Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A ^*)–B ^*(A+A ^*)B>0, whereB=–(P+L) ^–1 R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ converges if and only ifA+A ^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A _ij ] ₁ ² , whereA ₂₁=–A ₁₂ ^* ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ₁₁ ^* =A₂₂+A ₂₂ ^* ; and the proper values ofA ₁₂ ^–1 A ₁₁,A ₁₂ ^*–1 A ₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.     

Elastic constants of monotropic plastic foams. 2. Deformation parallel to foam rise direction. Numerical calculations

In order to perform numerical calculations of the elastic constants of monotropic plastic foams with a pronounced strut-like structure, a package of computing programs has been developed. Numerical values of the elastic constants were determined using the Simpson method for the calculation of triple averaging integrals and stepwise unconditional minimization of the one-argument potential energy function. The parameters of the numerical calculation process providing acceptable accuracy of the results were evaluated. Dimensions of the structural elements in the model of isotropic plastic foams were determined and analyzed.Institute of Polymer Mechanics, Riga, LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 163–172, March–April, 1998.     

Bowen–Franks Groups for Subshifts and Ext-Groups for C*-Algebras

We generalize the Bowen–Franks groups for topological Markov shifts to general subshifts as the Ext-groups for the associated C ^*-algebras. The generalized Bowen–Franks groups for subshifts are shown to be invariant under flow equivalence and, hence, invariant under topological conjugacy. They are regarded as the indices of Fredholm operators related to extensions of the associated C ^*-algebras so that they are described in terms of symbolic dynamical systems. In particular, the group for a sofic subshift is determined by the adjacency matrix of its left Krieger cover graph. The Bowen–Franks groups for some non sofic subshifts are calculated, proving that certain subshifts with the same topological entropy are not flow equivalent.     

Some algebraic properties of cyclic homology groups

In [10], see also [8], a cyclic homology theory HC _* ^– was introduced. The purpose of this paper is to study algebraically the properties of this version of cyclic homology. First we study its relation to Connes cyclic cohomology theory HC ^* and to the usual cyclic homology theory HC _* studied by Loday and Quillen in [15]. We explain the precise sense in which HC _* ^– is dual to HC ^*. Next we study products and describe a general method for constructing product operations in cyclic homology and cohomology theories. Finally we examine the relation between the theory HC _* ^– and algebraic K-theory.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Generalized Hyers–Ulam–Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over -algebras

Assume that X is a left Banach module over a unital C^*-algebra A. It is shown that almost every n-sesquilinear-quadratic mapping h:X×X×Xⁿ→A is an n-sesquilinear-quadratic mapping when holds for all x,y,z₁,…,z_nX.Moreover, we prove the generalized Hyers–Ulam–Rassias stability of an n-sesquilinear-quadratic mapping on a left Banach module over a unital C^*-algebra.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

A splitting theorem for certain cohomology theories associated to BP* (−)

Let P(n)^*(–) be Brown-Peterson cohomology modulo I_n and put B(n)^*(–)=P(n)^*(–)[1/v_n]. In this note we construct a canonical multiplicative and idempotent operation _n in a suitable completion (n)^*(–) of B(n)^*(–) which has the property that its image is canonically isomorphic to the n-th Morava K-theory K(n)^*(–). In particular, the ring theory K(n)^*(–) is contained as a direct summand in the theory (n)^*(–). A similar result is not true before completing. pleting. Because the completion map B (n)^*(–) (n)^*(–) is injective, the above splitting theorem contains also information about B(n)^*(–). The proof of the theorem depends on a result about the behaviour of formal groups of finite height over complete graded F_p.     

Topologies on BP-algebras

The class of locally convex ^*-algebra topologies on a BP^*-algebra which possess the same bounded hermitian idempotent subsets is considered and is shown to have a finest element. The cone of positive elements of a symmetric BP^*-algebra is studied and is seen to be closed in this finest topology. Order-bounded BP^*-algebras are considered, and it is seen that the positive linear functionals span the dual of such an algebra. The class of equivalent topologies on an order-bounded commutative BP^*-algebra for which every positive linear functional is continuous is considered, and there are found to be such topologies which are neither barreled nor Q-topologies, so the results of [6.], 15–28) are extended.     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.     

Dunford–Pettis properties, Hilbert spaces and projective tensor products

We study complete continuity properties of operators onto ℓ₂ and prove several results in the Dunford–Pettis theory of JB^*-triples and their projective tensor products, culminating in characterisations of the alternative Dunford–Pettis property for where E and F are JB^*-triples.     

On the square root of a positive B( )-valued function

Let (Ω, , μ) be a measure space, a separable Banach space, and ^* the space of all bounded conjugate linear functionals on . Let f be a weak^* summable positive B( ^*)-valued function defined on Ω. The existence of a separable Hilbert space , a weakly measurable B( )-valued function Q satisfying the relation Q^*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺( ^*)-valued measures, the concepts of weak^*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

The K-Theory of Cuntz–Krieger Algebras for Infinite Matrices

We compute the K-theory groups of the Cuntz–Krieger C ^*-algebra O _A associated to an infinite matrix A of zeros and ones.     

A Parallel Algorithm for the Estimation of the Global Error in Runge–Kutta Methods

The object of this work is the estimate of the global error in the numerical solution of the IVP for a system of ODE's. Given a Runge–Kutta formula of order q, which yields an approximation y _n to the true value y(x _n ), a general, parallel method is presented, that provides a second value y _n ^* of order q+2; the global error e _n =y _n –y(x _n ) is then estimated by the difference y _n –y _n ^*. The numerical tests reported, show the very good performance of the procedure proposed. A comparison with the code GEM90 is also appended.     

Boundary values of a convergent sequence of J-contractive matrix-functions

In this note it is proved that if W_n(z) are J-contractive matrix-functions which are meromorphic in the disk ¦z¦<1 (j–w=">_n ^* (z)JW_n(z) W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)     

Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Rotating flow past an elliptic-cylindrical bump of large aspect ratio

Summary Very slow flow of a rapidly rotating fluid past a right cylindrical obstacle of heightd on one of two co-rotating parallel planes, separated by a distanceh, is studied. The elliptic boundaryvalue problem for the geostrophic flow is formulated quite generally, and solutions are given for an infinitely-long ridge and an elliptic cylinder. A modified Ekman number,E _*v/d ², where is the inverse aspect ratio of the cylinder, characterizes the latter solution. ForE _*<1, the fluid flows around a stagnant Taylor column over the elliptic cylinder; forE _*1, the fluid flows over the cylinder as though it were an infinitely-long ridge;E _*=O(1) shows part of the fluid flowing over, and part around, the cylinder. The drag increases from orderE ^–1/4 to orderE ^–1 asE _* increases from small to large values.

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Zusammenfassung In dieser Arbeit wird die sehr langsame Strömung in einem rasch rotierenden Fluid untersucht um einen zylindrischen Körper, dessen Höhed die viel kleiner ist als die Distanzh der zwei unendlich ausgedehnten mitrotierenden parallelen Platten, welche das Fluid begrenzen; der Körper sitzt auf der unteren Platte. Eine allgemeine Formulierung des elliptischen Randwertproblems wurde für die geostrophische Strömung angegeben. Lösungen wurden für ein unendlich langes Wehr und einen elliptischen Zylinder gefunden. Eine modifizierte Ekman-Zahl,E _*=v/d ² E, in welcher das reziproke Verhältnis von Länge zu Durchmesser darstellt, charakterisiert die Lösung für den elliptischen Zylinder. WennE _*1, umgeht die Flüssigkeit die stagnierende Taylorsäule, die auf dem elliptischen Zylinder ruht; wennE _*1, geht die Flüssigkeit über den Zylinder, als ob er ein unendlich langes Wehr wäre; wennE _*=0(1), geht ein Teil der Flüssigkeit über den Zylinder und ein anderer um ihn herum. Der Reibungswiderstand erhöht sich von 0(E ^–1/4) zu 0(E ^–1), wennE _* von kleinen zu grossen Werten ansteigt.

     

The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Let K be an eventually compact linear integral operator on L^p(Ω, μ), 1 p < ∞, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when P = 1. This work extends the previous analysis of the author who characterized the distinguished eigenvalues of K and K^*, and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of K or K^*. The characterizations of the support sets for the algebraic eigenspaces of K or K^* are phrased in terms of significant k-components which are maximal irreducible subsets of Ω and which yield a positive spectral radius for the integral operator defined by the restriction of k(x, y) to the Cartesian product of such sets. In this paper, we show that a basis for the functions, constituting the algebraic eigenspaces of K and K^* belonging to the spectral radius of K, can be chosen to consist of elements which are positive on their sets of support, except possibly on sets of measure less than some arbitrarily specified positive number. In addition, we present necessary and sufficient conditions, in terms of the significant k-components, for both K and K^* to possess a positive eigenfunction (a.e. μ) corresponding to the spectral radius, as well as necessary and sufficient conditions for the sequence γⁿKⁿg _p to converge whenever g 0, where − _p denotes the norm in L^p(Ω, μ), and γ₁ the smallest (in modulus) characteristic value of K. This analysis is made possible by introducing the concepts of chains, lengths of chains, height, and depth of a significant k-component as was done by U. Rothblum [Lin. Alg. Appl. 12 (1975), 281–292] for the matrix setting.