Zugl.: Münster (Westfalen), Univ., Diss., 2007

In this thesis we address the analytic and numerical aspects of isospectral flows. Such flows occur in mathematical physics and numerical linear algebra. Their main structural feature is to retain the eigenvalues in the solution space. We explore the solution of Isospectral flows and their stochastic counterpart using explicit generalisation of Magnus expansion. \par In the first part of the thesis we expand the solution of Bloch--Iserles equations, the matrix ordinary differential system of the form $ X'=[N,X^{2}],\ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),\ N\in \mathfrak{so}(n), $ where $\textrm{Sym}(n)$ denotes the space of real $n\times n$ symmetric matrices and $\mathfrak{so}(n)$ denotes the Lie algebra of real $n\times n$ skew-symmetric matrices. This system is endowed with Poisson structure and is integrable. Various important properties of the flow are discussed. The flow is solved using explicit Magnus expansion and the terms of expansion are represented as binary rooted trees deducing an explicit formalism to construct the trees recursively. Unlike classical numerical methods, e.g.\ Runge--Kutta and multistep methods, Magnus expansion respects the isospectrality of the system, and the shorthand of binary rooted trees reduces the computational cost of the exponentially growing terms. The desired structure of the solution (also with large time steps) has been displayed. \par Having seen the promising results in the first part of the thesis, the technique has been extended to the generalised double bracket flow $ X^{'}=[[N,X]+M,X], \ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),$ where $N\in \textrm{diag}(n)$ and $M\in \mathfrak{so}(n)$, which is also a form of an Isospectral flow. In the second part of the thesis we define the generalised double bracket flow and discuss its dynamics. It is noted that $N=0$ reduces it to an integrable flow, while for $M=0$ it results in a gradient flow. We analyse the flow for various non-zero values of $N$ and $M$ by assigning different weights and observe Hopf bifurcation in the system. The discretisation is done using Magnus series and the expansion terms have been portrayed using binary rooted trees. Although this matrix system appears more complex and leads to the tri-colour leaves; it has been possible to formulate the explicit recursive rule. The desired structure of the solution is obtained that leaves the eigenvalues invariant in the solution space.

Neste trabalho é apresentado um breve estudo sobre o processo de descoberta de conhecimento em banco de dados, com enfoque na etapa de mineração de dados através de regras de associação. Propostas por Agrawal em 1993, num estudo chamado análise de cesta de mercado, as regras de associação representam que com um certo grau de suporte e confiança um conjunto de itens pode estar presente numa transação visto que outro conjunto está presente. A necessidade de análise semelhante às realizadas por Agrawal surgiu em outros campos e estas foram estendidas a outras aplicações. Neste, são apresentadas as principais variações sobre o tema regras de associação encontradas na literatura. É proposta a mineração de dados através de regras de associação sobre filtros de mensagens e canais de informação do software de catálogo, agenda e correio eletrônico Direto. Para as pesquisas são utilizadas três ferramentas: Intelligent Miner, CBA e Magnus Opus. Elas foram aplicadas sobre uma lista de discussão da Linguagem Java, pois o projeto Direto ainda não possui mensagens públicas. As ferramentas possuem características distintas: o Intelligent Miner permite a definição de hierarquias sobre os dados que serão minerados; o Magnus Opus trabalha com diversos filtros e com a definição de intervalos para o tratamento de campos numéricos; o CBA permite que sejam especificados suportes múltiplos para os itens. / This work presents a brief review about knowledge discovery in database having association rules as the data mining process. Association rules were proposed by Agrawal in 1993 in a basket data analysis. Association rules have been extended to other applications because there is a necessity for similar Agrawal’s analysis in different domains. Here are presented some variations proposed in the literature about association rules along with the main algorithms. This work proposes the use of association rules over message filters and information channels from the Direto, which is a catalog, schedule and e-mail software. Three data mining tools were used: Intelligent Miner, CBA and Magnus Opus. They were applied over a Java discussion list because Direto project does not have public messages. Each tool has distinct features: Intelligent Miner allows to define a hierarchy over the data that will be mined; Magnus Opus works with many filters over the data and permits to define ranges over numeric fields and CBA allows to specify multiple minimum support over the items.

Syftet med denna uppsats var att undersöka hur Fem myror är fler än fyra elefanter inkräktade på den syn man hade på barn och barnprogram under 1970-talet. För att få en förståelse för debatten och kritiken som Fem myror är fler än fyra elefanter fick, undersökte vi även vilka värderingar och normer som var viktiga under denna period. Vi har även genomfört intervjuer med Magnus Härenstam, Brasse Brännström och Bengt Linné och analyserat fyra sketcher från programserien.

The early nineteenth century saw the emergence of institutional psychiatry across Europe. Aware that Sweden had fallen behind in this development, Parliament decreed in 1823 that a number of specialised institutions for the care of the insane were to be established. The Vadstena asylum, opened in 1826, was the first such institution in Sweden.   The aim of this study is to examine medical interpretation of and responses to erotic behaviour in psychiatric practice at the Vadstena asylum in the period 1849-1878. The book places the theme of the erotic, a topical subject in nineteenth-century public debate, in the context of psychiatry as an emerging specialty in Sweden. The book explores how erotic behaviour was conceptualised as disease, and the nature of therapeutic intervention in erotic cases, in order to present a more nuanced image of nineteenth-century medical attitudes to sexuality. By highlighting the superintendency of physician Ludvig Magnus Hjertstedt, and linking his account of an 1845 study tour through Europe to medical practice at Vadstena, the study situates responses to erotic patients in a period when psychiatry claims authority over human sexuality.   In methodological terms, the study applies critical questions inspired by revisionist scholarship to a body of empirical source material. Focusing on a single institution, and conducting in-depth readings of case notes – with regard to language, form, and function – allows the study to highlight the everyday practice of the asylum physician in his encounters with male and female erotic patients, including the use, importance and diagnostic integrity of the concepts nymphomania, erotomania and masturbation. Hjertstedt’s travel journal provides insight into the physician’s medical philosophy, informing the analysis of diagnostic and interpretive procedures, while connecting medical practice at Vadstena to its European paragons.     The results indicate that while the use of specific diagnostic terms to describe erotic behaviour was infrequent, therapeutic and managerial intervention shows that sexual acts and expressions of desire were considered disturbing and dangerous symptoms in both male and female patients. The analysis thus makes visible a gap between psychiatric theory and asylum practice, emphasising uncertainties and complexities inherent in the latter. While erotic behaviour could be considered indicative of illness, it might also be interpreted as a lack of character or a result of insufficient moral instruction. The asylum’s regime of work and moral instruction was designed to restore health as well as sound values and appropriate behaviour in its patients, indicating a medical culture at Vadstena which was both curative and normalising.

Neste trabalho é apresentado um breve estudo sobre o processo de descoberta de conhecimento em banco de dados, com enfoque na etapa de mineração de dados através de regras de associação. Propostas por Agrawal em 1993, num estudo chamado análise de cesta de mercado, as regras de associação representam que com um certo grau de suporte e confiança um conjunto de itens pode estar presente numa transação visto que outro conjunto está presente. A necessidade de análise semelhante às realizadas por Agrawal surgiu em outros campos e estas foram estendidas a outras aplicações. Neste, são apresentadas as principais variações sobre o tema regras de associação encontradas na literatura. É proposta a mineração de dados através de regras de associação sobre filtros de mensagens e canais de informação do software de catálogo, agenda e correio eletrônico Direto. Para as pesquisas são utilizadas três ferramentas: Intelligent Miner, CBA e Magnus Opus. Elas foram aplicadas sobre uma lista de discussão da Linguagem Java, pois o projeto Direto ainda não possui mensagens públicas. As ferramentas possuem características distintas: o Intelligent Miner permite a definição de hierarquias sobre os dados que serão minerados; o Magnus Opus trabalha com diversos filtros e com a definição de intervalos para o tratamento de campos numéricos; o CBA permite que sejam especificados suportes múltiplos para os itens. / This work presents a brief review about knowledge discovery in database having association rules as the data mining process. Association rules were proposed by Agrawal in 1993 in a basket data analysis. Association rules have been extended to other applications because there is a necessity for similar Agrawal’s analysis in different domains. Here are presented some variations proposed in the literature about association rules along with the main algorithms. This work proposes the use of association rules over message filters and information channels from the Direto, which is a catalog, schedule and e-mail software. Three data mining tools were used: Intelligent Miner, CBA and Magnus Opus. They were applied over a Java discussion list because Direto project does not have public messages. Each tool has distinct features: Intelligent Miner allows to define a hierarchy over the data that will be mined; Magnus Opus works with many filters over the data and permits to define ranges over numeric fields and CBA allows to specify multiple minimum support over the items.

Partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) are powerful tools in modeling real-world phenomena in many fields such as geo-engineering. For instance processes such as oil or gas recovery from hydrocarbon reservoirs and mining heat from geothermal reservoirs can be modeled by PDEs or SPDEs. An important task is to understand the behavior of such phenomena. This can be achieved through explicit solutions of equations. Since explicit solutions of many PDEs and SPDEs are rarely known, developing numerical schemes is a good alternative to provide approximations of these explicit solutions. The study of numerical solutions of PDEs and SPDEs is therefore an active research area and has attracted a lot of attentions since at least two decades. The aims of this dissertation is to develop numerical schemes to approximate semilinear parabolic PDEs and SPDEs in space and in time. The approximation in space is done via the standard Galerkin finite element method and the approximation in time, which is the core of our work is done via various numerical integrators. This dissertation consists of two general parts. The first part of this thesis deals with autonomous PDEs and SPDEs. Here, our main interest is on semilinear PDEs and SPDEs where the nonlinear part is stronger than the linear part also called (stochastic) reactive dominated transport equations, or stiff problems. For such problems, many numerical techniques in the current scientific literature lose their good stability properties. We develop a new explicit exponential integrator (called exponential Rosenbrock-type method) and a new semi-implicit method (called linear implicit Rosenbrock-type method), appropriate for such PDEs and SPDEs. We analyze the strong convergence of our novel fully discrete schemes towards the mild solution of the (S)PDE and obtain convergence orders similar to existing ones in the literature. The second part of this thesis focuses on numerical approximations of semilinear non-autonomous parabolic PDEs and SPDEs. Such equations are more realistic than the autonomous ones and find applications in many fields such as fluid mechanics, quantum field theory, electromagnetism, etc. Numerics of non-autonomous semilinear parabolic PDEs and SPDEs are far from being well understood in the literature. We fill that gap in this thesis by developing a new exponential integrator (called Magnus-type method) and the semi-implicit method for such problems and provide their strong convergence towards the mild solution. Moreover, for both autonomous and non-autonomous SPDEs driven by additive noise, we achieve optimal convergence order in time 1 or approximately 1. Numerical simulations are provided to illustrate our theoretical findings in both autonomous and non-autonomous cases.

Uppsatsen undersöker Statens informationsstyrelses fyra sammanställningar av folkberedskapens orts- och länsombuds årliga rapporter 1941-1944. Syftet är att ta reda på mer om vilken roll icke-statliga organisationer hade för roll i det kulturella folkberedskapsarbetet på lokal nivå runt omkring i Sverige under beredskapstiden. Genom kvalitativ textanalys lyfter studien fram vilka kopplingar Folkberedskapen (SIS) hade till icke-statliga organisationer och vilken verksamhet av folkberedskapskaraktär dessa organisationer bedrev. Resultatet visar att de redan etablerade icke-statliga organisationerna hade en mycket omfattande och tämligen autonom roll i arbetet.

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control. / Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716 / TESIS / Premios Extraordinarios de tesis doctorales

This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption.