The Tropical Eigenvalue-Vector Problem from Algebraic, Graphical, and Computational Perspectives

Tropical mathematics describes both the max-plus and min-plus algebras. In the former, we understand addition to be the component-wise maximum function, while the minimum function represents addition in the latter. For example, 1 plus 2 is equivalent to max{1, 2} = 2 in the max-plus framework, whereas in min-plus algebra 1 plus 2 equals min{1, 2} = 1. In both algebras, we multiply elements by performing standard addition; hence, 1 times 2 yields 1+2 = 3. This seemingly fanciful arithmetic actually provides a language through which we elegantly describe everyday phenomena such as the long-term behavior of discrete event systems (assembly lines, computer networks, train schedules, etc). Extending the notion of tropical algebra to matrices and vectors, we find that determining eigenvalues and associated eigenvectors, allows us to construct event systems that behave predictably and stably. First, we explore the graph theoretic underpinnings of the tropical eigenvalue and associated eigenspaces so that we may better understand how they are computed. Considering two classes of tropical matrices, irreducible and reducible, we then look at how Karp’s Algorithm computes eigenvalues of the former and what its output when given a reducible system can tell us about long-term behavior

Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators

This paper reports the analysis of the dynamics of a model of pulse-coupled oscillators with global inhibitory coupling. The model is inspired by experiments on colonies of bacteria-embedded synthetic genetic circuits. The total population can be either of finite (arbitrary) size or infinite, and is represented by a one-dimensional profile. Profiles can be discontinuous, possibly with infinitely many jumps. Their time evolution is governed by a singular differential equation. We address the corresponding initial value problem and characterize the dynamics' main features. In particular, we prove that trajectory behaviors are asymptotically periodic, with period only depending on the profile (and on the model parameters). A criterion is obtained for the existence of the corresponding periodic orbits, which reveals the existence of a sharp transition as the coupling parameter is increased. The transition separates a regime where any profile can be obtained in the limit of large times, to a situation where only trajectories with sufficiently large groups of synchronized oscillators perdure

Phaselocked patterns and amplitude death in a ring of delay coupled limit cycle oscillators

We study the existence and stability of phaselocked patterns and amplitude death states in a closed chain of delay coupled identical limit cycle oscillators that are near a supercritical Hopf bifurcation. The coupling is limited to nearest neighbors and is linear. We analyze a model set of discrete dynamical equations using the method of plane waves. The resultant dispersion relation, which is valid for any arbitrary number of oscillators, displays important differences from similar relations obtained from continuum models. We discuss the general characteristics of the equilibrium states including their dependencies on various system parameters. We next carry out a detailed linear stability investigation of these states in order to delineate their actual existence regions and to determine their parametric dependence on time delay. Time delay is found to expand the range of possible phaselocked patterns and to contribute favorably toward their stability. The amplitude death state is studied in the parameter space of time delay and coupling strength. It is shown that death island regions can exist for any number of oscillators N in the presence of finite time delay. A particularly interesting result is that the size of an island is independent of N when N is even but is a decreasing function of N when N is odd.Comment: 23 pages, 12 figures (3 of the figures in PNG format, separately from TeX); minor additions; typos correcte

Dynamic analysis of repetitive decision-free discreteevent processes: The algebra of timed marked graphs and algorithmic issues

A model to analyze certain classes of discrete event dynamic systems is presented. Previous research on timed marked graphs is reviewed and extended. This model is useful to analyze asynchronous and repetitive production processes. In particular, applications to certain classes of flexible manufacturing systems are provided in a companion paper. Here, an algebraic representation of timed marked graphs in terms of reccurrence equations is provided. These equations are linear in a nonconventional algebra, that is described. Also, an algorithm to properly characterize the periodic behavior of repetitive production processes is descrbed. This model extends the concepts from PERT/CPM analysis to repetitive production processes.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44155/1/10479_2005_Article_BF02248590.pd

Angles and devices for quantum approximate optimization

A potential application of emerging Noisy Intermediate-Scale Quantum (NISQ) devices is that of approximately solving combinatorial optimization problems. This thesis investigates a gate-based algorithm for this purpose, the Quantum Approximate Optimization Algorithm (QAOA), in two major themes. First, we examine how the QAOA resolves the problems it is designed to solve. We take a statistical view of the algorithm applied to ensembles of problems, first, considering a highly symmetric version of the algorithm, using Grover drivers. In this highly symmetric context, we find a simple dependence of the QAOA state’s expected value on how values of the cost function are distributed. Furthering this theme, we demonstrate that, generally, QAOA performance depends on problem statistics with respect to a metric induced by a chosen driver Hamiltonian. We obtain a method for evaluating QAOA performance on worst-case problems, those of random costs, for differing driver choices. Second, we investigate a QAOA context with device control occurring only via single-qubit gates, rather than using individually programmable one- and two-qubit gates. In this reduced control overhead scheme---the digital-analog scheme---the complexity of devices running QAOA circuits is decreased at the cost of errors which are shown to be non-harmful in certain regimes. We then explore hypothetical device designs one could use for this purpose.Eine mögliche Anwendung für “Noisy Intermediate-Scale Quantum devices” (NISQ devices) ist die näherungsweise Lösung von kombinatorischen Optimierungsproblemen. Die vorliegende Arbeit untersucht anhand zweier Hauptthemen einen gatterbasierten Algorithmus, den sogenannten “Quantum Approximate Optimization Algorithm” (QAOA). Zuerst prüfen wir, wie der QAOA jene Probleme löst, für die er entwickelt wurde. Wir betrachten den Algorithmus in einer Kombination mit hochsymmetrischen Grover-Treibern für statistische Ensembles von Probleminstanzen. In diesem Kontext finden wir eine einfache Abhängigkeit von der Verteilung der Kostenfunktionswerte. Weiterführend zeigen wir, dass die QAOA-Leistung generell von der Problemstatistik in Bezug auf eine durch den gewählten Treiber-Hamiltonian induzierte Metrik abhängt. Wir erhalten eine Methode zur Bewertung der QAOA-Leistung bei schwersten Problemen (solche zufälliger Kosten) für unterschiedliche Treiberauswahlen. Zweitens untersuchen wir eine QAOA-Variante, bei der sich die Hardware- Kontrolle nur auf Ein-Qubit-Gatter anstatt individuell programmierbare Ein- und Zwei-Qubit-Gatter erstreckt. In diesem reduzierten Kontrollaufwandsschema—dem digital-analogen Schema—sinkt die Komplexität der Hardware, welche die QAOASchaltungen ausführt, auf Kosten von Fehlern, die in bestimmten Bereichen als ungefährlich nachgewiesen werden. Danach erkunden wir hypothetische Hardware- Konzepte, die für diesen Zweck genutzt werden könnten