A modification of an approach to the solution of the Hilbert boundary-value problem for an analytic function in a multiconnected circular domain

We propose a modification of the approach proposed by us in Russian Mathematics (Iz. VUZ) 44 (2), 58–62 (2000) for the solution of the Hilbert boundary-value problem for an analytic function in a multiconnected circular domain. This approach implies the solution of the corresponding homogeneous problem including the determination of an analytic function from the known boundary values of its argument in a circular domain.     

An analytical solution for a nonlinear time-delay model in biology

In this paper, the homotopy analysis method is applied to develop a analytic approach for nonlinear differential equations with time-delay. A nonlinear model in biology is used as an example to show the basic ideas of this analytic approach. Different from other analytic techniques, the homotopy analysis method provides a simple way to ensure the convergence of the solution series, so that one can always get accurate approximations. A new discontinuous function is defined so as to express the piecewise continuous solutions of time-delay differential equations in a way convenient for symbolic computations. It is found that the time-delay has a great influence on the solution of the time-delay nonlinear differential equation. This approach has general meanings and can be applied to solve other nonlinear problems with time-delay.     

<Emphasis Type="Italic">N</Emphasis>=1 supersymmetric conformal block recursion relations

We present explicit recursion relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. We compare the results with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursion relations are an efficient tool for numerically studying the four-point correlation function in superconformal field theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 476–487, September, 2007.     

On a generalised<Emphasis Type="Italic">ε</Emphasis>-control problem in Banach space

In this paper a linearε-control problem for a general objective function is considered by functional analytic approach. Existence and uniqueness of solutions are shown. The characterization of the solution is given in terms of a hyperplane.     

An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

In this paper, the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by means of an analytic approach. Unlike perturbation techniques, this approach gives accurate analytic approximations uniformly valid for all dimensionless time. Besides, a simple but accurate analytic formula for the local skin friction is given, which agrees well with numerical results and thus is useful in the related industries. To the best of our knowledge, this type of analytic solutions has been never reported. Furthermore, the proposed analytic approach has general meaning and therefore may be applied in the similar way to other unsteady boundary-layer flows to get accurate analytic solutions valid for all time.     

An Analytic Approach to <Emphasis Type="Italic">Q</Emphasis>-Balls

We present the analytic approach for constructing Q-ball solutions in the general case of a sixth-order potential. In particular, we show that the symmetrized Woods-Saxon function describes the Q-ball profile function. The energy and charge can therefore be evaluated explicitly. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 342–347, August, 2005.     

Inversion of Analytic Functions via Canonical Polynomials: A Matrix Approach

An alternative to Lagrange inversion for solving analytic systems is our technique of dual vector fields. We implement this approach using matrix multiplication that provides a fast algorithm for computing the coefficients of the inverse function. Examples include calculating the critical points of the sinc function. Maple procedures are included which can be directly translated for doing numerical computations in Java or C. A preliminary version of this paper has been presented at AISC 2006.     

Projections of convex bodies and the fourier transform

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl _p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections ofl _p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.     

Variable 2-Microlocal Besov–Triebel–Lizorkin-Type Spaces

This article is devoted to the study of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of φ-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some related Sobolev-type embeddings and trace theorems of these spaces are also established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel–Lizorkin spaces.     

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.     

Sensitivity and Approximation of Coupled Fluid-Structure Equations by Virtual Control Method

The formulation of a particular fluid--structure interaction as an optimal control problem is the departure point of this work. The control is the vertical component of the force acting on the interface and the observation is the vertical component of the velocity of the fluid on the interface. This approach permits us to solve the coupled fluid--structure problem by partitioned procedures. The analytic expression for the gradient of the cost function is obtained in order to devise accurate numerical methods for the minimization problem. Numerical results arising from blood flow in arteries are presented. To solve the optimal control problem numerically, we use a quasi-Newton method which employs the analytic gradient of the cost function and the approximation of the inverse Hessian is updated by the Broyden, Fletcher, Goldforb, Shano (BFGS) scheme. This algorithm is faster than fixed point with relaxation or block Newton methods.     

Regular bilinear systems

This paper deals with (weakly) regular and abstract bilinearsystems which allow some degree of unboundedness in the controland observation operators for infinite-dimensional bilinearsystems. By adaption of the notion of abstract linear systemsintroduced by Salamon (1989) and Weiss (1989a), we give a bilinearversion of their approach which works in our setting. The paperinvestigates also the relation between the shift-invariant bilinearoperator and an analytic function. This latter can be consideredas a transfer function for bilinear systems. The results thusobtained are used to study the problem of observer design forsuch systems.     

On an approximative version of the notion of constructive analytic function

A constructive analytic function f is defined as a pair of form (A,), where. A is a fundamental sequence in some constructive metric space and is a regulator of its convergence into itself. The pointwise-defined function f corresponding to function f^* turns out to be Bishop-differentiable [2], while the domain of f^* is the limit of a growing sequence of compacta. The derivative of a constructive analytic function and the integral along a curve are defined approximatively. It is proved that the fundamental theorems of constructive complex analysis are valid for such functions. Eight items of literature are cited.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 50, pp. 49–58, 1976. Result announced November 1, 1973.The author thanks N. A. Shanin, who suggested the consideration of the approximative approach to the definition of a constructive analytic function.     

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.     

The <Emphasis Type="Italic">p</Emphasis>-adic sector of the adelic string

We study the construction of Lagrangians that can be considered the Lagrangians of the p-adic sector of an adelic open scalar string. Such Lagrangians are closely related to the Lagrangian for a single p-adic string and contain the Riemann zeta function with the d’Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach that combines all p-adic Lagrangians. This new Lagrangian is attractive because it is an analytic function of the d’Alembertian. Investigating the field theory with the Riemann zeta function is also interesting in itself.     

Mixed boundary value problem for some pairs of metaanalytic function and analytic function

In this article, some mixed boundary value problems (BVPs) on the unit circumference for some pairs of a metaanalytic function and an analytic function are discussed. Using the relationship between metaanalytic function and polyanalytic function, the expression of solution and the condition of solvability for the problem are obtained by reducing the problem to an equivalent system of a Haseman BVP for analytic function and a Hilbert BVP for analytic function. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimates for the p-Laplacian-type operator in the subcritical Sobolev exponent range

We obtain Calderón–Zygmund-type gradient estimates below the duality exponent of very weak solutions to p-Laplacian-type elliptic equations with non-divergence datum by providing an analytic approach without using the fractional maximal function of order 1 for the non-divergence datum.     

A periodic single species model with intermittent unilateral diffusion in two patches

In this paper, we consider a periodic single species model with intermittent unilateral diffusion in two patches. By using analytic method, Poincare mapping, Lyapunov function approach, sufficient and necessary conditions on the existence, uniqueness and global attractivity of positive periodic solution and the extinction of species for the considered system are established. Two examples and numerical simulations are presented to validate our theoretical results.     

Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method

A general analytic approach for nonlinear eigenvalue problems is described. Two physical problems are used as examples to show the validity of this approach for eigenvalue problems with either periodic or non-periodic eigenfunctions. Unlike perturbation techniques, this approach is independent of any small physical parameters. Besides, different from all other analytic techniques, it provides a simple way to ensure the convergence of series of eigenvalues and eigenfunctions so that one can always get accurate enough approximations. Finally, unlike all other analytic techniques, this approach provides great freedom to choose an auxiliary linear operator so as to approximate the eigenfunction more effectively by means of better base functions. This approach provides us a new way to investigate eigenvalue problems with strong nonlinearity.