Des extensions plus petites que leurs groupes de Galois

In this article we construct some Galois extensions L∕K with finite Galois groups and such that |Gal(L∕K)|>[L:K]. Using an analog of the Noether method, we explain how to obtain, with a fixed center, such a Galois curiosity with a Galois group as large as we want.

Résumé : Dans cet article, nous construisons une extension galoisienne L∕K à groupe de Galois fini et telle que |Gal(L∕K)|>[L:K]. En utilisant un analogue non commutatif de la méthode de Noether, nous expliquons ensuite comment, à centre fixé, l’on peut construire une telle curiosité galoisienne avec un groupe aussi gros que l’on veut.

Mots clés : Corps gauches; théorie de Galois.     

On non-surjective coarse isometries between Banach spaces

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that if

then there is a linear isometry U : X → Y so that

where is defined by

Representation properties of coarse isometries in free ultrafilter limits on are also discussed.     

A novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin–Bona–Mahony equation for modeling long surface gravity waves of small amplitude

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform and inverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized.

Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation.

Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations.

Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods.     

On the Inverse to the Harmonic Oscillator

Let b _d be the Weyl symbol of the inverse to the harmonic oscillator on R ^d . We prove that b _d and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for b _d . In the even-dimensional case we characterize b _d in terms of elementary functions.

In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions.     

The repercussion of José Anastácio da Cunha in Britain and USA in the nineteenth century

José Anastácio da Cunha (1744–87) is usually recognized as one of the most important Portuguese mathematicians ever. Nevertheless, his work does not seem to have had much repercussion abroad. His Principios mathematicos (Lisbon, 1790) were translated into French, but with very limited impact. However, thanks to a review by John Playfair, an English textbook by John Radford Young featured a proof by Anastácio da Cunha (of a proposition on parallels) and a definition of proportion influenced by the one used by Cunha. This proof on parallels also made its way to an American textbook by Benjamin Greenleaf. Young’s and Greenleaf’s are, so far, the only known cases of actual use (instead of mere reference to) of Cunha’s work outside Portugal.

To Maria Fernanda Estrada, with friendship.     

On problematic aspects in learning trigonometry

In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed.

The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle.     

A typological analysis: understanding pre-service teacher beliefs and how they are transformed

This article describes the beliefs and their transformations of members of a cohort of early-childhood, elementary and middle-level pre-service teachers (PSTs) as they professionally develop. A typological analysis of both quantitative and qualitative data collected between August 2011 and May 2013 was utilized to categorize how 40 PSTs’ beliefs transformed throughout their formal teacher preparation. Five typologies were identified, showing variation in how PST beliefs transform or remain static.

Among the findings, strong support related to the development of innovative beliefs during coursework coupled with at least one transformative experience where innovation was observed ‘working’ in the field were sufficient for the transformation to innovative beliefs, despite potential constraints by supervisors, cooperating teachers and/or mandated curricula (Typology 3). Another finding revealed disguised growth toward innovation among those in Typology 5, who reported being innovative and having productive beliefs but described extremely traditional practices. Implications call for improved connections between mathematics methods professors and field supervisors, particularly during clinical internships when PSTs are no longer enrolled in methods courses, to enhance PSTs’ productive struggle in their development of innovative beliefs (T3) and to increase opportunities for disconnects between innovative beliefs and traditional practices to be made explicit and negotiated (T5).     

Investigating undergraduate mathematics learners’ cognitive engagement with recorded lecture videos

The use of recorded lecture videos (RLVs) in mathematics instruction continues to advance. Prior research at the post-secondary level has indicated a tendency for RLV use in mathematics to be negatively correlated with academic performance, although it is unclear whether this is because regular users are generally weaker mathematics students or because RLV use is somehow depressing student learning. Through the lens of cognitive engagement, a quasi-experimental pre- and post-test design study was conducted to investigate the latter possibility.

Cognitive engagement was operationalized using the Revised Two-Factor Study Process Questionnaire (R-SPQ-2F), which measures learning approaches on two major scales: surface and deep. In two mathematics courses at two universities, in Australia and the UK, participants were administered the questionnaire near the course start and finish. Overall findings were similar in both contexts: a reduction in live lecture attendance coupled with a dependence on RLVs was associated with an increase in surface approaches to learning.

This study has important implications for future pedagogical development and adds to the sense of urgency regarding research into best practices using RLVs in mathematics.     

Weak set convergence and its application to projection methods

There are many examples in Numerical Analysis where convergence of approximate solutions to a solution of the original problem can not be shown in the sense of a norm topology but in the sense of weak convergence ([6], [9], [10]).

Moreover, (global) solutions are often not unique such that a concept of set convergence instead of convergence in the usual sense is more convenient and reasonable ([1], [2]). This particularly holds if weakly formulated problems are under consideration.

When dealing with problems where both situations coincide, a concept of weak set convergence seems to be adequate. Such a concept is developed and will be applied to certain projections methods.