Multilayer optics for XUV spectral region: technology fabrication and applications

We present research investigations in the field of multilayer optics in X-ray and extreme ultra-violet ranges (XUV), aimed at the development of optical elements for applications in experiments in physics and in scientific instrumentation. We discuss normal incidence multilayer optics in the spectral region of “water window”, multilayer optics for collimation and focusing of hard X-ray, multilayer dispersing elements for X-ray spectroscopy of high-temperature plasma, multilayer dispersing elements for analysis of low Z-elements. Our research pays special attention to optimization of multilayer optics for projection EUV-lithography (ψ-13nm) and short period multilayer optics.     

Precision of a dual energy X-ray absorptiometry device

The standard method to diagnose and follow-up osteoporosis is the measurement of bone mineral density (BMD) using dual X-ray absorptiometry (DEXA). Manufacturers' manuals of DEXA devices state the intrascanner coefficient of variance is less than 0.01 g/cm². The aim of this study was to evaluate the in vivo coefficient of variance of a Lunar DPX scanner in male and female healthy adult subjects. Average BMD for females and males were 1.170±0.091 g/cm² and 1.272±0.115 g/cm², respectively. Monthly phantom measurements provided and controlled by the manufacturer were 1.243±0.008 g/cm² (range 1.222 to 1.257) and the coefficient of variance was 0.006. It is concluded that the in vivo coefficient of variance of DEXA devices can slightly be higher than that proposed by the manufacturer. This revised version was published online in August 2006 with corrections to the Cover Date.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

The analysis of volatile trace compounds in landfill gases,compost heaps and forest air

Landfill gas, cryotrapped on a loop fashioned from a length of a capillary gas chromatography (GC) column, was examined for volatile organometallic compounds (VOMCs) and for volatile organic compounds (VOCs) by using GC–mass spectrometry (MS). A large number of organic components were present and many were identified, but the only VOMCs present in high enough concentrations to be detected were trimethylstibine and tetramethyltin. The use of inductively coupled plasma (ICP)‐MS as an element‐specific detector allowed the identification of a number of other organometallic species in the landfill gas, including trimethylarsine and trimethylbismuth, and, for the first time, butyltrimethyltin and dibutyldimethyltin. The presence of molybdenum hexacarbonyl was confirmed. Gas from a large‐scale compost heap and from compost incubated in the laboratory contained iodomethane but no common VOMCs (GC–ICP‐MS). Only VOCs were present in forest air (GC–MS). Copyright © 2003 John Wiley & Sons, Ltd.     

The Solow model improved through the logistic manpower growth law

In this paper the neo-classical economic Solow-Swan model (1956) has been improved replacing its Malthusian manpower law with the Verhulst (logistic) one. The relevant ordinary differential equation for the ratio capital/work has been then integrated in closed form via the Hypergeometric function₂ F ₁. The logistic growth injection for the manpower is detected to induce a more slow dynamics onto the Solow-Swan system, which keeps its stability. Increasing developments are displayed as the technologic progress rises. Further sceneries are tested and the congruence of the new solution with the classical one is shown switching to zero the selflimitation coefficent in the logistic law. Research supported by MURST grant:Metodi matematici in economia     

An Average-Stretch Full-Network Model for Rubber Elasticity

Two constitutive models that are based on the classical non-Gaussian, Kuhn-Grün probability distribution function are reviewed. It is shown that all chains of a network cell structure comprised of a finite number of identical chains in an affine deformation referred to principal axes may have the same invariant stretch, if and only if the chains are oriented initially along any of eight directions forming the diagonals of a unit cube. The 4-chain tetrahedral and the 8-chain cubic cell structures are familiar admissible models having this property. An easy derivation of the constitutive equation for the Wu and van der Giessen full-network model of initially identical chains arbitrarily oriented in the undeformed state is presented. The constitutive equations for the neo-Hookean model, the 3 -chain model, and the equivalent 4- and 8-chain models are then derived from the Wu and van der Giessen equation. The squared chain stretch of an arbitrarily directed chain averaged over a unit sphere surrounding all chains radiating from a cross-link junction as its center is determined. An average-stretch, full-network constitutive equation is then derived by approximation of the Wu and van der Giessen equation. This result, though more general in that no special chain cell morphology is introduced, is the same as the constitutive equation for the 4- and 8-chain models. Some concluding remarks on extensions to amended models are presented.     

On separation of gradient Young measures

Let be a linear subspace of real matrices without rank-one matrices and let be a finite set. Suppose is a bounded arcwise connected Lipschitz domain and is a sequence of bounded vector-valued mappings in such that in as , where is the closed -neighbourhood and the distance function to . We give estimates for such that up to a subsequence, in for some fixed . In other words, we give estimates on such that separates gradient Young measure. The two point set with is a special case of such sets up to a translation. Received: 20 July 2001 / Accepted: 23 January 2002 / Published online: 5 September 2002     

A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.