Free <Emphasis Type="Italic">G</Emphasis>-spaces and maximal equivariant compactifications

For a compact Lie group G we prove that every free (resp., semifree) G-space admits a based-free (resp., semifree) G-compactification. Examples of finite- and infinite-dimensional G-spaces are presented that do not admit a free or based-free G-compactification. We give several characterizations of the maximal G-compactification β_GX that are further applied to prove the formula (β_GX)/H=β_G/H(X/H) for arbitrary closed normal subgroup H⊂G. Mathematics Subject Classification (2000) 54H15, 54D35     

On <Emphasis Type="Italic">k</Emphasis>-pairable graphs from trees

The concept of the k-pairable graphs was introduced by Zhibo Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter p(G), called the pair length of a graph G, as the maximum k such that G is k-pairable and p(G) = 0 if G is not k-pairable for any positive integer k. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees G with p(G) = 1 and prove that p(G □ H) = p(G) + p(H) when both G and H are trees.     

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

A sample path relation for the sojourn times in G/G/1−PS systems and its applications

For the single server system under processor sharing (PS) a sample path result for the sojourn times in a busy period is proved, which yields a sample path relation between the sojourn times under PS and FCFS discipline. This relation provides a corresponding one between the mean stationary sojourn times in G/G/1 under PS and FCFS. In particular, the mean stationary sojourn time in G/D/1 under PS is given in terms of the mean stationary sojourn time under FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of these results suggest an approximation of the mean stationary sojourn time in G/GI/1 under PS in terms of the mean stationary sojourn time under FCFS. Mathematics Subject Classification (MSC 2000) 60K25· 68M20· 60G17· 60G10 This work was supported by a grant from the Siemens AG.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.     

Mackey structures on equivariant algebraic K-theory

We study the Mackey structure of the G-spectrum K _G C associated to a monoidal G-category C. It is proved that the coefficient system of K _G C coincides, as a (graded) Mackey functor, with the system of equivariant K-groups in the sense of Fröhlich and Wall. It is also shown that for any exact category U, there exists a G-spectrum Q _G U representing the equivariant K-theory of U in the sense of Dress and Kuku, and that Q _G U is naturally G-homotopy equivalent to K _G IsoU if every short exact sequence in U splits.     

<Emphasis Type="Italic">G</Emphasis>-compactness and groups

Lascar described E _KP as a composition of E _L and the topological closure of E _L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G _L . We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories. The first author is supported by the Polish Goverment grant N N201 384134. The second author is supported by the Polish Goverment grant N201 032 32/2231.     

G-theory of group rings for groups of square-free order

The formula for the G-theory of the group ring of a finite group G conjectured by Hambleton, Taylor, and Williams is shown to be valid for ¦G¦ square-free.     

G-spaces and Fixed Point Sets

For a compact connected group G, in a compact free G-space M every closed subset is the fixed point set of an autohomeomorphism if the weight of M is not bigger than that of G. In a compact group with nonsingleton components, every closed subset is the fixed point set of an autohomeomorphism if and only if the weight of the whole group is not bigger than that of the component of the identity.     

The Automorphism Groups of Finite Type G-Structures on Orbifolds

The automorphism group of a G-structure of finite type and order k on a smooth n-dimensional orbifold is proved to be a Lie group of dimension n+dim(g+g ₁+...+g _k-1), where g _i is the ith prolongation of the Lie algebra g of a given group G. This generalizes the corresponding result by Ehresmann for finite type G-structures on manifolds. The presence of orbifold points is shown to sharply decrease the dimension of the automorphism group of proper orbifolds. Estimates are established for the dimension of the isometry group and the dimension of the group of conformal transformations of Riemannian orbifolds, depending on the types of orbifold points.     

Primary decomposition over rings graded by finitely generated Abelian groups

Let A be a Noetherian ring which is graded by a finitely generated Abelian group G. In general, for G-graded modules there do not exist primary decompositions which are graded themselves. This is quite different from the case of gradings by torsion free group, for which graded primary decompositions always exists. In this paper we introduce G-primary decompositions as a natural analogue to primary decomposition for G-graded A-modules. We show the existence of G-primary decomposition and give a few characterizations analogous to Bourbaki's treatment for torsion free groups.     

Mackey systems of monoidal categories and G-spectra

We construct an equivariant infinite loop space machine denned on certain class of monoidal O _G -categories which have built-in Mackey structure. Applications include the equivariant infinite delooping of the classifying space BF(G) for stable spherical G-fibrations and also the construction of an infinite loop G-space E(X, G) with ₀ ^HE (X, G) naturally isomorphic to the equivariant Whitehead groups Wh _H (X) of given G-space X.Dedicated to Professor Shôrô Araki on his sixtieth birthday     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

Orbit projections as fibrations

The orbit projection π: M → M/G of a proper G-manifold M is a fibration if and only if all points in M are regular. Under additional assumptions we show that π is a quasifibration if and only if all points are regular. We get a full answer in the equivariant category: π is a G-quasifibration if and only if all points are regular.     

Approximations for the delay probability in the M/G/s queue

This paper develops approximations for the delay probability in an M/G/s queue. For M/G/s queues, it has been well known that the delay probability in the M/M/s queue, i.e., the Erlang delay formula, is usually a good approximation for other service-time distributions. By using an excellent approximation for the mean waiting time in the M/G/s queue, we provide more accurate approximations of the delay probability for small values of s. To test the quality of our approximations, we compare them with the exact value and the Erlang delay formula for some particular cases.     

Remainders in compactifications and generalized metrizability properties

When does a Tychonoff space X have a Hausdorff compactification with the remainder belonging to a given class of spaces? A classical theorem of Henriksen and Isbell and certain theorems, involving a new completeness type property introduced below, are applied to obtain new results on remainders of topological spaces and groups. In particular, some strong necessary conditions for a topological group to have a metrizable remainder, or a paracompact p-remainder, are established (the group itself turns out to be a paracompact p-space (Theorem 4.8)). It follows that if a non-locally compact topological group G is metrizable at infinity, then G is a Lindelöf p-space, and the Souslin number of G is countable (Corollary 4.10). This solves Problem 10.28 from [M. Hušek, J. van Mill (Eds.), Recent Progress in General Topology, vol. 2, North-Holland, 2002, pp. 1–57].     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

On general two-scale convergence and its application to the characterization of <Emphasis Type="Italic">G</Emphasis>-limits

We characterize some G-limits using two-scale techniques and investigate a method to detect deviations from the arithmetic mean in the obtained G-limit provided no periodicity assumptions are involved. We also prove some results on the properties of generalized two-scale convergence.