Low-flow estimates in regions of extrapolation of the regionalization equations: a new concept

ABSTRACT Knowledge of natural water availability, which is characterized by low flows, is essential for planning and management of water resources. One of the most widely used hydrological techniques to determine streamflow is regionalization, but the extrapolation of regionalization equations beyond the limits of sample data is not recommended. This paper proposes a new method for reducing overestimation errors associated with the extrapolation of regionalization equations for low flows. The method is based on the use of a threshold value for the maximum specific low flow discharge estimated at the gauging sites that are used in the regionalization. When a specific low flow, which has been estimated using the regionalization equation, exceeds the threshold value, the low flow can be obtained by multiplying the drainage area by the threshold value. This restriction imposes a physical limit to the low flow, which reduces the error of overestimating flows in regions of extrapolation. A case study was done in the Urucuia river basin, in Brazil, and the results showed the regionalization equation to perform positively in reducing the risk of extrapolation.

Word Equations and Related Topics. Independence, Decidability and Characterizations

The three main topics of this work are independent systems and chains of word equations, parametric solutions of word equations on three unknowns, and unique decipherability in the monoid of regular languages. The most important result about independent systems is a new method giving an upper bound for their sizes in the case of three unknowns. The bound depends on the length of the shortest equation. This result has generalizations for decreasing chains and for more than three unknowns. The method also leads to shorter proofs and generalizations of some old results. Hmelevksii’s theorem states that every word equation on three unknowns has a parametric solution. We give a significantly simplified proof for this theorem. As a new result we estimate the lengths of parametric solutions and get a bound for the length of the minimal nontrivial solution and for the complexity of deciding whether such a solution exists. The unique decipherability problem asks whether given elements of some monoid form a code, that is, whether they satisfy a nontrivial equation. We give characterizations for when a collection of unary regular languages is a code. We also prove that it is undecidable whether a collection of binary regular languages is a code.

Numerical study of Petrov-Galerkin formulations for the shallow water wave equations

The behavior of Petrov-Galerkin formulations for shallow water wave equations is evaluated numerically considering typical one-dimensional propagation problems. The formulations considered here use stabilizing operators to improve classical Galerkin approaches. Their advantages and disadvantages are pointed out according to the intrinsic time scale (free parameter) which has a particular importance in this kind of problem. The influence of the Courant number and the performance of the formulation in dealing with spurious oscillations are adressed.

Fluid dynamic modelling and simulation of circulating fluidized bed reactors: importance of the interface turbulence transfer

The main objective of this work is to analyze the importance of the gas-solid interface transfer of the kinetic energy of the turbulent motion on the accuracy of prediction of the fluid dynamic of Circulating Fluidized Bed (CFB) reactors. CFB reactors are used in a variety of industrial applications related to combustion, incineration and catalytic cracking. In this work a two-dimensional fluid dynamic model for gas-particle flow has been used to compute the porosity, the pressure, and the velocity fields of both phases in 2-D axisymmetrical cylindrical co-ordinates. The fluid dynamic model is based on the two fluid model approach in which both phases are considered to be continuous and fully interpenetrating. CFB processes are essentially turbulent. The model of effective stress on each phase is that of a Newtonian fluid, where the effective gas viscosity was calculated from the standard k-epsilon turbulence model and the transport coefficients of the particulate phase were calculated from the kinetic theory of granular flow (KTGF). This work shows that the turbulence transfer between the phases is very important for a better representation of the fluid dynamics of CFB reactors, especially for systems with internal recirculation and high gradients of particle concentration. Two systems with different characteristics were analyzed. The results were compared with experimental data available in the literature. The results were obtained by using a computer code developed by the authors. The finite volume method with collocated grid, the hybrid interpolation scheme, the false time step strategy and SIMPLEC (Semi-Implicit Method for Pressure Linked Equations - Consistent) algorithm were used to obtain the numerical solution.

Flows of Bingham Materials Through Ideal Porous Media: an Experimental and Theoretical Study

The flow of Bingham liquids through porous media has been studied. Experiments have been performed to determine the flow rate / pressure drop relationship for the flow of a grease of Binghamian rheological behavior through an array of rods of circular cross section. The yield stress and plastic viscosity of the grease have been determined with the aid of a controlled stress rotational rheometer. To investigate a wider range of the flow parameters, the mass and momentum conservation equations have been solved numerically, in conjunction with the generalized Newtonian constitutive law and the bi-viscosity model. The finite volume method has been employed to obtain the numerical solution. These numerical results also yielded a flow rate / pressure drop relationship, which is in very good agreement with the experimental results. A capillaric theory has been developed to determine an analytical relationship between the flow rate and pressure drop for flows of Bingham liquids through porous media. It is shown that the predictions of this theory are in good agreement with the experimental and numerical results.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes

In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis. The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares. From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations. The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.

Modelling of ultrasound assisted mixing in Newtonian and non-Newtonian liquids

The purpose of this work is to obtain a better understanding of behaviour of possible ultrasound appliance on fluid media mixing. The research is done in the regard to Newtonian and non-Newtonian fluids. The process of ultrasound appliance on liquids is modelled in COMSOL Multiphysics software. The influence of ultrasound using is introduced as waveform equation. Turbulence modelling is fulfilled by the k-ε model in Newtonian fluid. The modeling of ultrasound assisted mixing in non-Newtonian fluids is based on the power law. To verify modelling results two practical methods are used: Particle Image Velocimetry and measurements of mixing time. Particle Image Velocimetry allows capturing of velocity flow field continuously and presents detailed depiction of liquid dynamics. The second way of verification is the comparison of mixing time of homogeneity. Experimentally achievement of mixing time is done by conductivity measurements. In modelling part mixing time is achieved by special module of COMSOL Multiphysics – the transport of diluted species. Both practical and modelling parts show similar radial mechanism of fluid flow under ultrasound appliance – from the horn tip fluid moves to the bottom and along the walls goes back. Velocity profiles are similar in modelling and experimental part in the case of Newtonian fluid. In the case of non-Newtonian fluid velocity profiles do not agree. The development track of ultrasound-assisted mixing modelling is presented in the thesis.

Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.

Finite-Sample Inference Methods for Simultaneous Equations and Models with Unobserved and Generated Regressors

We propose finite sample tests and confidence sets for models with unobserved and generated regressors as well as various models estimated by instrumental variables methods. The validity of the procedures is unaffected by the presence of identification problems or \"weak instruments\", so no detection of such problems is required. We study two distinct approaches for various models considered by Pagan (1984). The first one is an instrument substitution method which generalizes an approach proposed by Anderson and Rubin (1949) and Fuller (1987) for different (although related) problems, while the second one is based on splitting the sample. The instrument substitution method uses the instruments directly, instead of generated regressors, in order to test hypotheses about the \"structural parameters\" of interest and build confidence sets. The second approach relies on \"generated regressors\", which allows a gain in degrees of freedom, and a sample split technique. For inference about general possibly nonlinear transformations of model parameters, projection techniques are proposed. A distributional theory is obtained under the assumptions of Gaussian errors and strictly exogenous regressors. We show that the various tests and confidence sets proposed are (locally) \"asymptotically valid\" under much weaker assumptions. The properties of the tests proposed are examined in simulation experiments. In general, they outperform the usual asymptotic inference methods in terms of both reliability and power. Finally, the techniques suggested are applied to a model of Tobin’s q and to a model of academic performance.

Reconstruction des structures magnéto-convectives solaires sous une région active, par l’utilisation conjointe d’un modèle de convection anélastique et d’une méthode d’assimilation de données.

L’utilisation d’une méthode d’assimilation de données, associée à un modèle de convection anélastique, nous permet la reconstruction des structures physiques d’une partie de la zone convective située en dessous d’une région solaire active. Les résultats obtenus nous informent sur les processus d’émergence des tubes de champ magnétique au travers de la zone convective ainsi que sur les mécanismes de formation des régions actives. Les données solaires utilisées proviennent de l’instrument MDI à bord de l’observatoire spatial SOHO et concernent principalement la région active AR9077 lors de l’ ́évènement du “jour de la Bastille”, le 14 juillet 2000. Cet évènement a conduit à l’avènement d’une éruption solaire, suivie par une importante éjection de masse coronale. Les données assimilées (magnétogrammes, cartes de températures et de vitesses verticales) couvrent une surface de 175 méga-mètres de coté acquises au niveau photosphérique. La méthode d’assimilation de données employée est le “coup de coude direct et rétrograde”, une méthode de relaxation Newtonienne similaire à la méthode “quasi-linéaire inverse 3D”. Elle présente l’originalité de ne pas nécessiter le calcul des équations adjointes au modèle physique. Aussi, la simplicité de la méthode est un avantage numérique conséquent. Notre étude montre au travers d’un test simple l’applicabilité de cette méthode à un modèle de convection utilisé dans le cadre de l’approximation anélastique. Nous montrons ainsi l’efficacité de cette méthode et révélons son potentiel pour l’assimilation de données solaires. Afin d’assurer l’unicité mathématique de la solution obtenue nous imposons une régularisation dans tout le domaine simulé. Nous montrons enfin que l’intérêt de la méthode employée ne se limite pas à la reconstruction des structures convectives, mais qu’elle permet également l’interpolation optimale des magnétogrammes photosphériques, voir même la prédiction de leur évolution temporelle.

Unfolded singularities of analytic differential equations

La thèse est composée d’un chapitre de préliminaires et de deux articles sur le sujet du déploiement de singularités d’équations différentielles ordinaires analytiques dans le plan complexe. L’article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity traite le problème de l’équivalence analytique de familles paramétriques de systèmes linéaires en dimension 2 qui déploient une singularité résonante générique de rang de Poincaré 1 dont la matrice principale est composée d’un seul bloc de Jordan. La question: quand deux telles familles sontelles équivalentes au moyen d’un changement analytique de coordonnées au voisinage d’une singularité? est complètement résolue et l’espace des modules des classes d’équivalence analytiques est décrit en termes d’un ensemble d’invariants formels et d’un invariant analytique, obtenu à partir de la trace de la monodromie. Des déploiements universels sont donnés pour toutes ces singularités. Dans l’article Confluence of singularities of non-linear differential equations via Borel–Laplace transformations on cherche des solutions bornées de systèmes paramétriques des équations non-linéaires de la variété centre de dimension 1 d’une singularité col-noeud déployée dans une famille de champs vectoriels complexes. En général, un système d’ÉDO analytiques avec une singularité double possède une unique solution formelle divergente au voisinage de la singularité, à laquelle on peut associer des vraies solutions sur certains secteurs dans le plan complexe en utilisant les transformations de Borel–Laplace. L’article montre comment généraliser cette méthode et déployer les solutions sectorielles. On construit des solutions de systèmes paramétriques, avec deux singularités régulières déployant une singularité irrégulière double, qui sont bornées sur des domaines «spirals» attachés aux deux points singuliers, et qui, à la limite, convergent vers une paire de solutions sectorielles couvrant un voisinage de la singularité confluente. La méthode apporte une description unifiée pour toutes les valeurs du paramètre.

Invariant discretizations of partial differential equations

Un algorithme permettant de discrétiser les équations aux dérivées partielles (EDP) tout en préservant leurs symétries de Lie est élaboré. Ceci est rendu possible grâce à l'utilisation de dérivées partielles discrètes se transformant comme les dérivées partielles continues sous l'action de groupes de Lie locaux. Dans les applications, beaucoup d'EDP sont invariantes sous l'action de transformations ponctuelles de Lie de dimension infinie qui font partie de ce que l'on désigne comme des pseudo-groupes de Lie. Afin d'étendre la méthode de discrétisation préservant les symétries à ces équations, une discrétisation des pseudo-groupes est proposée. Cette discrétisation a pour effet de transformer les symétries ponctuelles en symétries généralisées dans l'espace discret. Des schémas invariants sont ensuite créés pour un certain nombre d'EDP. Dans tous les cas, des tests numériques montrent que les schémas invariants approximent mieux leur équivalent continu que les différences finies standard.