Chapter devoted to the basic quantum properties of entanglement and separability. Introduces the Schmidt decomposition for pure states and the positive partial transpose criterion for mixed states as entanglement witnesses. Introduces the famous Einstein–Podolsky–Rosen paradox and its implementation in terms of qubits, then the Bell inequality, quickly reviewing the experimental demonstrations that quantum mechanics violates this inequality. Gives examples of the use of entanglement in a quantum algorithm to accelerate an information task, namely a database search (Grover algorithm) and the possibility of teleportation of a quantum state.

This chapter provides theoretical foundations for the Prioritarianism in Practice volume, by clarifying the features of prioritarian social welfare functions (SWFs). A prioritarian SWF sums up individuals’ well-being numbers plugged into a strictly increasing and strictly increasing transformation function. Prioritarian SWFs, like the utilitarian SWF, fall within the “generalized-utilitarian” class of SWFs.Generalized-utilitarian SWFs are additive and, hence, especially tractable for purposes of policy analysis.The chapter reviews the axiomatic properties of generalized-utilitarian SWFs and, specifically, of prioritarian SWFs. Prioritarianism satisfies the Pigou-Dalton axiom (a pure, gap-diminishing transfer of well-being from a better-off to a worse-off person is an ethical improvement), while utilitarianism does not. Pigou-Dalton is the axiomatic expression of the fact that a prioritarian SWF gives extra weight (priority) to well-being changes affecting worse-off individuals. The chapter also discusses the informational requirements of prioritarian SWFs (as regards interpersonal well-being comparisons).It reviews the various methodologies for applying a prioritarian SWF under uncertainty. And it describes the two main subfamilies of prioritarian SWFs, namely Atkinson and Kolm-Pollak SWFs.

In DA I.1, Aristotle asks whether nous (understanding or reason) is chōristē (separable) and presents a separability condition: the soul is separable if it has some activity proper to it that is not shared with the body. I argue that Aristotle is speaking here of separability in being, not separability in account or taxonomical separation. In the case of the soul, this sort of separability would allow the soul to exist apart from the body. Met. Λ.3, GA II.3, and DA III.4 suggest that Aristotle introduces the separability condition because understanding meets it. Reason is independent of the body in a way that no other power of the soul is. Nous alone is divine and separable. DA III.5 then situates this claim: there is an aspect of understanding that can only be active in connection with the body, but understanding is what it is and continues to be active apart from the body. This raises further questions about the life and ontological status of the sort of separated human nous Aristotle envisages. While figuring out its precise contours is difficult, Aristotle is, in fact, committed to the possibility of human intellectual activity continuing apart from the body.

An arbitration agreement is an agreement between parties to a legal relationship to submit an existing or any future disputes to arbitration. It follows that an agreement of the parties to arbitrate is the cornerstone of arbitration, as it contains the consent of the parties to submit their dispute to arbitration. There are two basic types of arbitration agreements. The first one is a submission agreement, which consists in an agreement to submit an existing dispute to arbitration. The second is an arbitration clause, which consists in an agreement to submit a future dispute to arbitration.1

Bulgaria has traditionally been an arbitration-friendly jurisdiction, both for domestic and international disputes.1 Arbitration as a means of dispute resolution was first implemented in Bulgaria at the end of the nineteenth century with the first Civil Procedure Act (1892). At this time arbitration was used mainly to solve commercial disputes between merchants with the assistance of local and national chambers of commerce. In the first half of the twentieth century, arbitration was widely used to resolve both civil and commercial cases, and the arbitrators had extensive powers, including resolving disputes ex aequo et bono. Also during this period, the first reported international commercial arbitration proceedings involving Bulgarian parties occurred, mainly in the international trade and infrastructure construction sectors. Throughout the socialist period (1944–1989), arbitration was allowed only in respect of legal disputes between Bulgarian socialist organisations (i.e., state-owned entities engaged in industry and trade, such as industrial plants, foreign trade enterprises, tourist companies, state-owned banks, etc.) and foreign enterprises or entities, and it was the privileged dispute resolution mechanism for such disputes. The existence of this sort of arbitration allowed Bulgarian practitioners to conserve and develop their knowledge in the field during the socialist period. In particular, the Arbitration Court at the Bulgarian Chamber of Commerce and Industry allowed some prominent scholars and practitioners to regularly engage in domestic and international arbitration proceedings and to develop, as a follow up, domestic legislation and academic materials on the topic.

Nigeria1 has a mixed legal system consisting of common law, sharia law,2 and customary law.3 At the pinnacle of these laws is the 1999 Constitution.4 The judiciary powers are vested in courts established by the constitution.5 Arbitration in Nigeria is governed by the federal statute Arbitration & Conciliation Act 1988 (ACA),6 which incorporates the 1985 UNCITRAL Model Law on International Commercial Arbitration (UNCITRAL Model Law), with some minor modifications. Nigeria is a federal system where several states have enacted their own arbitration legislation. One example of this is the Lagos State Arbitration Law of 2009 (LSAL).7 It applies to all arbitration proceedings in the state of Lagos that are not specifically governed by any other law. As compared to the federal statute, the Lagos Arbitration Act is more modern, having been adopted in 2009.

The popular image of the American legal system is that it is run amok with litigation-happy disputants. Whether this is true from an empirical perspective is debatable, but in fact American federal law and policy favors arbitration as the country’s preferred means of dispute resolution. This is a longstanding position dating back to the enactment of the 1925 Federal Arbitration Act (FAA). However, the history of this Act and its implementation has been inconsistent. It was forgotten at one point as individual American states enacted statutes to limit the use of mandatory arbitration in consumer contracts. Eventually, the US Supreme Court recognized the FAA as prevailing law and voided state laws limiting the use of arbitration under the federal preemption doctrine.1 In more recent times, the US Supreme Court expanded the scope of private arbitration clauses to include statutory claims, such as in the areas of antitrust, collective bargaining, and civil rights. Even more recently, it has begun to limit the availability of arbitration by placing restrictions on class action arbitration.

We give some important applications of the Hahn-Banach Theorem. We prove the existence of a support functional and hence that X^* separates points in X. Then we prove the existence of a functional that encodes the distance from a linear subspace, which is an important ingredient in a number of subsequent proofs. We show that separability of X^* implies separability of X, define the Banach adjoint of a linear map (between Banach spaces), and prove the existence of ‘generalised Banach limits’.

We define the notion of a norm and a normed space. We prove that various canonical definitions are indeed norms (e.g. the l^p norm, the L^p norm, and the supremum norm). We discuss convergence, equivalent norms, and various notions of isomorphism between normed spaces. Finally, we discuss separability in more detail.

We recall the definition of a metric sapce, along with definitions of convergence, continuity, separability, and compactness. The treatment is intentionally brisk, but proofs are included.

The severability of the arbitration agreement, a cornerstone principle of international arbitration that is considered in the first chapter of this book, is more firmly established now than it was three decades ago. Yet respondents still from time to time call that principle into question by attempting to vitiate the arbitral process by invoking a (supposed) defect in the underlying contract or treaty, while difficult issues also remain concerning the precise scope and limits of the principle. These issues underscore the merit in analyzing and revisiting what remains a jurisprudentially subtle and practically important question.