The many aspects of counting lattice points in polytopes

A wide variety of topics in pure and applied mathematics involve the problem of counting the number of lattice points inside a convex bounded polyhedron, for short called a polytope. Applications range from the very pure (number theory, toric Hilbert functions, Kostant’s partition function in representation theory) to the most applied (cryptography, integer programming, contingency tables). This paper is a survey of this problem and its applications. We review the basic structure theorems about this type of counting problem. Perhaps the most famous special case is the theory of Ehrhart polynomials, introduced in the 1960s by Eugène Ehrhart. These polynomials count the number of lattice points in the different integral dilations of an integral convex polytope. We discuss recent algorithmic solutions to this problem and conclude with a look at what happens when trying to count lattice points in more complicated regions of space.     

A New Derivation of the Quantum Navier–Stokes Equations in the Wigner–Fokker–Planck Approach

A quantum Navier–Stokes system for the particle, momentum, and energy densities is formally derived from the Wigner–Fokker–Planck equation using a moment method. The viscosity term depends on the particle density with a shear viscosity coefficient which equals the quantum diffusion coefficient of the Fokker–Planck collision operator. The main idea of the derivation is the use of a so-called osmotic momentum operator, which is the sum of the phase-space momentum and the gradient operator. In this way, a Chapman–Enskog expansion of the Wigner function, which typically leads to viscous approximations, is avoided. Moreover, we show that the osmotic momentum emerges from local gauge theory.     

Origin of the X‐ray magnetic circular dichroism at the L‐edges of the rare‐earths in RxR1−x′Al2 systems

An X‐ray magnetic circular dichroism (XMCD) study performed at the rare‐earth L_2,3‐edges in the R_xR_1?x′Al₂ compounds is presented. It is shown that both R and R′ atoms contribute to the XMCD recorded at the L‐edges of the selected rare‐earth, either R or R′. The amplitude of the XMCD signal is not directly correlated to the magnetization or to the value of the individual (R, R′) magnetic moments, but it is related to the molecular field acting on the rare‐earth tuned in the photoabsorption process. This result closes a longstanding study of the origin of the XMCD at the L‐edge of the rare‐earths in multi‐component systems, allowing a full understanding of the exact nature of these signals.     

On the Hahn-Banach Theorem in Certain Linear Quasi-uniform Structures

A Hahn-Banach type theorem in the frame of quasi-normed spaces (a class of linear spaces with a quasi-uniform structure that contains all linear lattices) is given. The classical result for positive functionals follows as a particular case.     

A Review of Synthetic Polymer Characterization by Pyrolysis–GC–MS

Pyrolysis–GC with mass spectrometry detection (Py–GC–MSD) study of the thermal degradation products of synthetic polymers is reviewed. Due to the high heating speed, accurate temperature reproducibility and a wide temperature range, Py–GC–MSD has been applied successfully for polymer characterization. Introduction of samples using the pyrolysis carrier gas through the split injection port, followed by sub-ambient cryofocusing of the pyrolysis products, has shown to give reproducible chromatograms (pyrograms). One of the advantages of this method is that all compositions of the polymers and additives can be investigated without any pretreatment, providing important compositional and structural information in a simple way. The method is a convenient method for compositional analysis of complex polymer materials. The aim of this review is to describe the kinds of applications for which Py–GC–MSD has been found to be suitable; to present guidelines for method optimization; to survey innovations that have recently been developed or are currently being researched; to point to problems in our understanding of the pyrograms; and to suggest areas in which research efforts might be most effective in realizing the full potential of this technique.     

A new concept in lightning parameters measurement

Nowadays the electrical activity measured in tall objects has been widely characterized, but the collected data of lightning rods installed in buildings do not amount to enough to obtain reliable statistics. The measures stored at relatively few points are not representative of the different types of installations. Furthermore, the protection model applied for lightning air terminal installations is based on theoretical models. This paper describes the design of a micro-station to measure some lightning parameters in real lightning air terminal installations correlated to the nature of emplacement instead of triggering lightning or choosing points with a high number of lightning impacts.     

Analytical Solution for Multi-Species Contaminant Transport Subject to Sequential First-Order Decay Reactions in Finite Media

Transport equations governing the movement of multiple solutes undergoing sequential first-order decay reactions have relevance in analyzing a variety of subsurface contaminant transport problems. In this study, a one-dimensional analytical solution for multi-species transport is obtained for finite porous media and constant boundary conditions. The solution permits different retardation factors for the various species. The solution procedure involves a classic algebraic substitution that transforms the advection-dispersion partial differential equation for each species into an equation that is purely diffusive. The new system of partial differential equations is solved analytically using the Classic Integral Transform Technique (CITT). Results for a classic test case involving a three-species nitrification chain are shown to agree with previously reported literature values. Because the new solution was obtained for a finite domain, it should be especially useful for testing numerical solution procedures.