Analytic continuation problem in CP

Rostov on the Don. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 4, pp. 80–98, July–August, 1988.     

Analytic continuation from a family of lines

Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables.     

Analytic continuation from a continuum to its neighborhood

Let be a positive number. A function f analytic in an open set ? ?? is called -analytic on the set E, E ??, if . THEOREM. Let K be a compact connected subset of the plane. For any >0 there exists an open neighborhood V of the set K such that any function -analytic on K coincides in some neighborhood of the set K with a function analytic in V. This theorem answers a question posed in the collection (RZhMat., 1979, 3B536, pp. 33–35 of the book).     

Analytic continuation of a class of dirichlet series

We consider Dirichlet series of the type Σ(logk)ⁿ(k)(logk)?^-s. We prove the existence of an analytic continuation to the cut plane and give exact information about the singularity. We use this to generalize results, which occur in Ramanujan’s second notebook.     

Analytic continuation ofs matrix in multichannel problems

The analytic continuation of thet ands matrices to the Riemann surface in multichannel problems is investigated. Explicit representations for the continuedt ands matrices on all sheets are obtained in terms of thet ands matrices on the physical sheet.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 427–438, June, 1993.     

Analytic continuation formulas and Jacobi-type relations for Lauricella function

An approach for constructing a complete system of formulas for the analytic continuation of the Lauricella generalized hypergeometric function F_D^(N) with any N beyond the boundary of the unit polydisk is proposed. The approach is exposed in detail for the continuation of the function under consideration in neighborhoods of points whose all N components equal 1 or ∞. For the Lauricella function, differential relations being analogues of Jacobi’s formula for the Gaussian hypergeometric function are also presented. The results can be applied to solve the crowding problem for the Schwarz–Christoffel integral and to the theory of the Riemann–Hilbert problem.     

A non-interior continuation algorithm for the CP based on a generalized smoothing function

Based on the generalized CP-function proposed by Hu et al. [S.L. Hu, Z.H. Huang, J.S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009) 69-82], we introduce a smoothing function which is a generalization of several popular smoothing functions. By which we propose a non-interior continuation algorithm for solving the complementarity problem. The proposed algorithm only needs to solve at most one system of linear equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate that the algorithm is effective.     

Virtual criterion for generalized Eagon-Northcott complexes

Given any map of finitely generated free modules, Buchsbaum and Eisenbud define a family of generalized Eagon-Northcott complexes associated to it [3]. We give sufficient criterion for these complexes to be virtual resolutions, thus adding to the known examples of virtual resolutions, particularly those not coming from minimal free resolutions.     

Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs

In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher-dimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, via a contraction argument, the existence and local uniqueness of solutions for a rather large class of nonlinear problems. We apply this technique to prove existence and local uniqueness of equilibrium solutions for the Cahn-Hilliard and the Swift-Hohenberg equations defined on two- and three-dimensional spatial domains.     

Analytic continuation of eigenvalues of the Lamé operator

Eigenvalues of the Lamé operator are studied as complex-analytic functions in period τ of an elliptic function. We investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues and the convergent radius of a perturbation series.     

On the continuation of the solutions of a generalized Cauchy-Riemann system in space

The problem of the analytic continuation of the solutions of a generalized Cauchy-Riemann system from its values on a part of the domain boundary is considered.     

Analytic continuation of the exponential transform from convex cavities

The analytic continuation of the exponential transform of a domain in Rn is proved under some global geometric assumptions on the boundary. Two approximation schemes of the continued transform (one based on a Taylor series truncation, the other on a global eigenfunction expansion) are also discussed.