The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

A Galerkin finite element method is developed for the two dimensional/three dimensional nonlinear time-dependent three-species Lotka–Volterra competition-diffusion equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation are proved. An error estimate for the numerical solution is obtained. Numerical computations are carried out to examine the expected orders of accuracy in the error estimates.     

On a model of sequential point patterns

A finite point process motivated by the cooperative sequential adsorption model is proposed. Analytical properties of the point process are considered in details. It is shown that the introduced point process is useful for modeling both aggregated and regular point patterns. A possible scheme of maximum likelihood estimation of the process parameters is briefly discussed. V. Shcherbakov is on leave from Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics, Moscow State University, Moscow.     

On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier–Stokes equations

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L_q framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss

The problem of estimating large covariance matrices of multivariate real normal and complex normal distributions is considered when the dimension of the variables is larger than the number of samples. The Stein–Haff identities and calculus on eigenstructure for singular Wishart matrices are developed for real and complex cases, respectively. By using these techniques, the unbiased risk estimates for certain classes of estimators for the population covariance matrices under invariant quadratic loss functions are obtained for real and complex cases, respectively. Based on the unbiased risk estimates, shrinkage estimators which are counterparts of the estimators due to Haff [L.R. Haff, Empirical Bayes estimation of the multivariate normal covariance matrix, Ann. Statist. 8 (1980) 586–697] are shown to improve upon the best scalar multiple of the empirical covariance matrix under the invariant quadratic loss functions for both real and complex multivariate normal distributions in the situation where the dimension of the variables is larger than the number of samples.     

Bayesian factor analysis with fat-tailed factors and its exact marginal likelihood

The traditional Bayesian factor analysis method is extended. In contrast to the case for previous studies, the matrix variate t-distribution is utilized to provide a prior density on the latent factors. This is a natural extension of the traditional model and yields many advantages. The crucial issue is the selection of the number of factors. The marginal likelihood, constructed by asymptotic and computational approaches, is generally utilized for this problem. However, both theoretical and computational problems have arisen.In this paper, the exact marginal likelihood is derived. It enables us to evaluate posterior model probabilities without inducing the above problems. Monte Carlo experiments were conducted to examine the performance of the proposed Bayesian factor analysis modelling methodology. The simulation results show that the proposed methodology performs well.     

Uniqueness results for nonlocal Hamilton-Jacobi equations

We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.     

Point estimation for multi-spectral distributed random matrices

In this communication, we consider a p×n random matrix which is normally distributed with mean matrix M and covariance matrix Σ, where the multivariate observation x_i=y_i+?_i with p dimensions on an object consists of two components, the signal y_i with mean vector μ and covariance matrix Σ_s and noise with mean vector zero and covariance matrix Σ_?, then the covariance matrix of x_i and x_j is given by Σ=Cov(x_i,x_j)=Γ⊗(B_|i-j|Σ_s+C_|i-j|Σ_?), where Γ is a correlation matrix; B_|i-j| and C_|i-j| are diagonal constant matrices. The statistical objective is to consider the maximum likelihood estimate of the mean matrix M and various components of the covariance matrix Σ as well as their statistical properties, that is the point estimates of Σ_s,Σ_? and Γ. More importantly, some properties of these estimators are investigated in slightly more general models.     

A maximum principle for smooth optimal impulsive control problems with multipoint state constraints

A nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered. Features of this problem are discussed from the viewpoint of the extension of the classical optimal control problem with the corresponding state constraints. A necessary optimality condition is formulated in the form of a smooth maximum principle; thorough comments are given, a short proof is presented, and examples are discussed.     

On estimates for the Fourier-Bessel integral transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Subscript>+</Subscript>)

Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L ₂(?₊) as applied to some classes of functions characterized by a generalized modulus of continuity.     

Error estimates in S P for cubature formulas exact for haar polynomials in the two-dimensional case

On the spaces S _p , an upper estimate is found for the norm of the error functional δ _N (f) of cubature formulas possessing the Haar d-property in the two-dimensional case. An asymptotic relation is proved for $ \left\| {\delta _N (f)} \right\|_{S_p^* } On the spaces S _p , an upper estimate is found for the norm of the error functional δ _N (f) of cubature formulas possessing the Haar d-property in the two-dimensional case. An asymptotic relation is proved for with the number of nodes N ∼ 2 ^d , where d → ∞. For N ∼ 2 ^d with d → ∞, it is shown that the norm of δ _N for the formulas under study has the best convergence rate, which is equal to N ^−1/p . Original Russian Text ? K.A. Kirillov, M.V. Noskov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 1, pp. 3–13.