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Partitions and Catalan Numbers

Thomas Koshy

in Catalan Numbers with Applications

This chapter investigates several occurrences of Catalan numbers in the theory of partitioning. Noncrossing partition is explained in detail.

EXPANDING THE TUTTE POLYNOMIAL OF A MATROID OVER THE INDEPENDENT SETS

Koko Kalambay Kayibi

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

This chapter provides direct combinatorial proof of an expansion of the Tutte polynomial by independent sets of the matroid. Another expansion of the Tutte polynomial is presented in terms of ... More

This chapter provides direct combinatorial proof of an expansion of the Tutte polynomial by independent sets of the matroid. Another expansion of the Tutte polynomial is presented in terms of spanning sets. In the process, it is shown that there is a partition of the set of independent sets of a matroid, such that if the independent set I and the basis B are contained in the same part of the partition, the external activity of I is equal to the external activity of B.Less

The Limits to Coercive Consociationalism in Northern Ireland

John McGarry and Brendan O'Leary

in The Northern Ireland Conflict: Consociational Engagements

The chapter, initially published in 1989, reviews efforts by the UK and Irish governments to achieve a settlement in Northern Ireland during the late 1980s. It bears the impress of its time; ... More

The chapter, initially published in 1989, reviews efforts by the UK and Irish governments to achieve a settlement in Northern Ireland during the late 1980s. It bears the impress of its time; normatively consociational, but somewhat bleak in its assessments. It contains an appraisal of constitutional options facing policymakers.Less

Parts of the Soul

Hendrik Lorenz

in The Brute Within: Appetitive Desire in Plato and Aristotle

This chapter raises and explains two central questions about Plato’s theory of the tripartite soul. First, what is involved in Plato’s claim that the embodied human soul is a thing of parts? ... More

This chapter raises and explains two central questions about Plato’s theory of the tripartite soul. First, what is involved in Plato’s claim that the embodied human soul is a thing of parts? Secondly, does Plato have a reasonable basis for holding that the soul is composed of precisely three parts, reason, spirit, and appetite?Less

The Argument for Tripartition

Hendrik Lorenz

in The Brute Within: Appetitive Desire in Plato and Aristotle

This chapter argues that Republic-style partition of the soul amounts to a position that can fairly be characterized in terms of two main claims. First, the proper subjects of desires and aversions ... More

This chapter argues that Republic-style partition of the soul amounts to a position that can fairly be characterized in terms of two main claims. First, the proper subjects of desires and aversions are not whole souls, but in each case some part or other of the soul in question. Secondly, the parts of the soul that are the proper subjects of desires and aversions are unified in the strong sense that it is impossible for a given soul-part to desire something and at the same time to be averse to it.Less

Partition

Hendrik Lorenz

in The Brute Within: Appetitive Desire in Plato and Aristotle

This chapter addresses the concern that for a Platonic soul to make sense, is not right to say that it has genuine parts. It also assesses the philosophical cost of soul-partition by considering its ... More

This chapter addresses the concern that for a Platonic soul to make sense, is not right to say that it has genuine parts. It also assesses the philosophical cost of soul-partition by considering its impact on the question of the soul’s immortality and by worrying, with Aristotle, about the soul’s unity.Less

Organizational Ecology as a Theory of Competition

Nikolaus Beck

in The Institutions of the Market: Organizations, Social Systems, and Governance

This chapter shows that organizational ecology is much more sociological and less biological than many critics of organizational ecology think. It argues that organizational ecology and its theory of ... More

This chapter shows that organizational ecology is much more sociological and less biological than many critics of organizational ecology think. It argues that organizational ecology and its theory of competition rests heavily on the seminal work of Emile Durkheim on the division of labour, and on the thoughts of Amos Hawley on human ecology. It suggests that the expressions used by the agents of organizational ecology are ‘Darwinian’ but that the theoretical argumentation is ‘Durkheimian’. The chapter highlights the theoretical foundations of organizational ecology and emphasizes that competition — and not the biological aspects of evolution — forms the core of the organizational ecology theory. The most the most important sub-concepts of organizational ecology that deal with competition and market formation are also discussed: density dependence and resource partitioning. The first concept deals with the impact of the number of competitors on the survival and founding chances of organizations. The latter is concerned with the influence of market concentration on the vital rates of specialist and generalist organizations.Less

The Tenability of Partition as a Mode of Conflict Regulation: Comparing Ireland With Palestine–Land of Israel

Sammy Smooha

in Northern Ireland and the Divided World: Post-Agreement Northern Ireland in Comparative Perspective

Smooha argues that Jews have settled in Palestine – Land of Israel in sufficient strength to establish a strong independent state in part of the area (pre‐1967 Israel), although their attempt to ... More

Smooha argues that Jews have settled in Palestine – Land of Israel in sufficient strength to establish a strong independent state in part of the area (pre‐1967 Israel), although their attempt to incorporate the rest (the West Bank and Gaza) by settlement has failed. As a result, partition has become the most feasible way to resolve the Jewish–Palestinian conflict. In Ireland, by contrast, the position of British settlers (Protestants) is said to be much weaker than their Jewish counterparts. The Protestants are weak demographically, lack international legitimacy, are not supported by Britain, and are faced with a confident Irish nationalism. As a consequence, the historical trend points to Britain's withdrawal and an end to the partition of Ireland as the most likely form of conflict resolution.Less

Senior Officials in the German Federal Administration: Institutional Change and Positional Differentiation

Klaus H. Goetz

in Bureaucratic Elites in Western European States: A Comparative Analysis of Top Officials

This account of institutional change and positional differentiation of senior officials in Germany's Federal administration is presented in five sections. Section I, ‘Senior Officials and the ... More

This account of institutional change and positional differentiation of senior officials in Germany's Federal administration is presented in five sections. Section I, ‘Senior Officials and the Reassertion of Political Authority’, is introductory and discusses the causes and implications of change in the organization and in the political and administrative roles of the senior civil service; changes identified include party politicization, parliamentarization and federalization of the national policy process, modernization initiatives, European integration, and unification. Section II gives a brief survey of the Federal senior ministerial personnel (looking at pay grade and rank, and centrality), and section III considers paths to the top, paying particular attention to the procedures for recruitment and promotion and the consequences of weak formal structures for personnel planning and development. Following on from the definition of political craft as a defining attribute of effective top officials, section IV highlights the central position of political coordination units as training grounds in the Federal administration and comments on the informal positional differentiation that they encourage. The discussion concludes in section V with an assessment of the implications of the partition of the ministerial bureaucracy between Bonn and Berlin.Less

Resizing and Reshaping the State: India from Partition to the Present

Gurharpal Singh

in Right-sizing the State: The Politics of Moving Borders

Gurharpal Singh traces the institutional dependence of the Indian state on conceptions of itself and its original borders as sacrosanct. But, the author argues that the state is, in reality, reshaped ... More

Gurharpal Singh traces the institutional dependence of the Indian state on conceptions of itself and its original borders as sacrosanct. But, the author argues that the state is, in reality, reshaped as a result of demands from linguistic and other groups of the dominant North Central Hindu/Indian populations, while demands from the periphery are ignored. The chapter examines how political elites deliberately entrenched the hegemonic beliefs about the state size after the partition, the centre's response to the challenges faced by Kashmir and Punjab to these beliefs, and various coalitions in favour and against reshaping of the Indian state.Less

Relativistic gases

Stephen J. Blundell and Katherine M. Blundell

in Concepts in Thermal Physics

This chapter repeats the derivation of the partition function for a gas, and hence of the other thermodynamic properties that can be obtained from it, but this time includes relativistic effects. It ... More

This chapter repeats the derivation of the partition function for a gas, and hence of the other thermodynamic properties that can be obtained from it, but this time includes relativistic effects. It shows that this leads to some subtle changes in these properties which have profound consequences. It first reviews the full relativistic dispersion relation for particles with non-zero mass and then derives the partition function for ultrarelativistic particles.Less

The analysis of biodiversity experiments: from pattern toward mechanism

Andy Hector, Thomas Bell, John Connolly, John Finn, Jeremy Fox, Laura Kirwan, Michel Loreau, Jennie McLaren, Bernhard Schmid, and Alexandra Weigelt

in Biodiversity, Ecosystem Functioning, and Human Wellbeing: An Ecological and Economic Perspective

Meta-analysis of the first generation of biodiversity experiments has revealed that there is a general positive relationship between diversity and ecosystem processes that is consistent across ... More

Meta-analysis of the first generation of biodiversity experiments has revealed that there is a general positive relationship between diversity and ecosystem processes that is consistent across trophic groups and ecosystem types. However, the mechanisms generating these general patterns are still under debate. While there are unresolved conceptual issues about the nature of diversity and complementarity, the debate is partly due to the difficulty of performing a full-factorial analysis of the functional effects of all species in a diverse community. However, there are now several different analytical approaches that can address mechanisms even when full factorial analysis is not possible. This chapter presents an overview and users' guide to these methods. This chapter concludes that the current toolbox of methods allows investigation of the mechanisms for most, if not all, biodiversity and ecosystem functioning experiments conducted to date that manipulate species within a single trophic level (e.g. plant biodiversity experiments). Methods that can address mechanisms in multitrophic studies are a key need for future research.Less

The Structure of Experience

Barbara Tversky, Jeffrey M. Zacks, and Bridgette Martin Hard

in Understanding Events: From Perception to Action

This chapter focuses on the structure of experience, covering language, objects, scenes, and events. It is argued that partitioning the world is the first step to comprehending it. Some partitioning ... More

This chapter focuses on the structure of experience, covering language, objects, scenes, and events. It is argued that partitioning the world is the first step to comprehending it. Some partitioning is so instantaneous and automatic that perception of the world is not of multimedia mixtures of continuously changing sensations but rather of coherent objects, events, and scenes. The mind goes on to parse those elements and to look for structure among the parts. Typically there is a perceptual basis for part structure. Truly understanding each of these elements of our lives requires assigning meaning to their parts.Less

TESTING FOR THE EXISTENCE OF HOMOMORPHISMS

Pavol Hell and Jaroslav Nešetřil

in Graphs and Homomorphisms

This chapter explores the algorithmic aspects of graph homomorphisms and of similar partition problems. The highlights include the dichotomy classification of graph homomorphisms to a fixed target ... More

This chapter explores the algorithmic aspects of graph homomorphisms and of similar partition problems. The highlights include the dichotomy classification of graph homomorphisms to a fixed target graph $H$; a proof of the fact that dichotomy for digraph homomorphisms would imply dichotomy for all constraint satisfaction problems; a presentation of consistency-based algorithms; and associated dualities that seem to be applicable to all known polynomial cases of the digraph homomorphism problem. The role of polymorphisms in the design of polynomial algorithms is highlighted, and it is shown that graphs with the same set of polymorphisms define polynomially equivalent problems. The polymorphism known as the majority function is shown to yield a polynomial time homomorphism testing algorithm. The dichotomy classification of list homomorphism problems for reflexive graphs is presented. List matrix partition problems are posed in the language of trigraph homomorphisms, and the richness of the associated algorithms is illustrated on the case of clique cutsets and generalized split graphs.Less

The Baʿth Facing Particularism and Secession

Ofra Bengio

in Saddam's Word: The Political Discourse in Iraq

This chapter addresses the internal weaknesses and the true motives of the Baʿths in using the concept of unity. They used the concept to help strengthen their position at the cost of other Iraqi ... More

This chapter addresses the internal weaknesses and the true motives of the Baʿths in using the concept of unity. They used the concept to help strengthen their position at the cost of other Iraqi groups. As the threat of partition arose, presumptuousness turned into anxiety. Husayn himself addressed the problem and compared Iraq to a “huge ship for Iraqi, Kurds and Arabs” and warned that he would not allow anyone to “drill a hole in his ship”. His opposition to the plan that might help Iraqi particularists and secessionists is also discussed here. The Shiʿi and Kurdish uprising is also detailed.Less