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13
A Column Generation Approach For Graph Coloring
 INFORMS Journal on Computing
, 1995
"... We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approac ..."
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Cited by 73 (2 self)
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We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented. 1 INTRODUCTION The graph coloring problem is one of the most useful models in graph theory. This problem has been used to solve problems in school timetabling [10], computer register allocation [7, 8], electronic bandwidth allocation [11], and many other areas. These applications suggest that effective algorithms for solving the graph coloring problem would be of great importance. D...
Mechanizing set theory: Cardinal arithmetic and the axiom of choice
 Journal of Automated Reasoning
, 1996
"... Abstract. Fairly deep results of ZermeloFrænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this resu ..."
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Cited by 16 (9 self)
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Abstract. Fairly deep results of ZermeloFrænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, orderisomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Wellordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are
The surprise examination or unexpected hanging paradox
 American Mathematical Monthly
, 1998
"... Many mathematicians have a dismissive attitude towards paradoxes. This is unfortunate, because many paradoxes are rich in content, having connections with serious mathematical ideas as well as having pedagogical value in teaching elementary logical reasoning. An excellent example is the socalled “s ..."
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Cited by 12 (0 self)
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Many mathematicians have a dismissive attitude towards paradoxes. This is unfortunate, because many paradoxes are rich in content, having connections with serious mathematical ideas as well as having pedagogical value in teaching elementary logical reasoning. An excellent example is the socalled “surprise examination paradox ” (described below), which is an argument that seems at first to be too silly to deserve much attention. However, it has inspired an amazing variety of philosophical and mathematical investigations that have in turn uncovered links to Gödel’s incompleteness theorems, game theory, and several other logical paradoxes (e.g., the liar paradox and the sorites paradox). Unfortunately, most mathematicians are unaware of this because most of the literature has been published in philosophy journals. In this article, I describe some of this work, emphasizing the ideas that are particularly interesting mathematically. I also try to dispel some of the confusion that surrounds the paradox and plagues even the published literature. However, I do not try to correct every error or explain every idea that has ever appeared in print. Readers who want more comprehensive surveys should see [30, chapters 7 and 8], [20], and [16].
Knowledge and Communication: A FirstOrder Theory
 Artificial Intelligence
, 2005
"... This paper presents a theory of informative communications among agents that allows a speaker to communicate to a hearer truths about the state of the world; the occurrence of events, including other communicative acts; and the knowledge states of any agent — speaker, hearer, or third parties — any ..."
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Cited by 5 (2 self)
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This paper presents a theory of informative communications among agents that allows a speaker to communicate to a hearer truths about the state of the world; the occurrence of events, including other communicative acts; and the knowledge states of any agent — speaker, hearer, or third parties — any of these in the past, present, or future — and any logical combination of these, including formulas with quantifiers. We prove that this theory is consistent, and compatible with a wide range of physical theories. We examine how the theory avoids two potential paradoxes, and discuss how these paradoxes may pose a danger when this theory are extended.
A Procedural Solution to the Unexpected Hanging and Sorites Paradoxes
, 1998
"... The paradox of the Unexpected Hanging, related prediction paradoxes, and the sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads in the case of the bald man versio ..."
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Cited by 3 (0 self)
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The paradox of the Unexpected Hanging, related prediction paradoxes, and the sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads in the case of the bald man version of the sorites. The reasoning then assigns each entity a value that depends on the previously assigned value of one of the neighboring entities. The final result is paradoxical because it conflicts with the obviously correct, commonsensical value. The paradox is due to the serial procedure of assigning a value based on the newly assigned value of the neighbor. An alternative procedure is to assign each value based only on the original values of neighborsa parallel procedure. That procedure does not give paradoxical answers. 1 The Paradox A version of the wellknown Unexpected Hanging Paradox is: Having been convicted of murder, Mr. Hyde hears the judge's sentence on Thursday, April 28:...
Fractional Statistics and ChernSimons Field Theory in 2+1 Dimensions
, 1999
"... The question of anyons and fractional statistics in field theories in 2+1 dimensions with ChernSimons (CS) term is discussed in some detail. Arguments are spelled out as to why fractional statistics is only possible in two space dimensions. This phenomenon is most naturally discussed within the fra ..."
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Cited by 3 (0 self)
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The question of anyons and fractional statistics in field theories in 2+1 dimensions with ChernSimons (CS) term is discussed in some detail. Arguments are spelled out as to why fractional statistics is only possible in two space dimensions. This phenomenon is most naturally discussed within the framework of field theories with CS term, hence as a prelude to this discussion I first discuss the various properties of the CS term. In particular its role as a gauge field mass term is emphasized. In the presence of the CS term, anyons can appear in two different ways i.e. either as soliton of the corresponding field theory or as a fundamental quanta carrying fractional statistics and both approaches are elaborated in some detail.
Carnival Finite State
"... It is hard to believe that two years of serving as president of PCTM will come to an end in June. I can still remember “taking the gavel ” from Ken Lloyd in June of 1998 and looking forward to an exciting and interesting experience working with all of the wonderful people who make up the Pennsylvani ..."
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It is hard to believe that two years of serving as president of PCTM will come to an end in June. I can still remember “taking the gavel ” from Ken Lloyd in June of 1998 and looking forward to an exciting and interesting experience working with all of the wonderful people who make up the Pennsylvania Council of Teachers of Mathematics. It has truly been that kind of experience and I want to again thank everyone for giving me the opportunity to have served the organization in this way. DON SCHEUER PRESIDENT, PCTM As I look toward the end of my service, I can only hope that we are all diligently working to recruit young members of our profession to get involved in PCTM and to assume leadership roles in the future. I think that anyone who has had this experience will testify that it is probably one of the richest parts of their professional life and that it is a key ingredient in being a better classroom teacher. Our Annual Conference in Harrisburg was a whopping success and it was personally
unknown title
, 2000
"... Many mathematicians have a dismissive attitude towards paradoxes. This is unfortunate, because many paradoxes are rich in content, having connections with serious mathematical ideas as well as having pedagogical value in teaching elementary logical reasoning. An excellent example is the socalled “su ..."
Abstract
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Many mathematicians have a dismissive attitude towards paradoxes. This is unfortunate, because many paradoxes are rich in content, having connections with serious mathematical ideas as well as having pedagogical value in teaching elementary logical reasoning. An excellent example is the socalled “surprise examination paradox ” (described below), which is an argument that seems at first to be too silly to deserve much attention. However, it has inspired an amazing variety of philosophical and mathematical investigations that have in turn uncovered links to Gödel’s incompleteness theorems, game theory, and several other logical paradoxes (e.g., the liar paradox and the sorites paradox). Unfortunately, most mathematicians are unaware of this because most of the literature has been published in philosophy journals. In this article, I describe some of this work, emphasizing the ideas that are particularly interesting mathematically. I also try to dispel some of the confusion that surrounds the paradox and plagues even the published literature. However, I do not try to correct every error or explain every idea that has ever appeared in print. Readers who want more comprehensive surveys should see [30, chapters 7 and 8], [20], and [16].
Northwestern University
, 2012
"... We model demand for noninstrumental information, drawing on the idea that people derive entertainment utility from suspense and surprise. A period has more suspense if the variance of the next period’s beliefs is greater. A period has more surprise if the current belief is further from the last per ..."
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We model demand for noninstrumental information, drawing on the idea that people derive entertainment utility from suspense and surprise. A period has more suspense if the variance of the next period’s beliefs is greater. A period has more surprise if the current belief is further from the last period’s belief. Under these definitions, we analyze the optimal way to reveal information over time so as to maximize expected suspense or surprise experienced by a Bayesian audience. We apply our results to the design of mystery novels, political primaries, casinos, game shows, auctions, and sports.
Chaos and Graphics Reptiles with woven horns
"... This paper describes a simple geometric construction for the visualization of Alexander’s horned sphere as a selfsimilar fractal curve in the plane. The construction is based on a recursive rep2 rectangle progression to a specified depth. Parameterized curve generation and rendering details are bri ..."
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This paper describes a simple geometric construction for the visualization of Alexander’s horned sphere as a selfsimilar fractal curve in the plane. The construction is based on a recursive rep2 rectangle progression to a specified depth. Parameterized curve generation and rendering details are briefly discussed.