Solution of Scott's Problem on the Number of Directions Determined by a Point Set in 3-Space

Let P be a set of n points in R³, not all in a common plane. We solve a problem of Scott (1970) by showing that the connecting lines of P assume at least 2n - 5 different directions if n is odd and at least 2n - 7 if n is even. The bound for odd n is sharp.     

Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators

A stochastic variational inequality is proposed to model a white noise excited elasto-plastic oscillator. The solution of this inequality is essentially a continuous diffusion process for which a governing diffusion equation is obtained to study the evolution in time of its probability distribution. The diffusion equation is degenerate, but using the fact that the degeneracy occurs on a bounded region we are able to show the existence of a unique solution satisfying the desired properties. We prove the ergodic properties of the process and characterize the invariant measure. Our approach relies on extending Khasminskii’s method (Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980), which in the present context leads to the study of degenerate Dirichlet problems with nonlocal boundary conditions. This research was partially supported by a grant from CEA, Commissariat à l’énergie atomique and by the National Science Foundation under grant DMS-0705247.     

Lectures on the functional renormalization group method

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.     

Sensitivity enhancement of beryllium determination in graphite furnace-atomic absorption spectrometry by use of atomization under pressure

Microchimica Acta - Using pressurized atomization both peak height and area sensitivity of beryllium is considerably improved. The enhancement may probably be explained by the fact that the Lorentz...     

The number of large prime divisors of consecutive integers

Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N^?(N) → ∞ as N → + ∞. Let Nν_N(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy N^Z?(N) ? p ? N^?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limv_N(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive.     

The Functional Callan-Symanzik Equation for the Coulomb Gas

A non-perturbative scheme, based on the functional generalization of the Callan-Symanzik equation is developed to treat the Coulomb interaction in an electron gas. The one-particle irreducible vertex functions are shown to satisfy an evolution equation whose initial condition is given by means of the classical action and the final point corresponds to the physical system. This equation is truncated by expanding it in momenta and excitation energies, leaving the electric charge as an arbitrary, not necessarily small parameter. Exact coupled partial differential equations up to first order in the frequencies and excitation energies are derived. The numerical integration of these equations is left to a later stage. Nevertheless, in order to demonstrate the relation with the perturbation expansion the one-loop Lindhard function and screening are reproduced in the independent mode approximation of the evolution equation.     

Molecular simulation of the Joule–Thomson inversion curve of hydrogen sulphide

The complete Joule–Thomson inversion curve for hydrogen sulphide is determined via molecular simulations in adiabatic ensembles. In addition to NpH Monte Carlo simulations, two new versions of the density-of-states simulations are applied for smoothly scanning relevant thermodynamic ranges. With the use of realistic site–site intermolecular potential models the calculations yield good agreement with the accessible experimental data.     

Practical Method for Analysis of Immiscible Displacement in Laboratory Core Tests

Abstract. An accurate analytical interpretation method to determine the Leverett function (f_w)and its derivative (f_w) from immiscible displacement data in core plugs is presented. Linear equations are developed to describe the displacement processes occurring before and after breakthrough. A quadratic function is introduced to represent the saturation distribution along the cores. The relationships derived in this study can be used for analysis of core tests involving constant injection rates and constant pressure differences. The applicability, practicality, and accuracy of the new analytical method are verified by means of the experimental data obtained in the present study and by those reported in the literature.     

Preparation and investigation of LC packing made by microwave‐assisted solid‐phase synthesis

The preparation of the so‐called bonded phase liquid chromatographic packings is usually carried out by heating the silica, the silane, a catalyst, or a scavenger in an appropriate solvent (often toluene.) Due to the longtime of boiling, the procedure is time and energy consuming, and solvent intensive. The goal of this work is to present a simple, environment‐friendly preparation method with reduced solvent consumption to synthetize RP liquid chromatographic stationary phases. The effects of reaction conditions (amount of reagents, composition of the reagent, microwave energy, reaction time, reproducibility of the synthesis) are discussed. Pore structure, surface coverage, the change of the pore structure and surface coverage upon reaction are demonstrated, the efficiency of the column (van Deemter plot for different solutes) is presented. A variety of applications (aromatic hydrocarbons, halobenzenes, bioactive peptides, resveratrol from red wine) demonstrates the separation power of the new phase.