Revistes Catalanes amb Accés Obert

Butlletí de la Societat Catalana de Matemàtiques
Vol. 21, núm. 1, 2006. Pàg. 157–159
English summaries
Pilar Bayer
Poincaré’s contributions to arithmetics
Poincaré’s research on fuchsian groups, automorphic functions, etc. is related to
that of the German school of his time. The influence of these topics on number
theory has increased over the years. Entire chapters of the mathematics of the
XIXth century and of the beginnings of the XXth century have found a natural
continuation in the present study of arithmetic models, in a way that makes
these difficult to understand without taking the classics into account. Poincaré
himself considered his contribution to arithmetics as reduced to the theory
of forms, but, as we shall see, many others of his works have an important
influence on the study of diophantine problems still today.
Keywords: fuchsian group, automorphic (fuchsian) function, abelian function,
quadratic form.
MSC2000 Subject Classification: 11-01, 11-03, 11-D, 11-E, 11-G, 14-G.
Sergi Elizalde
Combinatorics and biology: inference functions and sequence alignment
In this paper we show examples of applications of combinatorial tools to
some problems in computational biology. Statistical models are used to solve
158
important problems in biology, such as determining which parts of the genome
are translated to proteins, or how a DNA sequence evolved into another one
through a series of mutations, insertions and deletions. Each possible answer
has a certain probability that depends on the model parameters. When these
are known, the most likely answer, called explanation, is obtained by solving a
combinatorial optimization problem. The map that sends each observation to
its corresponding explanation is called an inference function.
In this paper we give an upper bound on the number of inference functions
of any directed graphical model. This bound is polynomial on the size of the
model, for a fixed number of parameters, thus improving an exponential upper
bound given by Pachter and Sturmfels. Then we apply this bound to a model
for sequence alignment that is used in computational biology, and we show
that in this case our bound is asymptotically tight.
Keywords: inference functions, graphical models, sequence alignment, Newton
polytope.
MSC2000 Subject Classification: 62F15, 52C45, 52B20, 62P10, 52B05.
Paul R. Halmos
How to write mathematics
In this celebrated article, Halmos exposes his personal view, full of common
sense, on several issues related with the writing activity in mathematics, either
of a global character (planning, organization, attitude towards readers) or of
more local import such as language usage, notation, punctuation symbols, etc.
Keywords: writing, style, exposition, communication, organization.
MSC2000 Subject Classification: 00A99.
Josep Pla i Carrera
The algebra of paper folding
This paper, following the example of the ruler and compass geometrical con-
structions, analyzes the lines and points that can be obtained with paperfolding.
The kernel of the work is a general definition of paperfolding geometrical
constructions. Three geometries of the plane are produced by appropriate
concretions of one of the constructions: the geometry of ruler and metric
English summaries 159
transporter, that of ruler and compass and, finally, the geometry of ruler,
circle and conic sections. Then, the paper proves that the latter corresponds to
numerical solutions of cubic and quartic equations.
Keywords: constructible geometry, ruler and compass, paper folding, algebra
of paper folding, origami.
MSC2000 Subject Classification: 51-01.


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Revistes Catalanes amb Accés Obert