On correspondence between tensors and bispinors

It is known that in the four-dimensional Riemannian space the complex bispinor generates a number of tensors: scalar, pseudo-scalar, vector, pseudo-vector, antisymmetric tensor. This paper solves the inverse problem: the above tensors are arbitrarily given, it is necessary to find a bispinor (bispinors) reproducing the tensors. The algorithm for this mapping constitutes construction of Hermitean matrix $M$ from the tensors and finding its eigenvalue spectrum. A solution to the inverse problem exists only when $M$ is nonnegatively definite. Under this condition a matrix $Z$ satisfying equation $M=ZZ^{+}$ can be found. One and the same system of tensor values can be used to construct the matrix $Z$ accurate to an arbitrary factor on the left-hand side, viz. unitary matrix $U$ in polar expansion $Z=HU$. The matrix $Z$ is shown to be expandable to a set of bispinors, for which the unitary matrix $U$ is responsible for the internal (gauge) degrees of freedom. Thus, a group of gauge transformations depends only on the Riemannian space dimension, signature, and the number field used. The constructed algorithm for mapping tensors to bispinors admits extension to Riemannian spaces of a higher dimension.Comment: 14 pages;LaTeX2e;to appear in the 9th Marcel Grossmann Meeting (MG9) Proceedings,Rome, July, 200

Bugs on a Slippery Plane : Understanding the Motility of Microbial Pathogens with Mathematical Modelling

Many pathogenic microorganisms live in close association with surfaces, typically in thin films that either arise naturally or that they themselves create. In response to this constrained environment, the cells adjust their behaviour and morphology, invoking communication channels and inducing physical phenomena that allow for rapid colonization of biomedically relevant surfaces or the promotion of virulence factors. Thus, it is very important to measure and theoretically understand the key mechanisms for the apparent advantage obtained from swimming in thin films. We discuss experimental measurements of flows around a peritrichously flagellated bacterium constrained in a thin film, derive a simplified mathematical theory and Green's functions for flows in a thin film with general slip boundary conditions, and establish connections between theoretical and experimental results. This article aims to highlight the importance of mathematics as a tool to unlock qualitative mechanisms associated with experimental observations in the medical and biological sciences

Pursuit Evasion: The Herding Noncooperative Dynamic Game-The Stochastic Model

This article proposes a solution to the herding problem, a class of pursuit evasion problem in a stochastic framework. The problem involves a pursuer agent trying to herd a stochastically moving evader agent into a pen. The problem is stated in terms of allowable sequential actions of the two agents. The solution is obtained by applying the principles of stochastic dynamic programming. Three algorithms for solution are presented with their accompanying results

Study of a zirconium getter for purification of xenon gas

Oxygen, nitrogen and methane purification efficiencies for a common zirconium getter are measured in 1050 Torr of xenon gas. Starting with impurity concentrations near 10^{-6} g/g, the outlet impurity level is found to be less than 120*10^{-12} g/g for O2 and less than 950*10^{-12} g/g for N2. For methane we find residual contamination of the purified gas at concentrations varying over three orders of magnitude, depending on the purifier temperature and the gas flow rate. A slight reduction in the purifier's methane efficiency is observed after 13 mg of this impurity has been absorbed, which we attribute to partial exhaustion of the purifier's capacity for this species. We also find that the purifier's ability to absorb N2 and methane can be extinguished long before any decrease in O2 performance is observed, and slower flow rates should be employed for xenon purification due to the cooling effect that the heavy gas has on the getter.Comment: 14 pages, 5 figure

Physical Vacuum Properties and Internal Space Dimension

The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization is not however unique. The case of 7-dimensional Riemannian space of signature 7(-) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of algebra E_{8}. Considerations are presented, from which it follows that the least-dimension space bearing on physics is the Riemannian 11-dimensional space of signature 1(-)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density.Comment: 19 pages, 1figure. Submitted to General Relativity and Gravitatio