Über Positive Resolventenwerte Positiver Operatoren

In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element xA is a resolvent value of a positive element yA if and only if the element x satisfies the negative principle: If aA, < 0 and xaa then xa 0.     

The Coordinatewise Uniformly Kadec–Klee Property in Some Banach Spaces

We introduce a new property UKK _c for a Banach space and show for an Orlicz sequence space the following: UKK _c H _{c 2}.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

Maximal Classes of Topological Spaces and Domains Determined by Function Spaces

In this paper the question of what classes A of T ₀-spaces should be paired with classes of domains in order that all function spaces [AB] for AA and B are -compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called RW-spaces (with finitely many components) is paired with the class of -compact pointed L-domains (L-domains).     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen

LetM be a multiplicative set with 1M andmnM if and only ifmM,nM for (m,n)=1. It is shown by elementary means that there exists the asymptotic density of the setM(M–1) for every multiplicative setM. The density is positive if and only ifM possesses a positive density and 2M for some . This result is slightly generalized to sums over multiplicative functionsf with |f|1.     

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

Joint convergence of conditional expectations

Let P_n, nIN{0}, be probability measures on a-fieldA; f_n, nIN{0}, be a family of uniformly boundedA-measurable functions andA _n, nIN, be a sequence of sub--fields ofA, increasing or decreasing to the-fieldA _o. It is shown in this paper that the conditional expectations converge in P_o-measure to with k, n, m , if P_n|A, nIN, converges uniformly to P_n|A and f_n, nIN, converges in P_o-measure to f_o.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

A finite characterization ofK-matrices in dimensions less than four

The class of realn × n matricesM, known asK-matrices, for which the linear complementarity problemw – Mz = q, w 0, z 0, w ^T z =0 has a solution wheneverw – Mz =q, w 0, z 0 has a solution is characterized for dimensionsn <4. The characterization is finite and practical. Several necessary conditions, sufficient conditions, and counterexamples pertaining toK-matrices are also given. A finite characterization of completelyK-matrices (K-matrices all of whose principal submatrices are alsoK-matrices) is proved for dimensions <4.Partially supported by NSF Grant MCS-8207217.Research partially supported by NSF Grant No. ECS-8401081.     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

On the Representation of Linear Continuous Functionals in Spaces of Analytic Functions of the Hardy–Sobolev Type in a Polydisk

We describe dual spaces of classes of the Hardy–Sobolev type of functions holomorphic in a polydisk for 0 < p 1 and q (0, ) and for p (1, ) and q = 1.     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.