HIGHER DEGREE HYPERBOLIC FORMS

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 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.

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}.

Powers of the Fundamental Ideal in the Witt Ring

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

Jones matrices and Sinclair matrices are 2/spl times/2 complex matrices that determine forward (transmission) and backward scattering, respectively. Under a unitary change of polarization basis they transform by ordinary unitary similarity and by unitary consimilarity, respectively, forming equivalence classes with common invariant polarimetric features. In most applications the Jones matrices T are normal and, thus, have orthogonal eigenvectors; whereas, Sinclair matrices S are symmetric and have orthogonal coneigenvectors. For forward scattering the term homogeneous is used in this case. Recently, inhomogeneous Jones and Sinclair matrices characterized by non-orthogonal eigenvectors and coneigenvectors have attracted attention. The present contribution considers some characteristics of these matrices. In particular it is shown that the graphical field-of-value representation of inhomogeneous Jones matrices leads to an interesting characterization of the degree of inhomogeneity, often characterized by the (cosine of the) angle between the normalized eigenvectors. Applications of the resulting theorems to both optical (transmission) and radar (backscattering) polarimetry are demonstrated, together with identifying the inherent mathematical intricacies and physical complexities which are integral to radar and optical polarimetry.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

Formally real fields from a Galois theoretic perspective

Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had crucial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736-1813) wrote his seminal memoir Reflections on the solution of algebraic equations. He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real/complex/positive/negative? The problem of algebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major efforts in this paper. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777-1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups4 Math Unlimitedappeared in different forms like the additive group Zn of integers modulo n, the multiplicative group Z∗n of integers modulo n relatively prime to n, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity. In 1872, Felix Klein delivered a famous lecture A Comparative Review of Recent Researches in Geometry. The aim of his (so-called) Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. So, the groups appear here “geometrically” as groups of rigid motions, of similarities, or as the hyperbolic group etc. During the analysis of the connections between the different geometries, the focus was on the study of properties of figures invariant under transformations. Soon after, the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations between figures got converted to the study of the associated transformations. In 1874, Sophus Lie introduced his “theory of continuous transformation groups” what we basically call Lie groups today. Poincaré and Klein began their work on the so-called automorphic functions and the groups associated with them around 1876. Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis. They are functions of a complex variable z, analytic in some domain, and invariant under the group of transformations x′ = ax+b(a, b, c, d real or complex and ad − bc 0), or under some subgroup of this group. We end this introduction with the ancient problem of finding all those positive integers (called congruent numbers) which are areas of right-angled triangles with rational sides. To this date, the general result describing all of them precisely is not proved. However, the problem can be re-stated in terms of certain groups called elliptic curves and this helps in getting a hold on the problem and in obtaining several partial results. Indeed, if n is a positive integer, then the rational numbers x, y satisfying y2 = x3 − n2x form a group law which can be described geometrically. Then, n is a congruent number if and only if, this group is infinite!

We investigate the asymptotics of the total number of simple $(4a+1)$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some nonzero $m \in [-X, X]$. Using Kearton and Levine’s classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of ${\mathbb{Z}}[1/m]$-equivalence classes of binary quadratic forms of discriminant $1-4m$, for $m$ running through the same range. Our heuristics, based on the Cohen–Lenstra heuristics, suggest that this total is asymptotic to $X^{3/2}/\log X$ and the largest contribution comes from the values of $m$ that are positive primes. Using sieve methods, we prove that the contribution to the total coming from $m$ positive prime is bounded above by $O(X^{3/2}/\log X)$ and that the total itself is $o(X^{3/2})$.

For a field F of characteristic not 2, let \({\widehat{F}}\) denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.

On the scaled trace forms and the transfer of a number field extension

Recovery of a manifold with boundary and its continuity as a function of its metric tensor