In this thesis a number of issues related to oceanographic problems have been dealt with on the basis of applying asymptotic methods.  The first study focused on the tidal generation of internal waves, a process which is quantifed by the conversion rates. These have traditionally been calculated by using the WKB approximation. However, the systematic imprecision of this theory for the lowest modes as well as turbulence at the seabed level affect the results. To handle these anomalies we introduced another asymptotic technique, homogenization theory, which led to signifcant improvements, especially for the lowest modes.  The second study dealt with the dynamical aspects of a nonlinear oscillator which can be interpreted as a variant of the classical two-box models used in oceanography. The system is constituted by two connected vessels containing a fluid characterised by a nonlinear equation of state and a large volume differences between the vessels is prescribed. It is recognised that the system, when performing relaxation oscillations, exhibits almost-discontinuous jumps between the two branches of the slow manifold of the problem. The lowest-order analysis yielded reasonable correspondence with the numerical results.  The third study is an extension of the lowest-order approximation of the relaxation oscillations undertaken in the previous paper. A Mandelstam condition is imposed on the system by assuming that the total heat content of the system is conserved during the discontinuous jumps.  In the fourth study an asymptotic analysis is carried out to examine the oscillatory behaviour of the thermal oscillator. It is found that the analytically determined corrections to the zeroth-order analysis yield overall satisfying results even for comparatively large values of the vessel-volume ratio. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.

In this thesis, analytical analysis and design techniques for wireless communications with diversity are studied. The impact of impairments such as correlated fading is analyzed using statistical models. Countermeasures designed to overcome, or even exploit, such effects are proposed and examined. In particular two applications are considered, satellite broadcast to vehicular terminals and communication using transmitters and receivers equipped with multiple antennas. Mobile satellite broadcast systems offer the possibility of high data rate services with reliability and ubiquitous coverage. The design of system architectures providing such services requires complex trade-offs involving technical, economical, and regulatory aspects. A satisfactory availability can be ensured using space, terrestrial, and time diversity techniques. The amount of applied diversity affects the spectral efficiency and system performance. Also, dedicated satellite and terrestrial networks represent significant investments and regulatory limitations may further complicate system design. The work presented in this thesis provides insights to the technical aspects of the trade-offs above. This is done by deriving an efficient method for estimating what resources in terms of spectrum and delay are required for a broadcast service to reach a satisfactory number of end users using a well designed system. The results are based on statistical models of the mobile satellite channel for which efficient analytical design and error rate estimation methods are derived. We also provide insight to the achievable spectral efficiency using different transmitter and receiver configurations. Multiple-element antenna communication is a promising technology for future high speed wireless infrastructures. By adding a spatial dimension, radio resources in terms of transmission power and spectrum can be used more efficiently. Much of the design and analysis work has focused on cases where the transmitter either has access to perfect channel state information or it is blind and the spatial channels are uncorrelated. Herein, systems where the fading of the spatial channels is correlated and/or the transmitter has access to partial channel state information are considered. While maintaining perfect channel knowledge at the transmitter may prove difficult, updating parameters that change on a slower time scale could be realistic. Here we formulate analysis and design techniques based on statistical models of the multichannel propagation. Fundamental properties of the multi-element antenna channel and limitations given by information theory are investigated under an asymptotic assumption on the number of antennas on either side of the system. For example, limiting normal distributions are derived for the squared singular values of the channel matrix and the mutual information. We also propose and examine a practical scheme capable of exploiting partial channel state information. In both applications outlined above, by using statistical models of the channel characteristics in the system design, performance can be improved. The main contribution of this thesis is the development of efficient techniques for estimating the system performance in different scenarios. Such techniques are vital to obtain insights to the impact of different impairments and how countermeasures against these should be designed. / QC 20101019

This report consider a system describing three competing species with populations x, y and z. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.

A composite likelihood consists of a combination of valid likelihood objects, and in particular it is of typical interest to adopt lower dimensional marginal likelihoods. Composite marginal likelihood appears to be an attractive alternative for modeling complex data, and has received increasing attention in handling high dimensional data sets when the joint distribution is computationally difficult to evaluate, or intractable due to complex structure of dependence. We present some aspects of methodological development in composite likelihood inference. The resulting estimator enjoys desirable asymptotic properties such as consistency and asymptotic normality. Composite likelihood based test statistics and their asymptotic distributions are summarized. Higher order asymptotic properties of the signed composite likelihood root statistic are explored. Moreover, we aim to compare accuracy and efficiency of composite likelihood estimation relative to estimation based on ordinary likelihood. Analytical and simulation results are presented for different models, which include multivariate normal distributions, times series model, and correlated binary data.

In order to introduce the investigation contemplated in this thesis, let us consider the differential equation d4y+ 3Σ zj (aj + bjzα) djy = 0(1)z4 dz4 j = 0 dzjHere, the variable z is regarded as complex, as are the coefficients aj, bj, (j=0,1,..., n-1) and a is an arbitrary positive integer. It is also assumed that the difference of no two roots of the indicial equation about z = 0 is congruent to zero modulo α .This problem arises in some current Bio-medical problems.Ball State UniversityMuncie, IN 47306

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

A composite likelihood consists of a combination of valid likelihood objects, and in particular it is of typical interest to adopt lower dimensional marginal likelihoods. Composite marginal likelihood appears to be an attractive alternative for modeling complex data, and has received increasing attention in handling high dimensional data sets when the joint distribution is computationally difficult to evaluate, or intractable due to complex structure of dependence. We present some aspects of methodological development in composite likelihood inference. The resulting estimator enjoys desirable asymptotic properties such as consistency and asymptotic normality. Composite likelihood based test statistics and their asymptotic distributions are summarized. Higher order asymptotic properties of the signed composite likelihood root statistic are explored. Moreover, we aim to compare accuracy and efficiency of composite likelihood estimation relative to estimation based on ordinary likelihood. Analytical and simulation results are presented for different models, which include multivariate normal distributions, times series model, and correlated binary data.

This thesis consists of three chapters. The first chapter focuses on adaptive smoothing splines for fitting functions with varying roughness. In the first part of the first chapter, we study an asymptotically optimal procedure to choose the value of a discretized version of the variable smoothing parameter in adaptive smoothing splines. With the choice given by the multivariate version of the generalized cross validation, the resulting adaptive smoothing spline estimator is shown to be consistent and asymptotically optimal under some general conditions. In the second part, we derive the asymptotically optimal local penalty function, which is subsequently used for the derivation of the locally optimal smoothing spline estimator. In the second chapter, we propose a Lipschitz regularity based statistical model, and apply it to coordinate measuring machine (CMM) data to estimate the form error of a manufactured product and to determine the optimal sampling positions of CMM measurements. Our proposed wavelet-based model takes advantage of the fact that the Lipschitz regularity holds for the CMM data. The third chapter focuses on the classification of functional data which are known to be well separable within a particular interval. We propose an interval based classifier. We first estimate a baseline of each class via convex optimization, and then identify an optimal interval that maximizes the difference among the baselines. Our interval based classifier is constructed based on the identified optimal interval. The derived classifier can be implemented via a low-order-of-complexity algorithm.

Battail [1989] shows that an appropriate criterion for the design of long block codes is the closeness of the normalized weight distribution to a Gaussian distribution. A subsequent work shows that iterated product of single parity check codes satisfy this criterion [1994]. Motivated by these earlier works, in this thesis, we study the effect of the interleaver on the performance of turbo codes for large block lengths, $N\rightarrow\infty$. A parallel concatenated turbo code that consists of two or more component codes is considered. We demonstrate that for $N\rightarrow\infty$, the normalized weight of the systematic $\widehat{w_1}=\displaystyle\frac{w_1}{\sqrt{N}}$, and the parity check sequences $\widehat{w_2}=\displaystyle\frac{w_2}{\sqrt{N}}$ and $\widehat{w_3}=\displaystyle\frac{w_3}{\sqrt{N}}$ become a set of jointly Gaussian distributions for the typical values of $\widehat{w_i},i=1,2,3$, where the typical values of $\widehat{w_i}$ are defined as $\displaystyle\lim_{N\rightarrow\infty}\frac{\widehat{w_i}}{\sqrt{N}}\neq 0,1$ for $i=1,2,3$. To optimize the turbo code performance in the waterfall region which is dominated by high-weight codewords, it is desirable to reduce $\rho_{ij}$, $i,j=1,2,3$ as much as possible, where $\rho_{ij}$ is the correlation coefficient between $\widehat{w_i}$ and $\widehat{w_j}$. It is shown that: (i)~$\rho_{ij}>0$, $i,j=1,2,3$, (ii)~$\rho_{12},\rho_{13}\rightarrow 0$ as $N\rightarrow\infty$, and (iii)~$\rho_{23}\rightarrow 0$ as $N\rightarrow\infty$ for "almost" any random interleaver. This indicates that for $N\rightarrow\infty$, the optimization of the interleaver has a diminishing effect on the distribution of high-weight error events, and consequently, on the error performance in the waterfall region. We show that for the typical weights, this weight distribution approaches the average spectrum defined by Poltyrev [1994]. We also apply the tangential sphere bound (TSB) on the Gaussian distribution in AWGN channel with BPSK signalling and show that it performs very close to the capacity for code rates of interest. We also study the statistical properties of the low-weight codeword structures. We prove that for large block lengths, the number of low-weight codewords of these structures are some Poisson random variables. These random variables can be used to evaluate the asymptotic probability mass function of the minimum distance of the turbo code among all the possible interleavers. We show that the number of indecomposable low-weight codewords of different types tend to a set of independent Poisson random variables. We find the mean and the variance of the union bound in the error floor region and study the effect of expurgating low-weight codewords on the performance. We show that the weight distribution in the transition region between Poisson and Gaussian follows a negative binomial distribution. We also calculate the interleaver gain for multi-component turbo codes based on these Poisson random variables. We show that the asymptotic error performance for multi-component codes in different weight regions converges to zero either exponentially (in the Gaussian region) or polynomially (in the Poisson and negative binomial regions) with respect to the block length, with the code-rate and energy values close to the channel capacity.

Battail [1989] shows that an appropriate criterion for the design of long block codes is the closeness of the normalized weight distribution to a Gaussian distribution. A subsequent work shows that iterated product of single parity check codes satisfy this criterion [1994]. Motivated by these earlier works, in this thesis, we study the effect of the interleaver on the performance of turbo codes for large block lengths, $N\rightarrow\infty$. A parallel concatenated turbo code that consists of two or more component codes is considered. We demonstrate that for $N\rightarrow\infty$, the normalized weight of the systematic $\widehat{w_1}=\displaystyle\frac{w_1}{\sqrt{N}}$, and the parity check sequences $\widehat{w_2}=\displaystyle\frac{w_2}{\sqrt{N}}$ and $\widehat{w_3}=\displaystyle\frac{w_3}{\sqrt{N}}$ become a set of jointly Gaussian distributions for the typical values of $\widehat{w_i},i=1,2,3$, where the typical values of $\widehat{w_i}$ are defined as $\displaystyle\lim_{N\rightarrow\infty}\frac{\widehat{w_i}}{\sqrt{N}}\neq 0,1$ for $i=1,2,3$. To optimize the turbo code performance in the waterfall region which is dominated by high-weight codewords, it is desirable to reduce $\rho_{ij}$, $i,j=1,2,3$ as much as possible, where $\rho_{ij}$ is the correlation coefficient between $\widehat{w_i}$ and $\widehat{w_j}$. It is shown that: (i)~$\rho_{ij}>0$, $i,j=1,2,3$, (ii)~$\rho_{12},\rho_{13}\rightarrow 0$ as $N\rightarrow\infty$, and (iii)~$\rho_{23}\rightarrow 0$ as $N\rightarrow\infty$ for "almost" any random interleaver. This indicates that for $N\rightarrow\infty$, the optimization of the interleaver has a diminishing effect on the distribution of high-weight error events, and consequently, on the error performance in the waterfall region. We show that for the typical weights, this weight distribution approaches the average spectrum defined by Poltyrev [1994]. We also apply the tangential sphere bound (TSB) on the Gaussian distribution in AWGN channel with BPSK signalling and show that it performs very close to the capacity for code rates of interest. We also study the statistical properties of the low-weight codeword structures. We prove that for large block lengths, the number of low-weight codewords of these structures are some Poisson random variables. These random variables can be used to evaluate the asymptotic probability mass function of the minimum distance of the turbo code among all the possible interleavers. We show that the number of indecomposable low-weight codewords of different types tend to a set of independent Poisson random variables. We find the mean and the variance of the union bound in the error floor region and study the effect of expurgating low-weight codewords on the performance. We show that the weight distribution in the transition region between Poisson and Gaussian follows a negative binomial distribution. We also calculate the interleaver gain for multi-component turbo codes based on these Poisson random variables. We show that the asymptotic error performance for multi-component codes in different weight regions converges to zero either exponentially (in the Gaussian region) or polynomially (in the Poisson and negative binomial regions) with respect to the block length, with the code-rate and energy values close to the channel capacity.