"Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.

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Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

A new continuum traffic flow model is developed in this paper based on an improved car-following model. In this new continuum model, the speed gradient replaces the density gradient in the equation of motion, and this replacement guarantees the property that the characteristic speeds are always less than or equal to the macroscopic flow speed. This new model also overcomes the backward travel problem that exists in many high-order continuum models. Shock waves, rarefaction waves, stop-and-go waves, and local cluster effects can be obtained from this new model and are consistent with the diverse nonlinear dynamical phenomena observed in the freeway traffic.

A new continuum model for traffic flow and numerical tests

In this work a functional form for the speed-density relationship is presented This functional form is made up of a nondimensional spacing, the equivalent spacing and of a function, the generating function, whose argument is the equivalent spacing This functional form is derived by means of two different arguments The first argument is based on the set of properties that the volume-speed-density relationships should satisfy The second one arises when applying a dimensional analysis to a generic car-following model Finally, several examples of generating functions are shown

On the functional form of the speed-density relationship—I: General theory

Time-delayed systems are found to display remarkable temporal patterns the dynamics of which split into regular and chaotic components repeating at the interval of a delay. This novel long-term behavior for delay dynamics results from strongly asymmetric nonlinear delayed feedback driving a highly damped harmonic oscillator dynamics. In the corresponding virtual space-time representation, the behavior is found to develop as a chimeralike state, a new paradigmatic object from the network theory characterized by the coexistence of synchronous and incoherent oscillations. Numerous virtual chimera states are obtained and analyzed, through experiment, theory, and simulations.

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Virtual chimera states for delayed-feedback systems.

This paper presents a new continuum traffic flow theory. The derivation of this new theory is based on both empirical evidence of traffic flow behavior and basic assumptions on drivers' reaction to stimuli. Central to the development of this theory is the existence of an equilibrium speed-concentration relationship and an introduction of a disturbance propagation speed. The new theory includes a well established continuum theory (the LWR theory) as a special case and removes certain deficiencies of this theory. Unlike existing higher order continuum models, the new theory does not exhibit the undesirable behavior of 'wrong-way travel' because in this theory traffic disturbances are always propagated against the traffic stream.

A theory of nonequilibrium traffic flow

Car following models and the fundamental diagram of road traffic

Vehicle accelerations in the hydrodynamic theory of traffic flow

This paper is intended to introduce the methods and literature of difference equations to engineers who are unaquainted with them or their wide applicability and power.

Difference equations and their applications

Two types of mathematical models are presented to explain the empirical observation that a form of “wave phenomenon” occurs when a long line of vehicles on a crowded highway is stopped or started by a signal. The interruption of the steady flow of vehicles initiates “waves of stopping” or “waves of starting.” The first model considered is a car-following model based on a system of linear differential-difference equations that contains two physically significant parameters. The second model adopts a macroscopic viewpoint in which a stream of traffic on a single-lane road may be represented by a compressible “traffic fluid” of density or concentration k and a rate of flow q. The analysis is based on the equation of continuity of the “traffic fluid” and a novel empirical “flow-concentration” relation. The behavior of small disturbances in the vicinity of steady flow is examined by the method of perturbations. The wave phenomena predicted by the first and second models are compared.

Wave theories of traffic flow

The parametrically excited transverse oscillations of a uniform straight column whose ends are pinned are investigated. The ends of the column are considered to be subjected to a pulsating load whose time graph is a square wave. By the use of matrix algebra, an expression giving the regions of instability is obtained. The effects of more general boundary conditions are studied and a nonlinear differential equation to account for the limitation of the amplitude of the oscillations is developed.

Louis A. Pipes

Papers

Car following models and the fundamental diagram of road traffic

Vehicle accelerations in the hydrodynamic theory of traffic flow

Difference equations and their applications

Wave theories of traffic flow

The dynamic stability of a uniform straight column excited by a pulsating load