There has been a lot of work done in recent decades in the field of symbolic dynamics. Much attention has been paid to the so-called "complexity" function, which gives a sense of the rate at which the number of words in the system grow. In this paper, we explore this and several notions of complexity of specific symbolic dynamical systems. In particular, we compute positive entropy and state some k-balancedness properties of a few specific (random) substitutions. We also view certain sequences as subsets of Z², stating several properties and computing bounds on entropy in a specific example. / Graduation date: 2011

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A secure system requires cryptography and effective cryptography requires high quality system entropy. Within a virtualized Linux environment the quality and the amount of system entropy can be over overestimated. These virtualized environments can also have difficulty generating entropy data. To address the problems with entropy in virtualized Linux environments, my thesis investigates and evaluates exposing a unique true random number generator via an entropy-sharing tool called Entropy Broker. Entropy Broker distributes entropy data generated by the true random number generator to several virtualized Linux guest systems to increase the entropy of each system and in turn, increase the security of their cryptographic libraries. Entropy Broker and the true random number generator are evaluated against the Linux pseudo random number generator, the Haveged pseudo random number generator, and an on chip random number generator developed by Intel. / May 2016

Ho Siu-wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 103-104). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Classical Information Theory --- p.2 / Chapter 1.1.1 --- Shannon Entropy --- p.3 / Chapter 1.1.2 --- "Shannon Joint Entropy, Conditional Entropy, Mutual Information and Conditional Mutual Information" --- p.5 / Chapter 1.1.3 --- Applications of Shannon Entropy --- p.7 / Chapter 1.2 --- Mathematical background for Quantum Mechanics --- p.8 / Chapter 1.2.1 --- Dirac Notation --- p.8 / Chapter 1.2.2 --- Linear Operators and Matrices --- p.11 / Chapter 1.2.3 --- Spectral Decomposition and Diagonalization --- p.11 / Chapter 1.2.4 --- Functions of Normal Matrices --- p.12 / Chapter 1.2.5 --- Trace --- p.13 / Chapter 1.2.6 --- Kronecker Product --- p.13 / Chapter 1.3 --- Elementary Quantum Mechanics --- p.14 / Chapter 1.3.1 --- State Space --- p.15 / Chapter 1.3.2 --- Evolution --- p.16 / Chapter 1.3.3 --- Quantum Measurements --- p.17 / Chapter 1.3.4 --- Joint Systems --- p.20 / Chapter 1.3.5 --- Quantum Mixtures --- p.22 / Chapter 1.3.6 --- Subsystems --- p.28 / Chapter 1.4 --- von Neumann Entropy --- p.31 / Chapter 1.4.1 --- Definition --- p.32 / Chapter 1.4.2 --- Applications of the von Neumann Entropy --- p.34 / Chapter 1.4.3 --- Conditional Entropy --- p.34 / Chapter 1.5 --- Organization of The Thesis --- p.36 / Chapter Chapter 2 --- Problem Formulations --- p.38 / Chapter 2.1 --- Measurements that Produce Pure States --- p.39 / Chapter 2.2 --- The Shannon Entropy of a Quantum States --- p.41 / Chapter 2.3 --- An Equivalent Density Matrix Obtained by Mixing Orthogonal States --- p.44 / Chapter Chapter 3 --- Pure Post-Measurement States (PPMS) Measurements --- p.46 / Chapter 3.1 --- Introduction --- p.46 / Chapter 3.2 --- Definition of PPMS measurements --- p.46 / Chapter 3.3 --- Properties of PPMS Measurement --- p.52 / Chapter 3.4 --- An Alternative Definition of von Neumann entropy in terms of PPMS Measurements --- p.73 / Chapter Chapter 4 --- Mental Measurement of a Quantum State --- p.75 / Chapter 4.1 --- Introduction --- p.75 / Chapter 4.2 --- An Alternative Definition of a Projective Measurement --- p.76 / Chapter 4.3 --- Characteristics of a Projective PPMS Measurement --- p.81 / Chapter 4.4 --- The Choice of the Mental Measurement --- p.84 / Chapter 4.5 --- An Alternative Definition of von Neumann Entropy by Means of a Mental Measurement --- p.86 / Chapter 4.6 --- Construction of the Mental Measurement --- p.86 / Chapter Chapter 5 --- Completeness of Density Matrix Postulate --- p.92 / Chapter 5.1 --- Introduction --- p.92 / Chapter 5.2 --- Complete Specification of Quantum Ensemble by Density Matrix --- p.93 / Chapter 5.3 --- An Alternative Definition of von Neumann Entropy by Shannon Entropy --- p.98 / Chapter Chapter 6 --- Conclusion and Future Works --- p.99 / Chapter 6.1 --- Conclusion --- p.99 / Chapter 6.2 --- Future Work --- p.101 / Reference --- p.103

Computational Fluid Dynamics (CFD) is a field that is constantly advancing. Its advances in terms of capabilities are a result of new theories, faster computers, and new numerical methods. In this thesis, advances in the computational fluid dynamic modeling of moving bodies and combustive flows are investigated. Thus, the basic theory behind CFD is being extended to solve a new class of problems that are generally more complex. The first chapter that investigates some of the results, chapter IV, discusses a technique developed to model unsteady aerodynamics with moving boundaries such as flapping winged flight. This will include mesh deformation and fluid dynamics theory needed to solve such a complex system. Chapter V will examine the numerical modeling of a combustive flow. A three dimensional single vane burner combustion chamber is numerically modeled. Species balance equations along with rates of reactions are introduced when modeling combustive flows and these expressions are discussed. A reaction mechanism is validated for use with in situ reheat simulations. Chapter VI compares numerical results with a laminar methane flame experiment to further investigate the capabilities of CFD to simulate a combustive flow. A new method of examining a combustive flow is introduced by looking at the solutions ability to satisfy the second law of thermodynamics. All laminar flame simulations are found to be in violation of the entropy inequality.

Entropy based design is a novel design method that incorporates the second law of thermodynamics with computational and experimental techniques to achieve the upper limits of performance and quality in engineering technologies. As the emerging technologies are pressing towards the theoretical limits of efficiency, the concept of entropy and entropy based design will have an increasing role of performance. Measuring entropy generation is a valuable diagnostic tool from which the areas with high destruction rates of available energy may be determined and re-designed. In this work, a general model is developed, based on previous analytical expressions for pressure drop and heat transfer, for predicting entropy generation in a microchannel. The model includes the effects due to developing and fully developed flow, entrance and exit geometries, cross-sectional shapes, aspect ratio, and different thermal boundary conditions. An experimental technique is presented that enables the measurement of the spatial istribution of entropy generation in a microchannel. The experimental method is a combination of Micro Particle Image velocimetry to measure the spatial distribution of velocity and Micro Laser Induced Fluorescence to determine the temperature data. This method provides certain advantages over conventional anemometry techniques. This method, offers the whole-field non-intrusive, and instantaneous measurement of entropy generation in the device; while, previous techniques are limited to single point, averaged measurements.

Entropy based design is a novel design method that incorporates the second law of thermodynamics with computational and experimental techniques to achieve the upper limits of performance and quality in engineering technologies. As the emerging technologies are pressing towards the theoretical limits of efficiency, the concept of entropy and entropy based design will have an increasing role of performance. Measuring entropy generation is a valuable diagnostic tool from which the areas with high destruction rates of available energy may be determined and re-designed. In this work, a general model is developed, based on previous analytical expressions for pressure drop and heat transfer, for predicting entropy generation in a microchannel. The model includes the effects due to developing and fully developed flow, entrance and exit geometries, cross-sectional shapes, aspect ratio, and different thermal boundary conditions. An experimental technique is presented that enables the measurement of the spatial istribution of entropy generation in a microchannel. The experimental method is a combination of Micro Particle Image velocimetry to measure the spatial distribution of velocity and Micro Laser Induced Fluorescence to determine the temperature data. This method provides certain advantages over conventional anemometry techniques. This method, offers the whole-field non-intrusive, and instantaneous measurement of entropy generation in the device; while, previous techniques are limited to single point, averaged measurements.

In hydraulics, velocity distribution is needed to determine flow characteristics, like discharge, sediment discharge, head loss, energy coefficient, moment coefficient, and scour. However, the complicated interaction between water and sediment causes great difficulties in the measurement of flow and sediment discharge. Thus, the development of a method which can simulate the velocity distribution and sediment discharge in open channels is designable. Traditional methods for the estimation of velocity distribution, such as the Prandtl-von Karman logarithmic velocity and of sediment concentration distribution, such as the Rouse equation, are generally invalid at or near the channel bed and are inaccurate at the water surface. Considering the limitations of traditional methods, entropy based models have been applied, yet the assumption on the cumulative distribution function made in these methods limits their application. The objective of this research is to develop an efficient method to estimate velocity distribution and suspended sediment discharge in open channels using the Tsallis entropy. This research focuses on a better-organized hypothesis on the cumulative probability distribution function under more applicable coordinates, which should be transformable in different dimensions. Velocity distribution and sediment distribution are derived using the Tsallis entropy under the hypothesis that the cumulative probability distribution follows a non-linear function, in which the value of the exponent is shown to be related to the width-depth ratio of channel cross-section. Three different combinations of entropy and empirical methods for velocity and sediment concentration distribution are applied to compute suspended sediment discharge. Then advantages and disadvantages of each method are discussed. The velocity distribution derived using the Tsallis entropy is expected to be easy to apply and valid throughout the whole cross-section of the open channel. This research contributes to the application of entropy theory and shows its advantages in hydraulic engineering.