Fractional diffusion equations have played an increasingly important role in ex- plaining long-range interactions, nonlocal dynamics and anomalous diffusion, pro- viding effective means of describing the memory and hereditary properties of such processes. This dissertation explores the uncertainty propagation in space fractional diffusion equations in one and multiple dimensions with variable diffusivity and order parameters. This is achieved by:(i) deploying accurate numerical schemes of the forward problem, and (ii) employing uncertainty quantifications tools that accelerate the inverse problem. We begin by focusing on parameter calibration of a variable- diffusivity fractional diffusion model. A random, spatially-varying diffusivity field is considered together with an uncertain but spatially homogeneous fractional operator order. Polynomial chaos (PC) techniques are used to express the dependence of the stochastic solution on these random variables. A non-intrusive methodology is used, and a deterministic finite-difference solver of the fractional diffusion model is utilized for this purpose. The surrogates are first used to assess the sensitivity of quantities of interest (QoIs) to uncertain inputs and to examine their statistics. In particular, the analysis indicates that the fractional order has a dominant effect on the variance of the QoIs considered. The PC surrogates are further exploited to calibrate the uncertain parameters using a Bayesian methodology. In the broad range of parameters addressed, the analysis shows that the uncertain parameters having a significant impact on the variance of the solution can be reliably inferred, even from limited observations. Next, we address the numerical challenges when multidimensional space-fractional diffusion equations have spatially varying diffusivity and fractional order. Significant computational challenges arise due to the kernel singularity in the fractional integral operator as well as the resulting dense discretized operators. Hence, we present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the first-order convergence rate allowed by the regularity of the solutions. In the last part, we explore the application of a Bayesian formalism to detect an anomaly in a fractional medium. Specifically, a computational method is presented for inferring the location and properties of an inclusion inside a two-dimensional domain. The anomaly is assumed to have known shape, but unknown diffusivity and fractional order parameters, and is assumed to be embedded in a fractional medium of known fractional properties. To detect the presence of the anomaly, the medium is forced using a collection of localized sources, and its response is measured at the source locations. To this end, the singularity-aware finite-difference scheme is applied. A non-intrusive regression approach is used to explore the dependence of the computed signals on the properties of the anomaly, and the resulting surrogates are first exploited to characterize the variability of the response, and then used to accelerate the Bayesian inference of the anomaly. In the regime of parameters considered, the computational results indicate that robust estimates of the location and fractional properties of the anomaly can be obtained, and that these estimates become sharper when high contrast ratios prevail between the anomaly and the surrounding matrix.

No description available.

Cardiovascular diseases (CVDs) remain the leading cause of death worldwide. Patients at risk of evolving CVDs are assessed by evaluating a risk factor-based score that incorporates different bio-markers ranging from age and sex to arterial stiffness (AS). AS depicts the rigidity of the arterial vessels and leads to an increase in the arterial pulse pressure, affecting the heart and vascular physiology. These facts have encouraged researchers to propose surrogate markers of cardiovascular risks and develop simple and non-invasive models to better understand cardiovascular system operations. This work thus fundamentally capitalizes on developing a novel class of low-dimensional physics-based fractional-order models of systemic arteries and exploring the feasibility of fractional differentiation order to portray the vascular stiffness. Fractional-order modeling is a successful paradigm to integrate multiscale and interconnected mechanisms of the complex arterial system. However, this type of modeling alone often fails to efficiently integrate altered variabilities in vascular physiology from various sources of large datasets, multi-modalities, and levels. In this regard, combining fractional-order-based approaches with machine learning techniques presents a unique opportunity to develop a powerful prediction framework that reveals the correlation between intertwined vascular events. This work is divided into three parts. The first part contributes to developing the fractional-order lumped parametric model of the arterial system. First, we propose fractional-order representations to model and characterize the complex and frequency-dependent apparent arterial compliance. Second, we propose fractional-order arterial Windkessel modeling the aortic input impedance and hemodynamic. Subsequently, the proposed models have been applied and validated using both human in-silico healthy datasets and real vascular aging and hypertension. The second part addresses the non-zero initial value problem for fractional differential equations (FDEs) and proposes an estimation technique for joint estimation of the input, parameters, and fractional differentiation order of non-commensurate FDEs. The performance of the proposed estimation techniques is illustrated on arterial and neurovascular hemodynamic response models. The third part explores the feasibility of using machine learning algorithms to estimate the gold-standard measurement of AS, carotid-to-femoral pulse wave velocity. Different modalities have been investigated to generate informative input features and reduce the dimensionality of the time series pulse waves.

This report deals with the mathematical model of the transient behaviour of an existing ethane dehydrogenation furnace which is composed of two main sections: a preheating convection section and a radiant-heated section. The correlation of pressure drop with time has been found from the available data. The fractional carbon deposition and the multiplier coefficient of a pressure drop equation have been determined by the direct search optimization technique of Hooke and Jeeves. An optimal policy for the cyclic operation of the furnace was determined by considering plant temperature profile and hydrocarbon/ steam ration as parameters for maximizing average ethylene produced per day. The effect of temperature profile on the distribution of carbon deposited along the reactor was also predicted and discussed. / Thesis / Master of Engineering (ME)

We reserve the first part of this thesis to a brief (and by far incomplete, but hopefully self-contained) introduction to the vast subject of quantum Hall physics. We dedicate the first chapter to a discursive broad introduction. The second one is instead used to introduce the integer and fractional quantum Hall effects, with an eye to the synthetic quantum matter platforms for their realization. In the third chapter we present famous Laughlin's wavefunction and discuss its basic features, such as the gapless edge modes and the gapped quasiparticle excitations in the bulk. We close this introductory part with a fourth chapter which presents a brief overview on the chiral Luttinger liquid theory. In the second part of this thesis we instead proceed to present our original results. In the fifth chapter we numerically study the linear and non-linear dynamics of the chiral gapless edge modes of fractional quantum Hall Laughlin droplets -- both fermionic and bosonic -- when confined by anharmonic trapping potentials with model short range interactions; anharmonic traps allow us to study the physics beyond Wen's low-energy/long-wavelength chiral Luttinger liquid paradigm in a regime which we believe is important for synthetic quantum matter systems; indeed, even though very successful, corrections to Wen's theory are expected to occur at higher excitation energies/shorter wavelengths. Theoretical works pointed to a modified hydrodynamic description of the edge modes, with a quadratic correction to Wen's linear dispersion $\omega_k=vk$ of linear waves; even though further works based on conformal field theory techniques casted some doubt on the validity of the theoretical description, the consequences of the modified dispersion are very intriguing. For example, in conjunction with non-linearities in the dynamics, it allowed for the presence of fractionally quantized solitons propagating ballistically along the edge. The strongly correlated nature of fractional quantum Hall liquids poses technical challenges to the theoretical description of its dynamics beyond the chiral Luttinger liquid model; for this reason we developed a numerical approach which allowed us to follow the dynamics of macroscopic fractional quantum Hall clouds, focusing on the neutral edge modes that are excited by applying an external weak time-dependent potential to an incompressible fractional quantum Hall cloud prepared in a Laughlin ground state. By analysing the dynamic structure factor of the edge modes and the semi-classical dynamics we show that the edge density evolves according to a Korteweg-de Vries equation; building on this insight, we quantize the model obtaining an effective chiral Luttinger liquid-like Hamiltonian, with two additional terms, which we believe captures the essential low-energy physics of the edge beyond Wen's highly successful theory. We then move forward by studying -- even though only partially -- some of the physics of this effective model and analyse some of its consequences. In the sixth chapter we look at the spin properties of bulk abelian fractional quantum Hall quasiparticles, which are closely related to their anyonic statistics due to a generalized spin-statistics relation - which we prove on a planar geometry exploiting the fact that when the gauge-invariant generator of rotations is projected onto a Landau level, it fractionalizes among the quasiparticles and the edge. We then show that the spin of Jain's composite fermion quasielectron satisfies the spin-statistics relation and is in agreement with the theory of anyons, so that it is a good anti-anyon for the Laughlin's quasihole. On the other hand, even though we find that the Laughlin’s quasielectron satisfies the spin-statistics relation, it carries the wrong spin to be the anti-anyon of Laughlin’s quasihole. Leveraging on this observation, we show how Laughlin's quasielectron is a non-local object which affects the system's edge and thus affecting the fractionalization of the spin. Finally, in the seventh chapter we draw our conclusions.

In this Ph.D. thesis we discuss results concerning variational convergences for functionals and differential operators on Lipschitz continuous vector fields. The convergences taken into account are gamma-convergence (for functionals) and H-convergence (for differential operators).

This Thesis is devoted to the study of topological phases of matter in optical platforms, focusing on non-Hermitian systems with gain and losses involving nonreciprocal elements, and fractional quantum Hall liquids where strong interactions play a central role.In the first part we investigated nonlinear Taiji micro-ring resonators in passive and active silicon photonics setups. Such resonators establish a unidirectional coupling between the two whispering-gallery modes circulating in their perimeter. We started by demonstrating that a single nonlinear Taiji resonator coupled to a bus waveguide breaks Lorentz reciprocity. When a saturable gain is added to a single Taiji resonator, a sufficiently strong unidirectional coupling rules out the possibility of lasing in one of the whispering-gallery modes with independence of the type of optical nonlinearity and gain saturation displayed by the material. This can be regarded as a dynamical time-reversal symmetry breaking. This effect is further enhanced by an optical Kerr nonlinearity. We showed that both ring and Taiji resonators can work as optical isolators over a broad frequency band in realistic operating conditions. Our proposal relies on the presence of a strong pump in a single direction: as a consequence four-wave mixing can only couple the pump with small intensity signals propagating in the same direction. The resulting nonreciprocal devices circumvent the restrictions imposed by dynamic reciprocity. We then studied two-dimensional arrays of ring and Taiji resonators realizing quantum spin-Hall topological insulator lasers. The strong unidirectional coupling present in Taiji resonator lattices promotes lasing with a well-defined chirality while considerably improving the slope efficiency and reducing the lasing threshold. Finally, we demonstrated that lasing in a single helical mode can be obtained in quantum spin-Hall lasers of Taiji resonators by exploiting the optical nonlinearity of the material. In the second part of this Thesis we dived into more speculative waters and explored fractional quantum Hall liquids of cold atoms and photons. We proposed strategies to experimentally access the fractional charge and anyonic statistics of the quasihole excitations arising in the bulk of such systems. Heavy impurities introduced inside a fractional quantum Hall droplet will bind quasiholes, forming composite objects that we label as anyonic molecules. Restricting ourselves to molecules formed by one quasihole and a single impurity, we find that the bound quasihole gives a finite contribution to the impurity mass, that we are able to ascertain by considering the first-order correction to the Born-Oppenheimer approximation. The effective charge and statistical parameter of the molecule are given by the sum of those of the impurity and the quasihole, respectively. While the mass and charge of such objects can be directly assessed by imaging the cyclotron orbit described by a single molecule, the anyonic statistics manifest as a rigid shift of the interference fringes in the differential scattering cross section describing a collision between two molecules.

No description available.

Background: Tractography has become a standard technique for planning neurosurgical operations in the past decades. This technique relies on diffusion magnetic resonance imaging. The cutoff value for the fractional anisotropy (FA) has an important role in avoiding false-positive and false-negative results. However, there is a wide variation in FA cutoff values. Methods: We analyzed a prospective cohort of 14 patients (six males and eight females, 50.1 ± 4.0 years old) with intracerebral tumors that were mostly gliomas. Magnetic resonance imaging (MRI) was obtained within 7 days before and within 7 days after surgery with T1 and diffusion tensor image (DTI) sequences. We, then, reconstructed the corticospinal tract (CST) in all patients and extracted the FA values within the resulting volume. Results: The mean FA in all CSTs was 0.4406 ± 0.0003 with the fifth percentile at 0.1454. FA values in right-hemispheric CSTs were lower (p < 0.0001). Postoperatively, the FA values were more condensed around their mean (p < 0.0001). The analysis of infiltrated or compressed CSTs revealed a lower fifth percentile (0.1407 ± 0.0109 versus 0.1763 ± 0.0040, p = 0.0036). Conclusion: An FA cutoff value of 0.15 appears to be reasonable for neurosurgical patients and may shorten the tractography workflow. However, infiltrated fiber bundles must trigger vigilance and may require lower cutoffs.

In this thesis, we provide a comprehensive overview of classical fractional derivatives and collect results on mapping properties. In particular, we discuss mapping properties e.g. we prove that the 𝛼 order fractional derivative maps the Sobolev space W_0^(p,s) to the fractional Sobolev-Slobodeckij space W^(p,s-α) for all 𝛼 < 𝑠 < 1. Further, we present several definitions of “Bernstein fractional derivatives” using the Bernstein function and in particular, we study the Bernstein censored fractional derivative by using the Picard method to get its inverse Bernstein censored fractional integral. Moreover, we use analytic tools to get the existence and uniqueness of the solution of the corresponding resolvent equation. Finally, we construct a stochastic process through Ikeda–Nagasawa–Watanabe (INW) piecing together procedure such that its generator is the Bernstein censored fractional derivative. Additionally, we show that this process gives a Feller semigroup.:Introduction 1 Basics 1.1 Some results in functional analysis 1.2 Fourier, Laplace and Mellin transforms 1.3 Regularly varying functions 1.4 Markov processes 1.5 Lévy processes and subordinators 2 Fractional derivatives and integrals 2.1 Classical fractional integrals and derivatives 2.2 Mapping properties of fractional integrals and derivatives 2.3 Bernstein functions 2.4 Fractional derivatives based on Bernstein Functions 2.5 Probabilistic interpretation of fractional derivatives 2.6 Fractional Laplace operator 3 Censored Bernstein fractional derivative and integral 3.1 Sonine pairs 3.2 Examples of Sonine pairs 3.3 Mapping properties of general fractional derivatives 3.4 Censored Bernstein fractional derivative and integral 3.5 Linear censored initial value problem 4 Censored process 4.1 Construction 4.2 Probabilistic representation 5 Application 5.1 Censored subordinator for a regularly varying kernel 5.2 Linear censored initial value problem for regularly varying kernels Bibliography