A higher-order numerical scheme for system of two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy

We give a modified block-by-block method for the nonlinear fractional order Volterra integral equation system by using quadratic Lagrangian interpolation based on the classical block-by-block method. The core of the method is that we divide its domain into a series of subdomains, that is, block it, and use piecewise quadratic Lagrangian interpolation on each subdomain to approximate $\mathit{\boldsymbol{\kappa}}(x, y, s, r, u(s, r))$. Our proposed method has uniform accuracy and its convergence order is $O(h_x^{4-\alpha}+h_y^{4-\beta})$. We give a strict proof for the error analysis of the method, and give several numerical examples to verify the correctness of the theoretical analysis

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

Development of a relativistic full-potential first-principles multiple scattering Green function method applied to complex magnetic textures of nano structures at surfaces

This thesis is concerned with the quantum mechanical investigation of a novel class of magnetic phenomena in atomic- and nanoscale-sized systems deposited on surfaces or embedded in bulk materials that result from a competition between the exchange and the relativistic spin-orbit interactions. The thesis is motivated by the observation of novel spin-textures of one- and two-dimensional periodicity of nanoscale pitchlength exhibiting a unique winding sense observed in ultra-thin magnetic lms on nonmagnetic metallic substrates with a large spin-orbit interaction. The goal is to extend this eld to magnetic clusters and nano-structures of nite size in order to investigate in how far the size of the cluster and the atoms at the edge of the cluster or ribbon that are particular susceptible to relativistic eects change the balance betweendierent interactions and thus lead to new magnetic phenomena. As an example, the challenging problem of Fe nano-islands on Ir(111) is addressed in detail as for an Fe monolayer on Ir(111) a magnetic nanoskyrmion lattice was observed as magnetic structure.To achieve this goal a new rst-principles all-electron electronic structure code based on density functional theory was developed. The method of choice is the Korringa-Kohn-Rostoker (KKR) impurity Green function method, resorting on a multiple scattering approach. This method has been conceptually further advanced to combine the neglect of any shape approximation to the full potential, with the treatment ofnon-collinear magnetism, of the spin-orbit interaction, as well as of the structural relaxation together with the perfect embedding of a nite size magnetic cluster of atoms into a surface or a bulk environment. For this purpose the formalism makes use of an expansion of the Green function involving explicitly left- and right-hand side scattering solutions. Relativistic eects are treated via the scalar-relativistic approximation and a spin-orbit coupling term treated self-consistently. This required the development of a new algorithm to solve the relativistic quantum mechanical scattering problem for a single atom with a non-spherical potential formulated in terms of the Lippmann-Schwinger integral equation. Prior to the investigation of the Fe nano-islands, the magnetic structure of an Fe monolayer is studied using atomistic spin-dynamics on the basis of a classical model Hamiltonian, which uses realistic coupling parameters obtained from rst principles. It is shown that this method is capable to nd the experimentally determined magnetic structure. [...

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

A computational study of several problems in stochastic modelling

This thesis investigates the computational aspects of two different approaches to the study of mathematical models consisting of systems of stochastic differential equations. These two approaches, presented as alternatives to the simple but usually expensive method of Monte Carlo simulation, are: (i) calculation of the appropriate probability distribution of the state variables by solution of the partial differential equation of a related diffusion model, and (ii) solution of systems of deterministic ordinary differential equations for the moments of the state variables. We investigate assumptions for the use of diffusion models and show how a diffusion model can be used in the study of the Interior First Passage Problem, wherein we are interested in the first passage time for a process to reach some point or sub-region within the region on which the process is defined. Since analytical solutions are not usually available for the partial differential equations associated with diffusion models we look in detail at finite difference methods of solution. We demonstrate the relationship between a finite difference solution and a random walk solution for a simple problem. A particular finite difference scheme is defined for which sufficient conditions are established for the convergence of several iterative methods of solution of the finite difference equations. In addition we discuss stability of the finite difference scheme. We demonstrate the use of a diffusion model in a study of the question of the settlement of Polynesia. The aim of the study is to calculate probabilities of ocean-drift voyages to and between various island groups in Polynesia, in order to test Heyerdahl’s hypothesis of the settlement of Polynesia from the Americas. In some cases of continuous stochastic systems it might be sufficient or more appropriate to solve for moments of the state variables. However, except for models consisting of completely linear systems of stochastic differential equations, the moment equations usually form an infinite coupled system in which equations for moments of any given order involve moments of higher orders. Then to facilitate solution it is necessary to approximate the infinite system with a finite closed system of equations. We investigate one method of achieving this which involves the use of quasi-moments, which are the expectations of multi- dimensional Hermite polynomials in the state variables. This method is shown to be satisfactory in theory but unattractive in practice due to the considerable algebraic manipulation involved in deriving the moment equations and the quasi-moment hierarchy truncation approximations. Therefore we describe and demonstrate an algorithm, written in the Snobol4 programming language, which acts as a pre­ processor to the continuous systems simulation language ACSL. This enables ACSL to be extended to allow for the definition of stochastic differential equations, from which the preprocessor automatically generates an ACSL program for the moment equations, involving hierarchy truncation approximations wherever necessary

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Gauge/Gravity duality

In der vorliegenden Arbeit wird mit Hilfe der verallgemeinerten Eichtheorie/Gravitations-Dualität, welche stark gekoppelte Eichtheorien mit schwach gekrümmten gravitativen Theorien verbindet, stark korrelierte Quantenzustände der Materie untersucht. Der Schwerpunkt liegt dabei in Anwendungen auf Systeme der kondensierten Materie, insbesondere Hochtemperatur-Supraleitung und kritische Quantenzustände bei verschwindender Temperatur. Die Eichtheorie/Gravitations-Dualität entstammt der Stringtheorie und erlaubt eine Umsetzung des holographischen Prinzips. Aus diesem Grund wird eine kurze Einführung in die Konzepte der Stringtheorie und ihre Auswirkungen auf das holographische Prinzip gegeben. Für das tiefere Verständnis der effektiven Niederenergie-Feldtheorien wird zusätzlich die Supersymmetrie benötigt. Ausgestattet mit einem robusten Stringtheorie-Hintergrund wird die unterschiedliche Interpretation der Dirichlet- oder D-Branen, ausgedehnte Objekte auf denen offene Strings/Fäden enden können, diskutiert: Zum einen als massive solitonische Lösungen der Typ II Supergravitation und auf der anderen Seite, ihre Rolle als Quelle für supersymmetrische Yang-Mills Theorien. Die Verbindung dieser unterschiedlichen Betrachtungsweise der D-Branen liefert eine explizite Konstruktion der Eichtheorie/Gravitations-Dualität, genauer der AdS_5/CFT_4 Korrespondenz zwischen der N=4 supersymmetrischen SU(N_c) Yang-Mills Theorie in vier Dimensionen mit verschwindender beta-Funktion in allen Ordnungen, also eine echte konforme Theorie, und Type IIB Supergravitation in der zehn dimensionalen AdS_5 X S^5 Raumzeit. Darüber hinaus wird das Wörterbuch, das zwischen den Operatoren der konformen Feldtheorie und den gravitativen Feldern übersetzt, im Detail eingeführt. Genauer gesagt, die Zustandssumme der stark gekoppelten N=4 supersymmetrischen Yang-Mills Theorie im Grenzwert großer N_c, ist identisch mit der Zustandssumme der Supergravitation unter Berücksichtigung der zugehörigen Lösungen der Bewegungsgleichungen, ausgewertet am Rand des AdS-Raumes. Die Anwendung der perturbativen Quantenfeldtheorie und die Verbindungen zur quantenstatistischen Zustandssumme erlaubt die Erweiterung des holographischen Wörterbuchs auf Systeme mit endlichen Dichten und endlicher Temperatur. Aus diesem Grund werden alle Aspekte der Quantenfeldtheorie behandelt, die für die Anwendung der ``Linear-Response''-Theorie, der Berechnung von Korrelationsfunktionen und die Beschreibung von kritischen Phänomenen benötigt werden, wobei die Betonung auf allgemeine Zusammenhänge zwischen Thermodynamik, statistischer Physik bzw. statistischer Feldtheorie und Quantenfeldtheorie liegt. Des Weiteren wird der Renormierungsgruppen-Formalismus zur Beschreibung von effektiven Feldtheorien und kritischen Phänomene im Kontext der verallgemeinerten Eichtheorie/Gravitations-Dualität ausführlich dargelegt. Folgende Hauptthemen werden in dieser Arbeit behandelt: Die Untersuchung der optischen Eigenschaften von holographischen Metallen und ihre Beschreibung durch das Drude-Sommerfeld Modell, ein Versuch das Homes'sche Gesetz in Hochtemperatur-Supraleitern holographisch zu beschreiben indem verschiedene Diffusionskonstanten und zugehörige Zeitskalen berechnet werden, das mesonische Spektrum bei verschwindender Temperatur und schlussendlich holographische Quantenzustände bei endlichen Dichten. Entscheidend für die Anwendung dieses Rahmenprogramms auf stark korrelierte Systeme der kondensierten Materie ist die Renormierungsgruppenfluss-Interpretation der AdS_5/CFT_4 Korrespondenz und die daraus resultierenden emergenten, holographischen Duale, welche die meisten Beschränkungen der ursprünglichen Theorie aufheben. Diese sogenannten ``Bottom-Up'' Zugänge sind besonders geeignet für Anwendungen auf Fragestellungen in der Theorie der kondensierten Materie und der ``Linear-Response''-Theorie, mittels des holographischen Fluktuations-Dissipations-Theorem. Die Hauptergebnisse der vorliegenden Arbeit umfassen eine ausführliche Untersuchung der R-Ladungs-Diffusion und der Impulsdiffusion in holographischen s- und p-Wellen Supraleitern, welche durch die Einstein-Maxwell Theorie bzw. die Einstein-Yang-Mills Theorie beschrieben werden, und eine Vertiefung des Verständnisses der universellen Eigenschaften solcher Systeme. Als zweites wurde die Stabilität der kalten holographischen Quantenzustände der Materie untersucht, wobei eine zusätzliche Diffusions-Mode entdeckt wurde. Diese Mode kann als eine Art ``R-Spin-Diffusion'' aufgefasst werden, die der Spin-Diffusion in Systemen mit frei beweglichen ``itineranten'' Elektronen ähnelt, wobei die Entkopplung der Spin-Bahn Kopplung die Spin-Symmetrie in eine globale Symmetrie überführt. Das Fehlen der Instabilitäten und die Existenz einer ``Zero-Sound'' Mode, bekannt von Fermi-Flüssigkeiten, deuten eine Beschreibung der kalten holographischen Materie durch eine effektive hydrodynamische Theorie an.In this dissertation strongly correlated quantum states of matter are explored with the help of the gauge/gravity duality, relating strongly coupled gauge theories to weakly curved gravitational theories. The main focus of the present work is on applications to condensed matter systems, in particular high temperature superconductors and quantum matter close to criticality at zero temperature. The gauge/gravity duality originates from string theory and is a particular realization of the holographic principle. Therefore, a brief overview of the conceptual ideas behind string theory and the ramifications of the holographic principle are given. Along the way, supersymmetry and supersymmetric field theories needed to understand the low energy effective field theories of superstring theory will be discussed. Armed with the string theory background, the double life of D-branes, extended object where open strings end, is explained as massive solitonic solutions to the type II supergravity equations of motion and their role in generating supersymmetric Yang-Mills theories. Connecting these two different pictures of D-branes will give an explicit construction of a gauge/gravity duality, the AdS_5/CFT_4 correspondence between N=4 supersymmetric SU(N_c) Yang-Mills theory in four dimensions with vanishing beta-function to all orders, describing a true CFT, and type IIB supergravity in ten-dimensional AdS_5 X S^5 spacetime. Furthermore, the precise dictionary relating operators of the conformal field theory to fields in the gravitational theory is established. More precisely, the partitions functions of the strongly coupled N=4 supersymmetric Yang-Mills theory in the large N_c limit is equal to the on-shell supergravity partitionevaluated at the boundary of the AdS space. Applying the knowledge of perturbative quantum field theory and its relation to the quantum partition function the dictionary may be extended to finite temperature and finite density states. Thus, all aspects of quantum field theory relevant for the application of linear response theory, the computation of correlation functions, and the description of critical phenomena are covered with emphasis on elucidating connections between thermodynamics, statistical physics, statistical field theory and quantum field theory. Furthermore, the renormalization group formalism in the context of effective field theories and critical phenomena will be developed explaining the critical exponents in terms of hyperscaling relations. The main topics covered in this thesis are: the analysis of optical properties of holographic metals and their relation to the Drude-Sommerfeld model, an attempt to understand Homes' law of high temperature superconductors holographically by computing different diffusion constants and related timescales, the mesonic spectrum at zero temperature and holographic quantum matter at finite density. Crucially for the application of this framework to strongly correlated condensed matter systems is the renormalization flow interpretation of the AdS_5/CFT_4 correspondence and the resulting emergent holographic duals relaxing most of the constraints of the original formulation. These so-called bottom up approaches are geared especially towards applications in condensed matter physics and to linear response theory, via the central operational prescription, the holographic fluctuation-dissipation theorem. The main results of the present work are an extensive analysis of the R-charge- and momentum diffusion in holographic s- and p-wave superconductors, described by Einstein-Maxwell theory and the Einstein-Yang-Mills model, respectively, and the lessons learned how to improve the understanding of universal features in such systems. Secondly, the stability of cold holographic quantum matter is investigated. So far, there are no instabilities detected in such systems. Instead, an interesting additional diffusion mode is discovered, which can be interpreted as an ``R-spin diffusion'', resembling spin diffusion in itinerant electronic systems where the spin decouples from the orbital momenta and becomes an internal global symmetry. The lack of instabilities and the existence of a zero sound and diffusion mode indicates that cold holographic matter is closely described by an effective hydrodynamic theory