Pavol Hell and Jaroslav Nesetril
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198528173
- eISBN:
- 9780191713644
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528173.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. This text is devoted entirely to the subject, bringing ...
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Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. This text is devoted entirely to the subject, bringing together the highlights of the theory and its many applications. It looks at areas such as graph reconstruction, products, fractional and circular colourings, and constraint satisfaction problems, and has applications in complexity theory, artificial intelligence, telecommunications, and statistical physics. It has a wide focus on algebraic, combinatorial, and algorithmic aspects of graph homomorphisms. A reference list and historical summaries extend the material explicitly discussed. The book contains exercises of varying difficulty. Hints or references are provided for the more difficult exercises.Less
Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. This text is devoted entirely to the subject, bringing together the highlights of the theory and its many applications. It looks at areas such as graph reconstruction, products, fractional and circular colourings, and constraint satisfaction problems, and has applications in complexity theory, artificial intelligence, telecommunications, and statistical physics. It has a wide focus on algebraic, combinatorial, and algorithmic aspects of graph homomorphisms. A reference list and historical summaries extend the material explicitly discussed. The book contains exercises of varying difficulty. Hints or references are provided for the more difficult exercises.
Geoffrey Grimmett and Colin McDiarmid (eds)
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.001.0001
- Subject:
- Mathematics, Probability / Statistics
Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and ...
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Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.Less
Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.
Jason Rosenhouse and Jennifer Beineke (eds)
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691171920
- eISBN:
- 9781400889136
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691171920.001.0001
- Subject:
- Mathematics, History of Mathematics
The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, ...
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The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books, research in recreational mathematics has often been neglected. This book returns with a brand-new compilation of fascinating problems and solutions in recreational mathematics. It gathers together the top experts in recreational math and presents a compelling look at board games, card games, dice, toys, computer games, and much more. The book is divided into five parts: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. Readers will discover what origami, roulette wheels, and even the game of Trouble can teach about math. Chapters contain new results, and include short expositions on the topic's background, providing a framework for understanding the relationship between serious mathematics and recreational games. Mathematical areas explored include combinatorics, logic, graph theory, linear algebra, geometry, topology, computer science, operations research, probability, game theory, and music theory.Less
The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books, research in recreational mathematics has often been neglected. This book returns with a brand-new compilation of fascinating problems and solutions in recreational mathematics. It gathers together the top experts in recreational math and presents a compelling look at board games, card games, dice, toys, computer games, and much more. The book is divided into five parts: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. Readers will discover what origami, roulette wheels, and even the game of Trouble can teach about math. Chapters contain new results, and include short expositions on the topic's background, providing a framework for understanding the relationship between serious mathematics and recreational games. Mathematical areas explored include combinatorics, logic, graph theory, linear algebra, geometry, topology, computer science, operations research, probability, game theory, and music theory.
Robin Wilson and John J. Watkins (eds)
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics
The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to ...
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The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: it constitutes the first book-length survey of the history of combinatorics, and it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th-century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections.Less
The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: it constitutes the first book-length survey of the history of combinatorics, and it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th-century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections.
Christophe Reutenauer
- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.001.0001
- Subject:
- Mathematics, Pure Mathematics
Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It ...
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Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.Less
Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.
Răzvan Gheorghe Gurău
- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198787938
- eISBN:
- 9780191829918
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198787938.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology
This book presents a self-contained, ab initio introduction to random tensors. The book is divided into two parts. The first part introduces the general framework and the main results on random ...
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This book presents a self-contained, ab initio introduction to random tensors. The book is divided into two parts. The first part introduces the general framework and the main results on random tensors. The second part presents in detail specific examples of random tensors models. The book presents both asymptotic results (or perturbative, in the physics literature) and constructive (non perturbative) results in full detail. The book is suitable for readers unfamiliar with the field. The material presented is divided into three broad categories of results. The first category connects random tensors to topological spaces, Euclidean dynamical triangulations and random geometry. The second category consists of perturbative results on random tensors. It contains the 1/N expansion, the enumeration of graphs of fixed degree, the continuum limit, the double scaling limit as well as the study of phase transitions and symmetry breaking in tensor models. The results in the third category are non perturbative: the proof of the universality of Gaussian tensor measures and the construction of quartically perturbed Gaussian measure. These results are obtained using methods from enumerative combinatorics, probability theory and constructive field theory. Random tensors generalize random matrices and provide a framework for the study of random geometries in any dimension relevant for conformal field theory, statistical physics and quantum gravity.Less
This book presents a self-contained, ab initio introduction to random tensors. The book is divided into two parts. The first part introduces the general framework and the main results on random tensors. The second part presents in detail specific examples of random tensors models. The book presents both asymptotic results (or perturbative, in the physics literature) and constructive (non perturbative) results in full detail. The book is suitable for readers unfamiliar with the field. The material presented is divided into three broad categories of results. The first category connects random tensors to topological spaces, Euclidean dynamical triangulations and random geometry. The second category consists of perturbative results on random tensors. It contains the 1/N expansion, the enumeration of graphs of fixed degree, the continuum limit, the double scaling limit as well as the study of phase transitions and symmetry breaking in tensor models. The results in the third category are non perturbative: the proof of the universality of Gaussian tensor measures and the construction of quartically perturbed Gaussian measure. These results are obtained using methods from enumerative combinatorics, probability theory and constructive field theory. Random tensors generalize random matrices and provide a framework for the study of random geometries in any dimension relevant for conformal field theory, statistical physics and quantum gravity.
Kanchan Chandra (ed.)
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199893157
- eISBN:
- 9780199980079
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199893157.001.0001
- Subject:
- Political Science, Comparative Politics
The book is motivated by a disjuncture in social science research on ethnicity, politics and economics: Although theories of the formation of ethnic groups are driven by the constructivist assumption ...
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The book is motivated by a disjuncture in social science research on ethnicity, politics and economics: Although theories of the formation of ethnic groups are driven by the constructivist assumption that ethnic identities can change over time, theories of the effect of ethnicity on economic and political outcomes are driven by the primordialist assumption that these identities are fixed. This book is a first cut at building—and rebuilding—our theories of politics and economics on a fortified constructivist foundation. It proposes a new conceptual framework for thinking about ethnic identity. It uses this framework to synthesize constructivist arguments into a set of testable propositions about how and why ethnic identities change. It translates this framework—and the propositions derived from it—into a new, combinatorial language. And it employs these conceptual, constructivist, and combinatorial tools to theorize about the relationship between ethnicity, politics and economics using a variety of methods. Taking the possibility of change in ethnic identity into account, this book shows, dismantles the theoretical logics linking ethnic diversity to negative outcomes and processes such as democratic destabilization, clientelism, riots and state collapse. Even more importantly, it changes the questions we can ask about the relationship between ethnicity, politics and economicsLess
The book is motivated by a disjuncture in social science research on ethnicity, politics and economics: Although theories of the formation of ethnic groups are driven by the constructivist assumption that ethnic identities can change over time, theories of the effect of ethnicity on economic and political outcomes are driven by the primordialist assumption that these identities are fixed. This book is a first cut at building—and rebuilding—our theories of politics and economics on a fortified constructivist foundation. It proposes a new conceptual framework for thinking about ethnic identity. It uses this framework to synthesize constructivist arguments into a set of testable propositions about how and why ethnic identities change. It translates this framework—and the propositions derived from it—into a new, combinatorial language. And it employs these conceptual, constructivist, and combinatorial tools to theorize about the relationship between ethnicity, politics and economics using a variety of methods. Taking the possibility of change in ethnic identity into account, this book shows, dismantles the theoretical logics linking ethnic diversity to negative outcomes and processes such as democratic destabilization, clientelism, riots and state collapse. Even more importantly, it changes the questions we can ask about the relationship between ethnicity, politics and economics
Kanchan Chandra
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199893157
- eISBN:
- 9780199980079
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199893157.003.0001
- Subject:
- Political Science, Comparative Politics
This chapter provides a complete sketch of the key concepts and arguments introduced in this book as well as the stakes attached to each. It introduces the conceptual framework for thinking about ...
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This chapter provides a complete sketch of the key concepts and arguments introduced in this book as well as the stakes attached to each. It introduces the conceptual framework for thinking about ethnic identity and ethnic identity change proposed in this book. It shows how this framework synthesizes constructivist arguments into a set of testable and logically connected propositions about why and how ethnic identities change; how this framework—and the propositions derived from it—into a new, combinatorial language; and how these conceptual, constructivist, and combinatorial tools are deployed to theorize about the relationship between ethnicity, politics and economics using a variety of methods.Less
This chapter provides a complete sketch of the key concepts and arguments introduced in this book as well as the stakes attached to each. It introduces the conceptual framework for thinking about ethnic identity and ethnic identity change proposed in this book. It shows how this framework synthesizes constructivist arguments into a set of testable and logically connected propositions about why and how ethnic identities change; how this framework—and the propositions derived from it—into a new, combinatorial language; and how these conceptual, constructivist, and combinatorial tools are deployed to theorize about the relationship between ethnicity, politics and economics using a variety of methods.
Stephen Connelly
- Published in print:
- 2021
- Published Online:
- September 2021
- ISBN:
- 9781474418065
- eISBN:
- 9781474495738
- Item type:
- chapter
- Publisher:
- Edinburgh University Press
- DOI:
- 10.3366/edinburgh/9781474418065.003.0006
- Subject:
- Law, Legal History
To investigate the relationship between the possible and law, Leibniz’s early combinatorial work, replete with legal examples is examined in technical detail. The chapter argues that Leibniz regards ...
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To investigate the relationship between the possible and law, Leibniz’s early combinatorial work, replete with legal examples is examined in technical detail. The chapter argues that Leibniz regards combinatorial activity as the sole way in which human mind operates, and thus regards it as an Urdoxa in the senses of Husserl and Deleuze.Less
To investigate the relationship between the possible and law, Leibniz’s early combinatorial work, replete with legal examples is examined in technical detail. The chapter argues that Leibniz regards combinatorial activity as the sole way in which human mind operates, and thus regards it as an Urdoxa in the senses of Husserl and Deleuze.
Roi Wagner, Naomi Aradi, Avinoam Baraness, David Garber, Stela Segev, Shai Simonson, Ilana Wartenberg, Menso Folkerts, Barnabas Hughes, Roi Wagner, and J. Lennart Berggren
Victor J. Katz (ed.)
- Published in print:
- 2016
- Published Online:
- January 2018
- ISBN:
- 9780691156859
- eISBN:
- 9781400883202
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156859.003.0003
- Subject:
- Mathematics, History of Mathematics
This chapter covers mathematics written in Hebrew between the eleventh and sixteenth centuries in Europe. It starts with the practical and scholarly—as well as earlier and later—Hebrew expositions of ...
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This chapter covers mathematics written in Hebrew between the eleventh and sixteenth centuries in Europe. It starts with the practical and scholarly—as well as earlier and later—Hebrew expositions of arithmetic, from Ibn Ezra's foundational twelfth-century The Book of Number, to Levi ben Gershon's early-fourteenth-century arithmetic. The chapter then follows with two discussions of combinatorics: Ibn Ezra's calculations of the number of possible conjunctions of a given number of planets from among the seven planets, and Ben Gershon's abstract and general discussion of permutations and combinations. Finally, this chapter discusses two important treatises that summarize geometric knowledge in a semi-practical style as well as measurements in a religious context.Less
This chapter covers mathematics written in Hebrew between the eleventh and sixteenth centuries in Europe. It starts with the practical and scholarly—as well as earlier and later—Hebrew expositions of arithmetic, from Ibn Ezra's foundational twelfth-century The Book of Number, to Levi ben Gershon's early-fourteenth-century arithmetic. The chapter then follows with two discussions of combinatorics: Ibn Ezra's calculations of the number of possible conjunctions of a given number of planets from among the seven planets, and Ben Gershon's abstract and general discussion of permutations and combinations. Finally, this chapter discusses two important treatises that summarize geometric knowledge in a semi-practical style as well as measurements in a religious context.
Roi Wagner
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780691171715
- eISBN:
- 9781400883783
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691171715.003.0005
- Subject:
- Mathematics, History of Mathematics
This chapter presents two case studies that highlight the problems of mathematical semiosis: how mathematical signs obtain and change their senses. The first case study follows the paradigmatic ...
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This chapter presents two case studies that highlight the problems of mathematical semiosis: how mathematical signs obtain and change their senses. The first case study follows the paradigmatic mathematical sign, x, as it is used in applications of powers series to combinatorics via generating functions. The second case study concerns gender role stereotypes involving the so-called “stable marriage problem.” Both case studies open up questions of how meaning is transferred within and across mathematical contexts and try to substantiate the book's claims about interpretation, formalization, and constraints over mathematical objects and statements. The chapter also considers gender-neutral mathematical language in the context of sexuality.Less
This chapter presents two case studies that highlight the problems of mathematical semiosis: how mathematical signs obtain and change their senses. The first case study follows the paradigmatic mathematical sign, x, as it is used in applications of powers series to combinatorics via generating functions. The second case study concerns gender role stereotypes involving the so-called “stable marriage problem.” Both case studies open up questions of how meaning is transferred within and across mathematical contexts and try to substantiate the book's claims about interpretation, formalization, and constraints over mathematical objects and statements. The chapter also considers gender-neutral mathematical language in the context of sexuality.
E. KEITH LLOYD
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.003.0013
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics
By the end of the 18th century, what is now called ‘enumerative combinatorics’ was emerging as a distinct discipline. The connections between certain combinatorial problems and algebraical expansions ...
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By the end of the 18th century, what is now called ‘enumerative combinatorics’ was emerging as a distinct discipline. The connections between certain combinatorial problems and algebraical expansions had already been recognized, but were now more extensively exploited. In the 20th century the theory of permutation groups was successfully used to solve many enumeration problems.Less
By the end of the 18th century, what is now called ‘enumerative combinatorics’ was emerging as a distinct discipline. The connections between certain combinatorial problems and algebraical expansions had already been recognized, but were now more extensively exploited. In the 20th century the theory of permutation groups was successfully used to solve many enumeration problems.
PETER J. CAMERON
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.003.0016
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics
This chapter presents a quick overview of the recent development of combinatorics and its current directions — as a discipline in its own right, as a part of mathematics and, more generally, as a ...
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This chapter presents a quick overview of the recent development of combinatorics and its current directions — as a discipline in its own right, as a part of mathematics and, more generally, as a part of science and of society.Less
This chapter presents a quick overview of the recent development of combinatorics and its current directions — as a discipline in its own right, as a part of mathematics and, more generally, as a part of science and of society.
Barbara Maria Stafford
- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780226630489
- eISBN:
- 9780226630656
- Item type:
- chapter
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226630656.003.0003
- Subject:
- Philosophy, Philosophy of Mind
Carnal knowledge contravenes body/mind dualism. The repetitive horizontal inlay presents an ancient binding format relying on compressed color as an invitation to sensory response. This essay argues ...
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Carnal knowledge contravenes body/mind dualism. The repetitive horizontal inlay presents an ancient binding format relying on compressed color as an invitation to sensory response. This essay argues that the variants of artisanal inlay {mosaics, intarsia, grids, even twitter snatches, and pointilliste memes} are chief among erotic types of compositional ordering. These initially hand-made, tightly interlocked, and obsessively repetitive patterns mime an accumulating, self-generating intensity of feelings. It’s notable that supposedly cooly abstract strains of Modern Art seductively explore this self-assembling, pavement-like combinatoric and the sensuousness of its varicolored tesselation. Consider Jasper Johns’ shadow-steeped crazy-work flagstone paths, David Hockney’s wet poolside terrazzo parquet, or Ellsworth Kelly’s ecstatic arrays of spectral light. Here, however, the central focus alights on a more sultry integration of perception, thought, and feeling into a single sensory-motor concept. Neither material symbol nor passionate artifact, but both, Cy Twombly’s sumptuous and worldly Peony Blossom Paintings (2007) presents the lateralized stages of an overwhelming physical relationship. This immersive, large-scale series configures the changing situation of desire from irresistable arousal, to entangled climax, to ebbing flicker, all exhibited by means of a chromatic format characterized by haiku containment.Less
Carnal knowledge contravenes body/mind dualism. The repetitive horizontal inlay presents an ancient binding format relying on compressed color as an invitation to sensory response. This essay argues that the variants of artisanal inlay {mosaics, intarsia, grids, even twitter snatches, and pointilliste memes} are chief among erotic types of compositional ordering. These initially hand-made, tightly interlocked, and obsessively repetitive patterns mime an accumulating, self-generating intensity of feelings. It’s notable that supposedly cooly abstract strains of Modern Art seductively explore this self-assembling, pavement-like combinatoric and the sensuousness of its varicolored tesselation. Consider Jasper Johns’ shadow-steeped crazy-work flagstone paths, David Hockney’s wet poolside terrazzo parquet, or Ellsworth Kelly’s ecstatic arrays of spectral light. Here, however, the central focus alights on a more sultry integration of perception, thought, and feeling into a single sensory-motor concept. Neither material symbol nor passionate artifact, but both, Cy Twombly’s sumptuous and worldly Peony Blossom Paintings (2007) presents the lateralized stages of an overwhelming physical relationship. This immersive, large-scale series configures the changing situation of desire from irresistable arousal, to entangled climax, to ebbing flicker, all exhibited by means of a chromatic format characterized by haiku containment.
Barbara Maria Stafford
- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780226630489
- eISBN:
- 9780226630656
- Item type:
- chapter
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226630656.003.0006
- Subject:
- Philosophy, Philosophy of Mind
Since Kant, there has been an intense debate around the question of the location of the Sublime: Is it in the object or in the subject? This essay tackles ineffability’s dark side: the fast-growing, ...
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Since Kant, there has been an intense debate around the question of the location of the Sublime: Is it in the object or in the subject? This essay tackles ineffability’s dark side: the fast-growing, even ubiquitous, socio-cultural phenomenon of not being able to speak or image what we create. These range from a dubious contemporary Cabinet of Wonders stocked with the unnameable products of transplant surgery, to the undecideables emerging from Synthetic Biology Laboratories, to the science-fiction Outback of wetware BioArt. If Neoplatonism linked ineffability with ultimate matters and mattering, such as awe in the presence of the sublime unity of God, light, cosmic harmony, the concept has now descended into shambling unspeakability. This current inexpressibility, evident in the prevalence of the popular non-descriptor, “stuff,” aligns with the imagistic, linguistic, and ontological inability to configure a rising tide of bewildering biological and online entities. These confounding products of an extreme combinatorial laboratory science, technology, and art are not only without a concept, but without the possibility of a concept.Less
Since Kant, there has been an intense debate around the question of the location of the Sublime: Is it in the object or in the subject? This essay tackles ineffability’s dark side: the fast-growing, even ubiquitous, socio-cultural phenomenon of not being able to speak or image what we create. These range from a dubious contemporary Cabinet of Wonders stocked with the unnameable products of transplant surgery, to the undecideables emerging from Synthetic Biology Laboratories, to the science-fiction Outback of wetware BioArt. If Neoplatonism linked ineffability with ultimate matters and mattering, such as awe in the presence of the sublime unity of God, light, cosmic harmony, the concept has now descended into shambling unspeakability. This current inexpressibility, evident in the prevalence of the popular non-descriptor, “stuff,” aligns with the imagistic, linguistic, and ontological inability to configure a rising tide of bewildering biological and online entities. These confounding products of an extreme combinatorial laboratory science, technology, and art are not only without a concept, but without the possibility of a concept.
DONALD E. KNUTH
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.003.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics
Early work on the generation of combinatorial patterns began as civilization itself was taking shape. The story is quite fascinating, and we will see that it spans many cultures in many parts of the ...
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Early work on the generation of combinatorial patterns began as civilization itself was taking shape. The story is quite fascinating, and we will see that it spans many cultures in many parts of the world, with ties to poetry, music, and religion. There is space here to discuss only some of the principal highlights; but perhaps a few glimpses into the past will stimulate us to dig deeper into the roots of the subject, as the world gets ever smaller and as global scholarship continues to advance.Less
Early work on the generation of combinatorial patterns began as civilization itself was taking shape. The story is quite fascinating, and we will see that it spans many cultures in many parts of the world, with ties to poetry, music, and religion. There is space here to discuss only some of the principal highlights; but perhaps a few glimpses into the past will stimulate us to dig deeper into the roots of the subject, as the world gets ever smaller and as global scholarship continues to advance.
Matt Clay
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0004
- Subject:
- Mathematics, Geometry / Topology
This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show ...
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This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.Less
This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.
Tomas Riad
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199543571
- eISBN:
- 9780191747168
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199543571.003.0010
- Subject:
- Linguistics, Phonetics / Phonology, Language Families
A major theme of this book is the lexcial specification of prosodic information, and the consequences that this has for the understanding of word formation and the shape of the lexicon in Swedish. It ...
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A major theme of this book is the lexcial specification of prosodic information, and the consequences that this has for the understanding of word formation and the shape of the lexicon in Swedish. It also help for the understanding of the stress system as such. Thus, there are four types of morphemes: tonic (i.e. stressed), pretonic, posttonic and unspecified. Given the previously introduced constraints on Culminativity within the minimal prosodic word, this system now predicts that certain combinations of morphemes should be better formed than others, and those predictions are shown to be borne out. Where no stress is in place, a phonological routine will provide stress at the right edge of the word, where however a posttonic morpheme can steer stress to the preceding syllable. This system explains the pattern of how “Germanic” and “foreign” morphemes combine or don’t combine in Swedish, as a purely prosodic phenomenon.Less
A major theme of this book is the lexcial specification of prosodic information, and the consequences that this has for the understanding of word formation and the shape of the lexicon in Swedish. It also help for the understanding of the stress system as such. Thus, there are four types of morphemes: tonic (i.e. stressed), pretonic, posttonic and unspecified. Given the previously introduced constraints on Culminativity within the minimal prosodic word, this system now predicts that certain combinations of morphemes should be better formed than others, and those predictions are shown to be borne out. Where no stress is in place, a phonological routine will provide stress at the right edge of the word, where however a posttonic morpheme can steer stress to the preceding syllable. This system explains the pattern of how “Germanic” and “foreign” morphemes combine or don’t combine in Swedish, as a purely prosodic phenomenon.
Daniel V. Schroeder
- Published in print:
- 2021
- Published Online:
- March 2021
- ISBN:
- 9780192895547
- eISBN:
- 9780191915000
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192895547.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Why are so many large-scale processes irreversible, happening in one direction but not the other as time passes? This chapter answers that question using three simple model systems: a collection of ...
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Why are so many large-scale processes irreversible, happening in one direction but not the other as time passes? This chapter answers that question using three simple model systems: a collection of two-state particles such as flipped coins or elementary magnetic dipoles; the Einstein model of a solid as a collection of identical quantum oscillators; and a monatomic ideal gas such as helium or argon. For each system we learn to calculate the multiplicity: the number of possible microscopic arrangements. Taking the logarithm of the multiplicity gives the entropy. And the laws of probability then imply the second law of thermodynamics: Entropy tends to increase.Less
Why are so many large-scale processes irreversible, happening in one direction but not the other as time passes? This chapter answers that question using three simple model systems: a collection of two-state particles such as flipped coins or elementary magnetic dipoles; the Einstein model of a solid as a collection of identical quantum oscillators; and a monatomic ideal gas such as helium or argon. For each system we learn to calculate the multiplicity: the number of possible microscopic arrangements. Taking the logarithm of the multiplicity gives the entropy. And the laws of probability then imply the second law of thermodynamics: Entropy tends to increase.
Paul Balister, Béla Bollobás, and Bhargav Narayanan
- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198809456
- eISBN:
- 9780191847073
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198809456.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The chapter “Reconstructing Random Jigsaws” examines the reconstruction problem for a family of discrete structures, asking whether it is possible to uniquely reconstruct a structure in this family ...
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The chapter “Reconstructing Random Jigsaws” examines the reconstruction problem for a family of discrete structures, asking whether it is possible to uniquely reconstruct a structure in this family from the “deck” of all its substructures of some fixed size. Reconstruction problems involving combinatorics and randomness have a very rich history. The oldest such problem is perhaps the graph reconstruction conjecture of Kelly and Ulam; analogous questions include reconstructing finite sets satisfying symmetry conditions, reconstructing finite abelian groups, and reconstructing finite subsets of the plane. A natural line of enquiry is to ask how the answer to the reconstruction problem changes when it is necessary to reconstruct a typical (as opposed to an arbitrary) structure in a family of discrete structures. This chapter presents a theoretical case study of interest for all the complex architectures of networks: a reconstruction problem connected with DNA sequencing via the shotgun-sequencing technique.Less
The chapter “Reconstructing Random Jigsaws” examines the reconstruction problem for a family of discrete structures, asking whether it is possible to uniquely reconstruct a structure in this family from the “deck” of all its substructures of some fixed size. Reconstruction problems involving combinatorics and randomness have a very rich history. The oldest such problem is perhaps the graph reconstruction conjecture of Kelly and Ulam; analogous questions include reconstructing finite sets satisfying symmetry conditions, reconstructing finite abelian groups, and reconstructing finite subsets of the plane. A natural line of enquiry is to ask how the answer to the reconstruction problem changes when it is necessary to reconstruct a typical (as opposed to an arbitrary) structure in a family of discrete structures. This chapter presents a theoretical case study of interest for all the complex architectures of networks: a reconstruction problem connected with DNA sequencing via the shotgun-sequencing technique.
