Results 1  10
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45
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 78 (5 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
Stochastic Optimal Growth with Unbounded Shock
 Journal of Economic Theory
, 2002
"... This paper considers a neoclassical optimal growth problem where the shock that perturbs the economy in each time period is potentially unbounded on the state space. Su#cient conditions for existence, uniqueness and stability of equilibria are derived in terms of the primitives of the model using ..."
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Cited by 18 (10 self)
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This paper considers a neoclassical optimal growth problem where the shock that perturbs the economy in each time period is potentially unbounded on the state space. Su#cient conditions for existence, uniqueness and stability of equilibria are derived in terms of the primitives of the model using recent techniques from the field of perturbed dynamical systems. Journal of Economic Literature Classification Numbers: C61, C62, O41
Characterization of dependence of multidimensional Lévy processes using Lévy copulas
 J. Multivariate Anal
, 2006
"... This paper suggests to use Lévy copulas to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a kind of Sklar’s theorem states that the law of a general multivariate Lévy ..."
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Cited by 16 (1 self)
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This paper suggests to use Lévy copulas to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a kind of Sklar’s theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multidimensional Lévy process X from the ordinary copulas of the random vectors Xt for fixed t.
Continuous Stochastic Logic Characterizes Bisimulation of Continuoustime Markov Processes
 J. of Logic and Alg. Progr
, 2002
"... In a recent paper Baier, Haverkort, Hermanns and Katoen [BHHK00], analyzed a new way of modelchecking formulas of a logic for continuoustime processes  called Continuous Stochastic Logic (henceforth CSL) { against continuoustime Markov chains { henceforth CTMCs. One of the important results o ..."
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Cited by 15 (3 self)
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In a recent paper Baier, Haverkort, Hermanns and Katoen [BHHK00], analyzed a new way of modelchecking formulas of a logic for continuoustime processes  called Continuous Stochastic Logic (henceforth CSL) { against continuoustime Markov chains { henceforth CTMCs. One of the important results of that paper was the proof that if two CTMCs were bisimilar then they would satisfy exactly the same formulas of CSL. This raises the converse question { does satisfaction of the same collection of CSL formulas imply bisimilarity? In other words, given two CTMCs which are known to satisfy exactly the same formulas of CSL does it have to be the case that they are bisimilar? We prove that the answer to the question just raised is \yes". In fact we prove a signi cant extension, namely that a subset of CSL suces even for systems where the statespace may be a continuum. Along the way we prove a result to the eect that the set of Zeno paths has measure zero provided that the transition rates are bounded.
Stochastic inequality constrained closedloop model predictive control  with application to chemical process operation
, 2004
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Admissible representations of probability measures
 Electronic Notes in Theoretical Computer Science
"... In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We ..."
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Cited by 9 (0 self)
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In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. This canonical representation turns out to be admissible with respect to the weak topology on M(X). Moreover, we prove that for countably based topological spaces X the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology on M(X).
Representing Probability Measures using Probabilistic Processes
 Journal of Complexity
, 2006
"... In the Type2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the repres ..."
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Cited by 7 (2 self)
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In the Type2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the represented space. Conversely, for several wellbehaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names ” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures. 1
Arithmetic Dynamical Systems
, 2000
"... The main objects of study in this thesis are Z d actions by automorphisms of compact abelian groups, which arise in a natural arithmetic setting. In particular, to a countable integral domain D and units 1 ; : : : ; d 2 D we associate a Z d action by automorphisms of the compact abelian ..."
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Cited by 7 (5 self)
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The main objects of study in this thesis are Z d actions by automorphisms of compact abelian groups, which arise in a natural arithmetic setting. In particular, to a countable integral domain D and units 1 ; : : : ; d 2 D we associate a Z d action by automorphisms of the compact abelian group b D. This generalises the `Sinteger dynamical systems' introduced by Chothi, Everest and Ward, where d = 1 and D is a ring of Sintegers in an A field. Familiar dynamical properties such as expansiveness and entropy are investigated in this setting, together with the emerging theory of expansive subdynamics introduced by Boyle and Lind. Homoclinic points are also examined. The main results are as follows. 1. Using results of Lind, Schmidt and Ward, an explicit entropy formula is given which applies whenever D is an integrally closed domain (Theorems 3.3.4 and 3.3.8). 2. The wellknown expansiveness criteria for toral automorphisms, involving the eigenvalues of associated integer...
Random channel assignment in the plane
 Random Structures & Algorithms
, 2003
"... ABSTRACT: In the model for random radio channel assignment considered here, points corresponding to transmitters are thrown down independently at random in the plane, and we must assign a radio channel to each point but avoid interference. In the most basic version of the model, we assume that there ..."
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Cited by 5 (0 self)
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ABSTRACT: In the model for random radio channel assignment considered here, points corresponding to transmitters are thrown down independently at random in the plane, and we must assign a radio channel to each point but avoid interference. In the most basic version of the model, we assume that there is a threshold d such that, in order to avoid interference, points within distance less than d must be assigned distinct channels. Thus we wish to color the nodes of a corresponding scaled unit disk graph. We consider the first n random points, and we are interested in particular in the behavior of the ratio of the chromatic number to the clique number. We show that, as n 3 �, in probability this ratio tends to 1 in the “sparse ” case [when d � d(n) is such that the average degree grows more slowly than ln n] and tends to 2�3/ � � 1.103 in the “dense ” case (when the average degree grows faster than ln n). We also consider related graph invariants, and the more general channel assignment model when assignments must satisfy “frequencydistance ” constraints. © 2003
Numerical methods for the stochastic LandauLifshitz NavierStokes equations
 Physical Review E
"... The LandauLifshitz NavierStokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several CFD approaches are considered (including MacCormack’s twostep Lax ..."
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Cited by 5 (2 self)
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The LandauLifshitz NavierStokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several CFD approaches are considered (including MacCormack’s twostep LaxWendroff scheme and the Piecewise Parabolic Method) and are found to give good results (about 10 % error) for the variances of momentum and energy fluctuations. However, neither of these schemes accurately reproduces the density fluctuations. We introduce a conservative centered scheme with a thirdorder RungeKutta temporal integrator that does accurately produce density fluctuations. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS PDE solver are compared with theory, when available, and with molecular simulations using a Direct Simulation Monte Carlo (DSMC) algorithm. 1 1