On the electrostatics of cubic crystals

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more “powerful” than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.

Catadioptric sensors refer to the combination of lens-based devices and reflective surfaces. These systems are useful because they may have a field of view which is greater than hemispherical, providing the ability to simultaneously view in any direction. Configurations which have a unique effective viewpoint are of primary interest, among these is the case where the reflective surface is a parabolic mirror and the camera is such that it induces an orthographic projection and which we call paracatadioptric. We present an algorithm for the calibration of such a device using only the images of lines in space. In fact, we show that we may obtain all of the intrinsic parameters from the images of only three lines and that this is possible without any metric information. We propose a closed-form solution for focal length, image center, and aspect ratio for skewless cameras and a polynomial root solution in the presence of skew. We also give a method for determining the orientation of a plane containing two sets of parallel lines from one uncalibrated view. Such an orientation recovery enables a rectification which is impossible to achieve in the case of a single uncalibrated view taken by a conventional camera. We study the performance of the algorithm in simulated setups and compare results on real images with an approach based on the image of the mirror's bounding circle.

Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω(r, θ) about a unique axis can be represented as a weighted integral of Ω over r and θ, (r, θ, φ) being spherical polar coordinates about the axis of rotation. For high-frequency acoustic modes, Δω/m collapses essentially to a function of ω — ω /(l+1/2) and M = m/(l + 1/2) alone, and the weighting kernel K(r,θ) becomes asymptotically degenerate, each factor being of essentially Abel type. Therefore, formally, the splitting integral can be inverted, once a procedure has been found for extending Δω over the domain of (ω, M) such that the turning points (rt,ωt), given by (c(rt)/rt, sinωt) = (ω, M) where c is sound speed, span the star. Obtaining that representation is the most difficult stage of the inversion. We report on a procedure that treats the inverted two-dimensional Abel integral as a repeated double integral, representing the data successively along a set of parallel lines M =constant. The method is illustrated by an inversion of artificial data which is compared with the angular velocity from which those data were computed.

The structure of a strange attractor

The desire to increase the amount of information that can be encoded onto a single photon has driven research into many areas of optics. One such area is optical orbital angular momentum (OAM) [1]. These beams have helical phasefronts and carry an orbital angular momentum of mbar per photon, where the integer m is unbounded, giving a large state space in which to encode information. We recently developed a telescope system comprising two bespoke refractive optical elements to transform OAM states into transverse momentum states [2]. This is achieved by mapping the azimuthal position of the input plane to the lateral position in the output [3]. A mapping of this type transforms a set of concentric rings at the input plane into a set of parallel lines in the output plane. A lens can then separate the resulting transverse momentum states into specified lateral positions, allowing for the efficient measurement of multiple OAM states simultaneously. Separating OAM states in this way presents an opportunity for this larger alphabet to improve the data capacity of a free space link and has potential application in both the classical and quantum regimes. We will present our latest design, increasing the bandwidth of measurable states to over 50 OAM modes. In such a system we study the crosstalk introduced by a thin phase turbulence, showing that turbulence similarly degrades the purity of all the modes within this range.

Synthesis, physico-chemical and DNA interaction studies of homo- and hetero-trinuclear complexes

Determining object shape from local velocity measurements

With a still developing and largely unpoliced code of ethics, the Internet forces users to renegotiate continually the boundaries and registers of publicity and privacy, and thus, it is the technic that constellates the ideal conditions for gossip, quite apart from its speed and efficiency in circulating tittle-tattle. Proceeding from observations about how the Internet challenges definitional certitudes about the practice and performance of gossip, this article considers the queer performances of Stephen Karam's darkly comic play Speech & Debate, in which three high-school misfits meet and bond over shared sexual secrets that begin to circulate online. Karam's representation suggests how the new platforms or stages for gossip that develop out of networking sites and technologies may create new counter-publics, capable, in their turn, of occasioning new forms of counter-intimacy. These platforms offer not only new potentialities for intertextual citation and public discourse but also mechanisms for self-promotion and, in this way, raise questions about the distinction between confession and gossip and between intimacy and publicity: two sets of parallel lines that have begun to bend toward one another.

Given an arbitrary simple polygon with N vertices we present an algorithm for shading the interior of the polygon with a set of parallel lines where the slope and the distance between lines are prespecified. If the number of shading line segments is M, the algorithm described in the paper runs in 0(N log N + M) time. The algorithm is generalizable to shade any region or regions of an arbitrary planar subdivision.

A geometry for simulation of polar surfaces of ionic crystals is described. The simulation cells form a two-dimensional periodic array with one side bounded by the polar surface of a large piece of rigid ionic crystal. To simulate a polar surface in this way, the surface must carry excess charge to ensure that the cells are a model of the surface of a piece of crystal in bulk thermodynamic equilibrium. The electrostatic energy of the simulation cells is presented as a sum of convergent two dimensional lattice sums. The cancellation of divergences implicit in this energy because of the excess charge distributions is described in detail.

The electrostatic potential which arises from a lattice array of point ions is computed in terms of a Taylor's series expansion for small distances from a lattice site. This expansion gives the change in electrostatic energy when an ion moves in the background of a perfect point ion lattice potential. The Taylor's series coefficients for terms up to fourth order in the ion displacement are evaluated for the NaCl and CaF2 lattice structures.

The electrostatic potential above the 100 surface plane of a f.c.c. ionic crystal has been calculated using a point charge distribution for the positive nuclei and a continuous charge density distribution for electrons. The continuous electron charge distribution has been obtained from the known results of a Fourier crystal structure analysis. The results for the electric field and its different gradients are compared with those obtained from the Lennard‐Jones and Dent model (point charge distributions for both the positive and negative ions). It has been found that the two models lead to significant differences of the electrostatic potentials and their derivatives.

Peculiar Variations of the Electrostatic Potential of Spike Protein N-terminal Domain Associated with the Emergence of Successive SARS-CoV-2 Omicron Lineages

In this paper we present a coupled thermodynamic formulation to predict stresses, in an ionic solid due to diffusion of charged defects in an electrochemical potential gradient. Chemical expansion is considered primarily by treating compositional strains as eigen strains. Two material properties are introduced a) A second order tensor that represents the eigen strains in the solid due to non-stoichiometry and b) A fourth order tensor that represents the variations in elastic properties due to non-stoichiometry. A general theory is first developed for a steady state, isothermal condition, while considering typical electrochemical reactions at the interface and the bulk of a typical oxide ion conductor by considering all the major defects that are known to operate. Two geometries typical of solid electrolytes (planar and tubular) are considered, involving diffusion of vacancies and electrons. The governing equations are solved for the resulting stresses due to chemically induced strains resulting from a deviation in the stoichiometric composition of the solid. The results show the influence of considering the coupled problem on the distribution of defects, electrostatic potential, and the current voltage characteristics.

A variational method is used to determine the polarizability of an electron trapped by a positive point defect in an ionic crystal. The model Hamiltonian describing the interaction between the point defect and an electron in an ionic crystal is related to the model introduced by Simpson to calculate the charge distribution and energy levels of trapped electrons in ionic crystals. However, in the problem considered here, an applied electric field is also present, and consequently a somewhat different model is proposed. The polarizabilities are calculated for both the new model and Simpson's original model, and the results are compared with the results of a theoretical calculation by Adamov, Ledovskaya, and Rebane of the polarizability of an $F$ center in alkali halide crystals. In atomic units, the polarizabilities obtained for the new model range from 227 for NaF to 4280 for LiI.