Theory Of Analytic Functions Research Articles

Weighted Harmonic Means

The purpose of this study is to develop a conception of the differential subordination involving harmonic means of the expressions \(\psi (p(z), zp'(z);z)\), where p is an analytic function in the unit disk, such that \(p(0)=1, p(z)\not \equiv 1\). Here, we discuss convex weighted harmonic means and we find some applications in the theory of analytic functions.

On a system of partial differential equations and the bivariate Hermite 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 partial differential equations, then, it can be expanded in terms of the product of the bivariate Hermite polynomials. This expansion theorem allows us to develop a systematic method to prove the identities involving the bivariate Hermite polynomials. With this expansion theorem, we can easily derive, among others, the Mehler formula, Nielsen's formulas, Doetsch's formula, the addition formula, Weisner's formulas, Carlitz's formulas for the bivariate Hermite polynomials.

Strict superharmonicity of Mityuk’s function for countably connected domains of simple structure

Strict superharmonicity of generalized reduced module as a function of a point (we call it Mityuk’s function) is established for the subclass of countably connected domains with unique limit point boundary component. The function just mentioned was first studied in detail by I.P. Mityuk and plays now an important role in the research of the exterior inverse boundary value problems of the theory of analytic functions in the multiply connected domains. At the heart of such a research one can see the fact that the critical points of Mityuk’s function are only maxima, saddles or semisaddles of corresponding surface. This fact is followed from the above strict superharmonicity.

Classical Dynamics of Fullerenes

The classical mechanics of large molecules and fullerenes is studied. The approach is based on the model of collective motion of these objects. The mixed Lagrangian (material) and Eulerian (space) description of motion is used. In particular, the Green and Cauchy deformation tensors are geometrically defined. The important issue is the group-theoretical approach to describing the affine deformations of the body. The Hamiltonian description of motion based on the Poisson brackets methodology is used. The Lagrange and Hamilton approaches allow us to formulate the mechanics in the canonical form. The method of discretization in analytical continuum theory and in classical dynamics of large molecules and fullerenes enable us to formulate their dynamics in terms of the polynomial expansions of configurations. Another approach is based on the theory of analytical functions and on their approximations by finite-order polynomials. We concentrate on the extremely simplified model of affine deformations or on their higher-order polynomial perturbations.

The boundary value problem for piezo-electric half space with thin elastic inclusion

The problem of finding mechanical and electric fields in a piezo-elastic half-space with elastic inclusion is considered. The inclusion is loaded with forces of constant intensity. The tangential contact stresses along the contact surface are determined and the behavior of the contact stresses in the neighborhood of singular points is established. By using methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite interval. Using an integral transformation a Riemann problem is obtained, and the solution is presented in its explicit form.

Kazimierz Abramowicz (1888−1936)

Kazimierz Abramowicz was a Polish mathematician, working in the theory of analytic functions. In 1922 he defended his Ph.D. thesis On hypergeometric functions with arbitrary amount of seemingly si ...

Elastoplastic problem for a plate with partially unknown boundary

In this paper there is considered the Elastoplastic problem for infinite plate, that is weakened by two identical square holes. The boundaries of the holes are partially unknown contours. The plate is in a stressed state, a region of plasticity contains only unknown parts of holes contours and does not spread inside of the plate. Applying the theory of functions of a complex variable and the conformal mapping theory the problem is reduced to a boundary value problem of the analytic function theory and the solution of this problem is obtained, the unknown parts of the holes contours are defined.

Contact problem of electroelasticity for a piecewise homogeneous piezoelectric plate with an elastic coating

Abstract The problem of determining the mechanical and electric fields in a piecewise homogeneous piezoelectric medium reinforced by a finite or semi-infinite coating is considered. Applying methods from the theory of analytic functions, the problem reduces to a singular integrodifferential equation with a fixed singularity. Using an integral transformation, a boundary value problem with shift (Riemann boundary value problem) is obtained, the solutions of which are represented in explicit form. On the basis of the solution obtained, an asymptotic study of contact stresses at end points (singular points) of the coating is performed.

Modeling the Movement of Groundwater from the Pits, Surrounded with Tongues of Zhukovsky

In the hydrodynamic statement the filtration low from ditches, walled tongues of Zhukovsky is considered. The fluid moves through the layer of soil, underlain by a well-permeable pressure aquifer, which is contained waterproof area on the roof. For study the infiltration to the free surface of groundwater is formulated a mixed multi-parameter boundary value problem of the theory of analytic function, which is solved by the Polubarinova-Kochina's method and ways the conformal mapping areas of a special kind, which are characteristic for tasks of an underground hydromechanics.

Modeling the Movement of Groundwater from the Pits, Surrounded with Piles of Zhukovsky

In the hydrodynamic statement the filtration low from ditches, walled tongues of Zhukovsky is considered. The fluid moves through the layer of soil, underlain by a well-permeable pressure aquifer, which is contained waterproof area on the roof. For study the infiltration to the free surface of groundwater is formulated a mixed multi-parameter boundary value problem of the theory of analytic function, which is solved by the Polubarinova-Kochina's method and ways the conformal mapping areas of a special kind, which are characteristic for tasks of an underground hydromechanics.

Circle packing with generalized branching

There is a fairly comprehensive theory of discrete analytic functions based on circle packing. In this theory, discrete analytic functions are represented as maps between circle packings that share combinatorial tangency patterns. Branching behavior, however, has until now been restricted by the need to place branch points at circle centers. In this paper, the authors introduce mechanisms for generalized branching which remove this restriction. Their use is illustrated by overcoming combinatorial obstructions to branching in the construction of a discrete Ahlfors function.

L2–Optimal Fopdt Models of High–Order Transfer Functions

AbstractNecessary conditions for the L2 optimality of a first order plus dead time (FOPDT) model of a high‐order plant are derived using classic analytic function theory. They are expressed as a set of three nonlinear equations that partly resemble the interpolation conditions valid for rational approximation. From these conditions a simple procedure to find the optimal FOPDT model is obtained. Examples taken from the relevant literature are worked out to show the performance of the method in comparison with alternative techniques.

Early-Age Stress and Pressure Developments in a Wellbore Cement Liner: Application to Eccentric Geometries

This paper introduces a predictive model for the stress and pressure evolutions in a wellbore cement liner at early ages. A pressure state equation is derived that observes the coupling of the elastic changes of the solid matrix, the eigenstress developments in the solid and porespaces, and the mass consumption of water in course of the reaction. Here, the transient constitution of the solid volume necessitates advancing the mechanical state of the poroelastic cement skeleton incrementally and at constant hydration degree. Next, analytic function theory is employed to assess the localization of stresses along the steel–cement (SC) and rock–cement (RC) interfaces by placing the casing eccentrically with respect to the wellbore hole. Though the energy release rate due to complete debonding of either interface is only marginally influenced by the eccentricity, the risk of evolving a microcrack along the thick portion of the sheath is substantially increased. Additionally, it is observed that the risk of microannulus formation is principally affected by the pressure rebound, which is engendered by the slowing reaction rate and amplified for rock boundaries with low permeability.

CORRECTNESS OF THE LOCAL BOUNDARY VALUE PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS

Correctness of boundary value problems in a plane for elliptical equations has been studied properly using the method of the theory of analytic functions. At investigation of analogous problems, when the number of independent variables is more than two, there arise principle difficulties. Quite good and convenient method of singular integral equations has to be abandoned because there is no complete theory of multidimensional singular integral equations. Boundary value problems for second-order elliptical equations in domains with edges have been studied properly earlier. Explicit classical solutions to Dirichlet and Poincare problems in cylindrical domains for one class of multidimensional elliptical equations can be found in the authors works. In this article,the author proved that the local boundary value problem, which is the generalization of Dirichet and Poincare problem, has only solution. Besides, the criterion of uniqueness of regular solution is obtained.

Determination of the Stress Field of the Brittle Rock Caused by Quasistatic Load

Abstract In the paper the mathematical model has been proposed according of which the rock is considered as an orthotropic half-plane. On the boundary of this half-plane the loads, moving at a constant velocity, exert a pressure. The problem was solved by means of the theory of analytical functions.

The contact problem for a piecewise-homogeneous orthotropic plate with a finite inclusion of variable cross-section

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite wedge-shaped inclusion, which meets the interface at a right angle and is loaded with normal forces is considered. The normal contact stresses along the contact line are determined and the behavior of the contact stresses in the neighborhood of singular points is established. By using methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite interval. Using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form.

The boundary contact problem of electroelasticity and related integral differential equations

The problem of finding mechanical and electrical fields in a homogeneous piezoelectric plate supported by a thin wedged-shaped cover plate is considered. Using the methods of the theory of analytic functions, the problem is reduced to a system of singular integro-differential equations. With the help of integral transformation, the exact solution of the posed by us problem is obtained.

Cyclic polynomials in two variables

We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for polynomials, and harmonic analysis on curves.

Radial fracture in a three-phase composite: Application to wellbore cement liners at early ages

Little understanding exists between the early-age stress developments in a wellbore cement sheath and its risk of impairment. During hydration, the cement morphology and pore-pressure changes induce eigenstresses in the solid and pore volumes. Utilizing these stresses as the driving mechanism of fracture, this paper formalizes the inspection of a radial crack in an elastic cement sheath constrained by an inner steel casing and an outer rock formation. The solution is constructed in the framework of analytic function theory and seeks the Green’s function for an edge dislocation in the intermediate cement phase. A dislocation pile-up along the line of fracture constructs a singular integral equation for the crack opening displacement derivative, from which the energy release rate is readily deduced.Under the uniform development of eigenstresses, the stiffness ratios of steel-to-cement and rock-to-cement generally predict the crack to initiate along the steel-cement interface. Here, the impacts of (i) a rigid bond and (ii) a sliding interface with no shear are assessed. This leads to the primary result of the paper: the potential for radial fracture is substantially mitigated by ensuring the shear connection between the steel casing and the cement sheath.

Решение второй основной задачи для упругого диска при неравномерном радиальном перемещении его границы

To solve this problem we apply the theory of analytic functions and formulas of Kolosov-Muskhelishvili in the case of non-uniform radial displacement of the boundary of the circle, represented as a Fourier series. The graphics and computational capabilities of Mathematica was shown. It allows building the distribution of stresses and displacements, using only the general Kolosov-Muskhelishvili formulas and coefficients of potentials calculated in complex form.