Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

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.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

Shape-preserving Kolmogorov widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ₀ and let ₊ ^s be the set of functions x defined on a finite interval I and such that, for all collections of s + 1 pairwise different points t ₀,..., t _s I, the corresponding divided differences [x; t ₀,...,t _s ] of order s are nonnegative. Let ₊ ^s B _{p +} ^s B _p, 1 p where B _p is a unit ball in the space L _p, and let ₊ ^s L _{q +} ^s L _q, 1 q . For every s 3 and 1 q p , we determine the exact orders of the shape-preserving Kolmogorov widths {x - y} \right\ L_q , $$]]>, where M ⁿ is the collection of all affine linear manifolds M ⁿ in L _q such that dim M ⁿ n and M ⁿ ₊ ^s L _q .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 901–926, July, 2004.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Non removable edges in 3-connected cubic graphs

LetG be a cyclicallyk-edge-connected cubic graph withk 3. Lete be an edge ofG. LetG be the cubic graph obtained fromG by deletinge and its end vertices. The edgee is said to bek-removable ifG is also cyclicallyk-edge-connected. Let us denote by S _k (G) the graph induced by thek-removable edges and by N _k (G) the graph induced by the non 3-removable edges ofG. In a previous paper [7], we have proved that N ₃(G) is empty if and only ifG is cyclically 4-edge connected and that if N ₃(G) is not empty then it is a forest containing at least three trees. Andersen, Fleischner and Jackson [1] and, independently, McCuaig [11] studied N ₄(G). Here, we study the structure of N _k (G) fork 5 and we give some constructions of graphs such thatN _k (G) = E(G). We note that the main result of this paper (Theorem 5) has been announced independently by McCuaig [11].

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Résumé SoitG un graphe cubique cyliquementk-arête-connexe, aveck 3. Soite une arête deG et soitG le graphe cubique obtenu à partir deG en supprimante et ses extrémités. L'arêtee est ditek-suppressible siG est aussi cycliquementk-arête-connexe. Désignons par S _k (G) le graphe induit par les arêtesk-suppressibles et par N _k (G) celui induit par les arêtes nonk-suppressibles. Dans un précédent article [7], nous avons montré que N ₃(G) est vide si et seulement siG est cycliquement 4-arête-connexe et que si N ₃(G) n'est pas vide alors c'est une forêt possédant au moins trois arbres. Andersen, Fleischner and Jackson [1] et, indépendemment, McCuaig [11] ont étudié N ₄(G). Ici, nous étudions la structure de N _k (G) pourk 5 et nous donnons des constructions de graphes pour lesquelsN _k (G) = E(G). Nous signalons que le résultat principal de cet article (Théorème 5) a été annoncé indépendamment par McCuaig [11].

     

On weakly mixing Markov operators and non-singular transformations

Summary Let P be a Markov operator on L (X, , m). Theorem 1: (i) P is weakly mixing (ii) For every fL there is a sequence {n_t} of density 1 such that all w ^*-cluster points of are constants (iii) For every fL there is a {k_j} with w ^*-convergent to a constant. Theorem 2: If P is induced by a non-singular transformation , P is weakly mixing For every A, { ^–n(A)} has a remotely trivial subsequence. The existence of a finite invariant measure is not required in these results.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

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.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

The Harish-Chandra theorem for the quantum algebrau q (sl (3))

A basis of a quantum universal enveloping algebraU is constructed; the following theorem is proved with the help of this basis: For any nonzero element U, there exists a finite-dimensional representation such that(u) 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45 No. 3, pp. 436–439, March, 1993.     

On meromorphic solutions of algebraic differential equations in angular domains

We obtain asymptotic estimates of meromorphic solutions to the differential equationP _n (z, , )=P _n–1 (z, , ,..., ^(m) ) in the angular domain P={z: arg z · }. Here P_n(z, w, w) is a polynomial in all variables, and of degree n with respect to w and w; P_n–1(z, w, w, ..., w^(m)) is a polynomial in all variables, and of degree n –1 with respect to w, w, ..., w^(m) In the particular case, when the solutions are entire functions, these estimates are more precise than the known estimates that are obtained by using the method of Wiman-Valiron, which cannot be applied to meromorphic solutions in the domain P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 514–523, April, 1992.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

Hardy spaces of predictable martingales

, P _p Q _p .

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This research was partially supported by the Hungarian Scientific Foundation.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

On Riesz means with respect to a cylindric distance function

(1–) ₊ , — R ⁿ =R ^j ×R ^k , ()=max{¦ ₁¦, ¦ ₁¦},=( ₁, ₂), ₁R ^J , ₂R ^k ,j,k1,n=j+k. n=3 , (1–) ₊ [L ¹(R ⁿ )]¹, >1/2; j=4, (1–) ₊ R L ^p (R ⁿ ). .

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The author would like to thank Professor W. Trebels for encouragement and valuable advice.