On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Theory of laser-induced ultrafast structural changes in solids

The present thesis is a contribution to the study of laser-solid interaction. Despite the numerous applications resulting from the recent use of laser technology, there is still a lack of satisfactory answers to theoretical questions regarding the mechanism leading to the structural changes induced by femtosecond lasers in materials. We provide here theoretical approaches for the description of the structural response of different solids (cerium, samarium sulfide, bismuth and germanium) to femtosecond laser excitation. Particular interest is given to the description of the effects of the laser pulse on the electronic systems and changes of the potential energy surface for the ions. Although the general approach of laser-excited solids remains the same, the potential energy surface which drives the structural changes is calculated with different theoretical models for each material. This is due to the difference of the electronic properties of the studied systems. We use the Falicov model combined with an hydrodynamic method to study photoinduced phase changes in cerium. The local density approximation (LDA) together with the Hubbard-type Hamiltonian (LDA+U) in the framework of density functional theory (DFT) is used to describe the structural properties of samarium sulfide. We parametrize the time-dependent potential energy surface (calculated using DFT+ LDA) of bismuth on which we perform quantum dynamical simulations to study the experimentally observed amplitude collapse and revival of coherent $A_{1g}$ phonons. On the basis of a time-dependent potential energy surface calculated from a non-orthogonal tight binding Hamiltonian, we perform molecular dynamics simulation to analyze the time evolution (coherent phonons, ultrafast nonthermal melting) of germanium under laser excitation. The thermodynamic equilibrium properties of germanium are also reported. With the obtained results we are able to give many clarifications and interpretations of experimental results and also make predictions.

Development and applications of density matrix functional theory of generalized Hubbard Models

In dieser Doktorarbeit wird eine akkurate Methode zur Bestimmung von Grundzustandseigenschaften stark korrelierter Elektronen im Rahmen von Gittermodellen entwickelt und angewandt. In der Dichtematrix-Funktional-Theorie (LDFT, vom englischen lattice density functional theory) ist die Ein-Teilchen-Dichtematrix γ die fundamentale Variable. Auf der Basis eines verallgemeinerten Hohenberg-Kohn-Theorems ergibt sich die Grundzustandsenergie Egs[γgs] = min° E[γ] durch die Minimierung des Energiefunktionals E[γ] bezüglich aller physikalischer bzw. repräsentativer γ. Das Energiefunktional kann in zwei Beiträge aufgeteilt werden: Das Funktional der kinetischen Energie T[γ], dessen lineare Abhängigkeit von γ genau bekannt ist, und das Funktional der Korrelationsenergie W[γ], dessen Abhängigkeit von γ nicht explizit bekannt ist. Das Auffinden präziser Näherungen für W[γ] stellt die tatsächliche Herausforderung dieser These dar. Einem Teil dieser Arbeit liegen vorausgegangene Studien zu Grunde, in denen eine Näherung des Funktionals W[γ] für das Hubbardmodell, basierend auf Skalierungshypothesen und exakten analytischen Ergebnissen für das Dimer, hergeleitet wird. Jedoch ist dieser Ansatz begrenzt auf spin-unabhängige und homogene Systeme. Um den Anwendungsbereich von LDFT zu erweitern, entwickeln wir drei verschiedene Ansätze zur Herleitung von W[γ], die das Studium von Systemen mit gebrochener Symmetrie ermöglichen. Zuerst wird das bisherige Skalierungsfunktional erweitert auf Systeme mit Ladungstransfer. Eine systematische Untersuchung der Abhängigkeit des Funktionals W[γ] von der Ladungsverteilung ergibt ähnliche Skalierungseigenschaften wie für den homogenen Fall. Daraufhin wird eine Erweiterung auf das Hubbardmodell auf bipartiten Gittern hergeleitet und an sowohl endlichen als auch unendlichen Systemen mit repulsiver und attraktiver Wechselwirkung angewandt. Die hohe Genauigkeit dieses Funktionals wird aufgezeigt. Es erweist sich jedoch als schwierig, diesen Ansatz auf komplexere Systeme zu übertragen, da bei der Berechnung von W[γ] das System als ganzes betrachtet wird. Um dieses Problem zu bewältigen, leiten wir eine weitere Näherung basierend auf lokalen Skalierungseigenschaften her. Dieses Funktional ist lokal bezüglich der Gitterplätze formuliert und ist daher anwendbar auf jede Art von geordneten oder ungeordneten Hamiltonoperatoren mit lokalen Wechselwirkungen. Als Anwendungen untersuchen wir den Metall-Isolator-Übergang sowohl im ionischen Hubbardmodell in einer und zwei Dimensionen als auch in eindimensionalen Hubbardketten mit nächsten und übernächsten Nachbarn. Schließlich entwickeln wir ein numerisches Verfahren zur Berechnung von W[γ], basierend auf exakten Diagonalisierungen eines effektiven Vielteilchen-Hamilton-Operators, welcher einen von einem effektiven Medium umgebenen Cluster beschreibt. Dieser effektive Hamiltonoperator hängt von der Dichtematrix γ ab und erlaubt die Herleitung von Näherungen an W[γ], dessen Qualität sich systematisch mit steigender Clustergröße verbessert. Die Formulierung ist spinabhängig und ermöglicht eine direkte Verallgemeinerung auf korrelierte Systeme mit mehreren Orbitalen, wie zum Beispiel auf den spd-Hamilton-Operator. Darüber hinaus berücksichtigt sie die Effekte kurzreichweitiger Ladungs- und Spinfluktuationen in dem Funktional. Für das Hubbardmodell wird die Genauigkeit der Methode durch Vergleich mit Bethe-Ansatz-Resultaten (1D) und Quanten-Monte-Carlo-Simulationen (2D) veranschaulicht. Zum Abschluss wird ein Ausblick auf relevante zukünftige Entwicklungen dieser Theorie gegeben.

Many-body theory of laser-induced ultrafast demagnetization and angular momentum transfer in ferromagnetic transition metals

An electronic theory is developed, which describes the ultrafast demagnetization in itinerant ferromagnets following the absorption of a femtosecond laser pulse. The present work intends to elucidate the microscopic physics of this ultrafast phenomenon by identifying its fundamental mechanisms. In particular, it aims to reveal the nature of the involved spin excitations and angular-momentum transfer between spin and lattice, which are still subjects of intensive debate. In the first preliminary part of the thesis the initial stage of the laser-induced demagnetization process is considered. In this stage the electronic system is highly excited by spin-conserving elementary excitations involved in the laser-pulse absorption, while the spin or magnon degrees of freedom remain very weakly excited. The role of electron-hole excitations on the stability of the magnetic order of one- and two-dimensional 3d transition metals (TMs) is investigated by using ab initio density-functional theory. The results show that the local magnetic moments are remarkably stable even at very high levels of local energy density and, therefore, indicate that these moments preserve their identity throughout the entire demagnetization process. In the second main part of the thesis a many-body theory is proposed, which takes into account these local magnetic moments and the local character of the involved spin excitations such as spin fluctuations from the very beginning. In this approach the relevant valence 3d and 4p electrons are described in terms of a multiband model Hamiltonian which includes Coulomb interactions, interatomic hybridizations, spin-orbit interactions, as well as the coupling to the time-dependent laser field on the same footing. An exact numerical time evolution is performed for small ferromagnetic TM clusters. The dynamical simulations show that after ultra-short laser pulse absorption the magnetization of these clusters decreases on a time scale of hundred femtoseconds. In particular, the results reproduce the experimentally observed laser-induced demagnetization in ferromagnets and demonstrate that this effect can be explained in terms of the following purely electronic non-adiabatic mechanism: First, on a time scale of 10–100 fs after laser excitation the spin-orbit coupling yields local angular-momentum transfer between the spins and the electron orbits, while subsequently the orbital angular momentum is very rapidly quenched in the lattice on the time scale of one femtosecond due to interatomic electron hoppings. In combination, these two processes result in a demagnetization within hundred or a few hundred femtoseconds after laser-pulse absorption.

Evaluation of sets of oriented and non-oriented receptive fields as local descriptors

Local descriptors are increasingly used for the task of object recognition because of their perceived robustness with respect to occlusions and to global geometrical deformations. We propose a performance criterion for a local descriptor based on the tradeoff between selectivity and invariance. In this paper, we evaluate several local descriptors with respect to selectivity and invariance. The descriptors that we evaluated are Gaussian derivatives up to the third order, gray image patches, and Laplacian-based descriptors with either three scales or one scale filters. We compare selectivity and invariance to several affine changes such as rotation, scale, brightness, and viewpoint. Comparisons have been made keeping the dimensionality of the descriptors roughly constant. The overall results indicate a good performance by the descriptor based on a set of oriented Gaussian filters. It is interesting that oriented receptive fields similar to the Gaussian derivatives as well as receptive fields similar to the Laplacian are found in primate visual cortex.

Core-Shell Assisted Bimetallic Assembly of Pt and Ru Nanoparticles by DNA Hybridization

We have discovered that the current protocols to assemble Au nanoparticles based on DNA hybridization do not work well with the small metal nanoparticles (e.g. 5 nm Au, 3.6 nm Pt and 3.2 nm Ru particles). Further investigations revealed the presence of strong interaction between the oligonucleotide backbone and the surface of the small metal nanoparticles. The oligonucleotides in this case are recumbent on the particle surface and are therefore not optimally oriented for hybridization. The nonspecific adsorption of oligonucleotides on small metal nanoparticles must be overcome before DNA hybridization can be accepted as a general assembly method. Two methods have been suggested as possible solutions to this problem. One is based on the use of stabilizer molecules which compete with the oligonucleotides for adsorption on the metal nanoparticle surface. Unfortunately, the reported success of this approach in small Au nanoparticles (using K₂BSPP) and Au films (using 6-mercapto-1-hexanol) could not be extended to the assembly of Pt and Ru nanoparticles by DNA hybridization. The second approach is to simply use larger metal particles. Indeed most reports on the DNA hybridization induced assembly of Au nanoparticles have made use of relatively large particles (>10 nm), hinting at a weaker non-specific interaction between the oligonucleotides and large Au nanoparticles. However, most current methods of nanoparticle synthesis are optimized to produce metal nanoparticles only within a narrow size range. We find that core-shell nanoparticles formed by the seeded growth method may be used to artificially enlarge the size of the metal particles to reduce the nonspecific binding of oligonucleotides. We demonstrate herein a core-shell assisted growth method to assemble Pt and Ru nanoparticles by DNA hybridization. This method involves firstly synthesizing approximately 16 nm core-shell Ag-Pt and 21 nm core-shell Au-Ru nanoparticles from 9.6 nm Ag seeds and 17.2 nm Au seeds respectively by the seed-mediated growth method. The core-shell nanoparticles were then functionalized by complementary thiolated oligonucleotides followed by aging in 0.2 M PBS buffer for 6 hours. The DNA hybridization induced bimetallic assembly of Pt and Ru nanoparticles could then be carried out in 0.3 M PBS buffer for 10 hours.

On the exhaustiveness of truncation and dropping strategies in many-to-many matching markets

We consider two–sided many–to–many matching markets in which each worker may work for multiple firms and each firm may hire multiple workers. We study individual and group manipulations in centralized markets that employ (pairwise) stable mechanisms and that require participants to submit rank order lists of agents on the other side of the market. We are interested in simple preference manipulations that have been reported and studied in empirical and theoretical work: truncation strategies, which are the lists obtained by removing a tail of least preferred partners from a preference list, and the more general dropping strategies, which are the lists obtained by only removing partners from a preference list (i.e., no reshuffling). We study when truncation / dropping strategies are exhaustive for a group of agents on the same side of the market, i.e., when each match resulting from preference manipulations can be replicated or improved upon by some truncation / dropping strategies. We prove that for each stable mechanism, truncation strategies are exhaustive for each agent with quota 1 (Theorem 1). We show that this result cannot be extended neither to group manipulations (even when all quotas equal 1 – Example 1), nor to individual manipulations when the agent’s quota is larger than 1 (even when all other agents’ quotas equal 1 – Example 2). Finally, we prove that for each stable mechanism, dropping strategies are exhaustive for each group of agents on the same side of the market (Theorem 2), i.e., independently of the quotas.

Publicness of goods and violent conflict : evidence from Colombia

How the degree of publicness of goods affect violent conflict? Based on the theoretical model in Esteban and Ray (2001) we find that the effect of the degree of publicness depends on the group size. When the group is small (large), the degree of publicness increases (decreases) the likelihood of conflict. This opens an empirical question that we tackle using microdata from the Colombian conflict at the municipality level. We use three goods with different publicness degree to identify the sign of the effect of publicness on conflict. These goods are coca crops (private good), road density (public good subject to congestion) and average education quality (a purer public good). After dealing with endogeneity issues using an IV approach, we find that the degree of publicness reduces the likelihood of both paramilitary and guerrilla attacks. Moreover, coca production exacerbates conflict and the provision of both public goods mitigates conflict. These results are robust to size, geographical, and welfare controls. Policies that improve public goods provision will help to fight the onset of conflict.

Children' s school study : an experimental theory of verbal information and social attitudes. 'Estudio de la escuela primaria: una teoría experimental de la información verbal y las actitudes sociales'.

Determinar los conceptos diferenciales alcanzados por los alumnos de segunda etapa de EGB con respecto a los alumnos de primera etapa en el area de ciencias sociales. Medir mediante una escala de Likertel cambio de actitudes producido como consecuencia de los aprendizajes en el area de humanidades.. La muestra aleatoria la componían 284 alumnos (145 chicos y 139 chicas) de 5õ y 8õ de EGB y 3õ de BUP, alumnos en centros de un centro urbano (Cáceres) y 3 núcleos rurales de la misma provincia.. La investigación se divide en dos bloques, el primero de ellos, de corte más teórico, estudia la cibernética y la teoría de la información como ciencias y su aplicación en el campo de la psicopedagogía. Se determinan también los esquemas generales para la construcción de escalas de medida de actitudes, y se trata de determinar experimentalmente la existencia de diferencias significativas en el caudal lingüístico entre alumnos procedentes de un medio rural y los procedentes de un medio urbano. El segundo bloque, de corte experimental, analiza el concepto de escala usado en ciencias sociales, y se trata de determinar las actitudes que representan un mayor grado de madurez en la dimensión socio-política. Para ello se aplica a los sujetos de la muestra la Social-responsability Scale.. Escala Likert. Quessing-game method (modificado). Social Responsability Scale de Berkowitz y Lutzeman.. Análisis de Varianza. Análisis de correlación. En terminos de transinsformación didáctica puede considerarse que durante la segunda etapa de EGB hay adquisición de conocimientos que suponen un substancial enriquecimiento del caudal lingüístico. El ambiente posibilita y condiciona el enriquecimiento o desarrollo de las facultades individuales. Pueden establecerse correlaciones parciales entre la adquisión de conceptos y la evolución de actitudes..

Guidelines for strategic national planning of training and education for museums, cultural heritage and related fields in Sweden and in international co-operation

The guidelines presented in this document are a preliminary strategy for establishing a comprehensive policy for the needs of training and education wiyhin the sector and adjoining areas, across fields of knowledge and professions concerned, on relevant levels and for the varies institutions and operators. The objective of these guidelines is to analysis the problems, objectives and goals for development of a far reaching system of educational and training programs and courses for museums, cultural heritage and related fields of activities. This objective comprises a close collaboration between museum, cultural heritage organizations and educating organizations, notably within universities and colleges, but also other kinds of educating bodies.