Abstract:
We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl-Dirac operators on $\mathbb{R}^3$. In particular we are interested in those operators $\mathcal{D}_{B}$ for which the associated magnetic field $B$ is given by pulling back a $2$-form $\beta$ from the sphere $\mathbb{S}^2$ to $\mathbb{R}^3$ using a combination of the Hopf fibration and inverse stereographic projection. If $\int_{\mathbb{S}^2}\beta\neq0$ we show that \[ \sum_{0\le t\le T}\mathrm{dim}\,\mathrm{Ker}\,\mathcal{D}_{tB} =\frac{T^2}{8\pi^2}\,\biggl\lvert\int_{\mathbb{S}^2}\beta\biggr\rvert\,\int_{\mathbb{S}^2}\lvert{\beta}\rvert+o(T^2) \] as $T\to+\infty$. The result relies on Erd\H{o}s and Solovej's characterisation of the spectrum of $\mathcal{D}_{tB}$ in terms of a family of Dirac operators on $\mathbb{S}^2$, together with information about the strong field localisation of the Aharonov-Casher zero modes of the latter.

Abstract:
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form \lambda_n(H+V)=\lambda_n(H)+W(\sqrt n)n^{-1/4}+O(n^{-1/2}\ln(n)) as n\to+\infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.

Abstract:
Let $\mathcal{P}_{\Omega,tA}$ denoted the Pauli operator on a bounded open region $\Omega\subset\mathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field $B=\mathrm{curl}\,A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula \[ \mathsf{N}_{\Omega,tA}(\lambda(t))=\frac{t}{2\pi}\int_{\Omega}\lvert B(x)\rvert\,dx\;+o(t) \] as $t\to+\infty$, whenever $\lambda(t)=Ce^{-ct^\sigma}$ for some $\sigma\in(0,1)$ and $c,C>0$. The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.

Abstract:
We study the Fredholm properties of a general class of elliptic differential operators on $\R^n$. These results are expressed in terms of a class of weighted function spaces, which can be locally modeled on a wide variety of standard function spaces, and a related spectral pencil problem on the sphere, which is defined in terms of the asymptotic behaviour of the coefficients of the original operator.

Abstract:
In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.

Abstract:
We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

Abstract:
We present a critical comparison of the dielectric properties of three models of water - TIP4P/2005, TIP4P/2005f and TTM3F. Dipole spatial correlation is measured using the distance dependent Kirkwood function along with one dimensional and two dimensional dipole correlation functions. We find that the introduction of flexibility alone does not significantly affect dipole correlation and only affects $\varepsilon(\omega)$ at high frequencies. By contrast the introduction of polarizability increases dipole correlation and yields a more accurate $\varepsilon(\omega)$. Additionally the introduction of polarizability creates temperature dependence in the dipole moment even at fixed density, yielding a more accurate value for $d \varepsilon / d T$ compared to non-polarizable models. To better understand the physical origin of the dielectric properties of water we make analogies to the physics of polar nanoregions in relaxor ferroelectric materials. We show that $\varepsilon(\omega,T)$ and $\tau_D(T)$ for water have striking similarities with relaxor ferroelectrics, a class of materials characterized by large frequency dispersion in $\varepsilon(\omega,T)$, Vogel-Fulcher-Tamann behaviour in $\tau_D(T)$, and the existence of polar nanoregions.

Abstract:
The local structure of liquid water as a function of temperature is a source of intense research. This structure is intimately linked to the dynamics of water molecules, which can be measured using Raman and infrared spectroscopies. The assignment of spectral peaks depends on whether they are collective modes or single molecule motions. Vibrational modes in liquids are usually considered to be associated to the motions of single molecules or small clusters. Using molecular dynamics simulations we find dispersive optical phonon-like modes in the librational and OH stretching bands. We argue that on subpicosecond time scales these modes propagate through water's hydrogen bond network over distances of up to two nanometers. In the long wavelength limit these optical modes exhibit longitudinal-transverse splitting, indicating the presence of coherent long range dipole-dipole interactions, as in ice. Our results indicate the dynamics of liquid water have more similarities to ice than previously thought.

Abstract:
We present a class of spherically symmetric random variables defined by the property that as dimension increases to infinity the mass becomes concentrated in a hyperspherical shell, the width of which is negligible compared to its radius. We provide a sufficient condition for this property in terms of the functional form of the density and then show that the property carries through to equivalent elliptically symmetric distributions, provided that the contours are not too eccentric, in a sense which we make precise. Individual components of such distributions possess a number of appealing Gaussian-like limit properties, in particular that the limiting one-dimensional marginal distribution along any component is Gaussian.

Abstract:
advances in aerial pesticide application technology of chemicals have been given in the direction of reducing the syrup volume, which can cause poor distribution and consequent irregular deposition. this study aimed to evaluate the quality of the syrup spray on aerial application in soybean crop (glycine max l.). for the application, an experimental agricultural aircraft was used, applying a spray volume of 20 l ha-1. a bright blue food coloring added to the syrup spray was used for the determination of the volumes stored in the leaves of the upper third, middle and bottom of the soybean plants. these leaves were washed, and the volume determined by spectrophotometry. to obtain the drops spectrum were used artificial targets set by water sensitive paper, distributed in the middle and upper third of the plants. data were submitted to analysis of variance with a single factor, considering the different positions in the plant, and control cards were made from the lower and upper control limits. the aerial application of syrup spray in the soybean crop had lower values of the volumetric median diameter, relative amplitude and coverage on the middle third in relation of the upper third of the soybean crop. there was less of the syrup spray deposition in the lower third. the coverage indicators of the syrup spray showed that aerial application with the evaluated agricultural experimental aircraft is not under statistical process control, in other words, outside of the quality standard.