Theory Of Analytic Functions Research Articles

On the Correctness of Polynomial Difference Operators

Difference equations arise in various areas of mathematics. Together with the method of generating functions, they yield a powerful machinery for studying enumeration problems in combinatorial analysis (see, e.g. [1]). Another source of difference equations is the discretization of differential equations. For instance, the discretization of Cauchy-Riemann equations led to the appearance of the theory of discrete analytic functions (see, e.g. [2, 3]) which found applications in the theory of Reimannian surfaces and combinatorial analysis (see, e.g. [4, 5]). The methods of discretization of a differential problem comprise an important part of the theory of difference schemes, and they also lead to difference equations (see, e.g. [6]). The theory of difference schemes is exploring ways of constructing difference schemes, explores the challenges posed difference and the convergence of the solution of the difference problem to the solution of the original differential problem, engaged justification of algorithms for solving problems of difference. An important place among these are correct. We define the shift operators δj with respect to the variables xj : δjf (x) = f (x1, . . . , xj−1, xj + 1, xj+1, . . . , xn), and polynomial difference operator of the form

On the Phragmén-Lindelöf principle for entire power series on a Banach algebra

Many authors have managed to successfully extend the classical theory of analytic functions to functions defined on more abstract spaces (see for example [3, 8, 9]). The purpose of this paper is to provide a bit in this direction. In this paper an extension of the Phragmen-Lindelof principles of the classical theory to the functions expressed as power series on a Banach algebra not necessarily commutative, of course, involving concepts such as harmonic and subharmonic, is introduced.

Interaction among <i>n</i> Parallel Dislocations in One-Dimensional Hexagonal Quasicrystals

By means of analytic function theory, the problems of interaction amongparallel dislocations in one-dimensional hexagonal quasicrystals are investigated. The interaction force of parallel dislocations in the material is obtained in forms of complex variable function firstly, which is the versions of well-known Peach-Koehler formula in one-dimensional hexagonal quasicrystals on parallel dislocations. These results are development of the corresponding parts of quasicrystals. Meanwhile, in this paper, we firstly give the equivalent action point of parallel dislocations in one-dimensional hexagonal quasicrystals, which be of important reference value to researching the interaction problems of many dislocations in fracture mechanics of quasicrystals.

On the Limiting Cases Related to Flow Past a Zhukovskii Sheet Pile

Within the framework of the Darcy′s theory of plane steady fi ltration of an incompressible liquid, two limiting schemes are considered that model fi ltration fl ows under a Zhukovskii sheet pile through a large mass of ground underlain by an impermeable base or by a permeable water-bearing pressure horizon. For purposes of analyzing these schemes, mixed boundary-layer problems of the theory of analytical functions are formulated that are solved with the aid of the Polubarinova-Kochina method. Based on these models, algorithms have been developed for calculating a saturation zone in those cases where, while considering the motion of water, one has to take into account the exposure of fl ow pattern to the combined effect of such important factors as backwater from the side of the impermeable base or of the underlying well permeable water-bearing bed, evaporation or infi ltration on the free groundwater surface, as well as the ground capillarity.

Periodic array of traction-free interface cracks subjected to far-field uniform heat flow

A periodic array of interface cracks is subjected to a uniform heat flow in the far field. The crack opening displacements and complex stress intensity factors are determined, using analytic function theory, for the case that the upper half-space is less distortive than the lower half-space.

The effective solution of two‐dimensional integro‐differential equations and their applications in the theory of viscoelasticity

The effective solutions for integro‐differential equations related to problems of interaction of an elastic thin finite inclusion with a plate, when the inclusion and plate materials possess the creep property are constructed. If the geometric parameter of the inclusion is measured along its length according to the parabolic and linear law we have managed to investigate the obtained boundary value problems of the theory of analytic functions and to get exact solutions and establish behavior of unknown contact stresses at the ends of an elastic inclusion.

Application of Chebyshev polynomials to classes of analytic functions

Our objective in this paper is to consider some basic properties of the familiar Chebyshev polynomials in the theory of analytic functions. We investigate some basic useful characteristics for a class H(t), t∈(1/2,1], of functions f, with f(0)=0, f′(0)=1, analytic in the open unit disc U={z:|z|<1} satisfying the condition that1+zf″(z)f′(z)≺H(z,t)=11−2tz+z2(z∈U), where H(z,t) is the generating function of the second kind of Chebyshev polynomials. The Fekete–Szegö problem in the class is also solved.

Analytic function theory for operator-valued free probability

Abstract It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plane. In this paper, we prove an analogous characterization of the Cauchy transforms, again, entirely in terms of their analytic and asymptotic behavior. We further characterize those functions which arise as the Voiculescu transform of ⊞ {\boxplus} -infinitely divisible operator-valued distributions. As consequences of these results, we provide a characterization of infinite divisibility in terms of the domain of the relevant Voiculescu transform, provide a purely analytic definition of the semigroups of completely positive maps associated to infinitely divisible distributions and provide a Nevanlinna representation for non-commutative functions with the appropriate asymptotic behavior.

WELL-POSEDNESS OF POINCARE PROBLEM IN THE CYLINDRICAL DOMAIN FOR A CLASS OF MULTI-DIMENSIONAL ELLIPTIC EQUATIONS

The boundary value problems for second order elliptic equations in domains with edges are well studied. For elliptic equations, boundary-value problems on the plane were shown to be well posed by using methods from the theory of analytic functions of complex variable. When the number of independent variables is greater than two, difficulties of fundamental nature arise. Highly attractive and convenient method of singular integral equations can hardly be applied, because the theory of multidimensional singular integral equations is still incomplete. In this paper with the help of the method suggested by the author, the unique solvability is shown and explicit form of classical solution of Poincare problem in a cylindrical domain for a one class of multidimensional elliptic equations is received.

Bounded real lemma and structured singular value versus diagonal scaling: the free noncommutative setting

The structured singular value (often referred to simply as $$\mu $$ ) was introduced independently by Doyle and Safonov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the structured singular value provides a necessary and sufficient criterion for robustness with respect to a structured ball of uncertainty, it is notoriously difficult to actually compute. The method of diagonal (or simply D) scaling, on the other hand, provides an easily computable upper bound (which we call $$\widehat{\mu }$$ ) for the structured singular value, but provides an exact evaluation of $$\mu $$ (or even a useful upper bound for $$\mu $$ ) only in special cases. However it was discovered in the 1990s that a certain enhancement of the uncertainty structure (i.e., letting the uncertainty parameters be freely noncommuting linear operators on an infinite-dimensional separable Hilbert space) resulted in the $$D$$ -scaling procedure leading to an exact evaluation of $$\mu _{\text {enhanced}}$$ ( $$\mu _{\text {enhanced}} = \widehat{\mu }$$ ), at least for the tractable special cases which were analyzed in complete detail. On the one hand, this enhanced uncertainty has some appeal from the physical point of view: one can allow the uncertainty in the plant parameters to be time-varying, or more generally, one can catch the uncertainty caused by the designer’s decision not to model the more complex (e.g. nonlinear) dynamics of the true plant. On the other hand, the precise mathematical formulation of this enhanced uncertainty structure makes contact with developments in the growing theory of analytic functions in freely noncommuting arguments and associated formal power series in freely noncommuting indeterminates. In this article we obtain the $$\widetilde{\mu } = \widehat{\mu }$$ theorem for a more satisfactory general setting.

A $$q$$ q -extension of a partial differential equation and the Hahn polynomials

Using the theory of analytic functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $$q$$ -partial differential equations, then, it can be expanded in terms of the product of the homogeneous Hahn polynomials. Some non-trivial applications of this expansion theorem to $$q$$ -series are discussed.

Solutions of helmholtz equation in complex domain

A harmonic equation and the Helmholtz equation are elliptic type equations and describe important physical processes (the first – stationary, the second - stationary and dynamic). Effective solutions of boundary value problems for harmonic equation (in different regions in the plane) are constructed by the methods of the theory of analytic functions of a complex variable. These methods can not be applied directly to solving problems for the Helmholtz equation. In the scientific literature, solutions of boundary value problems for this equation are known only in certain areas that are represented by cumbersome formulas. In the paper, using the solution of the Helmholtz equation in a circle through the functions (not analytical) of complex variable and conformal mapping of a given area on the circle, a general approach to building a solution of the corresponding boundary value problem is formulated. An important prerequisite for presenting this solution as functional series is finding the solution of harmonic equation in a given region that satisfies the given boundary conditions and an analytic function in this region respectively. The solutions of the Helmholtz equation in the plane with an elliptic hole and half-plane are constructed. For effective formulation of boundary value prob­lems and finding analytic functions in these areas, systems of basic functions in the corresponding spaces of analytic functions are found.

Back Analysis Methods in Geotechnical Engineering

The paper deals with the back analysis method in geotechnical engineering, that goal is evaluation the more objective and reliable parameters of the rock mass on the basis of in-situ measurements. Stress, deformational, strength and rheological parameters of the rock mass are usually determined by some inaccuracies and errors arising from the complexity and variability of the rock mass. This higher or lower degree of imprecision is reflected in the reliability of the mathematical modelling results. The paper presents the utilization of direct optimization back analysis method, based on the theory of analytical functions of complex variable and Kolosov-Muschelischvili relations, to the evaluation of initial stress state inside the rock massif.

COMPARING AND ZILBER’S EXPONENTIAL FIELDS: ZERO SETS OF EXPONENTIAL POLYNOMIALS

We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for $\mathbb{C}$ using analytic function theory, for example, the Identity Theorem.

General case of contact problems for a regular polygon weakened with full-strength hole

The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a regular polygon and the internal boundary is the required full-strength hole including the origin of coordinates. The full-strength hole is cycle symmetric. It is assumed that to every link of the broken line conforming the outer boundary of the given body are applied absolutely smooth rigid punches with rectilinear bases, which are under the action of the force P that applies to their middle points. There is no friction between the surface of the given elastic body and the punches. The uniformly distributed normal stress Q is applied to the hole boundary. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and the shape of the hole’s contour are determined under the condition that the tangential normal stress arising at it takes a constant value. A similar problem is considered for a square and an equilateral triangle, which are weakened with full-strength holes. Using the method developed here, the partially unknown boundary value problems under consideration is reduced to known boundary value problems of the theory of analytic functions. The solutions are presented in quadratures, and full-strength contours are constructed.

When an academician becomes professor: the case of Joseph-Louis Lagrange

With the French Revolution, the role of the mathematician gradually evolved from proteges of princes to official professor. Lagrange provides a remarkable illustration of this fundamental change. Until 1792, this famous but discreet scientist had a European career, first in Turin and then to Berlin and finally Paris, ignoring national boundaries and rivalries between powers but always subject to the system of enlightened despotism. His life was that of a court mathematician, working in the narrow confines of academic institutions. The French Revolution completely changed his career and relaunched his work. He actively participated in the creation of the metric system and taught analysis in new educational institutions established by the Convention: the Ecole Normale and the Ecole Polytechnique. This is where he explained his theory of analytical functions, where differential and integral calculus were reduced to the study of the expansion of functions.

Application of Faber polynomials to the approximate solution of a generalized boundary value problem of linear conjugation in the theory of analytic functions

In the present paper the method of successive approximations and Faber polynomials are used to derive the approximate solution of a generalized boundary value problem of linear conjugation on the Lyapunov curve. The conditions for the existence and uniqueness of the solution are presented in the L2 and H(α) spaces.

LEM for twisted re-entrant angle sections

In this paper an innovative numerical method named as line element-less method, LEM, for finding solution of torsion problem has been extended to all shaped sections, including sections possessing re-entrant angles at their boundary. The response solution in terms of shear stress field or Prandtl function or warping function in all domain and for any kind of domain with arbitrary contour, may be performed quickly, calculating line integrals only. The method takes full advantage of the theory of analytic complex function and is robust in the sense that returns exact solution if this exists. Numerical implementation of LEM has been developed using Mathematica software without resorting to any discretization neither in the domain nor in the boundary. The latter means that you can use the same program for all sections just by changing the first few lines of program where you declare the geometry of the section. Some numerical applications have been reported to demonstrate the efficiency and accuracy of the method.

Reproducing kernel Hilbert spaces generated by the binomial coefficients

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions, statistics, and with the quantum case.

Boundary value problems of electroelasticity for a plate with an inclusion and a half-space with a slit

Problems of determining the mechanical and electrical fields in a piezoelectric plate reinforced with an inclusion or in a half-space weakened by a cut are considered. Using the methods of the theory of analytic functions these problems are reduced to a system of singular integro-differential equations (for a plate) or to a singular integral equation with a fixed singularity (for a half-space). Approximate and exact solutions of the problems are obtained by the method of orthogonal polynomials and integral transforms.