Guest Editors' Foreword

This issue of Discrete & Computational Geometry contains the detailed proof by T. Hales and S.P. Ferguson of the Kepler conjecture that the densest packing of three-dimensional Euclidean space by equal spheres is attained by "cannonball" packing. This is a landmark result. This conjecture, formulated by Kepler in 1611, was stated in Hilbert's formulation of his 18th problem [8]. The proof consists of mathematical arguments and a massive computer verification of many inequalities.     

Synthesis and application of a fluorous Lawesson's reagent: convenient chromatography-free product purification

[reaction: see text] A fluorous analogue of Lawesson's reagent for thionation of carbonyl compounds has been developed and its use demonstrated on a series of amides, esters, and ketones. The separation of the Lawesson's reagent-derived byproducts can be achieved by a simple fluorous solid-phase extraction.     

How ∗ not ∗ to solve a Sudoku

We prove the NP-hardness of a consistency checking problem that arises in certain elimination strategies for solving Sudoku-type problems.     

A global bifurcation index for set-valued perturbations of Fredholm operators

In the paper the homotopy invariant detecting global bifurcations of solutions to multi-parameter equations involving general set-valued perturbations of linear Fredholm operators of non-negative index is introduced. Some applications to the existence problems for differential inclusions are provided.     

Endor for characterizing transition metal centres in LiNbO3

Abstract

ENDOR investigations show that in nearly stoichiometric LiNbO₃ Mn²⁺ substitutes for Li with a Li vacancy in the first Li shell for most centres. In heavily Mg codoped LiNbO₃ the Cr³⁺ dopant, characterized by a new isotropic ESR signal, is shown to substitute for Nb. Information on local disorder is obtained.     

On gradient field theories: gradient magnetostatics and gradient elasticity

In this work, the fundamentals of gradient field theories are presented and reviewed. In particular, the theories of gradient magnetostatics and gradient elasticity are investigated and compared. For gradient magnetostatics, non-singular expressions for the magnetic vector gauge potential, the Biot–Savart law, the Lorentz force and the mutual interaction energy of two electric current loops are derived and discussed. For gradient elasticity, non-singular forms of all dislocation key formulas (Burgers equation, Mura equation, Peach–Koehler stress equation, Peach–Koehler force equation, and mutual interaction energy of two dislocation loops) are presented. In addition, similarities between an electric current loop and a dislocation loop are pointed out. The obtained fields for both gradient theories are non-singular due to a straightforward and self-consistent regularization.     

The gauge theory of dislocations: conservation and balance laws

We derive conservation and balance laws for the translational gauge theory of dislocations by applying Noether's theorem. We present an improved translational gauge theory of dislocations including the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under the continuous group of transformations is studied. Through Lie's infinitesimal invariance criterion we obtain conserved translational and rotational currents for the total Lagrangian made up of an elastic and dislocation part. We calculate the broken scaling current. Looking only on one part of the whole system, the conservation laws are changed into balance laws. Because of the lack of translational, rotational and dilatation invariance for each part, a configurational force, moment and power appears. The corresponding J , L and M integrals are obtained. Only isotropic and homogeneous materials are considered and we restrict ourselves to a linear theory. We choose constitutive laws for the most general linear form of material isotropy. Also we give the conservation and balance laws corresponding to the gauge symmetry and the addition of solutions. From the addition of solutions we derive a reciprocity theorem for the gauge theory of dislocations. Also, we derive the conservation laws for stress-free states of dislocations.     

Spectral Investigation of Phosphorus Compounds in Spinacia Oleracea Using Fourier Transform Infrared Method

The study uses the Fourier transform infrared spectroscopy method to evaluate the influence of phosphogypsum from the soil on the growth and accumulation of nutrients in Spinacia oleracea. Attenuated total reflectance spectra of Spinacia oleracea roots and stems originated from plants cultivated on 28% to 52% phosphogypsum enriched soil have been studied in the 4000–600 cm^?1 area. The structure of the infrared absorption bands for spinach roots and stems and for the soil where plants were grown was analyzed. Thus, it was observed that phosphorus–oxygen bonds contribution can be identified both in roots and stems absorption spectra. The significance of the research is the possibility to use the infrared investigation as rapid technique, particularly useful for agriculture, to study the extraction and assimilation process of the compounds containing phosphorus and nitrogen from the soil in which the spinach seedlings grew. In the case of this study, it was observed that the absorption of phosphorus in the spinach roots and stems has not the same effect. The analysis of obtained data indicated that the assimilation of phosphorus from phosphogypsum enriched soil induces changes in spinach roots structure that are most likely due to an increase synthesis of proteins, rather than to a decrease in carbohydrates quantity.     

Modifications Induced by UV-Vis Irradiation on DNA Extract of Kiwifruit Investigated by Spectroscopic and Statistic Methods

ABSTRACT

Revealing molecular alterations induced on kiwifruit under UV-Vis irradiation requires a discussion of biochemical-cell infrared (IR) fingerprint (900 cm^?1–1800 cm^?1) bands characteristic of nucleic acids. FTIR-ATR spectroscopy and statistics and nondestructive methods for screening exposure effects induced by irradiation were used. There the irradiation influence on the main molecular bonds (i.e., ν(C-C), ν_s(PO₂?) and ν_as(PO₂?)) can be observed. Regression methods were used for statistical investigations. Two categories of variables were used: the absorbance measured at fixed wavenumber variables and the exposure dose. The bivariate correlations, partial correlations, and polynomial regression methods from SPSS were used for statistical investigations. The obtained results show that FTIR-ATR, in correlation with statistics techniques, might be useful to assess immediate radiation and oxidative-induced damage to nucleic acids. In this case IR spectroscopy can be used successfully to study conformational changes during DNA reversible denaturation especially on the sugar-phosphate vibrations domain.     

Local Central Limit Theorem for Multi-group Curie–Weiss Models

Journal of Theoretical Probability - We study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this...