O objetivo principal deste trabalho é o estudo da regularidade anallítica global de certos operadores diferenciais definidos no toro. Uma ferramenta fundamental utilizada neste estudo são as séries parciais de Fourier, que nos permitem caracterizar tanto as distribuições periódicas quanto as funções anallíticas reais periódicas através do comportamento assintótico de seus coeficientes parciais de Fourier. Neste sentido, apresentamos também um estudo detalhado das relações destes objetos com seus coeficientes parciais de Fourier / The main goal of this work is to study global analytic regularity properties of certain differential operators acting in the torus. A main tool that will be used to achieve our goals are the partial Fourier series, which allow us to characterize objects such as periodic distributions or periodic real analytic functions in terms of the growth of their partial Fourier coefficients

O objetivo principal deste trabalho é o estudo da regularidade anallítica global de certos operadores diferenciais definidos no toro. Uma ferramenta fundamental utilizada neste estudo são as séries parciais de Fourier, que nos permitem caracterizar tanto as distribuições periódicas quanto as funções anallíticas reais periódicas através do comportamento assintótico de seus coeficientes parciais de Fourier. Neste sentido, apresentamos também um estudo detalhado das relações destes objetos com seus coeficientes parciais de Fourier / The main goal of this work is to study global analytic regularity properties of certain differential operators acting in the torus. A main tool that will be used to achieve our goals are the partial Fourier series, which allow us to characterize objects such as periodic distributions or periodic real analytic functions in terms of the growth of their partial Fourier coefficients

The thesis is composed of three parts - theoretical, computational and a project part. The theoretical part deals with heat sharing through conduction, flow and radiation. The computational part is focused on the overall calculation of the heating system to operate smoothly and reliably. Three gas condensing boilers are designed as a source of heat. The heating of the water is solved as a reservoir. It's source of heat is one gas condensation boiler. The project part contains a technical report and the project documentation on the stage of the implementation dossier.

[ES] Esta tesis de doctorado está dedicada al problema de descomposición de señales no estacionarias en componentes modales, entendida como componentes oscilatorias independientes, con amplitud y fase dependientes del tiempo. Para este fin, se propone un enfoque metodológico basado en representaciones en espacio de estados diagonales en bloques. Una contribución teórica primaria de esta tesis consiste en demostrar que la respuesta de un sistema de espacio de estados diagonal en bloques puede ser representada en una forma modal con amplitudes y frecuencias dependientes del tiempo. Subsecuentemente, construyendo sobre este resultado, un marco de trabajo basado en filtros de Kalman se propone para la descomposición modal de señales no estacionarias. Como resultado, una familia de métodos paramétricos para la descomposición modal de señales no estacionarias univariadas y multivariadas basadas en representaciones de espacio de estados diagonales en bloques y filtros de Kalman ha sido postulada. La representación básica está construida en bloques de segundo orden, cada uno de los cuales representa los componentes en fase y en cuadratura de un único componente oscilatorio no estacionario. Así, la respuesta total es construida como la suma ponderada de cada uno de estos modos. La identificación de estos modelos requiere la estimación conjunta de las trayectorias y los parámetros modales dependientes del tiempo, así como los hiperparámetros del modelo, constituidos por la matriz de mezcla de modos, las matrices de covarianza del vector de estados, de parámetros y del ruido de medición, y las condiciones iniciales. Para este propósito, un algoritmo de Expectación-Maximización ha sido adaptado como parte de esta tesis. La metodología obtenida es entonces evaluada en la descomposición y eliminación de ruido de registros electrocardiográficos (ECG), los cuales consisten en componentes no-estacionarias pseudo-periódicas y son susceptibles a diferentes tipos de interferencias. La estructura de estas señales las hace susceptibles a las descomposiciones modales basadas propuestas en esta tesis. A diferencia de otros métodos populares de descomposición de señales, las descomposiciones obtenidas con la metodología propuesta proveen componentes oscilatorios con interpretabilidad física y que proveen resultados consistentes para señales multivariadas, como en el caso de registros de ECG con múltiples derivaciones. Otra estrategia que se desarrolló en este proyecto investigativo lo constituye la aplicación de la transformada delta u operador de Euler al filtro de Kalman, esto condujo a resultados de alta precisión en la extracción de componentes de banda angosta. La metodología propuesta constituye una herramienta confiable para la descomposición modal en línea de señales no estacionarias multicomponentes, con resultados excelentes / [CA] Esta tesi de doctorat està dedicada al problema de descomposició de senyals no-estacionaris en components modals, entesa com a components oscil·latòries independents amb amplitud i fase dependents del temps. Per a este fi, es proposa un enfocament metodològic basat en representacions en espai d'estats diagonals en blocs. Una contribució teòrica primària d'esta tesi consistix a demostrar que la resposta d'un sistema d'espai d'estats diagonal en blocs pot ser representada en una forma modal amb amplituds i freqüències dependents del temps. Subseqüentment, construint sobre este resultat, un marc de treball basat en filtres de Kalman es proposa per a la descomposició modal de senyals no estacionaris. Com a resultat, una família de mètodes paramètrics per a la descomposició modal de senyals no estacionaris univariadas i multivariades basades en representacions d'espai d'estats diagonals en blocs i filtres de Kalman ha sigut postulada. La representació bàsica està construïda en blocs de segon ordre, cadascun dels quals representa els components en fase i en quadratura d'un únic component oscil·latori no estacionari. Així, la resposta total és construïda com la suma ponderada de cadascun d'estos modes. La identificació d'estos models requerix l'estimació conjunta de les trajectòries i els paràmetres modals dependents del temps, així com els hiperparámetros del model, constituïts per la matriu de mescla de modes, les matrius de covariància del vector d'estats, de paràmetres i del soroll de mesurament, i les condicions inicials. Per a este propòsit, un algorisme d'Expectació-Maximització ha sigut adaptat com a part d'esta tesi. La metodologia obtinguda és llavors avaluada en la descomposició i eliminació de soroll de registres electrocardiogràfics (ECG), els quals consistixen en components no-estacionàries pseudo-periòdiques i són susceptibles a diferents tipus d'interferències. L'estructura d'estos senyals les fa susceptibles a les descomposicions modals basades propostes en esta tesi. A diferència d'altres mètodes populars de descomposició de senyals, les descomposicions obtingudes amb la metodologia proposada proveïxen components oscil·latoris amb interpretabilidad física i que proveïxen resultats consistents per a senyals multivariats, com en el cas de registres d'ECG amb múltiples derivacions. Una altra estratègia que es va desenvolupar en este projecte investigativo el constituïx l'aplicació de la transformada delta o operador d'Euler al filtre de Kalman, això va conduir a resultats d'alta precisió en l'extracció de components de banda estreta. La metodologia proposada constituïx una eina de confiança per a la descomposició modal en línia de senyals no estacionaris multicomponents, amb resultats excel·lents. / [EN] This PhD thesis is devoted to the problem of the decomposition of non-stationary signals in modal components, understood as independent oscillatory components with time-dependent amplitude and frequency. To this end, a methodological approach based on diagonal time-dependent state space models is postulated. A primary theoretical contribution of this work is to demonstrate that the response of a system in diagonal time-dependent state space form can be cast in a modal form characterized by time-dependent amplitudes and frequencies. Subsequently, building up on this result, a Kalman filter based framework for non-stationary modal decomposition is proposed. As a result, a family of parametric modal decomposition methods is postulated for univariate and multivariate non-stationary signals based on block-diagonal time-dependent state space representations and Kalman filtering/smoothing. The representation is built upon second order blocks, each representing the in-phase and quadrature components of a single non-stationary oscillatory component. The total response is then constructed as the weighted sum of each of these modes. Accordingly, the model identification involves the joint estimation of the modal trajectories and the time-dependent modal parameters, along with the model hyperparameters, constituted by the mode mixing matrix, the state, parameter and noise covariances, and initial conditions. A tailored Expectation-Maximization algorithm is designed for this purpose as part of this thesis. The obtained methodology is assessed in the decomposition and denoising of electrocardiographic (ECG) signals, which consist of pseudo-periodic non-stationary signals and are susceptible to significant interference. The ECG signal structure makes them amenable to the proposed non-stationary modal decompositions. In contrast to other popular non-stationary signal decomposition methods, the proposed method provides a physically meaningful decomposition of oscillatory components, with consistent results for multivariate signals, such as multi-lead ECG records. Another strategy that was developed in this research project is the application of the delta transform or Euler operator to the Kalman filter, which led to highly precise results in extracting narrowband components. The proposed methodology constitutes a reliable tool for on-line modal decomposition of multi-component non-stationary signals, with results comparable and even better than other state-of-the-art methods. / Avendaño, LE. (2024). Método de descomposición modal no estacionaria basado en representación de espacio de estados con aplicación al análisis de señales ECG [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/211185

In this thesis, we study statistical inference for entropy, divergence, and related functionals of one or two probability distributions. Asymptotic properties of particular nonparametric estimators of such functionals are investigated. We consider estimation from both independent and dependent observations. The thesis consists of an introductory survey of the subject and some related theory and four papers (A-D). In Paper A, we consider a general class of entropy-type functionals which includes, for example, integer order Rényi entropy and certain Bregman divergences. We propose U-statistic estimators of these functionals based on the coincident or epsilon-close vector observations in the corresponding independent and identically distributed samples. We prove some asymptotic properties of the estimators such as consistency and asymptotic normality. Applications of the obtained results related to entropy maximizing distributions, stochastic databases, and image matching are discussed. In Paper B, we provide some important generalizations of the results for continuous distributions in Paper A. The consistency of the estimators is obtained under weaker density assumptions. Moreover, we introduce a class of functionals of quadratic order, including both entropy and divergence, and prove normal limit results for the corresponding estimators which are valid even for densities of low smoothness. The asymptotic properties of a divergence-based two-sample test are also derived. In Paper C, we consider estimation of the quadratic Rényi entropy and some related functionals for the marginal distribution of a stationary m-dependent sequence. We investigate asymptotic properties of the U-statistic estimators for these functionals introduced in Papers A and B when they are based on a sample from such a sequence. We prove consistency, asymptotic normality, and Poisson convergence under mild assumptions for the stationary m-dependent sequence. Applications of the results to time-series databases and entropy-based testing for dependent samples are discussed. In Paper D, we further develop the approach for estimation of quadratic functionals with m-dependent observations introduced in Paper C. We consider quadratic functionals for one or two distributions. The consistency and rate of convergence of the corresponding U-statistic estimators are obtained under weak conditions on the stationary m-dependent sequences. Additionally, we propose estimators based on incomplete U-statistics and show their consistency properties under more general assumptions.

<p>In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as &lsquo / mean-unbiased&rsquo / and &lsquo / medianunbiased&rsquo / estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1).</p> <p><br /> However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to&nbsp / compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the &lsquo / medianunbiased&rsquo / estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed / the &lsquo / most-probably-unbiased&rsquo / estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed / (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model / (3) the exact variance and MSE of LS estimator is determined / (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort / (5) an exact method of evaluating the density of the three estimators is described / (6) their exact bias, mean, variance and MSE are determined and analysed / and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed.</p> <p><br /> The discussion and results show that the estimators are still biased in the usual sense: &lsquo / in expectation&rsquo / . However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation.</p>

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés. / Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

Für einen R^d-wertigen stochastischen Prozess X auf R bezeichne X_t den Segmentprozess X_t:={X(t+u): u = 0. Es wird folgende affine stochastische Differentialgleichung mit unendlichem Gedächtnis betrachtet: dX(t)=L(X_t)dt + dW(t) für t >= 0, X_0=F, (A) wobei L:B -> R^d ein lineares stetiges Funktional, W einen Wiener-Prozess mit Werten in R^d sowie B einen semi-normierten linearen Unterraum von {f:(-00, 0] -> R^d} bezeichnen. Die Anfangsbedingung F ist eine B-wertige Zufallsvariable. Die Lösung X der Gleichung (A) lässt sich mittels einer Formel der Variation der Konstanten darstellen. Für die Existenz einer stationären Lösung werden hinreichende und notwendige Bedingungen vorgestellt. Für eine spezielle Klasse von Funktionalen L kann Gleichung (A) auf ein System gewöhnlicher stochastischer Gleichungen ohne Gedächtnis reduziert werden. Diese Reduktion wird im Detail untersucht, insbesondere gewinnt man hierdurch ein einfaches äquivalentes Kriterium für die Existenz stationärer Lösungen von Gleichungen mit Funktionalen L dieser Klasse. Durch Einbettung der Gleichung (A) in den Bidualraum B** gelingt die Bestimmung der Lyapunov-Exponenten der Lösung. Hierzu wird ein neuer Zusammenhang der Lösung der sogenannten adjungierten Gleichung von (A) und einer Spektralzerlegung des Raumes B benutzt. Die Untersuchung der stetigen Abhängigkeit der Lösung von dem Funktional L und der Anfangsbedingung F ermöglicht die Behandlung anwendungsorientierter Aspekte. In Verbindung mit den Ergebnissen über reduzierbare Gleichungen wird ein Verfahren zur Approximation der Lösung von Gleichung (A) durch Ornstein-Uhlenbeck-Prozesse vorgestellt. Eine allgemeine Klasse von Ito-Differentialgleichungen mit nichtlinearen vergangenheitsabhängigen Drift- und Dispersionskoeffizienten wird eingeführt, in der die Gleichung (A) als eine spezielle affine Gleichung verstanden werden kann. Für diese allgemeinen Gleichungen wird ein Existenz- und Eindeutigkeitssatz nachgewiesen. / For an R^d-valued stochastic process X denote the segment process by X_t:={X(t+u): u = 0. We consider the following affine stochastic differential equation with infinite delay: dX(t)=L(X_t)dt + dW(t) for t >= 0, X_0= F, (A) where L:B -> R^d denotes a linear continuous functional, W denotes a Wiener process with values in R^d and B is a semi-normed linear subspace of {f: (-00, 0] -> R^d}. The initial condition F is a B-valued random variable. The solution X of equation (A) can be represented by a variation of constants formula. We provide sufficient and necessary conditions for the existence of a stationary solution. For a special class of functionals L the equation (A) can be reduced to a system of ordinary stochastic differential equations without memory. This reduction is studied in detail. In particular, we deduce a simple equivalent condition for the existence of stationary solutions of equations with functionals L in this class. The embedding of equation (A) into the bidualspace B** enables us to calculate the Lyapunov exponents of the solution. For this purpose we exploit a new connection between the solution of the so-called adjoint equation of (A) and a spectral decompositon of the space B. By considering the continuous dependence of the solution on the functional L and the initial condition F we obtain results useful in applications. In conjunction with results on reducible equations we establish an approximation scheme for the solution of equation (A) by Ornstein-Uhlenbeck processes. Moreover, we introduce a general class of Ito differential equations with non-linear drift and dispersion hereditary coefficients. We deduce a result on the existence of unique solutions for this general class of equations. Equation (A) can be regarded as a special affine equation in this class.

A necessidade de obter solução de grandes sistemas lineares resultantes de processos de discretização de equações diferenciais parciais provenientes da modelagem de diferentes fenômenos físicos conduz à busca de técnicas numéricas escaláveis. Métodos multigrid são classificados como algoritmos escaláveis.Um estimador de erros deve estar associado à solução numérica do problema discreto de modo a propiciar a adequada avaliação da solução obtida pelo processo de aproximação. Nesse contexto, a presente tese caracteriza-se pela proposta de reutilização das estruturas matriciais hierárquicas de operadores de transferência e restrição dos métodos multigrid algébricos para acelerar o tempo de solução dos sistemas lineares associados à equação do transporte de contaminantes em meio poroso saturado. Adicionalmente, caracteriza-se pela implementação das estimativas residuais para os problemas que envolvem dados constantes ou não constantes, os regimes de pequena ou grande advecção e pela proposta de utilização das estimativas residuais associadas ao termo de fonte e à condição inicial para construir procedimentos adaptativos para os dados do problema. O desenvolvimento dos códigos do método de elementos finitos, do estimador residual e dos procedimentos adaptativos foram baseados no projeto FEniCS, utilizando a linguagem de programação PYTHONR e desenvolvidos na plataforma Eclipse. A implementação dos métodos multigrid algébricos com reutilização considera a biblioteca PyAMG. Baseado na reutilização das estruturas hierárquicas, os métodos multigrid com reutilização com parâmetro fixo e automática são propostos, e esses conceitos são estendidos para os métodos iterativos não-estacionários tais como GMRES e BICGSTAB. Os resultados numéricos mostraram que o estimador residual captura o comportamento do erro real da solução numérica, e fornece algoritmos adaptativos para os dados cuja malha retornada produz uma solução numérica similar à uma malha uniforme com mais elementos. Adicionalmente, os métodos com reutilização são mais rápidos que os métodos que não empregam o processo de reutilização de estruturas. Além disso, a eficiência dos métodos com reutilização também pode ser observada na solução do problema auxiliar, o qual é necessário para obtenção das estimativas residuais para o regime de grande advecção. Esses resultados englobam tanto os métodos multigrid algébricos do tipo SA quanto os métodos pré-condicionados por métodos multigrid algébrico SA, e envolvem o transporte de contaminantes em regime de pequena e grande advecção, malhas estruturadas e não estruturadas, problemas bidimensionais, problemas tridimensionais e domínios com diferentes escalas. / The need for solving large linear systems arising from the discretization of partial differential equations modelling physical phenomena motivates the search for scalable numerical techniques. Multigrid algorithms are instances of such techniques.In order to provide a suitable assessment of the solution obtained by such algorithms, an error estimator must be associated to the numerical solution of the discretized problem. In this context, this thesis proposes the reutilization of the hierarchical matrix structures of transfer operators and the restriction to algebraic multigrid methods to speed up the process of solving the linear systems associated with the contaminant transport equation in saturated porous media. In addition, it features the implementation of residual estimates for problems involving constant or non-constant data, the regimes of small- or large-scale advection and the proposal of employing the residual estimates associated to the source term and to the initial condition to build adaptive procedures for the problem data. The development of the computer codes of the finite element method, residual estimator and adaptive procedures were based on the FEniCS project, using the programming language PYTHONR and developed on the Eclipse platform. The implementation of the algebraic methods with reutilization relied upon the libray PyAMG. Grounding on the idea of reutilizing the hierarchical structures, fixed and automatic parameters multigrid methods were proposed and extended to non-stationary iterative methods such as GMRES and BICGSTAB. The numerical results demonstrate that the residual estimator captures the behavior of the real error of the numerical solution, and provide adaptive algorithms for the data whose output mesh yields a numerical solution alike to that obtained from a uniform mesh with more elements. Moreover, the methods with reutilization are faster than those that do not reuse the structures. Besides, the efficiency of such methods can also be observed in the solution of an auxiliary problem, which is necessary for deriving the residual estimates in the regime of large-scale advection. These results encompass both the type SA algebraic multigrid method and those pre-conditioned by them. Moreover, they involve the transport of contaminants in regime of small- and large-scale advection, structured and non-structured meshes, bi- and tridimensional problems and domains with different scales.

As famílias Anosov foram introduzidas por P. Arnoux e A. Fisher, motivados por generalizar a noção de difeomorfismo de Anosov. A grosso modo, as famílias Anosov são sequências de difeomorfismos (fi)i&#8712Z definidos em uma sequencia de variedades Riemannianas compactas (Mi)i&#8712Z, em que fi: Mi ->Mi+1 para todo i &#8712 Z, tal que a composição fi+no· · ·ofi, para n >=1, tem comportamento assintoticamente hiperbólico. Esta noção é conhecida como um sistema dinâmico não-estacionário ou um sistema dinâmico não-autônomo. Sejam M a união disjunta de cada Mi, para i &#8712 Z, e Fm(M) o conjunto consistente das famílias de difeomorfismos (fi)i&#8712Z de classe Cm definidos na sequência (Mi)i&#8712Z. O propósito principal deste trabalho é mostrar algumas propriedades das famílias Anosov. Em particular, mostraremos que o conjunto destas famílias é aberto em Fm(M), em que Fm(M) é munido da topologia forte (ou topologia Whitney); a estabilidade estrutural de certa classe de famílias Anosov, considerando conjugações topológicas uniformes; e várias versões para os Teoremas de variedades estáveis e instáveis. Os resultados que serão apresentados aqui generalizam alguns outros resultados obtidos em Sistemas Dinâmicos Aleatórios, os quais serão mencionados ao longo do trabalho. Além do anterior, será introduzida a entropia topológica para elementos em Fm(M) e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua em Fm(M) munido da topologia forte. Porém, ela é descontínua em cada elemento de Fm(M) munido da topologia produto. Também apresentaremos um resultado que pode ser uma ferramenta de muita utilidade no estudo da continuidade da entropia topológica de difeomorfismos definidos em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao longo desta pesquisa e que serão analisados em um trabalho futuro. / Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the notion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms (fi)i&#8712Z dened on a sequence of compact Riemannian manifolds (Mi)i&#8712Z, where fi: Mi -> Mi+1 for all i &#8712 Z, such that the composition fi+n o · · · o fi, for n >=1, has asymptotically hyperbolic behavior. This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system. Let M be the disjoint union of each Mi, for each i &#8712 Z, and Fm(M) the set consisting of families of Cm-dieomorphisms (fi)i&#8712Z dened on the sequence (Mi)i&#8712Z. The main goal of this work is to explore some properties of Anosov families. In particular, we will show that the set consisting of these families is open in Fm(M), where Fm(M) is endowed with the strong topology (or Whitney topology); the structural stability of a certain class of Anosov families, considering uniform topological conjugacies; and some versions of stable and unstable manifold theorems. The results that will be presented here generalize some results obtained in Random Dynamical Systems, which will be mentioned throughout the work. In addition to the above mentioned theorems, the topological entropy for elements in Fm(M) will be introduced, and we will show some of its properties. We will prove that this entropy is continuous on Fm(M) endowed with strong topology. However, it is discontinuous at each element of Fm(M) endowed with the product topology. We will also present a result that can be a very useful tool in the study of the continuity of the topological entropy of dieomorphisms dened on compact manifolds. We will nish the work by giving a list of problems that have arisen throughout this research and that will be analyzed in a future work.