Periodic Solution of Nonlinear Conservative Systems

Conservative systems represent a large number of naturally occurring and artificially designed scientific and engineering systems. A key consideration in the theory and application of nonlinear conservative systems is the solution of the governing nonlinear ordinary differential equation. This chapter surveys the recent approximate analytical schemes for the periodic solution of nonlinear conservative systems and presents a recently proposed approximate analytical algorithm called continuous piecewise linearization method (CPLM). The advantage of the CPLM over other analytical schemes is that it combines simplicity and accuracy for strong nonlinear and large-amplitude oscillations irrespective of the complexity of the nonlinear restoring force. Hence, CPLM solutions for typical nonlinear Hamiltonian systems are presented and discussed. Also, the CPLM solution for an example of a non-Hamiltonian conservative oscillator was presented. The chapter is aimed at showcasing the potential and benefits of the CPLM as a reliable and easily implementable scheme for the periodic solution of conservative systems

Chebyshev cardinal functions for solving volterra-fredholm integro- differential equations using operational matrices

Abstract In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are described in detail. These matrices play the important role of reducing an integral equation to a system of algebraic equations. Illustrative examples are shown, which confirms the validity and applicability of the presented technique

Wavelet Methods for the Solutions of Partial and Fractional Differential Equations Arising in Physical Problems

The subject of fractional calculus has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It deals with derivatives and integrals of arbitrary orders. The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI D controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, control theory, neutron point kinetic model, anomalous diffusion, Brownian motion, signal and image processing, fluid dynamics and material science are well described by differential equations of fractional order. Generally, nonlinear partial differential equations of fractional order are difficult to solve. So for the last few decades, a great deal of attention has been directed towards the solution (both exact and numerical) of these problems. The aim of this dissertation is to present an extensive study of different wavelet methods for obtaining numerical solutions of mathematical problems occurring in disciplines of science and engineering. This present work also provides a comprehensive foundation of different wavelet methods comprising Haar wavelet method, Legendre wavelet method, Legendre multi-wavelet methods, Chebyshev wavelet method, Hermite wavelet method and Petrov-Galerkin method. The intension is to examine the accuracy of various wavelet methods and their efficiency for solving nonlinear fractional differential equations. With the widespread applications of wavelet methods for solving difficult problems in diverse fields of science and engineering such as wave propagation, data compression, image processing, pattern recognition, computer graphics and in medical technology, these methods have been implemented to develop accurate and fast algorithms for solving integral, differential and integro-differential equations, especially those whose solutions are highly localized in position and scale. The main feature of wavelets is its ability to convert the given differential and integral equations to a system of linear or nonlinear algebraic equations, which can be solved by numerical methods. Therefore, our main focus in the present work is to analyze the application of wavelet based transform methods for solving the problem of fractional order partial differential equations. The introductory concept of wavelet, wavelet transform and multi-resolution analysis (MRA) have been discussed in the preliminary chapter. The basic idea of various analytical and numerical methods viz. Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), First Integral Method (FIM), Optimal Homotopy Asymptotic Method (OHAM), Haar Wavelet Method, Legendre Wavelet Method, Chebyshev Wavelet Method and Hermite Wavelet Method have been presented in chapter 1. In chapter 2, we have considered both analytical and numerical approach for solving some particular nonlinear partial differential equations like Burgers’ equation, modified Burgers’ equation, Huxley equation, Burgers-Huxley equation and modified KdV equation, which have a wide variety of applications in physical models. Variational Iteration Method and Haar wavelet Method are applied to obtain the analytical and numerical approximate solution of Huxley and Burgers-Huxley equations. Comparisons between analytical solution and numerical solution have been cited in tables and also graphically. The Haar wavelet method has also been applied to solve Burgers’, modified Burgers’, and modified KdV equations numerically. The results thus obtained are compared with exact solutions as well as solutions available in open literature. Error of collocation method has been presented in this chapter. Methods like Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM) are very powerful and efficient techniques for solving nonlinear PDEs. Using these methods, many functional equations such as ordinary, partial differential equations and integral equations have been solved. We have implemented HPM and OHAM in chapter 3, in order to obtain the analytical approximate solutions of system of nonlinear partial differential equation viz. the Boussinesq-Burgers’ equations. Also, the Haar wavelet method has been applied to obtain the numerical solution of BoussinesqBurgers’ equations. Also, the convergence of HPM and OHAM has been discussed in this chapter. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and the necessity to solve such equations. The mathematical preliminaries of fractional calculus, definitions and theorems have been presented in chapter 4. Next, in this chapter, the Haar wavelet method has been analyzed for solving fractional differential equations. The time-fractional Burgers-Fisher, generalized Fisher type equations, nonlinear time- and space-fractional Fokker-Planck equations have been solved by using two-dimensional Haar wavelet method. The obtained results are compared with the Optimal Homotopy Asymptotic Method (OHAM), the exact solutions and the results available in open literature. Comparison of obtained results with OHAM, Adomian Decomposition Method (ADM), VIM and Operational Tau Method (OTM) has been demonstrated in order to justify the accuracy and efficiency of the proposed schemes. The convergence of two-dimensional Haar wavelet technique has been provided at the end of this chapter. In chapter 5, the fractional differential equations such as KdV-Burger-Kuramoto (KBK) equation, seventh order KdV (sKdV) equation and Kaup-Kupershmidt (KK) equation have been solved by using two-dimensional Legendre wavelet and Legendre multi-wavelet methods. The main focus of this chapter is the application of two-dimensional Legendre wavelet technique for solving nonlinear fractional differential equations like timefractional KBK equation, time-fractional sKdV equation in order to demonstrate the efficiency and accuracy of the proposed wavelet method. Similarly in chapter 6, twodimensional Chebyshev wavelet method has been implemented to obtain the numerical solutions of the time-fractional Sawada-Kotera equation, fractional order Camassa-Holm equation and Riesz space-fractional sine-Gordon equations. The convergence analysis has been done for these wavelet methods. In chapter 7, the solitary wave solution of fractional modified Fornberg-Whitham equation has been attained by using first integral method and also the approximate solutions obtained by optimal homotopy asymptotic method (OHAM) are compared with the exact solutions acquired by first integral method. Also, the Hermite wavelet method has been implemented to obtain approximate solutions of fractional modified Fornberg-Whitham equation. The Hermite wavelet method is implemented to system of nonlinear fractional differential equations viz. the fractional Jaulent-Miodek equations. Convergence of this wavelet methods has been discussed in this chapter. Chapter 8 emphasizes on the application of Petrov-Galerkin method for solving the fractional differential equations such as the fractional KdV-Burgers’ (KdVB) equation and the fractional Sharma-TassoOlver equation with a view to exhibit the capabilities of this method in handling nonlinear equation. The main objective of this chapter is to establish the efficiency and accuracy of Petrov-Galerkin method in solving fractional differential equtaions numerically by implementing a linear hat function as the trial function and a quintic B-spline function as the test function. Various wavelet methods have been successfully employed to numerous partial and fractional differential equations in order to demonstrate the validity and accuracy of these procedures. Analyzing the numerical results, it can be concluded that the wavelet methods provide worthy numerical solutions for both classical and fractional order partial differential equations. Finally, it is worthwhile to mention that the proposed wavelet methods are promising and powerful methods for solving fractional differential equations in mathematical physics. This work also aimed at, to make this subject popular and acceptable to engineering and science community to appreciate the universe of wonderful mathematics, which is in between classical integer order differentiation and integration, which till now is not much acknowledged, and is hidden from scientists and engineers. Therefore, our goal is to encourage the reader to appreciate the beauty as well as the usefulness of these numerical wavelet based techniques in the study of nonlinear physical system

Approximate Inference for Determinantal Point Processes

In this thesis we explore a probabilistic model that is well-suited to a variety of subset selection tasks: the determinantal point process (DPP). DPPs were originally developed in the physics community to describe the repulsive interactions of fermions. More recently, they have been applied to machine learning problems such as search diversification and document summarization, which can be cast as subset selection tasks. A challenge, however, is scaling such DPP-based methods to the size of the datasets of interest to this community, and developing approximations for DPP inference tasks whose exact computation is prohibitively expensive. A DPP defines a probability distribution over all subsets of a ground set of items. Consider the inference tasks common to probabilistic models, which include normalizing, marginalizing, conditioning, sampling, estimating the mode, and maximizing likelihood. For DPPs, exactly computing the quantities necessary for the first four of these tasks requires time cubic in the number of items or features of the items. In this thesis, we propose a means of making these four tasks tractable even in the realm where the number of items and the number of features is large. Specifically, we analyze the impact of randomly projecting the features down to a lower-dimensional space and show that the variational distance between the resulting DPP and the original is bounded. In addition to expanding the circumstances in which these first four tasks are tractable, we also tackle the other two tasks, the first of which is known to be NP-hard (with no PTAS) and the second of which is conjectured to be NP-hard. For mode estimation, we build on submodular maximization techniques to develop an algorithm with a multiplicative approximation guarantee. For likelihood maximization, we exploit the generative process associated with DPP sampling to derive an expectation-maximization (EM) algorithm. We experimentally verify the practicality of all the techniques that we develop, testing them on applications such as news and research summarization, political candidate comparison, and product recommendation

Knowledge-based automatic tolerance analysis system

Tolerance measure is an important part of engineering, however, to date the system of applying this important technology has been left to the assessment of the engineer using appropriate guidelines. This work offers a major departure from the trial and error or random number generation techniques that have been used previously by using a knowledge-based system to ensure the intelligent optimisation within the manufacturing system. A system to optimise manufacturing tolerance allocation to a part known as Knowledge-based Automatic Tolerance Analysis (KATA) has been developed. KATA is a knowledge-based system shell built within AutoCAD. It has the ability for geometry creation in CAD and the capability to optimise the tolerance heuristically as an expert system. Besides the worst-case tolerancing equation to optimise the tolerance allocation, KATA's algorithm is supported by actual production information such as machine capability, types of cutting tools, materials, process capabilities etc. KATA's prototype is currently able to analyse a cylindrical shape workpiece and a simple prismatic part. Analyses of tolerance include dimensional tolerance and geometrical tolerance. KATA is also able to do angular cuts such as tapers and chamfers. The investigation has also led to the significant development of the single tolerance reference technique. This method departs from the common practice of multiple tolerance referencing technique to optimise tolerance allocation. Utilisation of this new technique has eradicated the error of tolerance stackup. The retests have been undertaken, two of which are cylindrical parts meant to test dimensional tolerance and an angular cut. The third is a simple prismatic part to experiment with the geometrical tolerance analysis. The ability to optimise tolerance allocation is based on real production data and not imaginary or random number generation and has improved the accuracy of the expected result after manufacturing. Any failure caused by machining parameters is cautioned at an early stage before an actual production run has commenced. Thus, the manufacturer is assured that the product manufactured will be within the required tolerance limits. Being the central database for all production capability information enables KATA to opt for several approaches and techniques of processing. Hence, giving the user flexibility of selecting the process plan best suited for any required situation

Ab initio calculation and physical insights gained through linkage to continuum theories

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 2000.Includes bibliographical references (p. 159-165).We explore the use of ab initio, density-functional methods for the study of large-scale materials problems. Three examples are presented: (i) the interplay of surface and edge reconstructions in long silicon nanowires, where we examine the effects on electrical and mechanical properties; (ii) the calculation of solvation effects based on ab initio dielectric models, where we derive the dielectric treatment as a coarse-grained molecular description, apply our method to the hydrolysis reaction of methylene chloride, and examine simplifications to our ab initio method; and (iii) the study of screw dislocation cores in bcc molybdenum and tantalum, where we find core structures contrary to those commonly accepted and barriers to dislocation motion in better agreement with experiment. Methodologically, we present a new matrix-based, algebraic formalism for ab initio calculations which modularizes and isolates the roles played by the basis set, the energy functional, the algorithm used to achieve self-consistency, and the computational kernels. Development and implementation of new techniques amounts to derivation and transcription of algebraic expressions. Modularizing the computational kernels yields portable codes that are easily optimized and parallelized, and we present highly efficient kernels for scalar, shared, and distributed memory computers. We conclude with an analytical study of the spatial locality of the single-particle density matrix in solid-state systems. This locality reflects the localization of electronic states and is essential for real-space and O(N) methods. We derive new behavior for this spatial range contrary to previous proposals, and we verify our findings in model semiconductors, insulators, and metals.by Sohrab Ismail-Beigi.Ph.D

Computational discovery of gene modules, regulatory networks and expression programs

Thesis (Ph. D.)--Harvard-MIT Division of Health Sciences and Technology, 2007.Includes bibliographical references (p. 163-181).High-throughput molecular data are revolutionizing biology by providing massive amounts of information about gene expression and regulation. Such information is applicable both to furthering our understanding of fundamental biology and to developing new diagnostic and treatment approaches for diseases. However, novel mathematical methods are needed for extracting biological knowledge from high-dimensional, complex and noisy data sources. In this thesis, I develop and apply three novel computational approaches for this task. The common theme of these approaches is that they seek to discover meaningful groups of genes, which confer robustness to noise and compress complex information into interpretable models. I first present the GRAM algorithm, which fuses information from genome-wide expression and in vivo transcription factor-DNA binding data to discover regulatory networks of gene modules. I use the GRAM algorithm to discover regulatory networks in Saccharomyces cerevisiae, including rich media, rapamycin, and cell-cycle module networks. I use functional annotation databases, independent biological experiments and DNA-motif information to validate the discovered networks, and to show that they yield new biological insights. Second, I present GeneProgram, a framework based on Hierarchical Dirichlet Processes, which uses large compendia of mammalian expression data to simultaneously organize genes into overlapping programs and tissues into groups to produce maps of expression programs. I demonstrate that GeneProgram outperforms several popular analysis methods, and using mouse and human expression data, show that it automatically constructs a comprehensive, body-wide map of inter-species expression programs.(cont.) Finally, I present an extension of GeneProgram that models temporal dynamics. I apply the algorithm to a compendium of short time-series gene expression experiments in which human cells were exposed to various infectious agents. I show that discovered expression programs exhibit temporal pattern usage differences corresponding to classes of host cells and infectious agents, and describe several programs that implicate surprising signaling pathways and receptor types in human responses to infection.by Georg Kurt Gerber.Ph.D

Publications of the Jet Propulsion Laboratory, 1977

This bibliography cites 900 externally distributed technical reports released during calendar year 1977, that resulted from scientific and engineering work performed, or managed, by the Jet Propulsion Laboratory. Report topics cover 81 subject areas related in some way to the various NASA programs. The publications are indexed by: (1) author, (2) subject, and (3) publication type and number. A descriptive entry appears under the name of each author of each publication; an abstract is included with the entry for the primary (first-listed) author

Understanding Quantum Technologies 2022

Understanding Quantum Technologies 2022 is a creative-commons ebook that provides a unique 360 degrees overview of quantum technologies from science and technology to geopolitical and societal issues. It covers quantum physics history, quantum physics 101, gate-based quantum computing, quantum computing engineering (including quantum error corrections and quantum computing energetics), quantum computing hardware (all qubit types, including quantum annealing and quantum simulation paradigms, history, science, research, implementation and vendors), quantum enabling technologies (cryogenics, control electronics, photonics, components fabs, raw materials), quantum computing algorithms, software development tools and use cases, unconventional computing (potential alternatives to quantum and classical computing), quantum telecommunications and cryptography, quantum sensing, quantum technologies around the world, quantum technologies societal impact and even quantum fake sciences. The main audience are computer science engineers, developers and IT specialists as well as quantum scientists and students who want to acquire a global view of how quantum technologies work, and particularly quantum computing. This version is an extensive update to the 2021 edition published in October 2021.Comment: 1132 pages, 920 figures, Letter forma

Situated Analytics for Data Scientists

Much of Mark Weiser's vision of ``ubiquitous computing'' has come to fruition: We live in a world of interfaces that connect us with systems, devices, and people wherever we are. However, those of us in jobs that involve analyzing data and developing software find ourselves tied to environments that limit when and where we may conduct our work; it is ungainly and awkward to pull out a laptop during a stroll through a park, for example, but difficult to write a program on one's phone. In this dissertation, I discuss the current state of data visualization in data science and analysis workflows, the emerging domains of immersive and situated analytics, and how immersive and situated implementations and visualization techniques can be used to support data science. I will then describe the results of several years of my own empirical work with data scientists and other analytical professionals, particularly (though not exclusively) those employed with the U.S. Department of Commerce. These results, as they relate to visualization and visual analytics design based on user task performance, observations by the researcher and participants, and evaluation of observational data collected during user sessions, represent the first thread of research I will discuss in this dissertation. I will demonstrate how they might act as the guiding basis for my implementation of immersive and situated analytics systems and techniques. As a data scientist and economist myself, I am naturally inclined to want to use high-frequency observational data to the end of realizing a research goal; indeed, a large part of my research contributions---and a second ``thread'' of research to be presented in this dissertation---have been around interpreting user behavior using real-time data collected during user sessions. I argue that the relationship between immersive analytics and data science can and should be reciprocal: While immersive implementations can support data science work, methods borrowed from data science are particularly well-suited for supporting the evaluation of the embodied interactions common in immersive and situated environments. I make this argument based on both the ease and importance of collecting spatial data from user sessions from the sensors required for immersive systems to function that I have experienced during the course of my own empirical work with data scientists. As part of this thread of research working from this perspective, this dissertation will introduce a framework for interpreting user session data that I evaluate with user experience researchers working in the tech industry. Finally, this dissertation will present a synthesis of these two threads of research. I combine the design guidelines I derive from my empirical work with machine learning and signal processing techniques to interpret user behavior in real time in Wizualization, a mid-air gesture and speech-based augmented reality visual analytics system