Entropy Analysis of Industrial Accident Data Series

Complex industrial plants exhibit multiple interactions among smaller parts and with human operators. Failure in one part can propagate across subsystem boundaries causing a serious disaster. This paper analyzes the industrial accident data series in the perspective of dynamical systems. First, we process real world data and show that the statistics of the number of fatalities reveal features that are well described by power law (PL) distributions. For early years, the data reveal double PL behavior, while, for more recent time periods, a single PL fits better into the experimental data. Second, we analyze the entropy of the data series statistics over time. Third, we use the Kullback–Leibler divergence to compare the empirical data and multidimensional scaling (MDS) techniques for data analysis and visualization. Entropy-based analysis is adopted to assess complexity, having the advantage of yielding a single parameter to express relationships between the data. The classical and the generalized (fractional) entropy and Kullback–Leibler divergence are used. The generalized measures allow a clear identification of patterns embedded in the data.

Rényi entropy and the thermodynamic stability of black holes

Thermodynamic stability of black holes, described by the Rényi formula as equilibrium compatible entropy function, is investigated. It is shown that within this approach, asymptotically flat, Schwarzschild black holes can be in stable equilibrium with thermal radiation at a fixed temperature. This implies that the canonical ensemble exists just like in anti-de Sitter space, and nonextensive effects can stabilize the black holes in a very similar way as it is done by the gravitational potential of an anti-de Sitter space. Furthermore, it is also shown that a Hawking–Page-like black hole phase transition occurs at a critical temperature which depends on the q-parameter of the Rényi formula.

Relative information entropy in cosmology: The problem of information entanglement

The necessary information to distinguish a local inhomogeneous mass density field from its spatial average on a compact domain of the universe can be measured by relative information entropy. The Kullback-Leibler (KL) formula arises very naturally in this context, however, it provides a very complicated way to compute the mutual information between spatially separated but causally connected regions of the universe in a realistic, inhomogeneous model. To circumvent this issue, by considering a parametric extension of the KL measure, we develop a simple model to describe the mutual information which is entangled via the gravitational field equations. We show that the Tsallis relative entropy can be a good approximation in the case of small inhomogeneities, and for measuring the independent relative information inside the domain, we propose the R\'enyi relative entropy formula.

Stability and error analysis for a numerical scheme to approximate elastic flow

Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Univ., Dissertation, 2015

Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibriaThe Patterson-Sullivan embedding and minimal volume entropy for outer space

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t→ ∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L¹-norm, as well as various Sobolev norms.

The Patterson-Sullivan embedding and minimal volume entropy for outer space

"Vegeu el resum a l'inici del document del fitxer adjunt."

The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center

We prove that if f is a partially hyperbolic diffeomorphism on the compact manifold M with one dimensional center bundle, then the logarithm of the spectral radius of the map induced by f on the real homology groups of M is smaller or equal to the topological entropy of f. This is a particular case of the Shub's entropy conjecture, which claims that the same conclusion should be true for any C1 map on any compact manifold.

Chebyshev polynomial approximation to approximate partial differential equations

This paper suggests a simple method based on Chebyshev approximation at Chebyshev nodes to approximate partial differential equations. The methodology simply consists in determining the value function by using a set of nodes and basis functions. We provide two examples. Pricing an European option and determining the best policy for chatting down a machinery. The suggested method is flexible, easy to program and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations.

The analysis of approximate quickselect and related problems

Approximate Quickselect, a simple modification of the well known Quickselect algorithm for selection, can be used to efficiently find an element with rank k in a given range [i..j], out of n given elements. We study basic cost measures of Approximate Quickselect by computing exact and asymptotic results for the expected number of passes, comparisons and data moves during the execution of this algorithm. The key element appearing in the analysis of Approximate Quickselect is a trivariate recurrence that we solve in full generality. The general solution of the recurrence proves to be very useful, as it allows us to tackle several related problems, besides the analysis that originally motivated us. In particular, we have been able to carry out a precise analysis of the expected number of moves of the ith element when selecting the jth smallest element with standard Quickselect, where we are able to give both exact and asymptotic results. Moreover, we can apply our general results to obtain exact and asymptotic results for several parameters in binary search trees, namely the expected number of common ancestors of the nodes with rank i and j, the expected size of the subtree rooted at the least common ancestor of the nodes with rank i and j, and the expected distance between the nodes of ranks i and j.

Advanced mapping of environmental data: Geostatistics, Machine Learning and Bayesian Maximum Entropy

This book combines geostatistics and global mapping systems to present an up-to-the-minute study of environmental data. Featuring numerous case studies, the reference covers model dependent (geostatistics) and data driven (machine learning algorithms) analysis techniques such as risk mapping, conditional stochastic simulations, descriptions of spatial uncertainty and variability, artificial neural networks (ANN) for spatial data, Bayesian maximum entropy (BME), and more.

An approximate mathematical model for solidification of a flowing liquid in a microchannel

A mathematical model is developed to analyse the combined flow and solidification of a liquid in a small pipe or two-dimensional channel. In either case the problem reduces to solving a single equation for the position of the solidification front. Results show that for a large range of flow rates the closure time is approximately constant, and the value depends primarily on the wall temperature and channel width. However, the ice shape at closure will be very different for low and high fluxes. As the flow rate increases the closure time starts to depend on the flow rate until the closure time increases dramatically, subsequently the pipe will never close.

An approximate solution method for boundary layer flow of a power law fluid over a flat plate

The work in this paper deals with the development of momentum and thermal boundary layers when a power law fluid flows over a flat plate. At the plate we impose either constant temperature, constant flux or a Newton cooling condition. The problem is analysed using similarity solutions, integral momentum and energy equations and an approximation technique which is a form of the Heat Balance Integral Method. The fluid properties are assumed to be independent of temperature, hence the momentum equation uncouples from the thermal problem. We first derive the similarity equations for the velocity and present exact solutions for the case where the power law index n = 2. The similarity solutions are used to validate the new approximation method. This new technique is then applied to the thermal boundary layer, where a similarity solution can only be obtained for the case n = 1.

Active Learning of Very-High Resolution Optical Imagery with SVM: Entropy vs Margin Sampling

An active learning method is proposed for the semi-automatic selection of training sets in remote sensing image classification. The method adds iteratively to the current training set the unlabeled pixels for which the prediction of an ensemble of classifiers based on bagged training sets show maximum entropy. This way, the algorithm selects the pixels that are the most uncertain and that will improve the model if added in the training set. The user is asked to label such pixels at each iteration. Experiments using support vector machines (SVM) on an 8 classes QuickBird image show the excellent performances of the methods, that equals accuracies of both a model trained with ten times more pixels and a model whose training set has been built using a state-of-the-art SVM specific active learning method

Extremal principle of entropy production in the climate system

Paltridge found reasonable values for the most significant climatic variables through maximizing the material transport part of entropy production by using a simple box model. Here, we analyse Paltridge's box model to obtain the energy and the entropy balance equations separately. Derived expressions for global entropy production, which is a function of the radiation field, and even its material transport component, are shown to be different from those used by Paltridge. Plausible climatic states are found at extrema of these parameters. Feasible results are also obtained by minimizing the radiation part of entropy production, in agreement with one of Planck's results, Finally, globally averaged values of the entropy flux of radiation and material entropy production are obtained for two dynamical extreme cases: an earth with uniform temperature, and an earth in radiative equilibrium at each latitudinal point