Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let ₀ > ₁ ··· > _D denote the eigenvalues of and let q ^h _ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram = _h is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q ^h _ij 0. It turns out that in , the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D – h and no other vertices. We call 0, D the trivial vertices of . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths.     

Verallgemeinerung eines Satzes von F. Riesz

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: XR, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F.The purpose of this paper is to represent such a bounded linear operator A: C(X)F as an integral Af= d with respect to some content , defined on the algebraa(X) which is generated by the open sets UX.This representation is unique, if is required to be regular (p. 11).     

Exponential bounds for queues with Markovian arrivals

Exponential bounds [queueb]e ^b are found for queues whose increments are described by Markov Additive Processes. This is done by application of maximal inequalities to exponential martingales for such processes. Through a thermodynamic approach the constant is shown to be the decay rate for an asymptotic lower bound for the queue length distribution. The class of arrival processes considered includes a wide variety of Markovian multiplexer models, and a general treatment of these is given, along with that of Markov modulated arrivals. Particular attention is paid to the calculation of the prefactor .     

Weak convergence of probability measures relative to incompatible topology and σ-field with applications to renewal theory

Summary Necessary and sufficient conditions are given in order that a sequence of probability measures, weakly convergent relative to a given topology ₀ and associated -field ( ₀), are weakly convergent (and satisfy a continuity theorem) relative to the ( ₀)-measurable functions which are continuous in some finer topology ₁, even if does not extend to ( ₀). These conditions are shown to be applicable to a sequence of translated renewal measures. Alternate conditions (tightness, uniformity of weak convergence) are investigated and shown to be inappropriate.This research was partially supported by UMC Summer Faculty Research Fellowships     

Barth type vanishing theorems

Let X be a smooth, projective, d-dimensional subvariety of ⁿ (). Barth's theorem says that H ^q (X, ^p _X )=0 when pq and q+p2d–n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for H ^q (X, ^p _X (m)), m (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for H ^q (X, ^p _X (1)), for H ^q (X, ^p _X (m)) when m–1, and, if dim(X)=n–2, for H ^q (X, ² _X (m)) and H ^q (X, S ^{k 1} _X (m)). We use standard techniques and some of our previous results.     

Funktionalgleichungen in Vektorräumen,Kompositionsalgebren und Systeme partieller Differentialgleichungen

Ohne ZusammenfassungDie sich auf das Problem der Existenz elliptischer verallgemeinerter Cauchy-Riemannscher Differentialgleichungssysteme beziehenden Untersuchungen der vorliegenden Arbeit bildeten einen Abschnitt in der Habilitationsschrift (Würzburg 1965) des Verfassers, die bis auf diesen Abschnitt inzwischen publiziert ist: [10], [11]. Die vorliegende Arbeit wurde wührend eines Gastaufenthaltes des Verfassers an der Faculty of Mathematics der University of Waterloo (Kanada) im Herbstsemester 1967 abgeschlossen und zum Teil durch NRC Grant Nr. A-2972 gefördert.Herrn Professor Dr. A. M. Ostrowski zum 75. Geburtstag gewidmet     

Continuity of Stochastic Convolutions

Let B be a Brownian motion, and let be the space of all continuous periodic functions f with period 1. It is shown that the set of all f such that the stochastic convolution does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.     

Zum Einbettungsproblem für Markoff-Prozesse

Every elementary Markov process with a polish state space and with a discrete set of time parameter dense in ₊, whose finite dimensional distributions are derived from a semigroup (K _t) of Markov kernels continuous in 0 ₊ and whose initial distribution satisfies K _t, can be imbedded in an elementary Markov process with the same state space and with parameter set ₊ so that the corresponding finite dimensional distributions are equal.     

Perturbation of orthonormal bases with an application to diagonalization

Given an orthonormal basis {e _n } _n=1 in a Hilbert spaceH, and a dense linear manifoldDH, we show that there exists a unitary operatorV onH such thatI-V is a trace-class operator with arbitrarily small trace norm, andVe _jD for allj. This result can be used to simplify certain arguments of J. Xia concerning the simultaneous diagonalization of operators on a space of square integrable functions.     

On the lower Markov spectrum

C. Hightower found two infinite sequences of gaps in the Markov spectrum, ( _n , _n ) and ( _n , _n ) with _n and _n both Markov elements, converging to . This paper exhibits Markov elements _n ^* and _n ^* such that, for alln 1, ( _n ^* , _n ) and ( _{n n} ^* ) are gaps in the Markov spectrum. Other results include showing that, for alln 1, _n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.     

A splitting technique for Harris recurrent Markov chains

Summary A technique is presented, which enables the state space of a Harris recurrent Markov chain to be split in a way, which introduces into the split state space an atom. Hence the full force of renewal theory can be used in the analysis of Markov chains on a general state space. As a first illustration of the method we show how Derman's construction for the invariant measure works in the general state space. The Splitting Technique is also applied to the study of sums of transition probabilities.     

Singularly perturbed Markov control problem: Limiting average cost

In this paper we consider a singularly perturbed Markov decision process with the limiting average cost criterion. We assume that the underlying process is composed ofn separate irreducible processes, and that the small perturbation is such that it unites these processes into a single irreducible process. We formulate the underlying control problem for the singularly perturbed MDP, and call it the limit Markov control problem (limit MCP). We prove the validity of the the limit control principle which states that an optimal solution to the perturbed MDP can be approximated by an optimal solution of the limit MCP for any sufficiently small perturbation. We also demonstrate that the limit Markov control problem is equivalent to a suitably constructed nonlinear program in the space of long-run state-action frequencies. This approach combines the solutions of the original separated irreducible MDPs with the stationary distribution of a certain aggregated MDP and creates a framework for future algorithmic approaches.On leave from Main College of Planning and Statistics, Warsaw, Poland.Supported in part by the AFOSR and the NSF under the grant ECS-8704954.     

Inverse Problems in the Theory of Singular Perturbations

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation ² y'=(1+² (x, ))y with a small parameter , where is analytic and even with respect to . It is well known that it has two formal solutions of the form y^±(x,)=e^±x/h^±(x,), where h^±(x,) is a formal series in powers of whose coefficients are functions of x. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if satisfies certain conditions, in particular concerning its x-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations ^r y' = A(x,) y in the neighborhood of a nonturning point and for second-order (analytic) equations y' - 2xy'-g(x,) y=0 exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution such that the coefficients have no poles at x=0.     

Smooth Measures and Regular Strongly Supermedian Kernels Generating Sub-Markovian Resolvents

In the context of a transient Borel right Markov process with a fixed excessive measure , we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no -semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both -polar and -negligible (U being the potential part of ), answering to a problem of Revuz.     

A Stationary Rho-Mixing Markov Chain Which Is Not “Interlaced” Rho-Mixing

A strictly stationary, countable-state Markov chain is constructed which is -mixing (with arbitrarily fast mixing rate) but fails to be *-mixing (interlaced-mixing).     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

Mixed correlation functions in systems interacting with Bose fields

For a large class of systems interacting with Bose fields, exact expressions are found for correlation functions containing products of Bose operators with evolution law of the free field and operators of the medium. Hierarchical equations describing the evolution of the system are constructed by means of these expressions. The Born—Markov approximation is investigated. A causality principle associated with the absence of an influence on the evolution of the system of stochastic forces due to the free field at short times is established.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 142–156, April, 1992.     

The asymptotic behaviour of certain additive functionals of Brownian motion

We prove pathwise asymptotic stability for certain additive functionals of one- and two-dimensional Brownian motion.     

Inzidenzräume mit Süss'scher Kollineationsgruppe

The following theorem will be proved: THEOREM. Let (P,) be a linear space with dimension dim(P,) 2, such that each line contains at least three points. Let further Aut(P,) be a two — transitive group of collineations on the point set P, such that every automorphism fixing one flag (p,L), fixes all lines through p. Then (P,) is a desarguesian affine space and contains the whole group of dilatations of (P,). Moreover one can define an addition and a multiplication on the point set, such that (P,+,-) is a nearfield, the line joining the neutral elements is a normal subfield and consists of the affine transformations x a + mx, m 0 and= (K).

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Herrn Professor Dr. Helmut Karzel zum 60. Geburtstag