Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaves 81-82). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Motivated by the work of Pólya, Schur, and Turán, a complete characterization of multiplier sequences for the Hermite polynomial basis is given. Laguerre's theorem and a remarkable curve theorem due to Pólya are generalized. Sufficient conditions for the location of zeros in certain strips in the complex plane are determined. Results pertaining to multiplier sequences and complex zero decreasing sequences for other polynomial sets are established. / viii, 178 leaves, bound ; 29 cm. / Thesis (Ph. D.)--University of Hawaii at Manoa, 2007.

Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.

The purpose of this thesis is to add to the structure theory of Aronszajn orderings. We shall focus essentially in four topics. The first topic of discussion is about the relation between Lipschitz and coherent trees. I will demonstrate that the tree $T(\rho_0)$ is coherent without any extra set theoretic hypothesis. The second topic presents an application of Todorcevic's $\rho$ functions to provide some partial answers to an old question of Juhaz asking whether a standard weakening of Jensen's diamond principle implies the existence of Suslin trees. In the third topic we focus on providing a satisfactory rough classification result of the class of Aronszajn lines. Our main result is that, assuming PFA, the class of Aronszajn lines is well-quasi-ordered by embeddability. The last topic is an investigation of the gap structure of the class of coherent Aronszajn trees. I will show that, assuming PFA, the class of coherent Aronszajn trees quasi-ordered by embeddability is the unique saturated linear order of cardinality $\aleph_2$.

The purpose of this thesis is to add to the structure theory of Aronszajn orderings. We shall focus essentially in four topics. The first topic of discussion is about the relation between Lipschitz and coherent trees. I will demonstrate that the tree $T(\rho_0)$ is coherent without any extra set theoretic hypothesis. The second topic presents an application of Todorcevic's $\rho$ functions to provide some partial answers to an old question of Juhaz asking whether a standard weakening of Jensen's diamond principle implies the existence of Suslin trees. In the third topic we focus on providing a satisfactory rough classification result of the class of Aronszajn lines. Our main result is that, assuming PFA, the class of Aronszajn lines is well-quasi-ordered by embeddability. The last topic is an investigation of the gap structure of the class of coherent Aronszajn trees. I will show that, assuming PFA, the class of coherent Aronszajn trees quasi-ordered by embeddability is the unique saturated linear order of cardinality $\aleph_2$.

In order to introduce the study undertaken in this thesis, let us consider the differential equationz2 (μ 2 - 2μ zm + z2m) d2y/dz2 + z (b0 + b1zm + b2z2m) dy/dz+ (c0 + c1zm + c2z2m) y = 0The variable z and he coefficients μ, bi, ci (i = 0, 1, 2) are regarded as complex and m is an arbitrary positive integer. It is also assumed that b0 + b1u + b2p2 = 0 and that the difference of the two roots of the indicial equation about z = 0 is not congruent to zero modulo m.

Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.

In this paper, we study the influence of the National Telecom Business Volume by the data in 2008 that have been published in China Statistical Yearbook of Statistics. We illustrate the procedure of modeling “National Telecom Business Volume” on the following eight variables, GDP, Consumption Levels, Retail Sales of Social Consumer Goods Total Renovation Investment, the Local Telephone Exchange Capacity, Mobile Telephone Exchange Capacity, Mobile Phone End Users, and the Local Telephone End Users. The testing of heteroscedasticity and multicollinearity for model evaluation is included. We also consider AIC and BIC criterion to select independent variables, and conclude the result of the factors which are the optimal regression model for the amount of telecommunications business and the relation between independent variables and dependent variable. Based on the final results, we propose several recommendations about how to improve telecommunication services and promote the economic development.

The nonlinear dynamic analysis of aeroelastic systems is a topic that has been covered extensively in the literature. The two main sources of nonlinearities in such systems, structural and aerodynamic nonlinearities, have analyzed numerically, analytically and experimentally. In this research project, the aerodynamic nonlinearity arising from the stall behavior of an airfoil is analyzed. Experimental data was used to fit a piecewise linear curve to describe the lift versus angle of attack behavior for a NACA 0012 2 DOF airfoil. The piecewise linear system equilibrium points are found and their stability analyzed. Bifurcations of the equilibrium points are analyzed and applying continuation software the bifurcation diagrams of the system are shown. Border collision and rapid/Hopf bifurcations are the two main bifurcations of the system equilibrium points. Chaotic behavior represented in the intermittent route to chaos was also observed and shown as part of the system dynamic analysis. Finally, sets of initial conditions associated with the system behavior are defined. Numerical simulations are used to show those sets, their subsets and their behavior with respect to the system dynamics. Poincaré sections are produced for both the periodic and the chaotic solutions of the system. The proposed piecewise linear model introduced some interesting dynamics for such systems. The introduction of the border collision bifurcation and the existence of periodic and chaotic solutions for the system are some examples. The model also enables the understanding of the mapping of initial conditions as it defines clear boundaries with different dynamics that can be used as Poincaré sections to understand further the global system dynamics. One of the constraints of the system is its validity as it is dependent on the range of the experimental data used to generate the model. This can be addressed by adding more linear pieces to the system to cover a wider range of the dynamics.