A. F. Lavrik's truncated equations

By a new method, we obtain some known results of A. F. Lavrik (Dokl. Akad. Nauk SSSR,171, No. 2, 278–280 (1966); Mat. Zametki,2, No. 5, 475–482 (1967); Izv. Akad. Nauk SSSR, Ser. Mat.,30, No. 2, 433–448 (1966)) regarding the truncated functional equations of various L-functions. As an application, we give an estimate of Dedekind's zeta-function of an algebraic number field K of degree n4, _k(1/2+it)t^n/6log^ct,t>1 and a similar estimate for L-series with grössencharacters. The method of the paper allows us to consider fields of degree n12.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 124–158, 1978.     

Duality in spectral optimization and numerical ranges of a family of self-adjoint operators

One gives results which complement the author's investigation, published in Dokl. Akad. Nauk SSSR,255, No. 4, 777–780 (1980). One establishes the relationship of the fundamental condition ensuring the duality relation between the direct and the dual problems in spectral optimization problems with the geometry of the numerical ranges of certain families of self-adjoint operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 189–192, 1982.     

Exponents of elements of a normal basis of the ideal of algebraic functions on a three-sheeted Riemannian surface

On a three-sheeted Riemannian surfaceR of genus given by an irreducible algebraic equation, we construct normal bases of the ideal of algebraic functions that are multiples of some integer divisors. A method for constructing such normal bases was given in [V. E. Kruglov,Dokl. Akad. Nauk SSSR,321, No. 1, 11–13 (1991)]. The relations obtained for the exponents of the constructed elements enable one to determine the number of solutions of the Riemann problem for any integer divisor and to find partial indices in the problems of factorization of matrices of permutation type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1029–1037, August, 1995.     

Application of topology in optimization theory

In 1958 the author gave a proof of the Maximum Principle [Dokl. Akad. Nauk SSSR 119 (6) (1958) 1070–1073] in the Theory of Optimal Control. B. Pshenichni had written that the proof was a little sensational, since it used topology to obtain a result of variational calculus. Later the author worked out the Tent Method [Russian Math. Surveys 30 (3) (1975) 1–54] that is a general way to solve some extremal problems. In fact, main ideas of the Method were contained in the proof of the Maximum Principle [Dokl. Akad. Nauk SSSR 119 (6) (1958) 1070–1073]. In the talk a short survey of the Tent Method and the idea of the proof of the Maximum Principle are explained, paying attention to necessary topological tools.     

An extension theorem for a family of functional spaces

For Lipschitz spaces of functions (in the sense of Yu. A. Brudnyi, V. K. Shalashov, Lipschitz spaces of functions, Dokl. Akad. Nauk SSSR,197, No. 1, 18–20, 1971), defined in a bounded domain with a Lipschitz boundary, one gives a theorem for the existence of a linear operator of extension. As a consequence, one formulates some new results of Lipschitz spaces (interpolation, embedding, change of variables).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 170–173, 1976.     

Fixed Points of Cone-Compressing and Cone-Extending Operators in Frechet Spaces

A generalization of norm type cone-compression and -expansionresults due to Krasnosel'skii (see Dokl. Akad. Nauk SSSR NS135 (1960) 527–530) is presented here for single-valuedcompletely continuous maps defined on a Fréchet space.Applications to second-order differential equations on the halfline are presented, and the existence of nontrivial solutionsis established. 2000 Mathematics Subject Classification 47H10,34B40.     

On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.     

On the cardinality of n-Urysohn and n-Hausdorff spaces

Two variations of Arhangelskii’s inequality $\left| X \right| \leqslant 2^{\chi (X) - L(X)}$ for Hausdorff X [Arhangel’skii A.V., The power of bicompacta with first axiom of countability, Dokl. Akad. Nauk SSSR, 1969, 187, 967–970 (in Russian)] given in [Stavrova D.N., Separation pseudocharacter and the cardinality of topological spaces, Topology Proc., 2000, 25(Summer), 333–343] are extended to the classes with finite Urysohn number or finite Hausdorff number.     

Polynomial complexity algorithms for computational problems in the theory of algebraic curves

It is proved that the classical algorithm for constructing Newton-Puiseux expansions for the roots of polynomials using the method of Newton polygons is of polynomial complexity in the notation length of the expansion coefficients. This result is used, in the case of a ground field of characteristic O, to construct polynomial-time algorithms for factoring polynomials over fields of formal power series, and for fundamental computational problems in the theory of algebraic curves, such as curve normalization.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 176, pp. 127–150, 1989.     

Representations of symmetric groups with maximal dimension

Results of numerical modeling connected with the study of the asymptotics of representations of the symmetric group S_N for largeN which are of maximal dimension are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 160–166, 1989.     

Semyanistyi’s Integrals and Radon Transforms on Matrix Spaces

We introduce a new analytic family of intertwining operators which include the Radon transform over matrix planes and its inverse. These operators generalize integral transformations introduced by Semyanistyi (Dokl. Akad. Nauk SSSR 134:536–539, [1960]) in his research related to the hyperplane Radon transform in ℝ ⁿ . We obtain an extended version of Fuglede’s formula, connecting generalized Semyanistyi’s integrals, Radon transforms and Riesz potentials on the space of real rectangular matrices. This result combined with the matrix analog of the Hilbert transform leads to variety of new explicit inversion formulas for the Radon transform of functions of matrix argument. The authors were supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). The first author was also supported by Abraham and Sarah Gelbart Research Institute for Mathematical Sciences. The second author was also supported by the NSF grants EPS-0346411 (Louisiana Board of Regents) and DMS-0556157).     

A Φ-Entropy Contraction Inequality for Gaussian Vectors

A beautiful result of Sarmanov (Dokl. Akad. Nauk SSSR 121(1), 52–55, 1958) says that for a Gaussian vector (X,Y), \operatorname Var(\mathbb E[f(Y)|X]) ￡ r²\operatorname Var(f(Y))\operatorname {Var}(\mathbb {E}[f(Y)|X])\le \rho^{2}\operatorname {Var}(f(Y)) for all measurable functions f, where ρ is the (linear) correlation coefficient between X and Y. We generalize this result to a general Φ-entropy (a nonlinear version of his result) by means of a previous result of D. Chafai based on Bakry–Emery’s Γ ₂-technique and tensorization.     

Construction of the functional model of B. S.-Nagy-C. Foias

Methodological paper: quick and simple construction of the Nagy—Foias functional model, using only elementary information about Hilbert spaces and operators on them. A formula is derived for a characteristic function. The account is based on an idea of B. S. Pavlov (Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 123–148 (1975)) on symmetrization of the notation for a model space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 16–23, 1977.     

Distribution of integral points on some hyperboloids of one sheet

Results of B. F. Skubenko (Izv. Akad. Nauk SSSR, Ser. Mat.,26, 721–752 (1962)) are generalized to indefinite ternary quadratic formsf(x)=f ₀(Cx), which are contained in the simplest formf ₀(x)=x ₁ x ₃–x _r ^e We prove that the integral points on the hyperboloid of one sheetf(x)=m,m<0, are uniformly distributed over area (in the sense of hyperbolic metric) and over residue classes for given modulus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 83–93, 1983.     

Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω]

   Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below. Theorem Let p ∈ [1,∞) and ω ∈ (0, 1/2] . Suppose W is a weight function on [-ω,ω] such that W(ω cos t) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that

On construction of asymptotic expansions for Pólya distributions

In the paper [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dokl. Akad. Nauk SSSR,246, No. 3 (1979)], asymptotic formulas for multidimensional Pólya distributions, including both polynomial and multidimensional hypergeometric distributions, were deduced. The derivation of those formulas was rather bulky and constituted the paper [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dep. VINITI, No. 3155-79, Moscow (1979)]. The aim of the present paper is to realize another approach to constructing asymptotic expansions for multidimensional distributions of more general structure generated by plans of Pólya random walks [Ya. P. Lumel'skii,Dokl. Akad. Nauk SSSR,209, No. 6 (1973)]. This allows us not only to prove the results of [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dep. VINITI, No. 3155-79, Moscow (1979)] but to derive new asymptotic formulas as well. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 125–136, Perm, 1993.     

Weighted Polynomial Inequalities with Doubling and A ∞ Weights

We consider weighted inequalities such as Bernstein, Nikolskii, Remez, etc., inequalities under minimal assumptions on the weights. It turns out that in most cases this mimimal assumption is the doubling condition. Sometimes, however, as for the Remez and Nikolskii inequalities, one needs the slightly stronger A _∈ fty condition. We shall consider both the trigonometric and the algebraic cases. August 20, 1997. Date revised: April 19, 1998. Date accepted: May 26, 1998.     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains

Let c_k, k=1,...,4, be arbitrary distinct points of . LetD be the family of all systems of simply connected domains in. By R(D_k, c_k) we denote the conformal radius of the domain D_k, relative to the point c_k. We prove that in the familyD one has the sharp inequality, (1) where a=(+i)/(–1), being the cross-ratio of the points c₁,c₂, c₃, c₄: E(–1, 1,a) is the continuum of least capacity containing the points –1,1,a. An explicit expression for capE(–1,1,a.) in terms of elliptic Jacobi functions has been obtained earlier by the author [Tr. Mat. Inst. Akad. Nauk SSSR,94, 47–65, 1968]. On the basis of the well-known properties of continua of least capacity, one shows that the largest value of the right-hand side of (1) is attained for a=± i3 and it is equal to 4^–8/3·3². One gives all the configurations for which equality prevails in the obtained estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 131–145, 1980.     

Bernstein–Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds

Bernstein–Nikolskii inequalities and Riesz interpolation formula are established for eigenfunctions of Laplace operators and polynomials on compact homogeneous manifolds.     

Application of separability and independence notions for proving lower bounds of circuit complexity

This note consists of two independent parts. In the first part the concept of an (m,c)-system for a set of linear forms is introduced, and a lower bound is obtained for the algebraic complexity of the computation of (m,c)-systems on algebraic circuits of a special form. In the second part, the notion of an -independent set of boolean functions is introduced and a lower bound is obtained for a certain complexity measure for circuits of boolean functions computing -independent sets. As a corollary it is shown that the standard algorithm for multiplying matrices or polynomials may be realized by a circuit of boolean functions in a way that is optimal with respect to a selected complexity measure.Translated from Zapiski Nauchykh Seminarov Leningradskogo Otdeleniya Mathematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 38–48, 1976. Main results presented December 12, 1974 and May 29, 1975.The author would like to express his deep appreciation to A. O. Slisenko for his help.