Theory Of Analytic Functions Research Articles

Lagrange’s theory of analytical functions and his ideal of purity of method

We reconstruct essential features of Lagrange's theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange's notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem --- especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method --- the generality and internal organization of the discipline.

Global Optimal Solutions of Noncyclic Mappings in Metric Spaces

We study some minimization problems for noncyclic mappings in metric spaces. We then apply the solution to obtain some results in the theory of analytic functions.

Circle packings as differentiable manifolds

Circle packings are configurations of circles satisfying specified patterns of tangency and have emerged as the foundation for a fairly comprehensive theory of discrete analytic functions. Though many classical results found their counterpart in circle packing, other concepts have not yet been transferred, particularly those which require a linear structure. This paper puts circle packings in a framework of smooth manifolds, providing access to linear structures in their tangent spaces. Since we are especially interested in applications to boundary value problems (of Beurling and Riemann–Hilbert type), we do not only investigate the manifolds of circle packings and packing labels, but also manifolds formed by the centers of the boundary circles. The approach is elementary and rests on a detailed analysis of the contact equations which govern the tangency relation between neighboring circles.

Problem of interaction of thin rigid needle-shaped inclusions located between different elastic materials

We study a piecewise-homogeneous elastic plane composed of two half-planes with different elastic parameters and two thin rigid needle-shaped inclusions located between them. One inclusion is rigidly connected with the environment, and the other inclusion is not, while contacting with it like a smooth rigid punch. We consider the plane deformed state generated by stresses given at infinity. The problem is reduced to a combination of a matrix Riemann boundary-value problem from the theory of analytic functions and a matrix Hilbert problem, which can be solved in terms of integrals through the reduction to two separate scalar Riemann boundary-value problems on a twosheeted Riemann surface.We explicitly obtain the complex potentials of the composite elastic plane, the stress intensity factors near the tips of the inclusion, and the rotation angles of the inclusions. We also present numerical examples illustrating how the stresses near the inclusions depend on the elastic and geometric parameters of the problem.

On ground water seepage under hydraulic structures

This present paper continues paper [1] of the authors and is concerned with the following problems regarding seepage flows with unknown boundaries under hydraulic structures, with use being made of the theory of two-dimensional (2D) steady state seepage of incompressible fluid according to Darcy’s law in a uniform isotropic soil. Below we construct a smooth underground contour of a rectangular subsurface dam of constant seepage rate in the case when a pervious abutment is underlain by a blanket consisting of two curvilinear and one (middle) horizontal elements which are also characterized by constant incident velocity. We are concerned with the fluid flow below Joukowski’s cutoff wall through the watered soil mass underlain by a highly permeable aquifer with confined groundwater; the left semi-infinite part of its roof is simulated by impermeable inclusion. These motions are examined using multiparameter problems of the theory of analytic functions. Their solutions are obtained by the semi-inverse method of the velocity hodograph by P.Ya. Polubarinova-Kochina and I.N. Kochina, by Polubarinova-Kochina’s approach dependent on the analytic theory of linear Fuchsian equations, and using the available conformal mappings of domains of special kinds peculiar for underground hydrodynamics. The results of the numerical calculations are given and the hydrodynamic effect of the physical parameters of models on the flow pattern is ascertained. The limiting cases previously examined in [1] are singled out.

Certain subclasses of analytic functions associated with fractional $q$-calculus operators

We first define the $q$-analogue operators of fractional calculus which are then used in defining certain classes of functions analytic in the open disk. The results investigated for these classes of functions include the coefficient inequalities and some distortion theorems. The results provide extensions of various known results in the $q$-theory of analytic functions. Special cases of our results are pointed out briefly.

Dynamic noise perturbed generalized superior Mandelbrot sets

The invention of the latest tools and technology toward computer aided graphics and drawing completely change the thinking view of researchers in analyzing and studying the behavior of a dynamical system. Inspired by work already performed and by adopting the experimental mathematical methods of combining the theory of analytic function with computer aided drawing technology, we generated generalized superior Mandelbrot sets (SM-sets). Also, we analyzed the effect of dynamic noise on SM-sets.

Mathematical modeling of flows from canals

In the hydrodynamic formulation, two-dimensional steady filtration in homogeneous isotropic ground from canals through a soil layer to the underlying highly permeable pressure water-bearing stratum is considered in the presence of the ground capillarity and evaporation from the free surface. To study filtration, a combined multiparametric boundary-value problem of the theory of analytical functions is formulated, which is solved using the P. Ya. Polubarinova-Kochina method and procedures of conformal mapping of regions of a special kind that are characteristic of the problems of subsurface hydromechanics. On the basis of this model an algorithm of calculating the capillary water spreading and the filtration discharge is developed for the situations where in water filtration from canals provision is made for the ground capillarity, evaporation from the free-surface of groundwater, and the additional pressure from the side of water of the underlying wellpermeable bed. With the aid of the obtained accurate analytical relations and numerical calculations a hydrodynamic analysis is made of the structure and character of specific features of the modeled process as well as of the effect of all physical parameters of the scheme on the filtration characteristics.

Completeness theorem for singular differential pencils

A theorem completeness theorem of special vector functions induced by the products of the so-called Weyl solutions of a fourth-order differential equation and by their derivatives on the semiaxis is presented. We prove that such nonlinear combinations of Weyl solutions and their derivatives constitute a linear subspace of decreasing (at infinity) solutions of a linear singular differential system of Kamke type. We construct and study the Green function of the corresponding singular boundary-value problems on the semiaxis for operator pencils defining differential systems of Kamke type. The required completeness theorem is proved by using the analytic and asymptotic properties of the Green function, operator spectral theory methods, and analytic function theory.

Ordinary differential operator and some of its applications to certain meromorphically p-valent functions

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out.

Spectral theory for nonconservative transmission line networks

The global theory of transmission line networks withnonconservative junction conditions is developed from a spectraltheoretic viewpoint. The rather general junction conditionslead to spectral problems for nonnormal operators.The theory of analytic functions which are almost periodic in a stripis used to establish the existence of an infinite sequence of eigenvaluesand the completeness of generalized eigenfunctions. Simple eigenvalues are generic.The asymptotic behavior of an eigenvalue counting function is determined.Specialized results are developed for rational graphs.

Simulation of seepage flows from channels

Plane steady-state seepage in a homogeneous isotropic ground from channels through a layer of soil with an underlying highly permeable pressurized water-bearing layer when the ground possesses capillarity and there is evaporation from the free surface is considered in a hydrodynamic formulation. To investigate it, a mixed multiparametric boundary-value problem of the theory of analytical functions is formulated, which is solved using the Polubarinova-Kochina method and conformal mapping of the regions of a special form, typical of problems of underground hydromechanics. On the basis of this model, an algorithm for calculating the capillary spreading of water and seepage flow is developed in situations when the ground capillarity is taken into account in the seepage of water from channels, as well as evaporation from the free surface of the ground waters, and also backwater of the underlying highly permeable stratum. Using the exact analytical relations obtained and numerical calculations, a hydrodynamic analysis of the structure and characteristic features of the simulated process, and also of the effect of all the physical parameters of the system on the seepage characteristics, is carried out. Limit and special cases, related to the absence of one or two of the three factors, characterizing the simulated process are considered: the ground capillarity, evaporation from the free surface, and also backwater of the underlying highly permeable water-bearing layer. The results of the calculations are compared with similar seepage characteristics with a similar scheme, but in which the flow region is underlaid by an impenetrable base.

The contact problem for a piecewise-homogeneous plane with a finite inclusion

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

Composition operators on noncommutative Hardy spaces

In this paper we initiate the study of composition operators on the noncommutative Hardy space H ball 2 , which is the Hilbert space of all free holomorphic functions of the form f ( X 1 , … , X n ) = ∑ k = 0 ∞ ∑ | α | = k a α X α , ∑ α ∈ F n + | a α | 2 < 1 , where the convergence is in the operator norm topology for all ( X 1 , … , X n ) in the noncommutative operatorial ball [ B ( H ) n ] 1 and B ( H ) is the algebra of all bounded linear operators on a Hilbert space H . When the symbol φ is a free holomorphic self-map of [ B ( H ) n ] 1 , we show that the composition operator C φ f : = f ∘ φ , f ∈ H ball 2 , is bounded on H ball 2 . Several classical results about composition operators (boundedness, norm estimates, spectral properties, compactness, similarity) have free analogues in our noncommutative multivariable setting. The most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of B ( H ) n , the operator algebras generated by the left creation operators on the full Fock space with n generators, and the classical complex function theory in the unit ball of C n . In a more general setting, we establish basic properties concerning the composition operators acting on Fock spaces associated with noncommutative varieties V P 0 ( H ) ⊆ [ B ( H ) n ] 1 generated by sets P 0 of noncommutative polynomials in n indeterminates such that p ( 0 ) = 0 , p ∈ P 0 . In particular, when P 0 consists of the commutators X i X j − X j X i for i , j = 1 , … , n , we show that many of our results have commutative counterparts for composition operators on the symmetric Fock space and, consequently, on spaces of analytic functions in the unit ball of C n .

Influence of the ground capillarity and of evaporation from the free groundwater surface on filtration from canals

Mathematical models of certain groundwater flows from a canal in the presence of free-surface evaporation are considered on the basis of the two-dimensional theory of steady-state filtration. Mixed boundary-value problems of analytical-function theory are formulated and solved with the Polubarinova-Kochina method for studying such flows. Algorithms of calculation of the dimensions of the saturation zone in situations where, in filtration of water from a canal, the capillarity of its ground or the level of water in it are allowed for are based on these models. Using the obtained exact analytical dependences and numerical calculations, we made a detailed analysis of the structure and characteristic features of the modeled processes and of the influence of all physical characteristics of the models on the dimensions of the saturation zone.

Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an ( N + 1 ) (N+1) -connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetric Schottky group. We prove the existence theorem for the second (quasimultiplicative) kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem, which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with N + 1 N+1 circular holes with electrodes and dielectrics on the walls when the conductor is placed at a right angle to the applied magnetic field.

A summary of Euler’s work on the pentagonal number theorem

In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the pentagonal number theorem, we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions.

Average power input into a barrier discharge by pulsating electric current

The analytical function theory is used to derive formulas for calculating the average power delivered by electric current to a pulsating (filamentary) barrier discharge.

Expository Research Papers

The two papers in this issue deal with differential equations, one with the numerical solution of partial differential equations, and the other one with analytic solutions for ordinary differential equations. In his paper From Functional Analysis to Iterative Methods, Robert Kirby is concerned with linear systems arising from discretizations of partial differential equations (PDEs). Specifically, the PDEs are elliptic and describe boundary value problems; the discretizations are done via finite elements, and at issue is the convergence rate of iterative methods for solving the linear systems. The author's approach is to go back to the underlying variational problem in a Hilbert space, and to make ample use of the Riesz representation theorem. This point of view results in short and elegant proofs, as well as the construction of efficient preconditioners. The general theory is illustrated with two concrete model problems of PDEs for convection diffusion and planar elasticity. This paper will appeal to anybody who has an interest in the numerical solution of PDEs. In 1963 the mathematician/meteorologist Edward Lorenz formulated a system of three coupled nonlinear ordinary differential equations, whose long-term behavior is described by an attractor with fractal structure. You can see a beautiful rendition of the thus named Lorenz attractor on the cover of this issue. Although it is easy to plot solutions of the Lorenz system, it is much harder to determine them mathematically. This is what motivated the paper Complex Singularities and the Lorenz Attractor by Divakar Viswanath and Sonmez Sahutoglu. Their idea is to allow the time variable to be complex, rather than real; to focus on singular solutions; and to express these singular solutions in terms of so-called psi series. After all is said and done, the authors end up with a two-parameter family of complex solutions to the Lorenz system. This a highly readable and very enjoyable paper, with concrete steps for future research, and connections to thunderstorms and analytic function theory.

RESONANCE WAVE SCATTERING BY A STRIP GRATING ATTACHED TO A FERROMAGNETIC MEDIUM

The difiraction of a uniform unit-amplitude E-polarized plane wave is considered in the case of its normal incidence on a strip periodic metal grating placed on the anisotropic hyrotropic ferromagnetic half-space boundary. The Dirichlet boundary conditions on the grating strips, the medium interface conjugation conditions, the Meixner condition that the energy is flnite in any conflned volume and the radiation condition are applied, and the boundary value difiraction problem in terms of Maxwell's (Helmholtz) equations is equivalently reduced to the dual system of functional equations with exponential kernel. The system is shown to be the Riemann-Hilbert problem in analytic function theory with the conjugation coe-cient difiering, in general, from \i1 and dependent on the incident wave frequency. An analytical regularization procedure based on the Riemann-Hilbert boundary value problem solution with the following use of the Plemelle- Sokhotsky formulas is suggested, resulting in the system of linear algebraic equations of the second kind with a compact operator. For these systems, the truncation technique possibility has been shown.