Residual empirical processes and qualitatively robust GM-tests in autoregression

The local qualitative robustness of GM-tests against outliers in the autoregression model is studied in the paper. A local scheme of data contamination by independent outliers with the intensity O(n ^?1/2) is considered. The qualitative robustness in terms of power equicontinuity is obtained. The GM-tests asymptotically optimal in the maximin sense are constructed.     

Local robustness of sign tests in AR(1) against outliers

The paper deals with robustness of nonparametric sign tests against outliers in the autoregression AR(1) model. We consider the local scheme of data contamination by independent additive outliers with intensity O(n ^?1/2). The qualitative robustness of tests in terms of power equicontinuity is obtained.     

Monotone quantum stochastic processes and covariant dynamical hemigroups

Based on the Hilbert C^?-module structure we study non-stationary monotone quantum stochastic processes and general Markov processes constructed from quantum dynamical hemigroups indexed by a totally ordered set. We prove that the quantum stochastic monotone process implementing the weakly covariant process described by a covariant quantum dynamical hemigroup with respect to a symmetry semigroup is again covariant in the strong sense.     

The central limit theorem for a sequence of random processes with space-varying long memory*

In this paper, we investigate a sequence of square-integrable random processes with space-varying memory. We establish sufficient conditions for the central limit theorem in the space L ²(μ) for the partial sums of the sequence of random processes with space-varying long memory. Of particular interest is a nonstandard normalization of the partial sums in the central limit theorem.     

Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in ? ^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T ^m–n with fiber a real moment-angle manifold R and the other over a quotient of R by a finite group with fiber a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.     

Two-sided heat kernel estimates for censored stable-like processes

In this paper, we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C ^1,1 open sets in ${\mathbb R^d}$ , where d ≥ 1 and ${\alpha\in (1, 2)}$ . We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C ^1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C ^1,1 open sets.     

L 2-Decay rate for non-ergodic Jackson network

We establish the additive theorem of L^2-decay rate for multi- dimensional Markov process with independent marginal processes. Using this and the decomposition method, we obtain explicit upper and lower bounds for decay rate of non-ergodic Jackson network. In some cases, we get the exact decay rate.     

A finite element method scheme for boundary value problems with noncoordinated degeneration of input data

A finite element method scheme is constructed for boundary value problems with noncoordinated degeneration of input data and singularity of a solution. We look at a rate with which an approximate solution by the proposed finite element method converges toward an exact R _ν -generalized solution in the weight set W _2,ν*+β ^2+1/1 (Ω, δ), and establish estimates for the finite element approximation.     

Some semigroups of completely positive maps on the CCR algebra

The Fock construction used by Davies in his theory of quantum stochastic processes yields a semigroup of completely positive maps on the C¹-algebra of the CCR. We show how such semigroups may be constructed using an arbitrary representation of the CCR and we investigate some of their properties.     

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.     

Homoclinic bifurcations at the onset of pulse self-replication

We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray-Scott system: u^″=uv², v^″=v−uv², which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α→∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches.     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

On the dimension of the attractor for a perturbed 3d Ladyzhenskaya model

We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ?Δ³. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ? > 0.     

Algorithmic and structural aspects of the P 3-Radon number

The generalization of classical results about convex sets in ? ⁿ to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P ₃-convexity on graphs. P ₃-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radon’s classical convexity result. We establish hardness results and describe efficient algorithms for trees.     

Population models with diffusion, strong Allee effect, and nonlinear boundary conditions

We consider a population model with diffusion, a strong Allee effect per capita growth function, and constant yield harvesting. In particular, we focus our study on a population living in a patch, Ω⊆Rⁿ with n≥1, that satisfies a certain nonlinear boundary condition. We establish our existence results by the method of sub-supersolutions.     

Strong approximation of multidimensional ℙ-ℙ plots processes by Gaussian processes with applications to statistical tests

The present paper is mainly concerned with the strong approximation of ?-? plot processes in ? ^d by sequences of Gaussian processes. In order to evaluate the limiting laws, a general notion of bootstrapped multidimensional ?-? plots processes, constructed by exchangeably weighting sample, is presented and investigated. The applications discussed here are change-point detection in multivariate copula models and the law of iterated logarithm. Finally, we extend our framework to the K-sample problem and apply our results to derive the limiting laws of Kolmogorov-Smirnov and Cramér-von Mises statistics.     

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

We propose a new definition of the Neyman chi-square divergence between distributions. Based on convexity properties and duality, this version of the χ² is well suited both for the classical applications of the χ² for the analysis of contingency tables and for the statistical tests in parametric models, for which it is advocated to be robust against outliers.We present two applications in testing. In the first one, we deal with goodness-of-fit tests for finite and infinite numbers of linear constraints; in the second one, we apply χ²-methodology to parametric testing against contamination.     

A simple averaging process for approximating the solutions of certain optimal control problems

It is shown that the extremal solutions of fixed duration Mayer control problems with implicit terminal constraints can be interpreted as fixed points of certain function-valued operators F constructed by solving pairs of initial value problems in tandem. A class of simple recursive averaging processes is proposed for approximating the fixed points of F. Results from the theory of monotone Hilbert space operators are used to establish the convergence of the averaging processes for a general linear-quadratic curve follower problem with unbounded control inputs, and for a simple second order bounded control input problem.     

Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

In the present paper we prove a criterion of Lip ^k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Fréchet space F. We establish that a manifold is Lip ^k -paracompact if and only if the model space F is paracompact and Lip ^k -normal. We prove a sufficient condition for existence of Lip ^k -partition of a unity on a manifold of class Lip ^k .     

Polynomial Models of Degree 5 and Algebras of Their Automorphisms

In classifying and studying holomorphic automorphisms of surfaces, it is often convenient to pass to tangent model surfaces. This method is well developed for surfaces of type (n,K), where K≤ ² ; for such surfaces, tangent quadrics (i.e., surfaces determined by equations of degree 2) with a number of useful properties have been constructed. In recent years, for surfaces of higher codimensions, tangent model surfaces of degrees 3 and 4 with similar properties were constructed. However, this construction imposes new constraints on the codimension. In this paper, the same method is applied to surfaces of even higher codimension. Model surfaces of the fifth degree are constructed. It is shown that all the basic useful properties of model surfaces are preserved, in spite of a number of technical difficulties.