On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents

Letp=2N/(N –2),N 3 be the limiting Sobolev exponent and ^N a bounded smooth domain. We show that for H ^–1(),f satisfies some conditions then–u=c ₁ u ^p–1 +f(x,u) + admits at least two positive solutions.     

Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (², 0)(, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f _t : (², 0)(, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family     

Local extrema of analytic functions

We give a complete answer to the problem of the finite decidability of the local extremality character of a real analytic function at a given point, a problem that found partial answers in some works by Severi and ojasiewicz. Consider a real analytic functionf defined in a neighbourhood of a pointx ₀R ⁿ . Restrictf to the spherical surface centered inx ₀ and with radiusr0 and take its infimumm(r) and its supremumM(r). We establish some properties ofm(r) andM(r) for smallr>0. In particular, we prove that they have asymptotic expansions of the formf(x ₀)+c·(r +o(r )) asr0 for a realc and a rational 1 (of course the parameters will usually be different form andM).This work was supported by the Brazilian Fundação Carlos Chagas and by the Italian Consiglio Nazionale delle Ricerche.     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

Solving a nonlinear spectral problem for a matrix

This paper examines the solving of the eigenvalue problem for a matrix M () with a nonlinear occurrence of the spectral parameter. Two methods are suggested for replacing the equation dat M()=0 by a scalar equationf()=0. Here the functionf() is not written formally, but a rule for computingf() at a fixed point of the domain in which the desired roots lie is indicated. Müller's method is used to solve the equationf()=0. The eigenvalue found is refined by Newton's method based on the normalized expansion of matrix M(), and the linearly independent vectors corresponding to it are computed. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 54–66, 1976.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

A criterion for <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-rectifiability of sets

The following result is proved: Let be a n-dimensional C¹-submanifold of R^N which is domain of a given _nR^N-valued map of class C¹. Then the set of all points P such that (P) is non-zero, simple and enveloped by T_P is C²-rectifiable. As a corollary we get a criterion for the C²-rectifiability of a rectifiable set based on the rectifiability of some generalized Gauss lift to the Grassmanian bundle R^N×G(N,n). Mathematics Subject Classification (2000) Primary 49Q15, 53A07; Secondary 49Q20, 49N60     

On Shikata's distance between differentiable structures

Shikata proved: there is a number (n) with the following property: If two compact homeomorphic n-dimensional manifolds have a distance less than (n), then they are diffeomorphic. We improve the known lower bound (n!)^–n for (n) to 1/3n ^–2.This work was done under the program Sonderforschungsbereich Theoretische Mathematik (SFB 40) at Bonn University while Shikata was SFB-guest at Bonn.     

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

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Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

Supplementing Radicals and Decompositions of Near-Rings

If is a radical of near-rings and is its supplementing radical, then (N)(N) N. We address the issue when (N) (N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N N_c is a hereditary Kurosh–Amitsur radical, _c is characterized in terms of distributors and criteria are given for the decomposition N = c(N) c(N). In the subvariety A of all abstract affine near-rings, assigning the maximal torsion ideal (N) is a hereditary Kurosh–Amitsur radical. If such near-rings N A satisfy dcc on principal right ideals, then N splits into a direct sum N = (N) (N) where the additive group of (N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

A finitely based theory of a non-trivial language,with exactly ℵ0 subcovers

LetL=f, g be the language with two unary operation symbols. I prove that the finitely based equational theory =[f₀=₀] ofL covers exactly ₀ others.Presented by S. Burris.Dedicated to George McNulty, my mentor in equational logic.     

Similarity solutions of radially symmetric two-phase flow

When both the diffusivityD and fractional flow functionf have a power law dependence on the water content , i.e.D=D _o andf=⁺¹, the nonlinear transport equation for radially symmetric two phase flow can, in certain circumstances, be reduced to a weakly coupled system of two first order nonlinear ordinary differential equations. Numerical solutions of these equations for a constant flux boundary conditionV _wo and comparison with experimental data are given. In particular, when the fluxV _wo and a are related byV _wo( + 1)/D _o=2, a new fully explicit analytical solution is found as (r, t)=(1 – r ²/4D _ot)^1/ forr ² < 4D _ot/ and (r, t)=0 forr ² 4D _ot/ We show that the existence of this exact soution is due to the presence of a Lagrangian symmetry.