Absolutely continuous variational measures of Mawhin’s type

In this paper we study absolutely continuous and σ-finite variational measures corresponding to Mawhin, F- and BV -integrals. We obtain characterization of these σ-finite variational measures similar to those obtained in the case of standard variational measures. We also give a new proof of the Radon-Nikodym theorem for these measures.     

Jensen measures and boundary values of plurisubharmonic functions

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions inB-regular domain. This theorem implies that the two classes of Jensen measures coincide inB-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.     

First and second moments for self‐similar couplings and Wasserstein distances

We study aspects of the Wasserstein distance in the context of self‐similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self‐similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self‐similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non‐trivial upper and lower bounds for the 2nd Wasserstein distance for these self‐similar measures.     

Theorems on convergence of stochastic integrals distributions to signed measures and local limit theorems for large deviations

We study properties of symmetric stable measures with index α > 2, α ≠ 2m, m ? \mathbbN m \in \mathbb{N} . Such measures are signed ones, and hence they are not probability measures. For this class of measures, we construct an analogue of the Lévy–Khinchin representation. We show that, in some sense, these signed measures are limit measures for sums of independent random variables. Bibliography: 11 titles.     

Multivariate measures of concordance

In 1984 Scarsini introduced a set of axioms for measures of concordance of ordered pairs of continuous random variables. We exhibit an extension of these axioms to ordered n-tuples of continuous random variables, n ≥ 2. We derive simple properties of such measures, give examples, and discuss the relation of the extended axioms to multivariate measures of concordance previously discussed in the literature.     

Generic properties of invariant measures for simple piecewise monotonic transformations

We endow the set of all invariant measures of topologically transitive subsetsL of certain piecewise monotonic transformations on [0, 1] with the weak topology. We show that the set of periodic orbit measures is dense, that the sets of ergodic, of nonatomic, and of measures with supportL are dense-sets, that the se of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseGin/gd-set.     

The Lévy-Khinchin Representation of the One Class of Signed Stable Measures and Some Its Applications

We study properties of symmetric stable measures with index α∈(2,4)∪(4,6). Such measures are signed ones and hence they are not probability measures. For this class of measures we construct an analogy of the Lévy-Khinchin representation. We show that in some sense these signed measures are limit measures for sums of independent random variables.     

Carleson measures on planar sets

In this paper, we investigate what are Carleson measures on open subsets in the complex plane. A circular domain is a connected open subset whose boundary consists of finitely many disjoint circles. We call a domain G multi-nicely connected if there exists a circular domain W and a conformal map ψ from W onto G such that ψ is almost univalent with respect the arclength on δW. We characterize all Carleson measures for those open subsets so that each of their components is multinicely connected and harmonic measures of the components are mutually singular. Our results suggest the extension of Carleson measures probably is up to this class of open subsets     

Boundary Measures for Geometric Inference

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer’s curvature measures of a compact set, showing that they can be reliably estimated from point-cloud approximations.     

Consecutive expansions of -out-of- systems

We introduce consecutive expansions of k–out–of–n systems, which have the property that components are totally ordered by the node criticality relation and with respect to well-known structural importance measures. We propose some formulae to easily compute these measures and study the hierarchies induced for them for large systems.     

The mean value property for harmonic functions on graphs and trees

Following the Euclidean example, we introduce the strong and weak mean value property for finite variation measures on graphs. We completely characterize finite variation measures with bounded support on radial trees which have the strong mean value property. We show that for counting measures on bounded subsets of a tree with root o, the strong mean value property is equivalent to the invariance of the subset under the action of the stabilizer of o in the automorphism group. We finally characterize, using the discrete Laplacian, the finite variation measures on a generic graph which have the weak mean value property and we give a non-trivial example. Received: July 21, 2000; in final form: March 13, 2001?Published online: March 19, 2002     

Totally dissipative measures for the shift and conformal σ-finite measures for the stable holonomies

In this paper we investigate some results of ergodic theory with infinite measures for a subshift of finite type. We give an explicit way to construct σ-finite measures which are quasi-invariant by the stable holonomy and equivalent to the conditional measures of some σ-invariant measure. These σ-invariant measures are totally dissipative, σ-finite but satisfy a Birkhoff Ergodic-like Theorem. The constructions are done for the symbolic case, but can be extended for uniformly hyperbolic flows or diffeomorphisms.     

Remarks on irrationality of q-harmonic series

 We sharpen the known irrationality measures for the quantities , where z ? {±1} and p ?ℤ \ {0, ±1}. Our construction of auxiliary linear forms gives a q-analogue of the approach recently applied to irrationality problems for the values of the Riemann zeta function at positive integers. We also present a method for improving estimates of the irrationality measures of q-series. Received: 22 October 2001     

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ _f ⁺ isomorphic to (I, f) generalizing the \-shift and another shift Σ _M over an infinite alphabet, which is of finite type given by an infinite transition matrixM. Σ _M has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure.     

Mahler measures in a cubic field

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.     

On ordinary and standard products of infinite family of <Emphasis Type="Italic">σ</Emphasis>-finite measures and some of their applications

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on ℝ^∞ and Rogers-Fremlin measures on ℓ ^∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal (i.e., 2 ^c ). We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.     

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ _C is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ∈[0,λ _C ]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ∈[0,λ _C ]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ _C -transient. Some deep properties regarding λ _C -transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.     

The Lévy–Khinchin representation of a class of stable signed measures

We study properties of symmetric stable measures with index of stability α ∈ (2, 4) ∪ (4, 6). For such signed measures, we construct a natural analog of the Lévy-Khinchin representation. We show that, in some special sense, these measures are limit measures for sums of independent random variables. Bibliography: 6 titles. Translated from Zapiski Nauchnykh. Seminarov POMI, Vol. 361, 2008, pp. 145–166.     

Foliations of polynomial growth are hyperfinite

We define a class of equivalence relations with polynomial growth and show that such relations always support finite invariant measures and are hyperfinite. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. We also extend a result of Dye for countable groups to show that if a locally compact second countable groupG acts freely on a Lebesgue spaceX with finite invariant measure, so that the orbit relation onX is hyperfinite, thenG is amenable.     

A local Steiner–type formula for general closed sets and applications

We introduce support (curvature) measures of an arbitrary closed set A in ^d and establish a local Steiner–type formula for the localized parallel volume of A. We derive some of the basic properties of these support measures and explore how they are related to the curvature measures available in the literature. Then we use the support measures in analysing contact distributions of stationary random closed sets, with a particular emphasis on the Boolean model with general compact particles. Mathematics Subject Classification (2000): 53C65, 28A75, 52A22, 60D05; 52A20, 60G57, 60G55, 28A80.