A review of the outstanding work of Maslov Viktor Pavlovich, recipient of the 2014 Russian Federation National Award in Science and Technology.

We describe a simple implementation of the Takagi factorization of symmetric matrices A = U Lambda U-T with unitary U and diagonal Lambda >= 0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of Lambda. The method is based on an algebraically exact expression.For parameterized family A(epsilon) = A + epsilon R = U-epsilon Lambda U-epsilon(epsilon)T, epsilon >= 0 with distinct singular values, the unitary matrices U-epsilon are discontinuous at the point epsilon = 0, if the singular values of A are multiple, but the composition U-epsilon Lambda U-epsilon(epsilon)T remains numerically stable and converges to A.The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available on Internet. (c) 2014 Elsevier Inc. All rights reserved.

A general approach is considered to find wave function and density operator evolution of the systems including the arbitrary number of coupled optical parametric interactions described by the quadratic Hamiltonian. This approach is based on the time-dependent canonical transformations that define the evolution of the system in the Heisenberg picture or in the interaction picture. An application is illustrated with four-frequency entangled light fields, which can be produced at the nonlinear optical interactions in vapors and solid states. The process includes one non-degenerate parametric down-conversion followed by two parametric up-conversions occurring in the field of the same classical pumping wave.

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension.

A new approach is proposed to solve the quantum evolution problem for a system with an arbitrary number of coupled optical parametric processes. Our method is based on the canonical transformations which define the evolution of the system in the Heisenberg picture. This theory overcomes the difficulties arising in the Wei–Norman method. The application of the approach developed is illustrated with the example of generation of a three-mode entangled light field.

Дифференцирование функций от некоммутирующих операторов, зависящих от параметра, по которому производится дифференцирование, усложняется, если операторы и их производные не коммутируют. Для вычисления производной ?̇?t операторозначной экспоненты et в случае [Gt, ?̇?t] ̸= 0 Фейнман [1] использовал формулу, связывающую Gt с левой производной ?̇?t этого семейства (?̇?t : ?̇?t et = (d/(dt))et (см. [2]), а также обсуждение в [3; с. 275, формула (1.10)], и в [4]):

We consider the multimode generalization of the normally ordered factorization formula of squeezings. This formula allows us to establish relationships between various representations of squeezed states, to calculate partial traces, mean values, and variations. The main results are expressed in terms of the matrix representation of canonical transformations, which is a convenient and numerically stable mathematical tool. Explicit representations are given for the inner product and the composition of generalized multimode squeezings. Explicitly solvable evolution problems are considered.

In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.

Professor Oleg Georgievich Smolyanov of the Department of Function Theory and Functional Analysis in the Faculty of Mechanics and Mathematics at Moscow State University, a doctor of the physical and mathematical sciences and one of the best-known experts in infinite-dimensional analysis, turned 70 years of age on 8 February 2008. Smolyanov was born in Russia’s northern capital, but spent his childhood years (including the first years of World War II) in Yaroslavl’, to which his family was evacuated at the beginning of the war. After moving to Moscow and graduating from secondary school there with a gold medal, he entered the Moscow Aviation Institute, which in those first years of triumphs of the Russian space programme was among the most popular institutions of higher education in the country and an object of aspiration for talented youth. In 1960, while still a student at the Aviation Institute, he began studies as an external student of the Faculty of Mechanics and Mathematics at Moscow State University (MSU). After completing his courses there in two and a half years (while working at the same time at a radio engineering factory), he began graduate studies in 1963 in the Department of Mathematics of the Faculty of Mechanics and Mathematics (with G.E. Shilov as advisor). That same year the journal Radiotekhnika i Èlektronika published his first two research papers, concerned with applied problems. But already then Smolyanov was intrigued by problems relating to analysis in infinite-dimensional spaces. He developed into a mathematician by participating in the seminars of S. V. Fomin and G. E. Shilov. In his first two mathematical papers, published in 1964 and 1965, he solved a difficult problem in the theory of topological vector spaces. In 1966 he defended his Ph.D. thesis, in which he obtained the first results in a new area of infinite-dimensional analysis which is now actively developing. In the same year he started teaching in the Faculty of Mechanics and Mathematics of MSU. In 1983 Smolyanov defended his D.Sc. thesis. He is the author of more than 180 research papers, including 5 surveys in Uspekhi Matematichaskikh Nauk (translated as Russian Mathematical Surveys) and 3 monographs (O. G. Smolyanov, Analysis in linear topological spaces and applications, MSU, Moscow 1979; O.G. Smolyanov and E.T. Shavgulidze, Path

This study considers a model of the income distribution of agents whose pairwise interaction is asymmetric and price-invariant. Asymmetric transactions are typical for chain-trading groups who arrange their business such that commodities move from senior to junior partners and money moves in the opposite direction. The price-invariance of transactions means that the probability of a pairwise interaction is a function of the ratio of incomes, which is independent of the price scale or absolute income level. These two features characterize the hierarchical model. The income distribution in this class of models is a well-defined double-Pareto function, which possesses Pareto tails for the upper and lower incomes. For gross and net upper incomes, the model predicts definite values of the Pareto exponents, $a_{\rm gross}$ and $a_{\rm net}$, which are stable with respect to quantitative variation of the pair-interaction. The Pareto exponents are also stable with respect to the choice of a demand function within two classes of status-dependent behavior of agents: linear demand ($a_{\rm gross}=1$, $a_{\rm net}=2$) and unlimited slowly varying demand ($a_{\rm gross}=a_{\rm net}=1$). For the sigmoidal demand that describes limited returns, $a_{\rm gross}=a_{\rm net}=1+\alpha$, with some $\alpha>0$ satisfying a transcendental equation. The low-income distribution may be singular or vanishing in the neighborhood of the minimal income; in any case, it is $L_1$-integrable and its Pareto exponent is given explicitly. The theory used in the present study is based on a simple balance equation and new results from multiplicative Markov chains and exponential moments of random geometric progressions.

New nonexplosion conditions for Markov processes are derived from the general operator form of the conservativity condition for a quantum dynamical semigroup.

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products (·, ⋆) and the maps K, K −1 which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): F(γ) → F⋃(γ) = {F(α⋃β)}α,β⊂γ. For the operators F⋃(γ) and F⋂(γ), we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function F(γ).