Learning to rank is utilized in many scenarios, such as disease-gene association, information retrieval and recommender system. Improving the prediction accuracy of the ranking model is the main target of existing works. Contextual information has a significant influence in the ranking problem, and has been proved effective to increase the prediction performance of ranking models. Then we construct similarities for different types of entities that could utilize contextual information uniformly in an extensible way. Once we have the similarities constructed by contextual information, how to uti- lize them for different types of ranking models will be the task we should tackle. In this thesis, we propose four algorithms for learning to rank with contextual informa- tion. To refine the framework of matrix factorization, we propose an area under the ROC curve (AUC) loss to conquer the sparsity problem. Clustering and sampling methods are used to utilize the contextual information in the global perspective, and an objective function with the optimal solution is proposed to exploit the contex- tual information in the local perspective. Then, for the deep learning framework, we apply the graph convolutional network (GCN) on the ranking problem with the combination of matrix factorization. Contextual information is utilized to generate the input embeddings and graph kernels for the GCN. The third method in this thesis is proposed to directly exploit the contextual information for ranking. Laplacian loss is utilized to solve the ranking problem, which could optimize the ranking matrix directly. With this loss, entities with similar contextual information will have similar ranking results. Finally, we propose a two-step method to solve the ranking problem of the sequential data. The first step in this two-step method is to generate the em- beddings for all entities with a new sampling strategy. Graph neural network (GNN) and long short-term memory (LSTM) are combined to generate the representation of sequential data. Once we have the representation of the sequential data, we could solve the ranking problem of them with pair-wise loss and sampling strategy.

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

Spaceborne sensors, which collect imagery of the Earth in various spectral bands, are limited by the data transmission rates. As a result the multispectral bands are transmitted at a lower resolution and only the panchromatic band is transmitted at its full resolution. The information contained in the multispectral bands is an invaluable tool for land use mapping, urban feature extraction, etc. However, the limited spatial resolution reduces the appeal and value of this information. Pan sharpening techniques enhance the spatial resolution of the multispectral imagery by extracting the high spatial resolution of the panchromatic band and adding it to the multispectral images. There are many different pan sharpening methods available like the ones based on the Intensity-Hue-Saturation and the Principal Components Analysis transformation. But these methods cause heavy spectral distortion of the multispectral images. This is a drawback if the pan sharpened images are to be used for classification based applications. In recent years, multiresolution based techniques have received a lot of attention since they preserve the spectral fidelity in the pan sharpened images. Many variations of the multiresolution based techniques exist. They differ based on the transform used to extract the high spatial resolution information from the images and the rules used to synthesize the pan sharpened image. The superiority of many of the techniques has been demonstrated by comparing them with fairly simple techniques like the Intensity-Hue-Saturation or the Principal Components Analysis. Therefore there is much uncertainty in the pan sharpening community as to which technique is the best at preserving the spectral fidelity. This research investigates these variations in order to find an answer to this question. An important parameter of the multiresolution based methods is the number of decomposition levels to be applied. It is found that the number of decomposition levels affects both the spatial and spectral quality of the pan sharpened images. The minimum number of decomposition levels required to fuse the multispectral and panchromatic images was determined in this study for image pairs with different resolution ratios and recommendations are made accordingly.

We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

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In this dissertation, we study the existence and multiplicity of positive solutions to classes of one-dimensional singular p-Laplacian problems with nonlinear and intergral boundary conditions when the reaction termis p-superlinear or p-sublinear at infinity. In the p-superlinear case, we prove the existence of a large positive solution when a parameter is small and if, in addition, the reaction term satisfies a concavity-like condition at the origin, the existence of two positive solutions for a certain range of the parameter. In the p-sublinear case, we establish the existence of a large positive solution when a parameter is large. We also investigate the number of positive solutions for the general PHI-Laplacian with nonlinear boundary conditions when the reaction term is positive. Our results can be applied to the challenging infinite semipositone case and complement or extend previous work in the literature.Our approach depends on Amann's fixed point in a Banach space, degree theory, and comparison principles.

The goals of the research presented in this dissertation were to create algorithms that produce images of complex phenomena, to study the efficacy of the algorithms, and to apply these algorithms to important mineralogical problems. The algorithms that were created include the Sphere Projection method, the Chicken Wire method, and methods for calculating the curvature at any point on a surface. The Sphere Projection method is best applied to roughly spherical surfaces. A theorem about the "fit" to a sphere determines the accuracy of the model in this special case and gives some insight into the limitations of this method. The Chicken Wire method was developed to model those surfaces for which the Sphere Projection method was ineffective. The effectiveness of the Chicken Wire method was also determined. The algorithms were used to produce images of equi-value surfaces of the Laplacian of the electron density function in selected molecules. The water molecule, H2O, was studied to demonstrate that these new methods are capable of reproducing known features. The disiloxane molecule, H6Si2O7, was studied because it serves as a model for bonding in quartz and other important silicates. Lastly, the molecule NaLi2Si2OF9 was examined as a molecular model for low albite. A new discovery suggests that these algorithms will be an important tool in mineralogy. / Ph. D.

Le but de cette thèse est de trouver des bornes supérieures pour les valeurs propres des opérateurs naturels agissant sur les fonctions d’une variété compacte (M; g). Nous étudions l’opérateur de Laplace–Beltrami et des opérateurs du type laplacien. Dans le cas du laplacien, deux aspects sont étudiés. Le premier aspect est d’étudier des relations entre la géométrie intrinsèque et les valeurs propres du laplacien. Nous obtenons des bornes supérieures ne dépendant que de la dimension et d’un invariant conforme qui s’appelle le volume conforme minimal. Asymptotiquement, ces bornes sont consistantes avec la loi de Weyl. Elles améliorent également les résultats de Korevaar et de Yang et Yau. La méthode employée est intéressante en soi. Le deuxième aspect est d’étudier la relation entre la géométrie extrinsèque et les valeurs propres du laplacien agissant sur des sous-variétés compactes de RN et de CPN. Nous étudions un invariant extrinsèque qui s’appele l’indice d’intersection. Pour des sous-variétés compactes de RN, nous généralisons les résultats de Colbois, Dryden et El Soufi et obtenons des bornes supérieures qui sont stables par des petites perturbations. Pour des sous-variétés de CPN, nous obtenons une borne supérieure ne dépendant que du degré des sous-variétés. Pour des opérateur du type laplacien, une modification de notre méthode donne des bornes supérieures pour les valeurs propres des opérateurs de Schrödinger en termes du volume conforme minimal et de l’intégrale du potentiel. Nous obtenons également les bornes supérieures pour les valeurs propres du laplacien de Bakry–Émery dépendant d’invariants conformes. / The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian manifold (M; g) such as the Laplace–Beltrami operator and Laplace-type operators. In the case of the Laplace-Beltrami operator, two aspects are investigated: The first aspect is to study relationships between the intrinsic geometry and eigenvalues of the Laplacian operator. In this regard, we obtain upper bounds depending only on the dimension and a conformal invariant called min-conformal volume. Asymptotically, these bounds are consistent with the Weyl law. They improve previous results by Korevaar and Yang and Yau. The method which is introduced to obtain the results, is powerful and interesting in itself. The second aspect is to study the interplay of the extrinsic geometry and eigenvalues of the Laplace–Beltrami operator acting on compact submanifolds of RN and of CPN. We investigate an extrinsic invariant called the intersection index studied by Colbois, Dryden and El Soufi. For compact submanifolds of RN, we extend their results and obtain upper bounds which are stable under small perturbation. For compact submanifolds of CPN, we obtain an upper bound depending only on the degree of submanifolds. For Laplace type operators, a modification of our method lead to have upper bounds for the eigenvalues of Schrödinger operators in terms of the min-conformal volume and integral quantity of the potential. As another application of our method, we obtain upper bounds for the eigenvalues of the Bakry–Émery Laplace operator depending on conformal invariants.

La thèse consiste en l’étude spectrale d’opérateurs partiellement semi-classiques. Quand la géométrie du problème suggère une localisation anisotrope des fonctions propres associées aux basses énergies (bord du domaine, lieu d’annulation du champs magnétique), le développement local de l’opérateur amène naturellement à une structure à double échelle. Il s'agit, via un schéma de réduction "à la Born-Oppenheimer", utilisant le formalisme du calcul pseudodifférentiel pour des symboles à valeur opérateur, de montrer l’existence d’un opérateur effectif à symbole scalaire. On en déduit ensuite des formules de Weyl pour le comptage des basses valeurs propres. Cette stratégie est appliquée : au Laplacien de Robin sur un domaine borné, en dimension quelconque et au Laplacien magnétique dans R², dans le cas où le champ magnétique s’annule sur une courbe fermée. / The thesis consists in the spectral study of partially semiclassical operators. When the geometry of the problem suggests an anisotropic localization of the eigenfunctions associated to low energies (boundary of the domain, vanishing magnetic field), the local expansion of the operator naturally brings to a doublescale structure. Via a reduction scheme "à la Born-Oppenheimer", using the formalism of pseudodifferential calculus for operator-valued symbols, we can show the existence of an effective operator, with scalar symbol. Then, we deduce Weyl formulae for the number of low-lying eigenvalues. This strategy is applied : to the Robin Laplacian on a bounded domain, in any dimension and to the magnetic Laplacian in R², in the case where the magnetic field vanishes on a closed curve.