Chain Decomposition Theorems for Ordered Sets (and Other Musings)

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets

Optimal Pebbling in Products of Graphs

We prove a generalization of Graham's Conjecture for optimal pebbling with arbitrary sets of target distributions. We provide bounds on optimal pebbling numbers of products of complete graphs and explicitly find optimal $t$-pebbling numbers for specific such products. We obtain bounds on optimal pebbling numbers of powers of the cycle $C_5$. Finally, we present explicit distributions which provide asymptotic bounds on optimal pebbling numbers of hypercubes.Comment: 28 pages, 1 figur

A structure theorem for posets admitting a “strong” chain partition: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)

AbstractSuppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q∈P lie in distinct chains and p<q, then every other element of P is either above p or below q.In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets ⊕i=1nRi such that, for 1⩽i⩽n, one of the following occurs:(1)Ri is disjoint from one of the chains of the partition; or(2)if p, q∈Ri are in distinct chains, then they are incomparable.The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets.In 1996, a proof of the Daykin–Daykin conjecture was announced (by two other mathematicians), but their proof needs to be rectified.In this note, a generalization of the conjecture is proven that applies to finite or infinite posets partitioned into a (possibly infinite) number of chains with the same property. In particular, it is shown that a poset admits such a partition if and only if it is an ordinal sum of posets, each of which is either a width 2 poset or else a disjoint sum of chains. A forbidden subposet characterization of these partial orders is also obtained

History Vs. Expectations

In models with external economies, there are often two or more long run equilibria. Which equilibrium is chosen? Much of the literature presumes that "history" sets initial conditions which determine the outcome, but an alternative view stresses the role of "expectations", i.e. of self-fulfilling prophecy. This paper uses a simple trade model with both external economies and adjustment costs to show how the parameters of the economy determine the relative importance of history and expectations in determining equilibrium.

Symmetric function generalizations of graph polynomials

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.Includes bibliographical references (p. 67-70).by Timothy Yi-Chung Chow.Ph.D

Social Interactions

Prepared for Annual Reviews of Economics.

The North Carolina Speaker Ban Commission hearing : a study of evidence

The purpose of this thesis is to analyze the evidence presented by the American Association of University Professors and the American Legion before the North Carolina Speaker Ban Study Commission. Chapter One traces the history of the criteria used to evaluate rhetorical evidence from Aristotle to the present. Chapter Two gives a history of the North Carolina Speaker Ban Law and emphasizes the Commission study. Chapter Three applies selected evidential criteria to the presentation of the American Association of University Professors, and Chapter Four applies the criteria to the presentation of the American Legion. Finally, Chapter Five summarizes the study, draws conclusions about the treatment of evidence by the two organizations, and recommends subjects for additional research. This study finds that the American Association of University Professors spokesmen present evidence based on their expertise in education, law, and AAUP data. They have recently studied the Speaker Ban controversy even though they have not actually participated in the controversy within the North Carolina state system of higher education. Their evidence shows little speaker bias

Distance matrices of trees

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1986.MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.Bibliography: leaf 56.by Karen L. Collins.Ph.D

The von Neumann Model and the Early Models of General Equilibrium

The paper reconstructs the von Neumann model, comments on its salient features and critically reviews some of its generalisations. The issues related to thetreatment of consumption, decomposability and uniqueness of the rate of growth and interest will be especially scrutinised. The most prominent models of general equilibrium that appeared before or roughly at the same time as von Neumann's model will be also reviewed in the paper and compared with it. It will be demonstrated that none of them had any noticeable influence on von Neumann's model, which is genuinely distinct, ideologically free and methodologically fresh and forward-looking. It will be argued that the model can be viewed as a brilliant mathematical metaphor of some deep-rooted old vision, pertaining to the core issues of commodity production

Random Keys Genetic Algorithms Scheduling and Rescheduling Systems for Common Production Systems

The majority of scheduling research deals with problems in specific production environments with specific objective functions. However, in many cases, more than one problem type and/or objective function exists, resulting in the need for a more generic and flexible system to generate schedules. Furthermore, most of the published scheduling research focuses on creating an optimal or near optimal initial schedule during the planning phase. However, after production processes start, circumstances like machine breakdowns, urgent jobs, and other unplanned events may render the schedule suboptimal, obsolete or even infeasible resulting in a rescheduling problem, which is typically also addressed for a specific production environment, constraints, and objective functions. This dissertation introduces a generic framework consisting of models and algorithms based on Random Keys Genetic Algorithms (RKGA) to handle both the scheduling and rescheduling problems in the most common production environments and for various types of objective functions. The Scheduling system produces predictive (initial) schedules for environments including single machines, flow shops, job shops and parallel machine production systems to optimize regular objective functions such as the Makespan and the Total Tardiness as well as non-regular objective functions such as the Total Earliness and Tardiness. To deal with the rescheduling problem, and using as a basis the same RKGA, a reactive Rescheduling system capable of repairing initial schedules after the occurrence of unexpected events is introduced. The reactive Rescheduling system was designed not only to optimize regular and non-regular objective functions but also to minimize the instability, a very important aspect in rescheduling to avoid shop chaos due to disruptions. Minimizing both schedule inefficiency and instability, however, turns the problem into a multi-objective optimization problem, which is even more difficult to solve. The computational experiments for the predictive model show that it is able to produce optimal or near optimal schedules to benchmark problems for different production environments and objective functions. Additional computational experiments conducted to test the reactive Rescheduling system under two types of unexpected events, machine breakdowns and the arrival of a rush job, show that the proposed framework and algorithms are robust in handling various problem types and computationally reasonable