João Nuno Tavares

CHEN INTEGRALS, GENERALIZED LOOPS AND LOOP CALCULUS

We use Chen iterated line integrals to construct a topological algebra of separating functions on the group of loopsLℳp. has a Hopf algebra structure which allows the construction of a group structure on its spectrum. We call this topological group the group of generalized loops Then we develop a loop calculus, based on the end point and area derivative operators, providing a rigorous mathematical treatment of the early heuristic ideas of Gambini, Trias and also Mandelstam, Makeenko and Migdal. Finally, we define a natural action of the “pointed” diffeomorphism group Diffp(ℳ) on , and consider a variational derivative which allows the construction of homotopy invariants. This formalism is useful for constructing a mathematical theory of loop representation of gauge theories and quantum gravity.

About Cartan geometrization of non-holonomic mechanics

Abstract Following the ideas of Elie Cartan (1928), we use Cartan’s equivalence method and the notion of Cartan’s affine generalized space and development to geometrize non-holonomic mechanics.

Recursive circle packing problems

This paper presents a class of packing problems where circles may be placed either inside or outside other circles, the whole set being packed in a rectangle. This corresponds to a practical problem of packing tubes in a container. Before being inserted in the container, tubes may be put inside other tubes in a recursive fashion. A variant of the greedy randomized adaptive search procedure is proposed for tackling this problem, and its performance is assessed in a set of benchmark instances.

A new approach to sex estimation using the mandibular canine index

Rao etu2002al.s mandibular canine index (MCI) is a simple odontometric method which uses the mandibular canine as the key to sex estimation. This index is defined as the ratio between the right canine mesiodistal dimension and the mandibular canine arch width. The aim of this study was to contribute to sex estimation using dental techniques by analysing the MCI efficiency, and to propose a new approach for its use. Measurements were taken from 120 plaster casts (70 females) in the 16–30 year age group. Although statistically significant sexual dimorphism was observed in both the mesiodistal dimension and the mandibular canine arch width, the MCI showed a low accuracy in sex classification (54.2% correct identifications). This accuracy was improved to 64.2% using receiver operating characteristics curve analysis. Yet, despite the better accuracy, these results reinforce the idea that the MCI may not be particularly useful in sex prediction, since it may not reflect the same degree of sexual dimorphism as its absolute measures.

Complexification of gauge theories

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification Cc of G. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.

Mathematical models in cancer therapy

In this article, deterministic mathematical models are derived from biochemical models within a human cell in two distinct cases, for comparison: healthy cell and cancerous cell. The former model is based in the cell cycle model by Novak and Tyson and its adaptation by Conradie, and makes use of the MAPK cascade pathway and the PI3K/AKT pathway for signalling transduction, to create a wider updated model for the regulation of a healthy cell. The latter model, for the cancer cell, is derived from the healthy cell model by altering specific pathways and interpreting the outcome in the light of literature in cancer. This last study is done in two approaches: simulation of common deregulations and specific cancer simulation, colon cancer. After studying both models, we propose targeting therapies and simulate their consequences. We thus explore mathematical modelling efficacy and usefulness in providing enough information from which to derive ideas for therapies. The purpose is to validate mathematics, once again, as a powerful tool with which one can model the underlying nature of chaotic systems and extract useful conclusions to real-life problems.

Minisymposium: 10 Years of Portuguese Study Groups with Industry

This minisymposium provided an overview of the implementation and evolution of European Study Groups (ESGI) in Portugal, describing our experience and some industrial problems dealt with, the challenges that had to be overcome and examples of successful and less successful stories.

Generalized coordinates on the phase space of Yang-Mills theory

We study the suitability of complex Wilson loop variables as (generalized) coordinates on the physical phase space of SU(2) Yang-Mills theory. To this end, we construct a natural one-to-one map from the physical phase space of the Yang-Mills theory with compact gauge group G to a subspace of the physical configuration space of the complex -Yang-Mills theory. Together with a recent result by Ashtekar and Lewandowski this implies that the complex Wilson loop variables form a complete set of generalized coordinates on the physical phase space of SU(2) Yang-Mills theory. They also form a generalized canonical loop algebra. Implications for both general relativity and gauge theory are discussed.