Boutet de Monvel`s calculus and groupoids I

Can Boutet de Monvel`s algebra on a compact manifold with boundary be obtained as the algebra Psi(0)(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal I isomorphic to G. In fact, we prove first that G similar or equal to Psi circle times K with the C*-algebra Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both Psi circle times K and I are extensions of C(S*Y) circle times K by K (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS IN SEMI-RIEMANNIAN GEOMETRY VIA MASLOV INDEX

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

On the geometry of the spin-statistics connection in quantum mechanics

The Spin-Statistics theorem states that the statistics of a system of identical particles is determined by their spin: Particles of integer spin are Bosons (i.e. obey Bose-Einstein statistics), whereas particles of half-integer spin are Fermions (i.e. obey Fermi-Dirac statistics). Since the original proof by Fierz and Pauli, it has been known that the connection between Spin and Statistics follows from the general principles of relativistic Quantum Field Theory. In spite of this, there are different approaches to Spin-Statistics and it is not clear whether the theorem holds under assumptions that are different, and even less restrictive, than the usual ones (e.g. Lorentz-covariance). Additionally, in Quantum Mechanics there is a deep relation between indistinguishabilty and the geometry of the configuration space. This is clearly illustrated by Gibbs' paradox. Therefore, for many years efforts have been made in order to find a geometric proof of the connection between Spin and Statistics. Recently, various proposals have been put forward, in which an attempt is made to derive the Spin-Statistics connection from assumptions different from the ones used in the relativistic, quantum field theoretic proofs. Among these, there is the one due to Berry and Robbins (BR), based on the postulation of a certain single-valuedness condition, that has caused a renewed interest in the problem. In the present thesis, we consider the problem of indistinguishability in Quantum Mechanics from a geometric-algebraic point of view. An approach is developed to study configuration spaces Q having a finite fundamental group, that allows us to describe different geometric structures of Q in terms of spaces of functions on the universal cover of Q. In particular, it is shown that the space of complex continuous functions over the universal cover of Q admits a decomposition into C(Q)-submodules, labelled by the irreducible representations of the fundamental group of Q, that can be interpreted as the spaces of sections of certain flat vector bundles over Q. With this technique, various results pertaining to the problem of quantum indistinguishability are reproduced in a clear and systematic way. Our method is also used in order to give a global formulation of the BR construction. As a result of this analysis, it is found that the single-valuedness condition of BR is inconsistent. Additionally, a proposal aiming at establishing the Fermi-Bose alternative, within our approach, is made.

Asymptotic bifurcation results for quasilinear elliptic operators

We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.

An Lp − Lq - Version of Morgan's Theorem Associated with Partial Differential Operators

2000 Mathematics Subject Classification: 42B10, 43A32.

Lp Microlocal Properties for Multi-Quasi-Elliptic Pseudodifferential Operators

2010 Mathematics Subject Classification: Primary 35S05; Secondary 35A17.

POSITIVITY PROPERTIES OF THE FOURIER TRANSFORM AND THE STABILITY OF PERIODIC TRAVELLING-WAVE SOLUTIONS

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg-de Vries-type u(t) + u(p)u(x) - Mu(x) = 0, with M being a general pseudodifferential operator and where p >= 1 is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg-de Vries and modified Korteweg-de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

Pseudodifferential operators with C*-algebra-valued symbols: Abstract characterizations

Given a separable unital C*-algebra C with norm parallel to center dot parallel to, let E-n denote the Banach-space completion of the C-valued Schwartz space on R-n with norm parallel to f parallel to(2)=parallel to < f, f >parallel to(1/2), < f, g >=integral f(x)* g(x)dx. The assignment of the pseudodifferential operator A=a(x,D) with C-valued symbol a(x,xi) to each smooth function with bounded derivatives a is an element of B-C(R-2n) defines an injective mapping O, from B-C(R-2n) to the set H of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E-n. In this paper, we construct a left-inverse S for O and prove that S is injective if C is commutative. This generalizes Cordes' description of H in the scalar case. Combined with previous results of the second author, our main theorem implies that, given a skew-symmetric n x n matrix J and if C is commutative, then any A is an element of H which commutes with every pseudodifferential operator with symbol F(x+J xi), F is an element of B-C(R-n), is a pseudodifferential operator with symbol G(x - J xi), for some G is an element of B-C(R-n). That was conjectured by Rieffel.

Gender differences in bone geometry during the adolescent growth spurt

To investigate whether there are gender differences in the bone geometry of the proximal femur during the adolescent years we used an interactive computer program ?Hip Strength Analysis? developed by Beck and associates (Beck et al., Invest Radiol. 1990,25:6-18.) to derive femoral neck geometry parameters from DXA bone scans (Hologic 2000, array mode). We analyzed a longitudinal data-set collected on 70 boys and 68 girls over a seven year period. Distance and velocity curves for height were fitted for each child utilizing a cubic spline procedure and the age of peak height velocity (PHV) was determined. To control for maturational differences between children of the same chronological age and between boys and girls, section modulus (Z) an index of bending strength, cross sectional area of bone (CSA), sub-periosteal width (SPW), and BMD values at the neck and shaft of the proximal femur were determined for points on each individual?s curve at the age of PHV and one and two years on either side of peak. To control for size differences, height and weight were introduced as co-variates in the two-way analyses of variance looking at gender over time measured at the maturational age points (-2, -1, age of PHV, +1, +2). The following figure presents the results of the analyses on two variables, BMD and Z at neck and shaft regions:After the age of peak linear growth (PHV), independent of body size, there was a gender difference in BMD at the shaft but not at the neck. Section modulus at both sites indicated that male bones became significantly stronger after PHV. Underlying these maturational changes, male bones became wider (SPW) after PHV in both the neck and shaft and enclosed more material (CSA) at all maturational age points at both regions. These results call into question the emphasis on using BMD as a measure of skeletal integrity in growing children

Factorization by invariant embedding of elliptic problems: circular and star-shaped domains

Dissertação apresentada para obtenção do grau de Doutor em Matemática na especialidade de Equações Diferenciais, pela Universidade Nova de Lisboa,Faculdade de Ciências e Tecnologia

Reinforcement Learning Based on the Bayesian Theorem for Electricity Markets Decision Support

This paper presents the applicability of a reinforcement learning algorithm based on the application of the Bayesian theorem of probability. The proposed reinforcement learning algorithm is an advantageous and indispensable tool for ALBidS (Adaptive Learning strategic Bidding System), a multi-agent system that has the purpose of providing decision support to electricity market negotiating players. ALBidS uses a set of different strategies for providing decision support to market players. These strategies are used accordingly to their probability of success for each different context. The approach proposed in this paper uses a Bayesian network for deciding the most probably successful action at each time, depending on past events. The performance of the proposed methodology is tested using electricity market simulations in MASCEM (Multi-Agent Simulator of Competitive Electricity Markets). MASCEM provides the means for simulating a real electricity market environment, based on real data from real electricity market operators.

In situ measured cross section geometry of old timber structures and its influence on structural safety

Old timber structures may show significant variation in the cross section geometry along the same element, as a result of both construction methods and deterioration. As consequence, the definition of the geometric parameters in situ may be both time consuming and costly. This work presents the results of inspections carried out in different timber structures. Based on the obtained results, different simplified geometric models are proposed in order to efficiently model the geometry variations found. Probabilistic modelling techniques are also used to define safety parameters of existing timber structures, when subjected to dead and live loads, namely self-weight and wind actions. The parameters of the models have been defined as probabilistic variables, and safety of a selected case study was assessed using the Monte Carlo simulation technique. Assuming a target reliability index, a model was defined for both the residual cross section and the time dependent deterioration evolution. As a consequence, it was possible to compute probabilities of failure and reliability indices, as well as, time evolution deterioration curves for this structure. The results obtained provide a proposal for definition of the cross section geometric parameters of existing timber structures with different levels of decay, using a simplified probabilistic geometry model and considering a remaining capacity factor for the decayed areas. This model can be used for assessing the safety of the structure at present and for predicting future performance.

Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems

Dissertação para obtenção do Grau de Doutor em Matemática