Search Results

Finding the Roots of Algebraic Equations

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

This chapter highlights some of the struggles to solve fifth-degree polynomials as well as the development of the idea of a group and its application to answering the question of solvability of ... More

This chapter highlights some of the struggles to solve fifth-degree polynomials as well as the development of the idea of a group and its application to answering the question of solvability of polynomial equations. It begins by surveying some of the methods by which mathematicians attempted to solve—and sometimes did solve—various algebraic equations of degree higher than four. The chapter then turns to the theory of permutations as well as group theory. It turned out that the key to proving the impossibility of the fifth-degree polynomials and, more generally, the key to determining which polynomial equations were, in fact, solvable algebraically, was a totally new construct, the group.Less

Human Groups and Genocide

Dr. David Nersessian

in Genocide and Political Groups

This chapter analyses whether political groups merit equal treatment to the four groups enumerated in the Genocide Convention. It first discusses what human ‘groups’ really are and the inherent ... More

This chapter analyses whether political groups merit equal treatment to the four groups enumerated in the Genocide Convention. It first discusses what human ‘groups’ really are and the inherent difficulties of defining them for legal and social purposes. It details the linkages and interactions between international law and broader understandings of human groups as social collectives. It then proposes a new theoretical understanding of genocide as a rights-based offence. This theory links the legal protection of groups to certain individual rights of a collective nature that give rise to the group's existence in this context. This sets the stage for the subsequent analysis demonstrating that political collectives have many similar characteristics to the existing four groups, such that they likewise deserve specific protection from physical and biological destruction ‘as such’ through a separate crime of political genocide.Less

CONCRETE GROUP THEORY

A. A. Ivanov

in The Fourth Janko Group

This chapter discusses concrete group theory. Topics covered include symplectic and orthogonal GF(2)-forms, transvections and Siegel transformations, Witt's theorem, and the space of forms.

GROUP ORBITALLY CONSTRAINED APPROXIMATION

Moody T. Chu and Gene H. Golub

in Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

Group actions together with properly formulated objective functions can offer a channel to tackle various classical or new and challenging problems rising from applied linear algebra. This chapter ... More

Group actions together with properly formulated objective functions can offer a channel to tackle various classical or new and challenging problems rising from applied linear algebra. This chapter presents a framework to bring together the notions of group theory, linear transformations, and dynamical systems as a tool to undertake the task of system identification by canonical forms.Less

Judges as Members of Small Groups

Wendy L. Martinek

in The Psychology of Judicial Decision Making

Virtually all appellate courts use groups of judges to decide cases. Accordingly, small group theory is a useful perspective for investigating appellate court decision making. In particular, ... More

Virtually all appellate courts use groups of judges to decide cases. Accordingly, small group theory is a useful perspective for investigating appellate court decision making. In particular, scholarship about group cognition can help make sense of when we can expect judges to rely on legal (e.g., precedent) versus nonlegal (e.g., political preference) factors in the choices they make. In addition, research on the roles and status of group members in relationship to leadership and conformity can help us understand when certain formal (e.g., chief judge) and informal (e.g., freshman judge) roles are likely to structure judicial choice. More generally, placing appellate courts in a small group framework can aid in better understanding the quality of adjudication by focusing on how groups aggregate the expertise of individual group members to produce group outcomes.Less

Frenkel Excitonic States in The Heitler-London Approximation

Vladimir M. Agranovich

in Excitations in Organic Solids

This chapter discusses Frenkel excitonic states in Heitler-London approximation. The Coulomb and mechanical excitons in molecular crystals are introduced using the crystal Hamiltonian. The excitons ... More

This chapter discusses Frenkel excitonic states in Heitler-London approximation. The Coulomb and mechanical excitons in molecular crystals are introduced using the crystal Hamiltonian. The excitons symmetry properties are considered via group-theoretical methods. Applications to crystals of naphthalene type are given.Less

Broken Symmetry and Physicists' QFT

Laura Ruetsche

in Interpreting Quantum Theories

Chapter 12 argued that quantum statistical mechanics puts unitarily inequivalent representations to use in ways no rigid interpretation can make sense of. Two features of working QFTs which promise a ... More

Chapter 12 argued that quantum statistical mechanics puts unitarily inequivalent representations to use in ways no rigid interpretation can make sense of. Two features of working QFTs which promise a quantum field theoretic realization of Chapter 12's argument are Goldstone bosons and the Higgs mechanism. This chapter explains why they're promising by presenting them as instance of broken symmetry. Then it tempers the promise by admitting that the working QFTs in which these features occur are less mathematically explicit than they need to be to persuasively realize the argument of Chapter 12. The chapter closes by extracting from this very circumstance a non-conclusive reason to lend the argument of Chapter 12 interpretive weight. The reason is that our best theories of physics are still under construction, and their successors could share with the models presented in Chapter 12 the features on which the argument of Chapter 12 hinged.Less

Implementation of the renormalization group

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the ... More

Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.Less

Cohesion Theory: Explaining a Military’s Will to Fight

Jasen J. Castillo

in Endurance and War: The National Sources of Military Cohesion

In this chapter cohesion theory is introduced to explain why national militaries differ in two important ways: (1) the ability of most units to fight hard on the battlefield and, (2) the capacity of ... More

In this chapter cohesion theory is introduced to explain why national militaries differ in two important ways: (1) the ability of most units to fight hard on the battlefield and, (2) the capacity of the armed forces as a whole to avoid disintegration when facing the possibility of losing a war. The first section defines military cohesion, the study's dependent variable. The second section outlines the collective action problem associated with motivating groups to sustain cooperation even though individual members risk high costs to achieve a common goal. Fighting in a war represents an extreme example of collective action. This chapter explains that group cohesion depends on the ability of a group's leaders to promote and enforce norms of unconditional loyalty. This core idea applies to groups of any size and function, but will be applied to a country's armed forces in subsequent chaptersLess

Communist Neo-Traditionalism: An Introductory Essay

Andrew G. Walder

in Communist Neo-Traditionalism: Work and Authority in Chinese Industry

The neo-traditional image of communist society differs fundamentally from the images of totalitarianism and group theory. It shares with the totalitarian idea a focus on the distinctive communist ... More

The neo-traditional image of communist society differs fundamentally from the images of totalitarianism and group theory. It shares with the totalitarian idea a focus on the distinctive communist institutions that foster organized political control, and it shares the premise that these forms of organization shape patterns of association and political behavior in distinctive ways. The neo-traditional image stresses the social network as its main structural concept. The elements of workplace organization—generic features of modern communism—give rise to several other features of factory political life and authority relations that complete the definition of the type. Communist neo-traditionalism guides comparisons of industrial authority by focusing on organized dependence and institutional culture of the factory. The chapters included in this book explore the characteristics of neo-traditionalism in Chinese factories. They also determine the ways that the Chinese variant has diverged from the Soviet.Less

Applying New Mathematics: Group Theory and Quantum Mechanics

Otávio Bueno and Steven French

in Applying Mathematics: Immersion, Inference, Interpretation

Here we present our first case study: the introduction of group theory into quantum mechanics in the 1920s and 1930s. It is helpful in this context to distinguish the ‘Weyl’ and ‘Wigner’ programmes, ... More

Here we present our first case study: the introduction of group theory into quantum mechanics in the 1920s and 1930s. It is helpful in this context to distinguish the ‘Weyl’ and ‘Wigner’ programmes, where the former is concerned with using group theory to provide secure foundations for the emerging quantum physics and the latter emphasizes its practical applications. We suggest the application of the mathematics to the physics depended on certain structural ‘bridges’ within the mathematics itself and also that both this mathematics and the physics were in a state of flux. Given those features, we argue that the partial structures approach offers a suitable framework for representing these developments. One can resist Steiner’s claim that the mathematics is doing all the work in these cases, as it is only because of prior idealizing moves on the physics side that the mathematics can be brought into play to begin with.Less

Democratic expectations: the representation account

Darren Halpin

in Groups, Representation and Democracy: Between Promise and Practice

This chapter scrutinizes the dominant lens through which the democratic expectations of groups are viewed. It argues that group scholars have implicitly, but sometimes explicitly, viewed groups as ... More

This chapter scrutinizes the dominant lens through which the democratic expectations of groups are viewed. It argues that group scholars have implicitly, but sometimes explicitly, viewed groups as though they were all engaged in democratic representation. In turn, this has informed expectations that groups should be democratically accountable to their affiliates. Where group practices have not conformed to this picture, this is read as a group deficiency. The orthodox frameworks and lenses used to appraise groups have supported a rather pessimistic view of the potential for groups to be agents of democracy, because they do not practise it themselves. The chapter reviews a range of arguments that seek to weaken or outright challenge this representation account. It concludes that group theory is poorly served by an insistence on a relatively constrained account of representation.Less

Holistic Darwinism: Synergistic Selection and the Evolutionary Process

Peter A. Corning

in Holistic Darwinism: Synergy, Cybernetics, and the Bioeconomics of Evolution

This chapter presents a critique of Neo-Darwinism and the case for Holistic Darwinism. It discusses the realization of the founding fathers of modern genetics and population biology concerning the ... More

This chapter presents a critique of Neo-Darwinism and the case for Holistic Darwinism. It discusses the realization of the founding fathers of modern genetics and population biology concerning the flaws of Neo-Darwinism, particularly in relation to group selection theory. These attacks on group selection theory began with William D. Hamilton's The Genetical Evolution of Social Behavior and was fully elaborated in George C. Williams' Adaptation and Natural Selection. This chapter also highlights the key features of Holistic Darwinism and its theoretical superiority over Neo-Darwinism.Less

Grid-Group Theory and Corporal Punishment

Jean Giles-Sims and Charles Lockhart

in Corporal Punishment of Children in Theoretical Perspective

This chapter demonstrates the utility of one specific theory of culture—grid-group theory—for explaining variations in parental disciplinary practices. Accordingly, this chapter introduces grid-group ... More

This chapter demonstrates the utility of one specific theory of culture—grid-group theory—for explaining variations in parental disciplinary practices. Accordingly, this chapter introduces grid-group theory, derives profiles of four distinct parenting orientations from grid-group theory, explores empirical support for these distinctive cultural orientations toward corporal punishment, briefly discusses the policy implications of grid-group theory with respect to reducing parental use of corporal punishment, and suggests priorities for future research. Grid-group theory, in short, is a means for explaining the origins of social preferences. It offers a means for deriving a diverse but limited number of cultural types based on persons' answers to basic social questions. Grid-group theorists argue that persons' answers to these questions depend on their beliefs and values with respect to two fundamental social dimensions: the need for and legitimacy of external prescription to control behavior (grid) and the desirability of affiliation with others (group).Less

The Presentation of Objects and the Representation of Structure

Steven French

in The Structure of the World: Metaphysics and Representation

Here the commitments of OSR are clarified by drawing on the following useful distinction: the structure of the world is presented to us in the theoretical context under consideration by means of the ... More

Here the commitments of OSR are clarified by drawing on the following useful distinction: the structure of the world is presented to us in the theoretical context under consideration by means of the relevant laws and symmetries, as informed group-theoretically. Philosophers of science then represent that structure by means of various meta-level resources, such as the Semantic Approach. This is not the only such resource available. However, the main alternative—the Ramsey sentence formulation—is bedevilled by the so-called Newman problem. This has been dealt with by Melia and Saatsi and their emphasis on the intensional character of laws points the way to an appropriate understanding of structure that is articulated in the rest of the book.Less