Computing multiple pitchfork bifurcation points

A point (x*,λ*) is called apitchfork bifurcation point of multiplicityp≥1 of the nonlinear systemF(x, λ)=0,F:ℝⁿ×ℝ¹→ℝⁿ, if rank∂ _xF(x*, λ*)=n−1, and if the Ljapunov-Schmidt reduced equation has the normal formg(ξ, μ)=±ξ ²⁺ p±μξ=0. It is shown that such points satisfy a minimally extended systemG(y)=0,G:ℝ ⁿ⁺²→ℝⁿ⁺² the dimensionn+2 of which is independent ofp. For solving this system, a two-stage Newton-type method is proposed. Some numerical tests show the influence of the starting point and of the bordering vectors used in the definition of the extended system on the behavior of the iteration.     

MBIS: Multivariate Bayesian Image Segmentation tool

We present MBIS (Multivariate Bayesian Image Segmentation tool), a clustering tool based on the mixture of multivariate normal distributions model. MBIS supports multichannel bias field correction based on a B-spline model. A second methodological novelty is the inclusion of graph-cuts optimization for the stationary anisotropic hidden Markov random field model. Along with MBIS, we release an evaluation framework that contains three different experiments on multi-site data. We first validate the accuracy of segmentation and the estimated bias field for each channel. MBIS outperforms a widely used segmentation tool in a cross-comparison evaluation. The second experiment demonstrates the robustness of results on atlas-free segmentation of two image sets from scan–rescan protocols on 21 healthy subjects. Multivariate segmentation is more replicable than the monospectral counterpart on T1-weighted images. Finally, we provide a third experiment to illustrate how MBIS can be used in a large-scale study of tissue volume change with increasing age in 584 healthy subjects. This last result is meaningful as multivariate segmentation performs robustly without the need for prior knowledge.     

On the linear advection equation subject to random velocity fields

This paper deals with the random linear advection equation for which the time-dependent velocity and the initial condition are independent random functions. Expressions for the density and joint density functions of the solution are given. We also verify that in the Gaussian time-dependent velocity case the probability density function of the solution satisfies a convection-diffusion equation with a time-dependent diffusion coefficient. Some exact examples are presented.     

On convergence of splitting iteration methods for non-Hermitian positive-definite linear systems

A new splitting iteration method is presented for the system of linear equations when the coefficient matrix is a non-Hermitian positive-definite matrix. The spectral radius, the optimal parameter, and some norm properties of the iteration matrix for the new method are discussed in detail. Based on these results, the new method is convergent under reasonable conditions for any non-Hermitian positive-definite linear system. Finally, the numerical examples show that the new method is more effective than the Hermitian and skew-Hermitian splitting iterative (or positive-definite and skew-Hermitian splitting iterative) method in central processing unit time.     

Descent Iterations for Improving Approximate Eigenpairs of Polynomial Eigenvalue Problems with General Complex Matrices

In this paper, we consider descent iterations with line search for improving an approximate eigenvalue and a corresponding approximate eigenvector of polynomial eigenvalue problems with general complex matrices, where an approximate eigenpair was obtained by some method. The polynomial eigenvalue problem is written as a system of complex nonlinear equations with nondifferentiable normalized condition. Convergence theorems for iterations are established. Finally, some numerical examples are presented to demonstrate the effectiveness of the iterative methods. Received April 9, 2001; revised October 2, 2001 Published online February 18, 2002     

The k-fundamental group of a closed k-surface

In this paper, we deal with the problem of computing the digital fundamental group of a closed k-surface by using various properties of both a (simple) closed k-surface and a digital covering map. To be specific, let be a simple closed k_i-curve with l_i elements in Z^n_i, i∈{1,2}. Then, the Cartesian product is not always a closed k-surface with some k-adjacency of Z^n₁+n₂. Thus, we provide a condition for to be a (simple) closed k-surface with some k-adjacency depending on the k_i-adjacency, i∈{1,2}. Besides, even if is not a closed k-surface, we show that the k-fundamental group of can be calculated by both a k-homotopic thinning and a strong k-deformation retract.     

On the independence between risk profiles in the compound collective risk actuarial model

This paper examines a compound collective risk model in which the primary distribution comprised the Poisson–Lindley distribution with a λ parameter, and where the secondary distribution is an exponential one with a θ parameter. We consider the case of dependence between risk profiles (i.e., the parameters λ and θ), where the dependence is modelled by a Farlie–Gumbel–Morgenstern family. We analyze the consequences of the dependence on the Bayes premium. We conclude that the consequences of the dependence on the Bayes premium may vary considerably.     

Improved confidence regions for a mean vector under general conditions

In this paper, we propose two different confidence regions for a mean vector under general conditions when the population of interest is nonnormal and the sample size is moderate. It is shown that both confidence regions will improve the accuracy of the approximation in the sense that the coverage error is of order o(n^-1). Monte Carlo examples are demonstrated to show the performance with comparison to the classical methods.     

PHoMpara – Parallel Implementation of the Polyhedral Homotopy Continuation Method for Polynomial Systems

The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral homotopy functions, tracing the solution curves of the homotopy equations, and verifying the obtained solutions. A software package PHoMpara parallelizes PHoM to solve a polynomial system of large size. Many characteristics of the polyhedral homotopy continuation method make parallel implementation efficient and provide excellent scalability. Numerical results include some large polynomial systems that had not been solved.     

On the computation of controllability subspaces

A procedure to determine parametric expressions for the basis matrices of all possible controllability subspaces (c.s.)  of a pair (A, B) is presented. Based on this procedure the paramotrization of the basis matrices corresponding to the family of c.s. of (A, B) that are contained in the kernel of the output map c becomes possible. This result leads to an alternative method of computing the maximal c.s.  of )A, B) contained in the kernel of c, which avoids the intermediate computation of the maximal (A, B)-invarianfc subspaco contained in ker C.     

On the computation of Kalman gains

Modern filtering methods often require high order matrix differential equations to be solved. Standard numerical methods are traditionally slow and prone to be unstable. A numerical approach to the problem of computing the Kalman gain matrix is developed which is both numerically efficient and stable. If a piecewise approximation of the Kalman gain matrix is desired, efficiencies of many orders of magnitude can be realized.     

On computing derivatives forC 1 interpolating schemes: an optimization

The application of Powell-Sabin’s or Clough-Tocher’s schemes to scattered data problems, as known requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. We study alocal method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a node that we called acell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines theglobal Method II by Renka and Cline (cf. [16, pp. 230–231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency in the computations. The locality together with some implementation strategies produces a method well suited for the treatment of a big amount of data. An improvement of the estimates is also proposed.     

A Linear–time Tissue P System Based Solution for the 3–coloring Problem

In the literature, several examples of the efficiency of cell-like P systems regarding the solution of NP-complete problems in polynomial time can be found (obviously, trading space for time). Recently, different new models of tissue-like P systems have received important attention from the scientific community. In this paper we present a linear-time solution to an NP-complete problem from graph theory, the 3-coloring problem, and we discuss the suitability of tissue-like P systems as a framework to address the efficient solution to intractable problems.     

On the computation of the gap metric

This paper deals with the computation of the gap metric introduced by Zames and El-Sakkary [17]. It is shown that the gap between two systems (P₁, P₂) is precisely the maximum of the two expressions for (i, j) equal to (1, 2) and (2, 1), and (N_i, D_i) being normalized right coprine factorizations of P_i, I = 1, 2, in the sense of Vidyasagar [12]. This expression is computable using well-known techniques from interpolation theory.     

On the analysis of nonlinear nilpotent switched systems using the Hall–Sussmann system

We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. We show that a nice reachability analysis may be reduced to the reachability analysis of a specific canonical system, the nilpotent Hall–Sussmann system. Furthermore, local nice reachability properties for this specific system imply global nice reachability for general nilpotent systems. We derive several new results revealing the elegant Lie-algebraic structure of the nilpotent Hall–Sussmann system.     

On the computation of the exponential integral

A simple high-accuracy algorithm for calculating the exponential integral is developed. The exponential integral is represented in the form of a convergent series when x is small and in the form of an asymptotically convergent continued fraction when x is large. It is shown that the optimal boundary between these representations is x = 1. The use of 18 series terms and 220 continued fraction terms guarantees that the relative error is smaller than 2 × 10^?15, which exceeds what is needed in practice.