July 15, 2011

14:30-15:30, Marisa Resende, The
Conjugate Gradient Method and some applications

Abstract: Conjugate-Gradient Methods are a class of numerical optimization procedures developed with the aim of minimizing a given objective function. The methods in this class have had a widespread use with powerful, efficient and robust performances, since their introduction in 1952 till the end of the last century with the establishment of important global convergence results. In this talk we present some of their relevant aspects concerning the algorithmic structure, properties and applications, both for the linear and the non-linear cases, focusing also their relation towards the original method of Steepest Descent.

Abstract: Conjugate-Gradient Methods are a class of numerical optimization procedures developed with the aim of minimizing a given objective function. The methods in this class have had a widespread use with powerful, efficient and robust performances, since their introduction in 1952 till the end of the last century with the establishment of important global convergence results. In this talk we present some of their relevant aspects concerning the algorithmic structure, properties and applications, both for the linear and the non-linear cases, focusing also their relation towards the original method of Steepest Descent.

In
particular we study the
Polak-Ribière Method, a non-linear variant of the
Conjugate-Gradient Methods, and its application to machine learning on
a recent algorithm developed by Rasmussen aiming at estimate parameters
associated to a Gaussian Process Model.

[Seminar
supervised by Marta
Pascoal.]

15:30-16:30, Serkan Karaçuha, The Yang-Baxter Equation and Its Solutions By Hopf Algebras

15:30-16:30, Serkan Karaçuha, The Yang-Baxter Equation and Its Solutions By Hopf Algebras

Abstract: The Yang-Baxter Equation (YBE) was first introduced in the field of statistical mechanics independently by C.N. Yang (1967) and R.J. Baxter (1971). The term YBE has become a common name to denote a principle of integrability, i.e. exact solvability, in several fields of physics and mathematics. The solutions of the YBE are called the R-matrices. In this talk, it is shown that braided (quasi-triangular) Hopf algebras provides solutions of the YBE. To do this, the important concept of braided bialgebras is introduced. These bialgebras are the ones with a universal R-matrix inducing a solution of the YBE on any of their modules. At the end we explain how the R-matrix of a braided bialgebra produces solutions of the YBE.

[Seminar supervised by Christian Lomp.]

July 1, 2011

14:30-15:30, Nasim Karimi, Parking functions and
labeled trees

Abstract: Suppose that*n*
drivers want to park their cars in a one way street with *n*
empty
parking places and they have a preference for a special
place: the *i*'th driver that enters the street wants to park his
car in
place *f(i)*. Now suppose drivers enter to the street one by one
and the
*i*'th driver does park in place *f(i)* in case this place
is empty;
otherwise he probes the first next empty place and parks there.
We say that *f* is a parking function if finally all drivers can
park
their cars in the street. It is well-known that the set of parking
functions of size *n* is in bijection with the set of labeled
forests
with *n* vertices.

In this talk we define a bijection between these two sets. This bijection is not recursive.

[Seminar supervised by António Guedes de Oliveira.]

Abstract: Suppose that

In this talk we define a bijection between these two sets. This bijection is not recursive.

[Seminar supervised by António Guedes de Oliveira.]

June 3, 2011

14:30-15:30,
Mohammad
Ahmadi, Parallel splitting-up methods for
elliptic boundary value problems

Abstract: Partial differential equations arise in the mathematical modelling of many physical, chemical and biological phenomena. Most of the time for solving these PDEs we need numerical methods, but the main problem is time limitation for calculation. So the alternating direction methods and a while later parallel splitting-up methods were proposed. With parallel methods we can reduced multidimensional problems into the fractional steps of independent one dimensional problems and therefore their computation can be carried out by parallel processors.

15:30-16:30, Mahdi Dodangeh, Degenerate elliptic problems in a class of free domains

Abstract: In this talk we study a mixed boundary value problem for an operator of p-Laplacian type. The main feature of the problem is the fact that the exact domain where it is considered is not known a priori and is to be determined so that a certain integral condition is satisfied. We establish the existence of a unique solution to the problem, by means of the analysis of the range of an appropriate real function, and we show the continuous dependence with respect to a family of operators.

[Seminar supervised by José Miguel Urbano.]

May 20, 2011Abstract: Partial differential equations arise in the mathematical modelling of many physical, chemical and biological phenomena. Most of the time for solving these PDEs we need numerical methods, but the main problem is time limitation for calculation. So the alternating direction methods and a while later parallel splitting-up methods were proposed. With parallel methods we can reduced multidimensional problems into the fractional steps of independent one dimensional problems and therefore their computation can be carried out by parallel processors.

This talk introduces
parallel algorithms for solving elliptic problems and discusses their
convergence and efficiency.

[Seminar supervised by
Adérito Araújo.]15:30-16:30, Mahdi Dodangeh, Degenerate elliptic problems in a class of free domains

Abstract: In this talk we study a mixed boundary value problem for an operator of p-Laplacian type. The main feature of the problem is the fact that the exact domain where it is considered is not known a priori and is to be determined so that a certain integral condition is satisfied. We establish the existence of a unique solution to the problem, by means of the analysis of the range of an appropriate real function, and we show the continuous dependence with respect to a family of operators.

[Seminar supervised by José Miguel Urbano.]

14:30-15:30, Maria
de
Fátima Pina, Rolling
Pseudo-Riemannian
Manifolds

Abstract:We will present the concept of a rolling map for manifolds that are embedded in a pseudo-Riemannian manifold. This talk is based on recent work of F. Silva Leite and P. Crouch which, in turn, generalizes the notion of a rolling map given by Sharpe, for Euclidean manifolds. One particular case that will be studied in detail is the hyperbolic*n*-sphere
rolling
on the affine tangent space at a point, both embedded in the
generalized Minkowski space R_{1}^{n+1}.

[Seminar supervised by Fátima Leite.]

15:30-16:30, Jahed Naghipoor, Studying a mathematical model for biodegradable polymeric drug delivery system

Abstract: Biodegradable polymeric coatings on cardiovascular stents can be used for local delivery of therapeutic agents to diseased coronary arteries after stenting procedures. In this seminar, a mathematical model will be presented to design and simulate of such coating drug delivery. This mathematical model can be used as a tool for predicting drug delivery from other coatings using the same polymer-drug combination. The studied model can be used to develop mathematical models for predicting the degradation and drug release kinetics for other polymeric drug delivery system. The linearization of general nonlinear mathematical models also will be discussed.

[Seminar supervised by José Augusto Ferreira.]

Abstract:We will present the concept of a rolling map for manifolds that are embedded in a pseudo-Riemannian manifold. This talk is based on recent work of F. Silva Leite and P. Crouch which, in turn, generalizes the notion of a rolling map given by Sharpe, for Euclidean manifolds. One particular case that will be studied in detail is the hyperbolic

[Seminar supervised by Fátima Leite.]

15:30-16:30, Jahed Naghipoor, Studying a mathematical model for biodegradable polymeric drug delivery system

Abstract: Biodegradable polymeric coatings on cardiovascular stents can be used for local delivery of therapeutic agents to diseased coronary arteries after stenting procedures. In this seminar, a mathematical model will be presented to design and simulate of such coating drug delivery. This mathematical model can be used as a tool for predicting drug delivery from other coatings using the same polymer-drug combination. The studied model can be used to develop mathematical models for predicting the degradation and drug release kinetics for other polymeric drug delivery system. The linearization of general nonlinear mathematical models also will be discussed.

[Seminar supervised by José Augusto Ferreira.]

May 6, 2011

14:30-15:30,
Manuela
Sobral,
Profinite
structures
and
profinite
completions

Abstract: Profinite algebras and profinite completions first appeared in Galois theory and algebraic number theory. Profinite topological spaces as well as profinite ordered topological spaces form a part (half of) famous dualities: Stone duality and Priestley duality, respectively. In various other settings profinite structures and profinite completions played and play an important role.

In this talk we use results and problems in this area to exhibit the power and usefulness of some categorical ideas, tools and techniques.

April 29, 2011Abstract: Profinite algebras and profinite completions first appeared in Galois theory and algebraic number theory. Profinite topological spaces as well as profinite ordered topological spaces form a part (half of) famous dualities: Stone duality and Priestley duality, respectively. In various other settings profinite structures and profinite completions played and play an important role.

In this talk we use results and problems in this area to exhibit the power and usefulness of some categorical ideas, tools and techniques.

14:30-15:30,
Rui
Sá
Pereira,
A
tool
for
characterizing
Vector
bundles

Abstract: The classification of mathematical objects arises as a powerful tool for reducing the exhaustive study of the objects in general theories to smaller classes where each object represents a bigger, possibly infinite, class of objects, each sharing the same “unifying” characteristic, where unifying means in almost every case “up to isomorphism”. The classification of finite simple groups in algebra, the classification of a surface by the genus in geometry, are both stark examples of the advantage of reducing the study of a potential infinitude of different objects, to the study of a “handful” “interesting” ones . In algebraic topology full classification remains an elusive task, but nevertheless there are discrete algebraic invariants providing powerful tools for characterizing a central object naturally emerging in geometry, the vector bundle which has a multitude of applications in mathematics and theoretical physics. In this seminar, we introduce the Chern class of a vector bundle, an algebraic invariant that can be assigned to every vector bundle, thus providing a way to see whether two given vector bundles are non-isomorphic.

[Seminar supervised by Peter Gothen.]

Abstract: The classification of mathematical objects arises as a powerful tool for reducing the exhaustive study of the objects in general theories to smaller classes where each object represents a bigger, possibly infinite, class of objects, each sharing the same “unifying” characteristic, where unifying means in almost every case “up to isomorphism”. The classification of finite simple groups in algebra, the classification of a surface by the genus in geometry, are both stark examples of the advantage of reducing the study of a potential infinitude of different objects, to the study of a “handful” “interesting” ones . In algebraic topology full classification remains an elusive task, but nevertheless there are discrete algebraic invariants providing powerful tools for characterizing a central object naturally emerging in geometry, the vector bundle which has a multitude of applications in mathematics and theoretical physics. In this seminar, we introduce the Chern class of a vector bundle, an algebraic invariant that can be assigned to every vector bundle, thus providing a way to see whether two given vector bundles are non-isomorphic.

[Seminar supervised by Peter Gothen.]

15:45-16:45,
Maria Manuel
Clementino, From sets to elementary toposes

Abstract:
As
a
naive
introduction
to
notions and techniques of
Category Theory, we will try to guide the students, through a careful
analysis of the categorical behaviour of sets and maps, to the
definition of elementary topos.

[No Category Theory background is assumed.]

[No Category Theory background is assumed.]

March 25, 2011

14:30-15:30,
Ronald
Alberto
Zúñiga
Rojas,
The
fundamental
group
and
the
van
Kampen
Theorem

Abstract: The main idea of the presentation is to talk about the importance of the fundamental group pi_{1}(X; x) of a topological space X
and how it is related with coverings of X. To do that, it is necessary
to introduce the concept of covering map, and some of their basic
properties. Many coverings, including the particular case of the polar
coordinate map, are examples of G-coverings, arising from an action of
a group G on a topological space.

The study of closed path, and homotopic paths are also very important subjects to introduce and define, formally, the fundamental group, which is the group of equivalence classes of closed paths starting and ending at a fixed point, with the equivalence relation given by homotopy.

The Van Kampen Theorem is a very important result that describes the fundamental group of a union of two spaces in terms of the fundamental group of each and of their intersection, under suitable hypotheses.

At the end, if it is possible, there is an application of the Van Kampen Theorem that could be interesting to the audience.

[Seminar supervised by Peter Gothen.]

Abstract: The main idea of the presentation is to talk about the importance of the fundamental group pi

The study of closed path, and homotopic paths are also very important subjects to introduce and define, formally, the fundamental group, which is the group of equivalence classes of closed paths starting and ending at a fixed point, with the equivalence relation given by homotopy.

The Van Kampen Theorem is a very important result that describes the fundamental group of a union of two spaces in terms of the fundamental group of each and of their intersection, under suitable hypotheses.

At the end, if it is possible, there is an application of the Van Kampen Theorem that could be interesting to the audience.

[Seminar supervised by Peter Gothen.]

March 18, 2011

14:30-15:15,
Maria
de
Fátima
Carvalho,
Generalized
ergodic
theorems

Abstract: Given a measurable space X with a probability measure which is invariant and ergodic by a dynamics T: X --> X, the classical Birkhoff's theorem states that, for any integrable test function, its sequence of Cesàro time averages converges almost everywhere to the space mean. We will discuss an extension of this result to a class of non-invariant sigma-finite measures.

Abstract: Given a measurable space X with a probability measure which is invariant and ergodic by a dynamics T: X --> X, the classical Birkhoff's theorem states that, for any integrable test function, its sequence of Cesàro time averages converges almost everywhere to the space mean. We will discuss an extension of this result to a class of non-invariant sigma-finite measures.

March 11, 2011

14:30-15:15, Olga
Azenhas,
Littlewood-Richardson
Miscellany

Abstract: Schur functions constitute one of the most important basis for the space of symmetric functions but their importance is due mainly to their ubiquitous nature. In fact, Schur has identified these functions - they would later bear his name - as characters of certain irreducible polynomial representations. On other hand, in combinatorics, they are the generating functions for semistandard Young tableaux. A function is said to be Schur positive if it can be written as a linear combination of Schur functions with all coefficients non negative integers. Examples are the Schur function product and skew Schur functions where coefficients are the famous Littlewood-Richardson coefficients. For combinatorialists Littlewood-Richardson coefficients are particularly interesting because they enumerate various combinatorial objects. I shall illustrate these ideas by showing the appearance of Littlewood-Richardson coefficients in several problems.

15:15-16:00, Alessandro Conflitti, How to study Coxeter systems and live to tell the tale

Abstract: An accessible introduction to Coxeter systems presented as combinatorics of words.

[No prerequisite required, all are welcome.]

Abstract: Schur functions constitute one of the most important basis for the space of symmetric functions but their importance is due mainly to their ubiquitous nature. In fact, Schur has identified these functions - they would later bear his name - as characters of certain irreducible polynomial representations. On other hand, in combinatorics, they are the generating functions for semistandard Young tableaux. A function is said to be Schur positive if it can be written as a linear combination of Schur functions with all coefficients non negative integers. Examples are the Schur function product and skew Schur functions where coefficients are the famous Littlewood-Richardson coefficients. For combinatorialists Littlewood-Richardson coefficients are particularly interesting because they enumerate various combinatorial objects. I shall illustrate these ideas by showing the appearance of Littlewood-Richardson coefficients in several problems.

15:15-16:00, Alessandro Conflitti, How to study Coxeter systems and live to tell the tale

Abstract: An accessible introduction to Coxeter systems presented as combinatorics of words.

[No prerequisite required, all are welcome.]

March 07, 2011

16:45-17:30,
Raquel Caseiro, Dirac structures

Abstract: Dirac structures were introduced by T. Courant and A. Weinstein as a unified approach to Poisson and pre-symplectic geometry. Instead of considering linear transformations on V or on V*, the key idea is to work on the direct sum V⊕V*.

The aim of this seminar is to introduce these structures and review some of their basic properties. Then we will look to smooth Dirac structures and we will see some applications.

17:30-18:15, Camille Laurent-Gengoux, The Toda lattice: from theory to practice

Abstract: We will show how sophisticated tools of Poisson geometrie and Lie algebra theory can end up solving a relatively simple differential equation: the Toda lattice, and that abstract non-sense and concrete problems are not that far away one from the other.

Abstract: Dirac structures were introduced by T. Courant and A. Weinstein as a unified approach to Poisson and pre-symplectic geometry. Instead of considering linear transformations on V or on V*, the key idea is to work on the direct sum V⊕V*.

The aim of this seminar is to introduce these structures and review some of their basic properties. Then we will look to smooth Dirac structures and we will see some applications.

17:30-18:15, Camille Laurent-Gengoux, The Toda lattice: from theory to practice

Abstract: We will show how sophisticated tools of Poisson geometrie and Lie algebra theory can end up solving a relatively simple differential equation: the Toda lattice, and that abstract non-sense and concrete problems are not that far away one from the other.

February 25, 2011

14:30-15:15,
Alexander Kovacec, Polynomial Inequalities and Minimization

Abstract: The problem to find good approximations to global minima of multivariate real polynomials defined on basic semialgebraic sets leads to profound algebraic questions and solutions have surprising applications, e.g. approximation schemes for NP-complete problems like

Abstract: The problem to find good approximations to global minima of multivariate real polynomials defined on basic semialgebraic sets leads to profound algebraic questions and solutions have surprising applications, e.g. approximation schemes for NP-complete problems like

a) the
partition problem: given a sequence a_{1},a_{2},..., a_{n}
of positive
integers, can these be partitioned into two sets whose sums of elements
are equal?

b) the stable set problem: what is the maximum cardinality of a set of vertices in a graph so that no two of them are neighbours?

b) the stable set problem: what is the maximum cardinality of a set of vertices in a graph so that no two of them are neighbours?

In more precise terms the request is, given polynomials p, g

p_{*}:= inf_{x in K}
p(x), K
={x: g_{1}(x)>= 0, ..., g_{m}(x)
>= 0}.

The talk concentrates on the algebraic questions arising: to find algebraic nonnegativity certificates; to decide whether a polynomial is a sum of squares of polynomials; to give good lower bounds for p

February 18, 2011

14:30-15:15,
Carlos
Tenreiro,
Combining
cross-validation
and
plug-in
methods
for
kernel
density
bandwidth
selection

Abstract:The cross-validation and
multistage plug-in methods are
two of the most widely used procedures for choosing the bandwidth in a
kernel density estimation setting. In this talk we review the basic
ideas and results about kernel density bandwidth selection and we
propose a combination of these well-known procedures in order to obtain
a data-based bandwidth selector that presents an overall good
performance for a large set of underlying densities.

15:15-16:00,
Paulo
Eduardo
Oliveira,
Smoothing
sparsely
observed
discrete
distributions

Abstract: In categorical models it is often reasonable to assume some adjacency and contiguity relations between neighboring cells. In such cases it becomes justifiable the use of smoothing to improve upon simple histogram approximations of the probabilities. This is particularly convenient when in presence of a sparse number of observations. We will discuss approaches to this kind of problem, using kernel methods and local polynomials, with respect to usual least squares error criterium and a relative least squares that is inspired on the approaches suggested by chi-square tests.

Abstract: In categorical models it is often reasonable to assume some adjacency and contiguity relations between neighboring cells. In such cases it becomes justifiable the use of smoothing to improve upon simple histogram approximations of the probabilities. This is particularly convenient when in presence of a sparse number of observations. We will discuss approaches to this kind of problem, using kernel methods and local polynomials, with respect to usual least squares error criterium and a relative least squares that is inspired on the approaches suggested by chi-square tests.

February 11, 2011

14:30-15:15,
Jorge
Almeida,
Profinite
Algebra

Abstract: What do the construction of reals (from the rationals), of p-adic integers (from the integers), of power series (from polynomials), of absolute Galois groups, of free profinite semigroups, and so on, have in common? They are all obtained by completion of more elementary structures. In all these cases, the completion is a topological operation, but there is a very strong interaction with an algebraic structure. In fact, in all but the first case, the completion can be realized as an inverse limit of quotients of the more elementary structure. Moreover, with the additional exception of power series with coefficients in infinite rings, the quotients are finite. Be it in Number Theory, in Field Theory, in Group Theory or in Semigroup Theory, the latter motivated by applications in Computer Science, finite quotients are of special interest simply because it is in principle easy to compute in finite structures. Inverse limits of finite algebras are called profinite algebras. The aim of this talk is to introduce such structures and to explain why they have been playing an increasingly important role in Algebra.

15:15-16:00, Manuel Delgado, An algorithm to compute generalised Feng-Rao numbers of a numerical semigroup

Abstract: A numerical semigroup is a co-finite submonoid of the non-negative integers under addition.

In the framework of the Theory of Error-Correcting Codes, Feng and Rao introduced a notion of distance for the Weierstrass semigroup at a rational point of an algebraic curve, with decoding purposes. It is a purely combinatorial concept that can be defined for any numerical semigroup. Later on, that notion has been generalised and is used not only in the theory of error correcting codes, but also in cryptography.

Let s be an element of a numerical semigroup S. An element a of S is said to divide s if there exists b in S such that s=a+b. The set of divisors of s is denoted by D(s).

The (classical) Feng-Rao distance is a function d from S into the non negative integers defined by d(m) = min{#(D(n)): n>=m, n in S}.

Replacing the element n in the preceding definition by a set of r elements of S greater than m, one obtains the definition of the r^{th} Feng-Rao distance.

For a sufficiently large m, there exists a constant, the so-called r^{th} Feng-Rao number,
depending only
on r and S, such that the r^{th} Feng-Rao distance is the
classical
Feng-Rao distance plus that constant.

An algorithm to compute generalised Feng-Rao numbers will be presented. It can be used in practice and therefore can be extremely useful in the search for formulas for the generalised Feng-Rao numbers of numerical semigroups of certain classes.

(Joint work with J.I. Farrán, P.A. García-Sánchez and D. Llena.)

February 8, 2011Abstract: What do the construction of reals (from the rationals), of p-adic integers (from the integers), of power series (from polynomials), of absolute Galois groups, of free profinite semigroups, and so on, have in common? They are all obtained by completion of more elementary structures. In all these cases, the completion is a topological operation, but there is a very strong interaction with an algebraic structure. In fact, in all but the first case, the completion can be realized as an inverse limit of quotients of the more elementary structure. Moreover, with the additional exception of power series with coefficients in infinite rings, the quotients are finite. Be it in Number Theory, in Field Theory, in Group Theory or in Semigroup Theory, the latter motivated by applications in Computer Science, finite quotients are of special interest simply because it is in principle easy to compute in finite structures. Inverse limits of finite algebras are called profinite algebras. The aim of this talk is to introduce such structures and to explain why they have been playing an increasingly important role in Algebra.

15:15-16:00, Manuel Delgado, An algorithm to compute generalised Feng-Rao numbers of a numerical semigroup

Abstract: A numerical semigroup is a co-finite submonoid of the non-negative integers under addition.

In the framework of the Theory of Error-Correcting Codes, Feng and Rao introduced a notion of distance for the Weierstrass semigroup at a rational point of an algebraic curve, with decoding purposes. It is a purely combinatorial concept that can be defined for any numerical semigroup. Later on, that notion has been generalised and is used not only in the theory of error correcting codes, but also in cryptography.

Let s be an element of a numerical semigroup S. An element a of S is said to divide s if there exists b in S such that s=a+b. The set of divisors of s is denoted by D(s).

The (classical) Feng-Rao distance is a function d from S into the non negative integers defined by d(m) = min{#(D(n)): n>=m, n in S}.

Replacing the element n in the preceding definition by a set of r elements of S greater than m, one obtains the definition of the r

For a sufficiently large m, there exists a constant, the so-called r

An algorithm to compute generalised Feng-Rao numbers will be presented. It can be used in practice and therefore can be extremely useful in the search for formulas for the generalised Feng-Rao numbers of numerical semigroups of certain classes.

(Joint work with J.I. Farrán, P.A. García-Sánchez and D. Llena.)

09.45-10:45,
Ezgi
Iraz
Su,
Doing
Topology
in
the
category
of
locales:
Sublocale
lattices

Abstract: The lattices of subobjects in Loc (sublocale lattices) are much more complicated than their counterparts in Top (which are nice complete atomic Boolean algebras).

Some of the main differences are that

*2*^{c} many
non-isomorphic sublocales
[J. Isbell, Some problems in descriptive locale theory, Canad. Math.
Soc. Conf. Proc. 13 (1992) 243-265]).

In this talk we describe the basic structure of sublocale lattices.

[Seminar supervised by Jorge Picado.]

Abstract: The lattices of subobjects in Loc (sublocale lattices) are much more complicated than their counterparts in Top (which are nice complete atomic Boolean algebras).

Some of the main differences are that

(1)
most
sublocales
are
not
complemented,
and

(2) each locale has a smallest dense sublocale.

Even the lattice of
sublocales of a topological space can be much larger than the Boolean
algebra of its subspaces (e.g. Q has (2) each locale has a smallest dense sublocale.

In this talk we describe the basic structure of sublocale lattices.

[Seminar supervised by Jorge Picado.]

17:00-18:00,
Ebrahim
Azhdari,
Hausdorff
Measures
and
Dimension
on
*R*^{∞}

Abstract: In this seminar we study the Hausdorff measures*H*^{s},
0<=s
<∞,
and
the
topology
induced
by
them.
We
define
the
Hausdorff
dimension
on
*R*^{∞}
showing that
the Lebesgue measure, defined on *R*^{∞} by R. Baker,
operates as a
measure *H*^{∞}
. Then we will present some properties of these
Hausdorff measures and dimension. Finally some examples will be given.

[Seminar supervised by Susana Moura.]

December 15, 2010Abstract: In this seminar we study the Hausdorff measures

[Seminar supervised by Susana Moura.]

14:30-15:15,
Júlio Neves, Optimal
embeddings
of
Bessel-potential-type
spaces
into
generalized
Hölder
spaces

Abstract: Starting with the Theorem of Sobolev, we give a short survey on the results of embeddings of Sobolev type spaces into Hölder type spaces, including as well the famous result of Brézis and Wainger about almost Lipschitz continuity of elements of the Sobolev space with super-critical exponent of smoothness. Afterwards, we give necessary and sufficient conditions for embeddings of Bessel potential spaces modelled upon rearrangement invariant Banach function spaces X into generalized Hölder spaces. We also apply our results to the case when X is a Lorentz-Karamata space and, in particular, we present better target spaces than the ones given by the Brézis and Wainger result.

15:15-16:00, Susana Moura, Envelopes and sharp embeddings in function spaces

Abstract: We describe the concept of (growth and continuity) envelopes in function spaces, present some basic properties and give a short survey on the results in the context of the Besov and Triebel Lizorkin spaces of generalized smoothness. In general the knowledge of the envelope of a function space gives sharp results regarding embeddings, but we will show that this might not be the case in critical situations.

Abstract: Starting with the Theorem of Sobolev, we give a short survey on the results of embeddings of Sobolev type spaces into Hölder type spaces, including as well the famous result of Brézis and Wainger about almost Lipschitz continuity of elements of the Sobolev space with super-critical exponent of smoothness. Afterwards, we give necessary and sufficient conditions for embeddings of Bessel potential spaces modelled upon rearrangement invariant Banach function spaces X into generalized Hölder spaces. We also apply our results to the case when X is a Lorentz-Karamata space and, in particular, we present better target spaces than the ones given by the Brézis and Wainger result.

15:15-16:00, Susana Moura, Envelopes and sharp embeddings in function spaces

Abstract: We describe the concept of (growth and continuity) envelopes in function spaces, present some basic properties and give a short survey on the results in the context of the Besov and Triebel Lizorkin spaces of generalized smoothness. In general the knowledge of the envelope of a function space gives sharp results regarding embeddings, but we will show that this might not be the case in critical situations.

November 24, 2010

15:00-16:00,
Colloquium: Richard Tsai (Univ. Texas at Austin, USA), Point source discovery in
complicated domains

November 17, 201014:30-15:15,
Marta Pascoal, Dealing
with
uncertainty
in
network
optimisation

Abstract: Network optimisation is a branch of optimisation the problems of which are modelled over a valued graph, that is, a network.

We briefly present introductory concepts in this field and discuss connections between network optimisation and related subjects.

Classical methods applied in this area assume that deterministic information is associated with the graph structure, however, in real problems these parameters are often incomplete, inaccurate or stochastic.

We describe some of the possible formulations of network optimisation problems when uncertainty is present.

15:15-16:00, Sílvia Barbeiro, Splitting methods in the design of coupled flow and mechanics simulators

Abstract: Network optimisation is a branch of optimisation the problems of which are modelled over a valued graph, that is, a network.

We briefly present introductory concepts in this field and discuss connections between network optimisation and related subjects.

Classical methods applied in this area assume that deterministic information is associated with the graph structure, however, in real problems these parameters are often incomplete, inaccurate or stochastic.

We describe some of the possible formulations of network optimisation problems when uncertainty is present.

15:15-16:00, Sílvia Barbeiro, Splitting methods in the design of coupled flow and mechanics simulators

Abstract: The
dynamics of coupled flow and
mechanics are of interest in many areas of science. Developments in
this field are contributing to important achievements not only in soil
mechanics but also in civil, petroleum and even biomedical engineering.

The interactions between flow and mechanics can be modeled using various coupling schemes. In this seminar we analyse different operator-split strategies which lead to iteratively coupled schemes. The resulting sequential procedures are iterated at each time step until the solution converges within an acceptable tolerance. If the sequential solution strategies have stability and convergence properties that are closed to those of the fully coupled approach, they can be very competitive for solving problems of practical interest.

We will discuss the development of numerical solutions and give some insight into the theoretical basis of the underlying methods. Both the theory and the numerics will be illustrated via some examples.

The interactions between flow and mechanics can be modeled using various coupling schemes. In this seminar we analyse different operator-split strategies which lead to iteratively coupled schemes. The resulting sequential procedures are iterated at each time step until the solution converges within an acceptable tolerance. If the sequential solution strategies have stability and convergence properties that are closed to those of the fully coupled approach, they can be very competitive for solving problems of practical interest.

We will discuss the development of numerical solutions and give some insight into the theoretical basis of the underlying methods. Both the theory and the numerics will be illustrated via some examples.

November 10, 2010

14:30-15:15,
Fátima Leite, Introduction
to
geometric
control
theory
–
controllability
and
Lie
bracket

Abstract:
We
will
present
an
introduction
to
the
theory
of
nonlinear
control
systems,
with
emphasis
on
controllability
properties
of
such
systems.
The
basic
tools
in geometric control come from differential
geometry. A control system can be seen as a family of vector fields and
the most basic theoretical tool of the geometric view point is the Lie
bracket. We first introduce the differential geometric language of
vector fields, Lie bracket, distributions, integrability etc., and then
analyze basic controllability problems and give criteria for complete
controllability.

November 03, 2010

14:30-15:15,
Adérito
Araújo, Numerical
Approximation
of
Mean
Curvature
Flow

Abstract: The mean curvature flow problem for graphs is closely related to the mean curvature of level sets. Taking into account this fact, we consider

a level set algorithm in the context of finite differences together with a semi implicit time discretization. The main goal of this work is to study the qualitative

and quantitative properties of the numerical solution as well as the efficiency of the algorithm.

We also provide some numerical tests.

15:15-16:00, Ercília Sousa, Anomalous diffusion equations

Abstract: Fractional space derivatives are used to model anomalous diffusion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion.

In this talk, a one dimensional diffusion model is considered, where the usual second-order derivative gives place to a fractional derivative of order alpha, with 1< alpha <= 2. Different definitions for the fractional derivative are shown.

Some insights on how to solve fractional diffusion models numerically will be given, focusing on its additional difficulties when compared with the classical models. These difficulties are related to the fact that the fractional derivative of order alpha at a certain point, x, is not a local property anymore, except when alpha is an integer. Therefore it is expected that the theory involves information of the function further out of the region close to the point at which we are computing the derivative.

October 27, 2010

14:30-15:15,
Amílcar Branquinho, Interpretation
of
some
integrable
systems
via
multiple
orthogonal
polynomials

Abstract: Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based on the analysis of the matrical moments

for the main operator of the pair. The solutions of these systems are studied in terms of properties of this operator, giving, under some conditions, explicit

expressions for the resolvent function.

15:15-16:00, Luís Daniel Abreu, The Uncertainty Principle in Time-Frequency Analysis

Abstract: Our departure point in this talk will be the Heisenberg Uncertainty Principle formulated as a proposition involving a real function f

(function of "time") and its Fourier transform (function of "frequencies").

Within this formulation, The Uncertainty Principle becomes quite intuitive: it provides a precise mathematical description of the impossibility

of finding a "instantaneous frequency".

Mathematically, this is rather intuitive, since to measure a frequency we need to look at a function over a certain period of time.

However, the pleasurable activity of listening to music requires our ears to constantly beat the uncertainty principle: in a given instant they are

able to regognize what frequency is being played!

How is this possible?-The answer to this question is at the heart of a modern branch of mathematics called "Time-Frequency Analysis", which had

a tremendous development in recent years, with an hitherto unseen collaboration between pure and applied mathematics which had a tremendous

impact in our lifes: from mobile communications and file compression to Banach Algebras and Noncomutative geometry.

I will give an idea of the outputs of this endeavour, like "Wavelets" and "Compressed Sensing" as well as possible entry points for research at the

Phd student level.

14:00-14:45,
Inês Cruz, Symplectic
Geometry
versus
Riemannian
Geometry

Abstract: This talk will be an introduction to Symplectic Geometry. Basic notions in symplectic geometry (symplectic volume, symplectomorphism, Hamiltonian vector field,

symplectic capacity) will be introduced while comparing them with their Riemannian counterparts.

14:45-15:30, Antonio de Nicola, An introduction to contact geometry

Abstract: We will introduce the first elements of contact geometry and briefly mention some of its applications to physics as well

as its relation with symplectic geometry.

October 13, 2010

14:30-15:15,
Dmitry
Vorotnikov,
Attractors
in the absence of a dynamical system

Abstract: An attractor of a dynamical system is a certain set to which every orbit eventually becomes close. When an autonomous differential equation (or boundary value problem) generates a dynamical system, the corresponding attractor characterizes the long-time behaviour of its solutions.

October 06, 2010

14:30-15:15,
António Leal
Duarte, Inverse
Eigenvalue
Problems
for
graphs

Abstract: The (real) Inverse Eigenvalue Problem IEP for a graph G (with vertices 1, ... , n) consists in describing the set of all n-tuples of real numbers that may occur

as eigenvalues of real symmetric matrices A with graph G, that is, with a fixed zero pattern: a non diagonal element of A in position (i, j) is nonzero if and only if

there is an edge between i and j in G; the set of such matrices is denoted by S(G).

This seems to be a very difficult problem even when the graph is a tree T; in this case any sequence of n distinct real numbers does occur as eigenvalues of one matrix in S(T),

so the problem is just to describe the possible multiple eigenvalues. But even the apparently simpler problem of just describing the possible lists of multiplicities that may occur

between the eigenvalues of matrices in S(T) seems very difficult, depending heavily on the graph). Some related questions are:

i) What is
the maximum possible
multiplicities
of eigenvalues for matrices in S(G);

ii) what are the minimum nunber of distinct eigenvalues for A in S(G);

iii) what is the minimum number of multiplicity one eigenvalues for A in S(T).

ii) what are the minimum nunber of distinct eigenvalues for A in S(G);

iii) what is the minimum number of multiplicity one eigenvalues for A in S(T).

Problem i) is solved when G
is a tree but not much is known about ii)
and iii).

Abstract: Many natural phenomena are described in terms of dynamical systems arising from systems of differential equations.

One successful method for studying these systems is to discretize them, thus producing "symbolic" dynamical systems.

One of the main driving forces in symbolic dynamics has been the classification of symbolic dynamical systems up to isomorphism. Linear Algebra plays a central role in the field.

In this talk focus is given to another algebraic perspective: sofic systems are closely related with finite automata, and via this perspective, with semigroup theory.

This is one of the aspects of the interplay between symbolic dynamics and semigroup theory.

The deduction of semigroup theoretic invariants of symbolic dynamical systems received attention in the past few years. Research in semigroup theory also benefited

from interplay with symbolic dynamics.

Some results in both directions will be presented in the talk.

Maria Manuel Clementino

e-mail: mmc@mat.uc.pt

URL: http://www.mat.uc.pt/~mmc