The paper deals with the power and robustness of the R/S type tests under contiguous alternatives. We briefly review the long memory models in levels and volatility, and describe the R/S-type tests used to test for the presence of long memory. The empirical power of the tests is investigated when replacing the fractional difference operator (1 - L)d by the mixed operator (1-rL)d in the ARFIMA, LARCH and ARCH time series models. We also investigate the Gegenbauer process with a pole of the spectral density at frequency close to zero.

It is well known that the weak Euler approximation of a stochastic differential equation has order one, provided the coefficients of the equation are sufficiently smooth. We prove that the order of the approximation is still one in the case where the drift coefficient is a Lipschitz function and the diffusion coefficient is constant.

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A⊂2S: W(A,P)≡∫Sϕ(d(x,A))P(dx)→min A∈A, where ϕ is a nondecreasing function. A special case where A is a parametric class A={A(Θ):Θ∈T} is considered in detail. Our main interest is to obtain convergence results for sequences {A*n}, where A*n is an optimal set for a measure Pn satisfying Pn⇒P, as n→∞.

The paper is devoted to the research of large deviation probabilities in the approximation by compound Poisson law.

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1) transversal homoclinic orbits, (2) Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite-dimensional autonomous systems, the author was able to prove the existence of chaos through a shadowing lemma. Around the Silnikov homoclinic orbits, the author was able to prove the existence of chaos through a horseshoe construction. Very recently, there has been a breakthrough by the author in finding Lax pairs for Euler equations of incompressible inviscid fluids. Further results have been obtained by the author and collaborators.

This paper introduces a notion of geometric completeness for spaces of distributions, modelled after the notion of a complete variety in algebraic geometry. It is related to the following elimination problem for systems of PDE: consider the set of homogeneous solutions of a system of PDE in some space of distributions. When is the projection of this set onto some of its coordinates also the set of homogeneous solutions of a system of PDE?

Let S be the set of scalings {n −1:n=1,2,3,...} and let L z =z Z 2, z∈S, be the corresponding set of scaled lattices in R 2. In this paper averaging operators are defined for plaquette functions on L z to plaquette functions on L z′ for all z′, z∈S, z′=dz, d∈{2,3,4,...}, and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D=1,2,3,... and not only for D=2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.

A coordinate-free proof of the maximum principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the use of a concept of lift over a bundle map. We further derive necessary and sufficient conditions for the existence of so-called abnormal extremals.

We define, and discuss some properties of, a form which bears a relation to the Caratheodory form for multiple integral problems in the calculus of variations similar to the relation that the Hilbert 1-form of Finsler geometry bears to the Poincare–Cartan form for one-dimensional variational problems.

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.