Finite difference methods and their convergence for a class of singular two point boundary value problems

Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x y)=f(x,y), y(0)=A, y(1)=B, 0<<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all (0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM ₁ based on just one evaluation off. For uniform mesh we obtain two methodsM ₂ andM ₃ each based on three evaluations off. For =0,M ₁ andM ₂ both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h ²)-convergence established and illustrated by numerical examples.     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

Bounds for optimalα-β binary trees

In this paper we consider a special kind of binary trees where each right edge is associated with a positive number and each left edge with a positive number( ). Given, and the number of nodesn, an optimal tree is one which minimizes the total weighted path length. An algorithm for constructing an optimal tree for given, , n is presented, based on which bounds for balances and total weighted path lengths of optimal trees are derived.     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.     

A 0-stable linear multistep formulas of theα-type

The-type linear multistep formulas are a generalization of the Adams-type formulas. This paper is concerned with completely characterizing theA ₀-stability of thek-step, orderk -type formulas. Specifically, all such formulas of orders 4 or less are identified and it is shown that no-type formulas of order 5 or more exist. These theorems generalize some previous results.     

On Some Interpolation Rules for Lattice Ordered Groups

Let be an infinite cardinal. In this paper we define an interpolation rule IR() for lattice ordered groups. We denote by C() the class of all lattice ordered groups satisfying IR(), and prove that C() is a radical class.     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean -groups, and with an infinite cardinal or , we consider X-hulls where X stands for any of the following classes of -groups: -projectable; laterally -complete; boundedly laterally -complete; conditionally -complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the -groups is relatively -complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.     

A remark on fixed points of functors in topological categories

For a topological category over Set we prove that if a functor T: has a fixed cardinal (i.e. for each object K with card (UK)= we have card (UTK)), then T has a least fixed point, and if T has a successive pair of fixed cardinals and ⁺, then T has a greatest fixed point. This extends results of Adámek and Koubek.Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.     

Continuity of the Multifractal Spectrum of a Random Statistically Self-Similar Measure

Until now [see Kahane;⁽¹⁹⁾ Holley and Waymire;⁽¹⁶⁾ Falconer;⁽¹⁴⁾ Olsen;⁽²⁹⁾ Molchan;⁽²⁸⁾ Arbeiter and Patzschke;⁽¹⁾ and Barral⁽³⁾] one determines the multifractal spectrum of a statistically self-similar positive measure of the type introduced, in particular by Mandelbrot,^{(26, 27)} only in the following way: let be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for 0, denote by E the set of the points where possesses a local Hölder exponent equal to , and dim E the Hausdorff dimension of E ; then, there exists a deterministic open interval I *₊ and a function f: I *₊ such that for all in I, with probability one, dim E =f(). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all I, dim E =f(). This holds also for a new type of statistically self-similar measures deduced from a result recently obtained by Liu.⁽²²⁾ We also study another problem left open in the previous works on the subject: if =inf(I) or =sup(I), one does not know whether E is empty or not. Under suitable assumptions, we show that E ø and calculate dim E .     

Optimal quadrature formula of Markov's and Locher's type with weight function

Summary A quadrature formula of Markov's type with a weight functionx (1–x), which has properties of formulas exact for polynomials of a given degree and properties of optimal formulas on some sets of functions, is given. The particular case of formula (where ==p=q=0) is the formula of Locher [1, 2].     

On the positive definiteness of n ↦ epnα for α close to 2

For >2, let Q ⁺() be the infimum of those q>0 for which the function n e^pn is positive definite on N ₀ for every pq. We shall prove that Q ⁺()0 as 2.     

α-Sober spaces via the orthogonal closure operator

Each ordinal equipped with the upper topology is a T ₀-space. It is well known that for =2 the reflective hull of in Top₀ is the subcategory of sober spaces. Here, we define -sober space for each 2 in such a way that the reflective hull of in Top₀ is the subcategory of -sober spaces. Moreover, we obtain an order-preserving bijective correspondence between a proper class of ordinals and the corresponding (epi)reflective hulls. Our main tool is the concept of orthogonal closure operator, first introduced in [12].The author acknowledges financial support from Instituto Politécnico de Viseu and from Centro de Matemática da Universidade de Coimbra.     

Extremal partition and estimates for the coefficients in the class σ(r)

Let be the known class of functions f(z)=z+₀+₁z^–1+..., that are meromorphic and univalent in the region U^*={z ¦z¦ < 1}, and let (r) be the class of functions of for which f(U^*) does not contain a singly connected domain D_f, 0 D_f, with conformal radius r with respect to the coordinate origin, 0 < r < 1. Sharp inequalities are obtained for certain functionals, and sharp bounds are obtained for ¦ ₁ ¦ and ¦ ₂ ¦ in the class (r). The proof illustrates the possibility of using results with a symmetrization character in problems on extremal partition of a Riemannian sphere to obtain sharp bounds for coefficients in the classes and (r).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 101–116, 1991.     

Applications of Rellich's perturbation theory to a classical boundary and eigenvalue problem

Zusammenfassung Mit Hilfe der Rellichschen Störungstheorie wird die analytische Abhängigkeit der Lösungen eines Randwert- und eines Eigenwertproblems gezeigt, sowie des tiefsten Eigenwertes ₁() von einem Parameter , der in den Randbedingungen auftritt.Als Anwendung ergeben sich Schranken für das Energieintegral und ₁(). Ferner wird die Definition der RobinfunktionR(P, Q; ) erweitert auf negative Werte von , insbesondere singuläre Werte –p _j .

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Neu-Technikum Buchs     

Backward perturbation analysis of certain characteristic subspaces

Summary This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=x, the generalized eigenvalue problem Ax=Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.This subject was supported by the Swedish Natural Science Research Council and the Institute of Information Processing of the University of Umeå.     

Interpolation in the class of generalized axisymmetric potential functions

We consider the interpolation of anx -harmonic function with an arbitrary real . We investigate the existence and uniqueness of thex -harmonic interpolation function and also convergence of the interpolation process for some particular node placements.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 30–39, 1988.     

A relationship between the Mahler measure and the discriminant of algebraic numbers

In this note we show that in the well-known Dobrowolski estimate lnM() (ln lnd/ lnd)3,d , where is a nonzero algebraic number of degreed that is not a root of unity andM() is its Mahler measure, the parameterd can be replaced by the quantity=d/() ^1/d, where () is the modulus of the discriminant of. To this end, must satisfy the condition deg ^p=deg for any primep.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 415–420, March, 1996.     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University