Differential equations of electrodiffusion: constant field solutions, uniqueness, and new formulas of Goldman-Hodgkin-Katz type

The equations governing one-dimensional, steady-state electrodiffusion are considered when there are arbitrarily many mobile ionic species present, in any number of valence classes, possibly also with a uniform distribution of fixed charges. Exact constant field solutions and new formulas of Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike the usual approximate ones. Corresponding boundary conditions on the ionic concentrations are identified. The question of uniqueness of constant field solutions with such boundary conditions is considered, and is re-posed in terms of an autonomous ordinary differential equation of order $n+1$ for the electric field, where $n$ is the number of valence classes. When there are no fixed charges, the equation can be integrated once to give the non-autonomous equation of order $n$ considered previously in the literature including, in the case $n=2$, the form of Painlev\'e's second equation considered first in the context of electrodiffusion by one of us. When $n=1$, the new equation is a form of Li\'enard's equation. Uniqueness of the constant field solution is established in this case.Comment: 29 pages, 5 figure

Airy series solution of Painlev\'e II in electrodiffusion: conjectured convergence

A perturbation series solution is constructed in terms of Airy functions for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, for two ionic species carrying equal and opposite charges. The solution includes a formal determination of the associated electric field, which is known to satisfy a form of the Painlev\'e II differential equation. Comparisons with the numerical solution of the boundary-value problem show excellent agreement following termination of the series after a sufficient number of terms, for a much wider range of values of the parameters in the model than suggested by previously presented analysis, or admitted by previously presented approximation schemes. These surprising results suggest that for a wide variety of cases, a convergent series expansion is obtained in terms of Airy functions for the Painlev\'e transcendent describing the electric field. A suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases. It is conjectured that the series does converge for this subset.Comment: 30 pages, 9 figures. Typos corrected, figures modified, extra references adde

Subtyping Nonsuicidal Self-Injurers: An Application of Latent Variable Mixture Modeling

Using latent variable mixture modeling (LVMM), we sought to identify subtypes of individuals who engage in nonsuicidal self-injury (NSSI). Specifically, this study replicated Klonsky and Olino’s (2008) investigation of undergraduate self-injurers in which they found four clinically distinct subtypes: “Experimental NSSI,” “Mild NSSI,” “Multiple Functions/Anxious,” and “Automatic Functions/Suicidal” groups. The current study was also an extension of Klonsky and Olino in two ways. First, analyses were conducted on a combined sample of undergraduates and internet users who endorsed NSSI. Also, differences in exposure to trauma, post-traumatic stress disorder, and alcohol use were investigated. Results revealed a similar four-class structure of NSSI, with an additional fifth “Multi-method” group

Non-positivity of Groenewold operators

A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Liouville densities, leading to what may be termed term Groenewold operators. We investigate the spectra of the Groenewold operators that correspond to Gaussian and to certain uniform Liouville densities. We show that when the classical coordinate-momentum uncertainty product falls below Heisenberg's limit, the Groenewold operators in the Gaussian case develop negative eigenvalues and eigenvalues larger than 1. However, in the uniform case, negative eigenvalues are shown to persist for arbitrarily large values of the classical uncertainty product.Comment: 9 pages, 1 figures, submitted to Europhysics Letter

Lights, Camera, Action! The role of movies and video in classroom learning

Increased technology support in classrooms allows for enhanced opportunities for faculty to make use of multimedia. This article focuses on how faculty can best utilize movies and video clips to enhance student learning. A review of the fundamentals from a technical perspective is provided, as well as techniques to incorporate movie clips in active learning instruction. Support for faculty in the use of technology in teaching is essential in today’s classroom

Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. We exploit the fact that quantum non-compact algebras such as $U_q(su(1,1))$ and type-I quantum superalgebras such as $U_q(gl(1|1))$ and $U_q(gl(2|1))$ are known to admit non-trivial one-parameter families of infinite-dimensional and finite dimensional irreps, respectively, even for generic $q$. We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples we work out the the $R$-matrices for the three quantum algebras mentioned above in certain representations.Comment: 13 page

Circular No. 59 - Control of Stinking Smut of Wheat with Copper Carbonate

Stinking smut or bunt of wheat is an ever-present and destructive disease in the wheat fields of Utah. During the past season (1925) this disease was especially prevalent, causing losses in certain fields of from 25 to 50 per cent, not counting the loss to the grower in reduced grade of grain. In the threshing of smutty wheat there is also the risk of loss from smut explosion. Almost every season cases of this sort are reported. In addition of all of the wheat tested by the U. S. Grain Inspector at Logan for Northern Utah and Southern Idaho 30 per cent showed smut infection in 1925. The average reduction for smut is near ten cents a bushel with a variation from five to twenty cents. The cost of producing a smutted crop may equal or even exceed the cost of producing a clean crop. Loss occurring from this disease, since it is preventable, can hardly be considered attached to the total gross returns; it is a subtraction from the net profit. Effective methods for the prevention of these losses by smut are now available to every grain grower

The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.Comment: 36 pages, 6 figures. Revised in light of referees' comments, and further references adde

The Weierstrass–Enneper System for Constant Mean Curvature Surfaces and the Completely Integrable Sigma Model

The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints. A new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples, including a multi-soliton solution