This thesis focuses on convex sets and convex cones defined using hyperbolic polynomials. We first review some of the theory of convex sets in $\R^d$ in general. We then review some classical algebraic theorems concerning polynomials in a single variable, as well as presenting a few more modern results about them. We then discuss the theory of hyperbolic polynomials in several variables and their associated hyperbolicity cones. We survey various ways to build and decompose hyperbolic cones and we prove that every nontrivial hyperbolic cone is the intersection of its derivative cones. We conclude with a brief discussion of the set of extreme rays of a hyperbolic cone.

This thesis focuses on convex sets and convex cones defined using hyperbolic polynomials. We first review some of the theory of convex sets in $\R^d$ in general. We then review some classical algebraic theorems concerning polynomials in a single variable, as well as presenting a few more modern results about them. We then discuss the theory of hyperbolic polynomials in several variables and their associated hyperbolicity cones. We survey various ways to build and decompose hyperbolic cones and we prove that every nontrivial hyperbolic cone is the intersection of its derivative cones. We conclude with a brief discussion of the set of extreme rays of a hyperbolic cone.

In this study two nonlinear analysis methods are proposed for investigation of nonlinear effects of high temperature superconductive(HTSC) thin film planar microwave circuits. The MoM-HB combination method is based on the combination formulation of the moment method(MoM) and the harmonic balance(HB) technique. It consists of linear and nonlinear solvers. The power series method treats the voltages at higher order frequencies as the excitations at the corresponding frequencies, and the higher order current distributions are then obtained by using the moment method again. The power series method is simple and fast for finding the output power at higher order frequencies. The MoM-HB combination method is suitable for strong nonlinearity, and it can be also used to find the fundamental current redistribution, conductor loss, and the scattering parameters variation at the fundamental frequency. These two proposed methods are efficient, accurate, and suitable for distributed-type HTSC nonlinearity. They can be easily incorporated into commercial EM CAD softwares to expand their capabilities. These two nonlinear analysis method are validated by analyzing a HTSC stripline filter and HTSC antenna dipole circuits. HTSC microstrip lines are then investigated for the nonlinear effects of HTSC material on the current density distribution over the cross section and the conductor loss as a function of the applied power. The HTSC microstrip patch filters are then studied to show that the HTSCinterconnecting line could dominate the behaviors of the circuits at high power. The variation of the transmission and reflection coefficients with the applied power and the third output power are calculated. The HTSC microstrip line structure with gilded edges is proposed for improving the power handling capability of HTSC thin film circuit based on a specified limit of harmonic generation and conductor loss. A general analysis approach suitable for any thickness of gilding layer is developed by integrating the multi-port network theory into aforementioned proposed nonlinear analysis methods. The conductor loss and harmonic generation of the gilded HTSC microstrip line are investigated.

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.

In this study two nonlinear analysis methods are proposed for investigation of nonlinear effects of high temperature superconductive(HTSC) thin film planar microwave circuits. The MoM-HB combination method is based on the combination formulation of the moment method(MoM) and the harmonic balance(HB) technique. It consists of linear and nonlinear solvers. The power series method treats the voltages at higher order frequencies as the excitations at the corresponding frequencies, and the higher order current distributions are then obtained by using the moment method again. The power series method is simple and fast for finding the output power at higher order frequencies. The MoM-HB combination method is suitable for strong nonlinearity, and it can be also used to find the fundamental current redistribution, conductor loss, and the scattering parameters variation at the fundamental frequency. These two proposed methods are efficient, accurate, and suitable for distributed-type HTSC nonlinearity. They can be easily incorporated into commercial EM CAD softwares to expand their capabilities. These two nonlinear analysis method are validated by analyzing a HTSC stripline filter and HTSC antenna dipole circuits. HTSC microstrip lines are then investigated for the nonlinear effects of HTSC material on the current density distribution over the cross section and the conductor loss as a function of the applied power. The HTSC microstrip patch filters are then studied to show that the HTSCinterconnecting line could dominate the behaviors of the circuits at high power. The variation of the transmission and reflection coefficients with the applied power and the third output power are calculated. The HTSC microstrip line structure with gilded edges is proposed for improving the power handling capability of HTSC thin film circuit based on a specified limit of harmonic generation and conductor loss. A general analysis approach suitable for any thickness of gilding layer is developed by integrating the multi-port network theory into aforementioned proposed nonlinear analysis methods. The conductor loss and harmonic generation of the gilded HTSC microstrip line are investigated.

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.

We investigate the famous Laguerre–Pólya class of entire functions and its subclass, the Laguerre–Pólya class of type I. The functions from these classes can be expressed in terms of the Hadamard Canonical Factorization (see Chapter 1, Definition 1.2 and 1.3). The prominent theorem by E. Laguerre and G. Pólya gives a complete description of the Laguerre–Pólya class and the Laguerre–Pólya class of type I, showing that these classes are the respective closures in the topology of uniform convergence on compact sets of the set of real polynomials having only real zeros (that is, the set of so-called hyperbolic polynomials) and the set of real polynomials having only real negative zeros. Both the Laguerre–Pólya class and the Laguerre–Pólya class of type I play an essential role in complex analysis. For the properties and characterizations of these classes, see, for example, [31] by A. Eremenko, [40] by I.I. Hirschman and D.V. Widder, [43] by S. Karlin, [57] by B.Ja. Levin, [66, Chapter 2] by N. Obreschkov, and [74] by G. Pólya and G. Szegö. In the thesis, we study entire functions with positive coefficients and with the monotonic sequence of their second quotients of Taylor coefficients. We find necessary and sufficient conditions under which such functions belong to the Laguerre–Pólya class (or the Laguerre–Pólya class of type I).:List of symbols Introduction 1 Background of research 1 1.1 The Laguerre–Pólya class .................... 1 1.2 The quotients of Taylor coefficients ............... 3 1.3 Hutchinson’s constant ...................... 4 1.4 Multiplier sequences ....................... 4 1.5 Apolar polynomials........................ 8 1.6 The partial theta function .................... 10 1.7 Decreasing second quotients ................... 13 1.8 Increasing second quotients ................... 14 2 A necessary condition for an entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre–Pólya class 15 2.1 Proof of Theorem 2.1....................... 16 2.2 The q-Kummer function ..................... 29 2.3 Proof of Theorem 2.10 ...................... 31 2.4 Proof of Theorem 2.11 ...................... 43 3 Closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre–Pólya I class 49 3.1 Proof of Statement 3.1 ...................... 50 3.2 Proof of Theorem 3.2....................... 53 3.3 Proof of Theorem 3.4....................... 61 3.4 Proof of Theorem 3.6....................... 66 4 Entire functions from the Laguerre–Pólya I class having the increasing second quotients of Taylor coefficients 69 4.1 Proof of Theorem 4.1....................... 70 4.2 Proof of Theorem 4.3....................... 76 5 Number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients 81 5.1 Proof of Theorem 5.1....................... 82 5.2 Proof of Corollary 5.2....................... 88 5.3 Proof of Theorem 5.4....................... 88 6 Further questions 95 Acknowledgements 97 Selbständigkeitserklärung 101 Curriculum Vitae 103 Bibliography 107