Common Tangents to Spheres in ℝ3

We prove that four spheres in ℝ³ have infinitely many real common tangents if and only if they have aligned centers and at least one real common tangent.     

On the Equivalence of Multiparameter Gaussian Processes

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.      

A theoretical study of the large Hg-Hg spin-spin coupling constants in Hg(2)(2+), Hg(3)(2+), and Hg(2)(2+)-crown ether complexes

Nuclear spin-spin coupling constants (1)J(Hg-Hg) in the systems Hg(2)(2+) and Hg(3)(2+) represent the largest coupling constants so far observed in NMR experiments. We have performed a computational study on these ions, on Hg(2)(2+) complexes with 18-crown-6 and 15-crown-5, and on Hg(3)(2+) with solvent molecules and counterions. The results obtained with our recently developed program for the density functional computation of heavy nucleus spin-spin coupling constants are in good agreement with experiments. The data reveal that the bare ions Hg(2)(2+) and Hg(3)(2+) would afford much larger coupling constants than those experimentally observed, with an upper limit of approximately 0.9 MHz for Hg(2)(2+). This limit is much larger than that previously estimated by Hückel theory. It is demonstrated that in solution or due to complexation the experimentally determined values are much smaller than the free ion's coupling constants. With the help of intuitive MO arguments, it is illustrated how the environment strongly reduces the coupling constants in Hg(2)(2+) and Hg(3)(2+). The two-bond coupling constant (2)J(Hg-Hg) in Hg(3)(2+) is also examined.     

Theoretical investigation of the apparently irregular behavior of pt-pt nuclear spin-spin coupling constants

One-bond Pt-Pt nuclear spin-spin coupling constants J(Pt-Pt) for closely related dinuclear Pt complexes can differ by an order of magnitude without any obvious correlation with Pt-Pt distances. As representative examples, the spin-spin couplings of the dinuclear Pt(I) complexes [Pt(2)(CO)(6)](2+) (1) and [Pt(2)(CO)(2)Cl(4)](2-) (2) have been computationally studied with a recently developed relativistic density functional method. The experimental values are (1)J((195)Pt-(195)Pt) = 5250 Hz for 2 but 551 Hz for 1. Many other examples are known in the literature. The experimental trends are well reproduced by the computations and can be explained based on the nature of the ligands that are coordinated to the Pt-Pt fragment. The difference for J(Pt-Pt) of an order of magnitude is caused by a sensitive interplay between the influence of different ligands on the Pt-Pt bond, and relativistic effects on metal-metal and metal-ligand bonds as well as on "atomic orbital contributions" to the nuclear spin-spin coupling constants. The results can be intuitively rationalized with the help of a simple qualitative molecular orbital diagram.     

Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

We develop the asymptotic expansion theory for vector-valued sequences ${\left(F_N\right)}_{N\ge 1}$ of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence $F_N$. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of $F_N$ and we illustrate our results by several examples.     

SPDE with generalized drift and fractional-type noise

We consider a stochastic partial differential equation involving a second order differential operator whose drift is discontinuous. The equation is driven by a Gaussian noise which behaves as a Wiener process in space and the time covariance generates a signed measure. This class includes the Brownian motion, fractional Brownian motion and other related processes. We give a necessary and sufficient condition for the existence of the solution and we study the path regularity of this solution.     

Factors that influence the tribo-charging of insulating ducts in suction-type dilute-phase pneumatic transport systems

The aim of this study is to refine the understanding of the tribo-electrostatic phenomena that can affect the efficiency of suction-type dilute-phase transport systems. The study was conducted with mm-size ABS-PC particles, a granular material originating from genuine information technology wastes. The tribo-charging process was modelled using the response surface method of experimental design. The controlled variables were the granular material feed rate and the aspirating air speed rate. The monitored variable was the charge imparted to a well-defined section of the duct and which is equal to the charge transferred to the particles passing through that section of the pneumatic transport system. An induction type sensor connected to an electrometer has been employed for the measurements. The data were processed by a virtual instrument developed in LabView. A commercial software package was employed for estimating the effects of the various factors and predicting the optimum operating conditions, which were then confirmed by a final experiment.