Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Synthese, Evaluierung und 18 F-Markierung von neuen Liganden zur Visualisierung der Strychnin-insensitiven Bindungsstelle des NMDA-Rezeptors

Glutamat ist der wichtigste exzitatorische Neurotransmitter im Gehirn. Folglich spielen Glutamat-kontrollierte Rezeptorsysteme eine entscheidende Rolle in neurologischen Vorgängen, wie beispielsweise in Lern- und Gedächtnisprozessen. Gerade der NMDA-Rezeptor ist in eine Vielzahl solcher Vorgänge involviert und wird vor allem mit neurodegenerativen Erkrankungen wie Chorea Huntington, Morbus Alzheimer, Morbus Parkinson und zerebraler Ischämie in Verbindung gebracht. Folglich stellt die Visualisierung des NMDA-Rezeptorstatus eine Möglichkeit dar, den Verlauf solcher Prozesse zu untersuchen.rnDie Positronen-Emissions-Tomographie (PET) ist eine leistungsstarke Anwendung in der molekularen Bildgebung und erlaubt die in vivo-Visualisierung sowie Quantifizierung biochemischer Prozesse. Durch die Verwendung geeigneter Tracer können bestimmte pathologische und neurologische Abläufe beurteilt werden. rnZurzeit sind keine geeigneten PET-Tracer zur Untersuchung des NMDA-Rezeptors verfügbar. Bisher dargestellte PET-Liganden zeichneten sich durch nicht zufriedenstellende Affinitäten und Selektivitäten aus und führten meist auf Grund der hohen Lipophilie zu einem hohen Maß an unspezifischer Bindung. rnDie Strychnin-insensitive Glycinbindungsstelle des NMDA-Rezeptors stellt ein vielversprechendes Target dar, spezifische Liganden für diese Bindungsstelle zu synthetisieren. Hier zeichnen sich einige Verbindungsklassen durch exzellente Affinitäten und Selektivitäten sowie durch vielversprechende in vivo-Eigenschaften aus. rnAuf Grundlage dieser biologischen Daten wurden zwei Substanzen der 2-Indolcarbonsäure, nämlich die 4,6-Dichlor-3-(2-oxo-3-phenylimidazolidin-1-ylmethyl)-1H-indol-2-carbonsäure (MDJ-114) und die (E)-4,6-Dichlor-3-(2-phenylcarbamoylvinyl)-1H-indol-2-carbonsäure (GV150526), als Leitstruktur gewählt. Ferner wurde das 7-Chlor-4-hydroxy-3-(3-phenoxyphenyl)-1H-chinolin-2-on (L-701,324) aus der Substanzklasse der 4-Hydroxy-1H-chinolin-2-one als dritte Leitstruktur gewählt.rnFür diese Substanzen wurden 19F-markierte Analogverbindungen synthetisiert, um als inaktive Referenzverbindungen auf ihre Eignung überprüft zu werden. Hierzu wurde eine Fluorethoxygruppierung im terminalen Phenylring der entsprechenden Leitstruktur eingeführt. Durch Variation der Fluorethoxysubstitution in ortho-, meta- und para-Stellung, konnten die besten Affinitäten in einem kompetitiven Rezeptorbindungsassay durch Verdrängung von [3H]MDL-105,519 bestimmt werden. Als Maß für die Lipophilie wurden die entsprechenden log D-Werte über die HPLC-Methode bestimmt. Basierend auf den Ergebnissen der Evaluierung wurden zwei Derivate identifiziert, welche zur 18F-Markierung genutzt werden sollten (GV150526-Derivat 34: log D = 0,23 ± 0,03, IC50 = 0,20 ± 0,25 µM, Ki = 0,13 ± 0,16 µM; L701,324-Derivat 55: log D = - 0,25 ± 0,01, IC50 = 78 ± 37 µM, Ki = 51 ± 24 µM). Die 18F-Markierung erfolgte durch die Reaktion des entsprechenden Markierungsvorläufers mit dem Markierungssynthon 2-[18F]Fluorethyltosylat, welches durch die Umsetzung von Ethylenditosylat mit [18F]Fluorid hergestellt wurde. Die Radiosynthesen der beiden 18F-markierten Verbindungen [18F]34 (4,6-Dichlor-3-{2-[4-(2-[18F]fluorethoxy)-phenylcarbamoyl]-vinyl}-1H-indol-2-carbonsäure) und [18F]55 (7-Chlor-3-{3-[4-(2-[18F]fluorethoxy)-phenoxy]-phenyl}-4-hydroxy-1H-chinolin-2-on) wurden optimiert sowie semipräparative Abtrennverfahren entwickelt. Beide Tracer wurden auf ihre in vivo-Eignung im µPET-Experiment untersucht. Die Zeitaktivitätskurven lassen erkennen, dass beide Tracer entgegen der Erwartung nicht die Blut-Hirn-Schranke überwinden können. Für das GV150526-Derivat ([18F]34) wurden zusätzlich Autoradiographiestudien durchgeführt. Die erhaltenen Aufnahmen zeigten ein heterogenes Verteilungsmuster der Aktivitätsanreicherung. Ebenso wurde ein hohes Maß an unspezifischer Bindung beobachtet. Möglicherweise sind Cross-Affinitäten zu anderen Rezeptorsystemen oder der recht hohe lipophile Rest des Moleküls hierfür verantwortlich. Ein Grund für die unzureichende Hirngängigkeit der Radioliganden kann sich in der Carboxylatfunktion des GV150526-Derivats bzw. in der 4-Hydroxy-1H-chinolin-2-on-Einheit des L-701,324-Derivats wiederspiegeln. rnAuf Grundlage dieser Resultate können Versuche unternommen werden, für die Verbindungsklasse der 2-Indolcarbonsäuren entsprechende Ester als Prodrugs mit einer verbesserten Bioverfügbarkeit darzustellen. Ebenso können neue Strukturen als Grundlage für neue PET-Tracer untersucht werden.rnrn

Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views

We investigate the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. We then use the affine invariant to derive new algebraic connections between perspective views. It is shown that three perspective views of an object are connected by certain algebraic functions of image coordinates alone (no structure or camera geometry needs to be involved).

Affine processes, arbitrage-Free Term structures of legendre polynomials,and option pricing

Multivariate Affine term structure models have been increasingly used for pricing derivatives in fixed income markets. In these models, uncertainty of the term structure is driven by a state vector, while the short rate is an affine function of this vector. The model is characterized by a specific form for the stochastic differential equation (SDE) for the evolution of the state vector. This SDE presents restrictions on its drift term which rule out arbitrages in the market. In this paper we solve the following inverse problem: Suppose the term structure of interest rates is modeled by a linear combination of Legendre polynomials with random coefficients. Is there any SDE for these coefficients which rules out arbitrages? This problem is of particular empirical interest because the Legendre model is an example of factor model with clear interpretation for each factor, in which regards movements of the term structure. Moreover, the Affine structure of the Legendre model implies knowledge of its conditional characteristic function. From the econometric perspective, we propose arbitrage-free Legendre models to describe the evolution of the term structure. From the pricing perspective, we follow Duffie et al. (2000) in exploring Legendre conditional characteristic functions to obtain a computational tractable method to price fixed income derivatives. Closing the article, the empirical section presents precise evidence on the reward of implementing arbitrage-free parametric term structure models: The ability of obtaining a good approximation for the state vector by simply using cross sectional data.

On Affine Connections in a Riemannian Manifold with a Circulant Metric and two Circulant Affinor Structures

Ива Р. Докузова, Димитър Р. Разпопов - В настоящата статия е разгледан клас V оттримерни риманови многообразия M с метрика g и два афинорни тензора q и S. Дефинирана е и друга метрика ¯g в M. Локалните координати на всички тези тензори са циркулантни матрици. Намерени са: 1) зависимост между тензора на кривина R породен от g и тензора на кривина ¯R породен от ¯g; 2) тъждество за тензора на кривина R в случая, когато тензорът на кривина ¯R се анулира; 3) зависимост между секционната кривина на прозволна двумерна q-площадка {x, qx} и скаларната кривина на M.

Tannakian Formalism Applied to the Category of Mixed Hodge Structures

Many problems in analysis have been solved using the theory of Hodge structures. P. Deligne started to treat these structures in a categorical way. Following him, we introduce the categories of mixed real and complex Hodge structures. Category of mixed Hodge structures over the field of real or complex numbers is a rigid abelian tensor category, and in fact, a neutral Tannakian category. Therefore it is equivalent to the category of representations of an affine group scheme. The direct sums of pure Hodge structures of different weights over real or complex numbers can be realized as a representation of the torus group, whose complex points is the Cartesian product of two punctured complex planes. Mixed Hodge structures turn out to consist of information of a direct sum of pure Hodge structures of different weights and a nilpotent automorphism. Therefore mixed Hodge structures correspond to the representations of certain semidirect product of a nilpotent group and the torus group acting on it.

Extraction of fibre network architecture by X-ray tomography and prediction of elastic properties using an affine analytical model

This paper proposes a method for extracting reliable architectural characteristics from complex porous structures using micro-computed tomography (μCT) images. The work focuses on a highly porous material composed of a network of fibres bonded together. The segmentation process, allowing separation of the fibres from the remainder of the image, is the most critical step in constructing an accurate representation of the network architecture. Segmentation methods, based on local and global thresholding, were investigated and evaluated by a quantitative comparison of the architectural parameters they yielded, such as the fibre orientation and segment length (sections between joints) distributions and the number of inter-fibre crossings. To improve segmentation accuracy, a deconvolution algorithm was proposed to restore the original images. The efficacy of the proposed method was verified by comparing μCT network architectural characteristics with those obtained using high resolution CT scans (nanoCT). The results indicate that this approach resolves the architecture of these complex networks and produces results approaching the quality of nanoCT scans. The extracted architectural parameters were used in conjunction with an affine analytical model to predict the axial and transverse stiffnesses of the fibre network. Transverse stiffness predictions were compared with experimentally measured values obtained by vibration testing. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

An existence theorem for G-structure preserving affine immersions

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by the main result in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149-181], and with such a submanifold M and a point x in M we associate a canonical homogeneous structure I" (x) (a certain bilinear map defined on a subspace of T (x) M x T (x) M). We prove that I" (x) , together with the second fundamental form alpha (x) , encodes all the information about M, and we deduce from this the rigidity result that M is completely determined by alpha (x) and (Delta alpha) (x) , thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149-181] to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of I" (this result also seems to close a gap in [U. Christ, J. Differential Geom., 62 (2002), 1-15]). Here an important tool is the introduction of affine root systems of isoparametric submanifolds.

Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

Automated wordlength optimization framework for multi-source statistical interval-based analysis of nonlinear systems with control-flow structures

El uso de aritmética de punto fijo es una opción de diseño muy extendida en sistemas con fuertes restricciones de área, consumo o rendimiento. Para producir implementaciones donde los costes se minimicen sin impactar negativamente en la precisión de los resultados debemos llevar a cabo una asignación cuidadosa de anchuras de palabra. Encontrar la combinación óptima de anchuras de palabra en coma fija para un sistema dado es un problema combinatorio NP-hard al que los diseñadores dedican entre el 25 y el 50 % del ciclo de diseño. Las plataformas hardware reconfigurables, como son las FPGAs, también se benefician de las ventajas que ofrece la aritmética de coma fija, ya que éstas compensan las frecuencias de reloj más bajas y el uso más ineficiente del hardware que hacen estas plataformas respecto a los ASICs. A medida que las FPGAs se popularizan para su uso en computación científica los diseños aumentan de tamaño y complejidad hasta llegar al punto en que no pueden ser manejados eficientemente por las técnicas actuales de modelado de señal y ruido de cuantificación y de optimización de anchura de palabra. En esta Tesis Doctoral exploramos distintos aspectos del problema de la cuantificación y presentamos nuevas metodologías para cada uno de ellos: Las técnicas basadas en extensiones de intervalos han permitido obtener modelos de propagación de señal y ruido de cuantificación muy precisos en sistemas con operaciones no lineales. Nosotros llevamos esta aproximación un paso más allá introduciendo elementos de Multi-Element Generalized Polynomial Chaos (ME-gPC) y combinándolos con una técnica moderna basada en Modified Affine Arithmetic (MAA) estadístico para así modelar sistemas que contienen estructuras de control de flujo. Nuestra metodología genera los distintos caminos de ejecución automáticamente, determina las regiones del dominio de entrada que ejercitarán cada uno de ellos y extrae los momentos estadísticos del sistema a partir de dichas soluciones parciales. Utilizamos esta técnica para estimar tanto el rango dinámico como el ruido de redondeo en sistemas con las ya mencionadas estructuras de control de flujo y mostramos la precisión de nuestra aproximación, que en determinados casos de uso con operadores no lineales llega a tener tan solo una desviación del 0.04% con respecto a los valores de referencia obtenidos mediante simulación. Un inconveniente conocido de las técnicas basadas en extensiones de intervalos es la explosión combinacional de términos a medida que el tamaño de los sistemas a estudiar crece, lo cual conlleva problemas de escalabilidad. Para afrontar este problema presen tamos una técnica de inyección de ruidos agrupados que hace grupos con las señales del sistema, introduce las fuentes de ruido para cada uno de los grupos por separado y finalmente combina los resultados de cada uno de ellos. De esta forma, el número de fuentes de ruido queda controlado en cada momento y, debido a ello, la explosión combinatoria se minimiza. También presentamos un algoritmo de particionado multi-vía destinado a minimizar la desviación de los resultados a causa de la pérdida de correlación entre términos de ruido con el objetivo de mantener los resultados tan precisos como sea posible. La presente Tesis Doctoral también aborda el desarrollo de metodologías de optimización de anchura de palabra basadas en simulaciones de Monte-Cario que se ejecuten en tiempos razonables. Para ello presentamos dos nuevas técnicas que exploran la reducción del tiempo de ejecución desde distintos ángulos: En primer lugar, el método interpolativo aplica un interpolador sencillo pero preciso para estimar la sensibilidad de cada señal, y que es usado después durante la etapa de optimización. En segundo lugar, el método incremental gira en torno al hecho de que, aunque es estrictamente necesario mantener un intervalo de confianza dado para los resultados finales de nuestra búsqueda, podemos emplear niveles de confianza más relajados, lo cual deriva en un menor número de pruebas por simulación, en las etapas iniciales de la búsqueda, cuando todavía estamos lejos de las soluciones optimizadas. Mediante estas dos aproximaciones demostramos que podemos acelerar el tiempo de ejecución de los algoritmos clásicos de búsqueda voraz en factores de hasta x240 para problemas de tamaño pequeño/mediano. Finalmente, este libro presenta HOPLITE, una infraestructura de cuantificación automatizada, flexible y modular que incluye la implementación de las técnicas anteriores y se proporciona de forma pública. Su objetivo es ofrecer a desabolladores e investigadores un entorno común para prototipar y verificar nuevas metodologías de cuantificación de forma sencilla. Describimos el flujo de trabajo, justificamos las decisiones de diseño tomadas, explicamos su API pública y hacemos una demostración paso a paso de su funcionamiento. Además mostramos, a través de un ejemplo sencillo, la forma en que conectar nuevas extensiones a la herramienta con las interfaces ya existentes para poder así expandir y mejorar las capacidades de HOPLITE. ABSTRACT Using fixed-point arithmetic is one of the most common design choices for systems where area, power or throughput are heavily constrained. In order to produce implementations where the cost is minimized without negatively impacting the accuracy of the results, a careful assignment of word-lengths is required. The problem of finding the optimal combination of fixed-point word-lengths for a given system is a combinatorial NP-hard problem to which developers devote between 25 and 50% of the design-cycle time. Reconfigurable hardware platforms such as FPGAs also benefit of the advantages of fixed-point arithmetic, as it compensates for the slower clock frequencies and less efficient area utilization of the hardware platform with respect to ASICs. As FPGAs become commonly used for scientific computation, designs constantly grow larger and more complex, up to the point where they cannot be handled efficiently by current signal and quantization noise modelling and word-length optimization methodologies. In this Ph.D. Thesis we explore different aspects of the quantization problem and we present new methodologies for each of them: The techniques based on extensions of intervals have allowed to obtain accurate models of the signal and quantization noise propagation in systems with non-linear operations. We take this approach a step further by introducing elements of MultiElement Generalized Polynomial Chaos (ME-gPC) and combining them with an stateof- the-art Statistical Modified Affine Arithmetic (MAA) based methodology in order to model systems that contain control-flow structures. Our methodology produces the different execution paths automatically, determines the regions of the input domain that will exercise them, and extracts the system statistical moments from the partial results. We use this technique to estimate both the dynamic range and the round-off noise in systems with the aforementioned control-flow structures. We show the good accuracy of our approach, which in some case studies with non-linear operators shows a 0.04 % deviation respect to the simulation-based reference values. A known drawback of the techniques based on extensions of intervals is the combinatorial explosion of terms as the size of the targeted systems grows, which leads to scalability problems. To address this issue we present a clustered noise injection technique that groups the signals in the system, introduces the noise terms in each group independently and then combines the results at the end. In this way, the number of noise sources in the system at a given time is controlled and, because of this, the combinato rial explosion is minimized. We also present a multi-way partitioning algorithm aimed at minimizing the deviation of the results due to the loss of correlation between noise terms, in order to keep the results as accurate as possible. This Ph.D. Thesis also covers the development of methodologies for word-length optimization based on Monte-Carlo simulations in reasonable times. We do so by presenting two novel techniques that explore the reduction of the execution times approaching the problem in two different ways: First, the interpolative method applies a simple but precise interpolator to estimate the sensitivity of each signal, which is later used to guide the optimization effort. Second, the incremental method revolves on the fact that, although we strictly need to guarantee a certain confidence level in the simulations for the final results of the optimization process, we can do it with more relaxed levels, which in turn implies using a considerably smaller amount of samples, in the initial stages of the process, when we are still far from the optimized solution. Through these two approaches we demonstrate that the execution time of classical greedy techniques can be accelerated by factors of up to ×240 for small/medium sized problems. Finally, this book introduces HOPLITE, an automated, flexible and modular framework for quantization that includes the implementation of the previous techniques and is provided for public access. The aim is to offer a common ground for developers and researches for prototyping and verifying new techniques for system modelling and word-length optimization easily. We describe its work flow, justifying the taken design decisions, explain its public API and we do a step-by-step demonstration of its execution. We also show, through an example, the way new extensions to the flow should be connected to the existing interfaces in order to expand and improve the capabilities of HOPLITE.