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Introduction

Kanchan Chandra

in Constructivist Theories of Ethnic 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 ... More

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

Mathematics in Hebrew in Medieval Europe

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.)

in Sourcebook in the Mathematics of Medieval Europe and North Africa

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 ... More

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

Two Case Studies of Semiosis in Mathematics

Roi Wagner

in Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice

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 ... More

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

Enumeration (18th–20th centuries)

E. KEITH LLOYD

in Combinatorics: Ancient and Modern

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 ... More

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

A Personal View of Combinatorics

PETER J. CAMERON

in Combinatorics: Ancient and Modern

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 ... More

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

Lying Side by Side: Fitting Color to Eros

Barbara Maria Stafford

in Ribbon of Darkness: Inferencing from the Shadowy Arts and Sciences

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 ... More

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

From Communicable Matter to Incommunicable “Stuff”: Extreme Combinatorics and the Return of Ineffability

Barbara Maria Stafford

in Ribbon of Darkness: Inferencing from the Shadowy Arts and Sciences

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, ... More

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

Two Thousand Years of Combinatorics

DONALD E. KNUTH

in Combinatorics: Ancient and Modern

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 ... More

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

Free Groups and Folding

Matt Clay

in Office Hours with a Geometric Group Theorist

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 ... More

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

Prosodic status of morphemes in the lexicon: stress

Tomas Riad

in The Phonology of Swedish

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 ... More

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

Reconstructing Random Jigsaws

Paul Balister, Béla Bollobás, and Bhargav Narayanan

in Multiplex and Multilevel Networks

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 ... More

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

Free probability

Alice Guionnet

in Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School: Volume 104, July 2015

Free probability was introduced by D. Voiculescu as a theory of noncommutative random variables (similar to integration theory) equipped with a notion of freeness very similar to independence. In ... More

Free probability was introduced by D. Voiculescu as a theory of noncommutative random variables (similar to integration theory) equipped with a notion of freeness very similar to independence. In fact, it is possible in this framework to define the natural ‘free’ counterpart of the central limit theorem, Gaussian distribution, Brownian motion, stochastic differential calculus, entropy, etc. It also appears as the natural setup for studying large random matrices as their size goes to infinity and hence is central in the study of random matrices as their size go to infinity. In this chapter the free probability framework is introduced, and it is shown how it naturally shows up in the random matrices asymptotics via the so-called ‘asymptotic freeness’. The connection with combinatorics and the enumeration of planar maps, including loop models, are discussed.Less

The Probabilistic Turn (1876–1884)

Olivier Darrigol

in Atoms, Mechanics, and Probability: Ludwig Boltzmann's Statistico-Mechanical Writings - An Exegesis

This chapter deals with writings in which Boltzmann expressed the statistical nature of the entropy law and temporarily made the relation between entropy and combinatorial probability a basic ... More

This chapter deals with writings in which Boltzmann expressed the statistical nature of the entropy law and temporarily made the relation between entropy and combinatorial probability a basic constructive tool of his theory. In 1881, he discovered that this relation derived from what we now call the microcanonical distribution, and he approved Maxwell’s recent foundation of the equilibrium problem on the microcanonical ensemble. Boltzmann also kept working on problems he had tackled in earlier years. He proposed a new solution to the problem of specific heats, and he performed enormous calculations for the viscosity and diffusion coefficients in the hard-ball model. In a lighter genre, he conceived a new way of determining molecular sizes, and he speculated on a gas model in which the molecular forces would be entirely attractive.Less