Examples of uniform exponential growth in algebras

In this paper, we discuss the concept and examples of algebras of uniform exponential growth. We prove that Golod–Shafarevich algebras and group algebras of Golod–Shafarevich groups are of uniform exponential growth. We prove that uniform exponential growth of the universal enveloping algebra of a Lie algebra [Formula: see text] implies uniform exponential growth of [Formula: see text], and conversely should [Formula: see text] be graded by the natural numbers. We prove that a restricted Lie algebra is of uniform exponential growth if and only if its universal enveloping algebra is. We proceed to give several conditions equivalent to the uniform exponential growth of the graded algebra associated to a group algebra filtered by powers of its fundamental ideal.

Planning, Freedom, and the Rule of Law

Rule of law is widely considered to be an important element of a well-ordered society. It is an ideal of political morality that is realized to a greater or lesser extent in different legal systems. However, the rule of law is not a basic or fundamental ideal. Its normative significance is explained by its contribution to other, more fundamental, values. This chapter discusses the content of the rule of law (the institutional mechanisms and informal norms that comprise it) and the contribution that it makes to individual or personal freedom. The chapter presents an account of political freedom that relates freedom to the ability of persons to plan their lives. This planning account of freedom is just one component of a full theory of political freedom, but it is the component that best accounts for why the rule of law contributes to personal freedom.

Three Risks, One Solution? Exploring the Relationship between Risk and Regulation

The risks at the heart of regulation are most often understood as unwanted by-products of an essentially productive endeavor. They are to be captured and managed through dedicated technical and bureaucratic effort, the preserve of those skilled in the assessment of probability and impact and accomplished in the design of effective—and efficient—regulatory strategies. Yet such a view seriously misunderstands the complexity of risk that regulation must address. When regulation is viewed from a social and organizational vantage point, risks of a more social and political character emerge. This article teases apart three fundamental ideal types of risk inherent in regulatory processes, only one of which (labeled as actuarial risk) is apprehensible from a scientific or bureaucratic frame of reference. Of equal importance are sociocultural and political risks, risks that cannot be relegated to a lesser priority by an understanding that labels their impact on regulation variously as unwarranted, irrational, or emotional.

Morality and boundaries in Paul

In the Pauline communities, ethics, ethos and identity were closely intertwined. This essay analyses the way in which Paul emphasised the mental boundaries of the Christ communities to turn them into moral boundaries. In this process, the fencing off of these communities over against their past and their present was a fundamental feature of Paul’s reasoning. The communities thus became fenced off from their past, because the Christ event was seen as causing a major change in history. This change affected both Gentile and Jewish believers. At the same time, Paul stressed the boundaries with the outside world: he characterised the inside world as the loyal remnant of Israel, consisting of Jews and Gentiles alike, and pointed out that this group is the group of the elect ‘saints’. The perspective with which Paul looked at ethics and morality inside this group was strongly coloured by the assumed identity of this group as ‘Israel’. Even though the Mosaic Law was no longer the focal point for the identity of this eschatological Israel, the ethical demands Paul mentioned over against the members of this new Israel were highly influenced by the morality of the law. For Paul, sanctification was a fundamental ideal, and this ideal reflected the spirituality of the Holiness Code of Leviticus. This particular ethical model was framed by the awareness that Paul (and Christ before him) was ‘sent’ by God, much in the same way the prophets of Israel themselves had been sent.

The Elman-Lam-Krüskemper Theorem

For a (formally) real field K, the vanishing of a certain power of the fundamental ideal in the Witt ring of K(-1) implies that the same power of the fundamental ideal in the Witt ring of K is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of K. This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of K. We present an exposition of these results.

Assistive Technology for Deaf and Hard of Hearing Students

Although the majority of deaf and hard of hearing (d/hh) students are educated in the public school system (Turnball, Turnball, & Wehmeyer, 2010) there is limited research and literature regarding how educators can effectively meet their educational needs by implementing assistive and instructional technologies into their curriculum. This chapter provides an overview of the various assistive and instructional technologies available to d/hh students and outlines how these students access and use technology. This chapter contributes to the fundamental ideal that integrating assistive and instructional technologies can greatly enhance the academic and social outcomes for d/hh students. It should be noted, that the Deaf community does not adhere to person first language because they do not view deafness as a disability but as a culture.

ON HERMITIAN PFISTER FORMS

Let K be a field of characteristic different from 2. It is known that a quadratic Pfister form over K is hyperbolic once it is isotropic. It is also known that the dimension of an anisotropic quadratic form over K belonging to a given power of the fundamental ideal of the Witt ring of K is lower bounded. In this paper, weak analogues of these two statements are proved for hermitian forms over a multiquaternion algebra with involution. Consequences for Pfister involutions are also drawn. An invariant uα of K with respect to a nonzero pure quaternion of a quaternion division algebra over K is defined. Upper bounds for this invariant are provided. In particular an analogue is obtained of a result of Elman and Lam concerning the u-invariant of a field of level at most 2.

Open questions in the theory of spaces of orderings

Spaces of orderings provide an abstract framework in which to study spaces of orderings of formally real fields. Spaces of orderings of finite chain length are well understood [9, 11]. The Isotropy Theorem [11] and the extension of the Isotropy Theorem given in [13] are the main tools for reducing questions to the finite case, and these are quite effective. At the same time, there are many questions which do not appear to reduce in this way. In this paper we consider four such questions, for a space of orderings (X, G).1. Is it true that every positive primitive formula P(a) with parameters a in G which holds in every finite subspace of (X, G) necessarily holds in (X, G)?2. If f: X → ℤ is continuous and Σx∈Vf(x) ≡ 0 mod ∣V∣ holds for all fans V in X with ∣V∣ ≤ 2n, does there exist a form ϕ with entries in G such that mod Cont(X, 2nℤ)?3. Is it true that Cont(X, 2nℤ) ∩ Witt(X, G) = In(X, G), where I(X, G) denotes the fundamental ideal?4. Is the separating depth of a constructible set C in X necessarily bounded by the stability index of (X, G)?The unexplained terminology and notation is explained later in the main body of the paper. In a certain sense Question 1 is the main question. At the same time, Questions 2, 3 and 4 are of considerable interest, both from the point of view of quadratic form theory and from the point of view of real algebraic geometry.