invariant functionals and the uniqueness of invariant norms let be a representation of a compact group g on a banach space x the question we address is whether x carries a unique invariant norm in the sense that is the unique norm on x for which is a representation we characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when x is a dual banach space and is a x x continuous representation of g on x such that g consists of x x continuous operators we illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces lp where is a locally compact hausdorff space equipped with a positive radon measure and g acts on as a group of continuous invertible measure preserving transformations

relative group completions abstract we extend arbitrary group completions to the category of pairs g n where g is a group and n is a normal subgroup of g relative localizations are special cases our construction is a group theoretical analogue of fibrewise completion and fibrewise localization in homotopy theory and generalizes earlier work of hilton and others on relative localization at primes we use our approach to find conditions under which factoring out group radicals preserves exactness this has implications in the study of the effect of plus constructions on homotopy fibre sequences

matrix displacement decompositions and applications to toeplitz linear systems using the approach of bozzo di fiore and zellini new matrix displacement decomposition formulas are introduced it is shown how an arbitrary square matrix a can be expressed as sums of products of hessenberg algebra matrices and high level block matrices whose submatrices are hessenberg algebra matrices and have variable sizes in most cases these block factors are block diagonal matrices then these formulas are used in sequential and parallel solution of toeplitz systems

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

uniqueness of translation invariant norms let a be a banach function space and let m be a family of multipliers on a we provide conditions on m so that the original topology of a is the only complete norm topology on a making all of the maps from m continuous as a corollary we show that for a compact abelian group g and a circle group t for a lp t 1 p the lp norm is the only one that makes all translations continuous while for a c g a l g or a l1 g there are other norms with that property for noncompact groups the situation is different on the space l1 r the l1 norm is the only one that makes a single nontrivial translation continuous

central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the central configurations formed by two twisted regular polygons one n regular polygon and one l regular polygon we wish to extend the results of the symmetrical central configurations formed by two twisted n regular polygons however to our own surprise it will be proved that the necessary condition on the central configurations is l n

the a core and a cover of a group abstract this paper provides a comprehensive investigation of the cellular approximation functor cell a g in the category of groups approximating a group g by a group a we also study related notions such as a injection a generation and a constructibility of a group g and we find several interesting connections with the schur multiplier h 2 g z our constructions are direct and are given in a slow and detailed manner

postnikov pieces and bz p homotopy theory we present a constructive method to compute the cellularization with respect to b m z p for any integer m 1 of a large class of h spaces namely all those which have a finite number of non trivial b m z p homotopy groups the pointed mapping space map b m z p x is a postnikov piece we prove in particular that the b m z p cellularization of an h space having a finite number of b m z p homotopy groups is ap torsion postnikov piece along the way we characterize the bz p r cellular classifying spaces of nilpotent groups

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

homotopical localizations of spaces for a map f of spaces dror farjoun and the author have constructed an f localization functor where a space y is called f local when map f y is an equivalence this very general construction gives all known idempotent homotopy functors of spaces the main theorem of this paper shows that f localization functors always preserve fiber sequences of connected h spaces up to small error terms for instance the localization with respect to the n th morava k theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups this implies for example that a k 1 homology equivalence of h spaces must induce an isomorphism of v 1 periodic homotopy groups results are also obtained on the a nullification or a periodization functors which are just the f localization functors for the maps f from spaces a to points two spaces are said to have the same nullity when they give the same nullification functors and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds the a nullifications of nilpotent postnikov spaces are completely determined

ν torsion spaces and thick classes suppose that p is some chosen prime number a topological space is said to be v torsion if its homotopy groups are concentrated at p and its vn periodic homotopy groups vanish for n 1 1 1 a similar definition can be made for spectra and in this stable case it is easy to see that a spectrum is v torsion if and only if it can be built in the sense of 9 from any finite spectrum with nontrivial mod p homology the unstable analog of this statement is not true but in this paper we provide a close substitute for any space w let bw denote the class of spaces which can be built from w and t bw the smallest class of spaces which contains bw and is thick with respect to fibrations 1 6 we prove that a space x is v torsion if and only if it belongs to t bw for any finite complex w with nontrivial reduced mod p homology we go on to show that although the category of v torsion spaces is not closed under arbitrary homotopy colimits it is closed in many cases with respect to realizations of simplicial objects finally we consider the stoverblanc simplicial resolution rw x of a space x which comes up in a natural attempt to construct the w cellular approximation cellw x we show that for appropriate w the fibres of the map from the geometric realization of rw x to cellw x are v torsion since v torsion spaces are in some respects negligible objects this can be interpreted as saying that the realization of the stover blanc construction is closer to cellw x than one might suspect

are all localizing subcategories of stable homotopy categories coreflective abstract we prove that in a triangulated category with combinatorial models every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large cardinal axiom vop nka s principle is assumed true it follows that under the same assumptions orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories the existence of such a bijection was left as an open problem by hovey palmieri and strickland in their axiomatic study of stable homotopy categories and also by neeman in the context of well generated triangulated categories

uniqueness of norm on lp g and c g when g is a compact group let g be a compact group if the trivial representation of g is not weakly contained in the left regular representation of g on l 2 g and x is either l p g for 1oppn or c g then we show that every complete norm jj on x that makes translations from x jj into itself continuous is equivalent to jjjj p or jjjj n respectively if 1oppn and every left invariant linear functional on l p g is a constant multiple of the haar integral then we show that every complete norm jj on l p g that makes translations from l p g jj into itself continuous and that makes the map t lt from g into l l p g jj bounded is equivalent to jjjj p

discontinuous translation invariant functionals let g be an infinite a compact locally compact group we shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant frechet spaces of radon measures on g these spaces include the convolution measure algebra m g the lebesgue spaces lv g where i p oo and certain combinations thereof among other things it will be shown that c g has many discontinuous translation invariant functionals provided that g is amenable this solves a problem of g h meisters let g be a locally compact group in the present paper we shall consider various translation invariant spaces of functions and measures on g and prove the existence of discontinuous invariant functionals on such spaces among other things it will be shown that c g admits many discontinuous left invariant functionals provided that g is an infinite a compact locally compact amenable group our hypotheses are a little weaker than amenability this resolves the problem of g h meisters 11 as to whether there exists a discontinuous invariant functional on c qu where h denotes the circle group as far as the compact groups are concerned the problems for c g and for l g are equivalent see remark iv stated below as some basic references on this subject we refer to e granirer s papers 3 4 on invariant means the nice survey article 14 by meisters and g s woodward s paper 22 on discontinuous invariant functionals on a variety of invariant spaces let g be an arbitrary locally compact lc group all of the lebesgue spaces lp g 1 s p x are taken with respect to a fixed left haar measure ag on g if g is compact we normalize ag so that xg g 1 let c g and cc g denote the banach space of all continuous bounded complex valued functions on g and the subspace of all members of c g with compact supports respectively we denote by co g the uniform closure of cc g in c g thus the dual of co g may be identified with the convolution measure algebra m g of g cf hewitt and ross 7 a net fa in cc g converges to the zero function if and only if i iaf 0 uniformly on g and ii the f s are eventually supported by a fixed compact set depending on the net this topology for cjg is different from the uniform topology and will be used throughout the paper unless otherwise mentioned the continuous dual of cc g will be denoted c g an element of c g is often called received by the editors february 8 1983 and in revised form april 19 1983 1980 mathematics subject classification primary 43a15 secondary 43a05

shift invariant signal norms for fault detection and control two desirable properties of a signal norm to be used for the purpose of optimal control or fault detection are highlighted these properties are shift invariance and the permitting of persistent signals a class of norms called the window norms is analyzed among norms considered for control purposes they appear to be the only ones save the l norm that satisfy both properties the signal spaces defined by the window norms however include some interesting signals that are not in the l space the window norms have been found suitable for application in fault detection and are here also considered for optimal control it is shown that they can be taken as a support for the concept of l1 control but may also suggest a new class of optimal controllers

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

twisted angles for central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the twisted angles of the central configurations formed by two twisted regular polygons in particular we prove that for the 2n body problem the twisted angles must be 0 or n

solutions of regular polygon with an inner particle for newtonian n 1 body problem abstract in this paper we prove that if the solution of the planar n 1 body problem has n bodies rotating at the vertices of a regular n gon then the n bodies have equal mass and the remaining body must be at the center

central extensions and generalized plus constructions we describe the effect of homological plus constructions on the homotopy groups of eilenberg maclane spaces in terms of universal central extensions

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

applications of kazhdan s property t to automatic continuity let g be a compact group which has the strong kazhdan s property t and let be an isometric representation of g on l p for some measure space and 1 p we characterize the continuity of all the linear operators from l p which commute with translations and the property of p being the unique norm up to equivalence on l p with respect to which all translations are continuous in terms of the finite dimensionality of the subspace of all invariant func

idempotent transformations of finite groups we describe the action of idempotent transformations on finite groups we show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group this is done in terms of the first two homology groups we prove for example that except special linear groups such an orbit can have at most 7 elements we also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite

on similarity of quasinilpotent operators bounded linear operators on separable banach spaces algebraically similar to the classical volterra operator v acting on c 0 1 are characterized from this characterization it follows that v does not determine the topology of c 0 1 which answers a question raised by armando villena a sufficient condition for an injective bounded linear operator on a banach space to determine its topology is obtained from this condition it follows for instance that the volterra operator acting on the hardy space hp of the unit disk determines the topology of hp for any p 1

homotopical complexity and good spaces this paper is an exploration of two ideas in the study of closed classes the a complexity of a space x and the notion of good spaces spaces a for which c a c a a variety of formulae for the computation of complexity are given along with some calculations good spaces are characterized in terms of the functors cw a and p a the main result is a countable upper bound for a complexity when a is a good space