Dissertations / Theses on the topic 'Manifolds (Mathematics) Nonlinear theories'

In this thesis, we will study a class of fully nonlinear flows on Kähler manifolds. This family of flows generalizes the previously studied J-flow. We use the quotients of elementary symmetric polynomials or log of them to construct the flow. We obtain a necessary and sufficient condition in terms of positivity of certain cohomology class to guarantee the convergence of the flow. The corresponding limit metric gives rise to a critical metric satisfying a Hessian type equation on the manifold. We shall also discuss several geometric applications of our main result.

Thesis (MSc)--Stellenbosch University, 2000.
ENGLISH ABSTRACT: This study examined nonlinear modelling techniques as applied to dynamic systems, paying specific attention to the Method of Surrogate Data and its possibilities. Within the field of nonlinear modelling, we examined the following areas of study: attractor reconstruction, general model building techniques, cost functions, description length, and a specific modelling methodology. The Method of Surrogate Data was initially applied in a more conventional application, i.e. testing a time series for nonlinear, dynamic structure. Thereafter, it was used in a less conventional application; i.e. testing the residual vectors of a nonlinear model for membership of identically and independently distributed (i.i.d) noise. The importance of the initial surrogate analysis of a time series (determining whether the apparent structure of the time series is due to nonlinear, possibly chaotic behaviour) was illustrated. This study confrrmed that omitting this crucial step could lead to a flawed conclusion. If evidence of nonlinear structure in the time series was identified, a radial basis model was constructed, using sophisticated software based on a specific modelling methodology. The model is an iterative algorithm using minimum description length as the stop criterion. The residual vectors of the models generated by the algorithm, were tested for membership of the dynamic class described as i.i.d noise. The results of this surrogate analysis illustrated that, as the model captures more of the underlying dynamics of the system (description length decreases), the residual vector resembles Li.d noise. It also verified that the minimum description length criterion leads to models that capture the underlying dynamics of the time series, with the residual vector resembling Li.d noise. In the case of the "worst" model (largest description length), the residual vector could be distinguished from Li.d noise, confirming that it is not the "best" model. The residual vector of the "best" model (smallest description length), resembled Li.d noise, confirming that the minimum description length criterion selects a model that captures the underlying dynamics of the time series. These applications were illustrated through analysis and modelling of three time series: a time series generated by the Lorenz equations, a time series generated by electroencephalograhpic signal (EEG), and a series representing the percentage change in the daily closing price of the S&P500 index.
AFRIKAANSE OPSOMMING: In hierdie studie ondersoek ons nie-lineere modelleringstegnieke soos toegepas op dinamiese sisteme. Spesifieke aandag word geskenk aan die Metode van Surrogaat Data en die moontlikhede van hierdie metode. Binne die veld van nie-lineere modellering het ons die volgende terreine ondersoek: attraktor rekonstruksie, algemene modelleringstegnieke, kostefunksies, beskrywingslengte, en 'n spesifieke modelleringsalgoritme. Die Metode and Surrogaat Data is eerstens vir 'n meer algemene toepassing gebruik wat die gekose tydsreeks vir aanduidings van nie-lineere, dimanise struktuur toets. Tweedens, is dit vir 'n minder algemene toepassing gebruik wat die residuvektore van 'n nie-lineere model toets vir lidmaatskap van identiese en onafhanlike verspreide geraas. Die studie illustreer die noodsaaklikheid van die aanvanklike surrogaat analise van 'n tydsreeks, wat bepaal of die struktuur van die tydsreeks toegeskryf kan word aan nie-lineere, dalk chaotiese gedrag. Ons bevesting dat die weglating van hierdie analise tot foutiewelike resultate kan lei. Indien bewyse van nie-lineere gedrag in die tydsreeks gevind is, is 'n model van radiale basisfunksies gebou, deur gebruik te maak van gesofistikeerde programmatuur gebaseer op 'n spesifieke modelleringsmetodologie. Dit is 'n iteratiewe algoritme wat minimum beskrywingslengte as die termineringsmaatstaf gebruik. Die model se residuvektore is getoets vir lidmaatskap van die dinamiese klas wat as identiese en onafhanlike verspreide geraas bekend staan. Die studie verifieer dat die minimum beskrywingslengte as termineringsmaatstaf weI aanleiding tot modelle wat die onderliggende dinamika van die tydsreeks vasvang, met die ooreenstemmende residuvektor wat nie onderskei kan word van indentiese en onafhanklike verspreide geraas nie. In die geval van die "swakste" model (grootse beskrywingslengte), het die surrogaat analise gefaal omrede die residuvektor van indentiese en onafhanklike verspreide geraas onderskei kon word. Die residuvektor van die "beste" model (kleinste beskrywingslengte), kon nie van indentiese en onafhanklike verspreide geraas onderskei word nie en bevestig ons aanname. Hierdie toepassings is aan die hand van drie tydsreekse geillustreer: 'n tydsreeks wat deur die Lorenz vergelykings gegenereer is, 'n tydsreeks wat 'n elektroenkefalogram voorstel en derdens, 'n tydsreeks wat die persentasie verandering van die S&P500 indeks se daaglikse sluitingsprys voorstel.

It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait. In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas] The xi may generally be considered to be concentrations of species in a chemical reaction, in which case the k's are rate constants. In some cases the xi may be considered to be position and momentum variables in a mechanical system. We will divide the equations into two classes: those in which the evolution can be carried out by the action of one of Lie's transformation groups of the plane, and those for which this is not possible. Members of the first class can be integrated by quadrature either directly or by use of an integrating factor; those in the second class cannot. Of those in the first class the most interesting evolve by transformations of the projective group, and these, as well as the equations that cannot be integrated by quadrature, we study in some detail. We seek a qualitative analysis of systems which have no linear terms in their evolution equations when the origin from which the xi are measured is a critical point. The standard, linear, phase plane analysis is of course not adequate for our purposes.

Thesis (Ph. D.)--Ohio State University, 2004.
Title from first page of PDF file. Document formatted into pages; contains viii, 140 p.; also includes graphics. Includes bibliographical references (p. 139-140).

Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X* . Let G be a bounded open subset of X. Let T:X⊃ D(T)⇒ 2X* be maximal monotone and S: X ⇒ 2X* be bounded pseudomonotone and such that 0 notin cl((T+S)(D(T)∩partG)). Chapter 1 gives general introduction and mathematical prerequisites. In Chapter 2 we develop a homotopy invariance and uniqueness results for the degree theory constructed by Zhang and Chen for multivalued (S+) perturbations of maximal monotone operators. Chapter 3 is devoted to the construction of a new topological degree theory for the sum T+S with the degree mapping d(T+S,G,0) defined by d(T+S,G,0)=limepsilondarr 0+ dS+(T+S+ J,G,0), where dS+ is the degree for bounded (S+)-perturbations of maximal monotone operators. The uniqueness and homotopy invariance result of this degree mapping are also included herein. As applications of the theory, we give associated mapping theorems as well as degree theoretic proofs of known results by Figueiredo, Kenmochi and Le. In chapter 4, we consider T:X D(T)⇒ 2X* to be maximal monotone and S:D(S)=K⇒ 2X* at least pseudomonotone, where K is a nonempty, closed and convex subset of X with 0isinKordm. Let Phi:X⇒ ( infin, infin] be a proper, convex and lower-semicontinuous function. Let f* isin X* be fixed. New results are given concerning the solvability of perturbed variational inequalities for operators of the type T+S associated with the function f. The associated range results for nonlinear operators are also given, as well as extensions and/or improvements of known results by Kenmochi, Le, Browder, Browder and Hess, Figueiredo, Zhou, and others.

The method of multiple scales is used to study the nonlinear response of a relief valve under a constant static pressure and a sinusoidal dynamic pressure that can be generated by variations in the pneumatic transmission lines. Under certain conditions, large vibrations may occur, causing unsatisfactory performance as well as undesirable noise. In this theoretical analysis, the valve is modeled by a continuous spring fixed at one end and attached to a valve ball at the other end. The ball rests on a valve seat having nonlinear spring characteristics. The method of multiple scales is used to determine the non-linear response of the valve for the cases of harmonic, higher-harmonic, subharmonic and combination resonances. Conditions are determined for the onset and stability of large-amplitude oscillations.
M.S.

Indiana University-Purdue University Indianapolis (IUPUI)
Let f:X\rightarrow X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of \mathbb P^1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z\rightarrow z^b, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W^s_loc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by resenting two examples with a < b for which W^s_loc(S) is not real analytic in the neighborhood of any point.

Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods.
Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

Xie Chunjing.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.
Includes bibliographical references (leaves 65-71).
Abstracts in English and Chinese.
Acknowledgments --- p.i
Abstract --- p.ii
Introduction --- p.3
Chapter 1 --- Stability of Shock Waves in Viscous Conservation Laws --- p.10
Chapter 1.1 --- Cauchy Problem for Scalar Viscous Conservation Laws and Viscous Shock Profiles --- p.10
Chapter 1.2 --- Stability of Shock Waves by Energy Method --- p.15
Chapter 1.3 --- Nonlinear Stability of Shock Waves by Spectrum Anal- ysis --- p.20
Chapter 1.4 --- L1 Stability of Shock Waves in Scalar Viscous Con- servation Laws --- p.26
Chapter 2 --- Propagation of a Viscous Shock in Bounded Domain and Half Space --- p.35
Chapter 2.1 --- Slow Motion of a Viscous Shock in Bounded Domain --- p.36
Chapter 2.1.1 --- Steady Problem and Projection Method --- p.36
Chapter 2.1.2 --- Projection Method for Time-Dependent Prob- lem --- p.40
Chapter 2.1.3 --- Super-Sensitivity of Boundary Conditions --- p.43
Chapter 2.1.4 --- WKB Transformation Method --- p.45
Chapter 2.2 --- Propagation of a Stationary Shock in Half Space --- p.50
Chapter 2.2.1 --- Asymptotic Analysis --- p.50
Chapter 2.2.2 --- Pointwise Estimate --- p.51
Chapter 3 --- Nonlinear Stability of Viscous Transonic Flow Through a Nozzle --- p.58
Chapter 3.1 --- Matched Asymptotic Analysis --- p.58
Bibliography --- p.65

Chiu Ho Man Edward.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (leaves 36-39).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.4
Chapter 2 --- Some Preliminary Analysis --- p.11
Chapter 3 --- An Auxiliary Linear problem --- p.16
Chapter 4 --- Construction of natural constraint --- p.22
Chapter 5 --- Energy computation for reduced energy functional --- p.26
Chapter 6 --- Proof of Theorem 1.1 --- p.29
Bibliography --- p.36

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2016
An investigation and analysis of a collection of different techniques, for estimating the states of nonlinear systems, was undertaken. It was found that most of the existing literature on the topic could be organized into several groups of nonlinear observer design techniques, of which each group follows a specific concept and slight variations thereof. From out of this investigation it was discovered that a variation of the adaptive observer could be successfully applied to numerous nonlinear systems, given only limited output information. This particular technique formed the foundation on which a design procedure was developed in order to asymptotically estimate the states of nonlinear systems of a certain form, using only partial state information available. Lyapunov stability theory was used to prove the validity of this technique, given that certain conditions and assumptions are satisfied. A heuristic procedure was then developed to get a linearized model of the error transient behaviour that could form the upper bounds of the transient times of the observer. The technique above, characterized by a design algorithm, was then applied to three well-known nonlinear systems; namely the Lorenz attractor, the Rössler attractor, and the Van Der Pol oscillator. The results, illustrated through numerical simulation, clearly indicate that the technique developed is successful, provided all assumptions and conditions are satisfied.
MT2017

本博士论文对于有结构但又有相当一般性的约束条件下的非线性优化问题给出了系统性研究。比较经典的例子包指球面约束下的多重线性函数优化问题。这些模型已被广泛应用于数值线性代数、材料科学、量子物理学、信号处理、语音识别、生物医学工程以及控制论等。本论文着重探讨一类特定的方法来解这些广义模型，即块变量改进方法。具体地说，我们构造了一类块坐标下降型搜索算法来解带块变量结构的非线性优化问题。这类算法通过每次迭代中只更新一块变量以达到最大限度的目标函数值的改进(因而，这一新搜索算法命名为最优块改进算法(简称为MBI)。之后，我们重点研究了该算法在求解众多领域中实际问题的潜在能力。首先，这一算法可以直接应用于生物信息学中聚类基因表达数据的一种新模型的设计及求解。接着，我们把注意力转移到球面约束下的齐次多项式优化问题，此问题与张量的最优秩-1 逼近问题相关。对于这一优化问题， MBI 算法通常可以在较少的计算时间内找到全局最优解。第三，我们继续深入研究多项式优化问题，在双协半正定的新概念下建立了齐次多项式优化问题与其多重线性优化问题关系的一般性结果。最后，我们在Tucker 分解的框架下给出了求解高阶张量的最优多重线性秩的逼近问题的方法，并提出一种新的模型和算法来解未知变量数的Tucker 分解问题。本论文讨论并试验了一些应用实例，数值实验表明所提出的算法分别对于求解以上这些问题是可行并有效的。
In this thesis we present a systematic analysis for optimization of a general nonlinear function, subject to some fairly general constraints. A typical example includes the optimization of a multilinear tensor function over spherical constraints. Such models have found wide applications in numerical linear algebra, material sciences, quantum physics, signal processing, speech recognition, biomedical engineering, and control theory. This thesis is mainly concerned with a specific approach to solve such generic models: the block variable improvement method. Specifically, we establish a block coordinate descent type search method for nonlinear optimization, which accepts only a block update that achieves the maximum improvement (hence the name of our new search method: maximum block improvement (MBI)). Then, we focus on the potential capability of this method for solving problems in various research area. First, we demonstrate that this method can be directly used in designing a new framework for co-clustering gene expression data in the area of bioinformatics. Second, we turn our attention to the spherically constrained homogeneous polynomial optimization problem, which is related to best rank-one approximation of tensors. The MBI method usually finds the global optimal solution at a low computational cost. Third, we continue to consider polynomial optimization problems. A general result between homogeneous polynomials and multi-linear forms under the concept of co-quadratic positive semidefinite is established. Finally, we consider the problem of finding the best multi-linear rank approximation of a higher-order tensor under the framework of Tucker decomposition, and also propose a new model and algorithms for computing Tucker decomposition with unknown number of components. Some real application examples are discussed and tested, and numerical experiments are reported to reveal the good practical performance and efficiency of the proposed algorithms for solving those problems.
Detailed summary in vernacular field only.
Chen, Bilian.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 86-98).
Abstract also in Chinese.
Abstract --- p.iii
Acknowledgements --- p.vii
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Overview --- p.1
Chapter 1.2 --- Notations and Preliminaries --- p.5
Chapter 1.2.1 --- The Tensor Operations --- p.6
Chapter 1.2.2 --- The Tensor Ranks --- p.9
Chapter 1.2.3 --- Polynomial Functions --- p.11
Chapter 2 --- The Maximum Block Improvement Method --- p.12
Chapter 2.1 --- Introduction --- p.12
Chapter 2.2 --- MBI Method and Convergence Analysis --- p.14
Chapter 3 --- Co-Clustering of Gene Expression Data --- p.19
Chapter 3.1 --- Introduction --- p.19
Chapter 3.2 --- A New Generic Framework for Co-Clustering Gene Expression Data --- p.22
Chapter 3.2.1 --- Tensor Optimization Model of The Co-Clustering Problem --- p.22
Chapter 3.2.2 --- The MBI Method for Co-Clustering Problem --- p.23
Chapter 3.3 --- Algorithm for Co-Clustering 2D Matrix Data --- p.25
Chapter 3.4 --- Numerical Experiments --- p.27
Chapter 3.4.1 --- Implementation Details and Some Discussions --- p.27
Chapter 3.4.2 --- Testing Results using Microarray Datasets --- p.30
Chapter 3.4.3 --- Testing Results using 3D Synthesis Dataset --- p.32
Chapter 4 --- Polynomial Optimization with Spherical Constraint --- p.34
Chapter 4.1 --- Introduction --- p.34
Chapter 4.2 --- Generalized Equivalence Result --- p.37
Chapter 4.3 --- Spherically Constrained Homogeneous Polynomial Optimization --- p.41
Chapter 4.3.1 --- Implementing MBI on Multilinear Tensor Form --- p.42
Chapter 4.3.2 --- Relationship between Homogeneous Polynomial Optimization over Spherical Constraint and Tensor Relaxation Form --- p.43
Chapter 4.3.3 --- Finding a KKT point for Homogeneous Polynomial Optimization over Spherical Constraint --- p.45
Chapter 4.4 --- Numerical Experiments on Randomly Simulated Data --- p.47
Chapter 4.4.1 --- Multilinear Tensor Function over Spherical Constraints --- p.49
Chapter 4.4.2 --- Tests of Another Implementation of MBI --- p.49
Chapter 4.4.3 --- General Polynomial Function over Quadratic Constraints --- p.51
Chapter 4.5 --- Applications --- p.53
Chapter 4.5.1 --- Rank-One Approximation of Super-Symmetric Tensors --- p.54
Chapter 4.5.2 --- Magnetic Resonance Imaging --- p.55
Chapter 5 --- Logarithmically Quasiconvex Optimization --- p.58
Chapter 5.1 --- Introduction --- p.58
Chapter 5.2 --- Logarithmically Quasiconvex Optimization --- p.60
Chapter 5.2.1 --- A Simple Motivating Example --- p.61
Chapter 5.2.2 --- Co-Quadratic Positive Semide nite Tensor Form --- p.61
Chapter 5.2.3 --- Equivalence at Maxima --- p.64
Chapter 6 --- The Tucker Decomposition and Generalization --- p.68
Chapter 6.1 --- Introduction --- p.68
Chapter 6.2 --- Convergence of Traditional Tucker Decomposition --- p.71
Chapter 6.3 --- Tucker Decomposition with Unknown Number of Components --- p.73
Chapter 6.3.1 --- Problem Formulation --- p.74
Chapter 6.3.2 --- Implementing the MBI Method on Tucker Decomposition with Unknown Number of Components --- p.75
Chapter 6.3.3 --- A Heuristic Approach --- p.79
Chapter 6.4 --- Numerical Experiments --- p.80
Chapter 7 --- Conclusion and Recent Developments --- p.83
Bibliography --- p.86