The value functions of singularly perturbed time-optimal control problems in the framework of Lyapunov functions method

The Dirichlet problems for singularly perturbed Hamilton–Jacobi–Bellman equations are considered. Some impulse variables in the Hamiltonians have coefficients with a small parameter of singularity ε in denominators.The research appeals to the theory of minimax solutions to HJEs. Namely, for any ε>0, it is known that the unique lower semi-continuous minimax solution to the Dirichlet problem for HJBE coincides with the value function u^ε of a time-optimal control problem for a system with fast and slow motions.Effective sufficient conditions based on the fact are suggested for functions u^ε to converge, as ε tends to zero. The key condition is existence of a Lyapunov type function providing a convergence of singularly perturbed characteristics of HJBEs to the origin. Moreover, the convergence implies equivalence of the limit function u⁰ and the value function of an unperturbed time-optimal control problem in the reduced subspace of slow variables.     

Classes of meromorphic functions characterizable by growth of the spherical derivative

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 1, pp. 96–104, January–February, 1994.     

Intrinsic metrics and boundary values of functions of the Zygmund classes

Novosibirsk. Translated fromSibirskii Matematicheskii Zhurnal, Vol. 32, No. 2, pp. 3–12, March–April, 1991.     

Multiplier family of harmonic univalent functions

The aim of this paper is to study a multiplier family of harmonic univalent functions using the sequences {c_n} and {d_n} of positive real numbers. By specializing {c_n} and {d_n}, the generalized Bernardi–Libera–Livingston integral operator is modified for such functions and the closure of the multiplier family under the modified integral operator is determined. Also, convolution products, closure properties, distortion theorems, convex combinations and neighborhoods for such functions are given.     

Estimates from below for the functions with bounded specific oscillation

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 2, pp. 305–309, March–April, 1994.     

Multiplying Schur functions

A new combinatorial rule for expanding the product of Schur functions as a sum of Schur functions is formulated. The rule has several advantages over the Littlewood-Richardson rule (D. E. Littlewood and A. R. Richardson, Philos. Trans. Roy. Soc. London Ser. A233 (1934), 49–141). First this rule allows for direct computation of the expansion of the product of any number of Schur functions, not just the product of two Schur functions. Also, the rule is easily stated and is well suited to computer implementation. It is shown that the rule implies the Littlewood-Richardson rule and gives a combinatorial proof that the coefficient of S_λ in the product S_μS_ν equals the coefficient of S_ν in the expansion of the skew Schur function S_λ/μ. The rule is derived from some results proved independently by A. P. Hillman and R. M. Grassl (J. Combin. Comput. Sci. Systems5 (1980), 305–316) and by D. White (J. Combin. Theory Ser. A30 (1981), 237–247) on the Robinson-Schensted-Knuth correspondence.     

On the Cauchy problem for holomorphic functions of Lebesgue classL 2 in domains

Krasnoyarsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 186–195, September–October, 1992.     

Partially-bent functions

We study a conjecture stated in [6] about the numbers of non-zeros of, respectively, the auto-correlation function and the Walsh transform of the function (–1) ^f(x) , wheref(x) is any boolean function on {0, 1} ⁿ . The result that we obtain leads us to introduce the class of partially-bent functions. We study within these functions the propagation criterion. We characterize those partially-bent functions which are balanced and prove a relation between their number (which is unknown) and the number of non-balanced partially-bent functions on {0, 1} ^n–1. Eventually, we study their correlation immunity.     

Summation of analytic functions

For analytic functions of a certain class we establish the existence of a regular matrix summing to f(z) in the whole domain of analyticity off(z). In particular, the class under consideration contains the function (1 – z)^–1.Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 3–10, January, 1976.     

Two sufficient conditions for the univalence of analytic functions

In this article we obtain sufficient conditions for the univalence of n-symmetric analytic functions in the region ¦¦>–1 and in the disk ¦¦<–1. We examine the question of univalent variation of functions analytic in ¦¦<–1 and mapping ¦¦=1 onto a contour with two zero angles. We give an application of these results to the fundamental converse boundary-value problems.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 331–346, March, 1976.In conclusion the author would like to thank L. A. Aksent'ev for his guidance, and those who took part in his seminar for their useful advice.     

Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.     

Solution of a seepage problem in a nonhomogeneous medium by the method of p-analytical functions

We consider the problem of plane seepage in a nonhomogeneous medium of infinite thickness under reservoirs separated by Zhukovskii cutoff walls. The seepage coefficient of the permeable foundation is assumed to decrease with depth as a power function k(x, y)=(–y) ^–k (k=const > 0). For the case when k is a positive integer, the solution of the problem by the method of p-analytical functions is reduced to the solution of a system of Fredholm integral equations of the second kind. Some particular cases are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 55–66, 1988.     

Algebras of rapid decay functions on groups and cocycles of polynomial growth

Omsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 97–103, July–August, 1992.     

On obtaining close estimates in the approximation of functions of many variables by sums of functions of a fewer number of variables

We present a simple method for finding the values of the best approximation of a function of n variables of a given class by means of sums of two functions of a fewer number of variables; we establish close upper and lower bounds for the value of the best approximation to the functionf(x₁, ..., x_n), having the mixed derivativef x₁ ... x_n, by means of sums of a function of n–1 variables.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 105–114, July, 1972.     

On integral representation of functions determined by one class of pseudodifferential operators

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 5, pp. 1179–1193, September–October, 1995.     

Function evaluation using expansions in Bessel functions of the first kind

An iterative method is proposed for evaluation of fonctions that are expandable in series of Bessel functions of the first kind. The Bessel functions are evaluated by Miller's method, avoiding the need to determine their exact values. As an example, we describe algorithms for evaluation of the integral sine and the normal probability integral with an accuracy of to 10–12 significant digits.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 37–42, 1987.     

Reduction of the Gibbs phenomenon for smooth functions with jumps by the -algorithm

Recently, Brezinski has proposed to use Wynn's ε-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Padé–Fourier and Padé–Chebyshev approximants, including those recently studied by Kaber and Maday.     

Enumeration of lattice paths and generating functions for skew plane partitions

n-dimensional lattice paths not touching the hyperplanesX _i–X _i+1=–1,i=1,2,...,n, are counted by four different statistics, one of which is MacMahon's major index. By a reflection-like proof, heavily relying on Zeilberger's (Discrete Math. 44(1983), 325–326) solution of then-candidate ballot problem, determinantal expressions are obtained. As corollaries the generating functions for skew plane partitions, column-strict skew plane partitions, reverse skew plane plane partitions and column-strict reverse skew plane partitions, respectively, are evaluated, thus establishing partly new results, partly new proofs for known theorems in the theory of plane partitions.     

On monotone dominated sublinear functionals on the space of measurable functions

Moscow. Translated fromSibirski Matematicheski Zhurnal, Vol. 33, No. 6, pp. 94–105, November–December, 1992.     

Regular functions in a semiplane which are topologically equivalent to quasi-isometric mappings

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 6, pp. 1305–1313, November–December, 1994.