Aspects of mathematical modelling related to optimization

Many practical optimization problems involved mathematical models of complex real-world phenomena. This paper discusses some aspects of modelling that influence the performance of optimization methods. Information and advice are given concerning the construction of smooth models, the transformation of an optimization problem from one category to another, scaling, formulation of constraints, and techniques for special types of models.     

Some mathematical aspects of reinsurance

In two earlier papers, Waters (1979) and Andreadakis and Waters (1980), the effect on an insurer as a result of varying his retention limits has been studied. This present paper represents an extension of this earlier work. In particular we investigate whether it is possible to prove some of the results of the earlier papers without making restrictive assumptions about the distribution of annual claims or about the way in which the reinsurance premium is calculated. It will be shown that this is possible in the case of proportional but not possible in the case of non-proportional reinsurance.     

Some mathematical aspects of wave motion

A survey is made of some mathematical techniques and theorems which can have relevance at a moderately advanced level to certain classes of wave propagation problems.     

Ethnomathematics and mathematical education

From the epistemic point of view, mathematics has various aspects and the term ethnomathematics is the most suitable to express this diversity. Ethnomathematics may be defined as:

1. Prototypical activities that take place within a given group and have elements in common such as counting, representing the space, establishing and symbolising relations, reasoning, infering, etc. These activities give the members of the group an insight of the environment in which they live and the ways of interacting with other human beings.
2. A method for interpreting or thinking within a culture and a microculture whose members relate to each other by using a common method of communication. This method is influenced by physical, social and temporal elements that affect and render possible the existence and ability to think of those who share a similar background.

If mathematics is the accepted or protoypical activities of a group of scientist called mathematicians, then ethnomathematics could be defined as a discipline that comprises mathematics. There may be conflicts if a social group rejects the others and assumes the role of authority of mathematical knowledge. However, the debate should lead to the consolidation of a diverse and enriching point of view in a global future where mathematics need to be adapted to the peculiarities of every different culture. Every social group has implicit ways and explicit methods to acquire a culture. This is also the case in ethnomatematics. There are many signs in history that reveal a transformation in the enculturation process. This change may be related to politics, the family or society. Although the mathematical enculturation process has not been studied in depth from the ethnomathematical point of view, we think it largely depends on the characteristics of the environment where it takes place. However, some of the aspects of a particular environment may be shared by all cultures. I have called ethnodidactics the implementation of the different methods of current mathematics enculturation and the study of these methods in the environment where they take place (i.e. official and unofficial mathematics curricula; geographical, political, social and economic elements or conditions; the instructors, their professional attitude and their education, etc.) Ethnodidactics also includes the study of different forms of evaluation, the beliefs about education—or guided enculturation—and its goals. It might be concluded that equity is possible as far as mathematical enculturation is concerned. However, equity depends on the conditions of every different social environment. Finally, I have established a relation between ethnomathematics and ethnodidactics, by assessment of certain aspects of the professional knowledge acquired by education students. In order to do this I have drawn on an application of L.A. Zadeh's Fuzzy Theory.     

Ethics in mathematical education

The rising of non-Euclidean geometries forces mathematicians to go beyond the criterion of truth and to pose as priority the criterion of freedom. This criterion gains meaning in the circular path that links ethos with logos and it makes us able of experimenting with the wealth of the values of theory. Mathematical education thus finds a new dimension it abandons the ambit of only formal constructions and finds again its role in the determination of the frames on which the originary ethos constructs its “intrinsic geometry”. The role of Bayes’ law in edueation to tolerance is an example of these frames.     

Declining gender differences from FIMS to TIMSS

This paper examines the findings of the three IEA studies (FIMS, SIMS and TIMSS) on the variation among countries of gender differences in mathematics achievement, and places them in the context of the increasing presence of women in mathematics and the overall decrease in gender differences in mathematics achievement over the period examined, from 1964 to 1995.     

A mathematical model of immune competition related to cancer dynamics

This paper deals with the qualitative analysis of a model describing the competition among cell populations, each of them expressing a peculiar cooperating and organizing behavior. The mathematical framework in which the model has been developed is the kinetic theory for active particles. The main result of this paper is concerned with the analysis of the asymptotic behavior of the solutions. We prove that, if we are in the case when the only equilibrium solution if the trivial one, the system evolves in such a way that the immune system, after being activated, goes back toward a physiological situation while the tumor cells evolve as a sort of progressing travelling waves characterizing a typical equilibrium/latent situation. Copyright © 2009 John Wiley & Sons, Ltd.     

Special functions related to various types of root systems

This paper describes methods that allow one to obtain special functions and study their properties. Using the differential-difference operator theory developed by Ch. Dunkl, the author constructs the family of functions associated with the root system of type G ₂. It is shown that it contains the subset consisting of the well-known Gegenbauer polynomials. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Some mathematical aspects of the political business cycle

Under the hypothesis of a myopic electorate, vote-loss-minimizing behavior by the party in power, subject to a dynamic inflation-unemployment relation, is shown to generate an attractive, stable electoral policy cycle. The model presented is derived, with some improvements, from the analogous models of MacRae and Nordhaus. Furthermore, an attempt is made to specify the mathematical aspects of the problem by the Poincaré mapping.This work was realized within the activities of CNR, Gruppo Nazionale Analisi Funzionale (GNAFA). The author wishes to thank Professor M. D. Intriligator for his many comments, suggestions, and critique.     

Pauli problem and related mathematical problems

We formulate several mathematical modifications of the Pauli problem of recovering the particle state from the results of measuring its coordinates and momenta, including the most abstract statement of the problem for two projection-valued measures. We describe how the Pauli problem reduces to the problem of finding the eigenvectors of certain operators. The performed analysis introduces the problems of modifications of Pauli pairs, and the considered topics and formulated problems require further study. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 3–7, October, 2008.     

Predicting mathematical aptitude for higher education

This present study seeks to predict mathematical aptitude for higher education by examining the relationship between mathematics results from the Caribbean Examinations Council (CXC) general proficiency examination and the results from the General Certificate of Education (GCE) advanced level examination. This present study arose from a more extensive study that used a random sample of 177 high school students in the age cohort 17?19+ from a 7-year state-run, co-educational institution with a broad-based curriculum and a wide catchment area of student intake in The Republic of Trinidad and Tobago, West Indies. Semi-structured interviews and student academic records from the school comprised the survey instruments used for the study. Student grades were quantified using numerical scales. Findings suggest that the comprehension profile (Profile 2) of the CXC general proficiency level mathematics serves as a good predictor of mathematical aptitude for GCE advanced level mathematics. Implications for predicting mathematical aptitude for higher education and related issues are discussed.     

Special elements in a ring related to Drazin inverses

In this article, the existence of the Drazin (group) inverse of an element a in a ring is analysed when a ^m k?=?ka ⁿ , for some unit k and m, n?∈??. The same problem is studied for the case when a*?=?ka ^m k ^?1 and for the {k, s?+?1}-potent elements. In addition, relationships with other special elements of the ring are also obtained.     

The mathematical description of uniformity and related theorems

Uniform index is a conception that can describe the uniformity of a finite point set in a polyhedron, and is closely related to chaos. In order to study uniform index, the concept of contained uniform index is defined, which is similar to uniform index and has good mathematical properties. In this paper, we prove the convergence of the contained uniform index, and develop the base of proving the convergence of uniform index.