Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Let T be a locally compact Hausdorff space and let C ₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C ₀(T) X when c ₀ X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].     

On <Emphasis Type="Italic">L</Emphasis><Subscript>α,ω</Subscript> complete extensions of complete theories of Boolean algebras

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal = (T) such that We show that for some and for all other Ts, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants similar to the Tarskis invariants.Mathematics Subject Classification (2000): Primary 03C15, Secondary 03C35, 06E05Revised version: 2 February 2004     

Tanaka Formulae for (α, d, β)-Superprocesses

We establish Tanaka like formulae for the local time of (, d, )-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L ¹⁺ convergence argument (with 0<<1) rather than L ² convergence, for the derivation of the Tanaka formulae.     

A Stationary Rho-Mixing Markov Chain Which Is Not “Interlaced” Rho-Mixing

A strictly stationary, countable-state Markov chain is constructed which is -mixing (with arbitrarily fast mixing rate) but fails to be *-mixing (interlaced-mixing).     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

On Decomposing an Abelian p-Group Under a p′-Operator Group

Let A be a p-group acting on a finite Abelian p-group P. We give a necessary and sufficient condition on an A-irreducible decomposition of ₁(P) = _i1U_i for the existence of an A-indecomposable decomposition of P = _i1R_i such that ₁(R_i) = U_i for all i I. This readily implies a main result of [1].2000 Mathematics Subject Classification: 20D15     

Asymptotic Expansions for the Congestion Period for the M/M/∞ Queue

We consider the M/M/ queue with arrival rate , service rate and traffic intensity =/. We analyze the first passage distribution of the time the number of customers N(t) reaches the level c, starting from N(0)=m>c. If m=c+1 we refer to this time period as the congestion period above the level c. We give detailed asymptotic expansions for the distribution of this first passage time for , various ranges of m and c, and several different time scales. Numerical studies back up the asymptotic results.     

Some Congruences on S (X)

Some congruences on S (X) are investigated in this paper. A unique atom in the sublattice [C(E), C (E)] is determined for some spaces X and S-equivalences E and X.AMS Subject Classification (1991): 20M20 54H15     

K-theory of braided tensor ring categories with higher commutativity

We show that theK-theory of a Waldhausen categoryC with anA-ring structure is anA ring spectrum. If theA structure onC supports anE _n structure, so thatBC group completes to ann-fold loop space, thenK (C) is anE _n ring spectrum. In particular, theK-theory of the category of crossedG-sets,G a finite group, is anE ₂ ring spectrum.     

Groups with many subgroups having a transitive normality relation

A group is said to be aT-group if all its subnormal subgroups are normal. The structure of groups satisfying the minimal condition on subgroups that do not have the propertyT is investigated. Moreover, locally soluble groups with finitely many conjugacy classes of subgroups which are notT-groups are characterized.     

Cesàro α-Integrability and Laws of Large Numbers I

For a sequence of integrable random variables, we introduce a new set of conditions called Cesàro -Integrability and Strong Cesàro -Integrability and show that, for <1/2, these conditions that are strictly weaker than Cesàro Uniform Integrability and Strong Cesàro Uniform Integrability respectively, are sufficient for WLLN and SLLN to hold for a sequence of pairwise independent random variables. For some special kinds of dependent sequences of random variables also, Cesàro -integrability for appropriate is shown to be sufficient for WLLN to hold.     

L ∞ Algebra Representations

It is shown that the direct sum of an L algebra and a module over it has a natural L algebra structure.     

Stability of Numerical Methods for Ordinary Differential Equations

Variable stepsize stability results are found for three representative multivalue methods. For the second order BDF method, a best possible result is found for a maximum stepsize ratio that will still guarantee A(0)-stability behaviour. It is found that under this same restriction, A()-stability holds for 70°. For a new two stage two value first order method, which is L-stable for constant stepsize, A(0)-stability is maintained for stepsize ratios as high as aproximately 2.94. For the third order BDF method, a best possible result of (1/2)(1+ ) is found for a ratio bound that will still guarantee zero-stability.     

State-Homomorphisms on MV-Algebras

Riecan [12] and Chovanec [1] investigated states in MV-algebras. Earlier, Riecan [11] had dealt with analogous ideas in D-posets. In the monograph of Riecan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on MV-algebras. We remark that a different definition of a state in an MV-algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term "state-homomorphism" (instead of "state"). The author is indebted to the referee for his suggestion concerning terminology. Let be an MV-algebra which is defined on a set A with card A>1. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on and the system of all -closed maximal ideals of . For MV-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between MV-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.     

Ore Extensions of Hopf Algebras

In this paper, Ore extensions in the class of Hopf algebras are studied. The classification theorem enables one to describe the Hopf--Ore extensions for the group algebras, for the algebras and , and for the quantum ax + b group.     

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {V _nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S _n –A _n to be stochastically bounded or for the weak law of large numbers to hold where {A _n ,n1} is a (nonrandom) sequence in .     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

Iterating the Bar Construction

For a 1-connected space X Adams's bar construction B(C*(X)) describes H*(X) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces ⁱ X. In this paper for an n-connected pointed space X a sequence of A()-algebra structures , is constructed, such that for each there exists an isomorphism of graded algebras      

Testing optimality for quadratic 0–1 unconstrained problems

This paper analyses a necessary and sufficient optimality condition for quadratic pseudo-Boolean unconstrained problems. It is proved that in general testing any necessary and sufficient optimality condition is a difficult task for anyNP-hard problem. An-optimality condition is derived together with an approximation scheme to test it.This work has been supported by the Progetto Finalizzato Trasporti 2, 93.01799.PF74.Professor Paolo Carraresi died unexpectedly on March 5, 1994. At the time of his death this paper had been completed. While undertaking the final revision, the other two authors were reminded just how much they were indebted to Professor Carraresi after many years of common work together.