A scheme for searching for holomorphically homogeneous Levi-nondegenerate real hypersurfaces in C 4 \mathbb {C}^4 that are not reducible to tubular manifolds and are orbits of 7-dimensional real Lie algebras is proposed. The family of such algebras contains more than a thousand different types, but the class of homogeneous hypersurfaces under discussion turns out to be meagre. Thus, in previous (including joint) works of the author of the present article, it was shown that, out of 594 types of solvable indecomposable Lie algebras of dimension 7 that have 6-dimensional nilradicals, orbits with the named properties can have no more than 104 types of algebras. The scheme under discussion is applied to 59 of these types, characterized by the presence of exactly two 4-dimensional Abelian subalgebras in each algebra under consideration. It is shown that a large number of such 7-dimensional algebras cannot have orbits with the required characteristics. At the same time, the article provides examples of the required orbits obtained using the scheme and based on simplified representations of the basis holomorphic vector fields of 6-dimensional nilradicals of the original 7-dimensional Lie algebras.
A necessary and sufficient condition is established for the defining matrices under which the matrix soliton equation of parabolic type defined by them is local, i.e., described by differential polynomials. A similar result is also valid in the more general case of Drinfeld–Sokolov hierarchies of mKdV type.
In memory of Vladimir Igorevich Arnold Here we consider and connect some facts related to thermodynamics, known to physicists and mathematicians separately. We recall the connection between classical thermodynamics and the geometry of distributions, and of statistical thermodynamics with multidimensional geometry. We recall something about the specific properties of functions of very many variables and demonstrate how these properties are realized in physical phenomena.
This paper studies a class of three-dimensional integral equations with a nonlinear monotone integral operator of the Hammerstein–Nemytsky type. This class of equations has applications in many areas of mathematical physics and mathematical biology. In particular, such equations arise in gamma-quantum theory, in problems of neutron moderation in nuclear reactors, in queuing theory, and in the mathematical theory of spatiotemporal spread of epidemics. Under certain restrictions on the kernel of the equation and on the functions describing the nonlinearity (that are natural from the point of view of applications in the indicated areas), the existence of a one-parameter family of unbounded positive solutions is proved, and the set of corresponding parameters is explicitly described. It is also possible to study the asymptotic behavior of the constructed solutions at infinity. At the end, specific applied examples of the indicated equations are given to illustrate the importance of the results obtained.
A review of topological methods applied in solid-state theory is given. First, we recall the basic provisions of Bloch theory describing the properties of solids with a crystal lattice. Then we construct an algebra of observables of a topological dielectric and the resulting classes of symmetries and pseudosymmetries. Next, a description of the algebras of observables is given in terms of the K K -theory of graded C ∗ C^* -algebras and the accompanying topological invariants of a solid. The algebra of boundary observables is defined in terms of the K K -theory proposed by Kasparov. In conclusion, we describe the correspondence between the topological invariants of the body and its boundary (BB-correspondence). In the particular case of a periodic unitary model, this correspondence can be described explicitly.
