Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogeneous media

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through interfaces, which is mediated by a suitable Laplace-Beltrami type equation.     

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

On generalized perfect difference sets constructed from Sidon sets

Let $\mathbb{N}$ be the set of positive integers. For a nonempty set A of integers and every integer u, denote by $d_A(u)$ the number of $(a,a^{\prime })$ with $a,a^{\prime }\in A$ such that $u=a-a^{\prime }$. For a sequence S of positive integers, let $S(x)$ be the counting function of S. The set $A\subseteq \mathbb{N}$ is called a perfect difference set if $d_A(u)=1$ for every positive integer u. In 2008, Cilleruelo and Nathanson (2008) [4] constructed dense perfect difference sets from dense Sidon sets. In this paper, as a main result, we prove that: let $f:\mathbb{N}\to \mathbb{N}$ be an increasing function satisfying $f(n)\ge 2$ for any positive integer n, then for every Sidon set B and every function $\omega (x)\to \infty$, there exists a set $A\subseteq \mathbb{N}$ such that $d_A(u)=f(u)$ for every positive integer u and $B(x/3)-\omega (x)\le A(x)\le B(x/3)+\omega (x)$ for all $x\ge C_{f,B,\omega }$.     

Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.