Julia lines of random Dirichlet series

For the random entire functions defined by some random Dirichlet series almost surely every horizontal line is a Julia line without exceptional values. These results contain and generalize partially a theorem of P.L. Davies. The proofs are different.     

Almost continuability of solutions of differential equations

We introduce the notion of almost continuability of the solution of the differential equation of first order dy/dx = f(x, y) to the whole real axis. We give a criterion for the almost continuability of solutions for the case in which the right-hand side of the equation is a meromorphic function of one variable y: f(x, y) = g(y). As an example, we work out the case of a rational and, in particular, an entire function g(y).     

A linear boundary value problem for weakly quasiregular mappings in space

Given a number a weakly L-quasiregular map on a domain in space is a map u in a Sobolev space that satisfies almost everywhere in In this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space with dimension It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in can only assume a quasiregular linear boundary value; however, for all and , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in Received July 20, 2000 / Accepted September 22, 2000 / Published online December 8, 2000     

Computation of alternated reachability tubes for linear control systems under uncertainty

The problem of constructing the reachability domain for linear control system in a presence of geometrically bounded unknown disturbance is considered. A high actuality of the problem for engineering applications requires an efficient calculation technique for the reach sets in a class of closed-loop control. The technique suggested in the article is based on the ellipsoidal approximations developed by A.B. Kurzhansky for the alternated reachability domains. In the article these estimates are complemented with an adaptive regularization to guarantee the continuability of the ellipsoidal estimates. The quadratic structure of the regularization combines well with an ellipsoidal nature of the estimates thus making it possible to adjust the existing ellipsoidal estimation schema in a transparent fashion for achieving the continuable and non-singular estimates via adaptive choice of regularization parameters.     

Rational functions of best L p- approximation and analytic continuability

Sufficient conditions for the analytic (resp. meromorphic) continuability of a function f ∈ L _p([?1, l]), p > 0, in terms of rational functions of best weighted L _p- approximation with an unbounded number of finite poles are established.     

Almost coreflective and projective classes

This paper summarizes the situation around the problem of when classes of projective objects are almost coreflective, both in general categories and in Top or similar categories. In addition to known results, several new contributions and examples are added.For the sixtieth birthday of D. PumplünThe paper was written while the second author was visiting the University of Toledo, Ohio.     

On the analytic continuation of functions defined by regrouped power series

We consider functions defined by regrouped power series \(f(z) = \sum\nolimits_{n = 0}^\infty z ^{\lambda _{n_{P_{k_n } } } } (z)\) in the disk |z|<1 and also in some domain D outside of this disk. We obtain conditions under whichf(z) is analytically continuable outside of the disk |z|<1, the analytic continuation being effected with the help of the given series. We also consider the analytic continuability of functionsf(z, w).     

Investigation of solutions to one family of mathematical models of living systems

We consider a family of integral equations used as models of some living systems. We prove that an integral equation is reducible to the equivalent Cauchy problem for a non-autonomous differential equation with point or distributed delay dependently on the choice of the survival function of system elements. We also study the issues of the existence, uniqueness, nonnegativity, and continuability of solutions. We describe all stationary solutions and obtain sufficient conditions for their asymptotic stability. We have found sufficient conditions for the existence of a limit of solutions on infinity and present an example of equations where the rate of generation of elements of living systems is described by a unimodal function (namely, the Hill function).     

Hecke operators of the symplectic group of genus two over a real field

We investigate the arithmetic character of the Fourier coefficients of a Hilbert-Siegel modular form which is an eigenfunction of all Hecke operators. We prove the analytic continuability and a functional equation for the corresponding zeta function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 48–58, 1980.     

Baxter-Guttmann-Jensen conjecture for power series in directed percolation problem

A probabilistic model of a flow of fluid through a random medium,percolation model, provides a typical example of statistical mechanical problems which are easy to describe but difficult to solve. While the percolation problem on undirected planar lattices is exactly solved as a limit of the Potts models, there still has been no exact solution for the directed lattices. The most reliable method to provide good approximations is a numerical estimation using finite power-series expansion data of the infinite formal power series for percolation probability. In order to calculate higher-order terms in power series, Baxter and Guttmann [6] and Jensen and Guttmann [33] proposed an extrapolation procedure based on an assumption that thecorrection terms, which show the difference between the exact infinite power series and approximate finite series, are expressed as linear combinations of the Catalan numbers.In this paper, starting from a brief review on the directed percolation problem and the observation by Baxter, Guttmann, and Jensen, we state some theorems in which we explain the reason why the combinatorial numbers appear in the correction terms of power series. In the proof of our theorems, we show several useful combinatorial identities for the ballot numbers, which become the Catalan numbers in a special case. These identities ensure that a summation of products of the ballot numbers with polynomial coefficients can be expanded using the ballot numbers. There is still a gap between our theorems and the Baxter-Guttmann-Jensen observation, and we also give some conjectures.As a generalization of the percolation problem on a directed planar lattice, we present two topics at the end of this paper: The friendly walker problem and the stochastic cellular automata in higher dimensions. We hope that these two topics as well as the directed percolation problem will be of much interest to researchers of combinatorics.     

On grid generation for numerical models of geophysical fluid dynamics

A simple geometric condition that defines the class of classical (stereographic, conic and cylindrical) conformal mappings from a sphere onto a plane is derived. The problem of optimization of computational grid for spherical domains is solved in an entire class of conformal mappings on spherical (geodesic) disk. The characteristics of computational grids of classical mappings are compared for different spherical radii of geodesic disk. For a rectangular computational domain, the optimization problem is solved in the class of classical mappings and respective area of the spherical domain is evaluated.     

Global existence of solutions for perturbed differential equations

In this paper we consider sufficient conditions for the continuability of solutions for perturbed differential equations. We obtain also some results for the global existence of solutions for differential inclusions and for stochastic differential equations of McShane and Ito type. We give an application to the global inversion of local diffeomorphisms.     

ON THE CONTINUABILITY OF SOLUTIONS IN THE FUTURE FOR THE LIENARD SYSTEM

ＯＮＴＨＥＣＯＮＴＩＮＵＡＢＩＬＩＴＹＯＦＳＯＬＵＴＩＯＮＳＩＮＴＨＥＦＵＴＵＲＥＦＯＲＴＨＥＬＩＥＮＡＲＤＳＹＳＴＥＭＨｕａｎｇＬｉｈｏｎｇ（黄立宏）（ＨｕｎａｎＵｎｉｖｅｒｓｉｔｙ，湖南大学，邮编：４１００８２）Ａｂｓｔｒａｃｔ：Ｔｈｅｍａｉｎｐ...     

Shatalov‐Sternin's construction of complex WKB solutions and the choice of integration paths

We re‐examine Shatalov‐Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the “Riemann surface” (actually, a two‐dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the “Riemann surface” relevant for the exact WKB method.     

Estimates for eigenvalues of quasilinear elliptic systems

In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov's inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve.     

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.     

Fixed zeros in the model matching problem for systems over semirings

This paper studies “fixed zeros” of solutions to the model matching problem for systems over semirings. Such systems have been used to model queueing systems, communication networks, and manufacturing systems. The main contribution of this paper is the discovery of two fixed zero structures, which possess a connection with the extended zero semimodules of solutions to the model matching problem. Intuitively, the fixed zeros provides an essential component that is obtained from the solutions to the model matching problem. For discrete event dynamic systems modeled in max-plus algebra, a common Petri net component constructed from the solutions to the model matching problem can be discovered from the fixed zero structure.     

Obstructions to embeddability into hyperquadrics and explicit examples

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstnerič. The author was supported in part by the RCBS grant of Trinity College Dublin and by the Science Foundation Ireland.     

An algorithm for evolutionary surfaces

Summary An Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateau's problem forH-surfaces.     

Error analysis for optimal control problem governed by convection diffusion equations: DG method

In the current paper, we study the convergence properties of the DGFE approximation of optimal control problem governed by convection-diffusion equations. We derive a posteriori error estimates and a priori error estimates for both the states, ad-joint and the control variable approximation. For the optimal control problem, these estimates are apparently not available in the literature.