Inverse Problem for a Class of Two-Dimensional Diffusion Equations with Piecewise Constant Coefficients

In this paper, we consider an inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients. This problem is studied using an explicit formula for the relevant spectral measures and an asymptotic expansion of the solution of the diffusion equations. A numerical method that reduces the inverse problem to a sequence of nonlinear least-square problems is proposed and tested on synthetic data.     

Linear restrictions and two step least squares with applications

In this paper we consider the full rank regression model with arbitrary covariance matrix: Y = Xß + ε. It is shown that the effect of restricting the information Y to T = A′Y may be analyzed through an associatedi regression problem which is amenable to solution by two step least squares. The results are applied to the important case of missing observations, where some classical results are rederived.     

On the robustness of least squares procedures in regression models

The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.     

The use of multivariate MR imaging intensities versus metabolic data from MR spectroscopic imaging for brain tumour classification

This study investigated the value of information from both magnetic resonance imaging and magnetic resonance spectroscopic imaging (MRSI) to automated discrimination of brain tumours. The influence of imaging intensities and metabolic data was tested by comparing the use of MR spectra from MRSI, MR imaging intensities, peak integration values obtained from the MR spectra and a combination of the latter two. Three classification techniques were objectively compared: linear discriminant analysis, least squares support vector machines (LS-SVM) with a linear kernel as linear techniques and LS-SVM with radial basis function kernel as a nonlinear technique. Classifiers were evaluated over 100 stratified random splittings of the dataset into training and test sets. The area under the receiver operating characteristic (ROC) curve (AUC) was used as a global performance measure on test data. In general, all techniques obtained a high performance when using peak integration values with or without MR imaging intensities. For example for low- versus high-grade tumours, low- versus high-grade gliomas and gliomas versus meningiomas, the mean test AUC was higher than 0.91, 0.94, and 0.99, respectively, when both MR imaging intensities and peak integration values were used. The use of metabolic data from MRSI significantly improved automated classification of brain tumour types compared to the use of MR imaging intensities solely.     

Incomplete Orthogonal Distance Regression

A common method of fitting curves and surfaces to data is to minimize the sum of squares of the orthogonal distances from the data points to the curve or surface, a process known as orthogonal distance regression. Here we consider fitting geometrical objects to data when some orthogonal distances are not available. Methods based on the Gauss–Newton method are developed, analyzed and illustrated by examples. AMS subject classification (2000) 65D10, 65K05.     

Synthesis and stereochemical properties of chiral square complexes of iron(II)

The hexadentate, and ditopic ligand 2,5-bis([2,2']bipyridin-6-yl)pyrazine yields a chiral, tetrameric, square-shaped, self-assembled species upon complexation with Fe2+ ions. The racemate of this complex was resolved with antimonyl tatrate as the chiral auxiliary. The purity of the enantiomer was determined by NMR spectroscopy, by using a chiral, diamagnetic shift reagent, and by circular dichroism (CD). The CD spectrum was also calculated by time-dependent density functional theory, and the correlation that was found between CD spectrum and configuration was confirmed by X-ray cristallography. When a "chiralised" version of the ligand was used instead, the corresponding iron complex was obtained in diastereomerically pure form.     

Statistical Measures for Ordinary Least Squares Using the αQ Algorithm

This paper shows how the dynamic program algorithm called the Q algorithm can be used as an alternative algorithm to produce the coefficients of a least squares problem. It shows also how the output of the algorithm can be used to calculate various statistical quantities needed to evaluate linear models. In particular, we show how to calculate standard statistical quantities like the coefficient of determination R², the t statistics, and the F statistics. These quantities serve as a measure of how well the model fits the data.     

A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1?b<q, there is a unique integer c coprime to q with 1?c<q such that . Let N(a,q) denote the number of solutions of the congruence equation with 1?b,c<q such that b,c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value , and give a sharp asymptotic formula.     

Densité des Points à Coordonnées Multiplicativement Indépendantes

Nous montrons que pour toute sous-variété algébrique d'un tore multiplicatif (non contenue dans un sous-groupe algébrique propre), on peut choisir un ensemble Zariski dense de points algébriques de hauteur contrôlée, dont toutes les coordonnées sont multiplicativement indépendantes. Cet énoncé précise et généralise un théorème de S. Zhang qui lie la hauteur projective d'une varété au minimum essentiel de la hauteur des points algébriques de celle-ci. En tenant compte d'un résultat précédent des auteurs sur le problème de Lehmer généralisé à un tore, nous en déduisons une minoration pour la hauteur normalisée d'une sous-variété d'un tore. Cette dernière est optimale à un «-prés» en le degré géométrique de la variété étudiée (confer une conjecture du second auteur avec P. Philippon).In this article, we prove that on any subvariety of a multiplicative torus which is not contained in a proper algebraic subgroup, one can find a Zariski dense set of algebraic points of small height whose coordinates are multiplicatively independent. This statement generalizes an earlier result of S. Zhang which links the projective height of a variety with the essential minimum of its algebraic points. Taking into account an earlier result of the authors on the Lehmer problem generalized to a multiplicative torus, one deduces a lower bound for the normalized height of subvarieties of multiplicative groups. This lower bound is optimal up to an in the geometric degree of the variety studied (confer a conjecture by the second author and P. Philippon).     

A QR Decomposition for Matrix Pencils

This paper describes an efficient and numerically stable modification of the QR decomposition for solving a parametric set of linear least squares problems with a parametric matrix A + B for several values of the parameter . The method is demonstrated on a typical application.