Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Static analysis of functionally graded cylindrical and conical shells or panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded conical and cylindrical shells subjected to mechanical loadings. Several types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the conical or cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally conical and cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.

Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics

Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years. DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method. However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).

Modelling and analysis of thin-walled beams in the context of the Generalized Beam Theory

In this work, the Generalized Beam Theory (GBT) is used as the main tool to analyze the mechanics of thin-walled beams. After an introduction to the subject and a quick review of some of the most well-known approaches to describe the behaviour of thin-walled beams, a novel formulation of the GBT is presented. This formulation contains the classic shear-deformable GBT available in the literature and contributes an additional description of cross-section warping that is variable along the wall thickness besides along the wall midline. Shear deformation is introduced in such a way that the classical shear strain components of the Timoshenko beam theory are recovered exactly. According to the new kinematics proposed, a reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition. Later, a procedure for a posteriori reconstruction of all the three-dimensional stress components in the finite element analysis of thin-walled beams using the GBT is presented. The reconstruction is simple and based on the use of three-dimensional equilibrium equations and of the RCP procedure. Finally, once the stress reconstruction procedure is presented, a study of several existing issues on the constitutive relations in the GBT is carried out. Specifically, a constitutive law based on mirroring the kinematic constraints of the GBT model into a specific stress field assumption is proposed. It is shown that this method is equally valid for isotropic and orthotropic beams and coincides with the conventional GBT approach available in the literature. Later on, an analogous procedure is presented for the case of laminated beams. Lastly, as a way to improve an inherently poor description of shear deformability in the GBT, the introduction of shear correction factors is proposed. Throughout this work, numerous examples are provided to determine the validity of all the proposed contributions to the field.

Spectral theory of differential operators on finite metric graphs and on bounded domains

Die vorliegende Arbeit widmet sich der Spektraltheorie von Differentialoperatoren auf metrischen Graphen und von indefiniten Differentialoperatoren auf beschränkten Gebieten. Sie besteht aus zwei Teilen. Im Ersten werden endliche, nicht notwendigerweise kompakte, metrische Graphen und die Hilberträume von quadratintegrierbaren Funktionen auf diesen betrachtet. Alle quasi-m-akkretiven Laplaceoperatoren auf solchen Graphen werden charakterisiert, und Abschätzungen an die negativen Eigenwerte selbstadjungierter Laplaceoperatoren werden hergeleitet. Weiterhin wird die Wohlgestelltheit eines gemischten Diffusions- und Transportproblems auf kompakten Graphen durch die Anwendung von Halbgruppenmethoden untersucht. Eine Verallgemeinerung des indefiniten Operators $-tfrac{d}{dx}sgn(x)tfrac{d}{dx}$ von Intervallen auf metrische Graphen wird eingeführt. Die Spektral- und Streutheorie der selbstadjungierten Realisierungen wird detailliert besprochen. Im zweiten Teil der Arbeit werden Operatoren untersucht, die mit indefiniten Formen der Art $langlegrad v, A(cdot)grad urangle$ mit $u,vin H_0^1(Omega)subset L^2(Omega)$ und $OmegasubsetR^d$ beschränkt, assoziiert sind. Das Eigenwertverhalten entspricht in Dimension $d=1$ einer verallgemeinerten Weylschen Asymptotik und für $dgeq 2$ werden Abschätzungen an die Eigenwerte bewiesen. Die Frage, wann indefinite Formmethoden für Dimensionen $dgeq 2$ anwendbar sind, bleibt offen und wird diskutiert.

Laplace operator on finite graphs and a network diffusion model for the progression of the Alzheimer disease

Nella tesi viene descritto il Network Diffusion Model, ovvero il modello di A. Ray, A. Kuceyeski, M. Weiner inerente i meccanismi di progressione della demenza senile. In tale modello si approssima l'encefalo sano con una rete cerebrale (ovvero un grafo pesato), si identifica un generale fattore di malattia e se ne analizza la propagazione che avviene secondo meccanismi analoghi a quelli di un'infezione da prioni. La progressione del fattore di malattia e le conseguenze macroscopiche di tale processo(tra cui principalmente l'atrofia corticale) vengono, poi, descritte mediante approccio matematico. I risultati teoretici vengono confrontati con quanto osservato sperimentalmente in pazienti affetti da demenza senile. Nella tesi, inoltre, si fornisce una panoramica sui recenti studi inerenti i processi neurodegenerativi e si costruisce il contesto matematico di riferimento del modello preso in esame. Si presenta una panoramica sui grafi finiti, si introduce l'operatore di Laplace sui grafi e si forniscono stime dall'alto e dal basso per gli autovalori. Al fine di costruire una cornice matematica completa si analizza la relazione tra caso discreto e continuo: viene descritto l'operatore di Laplace-Beltrami sulle varietà riemanniane compatte e vengono fornite stime dall'alto per gli autovalori dell'operatore di Laplace-Beltrami associato a tali varietà a partire dalle stime dall'alto per gli autovalori del laplaciano sui grafi finiti.

On the experimental behaviour of a silty sand and its modelling through Generalized Plasticity.

La tesi contiene uno studio sperimentale sul comportamento di una sabbia limosa del sottosuolo della laguna veneta e propone un'interpretazione dei risultati sperimentali ottenuti alla luce dei presupposti teorici di un approccio costitutivo avanzato noto come "Plasticità Generalizzata". Il programma sperimentale è consistito nella realizzazione di prove edometriche e prove triassiali su campioni di sabbia provenienti dal sito di Treporti, situato in prossimità della bocca di Lido. La risposta sperimentale, in termini di modulo volumetrico, è stata messa a confronto con i risultati di alcuni studi di letteratura, con particolare riferimento a quelli condotti da Jefferies & Been (2000). La disponibilità di prove di compressione edometrica realizzate nella cella K0 e la conseguente possibilità di valutare il coefficiente di spinta a riposo ha permesso di interpretare le prove in termini di tensione media efficace p' e di verificare l'applicabilità al caso in esame degli approcci di letteratura disponibili, spesso sviluppati a partire da prove di compressione isotropa effettuate in cella triassiale. Il comportamento tenso-deformativo osservato è stato successivamente simulato con un modello costitutivo per sabbie sviluppato nell'ambito della Plasticità Generalizzata. In particolare sono state utilizzate tre diverse formulazioni, che costituiscono un avanzamento dell'iniziale modello costitutivo proposto da Pastor, Zienkiewicz e Chan (1990), basate sull'uso di un parametro di stato del materiale definito rispetto alle condizioni di Stato Critico. Dal confronto tra previsioni del modello e risposta sperimentale è stato possibile individuare la formulazione che meglio simula il comportamento meccanico osservato sia in compressione edometrica sia in prove di taglio ed è stato proposto un set di parametri costitutivi ritenuti rappresentativi del terreno studiato.

Evidence of a Generalized Defect of Acinar Cell Function in Shwachman-Diamond Syndrome

: Because the acinar cells of the exocrine pancreas in patients with Shwachman-Diamond syndrome (SDS) are severely depleted, we hypothesized that a similar deficiency may be present in acinar cells of the parotid gland.

Generalized expansion of nociceptive reflex receptive fields in chronic pain patients

Widespread central hypersensitivity is present in chronic pain and contributes to pain and disability. According to animal studies, expansion of receptive fields of spinal cord neurons is involved in central hypersensitivity. We recently developed a method to quantify nociceptive receptive fields in humans using spinal withdrawal reflexes. Here we hypothesized that patients with chronic pelvic pain display enlarged reflex receptive fields. Secondary endpoints were subjective pain thresholds and nociceptive withdrawal reflex thresholds after single and repeated (temporal summation) electrical stimulation. 20 patients and 25 pain-free subjects were tested. Electrical stimuli were applied to 10 sites on the foot sole for evoking reflexes in the tibialis anterior muscle. The reflex receptive field was defined as the area of the foot (fraction of the foot sole) from which a muscle contraction was evoked. For the secondary endpoints, the stimuli were applied to the cutaneous innervation area of the sural nerve. Medians (25-75 percentiles) of fraction of the foot sole in patients and controls were 0.48 (0.38-0.54) and 0.33 (0.27-0.39), respectively (P=0.008). Pain and reflex thresholds after sural nerve stimulation were significantly lower in patients than in controls (P<0.001 for all measurements). This study provides for the first time evidence for widespread expansion of reflex receptive fields in chronic pain patients. It thereby identifies a mechanism involved in central hypersensitivity in human chronic pain. Reverting the expansion of nociceptive receptive fields and exploring the prognostic meaning of this phenomenon may become future targets of clinical research.

Operator variability using different polishing methods and surface geometry of a nanohybrid composite

This study evaluated the operator variability of different finishing and polishing techniques. After placing 120 composite restorations (Tetric EvoCeram) in plexiglassmolds, the surface of the specimens was roughened in a standardized manner. Twelve operators with different experience levels polished the specimens using the following finishing/polishing procedures: method 1 (40 ?m diamond [40D], 15 ?m diamond [15D], 42 ?m silicon carbide polisher [42S], 6 ?m silicon carbide polisher [6S] and Occlubrush [O]); method 2 (40D, 42S, 6S and O); method 3 (40D, 42S, 6S and PoGo); method 4 (40D, 42S and PoGo) and method 5 (40D, 42S and O). The mean surface roughness (Ra) was measured with a profilometer. Differences between the methods were analyzed with non-parametric ANOVA and pairwise Wilcoxon signed rank tests (?=0.05). All the restorations were qualitatively assessed using SEM. Methods 3 and 4 showed the best polishing results and method 5 demonstrated the poorest. Method 5 was also most dependent on the skills of the operator. Except for method 5, all of the tested procedures reached a clinically acceptable surface polish of Ra?0.2 ?m. Polishing procedures can be simplified without increasing variability between operators and without jeopardizing polishing results.

A two-centre study on facial morphology in patients with complete bilateral cleft lip and palate at nine years of age

The aim of this study was to compare craniofacial morphology and soft tissue profiles in patients with complete bilateral cleft lip and palate at 9 years of age, treated in two European cleft centres with delayed hard palate closure but different treatment protocols. The cephalometric data of 83 consecutively treated patients were compared (Gothenburg, N=44; Nijmegen, N=39). In total, 18 hard tissue and 10 soft tissue landmarks were digitized by one operator. To determine the intra-observer reliability 20 cephalograms were digitized twice with a monthly interval. Paired t-test, Pearson correlation coefficients and multiple regression models were applied for statistical analysis. Hard and soft tissue data were superimposed using the Generalized Procrustes Analysis. In Nijmegen, the maxilla was protrusive for hard and soft tissue values (P=0.001, P=0.030, respectively) and the maxillary incisors were retroclined (P<0.001), influencing the nasolabial angle, which was increased in comparison with Gothenburg (P=0.004). In conclusion, both centres showed a favourable craniofacial form at 9-10 years of age, although there were significant differences in the maxillary prominence, the incisor inclination and soft tissue cephalometric values. Follow-up of these patients until facial growth has ceased, may elucidate components for outcome improvement.