Dispersive wave solutions of the klein-gordon equation in cosmology

L'equazione di Klein-Gordon descrive una ampia varietà di fenomeni fisici come la propagazione delle onde in Meccanica dei Continui ed il comportamento delle particelle spinless in Meccanica Quantistica Relativistica. Recentemente, la forma dissipativa di questa equazione si è rivelata essere una legge di evoluzione fondamentale in alcuni modelli cosmologici, in particolare nell'ambito dei cosiddetti modelli di k-inflazione in presenza di campi tachionici. L'obiettivo di questo lavoro consiste nell'analizzare gli effetti del parametro dissipativo sulla dispersione nelle soluzioni dell'equazione d'onda. Saranno inoltre studiati alcuni tipici problemi al contorno di particolare interesse cosmologico per mezzo di grafici corrispondenti alle soluzioni fondamentali (Funzioni di Green).

Master Equation: Biological Applications and Thermodynamic Description

It is well known that many realistic mathematical models of biological systems, such as cell growth, cellular development and differentiation, gene expression, gene regulatory networks, enzyme cascades, synaptic plasticity, aging and population growth need to include stochasticity. These systems are not isolated, but rather subject to intrinsic and extrinsic fluctuations, which leads to a quasi equilibrium state (homeostasis). The natural framework is provided by Markov processes and the Master equation (ME) describes the temporal evolution of the probability of each state, specified by the number of units of each species. The ME is a relevant tool for modeling realistic biological systems and allow also to explore the behavior of open systems. These systems may exhibit not only the classical thermodynamic equilibrium states but also the nonequilibrium steady states (NESS). This thesis deals with biological problems that can be treat with the Master equation and also with its thermodynamic consequences. It is organized into six chapters with four new scientific works, which are grouped in two parts: (1) Biological applications of the Master equation: deals with the stochastic properties of a toggle switch, involving a protein compound and a miRNA cluster, known to control the eukaryotic cell cycle and possibly involved in oncogenesis and with the propose of a one parameter family of master equations for the evolution of a population having the logistic equation as mean field limit. (2) Nonequilibrium thermodynamics in terms of the Master equation: where we study the dynamical role of chemical fluxes that characterize the NESS of a chemical network and we propose a one parameter parametrization of BCM learning, that was originally proposed to describe plasticity processes, to study the differences between systems in DB and NESS.

Numerical modelling of fluid-structure interaction with application in hemodynamics

The thesis deals with numerical algorithms for fluid-structure interaction problems with application in blood flow modelling. It starts with a short introduction on the mathematical description of incompressible viscous flow with non-Newtonian viscosity and a moving linear viscoelastic structure. The mathematical model consists of the generalized Navier-Stokes equation used for the description of fluid flow and the generalized string model for structure movement. The arbitrary Lagrangian-Eulerian approach is used in order to take into account moving computational domain. A part of the thesis is devoted to the discussion on the non-Newtonian behaviour of shear-thinning fluids, which is in our case blood, and derivation of two non-Newtonian models frequently used in the blood flow modelling. Further we give a brief overview on recent fluid-structure interaction schemes with discussion about the difficulties arising in numerical modelling of blood flow. Our main contribution lies in numerical and experimental study of a new loosely-coupled partitioned scheme called the kinematic splitting fluid-structure interaction algorithm. We present stability analysis for a coupled problem of non-Newtonian shear-dependent fluids in moving domains with viscoelastic boundaries. Here, we assume both, the nonlinearity in convective as well is diffusive term. We analyse the convergence of proposed numerical scheme for a simplified fluid model of the Oseen type. Moreover, we present series of experiments including numerical error analysis, comparison of hemodynamic parameters for the Newtonian and non-Newtonian fluids and comparison of several physiologically relevant computational geometries in terms of wall displacement and wall shear stress. Numerical analysis and extensive experimental study for several standard geometries confirm reliability and accuracy of the proposed kinematic splitting scheme in order to approximate fluid-structure interaction problems.

Monodromy calculations for some differential equations

In vielen Teilgebieten der Mathematik ist es w"{u}nschenswert, die Monodromiegruppe einer homogenen linearen Differenzialgleichung zu verstehen. Es sind nur wenige analytische Methoden zur Berechnung dieser Gruppe bekannt, daher entwickeln wir im ersten Teil dieser Arbeit eine numerische Methode zur Approximation ihrer Erzeuger.rnIm zweiten Abschnitt fassen wir die Grundlagen der Theorie der Uniformisierung Riemannscher Fl"achen und die der arithmetischen Fuchsschen Gruppen zusammen. Auss erdem erkl"aren wir, wie unsere numerische Methode bei der Bestimmung von uniformisierenden Differenzialgleichungen dienlich sein kann. F"ur arithmetische Fuchssche Gruppen mit zwei Erzeugern erhalten wir lokale Daten und freie Parameter von Lam'{e} Gleichungen, welche die zugeh"origen Riemannschen Fl"achen uniformisieren. rnIm dritten Teil geben wir einen kurzen Abriss zur homologischen Spiegelsymmetrie und f"uhren die $widehat{Gamma}$-Klasse ein. Wir erkl"aren wie diese genutzt werden kann, um eine Hodge-theoretische Version der Spiegelsymmetrie f"ur torische Varit"aten zu beweisen. Daraus gewinnen wir Vermutungen "uber die Monodromiegruppe $M$ von Picard-Fuchs Gleichungen von gewissen Familien $f:mathcal{X}rightarrow bbp^1$ von $n$-dimensionalen Calabi-Yau Variet"aten. Diese besagen erstens, dass bez"uglich einer nat"urlichen Basis die Monodromiematrizen in $M$ Eintr"age aus dem K"orper $bbq(zeta(2j+1)/(2 pi i)^{2j+1},j=1,ldots,lfloor (n-1)/2 rfloor)$ haben. Und zweitens, dass sich topologische Invarianten des Spiegelpartners einer generischen Faser von $f:mathcal{X}rightarrow bbp^1$ aus einem speziellen Element von $M$ rekonstruieren lassen. Schliess lich benutzen wir die im ersten Teil entwickelten Methoden zur Verifizierung dieser Vermutungen, vornehmlich in Hinblick auf Dimension drei. Dar"uber hinaus erstellen wir eine Liste von Kandidaten topologischer Invarianten von vermutlich existierenden dreidimensionalen Calabi-Yau Variet"aten mit $h^{1,1}=1$.

Analytische Ableitungen der Energie für den "Equation-of-Motion-Coupled-Cluster"-Ansatz zur Berechnung von Ionisationspotentialen

In der vorliegenden Arbeit wird die Theorie der analytischen zweiten Ableitungen für die EOMIP-CCSD-Methode formuliert sowie die durchgeführte Implementierung im Quantenchemieprogramm CFOUR beschrieben. Diese Ableitungen sind von Bedeutung bei der Bestimmung statischer Polarisierbarkeiten und harmonischer Schwingungsfrequenzen und in dieser Arbeit wird die Genauigkeit des EOMIP-CCSD-Ansatzes bei der Berechnung dieser Eigenschaften für verschiedene radikalische Systeme untersucht. Des Weiteren können mit Hilfe der ersten und zweiten Ableitungen vibronische Kopplungsparameter berechnet werden, welche zur Simulation von Molekülspektren in Kombination mit dem Köppel-Domcke-Cederbaum (KDC)-Modell - in der Arbeit am Beispiel des Formyloxyl (HCO2)-Radikals demonstriert - benötigt werden.rnrnDer konzeptionell einfache EOMIP-CC-Ansatz wurde gewählt, da hier die Wellenfunktion eines Radikalsystems ausgehend von einem stabilen geschlossenschaligen Zustand durch die Entfernung eines Elektrons gebildet wird und somit die Problematik der Symmetriebrechung umgangen werden kann. Im Rahmen der Implementierung wurden neue Programmteile zur Lösung der erforderlichen Gleichungen für die gestörten EOMIP-CC-Amplituden und die gestörten Lagrange-Multiplikatoren zeta zum Quantenchemieprogramm CFOUR hinzugefügt. Die unter Verwendung des Programms bestimmten Eigenschaften werden hinsichtlich ihrer Leistungsfähigkeit im Vergleich zu etablierten Methoden wie z.B. CCSD(T) untersucht. Bei der Berechnung von Polarisierbarkeiten und harmonischen Schwingungsfrequenzen liefert die EOMIP-CCSD-Theorie meist gute Resultate, welche nur wenig von den CCSD(T)-Ergebnissen abweichen. Einzig bei der Betrachtung von Radikalen, für die die entsprechenden Anionen nicht stabil sind (z.B. NH2⁻ und CH3⁻), liefert der EOMIP-CCSD-Ansatz aufgrund methodischer Nachteile keine aussagekräftige Beschreibung. rnrnDie Ableitungen der EOMIP-CCSD-Energie lassen sich auch zur Simulation vibronischer Kopplungen innerhalb des KDC-Modells einsetzen.rnZur Kopplung verschiedener radikalischer Zustände in einem solchen Modellpotential spielen vor allem die Ableitungen von Übergangsmatrixelementen eine wichtige Rolle. Diese sogenannten Kopplungskonstanten können in der EOMIP-CC-Theorie besonders leicht definiert und berechnet werden. Bei der Betrachtung des Photoelektronenspektrums von HCO2⁻ werden zwei Alternativen untersucht: Die vertikale Bestimmung an der Gleichgewichtsgeometrie des HCO2⁻-Anions und die Ermittlung adiabatischer Kraftkonstanten an den Gleichgewichtsgeometrien des Radikals. Lediglich das adiabatische Modell liefert bei Beschränkung auf harmonische Kraftkonstanten eine qualitativ sinnvolle Beschreibung des Spektrums. Erweitert man beide Modelle um kubische und quartische Kraftkonstanten, so nähern sich diese einander an und ermöglichen eine vollständige Zuordnung des gemessenen Spektrums innerhalb der ersten 1500 cm⁻¹. Die adiabatische Darstellung erreicht dabei nahezu quantitative Genauigkeit.

The Complex Monge-Ampère Equation

In questa trattazione si studia la regolarità delle soluzioni viscose plurisubarmoniche dell’equazione di Monge-Ampère complessa. Si tratta di un’equazione alle derivate parziali del secondo ordine completamente non lineare il cui termine del secondo ordine è il determinante della matrice hessiana complessa di una funzione incognita a valori reali u. Il principale risultato della tesi è un nuovo controesempio di tipo Pogorelov per questa equazione. Si prova cioè l’esistenza di soluzioni viscose plurisubarmoniche e non classiche per un equazione di Monge-Ampère complessa.

ABCtoolbox: a versatile toolkit for approximate Bayesian computations

Background The estimation of demographic parameters from genetic data often requires the computation of likelihoods. However, the likelihood function is computationally intractable for many realistic evolutionary models, and the use of Bayesian inference has therefore been limited to very simple models. The situation changed recently with the advent of Approximate Bayesian Computation (ABC) algorithms allowing one to obtain parameter posterior distributions based on simulations not requiring likelihood computations. Results Here we present ABCtoolbox, a series of open source programs to perform Approximate Bayesian Computations (ABC). It implements various ABC algorithms including rejection sampling, MCMC without likelihood, a Particle-based sampler and ABC-GLM. ABCtoolbox is bundled with, but not limited to, a program that allows parameter inference in a population genetics context and the simultaneous use of different types of markers with different ploidy levels. In addition, ABCtoolbox can also interact with most simulation and summary statistics computation programs. The usability of the ABCtoolbox is demonstrated by inferring the evolutionary history of two evolutionary lineages of Microtus arvalis. Using nuclear microsatellites and mitochondrial sequence data in the same estimation procedure enabled us to infer sex-specific population sizes and migration rates and to find that males show smaller population sizes but much higher levels of migration than females. Conclusion ABCtoolbox allows a user to perform all the necessary steps of a full ABC analysis, from parameter sampling from prior distributions, data simulations, computation of summary statistics, estimation of posterior distributions, model choice, validation of the estimation procedure, and visualization of the results.

Social cognition as a mediator variable between neurocognition and functional outcome in schizophrenia: empirical review and new results by structural equation modeling

Cognitive impairments are currently regarded as important determinants of functional domains and are promising treatment goals in schizophrenia. Nevertheless, the exact nature of the interdependent relationship between neurocognition and social cognition as well as the relative contribution of each of these factors to adequate functioning remains unclear. The purpose of this article is to systematically review the findings and methodology of studies that have investigated social cognition as a mediator variable between neurocognitive performance and functional outcome in schizophrenia. Moreover, we carried out a study to evaluate this mediation hypothesis by the means of structural equation modeling in a large sample of 148 schizophrenia patients. The review comprised 15 studies. All but one study provided evidence for the mediating role of social cognition both in cross-sectional and in longitudinal designs. Other variables like motivation and social competence additionally mediated the relationship between social cognition and functional outcome. The mean effect size of the indirect effect was 0.20. However, social cognitive domains were differentially effective mediators. On average, 25% of the variance in functional outcome could be explained in the mediation model. The results of our own statistical analysis are in line with these conclusions: Social cognition mediated a significant indirect relationship between neurocognition and functional outcome. These results suggest that research should focus on differential mediation pathways. Future studies should also consider the interaction with other prognostic factors, additional mediators, and moderators in order to increase the predictive power and to target those factors relevant for optimizing therapy effects.

Modeling Yield Surfaces of Various Structural Shapes

This study investigates the possibility of custom fitting a widely accepted approximate yield surface equation (Ziemian, 2000) to the theoretical yield surfaces of five different structural shapes, which include wide-flange, solid and hollow rectangular, and solid and hollow circular shapes. To achieve this goal, a theoretically “exact” but overly complex representation of the cross section’s yield surface was initially obtained by using fundamental principles of solid mechanics. A weighted regression analysis was performed with the “exact” yield surface data to obtain the specific coefficients of three terms in the approximate yield surface equation. These coefficients were calculated to determine the “best” yield surface equation for a given cross section geometry. Given that the exact yield surface shall have zero percentage of concavity, this investigation evaluated the resulting coefficient of determination (