Proton magic even-even isotopes and giant halos of Ca isotopes with relativistic continuum Hartree-Bogoliubov theory

We study the proton magic O, Ca, Ni, Zr, Sn, and Pb isotope chains from the proton drip line to the neutron drip line with the relativistic continuum Hartree-Bogoliubov (RCHB) theory. Particulary, we study in detail the properties of even-even Ca isotopes due to the appearance of giant halos in neutron rich Ca nuclei near the neutron drip line. The RCHB theory is able to reproduce the experimental binding energiesE _b and two neutron separation energiesS _2n very well. The predicted neutron drip line nuclei are²⁸O,⁷²Ca,⁹⁸Ni,¹³⁶Zr,¹⁷⁶Sn, and²⁶⁶Pb. Halo and giant halo properties predicted in Ca isotopes withA>60 are investigated in detail through analysis of two neutron separation energies, nucleon density distributions, single particle energy levels, and the occupation probabilities of energy levels including continuum states. The spin-orbit splitting and the diffuseness of nuclear potential in these Ca isotopes, as well as the neighboring lighter isotopes in the drip line Ca region and find certain possibilities of giant halo nuclei in the Ne−Na−Mg drip line nuclei are also studied.     

Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population

Summary Normalizing and variance stabilizing transformations of a sample correlation, multiple correlation and canonical correlation coefficients are obtained under an elliptical population. It is shown that the Fisher'sz-transformation is efficient for these statistics. A normalizing transformation is also studied for a latent root of a sample covariance matrix in an elliptical sample.     

On generalized convexity and duality with a square root term

We extend the duality theorems for a class of nondifferentiable problems with Mond-Weir type duals.     

Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context

We consider several aspects of the role and evaluation of the four-sexagesimal-place approximation to on the well-known Old Babylonian tablet YBC 7289. By referring to what is known about OB school texts, we show that this text is most probably a school exercise by a trainee scribe who got the approximation from a coefficient list. These coefficient lists are briefly described, with their use in geometrical problems. We consider other texts involving square roots and derive an algorithm for evaluating them, which complies with all known OB examples, from a simple geometrical construction of the type that seems to underlie many other OB procedures.Copyright 1998 Academic Press.Nous considérons plusieurs aspects du rôle et de l'évaluation de l'approximation à quatre places sexagésimales de sur cette tablette paléo-babylonienne. En nous référant à l'état des connaissances quant aux textes scolaires paléo-babyloniens, nous montrons comment ce texte est très probablement un exercice scolaire exécuté par un apprenti scribe qui a obtenu l'approximation à partir d'une liste de coéfficients. Nous décrivons en bref ces listes de coéfficients, ainsi que leur utilisation dans les problèmes de géométrie. Nous considérons d'autres textes traitant de racines carrées , et pour les évaluer nous dérivons un algorithme qui est en accord avec tous les exemples paléo-babyloniens connus; ceci à partir d'une construction géométrique simple du type qui semble être charactéristique de beaucoup d'autres procédures paléo-babyloniennes.Copyright 1998 Academic Press.MSC 1991 subject classifications: 01A17; 11-03     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P _G (q) for m×n rectangular subsets of the square lattice, with m8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin–Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B ₂,B ₃,B ₄,B ₅ are limiting points of partition-function zeros as n whenever the strip width m is 7 (periodic transverse b.c.) or 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B ₁₀) cannot be a chromatic root of any graph.     

A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system is given. Consider a meromorphic function on V =V+iV, the singular locus of which is a locally finite union of hyperplanes of the form V , = s, , s . Assume is of suitable decay in the imaginary directions, so that integrals of the form _+iV , d make sense for generic V. A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.     

Design, analysis, and implementation of a multiprecision polynomial rootfinder

We present the design, analysis, and implementation of an algorithm for the computation of any number of digits of the roots of a polynomial with complex coefficients. The real and the imaginary parts of the coefficients may be integer, rational, or floating point numbers represented with an arbitrary number of digits. The algorithm has been designed to deal also with numerically hard polynomials like those arising from the symbolic preprocessing of systems of polynomial equations, where the degree and the size of the coefficients are typically huge.The algorithm is based on an adaptive strategy which automatically exploits any specific feature of the input polynomial, like its sparsity or the conditioning of its roots, in order to speed up the computation. We introduce different concepts and tools suitably designed to arrive at an adaptive implementation, such as the concepts of root neighborhood, inclusion discs and some inclusion and conditioning theorems for their determination. The main engine for shrinking the inclusion discs is the simultaneous iteration method of Ehrlich–Aberth, complemented with a suitable technique for cluster analysis that is used for getting rid of the slow convergence in case of clustered or multiple roots.The algorithm, implemented in C, relies on the GNU multiprecision package GMP and allows many options. Counting, isolating and approximating all roots in a given set are the main goals that the algorithm provides. Automatic determination of multiplicities and the detection of real or imaginary roots can be selected as well. Polynomials having coefficients with a bounded precision may be processed too. Comparisons with the polynomial rootfinders of the packages Mathematica, Mapleand Pari, performed on a wide class of test polynomials show that our algorithm is generally much faster: in most cases the speedup factor is greater than 10 and, for certain polynomials, it is greater than 1000. The MPSolve package can be downloaded from the numeralgo library of netlib.     

Searching threshold effects in the interest rate: An application to Turkey case

This paper investigates the behaviour of interest rates in Turkey using a two-regime TAR model with an autoregressive unit root. This method recently developed by Caner and Hansen [Threshold autoregression with a unit roots, Econometrica 69 (6) (2001) 1555–1596] allows to simultaneously consider non-stationarity and non-linearity. Our finding indicates that the interest rate is a non-linear series and is characterized by a unit root process over the period 1990:1–2006:5.