Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties

Potential Analysis - We establish a Karhunen-Loève expansion for generic centered, second order stochastic processes, which does not rely on topological assumptions. We further investigate in...     

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

We address the issue of sample path simulation of Lévy-driven continuous-time autoregressive moving average (CARMA) processes. Approximate discrete-time simulation schemes are constructed along with quantifiable error analysis for stable, second-order and non-negative CARMA processes, based upon the so-called series representation of infinitely divisible laws and associated Lévy processes. We prove that under suitable conditions, the simulation scheme can be improved in terms of second-order structure, finite dimensional laws as well as sample path properties. The simulation procedure is often quite simple and allows one to conduct super-sampling without running the algorithm once again. The computational complexity of the proposed scheme is not affected much by the sampling scheme, such as sampling frequency and irregular spacing. Numerical results are presented throughout to illustrate the effectiveness of the proposed simulation scheme.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

The Singularity Spectrum f(α) of Some Moran Fractals

We show that the multifractal decomposition behaves as expected for a family of sets E known as homogeneous Moran fractals associated with the Fibonacci sequence , using probability measures () associated with the Fibonacci sequence . For each value of a parameter (_min, _max), we define multifractal components E of E, and show that they are fractals in the sense of Taylor. We give the explicit formula for the dimension of E. Also our method can be used for the Moran fractals associated with some more general sequences.     

Some Properties of Regular and α-Regular （LM）-Spaces

In this paper, we give some properties of regular (LM)-spaces and α-regular (LM)-spaces and investigate the relationship among them.     

Erratum to: “Uniqueness of Polyhedral Domains and p-Moduli of Path Families”

Doklady Mathematics -     

Revuz Measures, Energy Functionals and Capacities Under Girsanov Transform Induced by α-Excessive Function

The author investigates the relationships of some potential objects for a right Markov process and the same objects for the Girsanov transformed process induced byα-excessive function including Revuz measures, energy functionals, capacities and Lévy systems in this paper.     

Estimations of Hölder Regularities and Direction of Singularity by Hart Smith and Curvelet Transforms

Using Hart Smith’s and curvelet transforms, new necessary and new sufficient conditions for an L ²(?²) function to possess Hölder regularity, uniform and pointwise, with exponent α>0 are given. Similar to the characterization of Hölder regularity by the continuous wavelet transform, the conditions here are in terms of bounds of the transforms across fine scales. However, due to the parabolic scaling, the sufficient and necessary conditions differ in both the uniform and pointwise cases. We also investigate square-integrable functions with sufficiently smooth background. Specifically, sufficient and necessary conditions, which include the special case with 1-dimensional singularity line, are derived for pointwise Hölder exponent. Inside their “cones” of influence, these conditions are practically the same, giving near-characterization of direction of singularity.     

Some Properties of Artin-Rees Modules and Rings

The class of Artin-Rees modules is large.It contains modules satisfying (accr).In this paper, we investigate such modules and show that some properties of modules satisfying (accr) can be generalized to modules of this class.     

Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Algebras and Representation Theory - Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$ . In this...     

The Euclidean Bottleneck Steiner Path Problem and Other Applications of (α,β)-Pair Decomposition

We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s, t∈P, and an integer k>0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(nlog² n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. (in Wireless Networks 16, 1033–1043, 2010), who gave an O(n ²logn)-time algorithm. We also study the dual version of the problem, where a value λ>0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most λ. Our algorithms are based on two new geometric structures that we develop—an (α,β)-pair decomposition of P and a floor (1+ε)-spanner of P. For real numbers β>α>0, an (α,β)-pair decomposition of P is a collection $\mathcal{W}=\{(A_{1},B_{1}),\ldots,(A_{m},B_{m})\}$ of pairs of subsets of P, satisfying the following: (i) For each pair $(A_{i},B_{i}) \in\mathcal {W}$ , both minimum enclosing circles of A _i and B _i have a radius at most α, and (ii) for any p, q∈P, such that |pq|≤β, there exists a single pair $(A_{i},B_{i}) \in\mathcal{W}$ , such that p∈A _i and q∈B _i , or vice versa. We construct (a compact representation of) an (α,β)-pair decomposition of P in time O((β/α)³ nlogn). In some applications, a simpler (though weaker) grid-based version of an (α,β)-pair decomposition of P is sufficient. We call this version a weak (α,β)-pair decomposition of P. For ε>0, a floor (1+ε)-spanner of P is a (1+ε)-spanner of the complete graph over P with weight function w(p,q)=?|pq|?. We construct such a spanner with O(n/ε ²) edges in time O((1/ε ²)nlog² n), even though w is not a metric. Finally, we present two additional applications of an (α,β)-pair decomposition of P. In the first, we construct a strong spanner of the unit disk graph of P, with the additional property that the spanning paths also approximate the number of substantial hops, i.e., hops of length greater than a given threshold. In the second application, we present an O((1/ε ²)nlogn)-time algorithm for computing a one-sided approximation for distance selection (i.e., given k, $1 \le k \le{n \choose2}$ , find the k’th smallest Euclidean distance induced by P), significantly improving the running time of the algorithm of Bespamyatnikh and Segal.     

Geometry of ℑ-Spaces and Properties of Reversing Operators

Mathematical Notes -     

Fejér Sums and Fourier Coefficients of Periodic Measures

The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejér sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejér sums at any two points whose difference is incommensurable with π.     

Interpolation of Ideal Measures by Abstract K and J Spaces

We work with the abstract K and J interpolation method generated by a sequence lattice Г. We investigate the deviation of an interpolated operator from a given operator ideal by establishing formulae for the ideal measure of the interpolated operator in terms of the ideal measures of restrictions of the operator. Formulae are given in terms of the norms of the shift operators on Г.     

Shearer’s Measure and Stochastic Domination of Product Measures

Let G=(V,E) be a locally finite graph. Let $\vec{p}\in[0,1]^{V}$ . We show that Shearer’s measure, introduced in the context of the Lovász Local Lemma, with marginal distribution determined by $\vec{p}$ , exists on G if and only if every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionally, we derive a non-trivial uniform lower bound for the parameter vector of the dominated Bernoulli product field. This generalises previous results by Liggett, Schonmann, and Stacey in the homogeneous case, in particular on the k-fuzz of ?. Using the connection between Shearer’s measure and a hardcore lattice gas established by Scott and Sokal, we transfer bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem.     

Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of?Probability?Measures

We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.     

Some Properties of Fuzzy Filter and Its Category

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