Series with Commuting Terms in Topologized Semigroups

We show that the following general version of the Riemann–Dirichlet theorem is true: if every rearrangement of a series with pairwise commuting terms in a Hausdorff topologized semigroup converges, then its sum range is a singleton.

Normal Forms and Cohomology Classes at Single Resonances

This chapter discusses the single-resonance non-degeneracy conditions and normal forms. It then formulates Theorem 3.3, which covers the forcing equivalence in the single-resonance regime. The classical partial averaging theory indicates that after a coordinate change, the system has the normal form away from punctures. In order to state the normal form, one needs an anisotropic norm adapted to the perturbative nature of the system. The chapter also uses the idea of Lochak to cover the action space with double resonances. A double resonance corresponds to a periodic orbit of the unperturbed system. Finally, the chapter looks at a lemma which is an easy consequence of the Dirichlet theorem.

An inhomogeneous Dirichlet theorem via shrinking targets

We give an integrability criterion on a real-valued non-increasing function $\unicode[STIX]{x1D713}$ guaranteeing that for almost all (or almost no) pairs $(A,\mathbf{b})$, where $A$ is a real $m\times n$ matrix and $\mathbf{b}\in \mathbb{R}^{m}$, the system $$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$ is solvable in $\mathbf{p}\in \mathbb{Z}^{m}$, $\mathbf{q}\in \mathbb{Z}^{n}$ for all sufficiently large $T$. The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose $m=n=1$ case was recently solved, but whose general case remains open.

Fundamental Problems of the Electrodynamics of Heterogeneous Media

The consistent physic-mathematical model of propagation of an electromagnetic wave in a heterogeneous medium is constructed using the generalized wave equation and the Dirichlet theorem. Twelve conditions at the interfaces of adjacent media are obtained and justified without using a surface charge and surface current in explicit form. The conditions are fulfilled automatically in each section of counting schemes for calculations. A consistent physicomathematical model of interaction of nonstationaly electric and thermal fields in a layered medium with allowance or mass transfer is constructed. The model is based on the methods of thermodynamics and on the equations of an electromagnetic field and is formulated without explicit separation of the charge carriers and the charge of an electric double layer.