Mumford curves and Mumford groups in positive characteristic

A Mumford group is a discontinuous subgroup $\Gamma$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $\Gamma$ is a curve of genus 0. As abstract group $\Gamma$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to our paper "Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and K.~Kontogeorgis.Comment: 62 page

Topological noetherianity for cubic polynomials

Let $P_3(\mathbf{C}^{\infty})$ be the space of complex cubic polynomials in infinitely many variables. We show that this space is $\mathbf{GL}_{\infty}$-noetherian, meaning that any $\mathbf{GL}_{\infty}$-stable Zariski closed subset is cut out by finitely many orbits of equations. Our method relies on a careful analysis of an invariant of cubics introduced here called q-rank. This result is motivated by recent work in representation stability, especially the theory of twisted commutative algebras. It is also connected to certain stability problems in commutative algebra, such as Stillman's conjecture.Comment: 13 page

Miroirs de l’autorégulation de l’apprentissage : les dilemmes des formateurs d’enseignants

Les formateurs d’enseignants font face à des demandes croissantes concernant la promotion et l’autorégulation des apprentissages de leurs étudiants. De telles demandes peuvent entraîner plusieurs dilemmes professionnels résultant de conflits de croyances et de conditions d’enseignement qui prévalent. La présente étude examine comment les formateurs d’enseignants sont engagés dans la promotion de l’autorégulation de l’apprentissage, dans le contexte néerlandais de formation des enseignants. Cette étude examine le sens que revêt pour les formateurs d’enseignants, une utilisation active des approches de l’autorégulation de l’apprentissage. Elle dégage plusieurs dilemmes professionnels auxquels font face les formateurs d’enseignants. Les difficultés que ces derniers vivent lors de l’implantation d’une telle approche semblent être liées à leurs propres orientations, expériences et réflexions professionnelles.Teacher trainers face increasing demands concerning the promotion and the development of self-regulated learning by their students. These types of demands can produce professional dilemmas resulting from conflicts in beliefs and current teaching conditions. This study examined how teacher trainers are involved in the promotion of self-regulated learning in the context of training teachers in the Netherlands. This study examined the meaning that teacher trainers developed through the active use of self-regulated learning approaches. The author notes several professional dilemmas that confront these trainers. The difficulties experienced during the implementation of this approach seemed to be related to their own representations, experiences, and professional reflections.Los formadores de docentes se enfrentan a una demanda creciente en torno a la promoción y a la autoregulación de los aprendizajes de los estudiantes. Tales demandas pueden entrenar varios dilemas profesionales resultantes de conflictos de creencias y de condiciones de enseñanza que prevalen. El presente estudio demuestra cómo los formadores de docentes son comprometidos con la promoción de la autoregulación del aprendizaje, en el contexto neerlandés de formación de docentes. Analiza el significado que tiene una utilización activa de los enfoques de autoregulación y de aprendizaje para los formadores de docentes al mismo tiempo que destaca varios dilemas profesionales que los formadores de docentes enfrentan. Las dificultades que éstos viven al momento de implementar tales enfoques parecen ser ligadas a sus propias orientaciones, experiencias y reflexiones profesionales

Prediction error identification of linear dynamic networks with rank-reduced noise

Dynamic networks are interconnected dynamic systems with measured node signals and dynamic modules reflecting the links between the nodes. We address the problem of \red{identifying a dynamic network with known topology, on the basis of measured signals}, for the situation of additive process noise on the node signals that is spatially correlated and that is allowed to have a spectral density that is singular. A prediction error approach is followed in which all node signals in the network are jointly predicted. The resulting joint-direct identification method, generalizes the classical direct method for closed-loop identification to handle situations of mutually correlated noise on inputs and outputs. When applied to general dynamic networks with rank-reduced noise, it appears that the natural identification criterion becomes a weighted LS criterion that is subject to a constraint. This constrained criterion is shown to lead to maximum likelihood estimates of the dynamic network and therefore to minimum variance properties, reaching the Cramer-Rao lower bound in the case of Gaussian noise.Comment: 17 pages, 5 figures, revision submitted for publication in Automatica, 4 April 201

Variants of the Two Machine Flow Shop Problem connected with factorization of matrix functions

In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these variants the makespan is given and the problem is to find a schedule that meets this makespan, thereby minimizing the infeasibilities of the jobs in a prescribed sense: In the max-variant the maximum infeasibility of the jobs is to be minimized, whereas in the sum-variant the objective is to minimize the sum of the infeasibilities of the jobs. For both variants observations about the structure of the optimal schedules are presented. In particular, it is proved that every instance of these problems has an optimal permutation schedule. It is also shown that the max-variant can be solved by Johnson's Rule. For the sum-variant this is not the case: For solving this problem to optimality something quite different is necessary. Both variants are connected with factorization problems for certain rational matrix functions. The factorizations involved are optimal in some sense and generalize the notion of complete factorization. In this way a connection is established between job scheduling theory on one hand, and mathematical systems theory on the other

Companion based matrix functions: description and minimal factorization

Companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions. Necessary and sufficient conditions are given for a rational matrix function to be companion based. Minimal factorization of such functions is discussed in detail. It is shown that the property of being companion based is hereditary with respect to minimal factorization. Also, the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree. In this context, a connection with the Euclidean algorithm is made. The results apply to canonical Wiener-Hopf factorization as well as to complete factorization. The analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices. The algorithmic aspects of this problem are intimately related to the two machine flow shop problem and Johnson's rule from job scheduling theory