Quantum mechanics and partial differential equations

This paper develops the basic theory of pseudo-differential operators on Rⁿ, through the Calderón-Vaillancourt (0, 0) L²-estimate, as a natural part of the harmonic analysis on the Heisenberg group, the group-theoretic embodiment of Heisenberg's Canonical Commutation Relations. The symbol mapping is given a group-theoretic interpretation consistent with the Kirillov method of orbits. By comparing different well-known realizations of the unique irreducible representation of the Heisenberg group, the Toeplitz operators on the complex n-ball are shown essentially to be pseudo-differential operators. The proof of the Calderón-Vaillancourt estimate is almost purely group-theoretic. Criteria for positivity, and for compactness are also given.     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.     

Statistical mechanics of nonlinear wave equations

The recurrences of initial states for nonlinear wave equations of the form with odd f are studied. The main result refines and generalizes the investigations of Friedlander and is related to the Poincaré theorem for finite dimensional Hamiltonian systems. Bibliography: 7 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 287–294.     

Quartic functional equations

In this paper, we solve a new functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y)     

Conservation laws and constitutive equations in continuum mechanics

The authors examine one of the general problems of continuum mechanics, namely the problem of the constitutive equations.Moscow. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 14–21, January–February, 1969.     

Embedding problem and functional equations

In this paper we give a method for solving the functional equations arising from the differential embedding problem. We also obtain the conditions for embedding one-dimensional diffeomorphisms into differential flows.     

Orthogonality and quintic functional equations

Using the fixed point method, we prove the Hyers Ulam stability of an orthogonally quintic functional equation in Banach spaces and in non-Archimedean Banach spaces.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)