Classic Solutions of Boundary Value Problems for Partial Differential Equations with Operator of Finite Index in the Main Part of Equation

This paper is an attempt to give the review of a part of our results in the area of singular partial differential equations. Using the results of the theory of complete generalized Jordan sets we consider the reduction of the PDE with the irreversible linear operator $B$ of finite index in the main differential expression to the regular problems. Earlier we and other authors applied similar methods to the development of Lyapunov alternative method in singular analysis and numerous applications in mechanics and mathematical physics. In this paper, we show how the problem of the choice of boundary conditions is connected with the $B$-Jordan structure of coefficients of PDE. The estimation of various approaches shows that the most efficient approach for solving this problem is the functional approach combined with the alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator in the main part of the equation. On this base, the problem of the correct choice of boundary conditions for a wide class of singular PDE can be solved. The aggregated theorems of existence and uniqueness of classical solutions can be proved with continuously depending of experimental definite function. The theory is illustrated by considering the solution of some integro - differential equations with partial derivatives

Electromagnetic Potential in Pre-Metric Electrodynamics: Causal Structure, Propagators and Quantization

An axiomatic approach to electrodynamics reveals that Maxwell electrodynamics is just one instance of a variety of theories for which the name electrodynamics is justified. They all have in common that their fundamental input are Maxwell's equations $\textrm{d} F = 0$ (or $F = \textrm{d} A$) and $\textrm{d} H = J$ and a constitutive law H = # F which relates the field strength two-form $F$ and the excitation two-form $H$. A local and linear constitutive law defines what is called local and linear pre-metric electrodynamics whose best known application are the effective description of electrodynamics inside media including, e.g., birefringence. We analyze the classical theory of the electromagnetic potential $A$ before we use methods familiar from mathematical quantum field theory in curved spacetimes to quantize it in a locally covariant way. Our analysis of the classical theory contains the derivation of retarded and advanced propagators, the analysis of the causal structure on the basis of the constitutive law (instead of a metric) and a discussion of the classical phase space. This classical analysis sets the stage for the construction of the quantum field algebra and quantum states. Here one sees, among other things, that a microlocal spectrum condition can be formulated in this more general setting.Comment: 34 pages, references added, update to published version, title updated to published versio

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version

Invariants of Lie Algebras with Fixed Structure of Nilradicals

An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension $n<\infty$ restricted only by a required structure of the nilradical. Specifically, invariants are calculated here for families of real/complex solvable Lie algebras. These families contain, with only a few exceptions, all the solvable Lie algebras of specific dimensions, for whom the invariants are found in the literature.Comment: LaTeX2e, 19 page