An alternative to the Pythagorean rule? Reevaluating Problem 1 of cuneiform tablet BM 34 568

The first problem of the Seleucid mathematical cuneiform tablet BM 34 568 calculates the diagonal of a rectangle from its sides without resorting to the Pythagorean rule. For this reason, it has been a source of discussion among specialists ever since its first publication. but so far no consensus in relation to its mathematical meaning has been attained. This paper presents two new interpretations of the scribe`s procedure. based on the assumption that he was able to reduce the problem to a standard Mesopotamian question about reciprocal numbers. These new interpretations are then linked to interpretations of the Old Babylonian tablet Plimpton 322 and to the presence of Pythagorean triples in the contexts of Old Babylonian and Hellenistic mathematics. (C) 2007 Elsevier Inc. All rights reserved.

Singular Perturbation for the Discounted Continuous Control of Piecewise Deterministic Markov Processes

This paper deals with the expected discounted continuous control of piecewise deterministic Markov processes (PDMP`s) using a singular perturbation approach for dealing with rapidly oscillating parameters. The state space of the PDMP is written as the product of a finite set and a subset of the Euclidean space a""e (n) . The discrete part of the state, called the regime, characterizes the mode of operation of the physical system under consideration, and is supposed to have a fast (associated to a small parameter epsilon > 0) and a slow behavior. By using a similar approach as developed in Yin and Zhang (Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Applications of Mathematics, vol. 37, Springer, New York, 1998, Chaps. 1 and 3) the idea in this paper is to reduce the number of regimes by considering an averaged model in which the regimes within the same class are aggregated through the quasi-stationary distribution so that the different states in this class are replaced by a single one. The main goal is to show that the value function of the control problem for the system driven by the perturbed Markov chain converges to the value function of this limit control problem as epsilon goes to zero. This convergence is obtained by, roughly speaking, showing that the infimum and supremum limits of the value functions satisfy two optimality inequalities as epsilon goes to zero. This enables us to show the result by invoking a uniqueness argument, without needing any kind of Lipschitz continuity condition.

The distribution and morphological characteristics of catecholaminergic cells in the brain of monotremes as revealed by tyrosine hydroxylase immunohistochemistry

The present study describes the distribution and cellular morphology of catecholaminergic neurons in the CNS of two species of monotreme, the platypus (Ornithorhynchus anatinus) and the short-beaked echidna (Tachyglossus aculeatus). Tyrosine hydroxylase immunohistochemistry was used to visualize these neurons. The standard A1-A17, C1-C3 nomenclature was used for expediency, but the neuroanatomical names of the various nuclei have also been given. Monotremes exhibit catecholaminergic neurons in the diencephalon (All, A12, A13, A14, A15), midbrain (A8, A9, A10), rostral rhombencephalon (A5, A6, A7), and medulla (A1, A2, C1, C2). The subdivisions of these neurons are in general agreement with those of other mammals, and indeed other amniotes. Apart from minor differences, those being a lack of A4, A3, and C3 groups, the catecholaminergic system of monotremes is very similar to that of other mammals. Catecholaminergic neurons outside these nuclei, such as those reported for other mammals, were not numerous with occasional cells observed in the striatum. It seems unlikely that differences in the sleep phenomenology of monotremes, as compared to other mammals, can be explained by these differences. The similarity of this system across mammalian and amniote species underlines the evolutionary conservatism of the catecholaminergic system. Copyright (C) 2002 S. Karger AG, Basel.

The distribution and morphological characteristics of cholinergic cells in the brain of monotremes as revealed by ChAT immunohistochemistry

The present study employs choline acetyltransferase (ChAT) immunohistochemistry to identify the cholinergic neuronal population in the central nervous system of the monotremes. Two of the three extant species of monotreme were studied: the platypus (Omithorhynchus anatinus) and the short-beaked echidna (Tachyglossus aculeatus). The distribution of cholinergic cells in the brain of these two species was virtually identical. Distinct groups of cholinergic cells were observed in the striatum, basal forebrain, habenula, pontomesencephalon, cranial nerve motor nuclei, and spinal cord. In contrast to other tetrapods studied with this technique, we failed to find evidence for cholinergic cells in the hypothalamus, the parabigeminal nucleus (or nucleus isthmus), or the cerebral cortex. The lack of hypothalamic cholinergic neurons creates a hiatus in the continuous antero-posterior aggregation of cholinergic neurons seen in other tetrapods. This hiatus might be functionally related to the phenomenology of monotreme sleep and to the ontogeny of sleep in mammals, as juvenile placental mammals exhibit a similar combination of sleep elements to that found in adult monotremes. Copyright (C) 2002 S. Karger AG, Basel.

How to teach mathematics with MOODLE

The aim of this article is to present a Project in the Oporto’s Institute of Accounting and Administration, which pretends to contribute for a change in the way of teaching and learning Mathematics. One of the main objectives of this project is to innovate the teaching and learning processes, exploring technologies as a pedagogical resource and to induce higher motivation to students, improve the rate of success and make available to students a set of materials adapted to their needs. This concern is justified due to the fact that students have a weak preparation, without consolidated basis. Since the year 2007/2008 the courses were adjusted to the Bologna process, which requires several changes in teacher’s and student’s roles, methodologies and assessment. The number of weekly classes has been reduced, so it was necessary to develop new strategies and methodologies to support the student. With the implementation of the Bologna Process in the Accounting degree, we felt a great need to provide other types of activities to students. To complement our theoretical and practical classes we have developed a project called MatActiva based on the Moodle platform offered by PAOL - Projecto de Apoio On-Line (Online Support Project). Moodle allows us to use the language TEX to create materials that use mathematical symbols. Using this functionality, we created a set of easy to use interactive resources. In MatActiva project, the students have access to a variety of different materials. We have followed a strategy that makes the project compatible with the theoretical and practical subjects/classes, complementing them. To do so, we created some resources, for instance multiple-choice tests, which are the most accessed by the students. These tests can be realized and corrected on-line and for each wrong answer there is a feedback with the resolution. We can find other types of resources: diagnostic tests, theoretical notes. There are not only the pre-requirements for subjects mathematics, but also materials to help students follow up the programs. We also developed several lessons. This activity consists of a number of pages, where each page has contents and leads to other pages, based on the student's progress. The teacher creates the choices and determines the next page that the student will see, based upon their knowledge. There is also an area of doubts, where the students can place all the mathematical doubts they have, and a teacher gives the answers or clues to help them in their work. MatActiva also offers an area where we can find some humour, curiosities, contests and games including mathematical contents to test the math skills, as well as links to pages about mathematical contents that could be useful for the study. Since ISCAP receives ERASMUS students and some of them attend mathematics, we developed some materials in English, so they can also use MatActiva. The main objectives of our project are not only to bring success in the subjects of mathematics, but also to motivate the students, encourage them to overcome theirs difficulties through an auto-study giving them more confidence and improve their relationship with the mathematics as well as the communication between students and teachers and among students.

Entropy analysis of DNA code dynamics in human chromosomes

Deoxyribonucleic acid, or DNA, is the most fundamental aspect of life but present day scientific knowledge has merely scratched the surface of the problem posed by its decoding. While experimental methods provide insightful clues, the adoption of analysis tools supported by the formalism of mathematics will lead to a systematic and solid build-up of knowledge. This paper studies human DNA from the perspective of system dynamics. By associating entropy and the Fourier transform, several global properties of the code are revealed. The fractional order characteristics emerge as a natural consequence of the information content. These properties constitute a small piece of scientific knowledge that will support further efforts towards the final aim of establishing a comprehensive theory of the phenomena involved in life.

Mapping stability : real rational maps of degree zero

In memory of our beloved Professor José Rodrigues Santos de Sousa Ramos (1948-2007), who João Cabral, one of the authors of this paper, had the honor of being his student between 2000 and 2006, we wrote this paper following the research by experimentation, using the new technologies to capture a new insight about a problem, as him so much love to do it. His passion was to create new relations between different fields of mathematics. He was a builder of bridges of knowledge, encouraging the birth of new ways to understand this science. One of the areas that Sousa Ramos researched was the iteration of maps and the description of its behavior, using the symbolic dynamics. So, in this issue of this journal, honoring his memory, we use experimental results to find some stable regions of a specific family of real rational maps, the ones that he worked with João Cabral. In this paper we describe a parameter space (a,b) to the real rational maps fa,b(x) = (x2 −a)/(x2 −b), using some tools of dynamical systems, as the study of the critical point orbit and Lyapunov exponents. We give some results regarding the stability of these family of maps when we iterate it, specially the ones connected to the order 3 of iteration. We hope that our results would help to understand better the behavior of these maps, preparing the ground to a more efficient use of the Kneading Theory on these family of maps, using symbolic dynamics.

Experimental signatures of fermiophobic Higgs bosons

The most general Two Higgs Doublet Model potential without explicit CP violation depends on 10 real independent parameters. Excluding spontaneous CP violation results in two 7 parameter models. Although both models give rise to 5 scalar particles and 2 mixing angles, the resulting phenomenology of the scalar sectors is different. If flavour changing neutral currents at tree level are to be avoided, one has, in both cases, four alternative ways of introducing the fermion couplings. In one of these models the mixing angle of the CP even sector can be chosen in such a way that the fermion couplings to the lightest scalar Higgs boson vanishes. At the same time it is possible to suppress the fermion couplings to the charged and pseudo-scalar Higgs bosons by appropriately choosing the mixing angle of the CP odd sector. We investigate the phenomenology of both models in the fermiophobic limit and present the different branching ratios for the decays of the scalar particles. We use the present experimental results from the LEP collider to constrain the models.

Cognitive models for the concept of angle

Tese de doutoramento em Filosofia

Automorphisms of partial endomorphism semigroups

Publicationes Mathematicae Debrecen

Welfare effects of entry into international markets with licensing

We study the effects of entry of a foreign firm on domestic welfare in the presence of licensing, when the entrant is technologically inferior to the incumbent. We show that foreign entry increases domestic welfare for intermediate (respectively, sufficiently large) technological differences between the firms under licensing with fixed fee (respectively, output royalty).

Teaching mathematics using Massive Open Online Courses

Distance learning - where students take courses (attend classes, get activities and other sort of learning materials) while being physically separated from their instructors, for larger part of the course duration - is far from being a “new event”. Since the middle of the nineteenth century, this has been done through Radio, Mail and TV, taking advantage of the full educational potential that these media resources had to offer at the time. However, in recent times we have, at our complete disposal, the “magic wonder” of communication and globalization - the Internet. Taking advantage of a whole new set of educational opportunities, with a more or less unselfish “look” to economic interests, focusing its concern on a larger and collective “welfare”, contributing to the development of a more “equitable” world, with regard to educational opportunities, the Massive Open Online Courses (MOOCs) were born and have become an important feature of the higher education in recent years. Many people have been talking about MOOCs as a potential educational revolution, which has arrived from North America, still growing and spreading, referring to its benefits and/or disadvantages. The Polytechnic Institute of Porto, also known as IPP, is a Higher Education Portuguese institution providing undergraduate and graduate studies, which has a solid history of online education and innovation through the use of technology, and it has been particularly interested and focused on MOOC developments, based on an open educational policy in order to try to implement some differentiated learning strategies to its actual students and as a way to attract future ones. Therefore, in July 2014, IPP launched the first Math MOOC on its own platform. This paper describes the requirements, the resulting design and implementation of a mathematics MOOC, which was essentially addressed to three target populations: - pre-college students or individuals wishing to update their Math skills or that need to prepare for the National Exam of Mathematics; - Higher Education students who have not attended in High School, this subject, and who feel the need to acquire basic knowledge about some of the topics covered; - High School Teachers who may use these resources with their students allowing them to develop teaching methodologies like "Flipped Classroom” (available at http://www.opened.ipp.pt/). The MOOC was developed in partnership with several professors from several schools from IPP, gathering different math competences and backgrounds to create and put to work different activities such video lectures and quizzes. We will also try to briefly discuss the advertising strategy being developed to promote this MOOC, since it is not offered through a main MOOC portal, such as Coursera or Udacity.

Determinants of success of Nova SBE’s undergraduate students

Applied Policy Analysis major

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.