DSpace Community:http://dspace.cityu.edu.hk:80/handle/2031/7182015-11-23T18:21:51Z2015-11-23T18:21:51ZAsymptotic model equations and bifurcations analysis of compressible hyperelastic layerWang, Yuanbin (王元斌)http://dspace.cityu.edu.hk:80/handle/2031/80382015-09-25T02:20:48Z2013-01-01T00:00:00ZTitle: Asymptotic model equations and bifurcations analysis of compressible hyperelastic layer
Authors: Wang, Yuanbin (王元斌)
Abstract: In this dissertation, we study the equilibriumstates of a compressible hyperelastic layer
under compression after the primary and secondary bifurcations. This type of problem
is an old one and has been studied from different points of view. It is very difficult
to find analytical post-bifurcation solutions, especially for the secondary bifurcation
solutions. Bifurcation studies using two or three-dimensional continuum mechanics
formulations have already been presented for the case of axially load elastic material.
But the general post-bifurcation analysis of this problem for arbitrary hyperelastic material
is very few in the literature due to the complexity of the required calculations,
thus motivating the present work. It is worth noticing that for the more complicated
case of elastic material, numerical as well as asymptotic post-bifurcation analysis have
been presented in the literature. Of interesting here is to find the asymptotic analytical
bifurcation solutions for the compressible hyperelastic layer.
Starting from the two-dimensional field equations for a compressible hyperelastic
material, we use a methodology of coupled series-asymptotic expansions developed
earlier to derive two coupled nonlinear ordinary differential equations (ODEs) as the
model equations. The critical buckling stresses are determined by a linear bifurcation
analysis, which are in agreement with the results in literature. The method of multiple
scales is used to solve the model equations to obtain the second-order asymptotic solutions
after the primary bifurcations. An analytical formula for the post-buckling amplitudes
is derived. Two kinds of numerical solutions are also obtained, the numerical solutions of themodel equations by a differencemethod and those of the two-dimensional
field equations by the finite elements method. Comparisons among the analytical solutions,
numerical solutions and solutions obtained by the Lyapunov-Schmidt-Koiter
(LSK) method in literature are made and good agreements for the displacements are
found. It is also found that at some places the axial strain is tensile, although the layer
is under compression.
To consider the secondary bifurcation, we superimpose a small deformation on
the state after the primary bifurcation. With the analytical solution of the primary
bifurcation, we manage to reduce the problem of the secondary bifurcation to one of
the first bifurcation governed by a second order variable-coefficient ODE. Our analysis
identifies an explicit function (4.16) and from the existence/nonexistence of its zero
points one can immediately judge whether a secondary bifurcation can take place or
not. The zero corresponds to a turning point of the governing ODE, which leads to
nontrivial solutions. Further, by theWKBmethod, the equation (in a very simple form)
for determining the critical stress for the secondary bifurcation is derived. We further
use AUTO to compute the secondary bifurcation point numerically, which confirms
the validity of our analytical results. The numerical results in the secondary bifurcation
branch computed by AUTO indicate that the secondary bifurcation induces a "wave
number doubling" phenomenon and also the shape of the layer has a convexity change
along the axial direction.
Under the general three-dimensional pre-stress condition, for a new hyperelastic
material subjected to axially load, using the same method to derive a similar five order
asymptoticmodel equation. The impact of the pre-stretchm3 on the principal stretches
of the uniform pre-bifurcation state and the primary and secondary bifurcation of the
model equation have been studied. The results show that the pre-stretch m3 play a key
role in determining the solutions and bifurcation of model equation.
Notes: CityU Call Number: TA653 .W36 2013; v, 137 p. : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2013.; Includes bibliographical references (p. [129]-137)2013-01-01T00:00:00ZRegularization for regression and rankingZhao, Yulong (趙玉龍)http://dspace.cityu.edu.hk:80/handle/2031/80372015-09-25T02:20:45Z2013-01-01T00:00:00ZTitle: Regularization for regression and ranking
Authors: Zhao, Yulong (趙玉龍)
Abstract: Regularization is a method for learning and approximation which uses some additional
information to avoid overfitting in statistics and machine learning. The information
usually aims at improving the generalization ability by restrictions on regularity of
potential functions. In this thesis, we mainly focus on the elastic net for regression and regularized least squares ranking algorithms.
The elastic net regularization is analyzed in two settings, according to their hypothesis spaces. One assumes a data independent hypothesis space composed by features independent of samples. Within this setting, significant contributions are made
in several aspects. First, concentration estimates for sample error are presented by
introducing ℓ2-empirical covering number and utilizing an iteration process. Second,
a constructive approximation approach for estimating approximation error is presented.
Third, the elastic-net learning with infinite features is studied and the role that the tuning parameter ζ plays is also discussed. Finally, our learning rate is shown to be faster compared with existing results. The other assumes a data dependent hypothesis
space which is a subspace of a Reproducing Kernel Hilbert Space. Based on the
capacity condition of the Reproducing Kernel Hilbert Space, a learning rate for elastic
net is obtained by a stepping stone technique and an ℓ2-empirical covering number
technique. The role of parameters is also discussed.
The regularized least squares ranking algorithm is analyzed in Reproducing Kernel
Hilbert Spaces. By Hoeffding's decomposition, a U-statistic could be decomposed into an independent term and a degenerate U-statistic term. These two terms can be
analyzed individually. The optimal learning rate is achieved.
Notes: CityU Call Number: QA278.2 .Z47 2013; vii, 97 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2013.; Includes bibliographical references (leaves [87]-97)2013-01-01T00:00:00ZInfinite divisibility of interpolated gamma powers and variance : GGC financial modelYang, Dichuan (楊廸川)http://dspace.cityu.edu.hk:80/handle/2031/80362015-09-25T02:20:43Z2013-01-01T00:00:00ZTitle: Infinite divisibility of interpolated gamma powers and variance : GGC financial model
Authors: Yang, Dichuan (楊廸川)
Abstract: This thesis includes three main chapters–Chapter 2,3,4.
Chapter 2 is stochastic analysis. It is concerned with the distribution properties of
the binomial aX + bXα, where X is a Gamma random variable. We show in particular
that aX + bXα is infinitely divisible for all α ∈ [1, 2] and a, b ∈ R+, and that for
α = 2 the second order polynomial aX + bX2 is a generalized Gamma convolution whose
Thorin density and Wiener-Gamma integral representation are computed explicitly. As
a byproduct we show that fourth order multiple Wiener integrals are in general not infinitely divisible.
Chapter 3 is financial modeling. This chapter introduces the variance-GGC model which
is an extension of variance-Gamma model. First we calculate the skewness and kurtosis
of the variance-GGC model and get the relation of the skewness and kurtosis between
GGC processes and corresponding variance-GGC processes. Then the decomposition
property of variance-GGC processes is proved. At last, sensitivity analysis of this model has been conducted.
In Chapter 4, our goal is to relax a sufficient condition for the exponential almost sure stability of a certain class of stochastic differential equations. Compare to the existing theory, we prove the almost sure stability, replacing Lipschitz continuity and linear growth conditions by the existence of a strong solution of the underlying stochastic differential equation. This result is extendable for the regime-switching system. An explicit example is provided for the illustration purpose.
Notes: CityU Call Number: QA273 .Y326 2013; vi, 81 p. : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2013.; Includes bibliographical references (p. 78-81)2013-01-01T00:00:00ZRegime-switching control problems and related Markov chain approximation methodsShen, Jie (沈洁)http://dspace.cityu.edu.hk:80/handle/2031/80352015-09-25T02:20:41Z2013-01-01T00:00:00ZTitle: Regime-switching control problems and related Markov chain approximation methods
Authors: Shen, Jie (沈洁)
Abstract: The stochastic control problems and two-player stochastic differential game problems
with regime switching are two of the most important problems in control theory and
generally in mathematical finance. They have been received much attention for many
applications in different fields, such as pursuit-evasion games, queuing systems in
heavy traffic, risk-sensitive control, and constrained optimization problems, since formulated
in late 1970s. Various approximation methods for stochastic control have been
studied and corresponding convergence analyses have been done.
In particular, the Markov chain approximation method is a powerful and widely
applied in numerical problems for controlled stochastic processes. Generally speaking,
the approximating Markov chain starts by approximating the original controlled
process by an appropriate controlled Markov chain on a state space. The approximation
parameters are denoted by h and δ and the original cost functional is approximated
such that it's suitable for the defined Markov chain. Moreover, the Markov chain also
keeps track of the regimes when dealing with the stochastic problem with regimeswitching.
In many cases, the time interval in the stochastic control problem is bounded by
a finite number T, which makes the problem is a solution of a parabolic partial differential
equation. The main goal of this thesis is to make some contributions to the
Markov chain approximation methods to time-dependent regime-switching stochastic
control problems on a finite time horizon as well as the convergence of the algorithms by means of weak convergence methods. Furthermore, the application of this numerical
scheme expands to the underlying game problems. The sufficient conditions for the
existence of a saddle point of a discrete Markov game constructed by Markov chain
approximation of stochastic differential games in a general setup are provided. In addition, numerical solutions of several examples are provided for demonstration purpose.
Notes: CityU Call Number: QA402.37 .S53 2013; vi, 90 p. : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2013.; Includes bibliographical references (p. [83]-90)2013-01-01T00:00:00Z