Matemàtica i música amb ADN

Treball de recerca realitzat per un alumne d’ensenyament secundari i guardonat amb un Premi CIRIT per fomentar l'esperit científic del Jovent l’any 2005. Estudi sobre l’ADN que té com a finalitat conèixer introductòriament l’utilització del càlcul matemàtic computacional en les investigacions sobre aquest. Els objectius de l’estudi són per una part, conèixer què és l'ADN i quins són els seus mecanismes de duplicació i de transmissió de la informació genètica, així com el paper d'altres molècules que intervenen en aquest procés ; també s’estudia quins han estat els processos de la cèl·lula que l'ésser humà ha estat capaç de copiar o imitar. A partir d’aquesta introducció, es vol conèixer què s'entén concretament per computació amb ADN i alguns dels problemes matemàtics que s'han resolt, així com algunes aplicacions de l'ADN en altres camps. La recerca ha permès arribar a diverses conclusions. Primerament que l'ADN és un excel·lent candidat per poder fer càlculs matemàtics. En segon lloc, tot i que en el present treball no se solucionen problemes computacionalment difícils es mostra la capacitat de les molècules d'ADN per resoldre problemes. En tercer lloc, l'interès mostrat per importants empreses dedicades a la informàtica fa més esperançador que en un futur hi pugui haver ordinadors que funcionin amb molècules d'ADN. Finalment, es demostra que les matemàtiques, la informàtica i la biologia són tres camps que estan interrelacionats. Per tal de trencar una mica amb la serietat del treball, s'acaba descrivint una manera de posar música a les cadenes d'ADN, i es mostren alguns resultats com són les músiques associades als 24 cromosomes humans, així com les corresponents a 29 proteïnes.

Testing Cayley graph densities

We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: Given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an m-generated group is amenable if and only if the density of the corresponding Cayley graph equals to 2m. We test amenable and non-amenable groups, and also groups for which amenability is unknown. In the latter class we focus on Richard Thompson’s group F.

Pure braid subgroups of braided Thompson's groups

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Approximating subtree distances between Phylogenies

We give a 5-approximation algorithm to the rooted Subtree-Prune-and-Regraft (rSPR) distance between two phylogenies, which was recently shown to be NP-complete by Bordewich and Semple [5]. This paper presents the first approximation result for this important tree distance. The algorithm follows a standard format for tree distances such as Rodrigues et al. [24] and Hein et al. [13]. The novel ideas are in the analysis. In the analysis, the cost of the algorithm uses a \cascading" scheme that accounts for possible wrong moves. This accounting is missing from previous analysis of tree distance approximation algorithms. Further, we show how all algorithms of this type can be implemented in linear time and give experimental results.

Distortion of wreath products in some finitely presented groups

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Weak systems of Gandy, Jensen and Devlin

In Part I, we formulate and examine some systems that have arisen in the study of the constructible hierarchy; we find numerous transitive models for them, among which are supertransitive models containing all ordinals that show that Devlin's system BS lies strictly between Gandy's systems PZ and BST'; and we use our models to show that BS fails to handle even the simplest rudimentary functions, and is thus inadequate for the use intended for it in Devlin's treatise. In Part II we propose and study an enhancement of the underlying logic of these systems, build further models to show where the previous hierarchy of systems is preserved by our enhancement; and consider three systems that might serve for Devlin's purposes: one the enhancement of a version of BS, one a formulation of Gandy-Jensen set theory, and the third a subsystem common to those two. In Part III we give new proofs of results of Boffa by constructing three models in which, respectively, TCo, AxPair and AxSing fail; we give some sufficient conditions for a set not to belong to the rudimentary closure of another set, and thus answer a question of McAloon; and we comment on Gandy's numerals and correct and sharpen other of his observations.

On moduli of smoothness of fractional order

In this paper we consider the properties of moduli of smoothness of fractional order. The main result of the paper describes the equivalence of the modulus of smoothness and a function from some class.

Cat(0) and Cat (-1) dimensions of torsion free

We show that a particular free-by-cyclic group has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. We also classify the minimal proper 2-dimensional CAT(0) actions of this group; they correspond, up to scaling, to a 1-parameter family of locally CAT(0) piecewise Euclidean metrics on a fixed presentation complex for the group. This information is used to produce an infinite family of 2-dimensional hyperbolic groups, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.

Shellability of noncrossing partition lattices

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three.

Intersection Pairing and intersection motive of surfaces

We construct the Chow motive modelling intersection co-homology of a proper surface. We then study its functoriality properties. Using Murre's decompositions of the motive of a desingularization into KÄunneth components [Mr1], we show that such decompositions exist also for the intersection motive.

Limit Cycles for a class of three dimensional polynomial differential systems

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On the upper bound of the number of limit cycles obtained by the second order averaging method I

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On the upper bound of the number of limit cycles obtained by the second order averaging method II

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The density of injective endomorphisms of a free group

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Una Aproximació a la construcció significativa del coneixement matemàtic a l'ESO

This study displays and analyzes the contents of the Mathematics subject in ESO’s second cycle from a constructivist perspective. This analysis has been carried out by contrasting two groups of participants (control group and experimental group). These groups were formed by a sample of 240 students between the ages of 14 and 16 from four different educational centres of the Osona area. Research – Action methodology has been employed, combining quantitative techniques (statistical study with the SPSS package) with qualitative analysis (transcriptions of interviews and discussion group). This study has been carried out after years of classroom observation, reflection and action. The theoretical framework employed is a cognitive one, based on Ausubel’s Significative Learning Theory. Quantitative analysis shows how the researcher’s design improves, on the one hand, the students’ academic motivation and, on the other hand, their comprehensive memory, enabling them to achieve a more significant learning of the subjects’ contents. Furthermore, our analysis shows that the proposed method is more comprehensive than those employed by teachers collaborating with control groups. The main aim of the qualitative analysis is that of identifying the elements which configure the programme and contribute to an improvement of the aspects mentioned above. The key elements here are: co-operation as the basis of group dynamics; the employment, in some cases, of easily handled materials; the type of interaction between teacher and students, where, through open discussion, students are lead by teaching staff towards the course objectives; induction, that is, deducing formulae by initially using examples which are close to the students’ knowledge and experience or taken from everyday life (what we could call “down-top” mathematics). We should add here that the qualitative analysis does not only corroborate the results obtained by quantitative techniques, but also displays an increase of motivation in teaching staff. Teachers did show a positive attitude and welcomed the use and development of these materials in the next academic year. Finally, we discuss possible directions for further research.