Polynomials

Polynomial and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on polynomial and its applications has been done in mathematics, applied mathematics, and sciences. This book is based on recent results in all areas related to polynomial and its applications. This book provides an overview of the current research in the field of polynomials and its applications. The following papers have been published in this volume: ‘A Parametric Kind of the Degenerate Fubini Numbers and Polynomials’; ‘On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals’; ‘Fractional Supersymmetric Hermite Polynomials’; ‘Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation’; ‘Iterating the Sum of Möbius Divisor Function and Euler Totient Function’; ‘Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations’; ‘Truncated Fubini Polynomials’; ‘On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights’; ‘Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity’; ‘Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials’; ‘Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros.

Short distance constraints from HLbL contribution to the muon anomalous magnetic moment

Hadronic Light by Light (HLbL) scattering is not the biggest hadronic contribution to the muon's anomalous magnetic moment, but it has the biggest relative uncertainty of all the contributions to that observable. With the tension between the Standard Model value prediction and the measurement at 4.2 $\sigma$, theoretical physicists have set their sights on reducing the HLbL contribution's uncertainty to reduce the tension or push it beyond the discovery threshold. In such scenario, the high energy contribution of HLbL scattering to anomalous magnetic moment of the muon plays an important role. The aim of the research developed in this thesis is to study the HLbL leading order contribution in the maximally symmetric high energy region well above the hadronic threshold limit. This is achieved by performing an operator product expansion of the HLbL tensor, which we do systematically in the background field method. We consider our approach very efficient, also because it allows a straightforward renormalization of the field theoretical results. Our approach is also original and at the best of our knowledge not available in literature. The massless quark loop is the leading term and we compute it without neglecting its tensor structure. To this end, we use a tensor--loop--integral decomposition that does not introduce kinematic singularities. The resulting scalar loop integrals with shifted dimensions are computed with their full mass--dependence using a Mellin--Barnes representation. Our original method of computation for the quark loop provides an independent check of recent literature results. Furthermore, by conserving the full tensor structure of the amplitude, we are able to perform an explicit check of a proposed kinematic--singularity--free tensor decomposition for the HLbL scattering amplitude that plays a central role in the dispersive computation in the low--energy regime.Comment: 96 pages, 18 figures, Master's thesi

Límites de corta distancia de la contribución HLbL al momento magnético anómalo del muon

Hadronic Light by Light (HLbL) scattering is not the biggest hadronic contribution to the muon’s anomalous magnetic moment, but it has the biggest relative uncertainty of all the contributions to that observable. With the tension between the Standard Model value prediction and the measurement at 4.2 σ, theoretical physicists have set their sights on reducing the HLbL contribution’s uncertainty to reduce the tension or push it beyond the discovery threshold. In such scenario, the high energy contribution of HLbL scattering to anomalous magnetic moment of the muon plays an important role. The aim of the research developed in this thesis is to study the HLbL leading order contribution in the maximally symmetric high energy region well above the hadronic threshold limit. This is achieved by performing an operator product expansion of the HLbL tensor, which we do systematically in the background field method. We consider our approach very efficient, also because it allows a straightforward renormalization of the field theoretical results. Our approach is also original and at the best of our knowledge not available in literature. The massless quark loop is the leading term and we compute it without neglecting its tensor structure. To this end, we use a tensor–loop–integral decomposition that does not in- troduce kinematic singularities. The resulting scalar loop integrals with shifted dimensions are computed with their full mass dependence using a Mellin–Barnes representation. Our original method of computation for the quark loop provides an independent check of recent literature results. Furthermore, by conserving the full tensor structure of the amplitude, we are able to perform an explicit check of a proposed kinematic–singularity–free tensor decomposition for the HLbL scattering amplitude that plays a central role in the dispersive computation in the low–energy regime. (Texto tomado de la fuente)La dispersión HLbL no es la contribución hadrónica más grande para el momento magnético anómalo del muon, pero esta tiene la incertidumbre relativa más grande de todas las contribuciones a ese observable. Con la tensión entre la valor predicho por el Modelo Estándar y las mediciones actualmente en 4.2 σ, los físico teóricos se han centrado en reducir la incertidumbre de la contribución HLbL para reducir la tensión o llevarla más allá del umbral de descubrimiento. En tal escenario, la contribución de alta energía de la dispersión HLbL al momento magnético anómalo del muon juega un papel importante. El objetivo de la investigación desarrollada en esta tesis es estudiar la contribución HLbL de primer orden en la región de alta energía máximamente simétrica muy por encima del límite del umbral hadrónico. Esto se logra al realizar una expansión de productos de operadores del tensor HLbL, la cual realizamos sistemáticamente con el método de campos de fondo. Consideramos nuestra aproximación al problema muy eficiente, entre otras razones, porque esta permite la renormalización directa de los resultados de teoría de campos. Nuestro método es también original y, hasta nuestro mejor conocimiento, no se encuentra en la literatura. El quark loop sin masa es el primer término de la expansión y lo calculamos sin dejar de lado su estructura tensorial. Para lograrlo, usamos un método de descomposición tensorial de integrales de loop que no introduce singularidades cinemáticas. Las integrales escalares de loop resultantes con dimensiones modificadas son calculadas considerando toda su dependencia de la masa y utilizando la representación de Mellin-Barnes. Nuestro método original de cálculo para el quark loop proporciona una verificación independiente de los resultados publicados recientemente en la literatura. Más aún, al conservar la estructura tensorial completa de la amplitud, podemos llevar a cabo una verificación explícita de una descomposición libre de singularidades cinemáticas para la dispersión HLbL que juega un papel central en los cálculos dispersivos del régimen de baja energía.Maestrí

Approximation Theory and Related Applications

In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics