Algorithms and techniques for finding canonical differential equations of Feynman integrals

Modern day particle physics relies on perturbative Quantum Field Theory. Beyond the leading order, the computation of the appearing Feynman integrals is one of the most important and often also most complicated steps necessary to extract predictions from the theory. A prominent method for the latter is to derive a set of ordinary differential equations satisfied by the Feynman integrals and then subsequently transforming them into canonical form where the solution in terms of known functions can be obtained in a straightforward manner. In this thesis, we consider the crucial task of finding the appropriate basis change that transforms the set of differential equations into canonical form. To this end, we first discuss some of the basic properties of Feynman integrals and the special functions appearing in the solutions. We then show how the transcendental weight of these functions constitutes an essential guiding principle in the search for members of the canonical basis. In particular, we review the algorithmic determination of so-called dlog integrals and their leading singularities. Further, we provide a summary of heuristic methods that, in many cases, prove to be sufficient for finding a canonical basis with little effort. Through a simple example, we then also review how so-called balance transformations can be used to reach the properties of the canonical form step by step. As the main result of the thesis, we present a new algorithm for attaining the canonical form starting from a single canonical integral. In addition, this algorithm makes it possible to test the transcendental weight properties of individual integrals and can therefore also be seen as complementary to the other methods described in this thesis. Several univariate examples, as well as a multivariate example are used to demonstrate the power and flexibility of the algorithm and our public implementation. Finally, we use the methods discussed in the thesis in three state-of-the-art applications and highlight how our algorithm can find the canonical form in cases where existing methods fail to provide an answer. This includes differential equations with more than 500 basis integrals and a matrix involving elliptic functions.Die moderne Teilchenphysik stützt sich auf die störungstheoretische Quantenfeldtheorie. Jenseits der führenden Ordnung ist die Berechnung der auftretenden Feynman-Integrale einer der wichtigsten und oft auch kompliziertesten Schritte, die notwendig sind, um Vorhersagen aus der Theorie zu gewinnen. Eine häufig verwendete Methode zur Berechnung der Feynman-Integrale besteht darin, einen Satz gewöhnlicher Differentialgleichungen abzuleiten, die von den Feynman-Integralen erfüllt werden, und diese anschließend in eine kanonische Form zu transformieren, bei der die Lösung in Form bekannter Funktionen auf einfache Weise erhalten werden kann. Diese Arbeit befasst sich mit der entscheidenden Aufgabe, die geeignete Basisänderung zu finden, die den Satz der Differentialgleichungen in die kanonische Form bringt. Zu diesem Zweck werden zunächst einige der grundlegenden Eigenschaften von Feyn\-man-Integralen und der speziellen Funktionen, die in den Lösungen vorkommen, erläutert. Anschließend erörtern wir, wie das transzendentale Gewicht dieser Funktionen ein wesentliches Leitprinzip bei der Suche nach Teilen der kanonischen Basis darstellt. Insbesondere gehen wir auf die algorithmische Bestimmung der sogenannten dlog-Integrale und ihrer ``leading singularities'' ein. Außerdem geben wir einen Überblick über heuristische Methoden, die sich in vielen Fällen als ausreichend erweisen, um mit geringem Aufwand eine kanonische Basis zu finden. Anhand eines einfachen Beispiels wird anschließend gezeigt, wie man mit Hilfe von sogenannten ``balance''-Transformationen Schritt für Schritt zu den Eigenschaften der kanonischen Form gelangt. Als Hauptergebnis der Arbeit stellen wir einen neuen Algorithmus vor, mit dem die kanonische Form ausgehend von einem einzigen kanonischen Integral erreicht werden kann. Dieser Algorithmus ermöglicht es ebenfalls, die transzendentalen Gewichtseigenschaften einzelner Integrale zu testen und kann daher auch als Ergänzung zu den anderen in dieser Arbeit beschriebenen Methoden betrachtet werden. Anhand mehrerer univariater Beispiele, sowie eines multivariaten Beispiels, wird die Leistungsfähigkeit und Flexibilität des Algorithmus und der öffentlich zugänglichen Implementierung demonstriert. Schließlich verwenden wir die in dieser Arbeit diskutierten Methoden in drei hochmodernen Anwendungen und zeigen, wie unser Algorithmus die kanonische Form in Fällen finden kann, in denen bestehende Methoden nicht anwendbar sind. Dazu gehören unter anderem Differentialgleichungen mit mehr als 500 Basisintegralen und eine Matrix mit elliptischen Funktionen

Temperature control in air-heating set via direct search method and structured singular value

The contribution describes an application of the algebraic μ-synthesis to the control of a real plant with a nonlinear characteristic. The controller is designed through algebraic approach and subsequent minimization of the peak of the structured singular value denoted μ. The algebraic approach consists of the pole placement applied to the nominal plant and the algorithm of global optimization Differential Migration treating the problem of multimodality of the cost function. The results are compared with the D-K iteration, synthesis in the ring of proper and stable functions and the Naslin method. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

Control of uncertain time delay system with astatism and parametric and periodic uncertainties using SSV and factorization for two-degree-of-freedom-controller

Application of the Robust Control Toolbox for Time Delay Systems with Parametric and Periodic Uncertainties Using SSV (Structured Singular Value) for the Matlab system to uncertain time delay system with astatism is performed. The D-K iteration and the algebraic approach implemented in the toolbox are applied to 2nd order system with astatism and uncertain time delay and two other parameters in the numerator and denominator of the plant transfer function. Multiplicative uncertainty is used for treating uncertain time delay, the parametric uncertainty is modelled using general interconnection for the systems with parametric uncertainty in numerator and denominator. Copyright © 2019 by SCITEPRESS - Science and Technology Publications, Lda. All rights reserved.Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme project [LO1303 (MSM-7778/2014)

Application of the robust control toolbox for time delay systems with parametric and periodic uncertainties using SSV to uncertain time delay system with astatism

Application of the Robust Control Toolbox for Time Delay Systems with Parametric and Periodic Uncertainties Using SSV (Structured Singular Value) for the Matlab system to Uncertain Time Delay System with Astatism is performed. The D-K iteration and the algebraic approach implemented in the toolbox are applied to 2nd order system with astatism and uncertain time delay and two other parameters in the numerator and denominator of the plant transfer function. Multiplicative uncertainty is used for treating uncertain time delay, the parametric uncertainty is modelled using general interconnection for the systems with parametric uncertainty in numerator and denominator. (C) 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.Ministry of Education, Youth and Sports of the Czech RepublicMinistry of Education, Youth & Sports - Czech Republic [LO1303 (MSMT-7778/2014)

Simple robust controllers for delayed systems

A way for treating general delayed systems with uncertain delays in both the numerator and denominator is shown. The proposed procedure is demonstrated by an example. A simple controller is derived via algebraic theory and the structured singular value, which treats uncertain time delay in both the numerator and denominator of an anisochronic system. The overall performance is verified by simulations and compared with standard tool for robust control design

Dynamics of Binary Systems to Fourth Post-Minkowskian Order from the Effective Field Theory Approach

We present the contribution from potential interactions to the dynamics of non-spinning binaries to fourth Post-Minkowskian (4PM) order. This is achieved by computing the scattering angle to ${\cal O}(G^4)$ using the effective field theory approach and deriving the bound radial action through analytic continuation. We reconstruct the Hamiltonian and center-of-mass momentum in an isotropic gauge. The (three-loop) integrals involved in our calculation are computed via differential equations, including a sector yielding elliptic integrals. As a prelude of radiation-reaction effects, using the universal link between potential and tail terms, we also report: 1) The instantaneous energy flux at ${\cal O}(G^3)$; 2) The contribution to the 4PM unbound/bound radial action(s) depending on logarithms of the binding energy; 3) The (scheme-independent) logarithmic contribution to the 4PM non-local tail Hamiltonian for circular orbits. Our results in the potential region are in agreement with the recent derivation from scattering amplitudes. We also find perfect agreement in the overlap with the state-of-the-art in Post-Newtonian theory.Comment: 9 pages, 1 figure. 1 ancillary file (txt format

Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order

We obtain the total impulse in the scattering of non-spinning binaries in general relativity at fourth Post-Minkowskian order, i.e. ${\cal O}(G^4)$, including linear, nonlinear, and hereditary radiation-reaction effects. We derive the total radiated spacetime momentum as well as the associated energy flux. The latter can be used to compute gravitational-wave observables for generic (un)bound orbits. We employ the ("in-in") Schwinger-Keldysh worldline effective field theory framework in combination with modern "multi-loop" integration techniques from collider physics. The complete results are in agreement with various partial calculations in the Post-Newtonian/Minkowskian expansion.Comment: 6 pages + Refs + Supplemental. 1 figure and 1 table. 1 computer-readable ancillary fil

Precipitación y producción de agua en el Macizo del Caroig, Este de la Península Ibérica. Evento de escorrentía a escala de parcela durante una crecida torrencial en el barranco de Benacancil

[EN] Floods are a consequence of extreme rainfall events. Although surface runoff generation is the origin of discharge, flood research usually focuses on lowlands where the impact is higher. Runoff and sediment delivery at slope and pedon scale receiving much less attention in the effort to understand flood behaviour in time and space. This is especially relevant in areas where, due to climatic and hydrogeological conditions, streams are ephemeral, so-called dry rivers (¿wadis¿, "ramblas" or ¿barrancos¿) that are widespread throughout the Mediterranean. This paper researches the relationship between water delivery at pedon and slope scale with dry river floods in Macizo del Caroig, Eastern Iberian Peninsula. Plots of 1x1, 1x2, 1x4, and 2x8 m located in the ¿El Teularet¿ Soil Erosion and Degradation Research Station were monitored from 2004 to 2014 to measure soil and water delivery. Rainfall and flow at the dry river Barranco de Benacancil were also monitored. Results show that runoff and sediment discharge were concentrated in few events during the 11 years of research. A single flood event was registered in the channel on September 28, 2009, however, the runoff was registered 160 times at the plots. Runoff discharge was dependent on the size of the plots, with larger plots yielding lower runoff discharge per unit area, suggesting short runoff-travel distance and duration. Three rainfall events contributed with 26% of the whole runoff discharge, and five achieved 56% of the runoff. We conclude that the runoff generated at the plot scale is disconnected from the main channel. From a spatial point of view, there is a decrease in runoff coefficient along the slope. From a temporal point of view, the runoff is concentrated in a few rainfall events. These results show that the runoff generated at plot and slope scale does not contribute to the floods except for rainfall events with more than 100 mm day-1. The disconnection of the runoff and sediment delivery is confirmed by the reduction in the runoff delivery at plot scale due to the control of the length of the plot (slope) on the runoff and sediment delivery.[ES] . Lasinundacionesson consecuencia de lluvias extremas. Aunque la generación de escorrentía superficial es el origen de la descarga, la investigación de inundaciones generalmente se enfoca en las tierras bajas donde el impacto es mayor. La escorrentía y la distribución de sedimentos a escala de pendiente y pedón reciben mucha menos atención en la comprensión del comportamiento de las inundaciones en el tiempo y el espacio. Esto es especialmente relevante en zonas donde, debido a las condiciones climáticas e hidrogeológicas, los cauces son efímeros. Son los llamados ríos secos (¿wadis¿, ¿ramblas¿ o ¿barrancos¿) muy extendidos por todo el Mediterráneo. Este artículo investiga la relación entre el suministro de agua a escala de pedón y ladera con las crecidas de ríos secos en Macizo del Caroig, este de la Península Ibérica. Las parcelas de 1x1, 1x2, 1x4 y 2x8 m localizadas en la Estación de Investigación de Erosión y Degradación de Suelos ¿El Teularet¿ fueron monitoreadas de 2004 a 2014 para medir la producción de suelo y agua. También se monitorearon las precipitaciones y el caudal en el río seco Barranco de Benacancil. Los resultados muestran que la escorrentía y la descarga de sedimentos se concentraron en pocos eventos durante los 11 años de investigación. Se registró un solo evento de inundación en el canal el 28 de septiembre de 2009, sin embargo, la escorrentía se registró 160 veces en las parcelas. La descarga de escorrentía dependió del tamaño de las parcelas. Las parcelas más grandes produjeron una menor descarga de escorrentía por unidad de área, lo que sugiere una corta distancia y duración del recorrido de escorrentía. Tres eventos de lluvia contribuyeron con el 26% de la descarga total de la escorrentía y cinco lograron el 56% de la escorrentía. Se concluye que la escorrentía generada a escala de la parcela está desconectada del canal principal. Desde un punto de vista espacial, hay una disminución en el coeficiente de escorrentía a lo largo de la pendiente. Desde un punto de vista temporal, la escorrentía se concentra en unos pocos eventos de lluvia. Estos resultados muestran que la escorrentía generada a escala de parcela y pendiente no contribuyen a las inundaciones excepto para eventos de lluvia con más de 100 mm día-1 . La desconexión de la escorrentía y la entrega de sedimentos se confirma por la reducción de la escorrentía a escala de parcela debido al control de la longitud (pendiente) sobre la escorrentía y la entrega de sedimentos.Artemi Cerda thanks the Co-operative Research program from the OECD (Biological Resource Management for Sustainable Agricultural Systems) for its support with the 2016 CRP fellowship (OCDE TAD/CRP JA00088807), POSTFIRE Project (CGL2013-47862-C2-1 and 2-R), and POSTFIRE_CARE Project (CGL2016-75178-C2-2-R) sponsored by the Spanish Ministry of Economy and Competitiveness and AEI/FEDER, UE. This paper was written as a result of the collaboration that was initiated due to the COST ActionES1306: Connecting European Connectivity research and COST CA18135 FIRElinks: Fire in the Earth System. Science and Society. We wish to thank the Department of Geography secretariat team (Nieves Gomez, Nieves Dominguez, and Susana Tomas) for their support for three decades to our research at the Soil Erosion and Degradation Research team (SEDER), with special thanks to the scientific researchers that as visitors from other research teams contributed to the SEDER research. And we also thank the Laboratory for Geomorphology technicians (Leon Navarro) for the key contribution to our research. The collaboration of the Geography and Environmental Sciences students was fruitful and enjoyable.Cerda, A.; Novara, A.; Dlapa, P.; Lopez-Vicente, M.; Ubeda, X.; Popovic, Z.; Mekonnen, M.... (2021). Rainfall and water yield in Macizo del Caroig, Eastern Iberian Peninsula. Event runoff at plot scale during a rare flash flood at the Barranco de Benacancil. Cuadernos de Investigación Geográfica. 47(1):95-119. https://doi.org/10.18172/cig.48339511947