With the acceleration of economic globalization, the immature Chinese mainland stock market is gradually associated with the stock markets of other countries. This paper predict the return rate of Chinese mainland stock market using several models from GARCH family, test the predictability by calculating Value-at-Risk, also capture the dynamic correlation between other fifive countries or region and mainland China by DCC-GARCH model. The results indicate that E-ARMA-GARCH model fifits the best due to the signifificant heteroscedasticity and leverage effect of Chinese mainland stock market. It has the strongest positive correlation with HongKong while the weakest correlation with the United States.

We show that replicator equations follow naturally from the exponential affine structure of the simplex known from information geometry. It is then natural to call replicator equations linear if their fitness function is affine. For such linear replicator equations an explicit solution can be found. The approach is also demonstrated for the example of Eigen’s hypercycle, where some new analytic results are obtained using the explicit solution.

Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with. Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general. We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution.

We develop a theory concerning the reconstruction of multivariate exponential sums first over arbitrary fields and then consider the special cases of multivariate exponential sums over the fields of real and complex numbers.

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Two single-pass methods for fitting multiexponentials to experimental data are described. These methods rely on the construction of a matrix whose characteristic polynomial is used to determine the rates of decay. In the first method, which we call the multiple-delay method, the matrix is constructed using time delays of the experimental data. This method is fast and highly accurate even if the experimental signal contains exponential components with similar rates of decay. In the second method, which we call the successive-integral method, the matrix is constructed using integrals of the experimental data. This procedure yields good results for noisy signals and is a generalization of the method of Martin et al. ((1993) J. Neurosci. Methods, 51: 135-146). In addition, a particular instability of the multiexponential curve fitting problem is identified and a method for overcoming this instability is given.

A mixture model approach for clustering longitudinal data is introduced. The approach, which is based on mixtures of multivariate power exponential distributions, allows for varying tail-weight and peakedness in data. In the longitudinal setting, this corresponds to more or less concentration around the most central time course in a component. The models utilize a modified Cholesky decomposition of the component scale matrices and the associated maximum likelihood estimators are derived via a generalized expectation-maximization algorithm. / Thesis / Master of Science (MSc)

Methods featured are primarily conformal symplectic exponential time differencing methods, with a focus on families of methods, the construction of methods, and the features and advantages of methods, such as order, stability, and symmetry. Methods are applied to the problem of the damped harmonic oscillator. Construction of both exponential time differencing and integrating factor methods are discussed and contrasted. It is shown how to determine if a system of equations or a method is conformal symplectic with flow maps, how to determine if a method is symmetric by taking adjoints, and how to find the stability region of a method. Exponential time differencing Stormer-Verlet is derived and is shown as the example for how to find the order of a method using Taylor series. Runge-Kutta methods, partitioned exponential Runge-Kutta methods, and their associated tables are introduced, with versions of Euler's method serving as examples. Lobatto IIIA and IIIB methods also play a key role, as a new exponential trapezoid rule is derived. A new fourth order exponential time differencing method is derived using composition techniques. It is shown how to implement this method numerically, and thus it is analyzed for properties such as error, order of accuracy, and structure preservation.

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No description available.