Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all -rank one lattices in semisimple groups of -rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.     

An Outlook in Teaching and Research in Mathematics in Educational Institutions

The present paper outlines some basic principles upon which an outlook in teaching and research in mathematics in educational institutions can be developed. Defining applied mathematics, the author analyses the basic topics of mathematics which permeate throughout the wider fields and diversified aspects of this branch of mathematical science.

For the institutions of learning, a course curriculum has been proposed which will gradually motivate, orient and prepare individuals through the process of learning to understand the methods and techniques, so as to apply them to physical problems. This has been illustrated in the fields of fluid dynamics and aerodynamics through the analysis of complex function theory and differential equations.

Because of the frustrating situations prevailing amongst students and teachers in the institutions of learning in the present day, a critical analysis has been made to identify the respective roles of the academicians and administrators, and with a view to improving the situation, some remedy has been suggested regarding teaching and research in educational institutions.     

Introducing technology in algebra in Hong Kong: addressing issues in learning

A review of recent studies in Hong Kong suggests that possible problems in secondary school algebra may be due to the procedural paradigm orientation in the curriculum and the conventional style of teaching in the classroom which do not provide sufficient opportunities for students to develop conceptual understanding. Based on the works of a number of projects in the West, it is hypothesized that the introduction of technology in lessons which embody a cognitive model in their design and delivery will provide a viable alternative for enhancing algebraic thinking. The key features of the cognitive model are concrete preparation, cognitive conflict, construction, metacognition and bridging, imbedded in a ‘mediation’ style of teaching. These features are exemplified by detailed accounts of classroom episodes from a Hong Kong lesson supported by the use of graphics calculators. Results indicate that while the approach places particular demands on teachers (a tension between a transmission style of teaching and mediating) there is real potential for supporting more dynamic student constructions.     

Operational research in development banking in India

The paper defines development banking as practised by major Indian commercial banks since 1969. It then goes on to discuss how OR can be useful in tackling problems of development banking. Planning, operational and organisational design problems in relation to development banking are discussed. Several case studies in each area where OR has been applied are discussed. Potential areas of application of OR in each problem class are also identified. The paper then summarizes the experiences gained so far and recommends the action to be taken for the more effective application of OR in developing economies.     

Instabilities in elastomers and in soft tissues

Biological soft tissues exhibit elastic properties that canbe dramatically different from rubber-type materials (elastomers).To gain a better understanding of the role of constitutive relationshipsin determining material responses under loads we compare threedifferent types of instabilities (two in compression, one inextension) in hyperelasticity for various forms of strain-energyfunctions typically used for elastomers and for soft tissues.Surprisingly, we find that the strain-hardening property ofsoft tissues does not always stabilize the material. In particularwe show that the stability analyses for a compressed half-spaceand for a compressed spherical thick shell can lead to oppositeconclusions: a soft tissue material is more stable than an elastomerin the former case and less stable in the latter case.     

Convergenza in variazione e convergenza in lunghezza

Riassunto Viene stabilita, per le trasformazioni continue di una regione piana pluriconnessa, una maggiorazione per il numero delle imagini reciproche di un insieme dik punti del piano-imagine. Dalla disuguaglianza discende una limitazione superiore per il numero dei punti ?eccezionali?, ossia aventi un numero di modelli minore a quanti ne comporterebbe l'indice topologico.

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Résumé On établit, pour les transformations continues d'une région plane à plusieurs contours, une majoration du nombre des points de l'ensemble image-inverse d'un ensemble dek points du plane-image. Il en ressort une borne supérieure pour le nombre des points exceptionnels, c'e.-à-d. ayant moins d'images inverses qu'on s'attendrait de l'indice topologique.

     

Series in multiplicative systems in Lorentz spaces

Series in multiplicative systems χ with generalized monotone coefficients are studied. Necessary and sufficient Hardy–Littlewood type conditions for the sums of such series to belong to the Lorentz space are proved. As corollaries, we establish estimates of best approximation in the system χ and Konyushkov-type theorems on the equivalence of O- and ?-relations for the weighted sums of the Fourier coefficients in the system χ and for the best approximations.     

二值回归模型中自变量误差的处理

In this paper the binary regression model, in which the response Y is binary, i.e. Y takes the value 0 or 1 only, is considered. The conditional probability P{Y=1|X=x} is assumed to be of the form G(a₀+a₁x), where x is the measurement of the factor, a= (a₀,a₁)′ is the vector of the unknown parameter being estimated and G(a) = (1+e^a)^-1. The measurement x is assu-med to be contaminated when the value of x is out of certain limit. M-estimation is employed to estimate the unknown a of the model. In this paper, the consistent M-estimation is proved to be asymptotically normal and the optimal M-equation is obtained. Finally, a computation method is introduced to solve the M-equation and to get the optimal solution.