Jorge Soto-Andrade

Self-reference and fixed points: A discussion and an extension of Lawvere's Theorem

We consider an extension of Lawveres Theorem showing that all classical results on limitations (i.e. Cantor, Russel, Godel, etc.) stem from the same underlying connection between self-referentiality and fixed points. We first prove an even stronger version of this result. Secondly, we investigate the Theorems converse, and we are led to the conjecture that any structure with the fixed point property is a retract of a higher reflexive domain, from which this property is inherited. This is proved here for the category of chain complete posets with continuous morphisms. The relevance of these results for computer science and biology is briefly considered.

Understanding the parts in terms of the whole.

Abstract Metabolism is usually treated as a set of chemical reactions catalysed by separate enzymes. However, various complications, such as transport of molecules across membranes, physical association of different enzymes, giving the possibility of metabolite channelling, need to be taken into account. More generally, a proper understanding of the nature of life will require metabolism to be treated as a complete system, and not just as a collection of components. Certain properties of metabolic systems, such as feedback inhibition of the first committed step of a pathway, make sense only if one takes a broader view of a pathway than is usual in textbooks, so that one can appreciate ideas such as regulation of biosynthesis according to demand. More generally still, consideration of metabolism as a whole puts the emphasis on certain systemic aspects that are crucial but which can pass unnoticed if attention is always focussed on details. For example, a living organism, unlike any machine known or conceivable at present, makes and maintains itself and all of its components. Any serious investigation of how this can be possible implies an infinite regress in which each set of enzymes needed for the metabolic activity of the organism implies the existence of another set of enzymes to maintain them, which, in turn, implies another set, and so on indefinitely. Avoiding this implication of infinite regress represents a major challenge for future investigation.

A Bruhat decomposition of the group Sl∗(2,A)☆

Let F be a locally compact topological field, let V be a two-dimensional F -vector space equipped with a quadratic form Q and letG = SL(2,F ). Then, in [5] Weil gave a construction from which one may deduce the existence of a natural unitary represent σ of G on the spaceL2(V ), such that the operators σ(x), x ∈ G, commute with the natura action of the orthogonal group O(Q) onL2(V ). Using this fact to decompose σ and then varying (V ,Q), one may obtain all irreducible complex representations of G unlessF is a p-adic field withp = 2. Further, this parametrization of the complex dual of G is in all cases compatible with Langlands functoriality. The representation σ was originally constructed by using the representation theory of the Heisenberg group. However, be thought of, a posteriori, as simply a map from G to the space of bounded O(Q) intertwining operators onL2(V ) which preserves the usual Bruhat presentation of G. This point of view, originally communicated to the second author by P. Cartier, sugges possibility of replacingF by a locally compact topological ring A with involution ∗ and the groupSL(2,F ) by a “noncommutative∗-analogue” groupSl∗(2,A). For example, one may takeA to be the ringMn(F) with a∗ = at, a ∈ A, and, in that case, the group Sl∗(2,A) is just the symplectic group Sp(2n,F ). In the case that F is a finite field, this approach lead to an explicit construction of a complete set of irreducible complex representation groupSp(4,F ), as in [4]. If one wishes to apply this method to other rings with involution A, one needs at a min imum to check that the group Sl∗(2,A) admits what we call here a set of Bruhat genera

The effect of analogies on learning to solve algebraic equations

A total of 236 seventh grade students who had never been taught algebraic equations before, attending 10 Chilean schools of varying socio-economic status, were randomly divided into two groups at each school. The students in one group watched a 15-minute video teaching them how to solve five different first-degree linear equations using a traditional symbolic strategy, while in the other group, the students watched a 15-minute video teaching them how to solve the same equations using four analogies for solving an equation: a two-pan balance for the equals sign, a box for a variable, candies for numbers, and guessing the number of candies inside a box. The students were then tested on 12 equation solving problems, all of them written, using only symbolic notation. The group that watched the analogies video performed significantly better. Students with a below-average mathematics GPA who watched the analogies video did as well as students with an above-average GPA who watched the symbolic strategy video. Students who watched the analogies video also reached a better conceptual understanding, were better at making generalizations, did significantly better on reasoning problems involving equations, and had a better affective reaction.

Bruhat Presentations for ∗-Classical Groups

We introduce in a general setting, the unitary similitude groups associated to ϵ-hermitian form over a ring A with involution. We concentrate next in the rank 2 case. The classical presentation of GL(2, F), F a field, originating from its Bruhat decomposition can be extended to these unitary similitude groups. Furthermore, the multiplier associated to ϵ-hermitian forms affords a signed non commutative version of the classical determinant of 2 by 2 matrices, which we recover exploiting Grassmanns approach to determinants. Finally, we present various examples of our groups and give some categorical properties of them.

Strategies used by students on a massively multiplayer online mathematics game

We analyze the logs of an online mathematics game tournament, played simultaneously by thousands of students. Nearly 10,000 students, coming from 356 schools from all regions in Chile, registered to the fourth tournament instance. The children play in teams of 12 students from the same class, and send their personal bets to a central server every 2 minutes. Each competition lasts about one clock hour and takes place within school hours. Students are pre-registered and trained by their school teacher. The teacher is responsible for reviewing curriculum contents useful for improving performance at the game and coaches students participating in trial tournaments taking place a few weeks before the national tournament. All bets are recorded in a database that enables us to analyze later the sequence of bets made by each student. Using cluster analysis with this information, we have identified three types of players, each with a well-defined strategy.

Teaching modeling skills using a massively multiplayer online mathematics game

One important challenge in mathematics education is teaching modeling skills. We analyze the logs from a game-based learning system used in a massively multiplayer online tournament. Students had to detect an input–output pattern across 20 rounds. For each round, they received an input and had 2 minutes to predict the output by selecting a binary option (2 points if correct, −1 otherwise), or writing a model (4 points if model prediction was correct, −4 otherwise), or refraining (1 point). Thousands of 3rd to 10th grade students from hundreds of schools simultaneously played together on the web. We identified different types of players using cluster analysis. From 5th grade onwards, we found a cluster of students that wrote models with correct predictions. Half of the 7th to 10th grade students that detected patterns were able to express them with models. The analysis also shows diffusion within the teams of modeling strategies for simple patterns.

On the Role of Corporeality, Affect, and Metaphoring in Problem-Solving

We explore the role of corporeality, affect, and metaphoring in problem-solving. Our experimental research background includes average and gifted Chilean high school students, juvenile offenders, prospective teachers, and mathematicians, tackling problems in a workshop setting. We report on observed dramatic changes in attitude toward mathematics triggered by group working for long enough periods on problem-solving, and we describe ways in which (possibly unconscious) metaphoring determines how efficiently and creatively you tackle a problem. We argue that systematic and conscious use of metaphoring may significantly improve performance in problem-solving. The effect of the facilitator ignoring the solution of the problem being tackled is also discussed.

Intertwining operators for L 2(E)

Let (E,Q) be a finite dimensional quadratic vector space over a finite field. For the natural representation -π of the isometry group G of (E,Q) in the space L 2(E) of all complex valued functions on E, we analyse when the intertwining algebra of π is generated by just one averaging operator.

Programming Paradigms and Mind Metaphors: Convergence and Cross-fertilization in the Study of Cognition

This paper describes a notable convergence between biological organization and programming language abstractions. Our aim is to explore possibilities of cross-fertilization, at both conceptual and empirical levels, towards the understanding of what cognition and cognitive systems might be.