The hereditariness of the upper radical

Summary Starting from a regular classM, one can construct the upper radicalU M of the classM in a category which is like that of associative, alternative or not necessarily associative rings, or that of Lie rings. It turns out that in quite a few cases the upper radical is hereditary. (cf.Suliski [7], Rjabuhin [6], Armendariz [2], Szász—Wiegandt [8]).W. G. Leavitt has suggested the problem: Give a necessary and sufficient condition to be satisfied by the regular classM so that the upper radical classU M ofM is hereditary. In the present paper we shall give such a necessary and sufficient condition. If the classM satisfies an even stronger condition, then theU M-semisimple objects are subdirectly embeddable in a (direct) product ofM-objects. Also a necessary and sufficient condition is given which assures that eachU M-semisimple object can be subdirectly embedded in a (direct) product ofM-objects.This work was done when the second named author was in the University of Islamabad under UNESCO-UNDP Special Fund Pak. 47.     

A critically loaded multiclass Erlang loss system

We consider a generalization of the classical Erlang loss model to multiple classes of customers. Each of the J customer classes has an associated Poisson arrival process and an arbitrary holding time distribution. Classj customers requireM _j servers simultaneously. We determine the asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal. Asymptotically, the blocking probability of classj customers is proportional toM _j . This asymptotic result provides an approximation for the blocking probabilities which is reasonably accurate. We also consider the behavior of the Erlang fixed point (reduced load approximation) for this model under critical loading. Although the blocking probability of classj customers given by the Erlang fixed point is again asymptotically proportional toM _j , the Erlang fixed point typically gives the wrong limit. Next we show that under critical loading the throughputs have a pleasingly simple form of monotonicity with respect to arrival rates: the throughput of classi is increasing in the arrival rate of classi and decreasing in the arrival rate of classj forji. Finally, we compare two simple control policies for this system under critical loading.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].     

Asymptotic optimality of the generalized bayes estimator in multiparameter cases

The higher order asymptotic efficiency of the generalized Bayes estimator is discussed in multiparameter cases. For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the classA ₂ of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted classD of estimators.     

Alternating groups and moduli space lifting invariants

The genus of a curve discretely separates decidely different algebraic relations in two variables to focus us on the connected moduli space M _g . Yet, modern applications also require a data variable (function) on the curve. The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. A Nielsen class (§1.1) consists of r >- 3 conjugacy classes C in the data variable monodromy G. It generalizes the genus.     

Classification of gradient-like flows on dimensions two and three

We consider here gradient-like flows of classC ^r ( ^r ≥1) on a closed manifoldM of classC ^r+1 and dimension two or three. We study the classification of these flows by the relation of topological equivalence. In this sense, the flows which are more relevant are the polar flows (only one source and only one sink).     

Equivelar polyhedral manifolds inE 3

An equivelar polyhedral 2-manifold in the classM _p,q is one embedded inE ³ in which every face is a convexp-gon and every vertex isq-valent. Various constructions for equivelar manifolds are described, and it is shown that, in certain classesM _p,q, there is a manifold of given genusg≧2 for all but finitely manyg.     

Frostman shifts of inner functions

In this paper we study the classM of all inner functions whose non-zero Frostman shifts are Carleson-Newman Blaschke products. We present several geometric, measure theoretic and analytic characterizations ofM in terms of level sets, distribution of zeros, and behaviour of pseudohyperbolic derivatives and observe thatM is the set of all functions inH ^∞ whose range on the set of trivial points in the maximal ideal space is ∂D∪{0} The second author thanks the University of Metz for its support during a one-month research visit and acknowledges partial support by DGYCIT and CIRIT grants. Also both authors were partially supported by the European network HPRN-CT-2000-00116.     

Max-semistable laws

In this paper we give a characterization of the classMSS of all one-dimensional d.f.'s G satisfying the functional equation G^α(x)=G(L(x)), where α∈(0, 1] andL(x) is strictly monotone. Supported by the Bulgarian Ministry of Education and Science (grant MM-17/1993). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Solutions ofLu=u α dominated byL-harmonic functions

LetL be a second order elliptic differential operator on a differentiable manifoldM and let 1 <α≤2. We investigate connections bewween classU of all positive solutions of the equationLu=u ^α and classH of all positiveL-harmonic functions (i.e., solutions of the equationsLh=0). Putu∈U ⁰ ifu∈U and ifu≤h for someh∈H. To everyu∈U ⁰ there corresponds the minimalL-harmonic functionh _u which dominatesu andu→h _u is a 1–1 mapping fromU ⁰ onto a subsetH ⁰ ofH. The inverse mapping associates with everyh∈H ⁰ the maximal element ofU dominated byh. Supposeg(x, dy) is Green's kernel,k(x, y) is the Martin kernel and ?M is the Martin boundary associated withL. A subset Γ of ?M is calledR-polar if it is not hit by the rangeR of the (L, α)-superdiffusion. It is calledM-polar if $\int\limits_M {g\left( {c,dx} \right)[\int\limits_\Gamma {k(x,y)v(dy)]^\alpha } } $ is equal to 0 or ∞ for everyc∈M and every measure ρ. Everyh∈H has a unique representation $h(x) = \int\limits_{\partial M} {k\left( {x,y} \right)v\left( {dy} \right)} $ where ρ is a measure concentrated on the minimal partM ^* of ?M. We show that the condition:

1. ρ(Γ)=0 for allR sets Γ is necessary and the condition:
2. ρ(Γ)=0 for allM-polar sets Γ is sufficient forh to belong toH ⁰. IfM is a bounded domain of classC ^{2, λ} in ? ^d , then conditions (a) and (b) are equivalent and therefore each of them characterizesH ⁰. This was conjectured by Dynkin a few years ago and proved in a recent paper of Le Gall forL=Δ, α=2 and domains of classC ⁵.

     

Moduli compatti e linearmente topologizzati e dualità

Riassunto SianoR un anello commutativo con unità munito della topologia discreta edM unR-modulo topologico compatto e totalmente sconnesso. Nel lavoro si determinano condizioni necessarie e sufficienti, riguardanti sia l'anello che il modulo, affinchèM sia linearmente topologizzato comeR-modulo, sfruttando una condizione sufficiente individuata da Woodcock per il caso localmente compatto anzichè compatio (cfr. [6]). Si trova inoltre che i moduli compatti e linearmente topologizzati suR sono tutti e soli i duali secondo Pontrjagin di una sottoclasse ereditaria di pretorsione della categoria degliR-moduli, ed infine che, su tale classe, la dualità di Pontrjagin è equivalente alla dualità di Orsatti [3].

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Summary In this paper we give necessary and sufficient conditions for a compact topologicalR-moduleM to be linearly topologized (R is equipped with discrete topology). Moreover we give a characterization of linearly topologized compact modules in terms of their duals with respect to Pontrjagin duality forR-modules, thus identifying a classC F ⊆ ModR. Then we study the Haudorff completion ofR with respect to the topology of all ideals of finite index, and the dual of this completion with respect to Pontrjagin duality forR-modules. Eventually we show that for the classC F Pontrjagin duality is equivalent to other dualities studied in [2] and [3].

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del C.N.R.     

Reflexivity of N-tuples of contractions with rich joint left essential spectrum

It will be shown that WOT-closed algebra generated by N-tuple of double commuting contractions, for which the polydiscD ^N is a spectral set and whose joint left essential spectrum is dominating for the algebraH (D ^N ) is reflexive. The second version of our main result, instead of double commutativity, uses the membership of the classC ₀.     

On the uniqueness of solutions to linear complementarity problems

This paper characterizes the classU of all realn×n matricesM for which the linear complementarity problem (q, M) has a unique solution for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. It is shown that restricting the uniqueness property to the interior ofK(M) is necessary because whenU, the problem (q, M) has infinitely many solutions ifq belongs to the boundary of intK(M). It is shown thatM must have nonnegative principal minors whenU andK(M) is convex. Finally, it is shown that whenM has nonnegative principal minors, only one of which is 0, andK(M)≠R ⁿ , thenU andK(M) is a closed half-space.     

The torsion of a submanifold of a Riemannian manifold

For a submanifoldM ⁿ of a Riemannian manifoldM ^q, the concept of a torsion bivector at the point x M ⁿ for given one- and two-dimensional directions fromT _x M ⁿ is introduced using only the first and second fundamental forms ofM ⁿ. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM ⁿ is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM ⁿ into some totally geodesicM ⁿ⁺¹ inM ^q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM ² inM ^q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.     

Annihilator Primes and Foundation Primes

ABSTRACT

Let R be a Noetherian ring and M a finitely generated R -module. In this article, we introduce the set of prime ideals Fnd  M , the foundation primes of M . Using the fact that this set is nicely organized by foundation levels, we present an approach to the problem of understanding Annspec  M , the annihilator primes of M , via Fnd  M . We show: (1) Fnd  M is a finite set containing Annspec  M . Further, suppose that moreover every ideal of R has a centralizing sequence of generators; now, Annspec  M is equal to the set Ass  M of associated primes of M. Then: (2) For an arbitrary P  ∈ Fnd  M , P  ∈ Annspec  M if and only if there is no Q  ∈ Annspec  M such that P contains Q , and at the same time, the minimal foundation level on which appears P is greater than the minimal foundation level on which appears Q .     

M-harmonic Besovp-spaces and Hankel operators in the Bergman space on the ball in ℂ n

In this paper, we characterize the Hardy class ofM-harmonic functions on the unit ballB in ℂ ⁿ in terms of the Berezin transform. We define and study the Besovp-spaces ofM-harmonic functions. For anM-harmonic symbolf, we give various criteria for the Hankel operatorsH _f andH _f to be bounded, compact or in the Schatten-von-Neumann classS _p . These criteria establish a close relationship among Besovp-spaces, Berezin transform, the invariant Laplacian, and Hankel operators on the unit ballB.     

Dominating cliques in P5-free graphs

All induced connected subgraphs of a graphG contain a dominating set of pair-wise adjacent vertices if and only if there is no induced subgraph isomorphic to a path or a cycle of five vertices inG. Moreover, the problem of finding any given type of connected dominating sets in all members of a classG of graphs can be reduced to the graphsG∈G that have a cut-vertex or do not contain any cutsetS dominated by somes∈S. This research was supported in part by the “AKA” Research Fund of the Hungarian Academy of Sciences.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

Finite rank perturbations of contractions

We study finite rank perturbations of contractions of classC _.0 with finite defect indices. The completely nonunitary part of such a perturbation is also of classC _.0, while the unitary part is singular. When the defect indices of the original contraction are not equal, it can be shown that almost always (with respect to a suitable measure) the perturbation has no unitary part.     

L p estimates for the iterated Hardy-Littlewood maximal operator on\mathbb{R}^n and K n , k a local fieldand K n , k a local field

In light of two measure estimate inequalities from [4] for the iterated Hardy-Littlewood maximal operatorM ^k f, certain equivalence betweenM ^k f and the Zygmund classLLog ^a L are established on $\mathbb{R}^n $ , so that we generalize Stein'sLLogL theorem. In Section 3, a simple induction enables us to prove such extensions onK ⁿ, the n-dimensional linear space over a local fieldK, without recoursing to Leckband's result.