New uniqueness theorems for trigonometric series

A uniqueness theorem is proved for trigonometric series and another one is proved for multiple trigonometric series. A corollary of the second theorem asserts that there are two subsets of the -dimensional torus, the first having a countable number of points and the second having points such that whenever a multiple trigonometric series ``converges' to zero at each point of the former set and also converges absolutely at each point of the latter set, then that series must have every coefficient equal to zero. This result remains true if ``converges' is interpreted as any of the usual modes of convergence, for example as ``square converges' or as ``spherically converges.'

     

Coefficients of convergent multiple Haar series

It is well-known that if a multiple trigonometric series almost everywhere converges in the square or restricted rectangular sense to a finite function, then its coefficients grow slower than any exponential function. In this paper we prove the existence of a multiple Haar series that converges in the square or restricted rectangular sense to a finite function and contains a subsequence of coefficients that grows faster than any sequence defined in advance. Moreover, we show that for such series conditions of the Arutyunyan-Talalyan type can be violated at some points.     

SOME PROPERTIES OF MULTIPLE TAYLOR SERIES AND RANDOM TAYLOR SERIES

Some polar coordinates are used to determine the domain and the ball of convergence of a multiple Taylor series. In this domain and in this ball the series converges, converges absolutely and converges uniformly on any compact set. Growth and other properties of the series may also be studied. For some random multiple Taylor series there are some corresponding properties.     

Hyperbolic-parabolic deformations of rational maps Dedicated to the Memory of Professor Lei Tan

We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.     

Hyperbolic-parabolic deformations of rational maps

We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.     

Perverse Sheaves on Image Multiple Point Spaces

Using multiple point spaces some new examples of perverse sheaves on images of maps are described. Furthermore, suppose is a finite and proper map of complex analytic manifolds of dimension n and n+1 such that every multiple point space is nonsingular and has the dimension expected of a generic map. Then we can describe the composition series for the constant sheaf on the image in the category of perverse sheaves.     

On some continuity and differentiability properties of paths of Gaussian processes

The following path properties of real separable Gaussian processes ξ with parameter set an arbitrary interval are established. At every fixed point the paths of ξ are continuous, or differentiable, with probability zero or one. If ξ is measurable, then with probability one its paths have essentially the same points of continuity and differentiability. If ξ is measurable and not mean square continuous or differentiable at every point, then with probability one its paths are almost nowhere continuous or differentiable, respectively. If ξ harmonizable or if it is mean square continuous with stationary increments, then its paths are absolutely continuous with probability one if and only if ξ is mean square differentiable; also mean square differentiability of ξ implies path differentiability with probability one at every fixed point. If ξ is mean square differentiable and stationary, then on every interval with probability one its paths are either differentiable everywhere or nondifferentiable on countable dense subsets. Also a class of harmonizable processes is determined for which of the following are true: (i) with probability one paths are either continuous or unbounded on every interval, and (ii) mean square differentiability implies that with probability one on every interval paths are either differentiable everywhere or nondifferentiable on countable dense subsets.     

Bisection hardly ever converges linearly

Summary. For real functions that cross the unit interval, the method of bisection converges linearly if, but only if, the point of crossing is a diadic number where the function does not vanish, or, except for finitely many digits, its binary expansion coincides with that of one third or two thirds. Otherwise, the order of convergence remains undefined. If the point of crossing is one of Borel's normal real numbers (Lebesgue's measure of all of which equals one), then the sequence of ratios of two consecutive errors accumulates simultaneously at zero, one half, and negative infinity. Thus, in every finite sequence of estimates from the bisection, the last estimate need not be more accurate than the first one.     

An example of a zero series expansion in the Walsh system

We construct an example of a zero series expansion in the Walsh system which converges to zero outside some closed M set of zero measure and converges to + at each point of this set. This shows, in particular, that in the theorem which says that a Walsh series which converges everywhere to a finite symmetric function is a Fourier series it is impossible to omit the requirement of finiteness and allow convergence of the series on a set of zero measure to an infinity of specified sign.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 179–186, February, 1976.     

Uniqueness of double Walsh series

Summary We prove that under mild growth conditions, uniqueness holds for a multiple Walsh series whose square dyadic partial sums converge almost everywhere to an integrable function. We apply this result to obtain a new uniqueness result for Cesáro summable multiple Walsh series.     

On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity

This paper is concerned with a Cauchy problem where and is a nonnegative radially symmetric function in with compact support. Denote the solution of (P) by . Let if and $p^{\ast} = 1+6/(N-10) N \geq 11 p_{\ast} < p < p^{\ast} \lambda_{\varphi} > 0 $ such that: (i) If $ \lambda < \lambda_{\varphi} u_{\lambda} $ exists globally in time in the classical sense and converges to zero locally uniformly in as . (ii) If , then $ u_{\lambda} $ blows upincompletely in finite time. (iii) If , then blows upcompletely in finite time. Received: 20 December 1999; in final form: 26 May 2000 / Published online: 4 May 2001     

A near equitable 2-person cake cutting algorithm

Let the cake be represented by the unit interval of reals, with two players having possibly different valuations. We propose a finite algorithm that produces contiguous pieces for both players such that their values differ by at most ?, where ??>?0 is a given small number. Players are not required to reveal their complete value functions, they only have to announce the bisection points of a sequence of intervals. If both utility functions are everywhere positive then the algorithm converges to the unique equitable point.     

Perturbing PLA

We proved earlier that every measurable function on the circle, after a uniformly small perturbation, can be written as a power series (i.e., a series of exponentials with positive frequencies), which converges almost everywhere. Here, we show that this result is basically sharp: the perturbation cannot be made smooth or even Hölder. We also discuss a similar problem for perturbations with lacunary spectrum.     

关于正弦级数$\bm{L^{1}}$-收敛性研究中对单调递减条件的确切推广

本文在处理$L^1$-收敛性问题中给出了一个确切的条件和一种更直接的方式.     

用黎曼求和法(R,1)求和時的基卜斯現象

<正> 假設級數滿足下面兩個條件,即:(甲)在原點的某一鄰域內,對於h(≠0)的一切值級數收斂;     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,I

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.     

Convergence of the Magnus Series

The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results.     

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

l)ThisworkwassupportedbyNWOthroughgrantIBo7-3Go12.BOUNDAarv^LUEPRoBLEMFORELLIPTICEQUMIONwiTHMIXEDBOUNDAavCONDITION1.IntroductionInthispedwesketchavarietyofspecialmethodswhichareusedforconstructinge-unifornilyconvergelltschemes-WeshaJldemonstrateamethodwhichachieveshaprovedaccuracyforsolvingsingularlyperturbedb0undaryvalueproblemforeiliPicequatiouswithparabolicboundarylayers-InSecti0n4weshallintroduceanaturalclass,B,oftritefferenceschemes,inwhich(bytheabovementi0nedaP…     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,III

1.IntroductionThesolution0fpartialdifferentiaJequationsthataresingularlyperturbedand/orhavediscontinu0usboundaryconditionsgenerallyhave0nlylimitedsmoothness.DuetothisfaCtdndcultiesaPpearwhenwesolvethesepr0blemsbynumericalmethods.Forexampleforregularparab0licequationswithdiscontinuousboundaryconditions,classicalmethods(FDMorFEM)onregularrectangulargridsd0n0tconvergeintheIoo-normonadomainthatincludesaneighbourhood0fthediscontinulty[8,9,4].Iftheparametermultiplyingthehighest-orderderivativeva…     

关于一致凸Banach空间中渐近非扩张半群的几乎轨道的渐近行为

设Ｘ是具有Ｆｒｃｈｅｔ可微范数的一致凸Ｂａｎａｃｈ空间，Ｃ是Ｘ的有界闭凸子集，Ｓ＝｛Ｔ（ｔ）：ｔ≥０｝是Ｃ上渐近非扩张牛群．若ｕ（·）：［０，＋∞）→Ｃ是Ｓ的几乎轨道且关于ｔ∈［０，＋∞）一致连续，则｛ｕ（ｔ）｝几乎弱收敛到集合　　｛ｕ（ｒ）：ｒ≥ｔ｝∩Ｆ（ｓ）的唯一点。