As we realize that random walks, chain reactions, and recurrent events are all Markov chains, i.e., correlated processes without memory, in this chapter we derive a general theory, including classification and properties of the single states and of chains. In particular, we focus on the building blocks of the theory, i.e., irreducible chains, presenting and proving a number of fundamental and useful theorems. We end up deriving the balance equation for the limit probability and the approach to the limit for long times, developing and applying the Perron–Frobenius theory for non-negative matrices and the spectral decomposition for non-Hermitian matrices. Among the applications of the theory, we underline the sorting of Web pages by search engines.

We obtain bounds for an a priori unknown rate function. We prove the existence and uniqueness of invariant probability measures and the necessity of irreducibility.

National courts today find themselves ever more confronted with issues that are borderless, and sometimes global. This Chapter looks at the past of the common law and some aspects of its origins, evolution, and characteristics. The common law, which has come to define a conception of law standing in stark contrast to that of the Continental European civilian legal orders, including none more so than France’s, has been an irony of fate in its evolution, consolidated and strengthened as it was through the efforts of a Norman French king. If we think of matters calling most urgently for transnational solutions, do the origins of the common law suggest differences of a nature that would create insuperable obstacles to a new jus commune? Methodologically, the civil and common law systems continue to clash. Yet the questions their courts face today are increasingly issues of jurisdiction, of the reach of domestic courts beyond the territories that define national sovereignty. We already have seen innovative strides by courts in numerous states in recent years to meet the needs of transnationalizing law. With increasing comprehension can come harmony and fruitful mutual consultation, if humanity decides that it wishes to proceed towards this goal.

In this introductory chapter, we establish a theoretical framework for the book, drawing on the concept of 'queer entanglements' to argue for what a 'queer menagerie' might look like in terms of research, theory, and activism in regard to the intersections of gender, sexuality, and species. The chapter also provides definitions of the populations we focus on and outlines our reasons for our specific areas of focus. We also discuss our positionality as authors. In elaborating our theoretical framework, we focus on histories and presents of animal and LGBQTNB human lives, and we map out some potential ways of understanding why it would seem that such histories and presents take unique forms in the lives of LGBQTNB people and the animals they live with. We finish the chapter by outlining our two key concepts, ‘enmeshment’ and ‘irreducibility’, that help us to understand and represent the work of curating a queer menagerie. This introductory chapter concludes by providing an overview of the chapters included in this book.

In this introductory chapter, we establish a theoretical framework for the book, drawing on the concept of 'queer entanglements' to argue for what a 'queer menagerie' might look like in terms of research, theory, and activism in regard to the intersections of gender, sexuality, and species. The chapter also provides definitions of the populations we focus on and outlines our reasons for our specific areas of focus. We also discuss our positionality as authors. In elaborating our theoretical framework, we focus on histories and presents of animal and LGBQTNB human lives, and we map out some potential ways of understanding why it would seem that such histories and presents take unique forms in the lives of LGBQTNB people and the animals they live with. We finish the chapter by outlining our two key concepts, ‘enmeshment’ and ‘irreducibility’, that help us to understand and represent the work of curating a queer menagerie. This introductory chapter concludes by providing an overview of the chapters included in this book.

The A-hypergeometric or GKZ hypergeometric system of differential equations in the present form were introduced by Gel'fand, Zelevinsky, and Kapranov about 30 years ago. Series solutions are multivariable hypergeometric series defined by a matrix A. They found that affine toric ideals and their algebraic and combinatorial properties describe solution spaces of the A-hypergeometric differential equations, which also opened new research areas in commutative algebra, combinatorics, polyhedral geometry, and algebraic statistics. This chapter describes fundamental facts about the system and its solutions, and also gives pointers to recent advances. Applications of A-hypergeometric functions are getting broader. Early applications were mainly to period maps and algebraic geometry. The interplay with commutative algebra and combinatorics has been a source of new ideas for these two fields and for the theory of hypergeometric functions. Recent new applications are to multivariate analysis in statistics.