the study of a class of nonlinear chiral media exhibiting Kerr-like nonlinearities, for which the existence of bright and dark solitary waves is shown

On Generalized Linear Regular Delay Systems

AbstractIn this paper we study generalized linear regular delay systems. Such a system is reduced to a suitable homogeneous one. Using the Weierstrass canonical form, the latter is decomposed in two subsystems, whose solutions are obtained. Moreover, the form of the initial function is given, so that the corresponding initial value problem is uniquely solvable

On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work

Machine Learning Approaches on High Throughput NGS Data to Unveil Mechanisms of Function in Biology and Disease

In this review, the fundamental basis of machine learning (ML) and data mining (DM) are summarized together with the techniques for distilling knowledge from state-of-the-art omics experiments. This includes an introduction to the basic mathematical principles of unsupervised/supervised learning methods, dimensionality reduction techniques, deep neural networks architectures and the applications of these in bioinformatics. Several case studies under evaluation mainly involve next generation sequencing (NGS) experiments, like deciphering gene expression from total and single cell (scRNA-seq) analysis; for the latter, a description of all recent artificial intelligence (AI) methods for the investigation of cell sub-types, biomarkers and imputation techniques are described. Other areas of interest where various ML schemes have been investigated are for providing information regarding transcription factors (TF) binding sites, chromatin organization patterns and RNA binding proteins (RBPs), while analyses on RNA sequence and structure as well as 3D dimensional protein structure predictions with the use of ML are described. Furthermore, we summarize the recent methods of using ML in clinical oncology, when taking into consideration the current omics data with pharmacogenomics to determine personalized treatments. With this review we wish to provide the scientific community with a thorough investigation of main novel ML applications which take into consideration the latest achievements in genomics, thus, unraveling the fundamental mechanisms of biology towards the understanding and cure of diseases