Invariant Measure and a Limit Theorem for Some Generalized Gauss Maps

Continued fractions w.r.t. a specified class of numbers is considered. The invariant measures of the corresponding transformations are identified connecting the continued fractions with geodesics on the upper half plane. A problem of convergence in distribution of sums of the coefficients of the continued fraction is also considered.     

Highly sparse representations from dictionaries are unique and independent of the sparseness measure

The purpose of this paper is to study sparse representations of signals from a general dictionary in a Banach space. For so-called localized frames in Hilbert spaces, the canonical frame coefficients are shown to provide a near sparsest expansion for several sparseness measures. However, for frames which are not localized, this no longer holds true and sparse representations may depend strongly on the choice of the sparseness measure. A large class of admissible sparseness measures is introduced, and we give sufficient conditions for having a unique sparse representation of a signal from the dictionary w.r.t. such a sparseness measure. Moreover, we give sufficient conditions on a signal such that the simple solution of a linear programming problem simultaneously solves all the nonconvex (and generally hard combinatorial) problems of sparsest representation of the signal w.r.t. arbitrary admissible sparseness measures.     

A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

This work is devoted to the study of quadrature rules for integration with respect to (w.r.t.) general probability measures with known moments. Automatic calculation of the Clenshaw–Curtis rules is considered and analyzed. It is shown that it is possible to construct these rules in a stable manner for quadrature w.r.t. balanced measures. In order to make a comparison Gauss rules and their stable implementation for integration w.r.t. balanced measures are recalled. Convergence rates are tested in the case of binomial measures.     

Some remarks on absolute continuity and quantization of probability measures

We introduce a notion of monotonicity of dimensions of measures. We show that the upper and lower quantization dimensions are not monotone. We give sufficient conditions in terms of so-called vanishing rates such that νμ implies . As an application, we determine the quantization dimension of a class of measures which are absolutely continuous w.r.t. some self-similar measure, with the corresponding Radon–Nikodym derivative bounded or unbounded. We study the set of quantization dimensions of measures which are absolutely continuous w.r.t. a given probability measure μ. We prove that the infimum on this set coincides with the lower packing dimension of μ. Furthermore, this infimum can be attained provided that the upper and lower packing dimensions of μ are equal.     

Measuring the impact of primal heuristics

In modern MIP solvers, primal heuristics play a key role in finding high-quality solutions. However, classical performance measures reflect the impact of primal heuristics on the overall solving process badly. In this article, we introduce a new performance measure, the “primal integral”, which depends on the quality of solutions and on the time when they are found. We compare five state-of-the-art MIP solvers w.r.t. the newly proposed measure, and show that heuristics improve their performance by up to 80%.     

Adaptive dissimilarity index for measuring time series proximity

The most widely used measures of time series proximity are the Euclidean distance and dynamic time warping. The latter can be derived from the distance introduced by Maurice Fréchet in 1906 to account for the proximity between curves. The major limitation of these proximity measures is that they are based on the closeness of the values regardless of the similarity w.r.t. the growth behavior of the time series. To alleviate this drawback we propose a new dissimilarity index, based on an automatic adaptive tuning function, to include both proximity measures w.r.t. values and w.r.t. behavior. A comparative numerical analysis between the proposed index and the classical distance measures is performed on the basis of two datasets: a synthetic dataset and a dataset from a public health study.     

Stochastic scheduling problems I — General strategies

The paper contains an introduction to recent developments in the theory of non-preemptive stochastic scheduling problems. The topics covered are: arbitrary joint distributions of activity durations, arbitrary regular measures of performance and arbitrary precedence and resource constraints. The possible instability of the problem is demonstrated and hints are given on stable classes of strategies available, including the combinatorial vs. analytical characterization of such classes. Given this background, the main emphasis of the paper is on the monotonicity behaviour of the model and on the existence of optimal strategies. Existing results are presented and generalized, in particular w.r.t. the cases of lower semicontinuous performance measures or joint duration distributions having a Lebesgue density.     

Ergodicity of canonical Gibbs measures with respect to the diffeomorphism group

For general potentials we prove that every canonical Gibbs measure on configurations over a manifold X is quasi‐invariant w.r.t. the group of diffeomorphisms on X. We show that this quasi‐invariance property also characterizes the class of canonical Gibbs measures. From this we conclude that the extremal canonical Gibbs measures are just the ergodic ones w.r.t. the diffeomorphism group. Thus we provide a whole class of different irreducible representations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Lévy Measures for Symmetric Stable Semigroups on the Torus T∞

We investigate analytical properties of the Lévy measures fir symmetric stable semigroups on the compact Lie-projective group T. We apply these properties to describe the domain of the fractional powers of Laplacians on T. Among the analytical tools involved are the intrinsic metric and the scale of Hölder continuous functions w.r.t. this metric.     

Attracting properties for one dimensional flows of a general barotropic viscous fluid. Periodic flows

Summary We consider the motion of a barotropic compressible fluid in a one dimensional bounded region with impermeable boundary, see equation (1.1). Here, u(t, q) denotes the velocity and v(t, q) the specific volume. The quantity log v(t, q) measures the displacement of v(t, q) with respect to the equilibrium v 1. For the sake of brevity we denote here different norms by the simbol . We show that there is a positive constant r₀=r₀(), a small ball B₁ (r) (with radius R₁ (r), ), and a large ball B(r) (with radius R(r), ) such that the following holds, for each r [0, r₀ [(i) If f(t) < r for all t 0, and if (u(0), log v(0))R(r) (i.e. (u(0), log v(0)) B(r)) then, for sufficiently large values of t, (u(t), log v(t))R₁ (r); (ii) The solutions starting at time t=0 from the large ball B(r) have all the same asymptotic behaviour (see (1.11)); (iii) If f is T-periodic then there is a (unique) T-periodic solution (u(t), log v(t)) inside the small ball B₁ (r). This periodic solution atracts all solutions which intersect the large ball B(r). Periodic solutions had been previously studied only for very specific pressure laws, namely p(v)-log v and p(v)-v^–1.     

Zeros of the densities of infinitely divisible measures on R n

Let be an infinitely divisible probability measure onR ⁿ without Gaussian component and let be its Lévy measure. Suppose that is absolutely continuous with respect to the Lebesgue measure . We investigate the structure of the set ⁿ of admissible translates of . This yields a unified presentation of previously known results. We also show that if(S)>0 then is equivalent to , under the assumption that supp =R ⁿ , whereS is the closure of the semigroup generated by the support of .The research of this author is supported by KBN Grant.The research of this author is supported by AFSOR Grant No. 90-0168, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

C0-半群关于参数的可微性及其应用

考虑了C0-半群关于参数的可 微性,而参数含在半群的无穷小生成元中. 证明了：无穷小生成元关于参数的广义连续 性及强可微性蕴含着该C0-半群关于参数的可微性. 这些结果被应用于证明线性延 滞微分方程的解关于延滞量的可微性质.     

Asymptotic properties of local densities of measures generated by semimartingales

For families of probability measures (P , )) generated by semimartingales, we consider the local density)(y, )= _t (y, )) _t0 of a, measureP _y with respect to the measureP whose logarithm is the difference of a local martingale and a positive predictable increasing locally bounded process. Conditions are obtained under which the relations and hold, wherey _t depends in some way ont, while _t ast . Applications of these relations are exhibited and an example is given when the hypotheses of the theorems proved can be verified.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 48–55, 1986.     

Compactness in spaces of inner regular measures and a general Portmanteau lemma

This paper may be understood as a continuation of Topsoe's seminal paper [F. Topsoe, Compactness in spaces of measures, Studia Math. 36 (1970) 195-212] to characterize, within an abstract setting, compact subsets of finite inner regular measures w.r.t. the weak topology. The new aspect is that neither assumptions on compactness of the inner approximating lattices nor nonsequential continuity properties for the measures will be imposed. As a providing step also a generalization of the classical Portmanteau lemma will be established. The obtained characterizations of compact subsets w.r.t. the weak topology encompass several known ones from literature. The investigations rely basically on the inner extension theory for measures which has been systemized recently by König [H. König, Measure and Integration, Springer, Berlin, 1997; H. König, On the inner Daniell-Stone and Riesz representation theorems, Doc. Mat. 5 (2000) 301-315; H. König, Measure and integration: An attempt at unified systematization, Rend. Istit. Mat. Univ. Trieste 34 (2002) 155-214].     

Harnack inequalities for functional SDEs with multiplicative noise and applications

By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of Wang (1997) [14] is also investigated.     

A-priori estimates and existence of Gibbs measures: a simplified proof

We prove existence and uniform a-priori estimates for Gibbs states of certain classical lattice systems with unbounded spins and n-particle interactions. We use a characterization of Gibbs measures in terms of their Radon-Nikodym derivatives w.r.t. local shifts of the configuration space and the corresponding integration by parts formula. Detailed proofs are contained in [3].     

Co—半群关于参数的可微性及其应用

考虑了Co-半群关于参数的可微性,而参数含在半群在无穷小生成元中,证明了：无穷小生成元关于参数的广义连续性及强可微性蕴含着Co-半群关于参数的可微性。这些结果被应用于证明线性延滞微分方程的解关于延滞量的可微性质。     

Operator–valued Fourier multiplier theorems on Besov spaces

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports supp μ ? supp ν on the real line. This class of operators includes a generalization of the well‐known Sturm‐Liouville operator as well as of the measure geometric Laplacian given by . We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gauß‐Green‐formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.     

Allgemeine Zerlegungstheorie

The notion -disjoint decomposition is introduced as a common generalization of the elementary geometric decomposition of polyhedras and the disjoint decomposition of sets: Let Rø be a set, G a subgroup of the symmetric group S_R and 2^R with ø and ¦¦>1; then A, B 2^R are called -disjoint if ø. is the only set of contained in A B, and then A B A B is called the disjoint union. If fullfills the axioms (0, I, II, III) of §2, then can be provided in a natural way with certain binary relations. With respect to the decomposition equivalence can be turned in a partially ordered commutative semigroup with identity (cf. Satz §5).

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Dedicated to Prof.R.Artzy on the occasion of his 80. birthday