Ces travaux de thèse présentent une étude expérimentale des fluctuations d'une particule Brownienne soumise à deux différentes conditions hors-équilibre dans un fluide . Le but est de comprendre d'une manière générale la relation entre les fluctuations spontanées, la fonction de réponse linéaire et la production totale d'entropie des processus loin de l'équilibre thermique. La première partie est consacrée à l'étude du mouvement d'une particule colloïdale dans un état stationnaire périodique hors-équilibre induit par une force non-conservative et à sa réponse à une perturbation externe. Nous analysons la dynamique du système dans le contexte des différentes approches généralisées de fluctuation-dissipation. Nous montrons que ces relations théoriques sont satisfaites par les données expérimentales quand on prend en compte le rôle du courant du à la rupture du bilan détaillé. Dans une deuxième partie nous étudions les fluctuations et la réponse d'une particule Brownienne dans deux types de bains vieillissants qui relaxent vers l'équilibre thermique: un verre colloïdal de Laponite et une solution aqueuse de gélatine. Dans ce cas-là nous montrons que le flux de chaleur de la particule vers le bain pendant sa relaxation représente une correction hors-équilibre du théorème de fluctuation-dissipation. Donc, le flux de chaleur joue le même rôle que le courant dans un état stationnaire. En conséquence, les résultats de la thèse mettent en évidence l'importance générale de la production totale d'entropie pour quantifier les relations de fluctuation-dissipation généralisées dans les systèmes hors-équilibre. / This thesis describes an experimental study on fluctuations of a Brownian particle immersed in a fluid, confined by optical tweezers and subject to two different kinds of non-equilibrium conditions. We aim to gain a rather general understanding of the relation between spontaneous fluctuations, linear response and total entropy production for processes away from thermal equilibrium. The first part addresses the motion of a colloidal particle driven into a periodic non-equilibrium steady state by a nonconservative force and its response to an external perturbation. The dynamics of the system is analyzed in the context of several generalized fluctuation-dissipation relations derived from different theoretical approaches. We show that, when taking into account the role of currents due to the broken detailed balance, the theoretical relations are verified by the experimental data. The second part deals with fluctuations and response of a Brownian particle in two different aging baths relaxing towards thermal equilibrium: a Laponite colloidal glass and an aqueous gelatin solution. The experimental results show that heat fluxes from the particle to the bath during the relaxation process play the same role of steady state currents as a non-equilibrium correction of the fluctuation-dissipation theorem. Then, the present thesis provides evidence that the total entropy production constitutes a unifying concept which links the statistical properties of fluctuations and the linear response function for non-equilibrium systems either in stationary or non stationary states.

自Nelson和Plosser (1982)後，研究經濟資料是否具有單位根現象，已成為近二十年來熱門且重要的課題。因 為資料性質的不同(恆定或非恆定)，對實證計量模型的設定、統計推論以及原理論的發展有深遠的影響。與傳 統探討單一時間數列之單位根的論文不同的是，本篇論文將橫斷面的資料擴大，探討混合型資料的單位根現象 ( Panel Unit Root )。就此課題，文獻上已有兩個不同的檢定方法: Levin、Lin和Chu (1997)的LLC檢定法以及Im、 Pesaran和Shin (1995)的IPS檢定法。 我們的研究，有別於以上兩者，是從「概似比」的角度(likelihood ratio) 和應用檢定共積關係的Johansen (1988)「Trace檢定」，建構新的單位根檢定統計量。首先於文中推導出，「Trace檢定」可用於檢測單一時間數 列的單位根現象。進而，再將橫斷面資料擴大，採用mean group方法，加總平均每個橫斷面時間數列的「Trace 檢定」統計量，形成混合型資料之單位根檢定統計量 。根據中央極限定理，標準化後的 檢定統計量，極限上 收斂至標準常態分配。此外，我們也推導得出 檢定統計量與傳統ADF、LLC以及IPS檢定統計量極限上的關係。 最後，我們以「蒙地卡羅」模擬方法，分析小樣本下「型一誤差」與「檢定力」的表現。發現新的混合型資 料之單位根檢定統計量表現優良，近似於標準常態分配。故在做混合型資料的單位根分析時，採用 檢定統計 量，可得到較精確的推論。

Title: Fluid-structure interaction of compressible flow Author: RNDr. Jaroslava Hasnedlová Department: Department of Numerical Mathematics, Institute of Applied Mathematics Supervisors: Prof. RNDr. Miloslav Feistauer, DrSc., Dr. h. c., Prof. Dr. Dr. h. c. Rolf Rannacher Supervisors' e-mail addresses: feist@karlin.mff.cuni.cz, rannacher@iwr.uni-heidelberg.de Abstract: The presented work is split into two parts. The first part is devoted to the theory of the discontinuous Galerkin finite element (DGFE) method for the space-time discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The DGFE method is applied sep- arately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. The main result is the proof of error estimates in L2 (L2 )-norm and in DG-norm formed by the L2 (H1 )-seminorm and penalty terms. The second part of the thesis deals with the realization of fluid-structure interaction problem of the compressible viscous flow with the elastic structure. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method, when the compress- ible Navier-Stokes equations are formulated in...

The idea of time series forecasting techniques is that the past has certain information about future. So, the question of how the information is encoded in the past can be interpreted and later used to extrapolate events of future constitute the crux of time series analysis and forecasting. Several methods such as qualitative techniques (e.g., Delphi method), causal techniques (e.g., least squares regression), quantitative techniques (e.g., smoothing method, time series models) have been developed in the past in which the concept lies in establishing a model either theoretically or mathematically from past observations and estimate future from it. Of all the models, time series methods such as autoregressive moving average (ARMA) process have gained popularity because of their simplicity in implementation and accuracy in obtaining forecasts. But, these models were formulated based on certain properties that a time series is assumed to possess. Classical decomposition techniques were developed to supplement the requirements of time series models. These methods try to define a time series in terms of simple patterns called trend, cyclical and seasonal patterns along with noise. So, the idea of decomposing a time series into component patterns, later modeling each component using forecasting processes and finally combining the component forecasts to obtain actual time series predictions yielded superior performance over standard forecasting techniques. All these methods involve basic principle of moving average computation. But, the developed classical decomposition methods are disadvantageous in terms of containing fixed number of components for any time series, data independent decompositions. During moving average computation, edges of time series might not get modeled properly which affects long range forecasting. So, these issues are to be addressed by more efficient and advanced decomposition techniques such as Wavelets and Empirical Mode Decomposition (EMD). Wavelets and EMD are some of the most innovative concepts considered in time series analysis and are focused on processing nonlinear and nonstationary time series. Hence, this research has been undertaken to ascertain the predictability of nonstationary time series using wavelet and Empirical Mode Decomposition (EMD) based ARMA models. The development of wavelets has been made based on concepts of Fourier analysis and Window Fourier Transform. In accordance with this, initially, the necessity of involving the advent of wavelets has been presented. This is followed by the discussion regarding the advantages that are provided by wavelets. Primarily, the wavelets were defined in the sense of continuous time series. Later, in order to match the real world requirements, wavelets analysis has been defined in discrete scenario which is called as Discrete Wavelet Transform (DWT). The current thesis utilized DWT for performing time series decomposition. The detailed discussion regarding the theory behind time series decomposition is presented in the thesis. This is followed by description regarding mathematical viewpoint of time series decomposition using DWT, which involves decomposition algorithm. EMD also comes under same class as wavelets in the consequence of time series decomposition. EMD is developed out of the fact that most of the time series in nature contain multiple frequencies leading to existence of different scales simultaneously. This method, when compared to standard Fourier analysis and wavelet algorithms, has greater scope of adaptation in processing various nonstationary time series. The method involves decomposing any complicated time series into a very small number of finite empirical modes (IMFs-Intrinsic Mode Functions), where each mode contains information of the original time series. The algorithm of time series decomposition using EMD is presented post conceptual elucidation in the current thesis. Later, the proposed time series forecasting algorithm that couples EMD and ARMA model is presented that even considers the number of time steps ahead of which forecasting needs to be performed. In order to test the methodologies of wavelet and EMD based algorithms for prediction of time series with non stationarity, series of streamflow data from USA and rainfall data from India are used in the study. Four non-stationary streamflow sites (USGS data resources) of monthly total volumes and two non-stationary gridded rainfall sites (IMD) of monthly total rainfall are considered for the study. The predictability by the proposed algorithm is checked in two scenarios, first being six months ahead forecast and the second being twelve months ahead forecast. Normalized Root Mean Square Error (NRMSE) and Nash Sutcliffe Efficiency Index (Ef) are considered to evaluate the performance of the proposed techniques. Based on the performance measures, the results indicate that wavelet based analyses generate good variations in the case of six months ahead forecast maintaining harmony with the observed values at most of the sites. Although the methods are observed to capture the minima of the time series effectively both in the case of six and twelve months ahead predictions, better forecasts are obtained with wavelet based method over EMD based method in the case of twelve months ahead predictions. It is therefore inferred that wavelet based method has better prediction capabilities over EMD based method despite some of the limitations of time series methods and the manner in which decomposition takes place. Finally, the study concludes that the wavelet based time series algorithm could be used to model events such as droughts with reasonable accuracy. Also, some modifications that could be made in the model have been suggested which can extend the scope of applicability to other areas in the field of hydrology.

This project deals with solving differential equations. The aim is find the correct algorithm transforming differential equations of higher order with time variable coefficients to equivalent systems of differential equations of first order. Subsequently verify its functionality for equations containing the involutioin goniometrical functions and finally implement this algorithm. The reason for this transformation is requirement to solve these differential equations by programme TKSL (Taylor Kunovský simulation language).

The quadratic class of time-frequency distributions (TFDs) forms a set of tools which allow to effectively extract important information from a nonstationary signal. To determine which TFD best represents the given signal, it is a common practice to visually compare different TFDs' time-frequency plots, and select as best the TFD with the most appealing plot. This visual comparison is not only subjective, but also difficult and unreliable especially when signal components are closely-spaced in the time-frequency plane. To objectively compare TFDs, a quantitative performance measure should be used. Several measures of concentration/complexity have been proposed in the literature. However, those measures by being derived with certain theoretical assumptions about TFDs are generally not suitable for the TFD selection problem encountered in practical applications. The non-existence of practically-valuable measures for TFDs' resolution comparison, and hence the non-existence of methodologies for the signal optimal TFD selection, has significantly limited the use of time-frequency tools in practice. In this thesis, by extending and complementing the concept of spectral resolution to the case of nonstationary signals, and by redefining the set of TFDs' properties desirable for practical applications, we define an objective measure to quantify the quality of TFDs. This local measure of TFDs' resolution performance combines all important signal time-varying parameters, along with TFDs' characteristics that influence their resolution. Methodologies for automatically selecting a TFD which best suits a given signal, including real-life signals, are also developed. The optimisation of the resolution performances of TFDs, by modifying their kernel filter parameters to enhance the TFDs' resolution capabilities, is an important prerequisite in satisfying any additional application-specific requirements by the TFDs. The resolution performance measure and the accompanying TFDs' comparison criteria allow to improve procedures for designing high-resolution quadratic TFDs for practical time-frequency analysis. The separable kernel TFDs, designed in this way, are shown to best resolve closely-spaced components for various classes of synthetic and real-life signals that we have analysed.