Explicit MDS Codes with Complementary Duals

In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD codes have applications in communication system, side-channel attack (SCA) and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case

On some aspects of polynomial dynamical systems

The aim of this work is to study exact algebraic criteria local/global observability ([HK77], [Ino77]) for polynomial dynamical system by means of algebraic geometry and computational commutative algebra in the vein of [SR76], [Son79a], [Son79b], [Bai80], [Bai81], [Bar95], [Bar99], [Nes98], [Tib04], [KO13], [Bar16]. A key point in this topic is to work with polynomials with real coefficients and their real roots instead of their complex roots, as it is usually the case ([CLO15], [KR00]). A central concept is then the real radical of an ideal [BN93], [Neu98], [LLM+13], along with the Krivine- Dubois-Risler real nullstellensatz for polynomial rings [Kri64], [Dub70], [Ris70], [BCR98]. Underestimating this point leads to incorrect results (see, e.g. [Bar16] remark on [KO13]). This thesis is therefore devoted to set the necessary algebraic tools in the right context and level of generality (i.e. real algebra and real algebraic geometry) for applications to our dynamical systems and to further develop their exploit in this context. The first two chapters set the algebraic and algebraic geometry preliminaries. The third chapter is devoted to the applications of the previous algebraic concepts to the study of the ob- servability of polynomial dynamical systems. In the last chapter an approach to the construction of Lyapunov funtions to prove stability in estimation problems is presented

Design of teacher assistance tools in an exploratory learning environment for algebraic generalisation

The MiGen project is designing and developing an intelligent exploratory environment to support 11-14 year-old students in their learning of algebraic generalisation. Deployed within the classroom, the system also provides tools to assist teachers in monitoring students' activities and progress. This paper describes the architectural design of these Teacher Assistance tools and gives a detailed description of one such tool, focussing in particular on the research challenges faced, and the technologies and approaches chosen to implement the necessary functionalities given the context of the project

A Mathematical Framework for Agent Based Models of Complex Biological Networks

Agent-based modeling and simulation is a useful method to study biological phenomena in a wide range of fields, from molecular biology to ecology. Since there is currently no agreed-upon standard way to specify such models it is not always easy to use published models. Also, since model descriptions are not usually given in mathematical terms, it is difficult to bring mathematical analysis tools to bear, so that models are typically studied through simulation. In order to address this issue, Grimm et al. proposed a protocol for model specification, the so-called ODD protocol, which provides a standard way to describe models. This paper proposes an addition to the ODD protocol which allows the description of an agent-based model as a dynamical system, which provides access to computational and theoretical tools for its analysis. The mathematical framework is that of algebraic models, that is, time-discrete dynamical systems with algebraic structure. It is shown by way of several examples how this mathematical specification can help with model analysis.Comment: To appear in Bulletin of Mathematical Biolog