Solution of boundary-value problems for xk-analytical functions in a halfplane with two cuts

We consider two boundary-value problems for x^k-analytical fonctions (k=2m+1, m=1,2,3,...) of the variable z=x+iy in a halfplane with cuts along the intervals (0, 1) and (b, ) of the real line (b > a), when the imaginary or the real part of the x-analytical function is given at the ends of the finite cut and resp. the real or the imaginary part of the function is given at the ends of the semi-infinite cut. Closed-form solutions are obtained for this problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 3–10, 1986     

Mixed problem for x-analytical functions in a halfdisk

The mixed problem for x-analytical functions in a halfdisk with the real part of the x-analytical function defined on a part of the circle and the imaginary part defined on the rest of the circle is reduced to the problem of linear matching of p-analytical functions with different characteristics (p=x in the halfdisk and p=x/(x ² +y ² ) in the halfplane with the halfdisk cut out). The linear matching problem of p-analytical functions is analyzed and the class of p-analytical functions with the characteristic p=x/(x ² +y ² ) is shown to be equivalent to the class of x-analylical functions. An integral representation is obtained for p-analytical functions with the characteristic p=x/(x ² +y ² ) which is analogous to the standard integral representation for x-analytical functions. A procedure is developed for reducing the solution of the linear matching problem to the solution of the Fredholm integral equation of second kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 7–15, 1986.     

Solution of boundary-value problems of p-analytical functions with the characteristic p=x/(x2 + y2) on a halfplane with cuts

A method is developed for solution of boundary-value problems of p-analytical functions with the characteristic p=x/(x₂ + y₂) on a halfplane with parallel straight cuts. Two problems are considered, with the boundary values of the real or imaginary part of the p-analytical function specified on the lips of the cuts. Solution of these problems is reduced to solution of a system of Fredholm integral equations of the second kind. As an application of these boundary-value problems, we consider plane seepage in a nonhomogeneous medium under a hydrotechnical structure with two cutoff walls forming a subsurface circle.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 17–26, 1988.     

A class of Nevanlinna functions related to singular Sturm-Liouville problems

The class of Nevanlinna functions consists of functions which are holomorphic off the real axis, which are symmetric with respect to the real axis, and whose imaginary part is nonnegative in the upper halfplane. The Kac subclass of Nevanlinna functions is defined by an integrability condition on the imaginary part. In this note a further subclass of these Kac functions is introduced. It involves an integrability condition on the modulus of the Nevanlinna functions (instead of the imaginary part). The characteristic properties of this class are investigated. The definition of the new class is motivated by the fact that the Titchmarsh-Weyl coefficients of various classes of Sturm-Liouville problems (under mild conditions on the coefficients) actually belong to this class.

     

The Keldysh-Sedov problem for functions holomorphic in a bi-halfplane

We pose and solve the Keldysh-Sedov problem: determine a function holomorphic in a bi-halfplane where its real and imaginary parts are given on mutually nonintersecting rectangles which exhaust the plane and which are the frame of the boundary of the bi-halfplane. Necessary and sufficient conditions for this problem to be solvable are given.Translated from Matematicheskii Zametki, Vol. 19, No. 5, pp. 681–690, May, 1976.     

Some uniform approximation theorems of yk-analytical functions

The paper examines some issues of uniform approximation of y^k -analytical functions infinite simply connected regions in the upper halfplane y >0 whose boundary includes a section of the real axis. Polozhii's integral representation of y^k -analytical functions [3] makes it possible to extend a number of theorems of uniform approximation of analytical functions by polynomials to the class of y^k-analytical functions.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 3–12, 1985.     

Geometrical solution of some cubic equations with non-real roots

A general cubic equation ax ³ + bx ² + cx + d = 0 where a, b, c, d ∈R, a ≠ 0 has three roots with two possibilities—either all three roots are real or one root is real and the remaining two roots are imaginary. Dealing with the second possibility this paper attempts to give the geometrical locations of the imaginary roots of the equation under three different sets of conditions. These sets of conditions include: (i) the real root of the given cubic equation is given, (ii) the real part of an imaginary root is given, and (iii) the imaginary part of an imaginary root is given.     

Geometric convergence to e−z by rational functions with real poles

Summary In this paper, we show that there exists a sequence of rational functions of the formR _n(z)=p_n–1(z)/(1+z/n)ⁿ,n=1, 2, ..., with degp _n–1n–1, which converges geometrically toe ^–z in the uniform norm on [0, +), as well as on some infinite sector symmetric about the positive real axis. We also discuss the usefulness of such rational functions in approximating the solutions of heat-conduction type problems.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075.     

On the Coordinate of a Singular Point of the Time Correlation Function for a Spin System on a Simple Hypercubic Lattice at High Temperatures

Using the d ^–1 expansion method (d is the space dimension), we estimate the coordinate of the time-dependent autocorrelation function singular point on the imaginary time axis for the spin 1/2 Heisenberg model on a simple hypercubic lattice at high temperatures. We represent the coefficients of the time expansion (the spectral moments) for the autocorrelation function as the sums of the weighted lattice figures in which the trees constructed from the double bonds give the leading contributions with respect to d ^–1 and the same trees with the built-in squares from six bonds or diagrams with the fourfold bonds give the contribution of the next-to-leading order. We find the corrections to the coordinate of the autocorrelation function singular point that are due to the latter contributions.     

Factorization of Hilbert port operators with poles on the imaginary axis

A factorization method is given to extract poles located on the imaginary axis for J-biexpansive meromorphic operator-valued functions acting on an infinite-dimensional Hilbert space. Decomposition of a real operator in terms of real factors, applicable to Hilbert ports, is also described, thus generalizing synthesis techniques originally developed for passive n-ports.     

Typically real functions with singularities on the real axis

This article considers the class ^k of typically real functions in the region G=CK, where K is a closed set on the real axis. An integral representation is given for the class ^k and it is used to establish a number of extremal properties for functions of the class ^k and associated classes of functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 60–71, 1990.     

Antipodal Theorems for Sets in Rn Bounded by a Finite Number of Spheres

This is a continuation of paper in Adv. Appl. Math. 22 (1999), 219–226, on an antipodal theorem for sets Dⁿ in Rⁿ bounded by a finite number of spheres. Here this theorem is first applied to set-valued mappings from Dⁿ to the boundary of an (n + 1)-cube or a d- dimensional octahedron. Next, the antipodal theorem is reformulated in terms of real continuous functions on Dⁿ, together with applications to the classical theorems of Borsuk–Ulam and Lusternik–Schnirelmann–Borsuk.     

Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

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.     

Holonomic functions in a polycircle with nonnegative imaginary part

A well-known theorem of Nevanlinna on the representation of nonnegative measure of a function holomorphic in a circle and having nonnegative imaginary part is extended to functions of many complex variables, holomorphic in a polycircle and having there a nonnegative imaginary part.Translated from Matematicheskie Zametki, Vol. 15, No. 1, pp. 55–61, January, 1974.     

Modified Schwarz–Christoffel mappings using approximate curve factors

The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C¹ regularity, C^∞ regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C^∞) boundary curve can be achieved.     

Toward a high-frequency asymptotic limit of the solutions of the Helmholtz equation in the region of a caustic shadow. II.

Results of the first part of this work for the analytic index of refraction n(x, Z) where the complex eikonal in the shadow region behind a caustic is found by the method of characteristics in the two-dimensional complex space ², are applied for n(x, Z) of finite smoothness. The use of the quadratic approximation for n(x, Z) allows one to obtain the zeroth approximation of the asymptotic limit of the wave field behind a caustic in the boundary layer of width 0 (^–2/3).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 182–188, 1987.     

A system of quaternionic equations in four-dimensional complex space

The boundary properties are studied of a class of quaternionic functions containing the class of holomorphic functions of four variables. A condition is found in order for a function defined on some five- or six-dimensional part of the whole topological boundary D of a domain d⁴ of a special type to have a holomorphic continuation. The results obtained are used to solve singular integral equations in C⁴.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 41–46, January, 1978.The author thanks B. V. Shabat and V. S. Vinogradov for a discussion of this work and much useful advice.     

Critical points for least-squares problems involving certain analytic functions,with applications to sigmoidal nets

This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called single-hidden layer neural networks, which are built upon the standard sigmoidal activation tanh(x) (or equivalently (1 +e ^–x )^–1), a rough upper bound for this cardinality is provided as well.Supported in part by US Air Force Grant AFOSR-94-0293.     

Determination of a neighborhood of the imaginary axis which is disjoint from the spectrum of a real polynomial

The distance of the spectrum of f from the imaginary axis is estimated for a real polynomial with roots in the right (or as a corollary, in the left) half plane: f:min Resp(f) –1/tr(H₁H¹ > 0 where [a _n+i–2j]i, j=1,n] andH ₁:=[a _k]K:=n+i+2j+1 i, j=1,nTranslated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 161–166, August, 1977.In conclusion, the author thanks S. M. Lozinskii for assistance.