On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Die Weingarten-Regelflächen

Referring to articles of BELTRAMI (1865), DINI (1866) and CHARIAR (1978), but using a completely different approach, we determine allruled surfaces in Euclidean space ³, which are (at leastlocally) WEINGARTEN —-surfaces under theminimal assumption C². Theskew ruled WEINGARTEN —surfaces can be characterized by havingconstant invariants d O (parameter of distribution), k (skewness of distribution) and (striction angle); theirfunctional (WEINGARTEN-)relation between the mean curvature H and the Gaussian curvature K of is of the form H= (-K)^1/4 + (-K)^3/4 with arbitrary real constants ,. These facts allow various geometric interpretations.

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Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Das Zeitverhalten von einfachen offenen exponentiellen Warteschlangensystemen mit unendlich vielen Warteplätzen

Zusammenfassung Die zeitabhängige (instationäre) Lösung für die Zustandswahrscheinlichkeiten und für einige Kenngrößen von Warteschlangensystemen mit einer Bedienungsstation, unendlich vielen Warteplätzen, exponentiellem Zu- und Abgang und beliebigem Anfangszustand wird bestimmt. Die ZustandswahrscheinlichkeitenP _v (), d. h. die Wahrscheinlichkeiten für Einheiten im System zur Zeit, ergeben sich als Integrale, in denen modifizierteSessel-Funktionen 1. Art auftreten. Der ErwartungswertL () und die VarianzV() der Zahl von Einheiten im System lassen sich als Integrale darstellen, in denen nur die ZustandswahrscheinlichkeitP ₀() auftritt.Für<1 und erreichen die Systeme einen stationären Zustand (für den die Lösung bekannt ist); für1 und giltP _v ()0 für alle, L(),V().Ist>1, dann wachsenL() undV() für große linear mit; ihre Asymptoten werden berechnet. Ist=1, dann wachsenL() und die Standardabweichung() für große mit ; einfache Näherungsformeln werden gefunden.

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Summary The time dependent solution is determined for the state probabilities and for some characteristic values of queuing systems with a single server, an infinite number of waiting places, exponentially distributed inter-arrival and service times, and any initial state. The state probabilitiesP _v (), i.e. the probabilities for units in the system at time, are given in the form of integrals in which modifiedBessel functions of the first kind occur. Integrating the state probalityP ₀() over leads to the meanL() and the varianceV() of the number of units in the system.For<1 and the systems tend to a steady state (for which the solution is known); for1 and we haveP _v ()0 for all, L(),V().If>1 asymptotic expansions for large are found givingL() andV() proportional to. If=1 simple approximate formulas for large are obtained givingL() and the standard deviation() proportional to .

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Vorgel. v.:J. Nitsche.     

Абсолютная сходимос ть рядов Фурье—Хаара от суперпозиций функци й

A necessary and sufficient condition is found for the function to ensure absolute convergence of the Haar—Fourier series of all functions(f) provided that the Haar—Fourier series off converges absolutely. Absolute convergence means absolute convergence of the series of coefficients, and the condition is that should be in Lip 1.

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Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Cross-monotone subsequences

Two finite real sequences (a ₁,...,a _k ) and (b ₁,...,b _k ) are cross-monotone if each is nondecreasing anda _i+1–a _i b _i+1–b _i for alli. A sequence (₁,..., _n ) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (₁,..., _n ) is in CM(k), is bounded between aboutk ²/4 andk ₂/2. It also shows thatg(k), the smallestn for which all (₁,..., _n ) are in CM(k)and eithera _k b ₁ orb _k a ₁, equalsk(k–1)+2, and thath(k), the smallestn for which all (₁,..., _n ) are in CM(_k)and eithera ₁b ₁...a _k b _k orb ₁a ₁...b _k a _k , equals 2(k–1)²+2.The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark ²×k ² (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk ² is replaced byk ²–1.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Morera type theorems on the unit disc

, . . : , , , , .     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

The odd ordér analogue of Dempwolff's B-group problem

Let be a translation plane of odd order p^2R for p a prime. Let admit at least two p-groups, B_i, i=1, 2, B₁B₂ of orders > p^R/2 which fix Baer subplanes _i, i = 1,2, ₁₂ pointwise. It is shown that under these assumptions must be a Hall plane.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Generalized quadrangles with an ovoid that is translation with respect to opposite flags

The article [6] contains the result that if a finite generalized quadrangle of order s has an ovoid that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then is self-polar and is the set of absolute points of a polarity. In particular, if is the classical generalized quadrangle Q(4, q) then is a Suzuki-Tits ovoid. In this article, we remove the need to assume that is Q(4, q) in order to conclude that is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that is Q(4, q). At the same time, we also relax the requirement that have order s.Received: 14 May 2004     

An efficient heuristic algorithm for minimum matching

We give a new heuristic algorithm for minimum matching problems and apply it to the Euclidean problem with random vertices in 2 dimensions. The algorithm is based on simulated annealing and performs in practice faster than previous heuristic algorithms yielding suboptimal solutions of the same good quality. From configurations with up toN=20.000 vertices in the unit square we estimate that the length of a minimum matching scales asymptotically asLN with (=0.3123±0.0016.

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Zusammenfassung Wir stellen einen neuen heuristischen Algorithmus für minimale Matching-Probleme vor und wenden diesen auf das euklidische Problem mit zufÄlliger Punkteverteilung in 2 Dimensionen an. Auf Simulated Annealing basierend lÄuft der Algorithmus schneller als frühere heuristische Algorithmen und erreicht dabei suboptimale Lösungen gleich guter QualitÄt. Aus Konfigurationen mit bis zuN=20.000 Punkten im Einheitsquadrat schÄtzen wir, da\ für die LÄnge des minimalen Matchings asymptotischLN mit=0.3123±0.0016 gilt.

     

K-theory of twisted differential operators on flag varieties

Let be a semisimple Lie algebra overk, an algebraically closed field of characteristic zero, and let be a Cartan subalgebra inside a Borel subalgebra of . LetU be the enveloping algebra of . For letM() denote the corresponding Verma modúle and letU _u=U/AnnM(). LetW be the Weyl group and letW ⁰ be the stabiliser of inW. We prove the following theorem, which affirms a conjecture of T.J. Hodges.Oblatum 16-XII-1994