On unbounded eigenvalues in particle-transport theory

The dispersion function derived from particle-transport theory is analyzed for the specific case of a three-term redistribution law in order to define thosec>1 cases for which there can be either one or two pairs of unbounded eigenvalues, and the elementary solutions corresponding to the unbounded eigenvalues are reported.

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Zusammenfassung Die Dispersions-Funktion, die von der Teilchen-Transport-Theorie erhalten wurde, wird analysiert für den besonderen Fall eines dreigliedrigen Neuverteilungsgesetzes, um die Fällec>1 zu definieren, für die entweder ein oder zwei Paare von unbegrenzten Eigenwerten existieren. Die zugehörigen elementaren Lösungen werden angegeben.

     

On a problem in probability theory

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers $$N > 0, E > 0, 0 \leqslant \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _s $$ be given. Consider a random vector x = (x ₁, …, x _s), uniformly distributed on the set $$x_j \geqslant 0, j = 1, \ldots ,s; \sum\limits_{j = 1}^s {x_j = N} , \sum\limits_{j = 1}^s {\lambda _j x_j \leqslant E} .$$ We study the weak limit of x as s → ∞.     

On Borel probability measures and noncooperative game theory

In this article, the well-known minimax theorems of Wald, Ville and von Neumann are generalized under weaker topological conditions on the payoff function f and/or extended to the larger set of the Borel probability measures, instead of the set of mixed strategies.     

Approach theory meets probability theory

In this paper we reconsider the basic topological and metric structures on spaces of probability measures and random variables, such as e.g. the weak topology and the total variation metric, replacing them with more intrinsic and richer approach structures. We comprehensibly investigate the relationships among, and basic facts about these structures, and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorov?s theorem, can be recaptured in a considerably stronger form in the new setting.     

Unbounded probability theory compatible with the probability theory of numbers

We study the case in which the values of random variables increase without bound.     

Spectral theory of certain unbounded Jacobi matrices

Using the conjugate operator method of Mourre we study the spectral theory of a class of unbounded Jacobi matrices. We especially focus on the case where the off-diagonal entries an=nα(1+o(1)) and diagonal ones bn=λnα(1+o(1)) with α>0, λ∈R.     

On unbounded Bergman operators

This paper explores properties of the Bergman operator on unbounded open subsets of the plane. In addition to the characterization of the bounded commutant of such operators it proves the Berger-Shaw theorem and gives some general criteria under which the operator and its self-commutator are densely defined.     

Mourre's theory for some unbounded Jacobi matrices

We show a Mourre estimate for a class of unbounded Jacobi matrices. In particular, we deduce the absolute continuity of the spectrum of such matrices. We further conclude some propagation theorems for them.     

On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.     

On products of unbounded operators

We first investigate the lattice structure of the partially ordered group of all derivations on a lattice ordered ring (l-ring) R. Then, using the inner derivations on R, we prove some results regarding the commutativity of R. Finally, we study homo and anti-homo derivations on quasi semi-prime, semi-prime, reduced, and l-prime rings and l-rings; and, among other things, we generalize some of the results of Bell and Kappe [1] to l-rings. This revised version was published online in June 2006 with corrections to the Cover Date.     

On products of unbounded operators

A symmetric operator X^ is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of X^ are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX ₁ ... X _n are proved. The question of ^-complete positivity of the mapping p → Ap(X ₁,...,X _n) defined on complex polynomials in n variables is investigated. It is shown that the set ω is related to the McIntosh-Pryde joint spectrum of (X ₁,...,X _n) in case all the operators A, X ₁,...,X _n are bounded. Examples illustrating the theme of the paper are included. This revised version was published online in June 2006 with corrections to the Cover Date.     

On unbounded hyponormal operators II

The paper deals with unbounded hyponormal operators. Among others it is proved that any closed hyponormal operator with spectrum contained in a parabola generates a cosine function.     

Discrete problems in probability theory

A short survey is given of some directions in probability theory that have developed most intensively in recent years. Separable statistics and criteria for the verification of statistical hypotheses based on them, various schemes for distributing particles among cells, and problems connected with estimating the unknown size of a finite collection are considered.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 22, pp. 3–60, 1984.