The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytic stratifications and the cut locus of a class of distance functions

We study the analytic singularities of viscosity solutions of equations of eikonal type and obtain that the analytic singular support of these functions has an analytic stratification. The singular support can be identified with the cut locus of the distance to the boundary of an open set, when the interior is equipped with a degenerate Riemannian metric. We apply the result to elliptic equations as well as to model operators of Grušin type.     

Trajectory Planning in Robotics

Trajectory planning is a fundamental issue for robotic applications and automation in general. The ability to generate trajectories with given features is a key point to ensure significant results in terms of quality and ease of performing the required motion, especially at the high operating speeds necessary in many applications. The general problem of trajectory planning in Robotics is addressed in the paper, with an overview of the most significant methods, that have been proposed in the robotic literature to generate collision-free paths. The problem of finding an optimal trajectory for a given path is then discussed and some significant solutions are described.     

A branch-and-price approach for harvest scheduling subject to maximum area restrictions

Recently, research on exact methods has been undertaken to solve forest management problems subject to constraints on the maximum clearcut area by using the area restriction model approach. Three main basic integer programming models for these problems have been discussed in the literature: the so-called cluster, path and bucket formulations. Solving these models via branch-and-bound, where all variables and constraints are used a priori, is adequately suited for real problems of a small to medium size, but is not appropriate for larger problems. In this paper, we describe a branch-and-price approach for the cluster model, and we show that this formulation dominates the bucket model, by completing the results of the dominance relationships between the bounds of the three models. Branch-and-price was tested on real and hypothetical forests ranging from 45 to 2945 stands and temporal horizons ranging from three to twelve periods were employed. Results show that the solutions obtained by the proposed approach stood within 1% of the optimal solution and were achieved in a short computation time. It was found that branch-and-bound was unable to produce solutions for most forests from 850 stands with either eleven or an average number of stands per clearcut greater or equal than eight.     

Multiphasic Individual Growth Models in Random Environments

The evolution of the growth of an individual in a random environment can be described through stochastic differential equations of the form dY _t  = β(α − Y _t )dt + σdW _t , where Y _t  = h(X _t ), X _t is the size of the individual at age t, h is a strictly increasing continuously differentiable function, α = h(A), where A is the average asymptotic size, and β represents the rate of approach to maturity. The parameter σ measures the intensity of the effect of random fluctuations on growth and W _t is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient β assumes different values for different phases of the animal’s life. For simplicity, we consider two phases with growth coefficients β ₁ and β ₂. Results and methods are illustrated using bovine growth data.     

Weighted proximity search

Journal of Heuristics - Proximity search is an iterative method to solve complex mathematical programming problems. At each iteration, the objective function of the problem at hand is replaced by...     

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L ^p ?L ^q ” and Strichartz-type estimates for the solutions of the linearized equations.     

Damage spreading in a single-component irreversible reaction process: Dependence of the system's immunity on the Euclidean dimension

The spreading of a globally distributed damage, created in the stationary regime, is studied in a single-component irreversible reaction process, i.e., the BK model [Browne and Kleban,Phys. Rev. A 40, 1615 (1989)]. The BK model describes one variant of the A+AA₂ reaction process on a lattice in contact with a reservoir of A species. The BK model has a single parameter, namely the rate of arrival of A species to the lattice (Y). The model, exhibits an irreversible phase transition between a stationary reactive state with production of A₂ species and a poisoned state with the lattice fully covered by A species. The transition takes place at critical points (Y _C ) which solely depend on the Euclidean dimensiond. It is found that the system is immune ford=1 andd=2, in the sense that even 100% of initial damage is healed within a finite healing period (T _H ). Within the reactive regime,T _H diverges when approachingY _C according toT _H (Y _C –Y)^–, with 1.62 and 1.08 ford=1 andd=2, respectively. Ford=3 a frozen-chaotic transition is found close toY _s 0.4125, i.e., well inside the reactive regime 0YY _C 0.4985. Just atY _S the damageD(t) heals according toD(t) t ^–, with 0.71. For the frozen-chaotic transition atd=3 the order parameter critical exponent 0.997 is determined.     

A new carbonylhydridorhenium cluster

The preparation and properties of new polynuclear carbonylhydrido complexes (NEt₄)₃[H₄Re₄(OCH₃)(CO)₁₆] (I) and (NEt₄)₂[H₄Re₄(CO)₁₅] (II) are reported. Infrared and NMR spectra of these compounds are presented and discussed. The structure of compound II has been established by X-ray crystallographic analysis.