Renewal Theorems for Singular Differential Operators

Let * be the convolution on M( ₊) associated with a second order singular differential operator L on ]0, +[. If is a probability measure on ₊ with suitable moment conditions, we study how to normalize the measures * ⁿ ; n } (resp. ) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.     

On strong solutions of Poisson's equation in Beppo Levi spaces

LetG sun (n 2) be an unbounded open set having a compact complement and a smooth boundary G of classC ². InG we consider the equations — u=f,u¦G= and prove the existence of a solutionu L ^2,q(G) providedf L ^q(G) and W ^{2 —1/q-q}(G) (1 <q < ). HereL ^2,q(G) is the space of all functionsu L _Ioc ^q (G) having all second order distributional derivatives inL ^q(G). Concerning the uniqueness of this solution we show that the corresponding nullspace has dimensionn + 1 (n 2).

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Zusammenfassung SciG ⁿ (n 2) eine unbeschränkte offene Menge mit kompaktem Komplement und mit glattem Rand G der KlasseC ². InG betrachten wir das Randwertproblem — u=f,u¦g= und beweisen die Existenz einer Lösungu L ^2,q(G) für beliebigef L ^q(G) und Randwerte W ^2-1/q,q(G) (1 <q < ). Dabei istL ^2,q(G) der Raum aller Funktionenu L _Ioc ^q (G), die Distributionsableitungen zweiter Ordnung inL ^q (G) besitzen. Bezüglich der Eindeutigkeit solcher Lösungen zeigen wir, daß der entsprechende Nullraum die Dimensionn + 1 (n 2) besitzt.

     

Certain problems of vector analysis

In the paper one gives an explicit method of constructing in the domain ^m m2 a vector field, having a prescribed divergencef in and taking prescribed values on the boundary . The differential properties of the field are faithfully determined by the smoothness off, and Simultaneously, one constructs the solutions of a series of other problems of vector analysis, which present an independent interest.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 65–85, 1984.     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

On strong solutions of the Stokes equations in exterior domains

We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -u + p=f inG, ·u=g inG, u= on in an exterior domainG ⁿ (n3) with boundary of class C². Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ·=g inG, = 0 on , the exterior domain problem is reduced to the entire space problem and an interior problem.     

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

A Second Descent Problem for Quadratic Forms

Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied by Kahn and Laghribi. More precisely, we conjecture that for any quadratic form q over F and any Im(W(F) W(F(q))), there exists a quadratic form W(F) such that dim 2 dim and _F(q), where F(q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim 3 and any q, and get partial results for dim {4, 5,6}. We also give other related results.     

Comparison results for the lower tail of Gaussian seminorms

Let =( _n ) be i.i.d.N(0, 1) random variables andq(x), q(x):R [0, ) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q()<) andP(q()<) goes to a positive constant as 0⁺. We give satisfactory answers forl ₂-norms and also some results for sup-norms andl _p-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL ₂-norms.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Continuité de la densité de l'intégrale d'aire dans les espaces de Hardy

Nous estimons la dépendance en la fonction harmonique des fonctionnelles densité de l'intégrale d'aire. Nous obtenons notamment un contrôle en norme de sup _aR D u(,a)–D v(,a) ainsi qu'une inégalité aux bons- de type exponentielle.     

A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

LetDR ^N be a region with smooth boundaryD. Letp·q>1,p, q1. We consider the system:u _t=u+v ^p,v _t=u+u ^q inD×[0, ) withu=v=0 inD×[0, ) andu ₀,v ₀ nonnegative. Let=max(p, q). We show that ifD isR ^N, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (+1)/(pq–1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (+1)/(pq–1)<pc(D) both nontrivial global and nonglobal solutions exist. In caseD is a cone orD=R ^N, (a) holds with equality. An explicit formula forpc(D) is given.This research was supported in part by NSF Grant DMS-8822788 and in part by the Air Force Office of Scientific Research.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

The absolute convergence of weighted sums of dependent sequences of random variables

Summary The sum a _n X _n of a weighted series of a sequence {X _n } of identically distributed (not necessarily independent) random variables (r.v.s.) is a.s. absolutely convergent if for some in 0<1, ¦a _n ¦ < and E¦X _n ¦ < ; if a _n =z ⁿ for some ¦z¦<1 then it suffices that E(log¦X _n ¦)₊<. Examples show that these sufficient conditions are not necessary. For mutually independent {X _n } necessary conditions can be given: the a.s. absolute convergence of X _n z ⁿ (all ¦z¦<1) then implies E(log¦X _n ¦)₊ < , while if the X _n are non-negative stable r.v.s. of index , ¦a _n X _n ¦< if and only if ¦a _n ¦ < .     

The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Let denote a distance-regular graph with vertex set X, diameter D 3, valency k 3, and assume supports a spin model W. Write W = _{i = 0}^D t_i A_i where A_i is the ith distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and t_i {t₀, –t₀ } for 1 i D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters and q. We extend their results as follows. Fix any vertex x X and let T = T(x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T-module with endpoint r and diameter d. We obtain the intersection numbers c_i(U), b_i(U), a_i(U) as rational expressions involving r, d, D, and q. We show that the isomorphism class of U as a T-module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T-modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and is real.AMS 2000 Subject Classification: Primary 05E30     

Higher-Dimensional Representations of the Reflection Equation Algebra

We consider a new method for constructing finite-dimensional irreducible representations of the reflection equation algebra. We construct a series of irreducible representations parameterized by Young diagrams. We calculate the spectra of central elements s _k=Tr_q L ^k of the reflection equation algebra on q-symmetric and q-antisymmetric representations. We propose a rule for decomposing the tensor product of representations into irreducible representations.     

Asymptotics of the partial sums of a set of integral transforms

In this paper, we explore the asymptotic distribution of the zeros of the partial sums of the family of entire functions of order 1 and type 1, defined by G(,,z)=₀ ¹(t)t ^–1×(1–t)^–1e ^zt dt, where Re,Re>0, is Riemann-integrable on [0,1], continuous at t=0, 1 and satisfies (0)(1)0.     

Iterative refinement implies numerical stability

Suppose that a method computes an approximation of the exact solution of a linear systemAx=b with the relative errorq,q<1. We prove that if all computations are performed in floating point arithmeticfl and single precision, then with iterative refinement is numerically stable and well-behaved wheneverqA A ^–1 is at most of order unity.     

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.     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.