Soluções quase automórficas para equações diferenciais abstratas de segunda ordem

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Theoretical study of the absorption and nonradiative deactivation of 1-nitronaphthalene in the low-lying singlet and triplet excited states including methanol and ethanol solvent effects

The photophysics of the 1-nitronaphthalene molecular system, after the absorption transition to the first singlet excited state, is theoretically studied for investigating the ultrafast multiplicity change to the triplet manifold. The consecutive transient absorption spectra experimentally observed in this molecular system are also studied. To identify the electronic states involved in the nonradiative decay, the minimum energy path of the first singlet excited state is obtained using the complete active space self-consistent field//configurational second-order perturbation approach. A near degeneracy region was found between the first singlet and the second triplet excited states with large spin-orbit coupling between them. The intersystem crossing rate was also evaluated. To support the proposed deactivation model the transient absorption spectra observed in the experiments were also considered. For this, computer simulations using sequential quantum mechanic-molecular mechanic methodology was used to consider the solvent effect in the ground and excited states for proper comparison with the experimental results. The absorption transitions from the second triplet excited state in the relaxed geometry permit to describe the transient absorption band experimentally observed around 200 fs after the absorption transition. This indicates that the T-2 electronic state is populated through the intersystem crossing presented here. The two transient absorption bands experimentally observed between 2 and 45 ps after the absorption transition are described here as the T-1 -> T-3 and T-1 -> T-5 transitions, supporting that the intermediate triplet state (T-2) decays by internal conversion to T-1. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4738757]

Dissociative electron attachment to chloromethane.

A low energy electron may attach to a molecule, forming a metastable resonance, which may dissociate into a stable anion and a neutral radical. Chloromethane has been a good target for dissociative electron attachment studies, since it is a small molecule with a clear dissociative ‘sigma*’ shape resonance. We present potential energy curves for CH3Cl and its anion, as a function of the C-Cl distance. Due to the resonant nature of the anion, a correct description requires a treatment based on scattering calculations. In order to compute elastic cross sections and phase shifts we employed the Schwinger multichannel method, implemented with pseudopotentials of Bachelet, Hamann and Schlüter, at the static-exchange plus polarization approximation. At the equilibrium geometry, the resonance was found arround 3.3 eV, in accordance to experience. The incoming electron is captured by a ‘sigma*’ orbital located at the C-Cl bond, which will relax in the presence of this extra electron. We took this bond as the reaction coordinate, and performed several scattering calculations for a series of nuclear conformations. The phase shift obtained in each calculation was fitted by a two component function, consisting in the usual Breit-Wigner profile, which captures the resonant character, and a second order polynomial in the wave number, which accounts for the background contribution. That way, we obtained position and width of the resonance, which allowed us to build the potential energy curve. For larger distances, the anion becomes stable and usual electronic structure calculations suffice. Furthermore, the existence of a dipole-bound anion state is revealed when we employed a set of very diffuse functions. The knowledge on the behaviour of the neutral and anionic electronic states helps us in elucidating how the dissociation takes place.

Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

A new insight in stress recovery tecniques and hessian reconstruction methods

Stress recovery techniques have been an active research topic in the last few years since, in 1987, Zienkiewicz and Zhu proposed a procedure called Superconvergent Patch Recovery (SPR). This procedure is a last-squares fit of stresses at super-convergent points over patches of elements and it leads to enhanced stress fields that can be used for evaluating finite element discretization errors. In subsequent years, numerous improved forms of this procedure have been proposed attempting to add equilibrium constraints to improve its performances. Later, another superconvergent technique, called Recovery by Equilibrium in Patches (REP), has been proposed. In this case the idea is to impose equilibrium in a weak form over patches and solve the resultant equations by a last-square scheme. In recent years another procedure, based on minimization of complementary energy, called Recovery by Compatibility in Patches (RCP) has been proposed in. This procedure, in many ways, can be seen as the dual form of REP as it substantially imposes compatibility in a weak form among a set of self-equilibrated stress fields. In this thesis a new insight in RCP is presented and the procedure is improved aiming at obtaining convergent second order derivatives of the stress resultants. In order to achieve this result, two different strategies and their combination have been tested. The first one is to consider larger patches in the spirit of what proposed in [4] and the second one is to perform a second recovery on the recovered stresses. Some numerical tests in plane stress conditions are presented, showing the effectiveness of these procedures. Afterwards, a new recovery technique called Last Square Displacements (LSD) is introduced. This new procedure is based on last square interpolation of nodal displacements resulting from the finite element solution. In fact, it has been observed that the major part of the error affecting stress resultants is introduced when shape functions are derived in order to obtain strains components from displacements. This procedure shows to be ultraconvergent and is extremely cost effective, as it needs in input only nodal displacements directly coming from finite element solution, avoiding any other post-processing in order to obtain stress resultants using the traditional method. Numerical tests in plane stress conditions are than presented showing that the procedure is ultraconvergent and leads to convergent first and second order derivatives of stress resultants. In the end, transverse stress profiles reconstruction using First-order Shear Deformation Theory for laminated plates and three dimensional equilibrium equations is presented. It can be seen that accuracy of this reconstruction depends on accuracy of first and second derivatives of stress resultants, which is not guaranteed by most of available low order plate finite elements. RCP and LSD procedures are than used to compute convergent first and second order derivatives of stress resultants ensuring convergence of reconstructed transverse shear and normal stress profiles respectively. Numerical tests are presented and discussed showing the effectiveness of both procedures.

Inference on copula-based correlation structures

We propose an extension of the approach provided by Kluppelberg and Kuhn (2009) for inference on second-order structure moments. As in Kluppelberg and Kuhn (2009) we adopt a copula-based approach instead of assuming normal distribution for the variables, thus relaxing the equality in distribution assumption. A new copula-based estimator for structure moments is investigated. The methodology provided by Kluppelberg and Kuhn (2009) is also extended considering the copulas associated with the family of Eyraud-Farlie-Gumbel-Morgenstern distribution functions (Kotz, Balakrishnan, and Johnson, 2000, Equation 44.73). Finally, a comprehensive simulation study and an application to real financial data are performed in order to compare the different approaches.

Black-scholes type equations

In this work we are concerned with the analysis and numerical solution of Black-Scholes type equations arising in the modeling of incomplete financial markets and an inverse problem of determining the local volatility function in a generalized Black-Scholes model from observed option prices. In the first chapter a fully nonlinear Black-Scholes equation which models transaction costs arising in option pricing is discretized by a new high order compact scheme. The compact scheme is proved to be unconditionally stable and non-oscillatory and is very efficient compared to classical schemes. Moreover, it is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. In the next chapter we turn to the calibration problem of computing local volatility functions from market data in a generalized Black-Scholes setting. We follow an optimal control approach in a Lagrangian framework. We show the existence of a global solution and study first- and second-order optimality conditions. Furthermore, we propose an algorithm that is based on a globalized sequential quadratic programming method and a primal-dual active set strategy, and present numerical results. In the last chapter we consider a quasilinear parabolic equation with quadratic gradient terms, which arises in the modeling of an optimal portfolio in incomplete markets. The existence of weak solutions is shown by considering a sequence of approximate solutions. The main difficulty of the proof is to infer the strong convergence of the sequence. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution, but without additional regularity assumptions on the solution. The results are illustrated by a numerical example.

Large-scale coupled-cluster calculations

Coupled-cluster theory provides one of the most successful concepts in electronic-structure theory. This work covers the parallelization of coupled-cluster energies, gradients, and second derivatives and its application to selected large-scale chemical problems, beside the more practical aspects such as the publication and support of the quantum-chemistry package ACES II MAB and the design and development of a computational environment optimized for coupled-cluster calculations. The main objective of this thesis was to extend the range of applicability of coupled-cluster models to larger molecular systems and their properties and therefore to bring large-scale coupled-cluster calculations into day-to-day routine of computational chemistry. A straightforward strategy for the parallelization of CCSD and CCSD(T) energies, gradients, and second derivatives has been outlined and implemented for closed-shell and open-shell references. Starting from the highly efficient serial implementation of the ACES II MAB computer code an adaptation for affordable workstation clusters has been obtained by parallelizing the most time-consuming steps of the algorithms. Benchmark calculations for systems with up to 1300 basis functions and the presented applications show that the resulting algorithm for energies, gradients and second derivatives at the CCSD and CCSD(T) level of theory exhibits good scaling with the number of processors and substantially extends the range of applicability. Within the framework of the ’High accuracy Extrapolated Ab initio Thermochemistry’ (HEAT) protocols effects of increased basis-set size and higher excitations in the coupled- cluster expansion were investigated. The HEAT scheme was generalized for molecules containing second-row atoms in the case of vinyl chloride. This allowed the different experimental reported values to be discriminated. In the case of the benzene molecule it was shown that even for molecules of this size chemical accuracy can be achieved. Near-quantitative agreement with experiment (about 2 ppm deviation) for the prediction of fluorine-19 nuclear magnetic shielding constants can be achieved by employing the CCSD(T) model together with large basis sets at accurate equilibrium geometries if vibrational averaging and temperature corrections via second-order vibrational perturbation theory are considered. Applying a very similar level of theory for the calculation of the carbon-13 NMR chemical shifts of benzene resulted in quantitative agreement with experimental gas-phase data. The NMR chemical shift study for the bridgehead 1-adamantyl cation at the CCSD(T) level resolved earlier discrepancies of lower-level theoretical treatment. The equilibrium structure of diacetylene has been determined based on the combination of experimental rotational constants of thirteen isotopic species and zero-point vibrational corrections calculated at various quantum-chemical levels. These empirical equilibrium structures agree to within 0.1 pm irrespective of the theoretical level employed. High-level quantum-chemical calculations on the hyperfine structure parameters of the cyanopolyynes were found to be in excellent agreement with experiment. Finally, the theoretically most accurate determination of the molecular equilibrium structure of ferrocene to date is presented.

Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Higher-order perturbative relativistic corrections to energies and properties

Relativistic effects need to be considered in quantum-chemical calculations on systems including heavy elements or when aiming at high accuracy for molecules containing only lighter elements. In the latter case, consideration of relativistic effects via perturbation theory is an attractive option. Among the available techniques, Direct Perturbation Theory (DPT) in its lowest order (DPT2) has become a standard tool for the calculation of relativistic corrections to energies and properties.In this work, the DPT treatment is extended to the next order (DPT4). It is demonstrated that the DPT4 correction can be obtained as a second derivative of the energy with respect to the relativistic perturbation parameter. Accordingly, differentiation of a suitable Lagrangian, thereby taking into account all constraints on the wave function, provides analytic expressions for the fourth-order energy corrections. The latter have been implemented at the Hartree-Fock level and within second-order Møller-Plesset perturbaton theory using standard analytic second-derivative techniques into the CFOUR program package. For closed-shell systems, the DPT4 corrections consist of higher-order scalar-relativistic effects as well as spin-orbit corrections with the latter appearing here for the first time in the DPT series.Relativistic corrections are reported for energies as well as for first-order electrical properties and compared to results from rigorous four-component benchmark calculations in order to judge the accuracy and convergence of the DPT expansion for both the scalar-relativistic as well as the spin-orbit contributions. Additionally, the importance of relativistic effects to the bromine and iodine quadrupole-coupling tensors is investigated in a joint experimental and theoretical study concerning the rotational spectra of CH2BrF, CHBrF2, and CH2FI.

Chiral properties of two-flavour QCD at zero and non-zero temperature

Lattice Quantum Chromodynamics (LQCD) is the preferred tool for obtaining non-perturbative results from QCD in the low-energy regime. It has by nowrnentered the era in which high precision calculations for a number of phenomenologically relevant observables at the physical point, with dynamical quark degrees of freedom and controlled systematics, become feasible. Despite these successes there are still quantities where control of systematic effects is insufficient. The subject of this thesis is the exploration of the potential of todays state-of-the-art simulation algorithms for non-perturbativelyrn$\mathcal{O}(a)$-improved Wilson fermions to produce reliable results in thernchiral regime and at the physical point both for zero and non-zero temperature. Important in this context is the control over the chiral extrapolation. Thisrnthesis is concerned with two particular topics, namely the computation of hadronic form factors at zero temperature, and the properties of the phaserntransition in the chiral limit of two-flavour QCD.rnrnThe electromagnetic iso-vector form factor of the pion provides a platform to study systematic effects and the chiral extrapolation for observables connected to the structure of mesons (and baryons). Mesonic form factors are computationally simpler than their baryonic counterparts but share most of the systematic effects. This thesis contains a comprehensive study of the form factor in the regime of low momentum transfer $q^2$, where the form factor is connected to the charge radius of the pion. A particular emphasis is on the region very close to $q^2=0$ which has not been explored so far, neither in experiment nor in LQCD. The results for the form factor close the gap between the smallest spacelike $q^2$-value available so far and $q^2=0$, and reach an unprecedented accuracy at full control over the main systematic effects. This enables the model-independent extraction of the pion charge radius. The results for the form factor and the charge radius are used to test chiral perturbation theory ($\chi$PT) and are thereby extrapolated to the physical point and the continuum. The final result in units of the hadronic radius $r_0$ is rn$$ \left\langle r_\pi^2 \right\rangle^{\rm phys}/r_0^2 = 1.87 \: \left(^{+12}_{-10}\right)\left(^{+\:4}_{-15}\right) \quad \textnormal{or} \quad \left\langle r_\pi^2 \right\rangle^{\rm phys} = 0.473 \: \left(^{+30}_{-26}\right)\left(^{+10}_{-38}\right)(10) \: \textnormal{fm} \;, $$rn which agrees well with the results from other measurements in LQCD and experiment. Note, that this is the first continuum extrapolated result for the charge radius from LQCD which has been extracted from measurements of the form factor in the region of small $q^2$.rnrnThe order of the phase transition in the chiral limit of two-flavour QCD and the associated transition temperature are the last unkown features of the phase diagram at zero chemical potential. The two possible scenarios are a second order transition in the $O(4)$-universality class or a first order transition. Since direct simulations in the chiral limit are not possible the transition can only be investigated by simulating at non-zero quark mass with a subsequent chiral extrapolation, guided by the universal scaling in the vicinity of the critical point. The thesis presents the setup and first results from a study on this topic. The study provides the ideal platform to test the potential and limits of todays simulation algorithms at finite temperature. The results from a first scan at a constant zero-temperature pion mass of about 290~MeV are promising, and it appears that simulations down to physical quark masses are feasible. Of particular relevance for the order of the chiral transition is the strength of the anomalous breaking of the $U_A(1)$ symmetry at the transition point. It can be studied by looking at the degeneracies of the correlation functions in scalar and pseudoscalar channels. For the temperature scan reported in this thesis the breaking is still pronounced in the transition region and the symmetry becomes effectively restored only above $1.16\:T_C$. The thesis also provides an extensive outline of research perspectives and includes a generalisation of the standard multi-histogram method to explicitly $\beta$-dependent fermion actions.

Neuartige kryogene Penning-Falle für den Nachweis von Spin-Übergängen eines Protons und Bestimmung seines g-Faktors

In dieser Arbeit wird der Entwurf, der Aufbau, die Inbetriebnahme und die Charakterisierung einer neuartigen Penning-Falle im Rahmen des Experiments zur Bestimmung des g-Faktors des Protons präsentiert. Diese Falle zeichnet sich dadurch aus, dass die Magnetfeldlinien eines äußeren homogenen Magnetfeldes durch eine ferromagnetische Ringelektrode im Zentrum der Falle verzerrt werden. Der inhomogene Anteil des resultierenden Magnetfeldes, die sogenannte magnetische Flasche, lässt sich durch den Koeffizient B2 = 297(10) mT/mm2 des Terms zweiter Ordnung der Ortsabhängigkeit des Feldes quantifizieren. Eine solche ungewöhnlich starke Feldinhomogenität ist Grundvoraussetzung für den Nachweis der Spinausrichtung des Protons mittels des kontinuierlichen Stern-Gerlach-Effektes. Dieser Effekt basiert auf der im inhomogenen Magnetfeld entstehenden Kopplung des Spin-Freiheitsgrades des gefangenen Protons an eine seiner Eigenfrequenzen. Ein Spin-Übergang lässt sich so über einen Frequenzsprung detektieren. Dabei ist die nachzuweisende Änderung der Frequenz proportional zu B2 und zum im Fall des Protons extrem kleinen Verhältnis zwischen seinem magnetischen Moment nund seiner Masse. Die durch die benötigte hohe Inhomogenität des Magnetfeldes bedingten technischen Herausforderungen erfordern eine fundierte Kenntnis und Kontrolle der Eigenschaften der Penning-Falle sowie der experimentellen Bedingungen. Die in der vorliegenden Arbeit entwickelte Penning-Falle ermöglichte den erstmaligen zerstörungsfreien Nachweis von Spin-Quantensprüngen eines einzelnen gefangenen Protons, was einen Durchbruch für das Experiment zur direkten Bestimmung des g-Faktors mit der angestrebten relativen Genauigkeit von 10−9 darstellte. Mithilfe eines statistischen Verfahrens ließen sich die Larmor- und die Zyklotronfrequenz des Protons im inhomogenen Magnetfeld der Falle ermitteln. Daraus wurde der g-Faktor mit einer relativen Genauigkeit von 8,9 × 10−6 bestimmt. Die hier vorgestellten Messverfahren und der experimentelle Aufbau können auf ein äquivalentes Experiment zur Bestimmung des g-Faktors des Antiprotons zum Erreichen der gleichen Messgenauigkeit übertragen werden, womit der erste Schritt auf dem Weg zu einem neuen zwingenden Test der CPT-Symmetrie im baryonischen Sektor gemacht wäre.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

The Role of Anharmonicity in Hydrogen-Bonded Systems: The Case of Water Clusters

The nature of vibrational anharmonicity has been examined for the case of small water clusters using second-order vibrational perturbation theory (VPT2) applied on second-order Møller–Plesset perturbation theory (MP2) potential energy surfaces. Using a training set of 16 water clusters (H2O)n=2–6,8,9 with a total of 723 vibrational modes, we determined scaling factors that map the harmonic frequencies onto anharmonic ones. The intermolecular modes were found to be substantially more anharmonic than intramolecular bending and stretching modes. Due to the varying levels of anharmonicity of the intermolecular and intramolecular modes, different frequency scaling factors for each region were necessary to achieve the highest accuracy. Furthermore, new scaling factors for zero-point vibrational energies (ZPVE) and vibrational corrections to the enthalpy (ΔHvib) and the entropy (Svib) have been determined. All the scaling factors reported in this study are different from previous works in that they are intended for hydrogen-bonded systems, while others were built using experimental frequencies of covalently bonded systems. An application of our scaling factors to the vibrational frequencies of water dimer and thermodynamic functions of 11 larger water clusters highlights the importance of anharmonic effects in hydrogen-bonded systems.