An H2-Type Norm of a Discrete-Time Linear Stochastic System with Periodic Coefficients Simultaneously Affected by an Infinite Markov Chain and Multiplicative White Noise Perturbations

This article investigates the problem of the definition and computation of an H₂-type norm for discrete-time time-varying periodic stochastic linear systems simultaneously affected by multiplicative white noise perturbations and random jumping according to a Markov chain with an infinite countable number of states. Also, we solve an optimization problem that contains, as a special case, the H₂ optimal control problem for the considered class of stochastic systems under the assumption of perfect state measurements.     

Lie-trotter product formulas for nonlinear filtering

The nonlinear filtering problem for a diffusion process whose drift and diffusion coefficients depend parametrically on a finite-state jump process involves the solution of a vector system of linear, stochastic partial differential equations. A Lie-Trotter product formula is proven to hold for this system and a recursive implementation is discussed.     

Variable Selection in Varying-Coefficient Models Using P-Splines

In this article, we consider nonparametric smoothing and variable selection in varying-coefficient models. Varying-coefficient models are commonly used for analyzing the time-dependent effects of covariates on responses measured repeatedly (such as longitudinal data). We present the P-spline estimator in this context and show its estimation consistency for a diverging number of knots (or B-spline basis functions). The combination of P-splines with nonnegative garrote (which is a variable selection method) leads to good estimation and variable selection. Moreover, we consider APSO (additive P-spline selection operator), which combines a P-spline penalty with a regularization penalty, and show its estimation and variable selection consistency. The methods are illustrated with a simulation study and real-data examples. The proofs of the theoretical results as well as one of the real-data examples are provided in the online supplementary materials.     

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

Bounds on the largest Kronecker and induced multiplicities of finite groups

We give new bounds and asymptotic estimates on the largest Kronecker and induced multiplicities of finite groups. The results apply to large simple groups of Lie type and other groups with few conjugacy classes.     

Adjusting for High-Dimensional Covariates in Sparse Precision Matrix Estimation by Lasso Penalized D-Trace Loss

This paper develops a covariate-adjusted precision matrix estimation using a two-stage estimation procedure. Firstly, we identify the relevant covariates that affect the means by a joint l_1 penalization. Then, the estimated regression coefficients are used to estimate the mean values in a multivariate sub-Gaussian model in order to estimate the sparse precision matrix through a Lasso penalized D-trace loss. Under some assumptions, we establish the convergence rate of the precision matrix estimation under different norms and demonstrate the sparse recovery property with probability converging to one. Simulation shows that our methods have the finite-sample performance compared with other methods.     

Analysis of segregated boundary‐domain integral equations for variable‐coefficient problems with cracks

Segregated direct boundary‐domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Complete sets of solutions to partition–distribution equations

In the analysis of algebraic equations in discrete statistical systems, it is often necessary to find the complete sets of non-negative integer solutions k₁, k₂, k₃, … to the partition–distribution equations: k₁+k₂+k₃+…=m and k₁+2k₂+3k₃+…=n. In this paper, two efficient methods, indirect and recurrent, are derived to find the desired results for any given positive integers n and m. An example is provided to demonstrate the merits of these two methods.     

On the asymptotics of the number of reduced integral quadratic forms with a condition of divisibility of the first coefficients

For the quadratic forms mentioned in the title, we use the discrete ergodic method to obtain asymptotic formulas with remainders depending on the L-Dirichlet function L(s, χ) and its behavior under some hypotheses.