**PLENARY SESSIONS**

José Luis Balcázar (Universitat Politècnica de Catalunya y Universidad
de Cantabria)

Some recent research on Runge-Kutta methods

The study of Runge-Kutta methods has been a very active field of
research since the work of C. Runge (1895), K. Heun (1900) and W. Kutta
(1901). Nowadays, this class of methods has become a fundamental tool in the
practical solution of differential equations and it is present in the most used
packages of software. One can think that there is no much else to study in
relation with RK formulas but however, the development of new hardware and
software technologies that provide more powerful capabilities of computation,
and the numerical simulation of every time more complex real life problems lead
to new requirements to the numerical integrators that demand further
research. As an example we will consider some current problems, such as
those arising in computational aeroacoustic, and we will see how the research on
RK methods can contribute to improve their numerical
simulation.

José
Mourão (Instituto Superior Técnico)

Amílcar Sernadas (IT y Instituto Superior
Técnico)**
**

**THEMATIC SESSIONS**

Mathematical logic, foundations and computing

**Sophistication Revisited**

Kolmogorov complexity measures the amount of information in a string
as the size of the shortest program that computes the string. The Kolmogorov
structure function divides the smallest program producing a string in two parts:
the useful information present in the string, called sophistication if based on
total functions, and the remaining accidental information. We formalize a
connection between sophistication (due to Koppel) and a variation of
computational depth (intuitively the useful or nonrandom information in a
string), prove the existence of strings with maximum sophistication and show
that they are the deepest of all strings.

**Dynamical Systems and Pseudorandom Number
Generation**

There are many problems that are known to be dificult or impossible
to solve using deterministic algorithms. One example is the following: Given two
computer programs, prove that they are equivalent, i.e for same inputs, the
outputs are equals. The normal approach to solve this problems are the *Monte Carlo** methods*. A *Quasi-Monte
Carlo method*. The main purpose of this talk is to introduce a collection of
good pseudorandom sequences generated by dynamical systems from a purely
theoretical point of view as well as in view of diferent applications. We
analyze the pseudorandom sequences using mainly number theoretic methods as for
example exponential sum techniques.

**Towards a canonical classical natural deduction system **

This talk is about a new classical natural deduction system,
presented as a typed lambda-calculus. It is designed to be isomorphic to
Curien-Herbelin's calculus, both at the level of proofs and reduction, and the
isomorphism is based on the correct correspondence between cut (resp.
left-introduction) in sequent calculus, and substitution (resp. elimination) in
natural deduction. It is a combination of Parigot's calculus with the idea of
"coercion calculus" due to Cervesato-Pfenning, accommodating let-expressions in
a surprising way: they expand Parigot's syntactic class of named
terms.

This calculus aims to be the simultaneous answer to three problems.
The first problem is the lack of a canonical natural deduction system for
classical logic. The proposed calculus is not yet another classical calculus,
but rather a canonical reflection in natural deduction of the impeccable
treatment of classical logic by sequent calculus. The second problem is the lack
of a formalization of the usual semantics of Curien-Herbelin's calculus, that
explains co-terms and cuts as, respectively, contexts and hole-filling
instructions. The mentioned isomorphism is the required formalization, based on
the precise notions of context and hole-expression offered by the proposed
calculus. The third problem is the lack of a robust process of
"read-back'' into natural deduction syntax of calculi in the sequent calculus
format, that affects mainly the recent proof-theoretic efforts of derivation
of lambda-calculi for call-by-value. An isomorphic counterpart to the
Q-subsystem of Curien-Herbelin's calculus is derived, obtaining a new
lambda-calculus for call-by-value, combining control and
let-expressions.

Ignacio García-Marco (Universidad de la
Laguna)

**An algorithm for checking whether a simplicial toric ideal is a
complete intersection **

Let* k* be an arbitrary
field and *k*[*x*] = *k*[*x*_{1,} … ,*x _{n}*] and

The ideal IA is a *complete
intersection* if there exists a system of *A*-homogeneous binomials *g*_{1,…, }*g _{s}* such that

If *n* > *m* and *A* = {*d*_{1}*e*_{1,}…, *d _{m}e_{m}*;

The purpose of this work is to provide and implement an algorithm for
determining whether a simplicial toric ideal *I*_{A} is a complete intersection
without having to compute a minimal system of *A*-homogeneous generators of *I _{A}*. This
is a joint work with Isabel Bermejo.

**Applicative Theories and Computational
Complexity**

Abstract: We give a short survey on applicative theories and how they
can be used to characterize classes of computational complexity.

The presentation includes joint work with Isabel Oitavem (CMAF and
DM, FCT-UNL).

**Abstract Algebraic Logic Tools in Program
Development**

Hidden k-logics, as a natural generalization of k-deductive systems,
were introduced by Don Pigozzi and Manuel Martins in 2003 to specify object
oriented software systems. The use of hidden k-logics unifies the study of
several kinds of logics and provides a bridge between abstract algebraic logic
(AAL) and specification theory.

In this talk, we discuss a new application of AAL to program
development. We present an alternative approach to refinement of algebraic
specifications in which signature morphisms are replaced by logic
interpretations. Intuitively,an interpretation is a logic translation which
preserves meaning. Originally used as a tool for studying equivalent algebraic
semantics, this notion has proved to be an effective tool to capture a number of
transformations difficult to deal with in classical terms, such as data
encapsulation and the decomposition of operations into atomic
transactions.

**Effective homology of groups and discrete Morse
theory**

In the talk we will present several algorithms related with the computation of the homology of groups, by means of different techniques of Algebraic Topology. More concretely, we have developed some algorithms which, making use of the effective homology method, construct the homology groups of Eilenberg-MacLane spaces K(G,1) for different groups G, allowing one in particular to determine the homology groups of G. These results can be applied to the computation of homology groups of central extensions and 2-types. Moreover, our initial algorithms have been improved by using discrete Morse theory (concretely, by constructing discrete vector fields which describe the homology of finite cyclic groups).

Ana
Luísa Custódio (Universidade Nova de Lisboa)

**Building Improved Directions of Negative Curvature for Constrained
Optimization**

In this work, we provide an approach to the computation of improved
directions of negative curvature for nonlinearly constrained optimization
problems. In particular, we focus on the use of low cost procedures to improve
directions of negative curvature obtained from a direct factorization of the
Hessian matrix of the objective function. The key feature is that the directions
are computed within the null subspace of the Jacobian matrix of the constraints.
In addition, we show how to include the above procedure within an interior-point
algorithm for constrained optimization.Finally, some numerical experiments
showing the successful performance of our proposal are
presented.

This is a joint work with Javier Cano and Francisco J.
Prieto.

In this work we study a procedure to stabilize Galerkin finite
element approximations to linear evolutionary convection-reaction-diffusion
equations in the convection dominated regime. It is well known that standard
finite element approximations to this kind of equations develop spurious
oscillations when convection dominates diffusion. We propose a postprocessing
that is able to eliminate spurious oscillations at a fixed time. The idea is the
following. One first compute the standard Galerkin approximation at a given time
and then solve a steady convection-reaction-diffusion problem with data based on
the previously computed Galerkin approximation over the same finite element
space but using the SUPG stabilized method. In the second part of the talk we
propose an adaptive algorithm based on postprocessing that is able to compute an
oscillation-free Galerkin approximation over an automatically adapted mesh that
locates the smaller elements at the boundary layers.

This is a joint work with Javier de Frutos and Bosco García-Archilla

In this talk we will concentrate on necessary conditions of
optimality for free time optimal control problems involving control constraints.
We show how recent developed maximum principles for optimal control problems
involving mixed state-control constraints can be successfully applied to derive
new and interesting results for such problems. Other applications of
interest will also be treated.

The objective of this work is to analyze a time-dependent eddy
current problem defined in a 3D bounded domain including conducting and
dielectric materials and giving the current source in terms of current
intensities. The input current intensities will be introduced in the model as
non-local boundary conditions (see, for instance, [1]). Thus, by following [2],
we will analyze a formulation based on the magnetic field in the conductor
regions and a multivalued scalar magnetic potential in the dielectric
part.

From a mathematical point of view, we will obtain a parabolic problem
and prove its well posedness as well as some regularity results. We will propose
a finite element combined with an implicit Euler time discretization to
numerically solve the problem. Concerning the space discretization, the magnetic
field is approximated by the lowest N´ed´elec edge finite elements and the
magnetic potential by standard piecewise linear continuous elements. The current
intensities are imposed as jumps of the multivalued magnetic potential on some
prescribed cut surfaces. We will obtain convergence results for the main
physical quantities, namely, the magnetic field and the current density.
Finally, we will present some numerical results corresponding to the simulation
of an application of electromagnetic forming.

This is a joint work with A. Bermúdez, B. López-Rodríguez, R.
Rodríguez, P. Salgado.

**References**

[1] A. Bossavit, Most general non-local boundary conditions for the
Maxwell equation in a bounded region, COMPEL (2000), 19, pp.
239–245.

[2] A. Berm´udez, R. Rodr´ıguez and P. Salgado, Numerical solution of
eddy current problems in bounded domains using realistic boundary conditions,
Comput. Methods
Appl. Mech. Engrg.

This talk is devoted to the inverse problem of determining the shape
and location of a body immersed in a fluid, based on measured quantities on the
exterior boundary of the fluid domain. For certain fluid models, we suggest an
inverse algorithm that combines

This
is joint work with Nuno Martins (Faculdade de Ciências e Tecnologia,
Lisbon).

We consider convex problem of Semi-Infinite Programming (SIP) with
multi - dimensional index set. In study of these problems we apply a new
approach based on the notions of immobile indices and their immobility orders.
We formulate the first order optimality conditions for conves SIP that are
explicit and have the form of criterion. We compare this criterion with
other known optimality conditions for SIP and show its efficiency in the convex
case.

José Cariñena (Universidad de Zaragoza)

Michele Cirafici (Instituto Superior
Técnico)

Carlos
Herdeiro (Universidade do Porto)

Marco
Mackaay (Universidade do Algarve)

Juan Carlos Marrero (Uiversidad de La Laguna,
Tenerife)

Marc Mars (Universidad de Salamanca)

**Marginally outer trapped surfaces as quasi-local black
holes**

Marginally outer trapped surfaces (MOTS) are generally believed to be
suitablequasi-local replacements for the concept of black hole. In this talk I
willdiscuss a number or rigorous results obtained in the last few years that put
thisexpectation on firmer grounds. Specifically, I intend to mention existence
of MOTS in an initial data set, smooth local evolution of MOTS, jumps of
outermost MOTS, the Penrose inequality conjecture and uniqueness theorems of
static spacetimes containing a MOTS.

Filipe
Mena (Universidade do Minho)

Vicente Muñoz (CSIC,

**Moduli Spaces of Pairs and of Bundles**

Pairs are objects formed by a holomorphic bundle over a compact
Riemann surface, together with a holomorphic section. There is a concept of
stability for pairs depending on a real parameter tau and we can consider moduli
spaces of tau-stable pairs. We show how these moduli spaces are related to the
moduli spaces of stable bundles, and how they do depend on tau. This gives a
method to prove properties of the moduli spaces of bundles and of pairs, by
induction on the rank.

We show many instances of this technique: irreducibility,
birationality, Brauer class, stably rationality, Hodge structures, Torelli
theorems, Hodge numbers, motives, Hodge conjecture, ...

Ricardo Schiappa (Instituto Superior Técnico)

**Borel and Stokes Analysis of Instantons in Topological Strings **

I will review some recent developments concerning the nonperturbative
structure of topological string theory. In particular, I will describe how Borel
analysis and Stokes phenomena play fundamental roles in this
construction.

German Sierra (Universidad Autónoma de Madrid)

Luis Ugarte (Universidad de Zaragoza)**
**

**Special Hermitian structures and heterotic string compactifications
**

In this talk we will focus on the heterotic string equations with
non-zero fluxes in six dimensions. Solutions to these equations have an
SU(3)-structure for which the underlying almost complex structure is integrable,
the holonomy of the associated Bismut connection reduces to SU(3) and the Lee
form is a multiple of the differential of the dilaton function, that is, the
Hermitian structure is conformally balanced. We will show general results on the
balanced Hermitian geometry of 6-dimensional nilmanifolds, which lead to
explicit compact solutions with non-zero field strength, non-flat instanton and
constant dilaton of the heterotic string equations. This is a joint work with M.
Fernandez, S. Ivanov and R. Villacampa.

© 3º Encontro Ibérico de Matemática :: 2010