Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge

You cannot patch active plasma and collisionless sheath

The fluid equations for the plasma-sheath are examined over the range of collisionality from collisionless to fully collisional. The method of "patching" plasma to sheath is examined critically in comparison with matched asymptotic approximations and solutions computed without approximation. It is concluded that it is necessary to include a transition layer in order to smoothly join plasma and collisionless sheath-and then the Bohm Criterion applies. When the sheath is collisional, the orderings are different and there is no transitional layer but equally there is no collisionally modified Bohm Criterion. Nevertheless, a method is given for calculating the ion flux to the wall under conditions where the sheath is collisional

Shortcuts to Spherically Symmetric Solutions: A Cautionary Note

Spherically symmetric solutions of generic gravitational models are optimally, and legitimately, obtained by expressing the action in terms of the two surviving metric components. This shortcut is not to be overdone, however: a one-function ansatz invalidates it, as illustrated by the incorrect solutions of [1].Comment: 2 pages. Amplified derivation, accepted for publication in Class Quant Gra

Plasmas generated by ultra-violet light rather than electron impact

We analyze, in both plane and cylindrical geometries, a collisionless plasma consisting of an inner region where generation occurs by UV illumination, and an un-illuminated outer region with no generation. Ions generated in the inner region flow outwards through the outer region and into a wall. We solve for this system's steady state, first in the quasi-neutral regime (where the Debye length ${\lambda}_D$ vanishes and analytic solutions exist) and then in the general case, which we solve numerically. In the general case a double layer forms where the illuminated and un-illuminated regions meet, and an approximately quasi-neutral plasma connects the double layer to the wall sheath; in plane geometry the ions coast through the quasi-neutral section at slightly more than the Bohm speed $c_s$. The system, although simple, therefore has two novel features: a double layer that does not require counter-streaming ions and electrons, and a quasi-neutral plasma where ions travel in straight lines with at least the Bohm speed. We close with a pr\'{e}cis of our asymptotic solutions of this system, and suggest how our theoretical conclusions might be extended and tested in the laboratory.Comment: 10 pages, 3 figures, accepted by Physics of Plasma

Radio-frequency discharges in Oxygen. Part 1: Modeling

In this series of three papers we present results from a combined experimental and theoretical effort to quantitatively describe capacitively coupled radio-frequency discharges in oxygen. The particle-in-cell Monte-Carlo model on which the theoretical description is based will be described in the present paper. It treats space charge fields and transport processes on an equal footing with the most important plasma-chemical reactions. For given external voltage and pressure, the model determines the electric potential within the discharge and the distribution functions for electrons, negatively charged atomic oxygen, and positively charged molecular oxygen. Previously used scattering and reaction cross section data are critically assessed and in some cases modified. To validate our model, we compare the densities in the bulk of the discharge with experimental data and find good agreement, indicating that essential aspects of an oxygen discharge are captured.Comment: 11 pages, 10 figure

Photographic Assessment of Change in Trichotillomania: Psychometric Properties and Variables Influencing Interpretation

Although photographic assessment has been found to be reliable in assessing hair loss in Trichotillomania, the validity of this method is unclear, particularly for gauging progress in treatment. The current study evaluated the psychometric properties of photographic assessment of change in Trichotillomania. Photographs showing hair loss of adults with Trichotillomania were taken before and after participating in a clinical trial for the condition. Undergraduate college students (N = 211) rated treatment response according to the photos, and additional archival data on hair pulling severity and psychosocial health were retrieved from the clinical trial. Photographic assessment of change was found to possess fair reliability (ICC = 0.53), acceptable criterion validity (r = 0.51), good concurrent validity (r = 0.30–0.36), and excellent incremental validity (ΔR2 = 8.67, p \u3c 0.01). In addition, photographic measures were significantly correlated with change in quality of life (r = 0.42), and thus could be considered an index of the social validity of Trichotillomania treatment. Gender of the photo rater and pulling topography affected the criterion validity of photographic assessment (partial η2 = 0.05–0.11). Recommendations for improving photographic assessment and future directions for hair pulling research are discussed

Extraordinary sensitivity of the electronic structure and properties of single-walled carbon nanotubes to molecular charge-transfer

Interaction of single-walled carbon nanotubes with electron donor and acceptor molecules causes significant changes in the electronic and Raman spectra, the relative proportion of the metallic species increasing on electron donation through molecular charge transfer, as also verified by electrical resistivity measurements.Comment: 15 pages, 5 figurre

Electron surface layer at the interface of a plasma and a dielectric wall

We study the potential and the charge distribution across the interface of a plasma and a dielectric wall. For this purpose, the charge bound to the wall is modelled as a quasi-stationary electron surface layer which satisfies Poisson's equation and minimizes the grand canonical potential of the wall-thermalized excess electrons constituting the wall charge. Based on an effective model for a graded interface taking into account the image potential and the offset of the conduction band to the potential just outside the dielectric, we specifically calculate the potential and the electron distribution for magnesium oxide, silicon dioxide and sapphire surfaces in contact with a helium discharge. Depending on the electron affinity of the surface, we find two vastly different behaviors. For negative electron affinity, electrons do not penetrate into the wall and an external surface charge is formed in the image potential, while for positive electron affinity, electrons penetrate into the wall and a space charge layer develops in the interior of the dielectric. We also investigate how the electron surface layer merges with the bulk of the dielectric.Comment: 15 pages, 9 figures, accepted versio

Development of outbred CD1 mouse colonies with distinct standardized gut microbiota profiles for use in complex microbiota targeted studies.

Studies indicate that the gut microbiota (GM) can significantly influence both local and systemic host physiologic processes. With rising concern for optimization of experimental reproducibility and translatability, it is essential to consider the GM in study design. However, GM profiles can vary between rodent producers making consistency between models challenging. To circumvent this, we developed outbred CD1 mouse colonies with stable, complex GM profiles that can be used as donors for a variety of GM transfer techniques including rederivation, co-housing, cross-foster, and fecal microbiota transfer (FMT). CD1 embryos were surgically transferred into CD1 or C57BL/6 surrogate dams that varied by GM composition and complexity to establish four separate mouse colonies harboring GM profiles representative of contemporary mouse producers. Using targeted 16S rRNA amplicon sequencing, subsequent female offspring were found to have similar GM profiles to surrogate dams. Furthermore, breeding colonies of CD1 mice with distinct GM profiles were maintained for nine generations, demonstrating GM stability within these colonies. To confirm GM stability, we shipped cohorts of these four colonies to collaborating institutions and found no significant variation in GM composition. These mice are an invaluable experimental resource that can be used to investigate GM effects on mouse model phenotype