Cofibrations of algebras

Summary The chain complex of a twisted free product A_*t, FK, is chain homotopy equivalent to a differential graded algebra, which is identified to be a confibration of algebras as defined by Quillen. Under certain connectivity conditions we obtain a long exact sequence connecting the homologies of A, K, and A_*t FK. In particular we derive a long exact sequence connecting the homologies of Y, X and (Y U_g CX) (, C, are the loop, the cone and the suspension constructions respectively). A chain complex equivalent to the chain complex of the Milnor free group FX is recognized, from which results a theorem of Bott and Samelson that H(X) is freely generated as a graded algebra by H(X).     

Product Representations in ΠΣ-Fields

-fields are a very general class of difference fields that enable one to discover and prove multisum identities arising in combinatorics and special functions. In this article we focus on the problem how such multisums can be represented in terms of -fields. In particular we consider product representations and their simplifications in -fields.Received July 09, 2003     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

On 3-dimensional Riemannian Σ-spaces

The following result is proved: a 3-dimensional connected and simply connected Riemannian manifold admitting a reduced -structure (in the sense of O. Loos) is either a Riemannian symmetric space or it is isometric to a unimodular Lie group with a left-invariant Riemannian metric. At the same time, we give first nontrivial examples of Riemannian -spaces, which are not symmetric of finite order.     

The Symmetry of 2-Flat Homogeneous Spaces

A Riemannian manifold M is called 2-flat homogeneous if every geodesic is contained in some 2-flat , and if the group of isometries of M acts transitively on the set of pairs (p, ) with p . By a 2-flat we mean a closed, connected, flat, totally geodesic, 2-dimensional submanifold of M. It is proved in the paper that 2-flat homogeneous spaces are symmetric.     

Presentability of Groups of Σ-Definable Permutations over Admissible Sets

It is proved that a group of all -permutations of an arbitrary locally countable, recursively listed, admissible set is not -presentable over that set. An example is constructed showing that this group could be presentable provided that the condition of being locally countable is dropped.     

Necessary convergence conditions for series ΣαnSn in the case of identically distributed independent random quantities

Necessary (in some cases also sufficient) conditions are obtained for convergence of the series a _n S _n whereS _n = ₁ ⁿ _k k are independent random quantities. The cases in which k are symmetrical or identically distributed quantities are investigated in more detail.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 529–536, October, 1976.     

Zur Krümmungskennzeichnung der kinematischen Laguerre-Ketten

In the case of certain geometries () [1] over kinematic algebras [5],[6] the -trace of the system of chains can be characterized by constant curvature. The curvature allows an affine interpretation of non-affine points.     

Spreads admitting net generating regulizations

We say a spread S carries a regulization , if is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence S^c of S with respect to . If S^c is a hyperbolic or parabolic or elliptic linear congruence of lines, then is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.     

Manifolds of infinite topological type with integrable geodesic flows

Let (M,k) be a complete surface of constant negative curvature (resp. an -geometric 3-manifold). This paper constructs a complete riemannian 8-manifold (resp. 9-manifold) (,h) such that is homotopy equivalent to M, the geodesic flow of h is completely integrable and there is a riemannian embedding (M,k)(,h). This embeds the geodesic flow of (M,k) as a subsystem of an integrable geodesic flow. Amongst the manifolds is an 8-dimensional manifold whose fundamental group is the free group on countably many generators.Thanks to Keith Burns and Leo Jonker for comments. Research partially supported by the Natural Sciences and Engineering Research Council of Canada.Mathematics Subject Classification (2000): 58F17, 53D25, 37D40     

Über die mittelfläche einer isotropen geradenkongruenz

Let be the middle surface of an isotropic rectilinear congruence of class C³ in the real Euclidean space E³. When the spherical image of is parametrized by special isothermal coordinates (u,v) G ², can be described by a generating harmonic function A(u,v). Using such a C-representation of , the basic properties of regularity and curvature of are discussed. Moreover, the cases that be a minimal (regular) surface ₁, or a plane surface ₂ are solved explicitly. In connection with the latter results (which are already well-known from Ribaucour) several new characterizations for being a regular surface ₁ resp. ₂ are given: they are based on special properties (like: being asymptotic lines resp. lines of curvature of ) of those curves c (-Spurlinien) in the tangents of which form in each point Xc a minimal angle with the straight line of passing through X.

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Herrn Prof. Dr. Karl Strubecker zum 80. Geburtstag gewidmet     

On Tyler's M-Functional of Scatter in High Dimension

Let y¹, y²,..., yⁿ R^q be independent, identically distributed random vectors with nonsingular covariance matrix , and let S = S(y¹,..., yⁿ) be an estimator for . A quantity of particular interest is the condition number of ^-1 S. If the yⁱ are Gaussian and S is the sample covariance matrix, the condition number of ^-1 S, i.e. the ratio of its extreme eigenvalues, equals 1 + O^p((q/n)^1/2) as q and q/n 0. The present paper shows that the same result can be achieved with two estimators based on Tyler's (1987, Ann. Statist., 15, 234-251) M-functional of scatter, assuming only elliptical symmetry of (yⁱ) or less. The main tool is a linear expansion for this M-functional which holds uniformly in the dimension q. As a by-product we obtain continuous Fréchet-differentiability with respect to weak convergence.     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.     

Die zweiten Hüllbahnkrümmungen bei ebenen isotropen Zwangläufen

This paper deals with motions ₁/_o in the isotropic plane. To a given C⁴-curve q₁ in the moving plane ₁ we determine its enveloping curve q_o in the fixed plane _o und investigate its thirdorder properties. We derive necessary and sufficient conditions on q₁ and on the motion ₁/_o for q_o to have a point of stationary curvature. In special cases q_o has a cusp or a node. Furthermore envelopes of curves q₁ with singular points are discussed.     

Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure

Let w : [0, ) be a weight function on a set R. We assume that the associated extremal measure has density function v(t) with finitely many singularities of logarithmic type. We show that any continuous function f on which vanishes outside the set where v is positive or has a logarithmic singularity, is the uniform limit on of a sequence of weighted polynomials of the form wⁿ P_n, where P_n is of degree n. This extends previous results for continuous densities to densities having logarithmic singularities.     

The inverse hull of the free semigroup on a set X

In this paper the inverse hull of the free semigroup on a non-empty set X is characterized. It is shown that if F^O is a certain semilattice and (G,*) is the free group on X, then is isomorphic to a Rees factor semigroup of a subsemigroup of (F^O x G, #), where (A,w)#(B,u) = ((A*w B)*w^–1, w*u). Several properties of , including Green's relations on are also determined.     

On an optimum test of the equality of two covariance matrices

Let X: p × 1, Y: p × 1 be independently and normally distributed p-vectors with unknown means ₁, ₂ and unknown covariance matrices ₁, ₂ (>0) respectively. We shall show that Pillai's test, which is locally best invariant, is locally minimax for testing H ₀: ₁=₂ against the alternative H ₁: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaeiDaiaabkhacaqGOaWaaabmaeaadaaeqaqaaiabgkHiTiaadMea% caGGPaGaaiiiaiabg2da9iaacccacqaHdpWCcaGGGaGaeyOpa4Jaai% iiaiaaicdaaSqaaiaaigdaaeqaniabggHiLdaaleaacaqGYaaabaGa% aeylaiaabgdaa0GaeyyeIuoaaaa!4E3F!\[{\rm{tr(}}\sum\nolimits_{\rm{2}}^{{\rm{ - 1}}} {\sum\nolimits_1 { - I) = \sigma > 0} }\]as 0. However this test is not of type D among G-invariant tests.Research supported by the Canadian N.S.E.R.C. Grant.     

Krümmungskennzeichnung der Ketten einer allgemeinen Geometrie Σ(ℜ,\mathfrak{L})

Lie theory renders a concept of curvature for the geometries (,) introduced by W. Benz [1]. Among regular curves in the chains of (, ) are characterized by =0.     

4-Dimensional compact projective planes with small nilradical

We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL \{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Let X be a Banach space and (,,µ) be a -finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u₁,..., u_d, u_i >0, 1 i d} of linear contractions on L_p((,,µ); X), with 1 p<. In this paper differentiation theorems are proved for d-dimensional bounded processes in L_p((,,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on L_p((,,µ); R), that is, P(u) is a positive linear contraction on L_p((,,µ); R) such that T(u)f(w) P(u)f(·)() almost everywhere on for all f L_p((,,µ); X).