Hypergraphs and a functional equation of Bouwkamp and de Bruijn

Let . Bouwkamp and de Bruijn found that there exists a power series Ψ(u,v) satisfying the equation . We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Φ(u,0) and Ψ(u,0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversion to count hypertrees by number of vertices and number of edges of a specified size.     

On solutions of nonlinear Schrödinger equations with Cantor-type spectrum

The solvability of the Cauchy problem for the Nonlinear Nonfocusing Schrödinger equation (NNSE) with almost periodic initial data satisfying certain conditions is studied. It is shown that solutions are uniform almost periodic functions with respect to each variable. An example of initial data with Cantor-type spectrum is given. The Cauchy problem for NNSE is solved in the class of limit periodic functions which are well approximated by periodic ones.     

A quaternionic proof of the representation formula of a quaternary quadratic form

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper, we develop results analogous to Hurwitz's above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form.     

LINC-NIRVANA: MCAO toward Extremely Large Telescopes

LINC-NIRVANA is a Fizeau (imaging) interferometer exploiting the full spatial resolution of a 23 m class telescope in the combined beam of the Large Binocular Telescope supported through Multi-Conjugated Adaptive Optics (MCAO). By means of science cases, we show how LINC-NIRVANA takes advantage of the MCAO, increasing the sky coverage of the instrument and the field of view for the Fringe and Flexure tracker. We introduce the MCAO system of LINC-NIRVANA in detail, which in a first step will be installed with two deformable mirrors per arm and has the provision to be upgraded with a third mirror. The MCAO system implements several novel concepts proposed for extremely large telescopes, such as layer oriented MCAO, optical co-adding of guide stars, or Multiple Field of View sensing. LINC-NIRVANA will demonstrate some of the concepts for the first time on sky. To cite this article: W. Gaessler et. al., C. R. Physique 6 (2005).     

Optical backward mixing in sodium nitrite

We present an experimental study of backward mixing of argon laser lines and CO laser lines to generate visible light in sodium nitrite. Angle tuning has been measured. Some special aspects of the observed super narrow band-width of backward interactions are discussed.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

Outer-sphere oxidation of the superoxide radical anion

Efforts to use the Marcus model to describe oxidations of the superoxide radical anion (O(2-)) by transition-metal complexes have failed dramatically, with discrepancies between theory and experiment spanning 13 orders of magnitude. As a result, the prevailing view is that these reactions involve some complex interactions that are not yet understood. We now show that once the familiar form of the Marcus cross relation (MCR) is modified to allow for the relatively small size of O(2-), excellent agreement is obtained between theory and experiment. This simple finding dispels the decades of uncertainty surrounding these reactions and provides a reliable method for determining whether oxidations of (O2)- occur via inner- or outer-sphere pathways. More generally, the modified MCR is applicable to any homogeneous electron-transfer process characterized by significant differences in size between electron donors and acceptors.     

Size-dependent magnetic properties of CoFe2O4 nanoparticles prepared in polyol

Highly crystalline CoFe(2)O(4) nanoparticles with different diameters ranging from 2.4 to 6.1 nm have been synthesized by forced hydrolysis in polyol. The size can be controlled through adjusting the nominal water/metal molar ratio. X-ray diffraction, transmission electron microscopy, x-ray absorption spectroscopy and (57)Fe M?ssbauer spectrometry were employed to investigate the structure and the microstructure of the particles produced. Magnetic measurements performed on these particles show that they are superparamagnetic with a size-dependent blocking temperature. At 5 K, high saturation magnetization (~85 emu g(-1)) approaching that of the bulk was found for the larger particles, whereas a very large coercivity (14.5 kOe) is observed for the 3.5 nm sized particles.