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

H∞ Tracking Control for a Class of Nonlinear Systems

Develops the theory for tracking control using the nonlinear H∞ control design methodology for a class of nonlinear input affine systems. The authors use a two-step process of first designing the feedforward part of the controller to design for perfect trajectory following and then design the feedback part of the controller using nonlinear H∞ regulator theory. Results for infinite-time and finite-time horizons are presente

A Comprehensive Strategy for Longitudinal Vehicle Control with Fuzzy Supervisory Expert System

The main objectives of vehicle motion control on an automated highway system are stable and safe automatic longitudinal and/or lateral path following in a platoon of vehicles. Various controllers can be used to satisfy the same objectives, but they may require different variables to be sensed or different conditions to be met. Supervision can select a controller and switch to a different controller depending on the conditions. Specifically, a fuzzy supervisory expert system checks for various system conditions and chooses a controller from PID, PI, sliding mode and a fuzzy controller or gives a distress signal. The choice between PID and PI controllers is based on the availability of the error derivative. Robust and complex sliding mode control can counter the external disturbances which the PID and PI controllers cannot handle. The fuzzy controller is used when the sensors are not working perfectly but the sensor values are still reliable enough to define corresponding fuzzy linguistic variables

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

Pursuit Evasion: The Herding Non-cooperative Dynamic Game

A class of pursuit evasion problems is studied. This problem involves a dog agent herding a sheep agent in order to take the sheep to a pen. The problem is stated in terms of the allowable sequential actions of the two agents and the game being played because of the choices each agent has. The solution is obtained using the dynamic programming principle applied in the game setting. The algorithm is analyzed and simulation results are presented

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

Dynamic Programming Solution for a Class of Pursuit Evasion Problems: The Herding Problem

A herding dog and sheep problem is studied where the agent “dog” is considered the control action for moving the agent “sheep” to a fixed location using the dynamics of their interaction. The problem is solved for the deterministic case using dynamic programming. Proofs are provided for the correctness of the algorithms. The algorithm is analyzed for its complexity. A software package developed for experimentation is described

Image Processing Based Control for Scaled Automated Vehicles

This paper presents a way to design a lateral controller for an automated vehicle using information gained through image processing with the control objective being to stay on a desired path. Two possible ways to obtain the information necessary for lateral control by image processing are presented, one based on pixel intensity summation and the other on vanishing point calculations. The paper also describes two algorithms for the actual lateral control, one based on classical control theory and the other on modern H∞ control. The resulting control algorithms were implemented on a scaled autonomous vehicle system