Project (M.S.)--Brigham Young University. Dept. of Statistics, 2005. / Includes bibliographical references (p. 82-85).

No description available.

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.

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.

Thesis (Ph. D.)--State University of New York at Buffalo, 2001. / "August 2001." Includes bibliographical references (p. 80-87). Also available in print.

Thesis (M.Phil.)--City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Master of Philosophy." Includes bibliographical references (leaves 38-41)

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 14-15).

published_or_final_version / Mathematics / Master / Master of Philosophy

The objective of the thesis is to compare polynomials and finite Blaschke products, and demonstrate that they share many similar properties and hence we can establish a dictionary between these two kinds of finite maps for the first time. The results for polynomials were reviewed first. In particular, a special kind of polynomials was discussed, namely, Chebyshev polynomials, which can be defined by the trigonometric cosine function cos ?. Also, a complete classification for two polynomials sharing a set was given. In this thesis, some analogous results for finite Blaschke products were proved. Firstly, Chebyshev-Blaschke products were introduced. They can be defined by re- placing the trigonometric cosine function cos z by the Jacobi cosine function cd(u; ? ). They were shown to have several similar properties of Chebyshev polynomials, for example, both of them share the same monodromy, satisfy some differential equations and solve some minimization problems. In addition, some analogous results about two finite Blaschke products sharing a set were proved, based on Dinh's and Pakovich's ideas. Moreover, the density of prime polynomials was investigated in two different ways: (i) expressing the polynomials of degree n in terms of the zeros and the leading coefficient; (ii) expressing the polynomials of degree n in terms of the coefficients. Also, the quantitative version of the density of composite polynomials was developed and a density estimate on the set of composite polynomials was given. Furthermore, some analogous results on the the density of prime Blaschke products were proved. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy