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Edgereinforced random walk on a ladder
 Ann. Probab
, 2005
"... We prove that the edgereinforced random walk on the ladder Z × {1,2} with initial weights a> 3/4 is recurrent. The proof uses a known representation of the edgereinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal o ..."
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We prove that the edgereinforced random walk on the ladder Z × {1,2} with initial weights a> 3/4 is recurrent. The proof uses a known representation of the edgereinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyse this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edgereinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece. 1
Constructive Order Completeness
, 2004
"... Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an ..."
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Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an equivalent condition for the existence of the supremum, a condition of (upper) order locatedness which is vacuously true in the classical case. A generalization of this condition will provide a definition of upper locatedness for a partially ordered set. It turns out that the supremum of a set S exists if and only if S is upper located and has a weak supremum—that is, the classical least upper bound. A partially ordered set will be called order complete if each nonempty subset that is bounded above and upper located has a supremum. It can be proved that, as in the classical mathematics, R n is order complete. 1
Optimization with set relations: Conjugate Duality
 Optimization
"... The aim of this paper is to develop a conjugate duality theory for convex set–valued maps. The basic idea is to understand a convex set–valued map as a function with values in the space of closed convex subsets of R p. The usual inclusion of sets provides a natural ordering relation in this space. I ..."
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The aim of this paper is to develop a conjugate duality theory for convex set–valued maps. The basic idea is to understand a convex set–valued map as a function with values in the space of closed convex subsets of R p. The usual inclusion of sets provides a natural ordering relation in this space. Infimum and supremum with respect to this ordering relation can be expressed with aid of union and intersection. Our main result is a strong duality assertion formulated along the lines of classical duality theorems for extended real–valued convex functions. Acknowledgments This paper is part of the author’s Ph.D. thesis, written under supervision of Christiane Tammer and Andreas H. Hamel at Martin–Luther–Universität Halle–Wittenberg. The author wishes to express his deepest graditude. 1
On convex functions with values in semi–linear spaces; submitted to
 Journal of Nonlinear and Convex Analysis
, 2003
"... The following result of convex analysis is well–known [2]: If the function f: X → [−∞, +∞] is convex and some x0 ∈ core (dom f) satisfies f(x0)> −∞, then f never takes the value −∞. From a corresponding theorem for convex functions with values in semi–linear spaces a variety of results is deduced, a ..."
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The following result of convex analysis is well–known [2]: If the function f: X → [−∞, +∞] is convex and some x0 ∈ core (dom f) satisfies f(x0)> −∞, then f never takes the value −∞. From a corresponding theorem for convex functions with values in semi–linear spaces a variety of results is deduced, among them the mentioned theorem, a theorem of Deutsch and Singer on the single–valuedness of convex set–valued maps as well as a result on the compact–valuedness of convex set–valued maps. We also discuss the possibility of embedding the image points of such a convex function into a linear space.
Probabilistic PointtoPoint Information Leakage
"... Abstract—The outputs of a program that processes secret data may reveal information about the values of these secrets. This paper develops an information leakage model that can measure the leakage between arbitrary points in a probabilistic program. Our aim is to create a model of information leakag ..."
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Abstract—The outputs of a program that processes secret data may reveal information about the values of these secrets. This paper develops an information leakage model that can measure the leakage between arbitrary points in a probabilistic program. Our aim is to create a model of information leakage that makes it convenient to measure specific leaks, and provide a tool that may be used to investigate a program’s information security. To make our leakage model precise, we base our work on a simple probabilistic, imperative language in which secret values may be specified at any point in the program; other points in the program may then be marked as potential sites of information leakage. We extend our leakage model to address both nonterminating programs (with potentially infinite numbers of secret and observable values) and user input. Finally, we show how statistical approximation techniques can be used to estimate our leakage measure in realworld Java programs. Keywordsinformation leakage; probabilistic language; nontermination I.
Journal of Applied Mathematics and Stochastic Analysis, 13:2 (2000), 137146. FATOU’S LEMMA AND LEBESGUE’S CONVERGENCE THEOREM FOR MEASURES
, 1999
"... Analogues of Fatou’s Lemma and Lebesgue’s convergence theorems are established for f fd#r when {#n} is a sequence of measures. A "generalized" Dominated Convergence Theorem is also proved for the asymptotic behavior of f fnd#n and the latter is shown to be a special case of a more general result est ..."
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Analogues of Fatou’s Lemma and Lebesgue’s convergence theorems are established for f fd#r when {#n} is a sequence of measures. A "generalized" Dominated Convergence Theorem is also proved for the asymptotic behavior of f fnd#n and the latter is shown to be a special case of a more general result established in vector lattices and related tot he DunfordPettis property in Banach spaces.
URL: www.emis.de/journals/AFA/ IDEALTRIANGULARIZABILITY OF UPWARD DIRECTED SETS OF POSITIVE OPERATORS
"... Abstract. In this paper we consider the question when an upward directed set of positive idealtriangularizable operators on a Banach lattice is (simultaneously) idealtriangularizable. We prove that a majorized upward directed integral operators is idealtriangularizable. We also prove that a finit ..."
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Abstract. In this paper we consider the question when an upward directed set of positive idealtriangularizable operators on a Banach lattice is (simultaneously) idealtriangularizable. We prove that a majorized upward directed integral operators is idealtriangularizable. We also prove that a finite subset of an additive semigroup of positive power compact quasinilpotent operators is idealtriangularizable. Moreover, we prove that an additive semigroup of positive power compact quasinilpotent operators of bounded compactness index is idealtriangularizable. 1. Introduction and