1,720,981 research outputs found
A Time Splitting Method for the Three-Dimensional Linear Pauli Equation
Full text linkWe analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrodinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrodinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrodinger equation
A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle
Full text linkWe introduce and analyze a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kind with or without known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind
problems, where the former builds on connections with Toeplitz operators
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
A structural realist account of structure in physics theories on the basis of symmetries of solution spaces
No full textUnter den Begriff des wissenschaftlichen Realismus fallen alle jene philosophischen Meinungen und Auffassungen, welche in den Theorien der Wissenschaften (insbe-sondere der Naturwissenschaften) in einem nicht-trivialen Sinne Aussagen über die reale, extern existierende Welt sehen. Der strukturelle Realismus, welcher die über Theorienwechsel hinweg erhaltene Struktur wissenschaftlicher Theorien als Grund-stein für eine Art von Realismus sieht, ist eine der am weitesten verbreiteten und populärsten Formen des wissenschaftlichen Realismus. Trotz oder eher sogar auf Grund dieses Status gibt es zahlreiche Kritiken an dieser Position, allen voran der Vorwurf keine genauen bzw. konkreten Aussagen bezüglich der Bedeutung des Be-griffs der „Struktur“ anzubieten und daher leere Aussagen zu tätigen.
In dieser Arbeit wird ein Versuch der Charakterisierung der Struktur innerhalb der Physik in einem für den strukturellen Realismus relevanten Sinne formuliert, primär aufbauend auf dem Begriff dynamischer Symmetrien in physikalischen Lösungs- bzw. Zustandsräumen. Gemeinsam mit den typischen Argumenten für den struktu-rellen Realismus bildet dies eine Verteidigung des strukturellen Realismus vor anti-realistischen Argumenten einerseits und andererseits vor dem Vorwurf keine kon-kreten Aussagen über die Bedeutung des Begriffs der Struktur zu machen. Den Ab-schluss der Arbeit bildet eine Diskussion verschiedener möglicher Kritikpunkte an diesem Ansatz basierend auf relevanter philosophischer Literatur.Structural realism is a widely discussed form of scientific realism, sometimes dubbed “the most defensible form of scientific realism” (Ladyman, 2016). As a consequence several arguments against it have been raised, perhaps chief among them the con-tention that the notion of ‘structure’ that it is reliant on is ill-defined or too vague to be able to be able to even properly discuss its claims. In this thesis we present a de-fense of structural realism on this ground by providing a more graspable and con-crete candidate for the meaning of ‘structure’ in the theories of physics in particular. Together with previously established arguments relying on notions of structure, we make a case for structural realism based on the concept of dynamical symmetries of solution spaces to fundamental equations in physics
Sparse spectral methods for integral equations and equilibrium measures
Full text linkIn this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems.
These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost.
This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels.
The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
An operator splitting approach for the Pauli equation
No full textDiese Arbeit diskutiert numerische Ansätze zur Pauli Gleichung, welche ein semirelativistisches quantenphysikalisches Modell für geladene Fermionen wie z.B. Elektronen in einem elektromagnetischen Feld darstellt. Zunächst werden dafür die allgemeine Schrödinger Gleichung sowie der mathematische Formalismus der Quantenphysik und Spin sowie klassische Elektrodynamik basierend auf Lorentz-Kraft und Maxwell Gleichungen kurz eingeführt, gefolgt von einer kurzen Diskussion zweier fundamental relativistischer Quantenfeldgleichungen: der Klein-Gordon und Dirac Gleichung. Darauf aufbauend werden zwei verschiedene Motivationen für die Pauli Gleichung als sinnvolles Modell präsentiert: Ein Bottom-up Ansatz basierend auf experimentellen Resultaten und ein Top-down Ansatz als semi-relativistischer Limes der Dirac Gleichung. Nachdem die Pauli Gleichung auf diese Art motiviert und eingeführt wurde werden numerische Ansätze zum Pauli System wie auch der magnetischen Schrödinger Gleichung diskutiert. Dabei wird eine mögliche Skalierung dieser Gleichungen hergeleitet. Den Kern der Arbeit bildet die Erweiterung eines bekannten numerischen Ansatzes zur magnetischen Schrödinger Gleichung auf das Pauli Gleichungssystem, inklusive einer numerischen Implementation dieser Methode in der Programmiersprache Julia.In this thesis we discuss the Pauli equation which models the semi-relativistic evolution of electron states in an electromagnetic field. An introduction is given on the physics and mathematics of quantum mechanics and electromagnetism including core concepts such as the Schrödinger equation, potential couplings to Maxwell’s equations and the Lorentz force of classical electromagnetism and the relativistic Klein-Gordon and Dirac equations. From these, one finds two different approaches to arrive at the Pauli equation: A bottom-up approach which adds spin to conform to empirical results and a top-down approach which shows the Pauli equation to be the semi-relativistic limit of the fully relativistic Dirac equation. Once the Pauli equation’s modeling and relevance to modern physics is established, we move to discussing potential numerical approaches to finding solutions. We will present one sensible way to scale the Pauli equation for use in numerical procedures and present a coupled four operator-splitting approach to numerically solving the Pauli equation based on previous results for the magnetic Schrödinger equation in [1]. Most discussions are applied to both the magnetic Schrödinger equation and the Pauli equation. We conclude by presenting a handful of numerical experiments done in a Julia language implementation of this four operator-splitting method and by discussing potential applications and an outlook on potential future research
Numerical analysis and modeling of geometric structures in simulations of the IKKT matrix model
No full textSogenannte Matrixmodelle, darunter das IKKT Matrixmodell, sind Kandidaten für einen nicht-störungstheoretischen Ansatz zur Superstringtheorie. Von solchen Modellen werden Antworten auf fundamentale Fragen der Physik erwartet, wie zum Beispiel der Natur der Raumzeit auf Quantenskalen, also einer potentiellen Vereinugung von Gravitation und Quantenfeldtheorie.
Rezent publizierte numerische Monte-Carlo Berechnungen fanden eine (3 + 1) dimensionale und expandierende Substruktur, welche sich dynamisch aus dem (9 + 1) dimensionalen Hintergrundraum entwickelt, in einigen Konfigurationen des IKKT Matrixmodells und erinnerten damit an das physikalische, expandierende Universum. Ob diese Substruktur allerdings sinnvolle semiklassische Eigenschaften hat ist bisher nicht analysiert worden. Die Untersuchung dieser (3 + 1) dimensionalen und expandierenden Substruktur ist das Ziel dieser Arbeit, in welcher zuvor entwickelte numerische Methoden zur Analyse des semi-klassischen Grenzwerts von Matrix- oder Quantengeometrien weiterentwickelt und auf diese IKKT Konfigurationen angewandt werden.
In den untersuchten Matrixkonfigurationen finden sich tatsächlich charakteristische Eigenschaften dreier der neun räumlichen Dimensionen, allerdings zeigt die Struktur nach genauer numerischer Behandlung keine Zeitevolution in dem zuvor vermuteten Sinn. Stattdessen zeigt sich in diesen drei verschiedenen IKKT Matrixkonfigurationen eine zeitlich sehr stark an einen Punkt isolierte und hohle 2-Sphäre als semiklassischer Grenzwert der (3 + 1) dimensionale Struktur. Für die Struktur in den restlichen Dimensionen werden Argumente präsentiert welche diese einer Gaußschen Zufallsmatrixverteilung, speziell dem Gauß-verteilten unitären Zufallsmatrixensemble, zuweisen.
Diese Ergebnisse führen zu dem Schluss, dass zumindest mit dieser Wahl der Abschätzungen und Parameter der Monte Carlo Simulationen die gefundenen Matrixkonfigurationen keine Interpretation als (3+1)-dimensionale Raumzeit zulassen. Die Analyse von weiteren Parameterwahlen ist notwendig um zu überprüfen inwiefern es sich hierbei um universelle Resultate handelt. Im Appendix werden vorläufige Resultate von anderen nicht-Monte-Carlo Generierungsmethoden des IKKT Matrixmodells besprochen, welche vielversprechende zeitliche Entwicklung vorweisen.Recently published numerical Monte-Carlo generated sample configurations of the IKKT matrix model found that a (3+1) dimensional and expanding substructure emerges dynamically from the (9 + 1) dimensional background space. In the present paper we utilize and improve upon previously established numerical methods for the analysis of semi-classical limits of matrix- or quantum geometries to study the geometry of these IKKT model configurations. We indeed find characteristic properties distinguishing three among the nine spatial dimensions, though after a careful analysis this structure does not show time evolution in the expected sense. Instead, one finds that a temporally highly localized 2-sphere arises as the semi-classical limit in these three IKKT matrix configurations. We present arguments that show the structure in the remaining dimensions to be one corresponding to what would be expected of configurations of the so-called Gaussian unitary matrix ensemble (GUE) of random matrices. These results suggest that at least with the presently chosen Monte-Carlo parameters the obtained matrix configurations cannot be interpreted as containing a sensible physical spacetime. Further research and analysis is needed in order to determine whether this is a general property or a result of the specific assumptions, approximations and parameters used in the generation of the Monte-Carlo samples. We further present promising preliminary results of a different non-Monte-Carlo approach to the numerical generation of IKKT matrix model configurations in the appendix where one
does observe interesting time evolution
A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels
Full text linkWe present a sparse spectral method for nonlinear integro-differential
Volterra equations based on the Volterra operator's banded sparsity structure
when acting on specific Jacobi polynomial bases. The method is not restricted
to convolution-type kernels of the form but instead works for
general kernels at competitive speeds and with exponential convergence. We
provide various numerical experiments on problems with or without known
analytic solutions and comparisons with other methods
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