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This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely…
This work presents a novel approach to describe spectral properties of graphene layers with well defined edges. We microscopically analyze the boundary problem for the continuous Bogoliubov-de Gennes-Dirac (BdGD) equations and derive the…
The parqueting-reflection principle is shown to also work for constructing harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two arcs in $\mathbb{C}$ with a special intersecting angle…
Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation…
The quantum behavior of charge carriers in semiconductor structures is often described in terms of the effective mass Schr\"{o}dinger equation, neglecting the rapid fluctuations of the wave function on the scale of the atomic lattice. For…
Green's functions with continuum spectra are a way of avoiding the strong bounds on new physics from the absence of new narrow resonances in experimental data. We model such a situation with a five-dimensional model with two branes along…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure. The validity of the various distributivity laws is studied and forms the basis for a comparison…
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
Structural operational semantics can be studied at the general level of distributive laws of syntax over behaviour. This yields specification formats for well-behaved algebraic operations on final coalgebras, which are a domain for the…
We consider the Darboux transformation of the Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We obtained the Green functions of the transformed Dirac equation with the initial regular…
This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic…
In this paper, we consider a linear fractional differential equation with fractional boundary conditions. First, by obtaining Green's function, we derive the Lyapunov-type inequalities for such boundary value problems. Furthermore, we use…
In this work we revise the most recent developments concerning the study of first order problems regarding differential equations with involutions. We take into account two cases: the case of initial conditions and constant coefficients and…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
We explore the connections between Green's functions for certain differential equations, covariance functions for Gaussian processes, and the smoothing splines problem. Conventionally, the smoothing spline problem is considered in a setting…
In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
A key issue in the solution of partial differential equations via integral equation methods is the evaluation of possibly singular integrals involving the Green's function and its derivatives multiplied by simple functions over discretized…