Desenvolvimento da fluência no cálculo mental através do uso de estratégias de cálculo: um estudo no 4.º e no 6.º ano de escolaridade

Relatório de estágio de mestrado em Ensino do 1º e 2º Ciclo do Ensino Básico

New integrative computational approaches unveil the Saccharomyces cerevisiae pheno-metabolomic fermentative profile and allow strain selection for winemaking

During must fermentation by Saccharomyces cerevisiae strains thousands of volatile aroma compounds are formed. The objective of the present work was to adapt computational approaches to analyze pheno-metabolomic diversity of a S. cerevisiae strain collection with different origins. Phenotypic and genetic characterization together with individual must fermentations were performed, and metabolites relevant to aromatic profiles were determined. Experimental results were projected onto a common coordinates system, revealing 17 statistical-relevant multi-dimensional modules, combining sets of most-correlated features of noteworthy biological importance. The present method allowed, as a breakthrough, to combine genetic, phenotypic and metabolomic data, which has not been possible so far due to difficulties in comparing different types of data. Therefore, the proposed computational approach revealed as successful to shed light into the holistic characterization of S. cerevisiae pheno-metabolome in must fermentative conditions. This will allow the identification of combined relevant features with application in selection of good winemaking strains.

New FFT/IFFT Factorizations with Regular Interconnection Pattern Stage-to-Stage Subblocks

Les factoritzacions de la FFT (Fast Fourier Transform) que presenten un patró d’interconnexió regular entre factors o etapes son conegudes com algorismes paral·lels, o algorismes de Pease, ja que foren originalment proposats per Pease. En aquesta contribució s’han desenvolupat noves factoritzacions amb blocs que presenten el patró d’interconnexió regular de Pease. S’ha mostrat com aquests blocs poden ser obtinguts a una escala prèviament seleccionada. Les noves factoritzacions per ambdues FFT i IFFT (Inverse FFT) tenen dues classes de factors: uns pocs factors del tipus Cooley-Tukey i els nous factors que proporcionen la mateix patró d’interconnexió de Pease en blocs. Per a una factorització donada, els blocs comparteixen dimensions, el patró d’interconnexió etapa a etapa i a més cada un d’ells pot ser calculat independentment dels altres.

A new look at the Heston characteristic function

A new expression for the characteristic function of log-spot in Heston model is presented. This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function. This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot. We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced.

An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

New predictions for inclusive heavy-quarkonium P-wave decays

We show that some nonrelativistic quantum chromodynamics color-octet matrix elements can be written in terms of (derivatives of) wave functions at the origin and of nonperturbative universal constants once the factorization between the soft and ultrasoft scales is achieved by using an effective field theory where only ultrasoft degrees of freedom are kept as dynamical entities. This allows us to derive a new set of relations between inclusive heavy-quarkonium P-wave decays into light hadrons with different principal quantum numbers and with different heavy flavors. In particular, we can estimate the ratios of the decay widths of bottomonium P-wave states from charmonium data.

New predictions for inclusive heavy-quarkonium p-wave decays

We show that some nonrelativistic quantum chromodynamics color-octet matrix elements can be written in terms of (derivatives of) wave functions at the origin and of nonperturbative universal constants once the factorization between the soft and ultrasoft scales is achieved by using an effective field theory where only ultrasoft degrees of freedom are kept as dynamical entities. This allows us to derive a new set of relations between inclusive heavy-quarkonium P-wave decays into light hadrons with different principal quantum numbers and with different heavy flavors. In particular, we can estimate the ratios of the decay widths of bottomonium P-wave states from charmonium data.

A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Geometric stopping of a random walk and its applications to valuing equity-linked death benefits

We study discrete-time models in which death benefits can depend on a stock price index, the logarithm of which is modeled as a random walk. Examples of such benefit payments include put and call options, barrier options, and lookback options. Because the distribution of the curtate-future-lifetime can be approximated by a linear combination of geometric distributions, it suffices to consider curtate-future-lifetimes with a geometric distribution. In binomial and trinomial tree models, closed-form expressions for the expectations of the discounted benefit payment are obtained for a series of options. They are based on results concerning geometric stopping of a random walk, in particular also on a version of the Wiener-Hopf factorization.

Composição elementar do aerossol atmosférico na região central da Bacia Amazônica

The processes and sources that regulate the elemental composition of aerosol particles were investigated in both fine and coarse modes during the dry and wet seasons. One hundred and nine samples were collected from the biological reserve Cuieiras - Manaus from February to October 2008, and analyzed together with 668 samples that were previously collected at Balbina from 1998 to 2002. Particle induced X-ray emission technique was used to determine the elemental composition, while the concentration of black carbon was obtained from the measurement of optical reflectance. Absolute principal factor analysis and positive matrix factorization were performed for source apportionment, which was complemented with back trajectory analysis. A regional identity for the natural biogenic aerosol was found for the Central Amazon Basin and can be used in dynamical chemical region models.

Non-negative matrix decomposition approaches to frequency domain analysis of music audio signals

On étudie l’application des algorithmes de décomposition matricielles tel que la Factorisation Matricielle Non-négative (FMN), aux représentations fréquentielles de signaux audio musicaux. Ces algorithmes, dirigés par une fonction d’erreur de reconstruction, apprennent un ensemble de fonctions de base et un ensemble de coef- ficients correspondants qui approximent le signal d’entrée. On compare l’utilisation de trois fonctions d’erreur de reconstruction quand la FMN est appliquée à des gammes monophoniques et harmonisées: moindre carré, divergence Kullback-Leibler, et une mesure de divergence dépendente de la phase, introduite récemment. Des nouvelles méthodes pour interpréter les décompositions résultantes sont présentées et sont comparées aux méthodes utilisées précédemment qui nécessitent des connaissances du domaine acoustique. Finalement, on analyse la capacité de généralisation des fonctions de bases apprises par rapport à trois paramètres musicaux: l’amplitude, la durée et le type d’instrument. Pour ce faire, on introduit deux algorithmes d’étiquetage des fonctions de bases qui performent mieux que l’approche précédente dans la majorité de nos tests, la tâche d’instrument avec audio monophonique étant la seule exception importante.

Les polynômes orthogonaux matriciels et la méthode de factorisation

La méthode de factorisation est appliquée sur les données initiales d'un problème de mécanique quantique déja résolu. Les solutions (états propres et fonctions propres) sont presque tous retrouvés.

Modeling High-Dimensional Audio Sequences with Recurrent Neural Networks

Cette thèse étudie des modèles de séquences de haute dimension basés sur des réseaux de neurones récurrents (RNN) et leur application à la musique et à la parole. Bien qu'en principe les RNN puissent représenter les dépendances à long terme et la dynamique temporelle complexe propres aux séquences d'intérêt comme la vidéo, l'audio et la langue naturelle, ceux-ci n'ont pas été utilisés à leur plein potentiel depuis leur introduction par Rumelhart et al. (1986a) en raison de la difficulté de les entraîner efficacement par descente de gradient. Récemment, l'application fructueuse de l'optimisation Hessian-free et d'autres techniques d'entraînement avancées ont entraîné la recrudescence de leur utilisation dans plusieurs systèmes de l'état de l'art. Le travail de cette thèse prend part à ce développement. L'idée centrale consiste à exploiter la flexibilité des RNN pour apprendre une description probabiliste de séquences de symboles, c'est-à-dire une information de haut niveau associée aux signaux observés, qui en retour pourra servir d'à priori pour améliorer la précision de la recherche d'information. Par exemple, en modélisant l'évolution de groupes de notes dans la musique polyphonique, d'accords dans une progression harmonique, de phonèmes dans un énoncé oral ou encore de sources individuelles dans un mélange audio, nous pouvons améliorer significativement les méthodes de transcription polyphonique, de reconnaissance d'accords, de reconnaissance de la parole et de séparation de sources audio respectivement. L'application pratique de nos modèles à ces tâches est détaillée dans les quatre derniers articles présentés dans cette thèse. Dans le premier article, nous remplaçons la couche de sortie d'un RNN par des machines de Boltzmann restreintes conditionnelles pour décrire des distributions de sortie multimodales beaucoup plus riches. Dans le deuxième article, nous évaluons et proposons des méthodes avancées pour entraîner les RNN. Dans les quatre derniers articles, nous examinons différentes façons de combiner nos modèles symboliques à des réseaux profonds et à la factorisation matricielle non-négative, notamment par des produits d'experts, des architectures entrée/sortie et des cadres génératifs généralisant les modèles de Markov cachés. Nous proposons et analysons également des méthodes d'inférence efficaces pour ces modèles, telles la recherche vorace chronologique, la recherche en faisceau à haute dimension, la recherche en faisceau élagué et la descente de gradient. Finalement, nous abordons les questions de l'étiquette biaisée, du maître imposant, du lissage temporel, de la régularisation et du pré-entraînement.

Understanding Subsystems in Biology through Dimensionality Reduction, Graph Partitioning and Analytical Modeling

Biological systems exhibit rich and complex behavior through the orchestrated interplay of a large array of components. It is hypothesized that separable subsystems with some degree of functional autonomy exist; deciphering their independent behavior and functionality would greatly facilitate understanding the system as a whole. Discovering and analyzing such subsystems are hence pivotal problems in the quest to gain a quantitative understanding of complex biological systems. In this work, using approaches from machine learning, physics and graph theory, methods for the identification and analysis of such subsystems were developed. A novel methodology, based on a recent machine learning algorithm known as non-negative matrix factorization (NMF), was developed to discover such subsystems in a set of large-scale gene expression data. This set of subsystems was then used to predict functional relationships between genes, and this approach was shown to score significantly higher than conventional methods when benchmarking them against existing databases. Moreover, a mathematical treatment was developed to treat simple network subsystems based only on their topology (independent of particular parameter values). Application to a problem of experimental interest demonstrated the need for extentions to the conventional model to fully explain the experimental data. Finally, the notion of a subsystem was evaluated from a topological perspective. A number of different protein networks were examined to analyze their topological properties with respect to separability, seeking to find separable subsystems. These networks were shown to exhibit separability in a nonintuitive fashion, while the separable subsystems were of strong biological significance. It was demonstrated that the separability property found was not due to incomplete or biased data, but is likely to reflect biological structure.