In this paper we explore generalized “$r$-Fibonacci Numbers” using a combinatorial “tiling” interpretation. This approach allows us to provide simple, intuitive proofs to several identities involving $r$-Fibonacci Numbers presented by F.T. Howard and Curtis Cooper in the August, 2011, issue of the Fibonacci Quarterly. We also explore a connection between the generalized Fibonacci numbers and a generalized form of binomial coefficients.

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler's exponential generating function formula for the Eulerian polynomials. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics. The family of colored permutation groups includes the family of symmetric groups and the family of hyperoctahedral groups, also called the type A Coxeter groups and type B Coxeter groups, respectively. By specializing our formulas to these cases, they reduce to the Shareshian-Wachs q-analog of Euler's formula, formulas of Foata and Han, and a new generalization of a formula of Chow and Gessel.

Newton polygons are constructions over the p-adic numbers used to find information about the roots of a polynomial or power series. In this the- sis, we will first investigate the construction of the field Qp on the p-adic numbers. Then, we will use theorems such as Eisenstein’s Irreducibility Criterion, Newton’s Method, Hensel’s Lemma, and Strassman’s Theorem to build and justify Newton polygons.

Two basic approaches to reduce computational requirements for solving distillation problems have been studied: simplifications of the model based on physical approximations and order reduction techniques based on numerical approximations. Several problems have been studied using full and reduced-order techniques along with the following distillation models: Constant Molar Overflow, Constant Molar Holdup and Time-Dependent Molar Holdup. Steady-state results show excellent agreement in the profiles obtained using orthogonal collocation and demonstrate that with an order reduction of up to 54%, reduced-order models yield better results than physically simpler models. Step responses demonstrate that with a reduction in computing time of the order of 60% the method still provides better dynamic simulations than those obtained using physical simplifications. Frequency response data obtained from pulse tests has been used to verify that reduced-order solutions preserve the dynamic characteristics of the original full-order system while physical simplifications do not. The orthogonal collocation technique is also applied to a coupled columns scheme with good results. / Graduation date: 1992

Shack-Hartmann wavefront sensors are commonly used in optical alignment, ophthalmology, astronomy, adaptive optics and commercial optical testing. Wavefront error measurement yields Zernike polynomials which provide useful data for alignment correction calculations. In this thesis a practical alignment method of a helmet visor is proposed based on the wavefront error measurements. The optical system is modeled in Zemax software in order to collect the Zernike polynomial data necessary to relate the error measurements to the positioning of the visor. An artificial neural network based computer program is designed and trained with the data obtained from Zernike simulation in Zemax software and then the program is able to find how to invert the misalignments in the system. The performance of this alignment correction method is compared with the optical test setup measurements.

La memoria trata algunos aspectos de la teoría de casi-anillos de polinomios r(x) con coeficientes en un anillo r conmutativo y con unidad. En el capítulo I damos una descripción explicita de los elementos distributivos de r(x) y de la parte cero-simétrica r sub 0 (x). En los párrafos damos algunas caracterizaciones y propiedades del anillo formado por estos elementos distributivos. Obtenemos resultados similares en el casi-anillo de series de potencias formales. En el capítulo II está dedicado al estudio de subcasi-anillos que gozan de las dos propiedades distributivas en r (x) y de ideales de casi-anillos que dan cociente anillo particularizando esto para el caso del casi-anillo r(x). En el capítulo III encontramos todos los ideales maximales de z (x) (z el anillo de los enteros). Estudiamos también los ideales de composición del anillo de composición (r(x) + o) dando una descripción de todos los maximales. Acaba la memoria con un algoritmo para la descomposición de polinomios con coeficientes en cuerpo f es decir encontramos una descomposición de un polinomio en componentes indescomponibles / In this dissertation we study several aspects of near-rings. In the first chapter we give an explicit description of the distributive elements of the near-ring of polynomials R[x], over a commutative ring R a with identity. We also find the distributive elements in the near-ring of formal power series over a commutative rings with identity. In the second chapter, we search rings which are contained in R[x], we prove that if R is an integral domain, the set of distributive elements contains the subrings of the near-rings of polynomials. We also investigate ideals I of the near-ring such that the quotient is ring. In the next chapter we find all maximal ideals in Z[x] and maximal full ideals in the composition rings. The last section we provide the first polynomial time algorithm for decomposing polynomials into indecomposable ones.

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.

Computations with polynomials are at the heart of any computer algebra system and also have many applications in engineering, coding theory, and cryptography. Generally speaking, the low-level polynomial computations of interest can be classified as arithmetic operations, algebraic computations, and inverse symbolic problems. New algorithms are presented in all these areas which improve on the state of the art in both theoretical and practical performance. Traditionally, polynomials may be represented in a computer in one of two ways: as a "dense" array of all possible coefficients up to the polynomial's degree, or as a "sparse" list of coefficient-exponent tuples. In the latter case, zero terms are not explicitly written, giving a potentially more compact representation. In the area of arithmetic operations, new algorithms are presented for the multiplication of dense polynomials. These have the same asymptotic time cost of the fastest existing approaches, but reduce the intermediate storage required from linear in the size of the input to a constant amount. Two different algorithms for so-called "adaptive" multiplication are also presented which effectively provide a gradient between existing sparse and dense algorithms, giving a large improvement in many cases while never performing significantly worse than the best existing approaches. Algebraic computations on sparse polynomials are considered as well. The first known polynomial-time algorithm to detect when a sparse polynomial is a perfect power is presented, along with two different approaches to computing the perfect power factorization. Inverse symbolic problems are those for which the challenge is to compute a symbolic mathematical representation of a program or "black box". First, new algorithms are presented which improve the complexity of interpolation for sparse polynomials with coefficients in finite fields or approximate complex numbers. Second, the first polynomial-time algorithm for the more general problem of sparsest-shift interpolation is presented. The practical performance of all these algorithms is demonstrated with implementations in a high-performance library and compared to existing software and previous techniques.