Differential equations for the cuspoid canonical integrals

Differential equations satisfied by the cuspoid canonical integrals I_n(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(ɑ_1,ɑ_2,...,ɑ_(n−1)). A set of linear coupled ordinary differential equations is derived for each step in the sequence I_n(0,0,...,0,0) →I_n(0,0,...,0,ɑ_(n−1)) →I_n(0,0,...,ɑ_(n−2),ɑ_(n−1)) →...→I_n(0,ɑ_2,...,ɑ_(n−2),ɑ_(n−1)) →I_n(ɑ_1,ɑ_2,...,ɑ_n−2,ɑ_(n−1)). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives

Effect of current corrugations on the stability of the tearing mode

The generation of zonal magnetic fields in laboratory fusion plasmas is predicted by theoretical and numerical models and was recently observed experimentally. It is shown that the modification of the current density gradient associated with such corrugations can significantly affect the stability of the tearing mode. A simple scaling law is derived that predicts the impact of small stationary current corrugations on the stability parameter $\Delta'$. The described destabilization mechanism can provide an explanation for the trigger of the Neoclassical Tearing Mode (NTM) in plasmas without significant MHD activity.Comment: Accepted to Physics of Plasma

Theory of Semiclassical Transition Probabilities for Inelastic and Reactive Collisions. II Asymptotic Evaluation of the S Matrix

The asymptotic evaluation of the integral representation for an S matrix element in a previously developed semiclassical theory of molecular collisions is considered. The integral representation is evaluated asymptotically by the method of Chester, Friedman, and Ursell to give a uniform approximation for the S matrix element which is valid for classically accessible and classically inaccessible transitions. The results unify and extend those previously derived, which were restricted to the simple semiclassical and Airy function cases. A comparison is made with the simple, Airy, and uniform semiclassical approximations that occur in Miller's semiclassical theory of molecular collisions. Although the starting point of the two theories is different, it is concluded that their asymptotic results are essentially identical. In addition, a simpler derivation of the integral representation for an S matrix element from the semiclassical wavefunction is given, one which avoids the use of Green's theorem

Theory of Reactive Collisions: Conformal Transformation

Conformal mapping techniques are applied to the Schrödinger equation for a bimolecular exchange reaction, with all three atoms lying on a line. For the case of a very heavy central mass, the extension of the theory to three dimensions is indicated. An angle-shaped region of the potential-energy surface is mapped onto an infinite strip in order to simplify the theoretical treatment of the boundary conditions. The mapping function is determined with the help of the Schwarz–Christoffel formula, and its properties described. The transformed Schrödinger equation is converted into an integral equation using the method of Green's functions, and integral representations for the reflection and transmission coefficients are obtained

Dynamical Phase Transitions In Driven Integrate-And-Fire Neurons

We explore the dynamics of an integrate-and-fire neuron with an oscillatory stimulus. The frustration due to the competition between the neuron's natural firing period and that of the oscillatory rhythm, leads to a rich structure of asymptotic phase locking patterns and ordering dynamics. The phase transitions between these states can be classified as either tangent or discontinuous bifurcations, each with its own characteristic scaling laws. The discontinuous bifurcations exhibit a new kind of phase transition that may be viewed as intermediate between continuous and first order, while tangent bifurcations behave like continuous transitions with a diverging coherence scale.Comment: 4 pages, 5 figure

Assessment of motivation to control alcohol use: The motivational thought frequency and state motivation scales for alcohol control

publisher: Elsevier articletitle: Assessment of motivation to control alcohol use: The motivational thought frequency and state motivation scales for alcohol control journaltitle: Addictive Behaviors articlelink: http://dx.doi.org/10.1016/j.addbeh.2016.02.038 content_type: article copyright: © 2016 Elsevier Ltd. All rights reserved