On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.     

Asymptotic expansion of norm associated with conjugate trigonometric polynomial

LetT _n (x) be trigonometric polynomial of degreen, and be conjugation ofT _n (x). In this paper we obtain the complete asymptotic expansion for

Repleteness and regular lattice measures

Let X be an abstract set and L a lattice of subsets of X. To each lattice regular measure µ, we associate two induced measures and on suitable lattices of the Wallman space IR(L) and another measure µ on the spaceIR(L). We will investigate the reflection of smoothness properties of p onto , and µ; and try to set some new criterion for repleteness and measure repleteness.     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.     

A vector-sum theorem in two-dimensional space

Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x₁, , x_n} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.     

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL ¹ metric in the optimal order {n ^–1} if and only if ² f is of bounded variation where and , respectively.     

On Bergman-Toeplitz operators with commutative symbol algebras

Let be the unit disk in, be the Bergman space, consisting of all analytic functions from , and be the Bergman projection of onto . We constructC ^*-algebras , for functions of which the commutator of Toeplitz operators [T _a ,T _b ]=T _a T _b –T _b T _a is compact, and, at the same time, the semi-commutator [T _a ,T _b )=T _a T _b –T _ab is not compact.It is proved, that for each finite set =n ₀,n ₁, ...,n _m , where 1=n _{0 1} <... _m , andn _k {}, there are algebras of the above type, such that the symbol algebras Sym of Toeplitz operator algebras arecommutative, while the symbol algebras Sym of the algebras , generated by multiplication operators and , haveirreducible representations exactly of dimensions n ₀,n ₁,..., n _m .This work was partially supported by CONACYT Project 3114P-E9607, México.     

Note on irregularities of distribution

Letf be a periodic function on with period 1, piecewise continuously differentiable, satisfying . For an arbitrary sequence = ( _i ) in [0,1) put and . If then _n (f,) >c· logn holds for some positive constantc (depending onf only) and almost alln. In a certain sense the converse is also true: there is a class of functionsf with such that _n (f,) =o (logn).Support has been received from Netherlands Organization for the Advancement of Pure Research (Z. W. O.).     

Multiple recurrence theorem for nilpotent group actions

We prove that ifX is a compact space andT ₁, ...,T _t are homeomorphisms ofX generating a nilpotent group, then there existx X andn ₁,n ₂, ... N such that for eachj=1, ...t.Supported by British Technion Society.     

Some results on geodesic mappings of Riemannian manifolds satisfying the conditionR.R.=Q(S,R)

In this paper geodesically corresponding metricsg and on a manifoldM, dim 5, under the assumption that the tensorsR andS of the metricg satisfyR.R=Q(S, R), are considered. It is stated that the corresponding tensors and of not necessarily must satisfy . Certain relations between the curvatures ofg and are obtained.Supported by a post-doctoral fellowship of the researchcouncil of the KU Leuven; Bitnet FGBDA3O at BLEKUL11     

On T 3-Topological Space Omitting Many Cardinals

We prove that for every (infinite cardinal) there is a T ₃-space X with clopen basis, points such that every closed subspace of cardinality < has cardinality < .     

Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators

Suppose thatX _n =(X ₁,...X _n) is a collection ofm-dimensional random vectorsX _i forming a stochastic process with a parameter . Let be the MLE of . We assume that a transformationA( ) of has thek-thorder Edgeworth expansion (k=2,3). IfA extinguishes the terms in the Edgeworth expansion up tok-th-order (k2), then we say thatA is thek-th-order normalizing transformation. In this paper, we elucidate thek-th-order asymptotics of the normalizing transformations. Some conditions forA to be thek-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study thek-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

Reinforced random walk

Summary Leta _i,i1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion =X ₀,X ₁, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at timen an interval (i, i+1) has been crossed exactlyk times by the motion, its weight is . Given (X ₀,X ₁, ...,X _n)=(i₀, i₁, ..., i_n), the probability thatX _n+1 isi _n–1 ori _n+1 is proportional to the weights at timen of the intervals (i _n–1,i _n) and (i _n,ii_n+1). We prove that either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that X _n /n=0 a.s. For much more general reinforcement schemes we proveP ( visits all integers infinitely often)+P ( has finite range)=1.Supported by a National Science Foundation Grant     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

s-Prime elements in multiplicative lattices

Let be aC-lattice which is strong join principally generated. In this paper, we consider prime elements of for which every semiprimary element is primary. We show, for example, that a compact nonmaximal primep with this property is principal. We also show that if every primepm has this property, then is either a one dimensional domain or a primary lattice. It follows that if every primep satisfies the property, and if there are only a finite number of minimal primes in , then is the finite direct product of one-dimensional domains and primary lattices.     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.     

Spaces of compact operators and their dual spaces

Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒ^w' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T _n)⊂K(X, Y) such thatT=ω′−lim_nT_n and let . We show that is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in . This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces. Financial support from the Potchefstroom University and Maseno University is greatly acknowledged. Financial support from the NRF and Potchefstroom University is greatly acknowledged.     

Differential operators on homogeneous spaces. I

Summary In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation of the enveloping algebra of by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelI _x of _X is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grI _x , and relate it to the geometry of a generalizedSpringer-resolution, that is a map of the cotangent-bundle ofX onto a nilpotent variety in , as studied e.g. in [BM1]. We prove, for instance, that grI _x is prime if and only if _X is birational with normal image. In general, we show that is prime. Equivalently, the associated variety ofI _x in is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealI _y has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.