Formally real fields from a Galois theoretic perspective

Powers of the Fundamental Ideal in the Witt Ring

Three experiments identified factors that did and did not enhance the formation of two-node four-member equivalence classes when training and testing were conducted with trials presented in a trace stimulus pairing two-response (SP2R) format. All trials contained two separately presented stimuli. Half of the trials, called within-class trials, contained stimuli from the same class while the other half, called cross class trials, contained stimuli from different classes. On within class trials, making a YES response was correct and making a NO response was wrong. On cross class trials, making a NO response was correct and making a YES response was wrong. In Experiment 1, similar intermediate percentages of participants (about 50%) formed classes, regardless of whether the responses were labeled YES and NO or SAME and DIFF. Response labeling thus did not influence class formation. Regardless of response labels, failures of class formation were primarily due to failure of class-indicative responding produced by within-class transitivity probes. In Experiment 2, only 50% of participants formed classes without prior training, as in Experiment 1, but 100% of participants formed equivalence classes after the establishment of a generalized transitivity repertoire by use of a programmed transitivity induction protocol. Experiment 3 examined two components of the programmed transitivity induction protocol and found that the exclusion of AC trials had no effect on the percentage of participants who formed equivalence classes, while presenting the stimulus sets in randomized order interfered with equivalence class formation. A further analysis found that a number of stimulus control topographies differentiated between individuals who did and did not form equivalence classes. In general, then, these experiments demonstrate that equivalence classes can be formed reliably when training and testing are conducted in an SP2R format, supporting the view that equivalence class formation can account for the development of conceptual categories in natural settings.

Let F be a field with 2 = 0 and ϕ = ⟨⟨a 1,…,a n ⟩⟩ an n-fold anisotropic bilinear Pfister form over F with function field F(ϕ). In this paper we compute ker where I F ⊂ W(F) is the maximal ideal in the Witt ring W(F) of F. We use this computation to prove an n-linkage property of the subfields F 2(a 1,…,a n ).

A primary problem in the theory of quadratic forms over a field F of characteristic different from two is to prove that the rings H'F and GWF are isomorphic.Here H'F = H'(Ga\(Fq/F),Z/2Z)),where Fq is the quadratic closure of F , and GWF is the graded Witt ring associated to the fundamental ideal of even dimensional forms in the Witt ring WF of F .In this paper, we assume we are given a field extension K of F such that WK is 'close' to WF or H'K is 'close' to H'F .A method is developed to obtain information about these graded rings over F and its 2-extensions from information about the corresponding graded ring of K.This relative theory extends and includes the previously developed absolute case where K = Fq .Applications are also given to show that H'F and GWF are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

We define higher degree hyperbolic forms, analogous to the quadratic hyperbolic forms. We prove the following descent result. Let f be a form of degree d ≥ 3 over a field F of characteristic 0, and let K|f be a field extension. Then if f is equivalent over K to a hyperbolic form, f must already be equivalent to it over F. We also prove that in the monoid of equivalence classes of forms defined over F of a fixed degree d ≥ 3, under the tensor product, the submonoid generated by the equivalence classes of the hyperbolic forms is free. The proofs of these results involve the calculation of the centres and the Lie algebras of the higher degree hyperbolic forms. For the convenience of the reader we expound some of Harrison's seminal paper [5].

Canonical dimension of a smooth complete connected variety is the minimal dimension of image of its rational endomorphism. The $i$-th canonical dimension of a non-degenerate quadratic form is the canonical dimension of its $i$-th orthogonal grassmannian. The maximum of a canonical dimension for quadratic forms of a fixed dimension is known to be equal to the dimension of the corresponding grassmannian. This article is about the minima of the canonical dimensions of an anisotropic quadratic form. We conjecture that they equal the canonical dimensions of an excellent anisotropic quadratic form of the same dimension and we prove it in a wide range of cases.

Resumo Relações condicionais pré-experimentais que eliciam respostas emocionais, como ciúme, podem afetar a formação de classes de equivalência? Haveria diferenças entre homens e mulheres? Para abordar essas questões, 19 participantes, de ambos os sexos, foram expostos a treinos e testes para a formação de quatro classes de equivalência, uma delas constituída pelo nome do(a) namorado(a), o nome de mulher/homem desconhecida(o) e duas figuras não-familiares. Após os testes, aplicou-se a Escala de Ciúme Romântico. Os participantes classificados com maior grau de ciúme não formaram as classes de equivalência, independentemente de serem homens ou mulheres. Sugere-se que relações pré-experimentais com funções afetivas interferiram na formação das classes de equivalência em que o nome do namorado(a) deveria ser relacionado ao nome de desconhecido(a).

\n O processo de envelhecimento pode gerar perdas cognitivas que, se agravadas, podem acarretar o diagnóstico de demência, sendo a Demência de Alzheimer (DA) a mais comum. A DA é uma doença neurodegenerativas que afeta principalmente a memória e a linguagem. Procedimentos de ensino baseados no paradigma de equivalência de estímulos e no ensino por exclusão, apesar de testados com sucesso em diferentes populações com problemas de desenvolvimento, têm gerado poucas pesquisas com idosos com perdas cognitivas. Esta pesquisa apresenta três estudos, com objetivos distintos, mas complementares. O Estudo 1 teve como objetivo avaliar se idosos institucionalizados sem DA formariam classes de estímulos equivalente a partir de um procedimento de ensino de relações condicionais entre palavras ditadas, fotos, nomes impressos de profissões e de graus de parentesco; adicionalmente, pretendia-se verificar que essas relações se mantinham ao longo de um mês, em três testes de manutenção. Cinco idosos institucionalizados passaram pelo ensino de relações arbitrárias, em uma estrutura de treino linear com ensino de novas relações por exclusão. Testes de equivalência foram conduzidos em seguida à Fase de Ensino, e após cinco, 15 e 30 dias. Todos os idosos demonstraram desempenho por exclusão na Fase de Ensino e aprenderam todas as relações com o número mínimo de blocos de ensino programados (exceto uma participante). Todos os idosos apresentaram formação de classes de equivalência. Quatro dos idosos apresentaram manutenção dessas classes em todos os testes de manutenção. O procedimento, portanto, mostrou-se efetivo para esses idosos e gerou formação de equivalência, apesar da sua condição de institucionalização. O Estudo 2 teve como objetivo avaliar se idosos com DA, também institucionalizados, demonstrariam desempenho por exclusão e aprenderiam relações condicionais entre quatro nomes ditados e quatro fotos, em um procedimento de ensino por exclusão. Seis idosas com diagnóstico de DA passaram por tarefas de emparelhamento ao modelo, em que as relações entre modelos ditados e fotos foram ensinadas por exclusão. Apesar de todas as idosas terem respondido por exclusão na primeira apresentação de um estímulo modelo novo e seu comparação correspondente em pelo menos duas das três tentativas em que tal desempenho era possível, apenas duas mantiveram essa relação ao longo das demais tentativas e aprenderam as relações ensinadas. Possíveis motivos para esse desempenho são discutidos. O Estudo 3 avaliou se idosos institucionalizados, com DA e comprometimento cognitivo, formariam classes de equivalência a partir de um ensino de relações arbitrárias com estrutura de treino um-para-muitos, procedimento de dica atrasada e ensino por exclusão. Quatro idosas passaram pelo ensino de relações condicionais entre estímulos visuais (nomes escritos, fotos e nomes de profissões). Todas aprenderam as relações ensinadas com o número mínimo de blocos programados, porém apresentaram dificuldade em formar classes de equivalência e manter tais relações nos testes de manutenção após cinco e oito dias. Os procedimentos de ensino empregados foram eficientes em gerar a aprendizagem das relações, mas tal aprendizagem, nessas idosas, não sustentou a formação e manutenção de classes de equivalência.\n

In 1994, Moneyhun introduced and studied the concept of isoclinism in Lie algebras. Moghaddam and parvaneh in 2009 introduced and gave some properties of isoclinism of a pair of Lie algebras. In this paper we introduce the concept of relative n-isoclinism between two pairs of Lie algebras. They form equivalence classes and we show that each equivalent class contains an n-stem pair of Lie algebras. We also show that for a relative n-isoclinism family of Lie algebras C say, consists of (M, L) such that L is finitely generated and [M, n L] is finite dimension, then there exists an n-stem pair of Lie algebras (R, S) ∊ C such that ; for all (M, L) ∊ C}.

In this paper we give a detailed exposition of some of the results announced in [18]. The primary motivation for this work is Witt's observation [31, Satz 7] that if F is a field of characteristic #72, his ring 1V(F) of classes of anisotropic quadratic forms may be written as Z[G]/K where G is an abelian group of exponent two (actually G =F*/F*2), Z[G] is the integral group ring of G, and K is an ideal of Z[G] generated by elements of the form gl + g2-g3g4 and 1 + g5 with gi in G. Of course these elements can be described more explicitly, but for our purposes the only information we need about K is that any homomorphism of Z[G] -> Z sends K to 0 or to an ideal of the form 2nZ. In this introduction we shall call any ideal of Z[G] with this property admissible. In [26], Pfister proved certain structure theorems for IV(F) usilg his theory of multiplicative forms. His proofs were simplified in [11, 22, 23, 29, 30]. Harrison [11] and Leicht and Lorenz [22] gave important complements to Pfister's results concerning the ideal theory of WV(F). The main goal of our paper is to understand and generalize these structure theorems using standard techniques of commutative algebra. In [28] and [3], Scharlau and Belskii have introduced and studied Witt and Witt-Grothendieck rings for profinite groups. Their definitions generalize the usual notions of Wit.t rings, WV(F), and Witt-Grothendieck rings, K (F), of quadratic forms over fields of characteristic #72. These rings also have the form Z[G]/K with G an abelian group of exponent two and K an admissible ideal of Z[G]. Here we are mainly iinterested in another generalization of 1V(F) and K(F), namely the Witt rings, IV(C, J), and the Witt-Grothendieck rings, K(C,J) of classes of hermitian forms over a connected commutative semilocal ring C with involution J. Since J may be the identity these include the Witt and Witt-Grothendieck rings of classes of symmetric bilinear forms

The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n -2 terms and not less.The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.

On splitting the Knebusch–Milnor exact sequence

Let p be a prime. Iwasawa’s famous conjecture relating Kubota-Leopoldt p-adic L-functions to the structure of certain Galois groups has been proven by Mazur and Wiles in Mazur and Wiles (1984). Wiles later proved a far-reaching generalization involving p-adic L-functions for Hecke characters of finite order for a totally real number field in Wiles (1990). As we discussed in Greenberg (1983), an analogue of Iwasawa’s conjecture for p-adic Artin L-functions can then be deduced. The formulation again involves certain Galois groups. However, one can reformulate this result in terms of Selmer groups for the Artin representations. There are several advantages to such a reformulation. First of all, it fits perfectly into the much broader framework described in Greenberg (1989) which relates the p-adic L-function for a motive to the corresponding Selmer group. The crucial assumption in Greenberg (1989) that the motive be ordinary at p (or at least potentially ordinary) is satisfied by an Artin motive and all of its Tate twists.

Let q q be a power of an odd prime, and F q [ T ] \mathbb {F}_q[T] be the ring of polynomials over a finite field F q \mathbb {F}_q of q q elements. A quadratic form f f over F q [ T ] \mathbb {F}_q[T] is said to be regular if f f globally represents all polynomials that are represented by the genus of f f . In this paper, we study definite regular quadratic forms over F q [ T ] \mathbb {F}_q[T] . It is shown that for a fixed q q , there are only finitely many equivalence classes of regular definite primitive quadratic forms over F q [ T ] \mathbb {F}_q[T] , regardless of the number of variables. Characterizations of those which are universal are also given.