14 research outputs found
The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd
Mellin-Barnes Integrals: A Primer on Particle Physics Applications
We discuss the Mellin-Barnes representation of complex multidimensional
integrals. Experiments frontiered by the High-Luminosity Large Hadron Collider
at CERN and future collider projects demand the development of computational
methods to achieve the theoretical precision required by experimental setups.
In this regard, performing higher-order calculations in perturbative quantum
field theory is of paramount importance. The Mellin-Barnes integrals technique
has been successfully applied to the analytic and numerical analysis of
integrals connected with virtual and real higher-order perturbative corrections
to particle scattering. Easy-to-follow examples with the supplemental online
material introduce the reader to the construction and the analytic,
approximate, and numeric solution of Mellin-Barnes integrals in Euclidean and
Minkowskian kinematic regimes. It also includes an overview of the
state-of-the-art software packages for manipulating and evaluating
Mellin-Barnes integrals. These lecture notes are for advanced students and
young researchers to master the theoretical background needed to perform
perturbative quantum field theory calculations.Comment: This is a preprint of the following work: Ievgen Dubovyk, Janusz
Gluza and Gabor Somogyi, Mellin-Barnes Integrals: A Primer on Particle
Physics Applications, 2022, Springer reproduced with permission of Springer
Nature Switzerland AG. 280 page
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Dynamics and disorder in quantum antiferromagnets
La physique de la matière condensée, et notamment les systèmes fortement corrélés, amènent à des problèmes parmi les plus
stimulants et difficiles de la physique moderne. Dans ces systèmes, les interactions à plusieurs corps et les corrélations
entre les particules quantiques ne peuvent être négligées, sinon, les modèles échoueraient simplement à capturer les
mécanismes physiques en jeu et les phénomènes qui en découlent. En particulier, le travail présenté dans ce manuscrit traite
du magnétisme quantique et aborde plusieurs questions distinctes à l'aide d'approches computationnelles et méthodes
numériques à l'état de l'art. Les effets conjoints du désordre (i.e. impuretés) et des interactions sont étudiés concernant
un matériau magnétique spécifique : plutôt qu'une phase de la matière dite localisée, attendue à fort champ magnétique, une
phase ordonnée induite par le désordre lui-même est mise en lumière, avec une réapparition inattendue de la cohérence
quantique dans ledit composé. Par ailleurs, la réponse dynamique d'aimants quantiques à une perturbation externe, comme
celle mesurée dans des expériences de résonance magnétique nucléaire ou de diffusion inélastique de neutrons est étudiée.Condensed matter physics, and especially strongly correlated systems provide some of the most challenging problems of modern
physics. In these systems, the many-body interactions and correlations between quantum particles cannot be neglected;
otherwise, the models would simply fail to capture the relevant physics at play and phenomena ensuing. In particular, the
work presented in this manuscript deals with quantum magnetism and addresses several distinct questions through
computational approaches and state-of-the-art numerical methods. The interplay between disorder (i.e. impurities) and
interactions is studied regarding a specific magnetic compound, where instead of the expected many-body localized phase at
high magnetic fields, a novel disorder-induced ordered state of matter is found, with a resurgence of quantum coherence.
Furthermore, the dynamical response of quantum magnets to an external perturbation, such as it is accessed and measured in
nuclear magnetic resonance and inelastic neutron scattering experiments is investigated
Dynamics of the Standard Model
Describing the fundamental theory of particle physics and its applications, this book provides a detailed account of the Standard Model, focusing on techniques that can produce information about real observed phenomena. It begins with a pedagogic account of the Standard Model, introducing essential techniques such as effective field theory and path integral methods. It then focuses on the use of the Standard Model in the calculation of physical properties of particles. Rigorous methods are emphasized, but other useful models are also described. The second edition has been updated to include theoretical and experimental advances, such as the discovery of the Higgs boson, our understanding of neutrinos, and the major advances in CP violation and electroweak physics. This book is valuable to graduate students and researchers in particle physics, nuclear physics and related fields. This edition, first published in 2014, has been reissued as an Open Access publication