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We consider the heat equation ∂∙tu + u∇M(t) . v - ∆M(t)u = 0 u(x,0) = u0 x ∈ M(0) on an evolving curve which forms a "kink" in finite time. We describe the behaviour of the solution at the singularity and look to continue the solution past the singularity. We perturb the heat equation and study the effects of a deterministic perturbation and a stochastic perturbation on the solution, before the singularity. We then consider the heat equation on an evolving surface that forms a "cone" singularity in finite time and study the behaviour of the solution at the singularity. We then look to continue the solution past the singularity, in some probabilistic sense. Finally, we consider the heat equation on an evolving curve, where the evolution of the curve is coupled to the solution of the equation on the curve. We prove existence and uniqueness of the solution for small times, before any singularity can occur.

Image denoising is one of the most major steps in current image processing. It is a pre-processing step which aims to remove certain unknown, random noise from an image and obtain an image free of noise for further image processing, such as image segmentation. Image segmentation, as another branch of image processing, plays a significant role in connecting low-level image processing and high-level image processing. Its goal is to segment an image into different parts and extract meaningful information for image analysis and understanding. In recent years, methods based on PDEs and variational functional became very popular in both image denoising and image segmentation. These two branches of methods are presented and investigated in this thesis. In this thesis, several typical methods based on PDE are reviewed and examined. These include the isotropic diffusion model, the anisotropic diffusion model (the P-M model), the fourth-order PDE model (the Y-K model), and the active contour model in image segmentation. Based on the analysis of behaviours of each model, some improvements are proposed. First, a new coefficient is provided for the P-M model to obtain a well-posed model and reduce the “block effect”. Second, a weighted sum operator is used to replace the Laplacian operator in the Y-K model. Such replacement can relieve the creation of the speckles which is brought in by the Y-K model and preserve more details. Third, an adaptive relaxation method with a discontinuity treatment is proposed to improve the numerical solution of the Y-K model. Fourth, an active contour model coupling with the anisotropic diffusion model is proposed to build a noise-resistance segmentation method. Finally, in this thesis, three ways of deriving PDE are developed and summarised. The issue of PSNR is also discussed at the end of the thesis.

This thesis is divided into two parts. In the first part (Chapters 1, 2, 3) various Robinson-Schensted (RS) algorithms are discussed. An introduction to the classical RS algorithm is presented, including the symmetry property, and the result of the algorithm Doob h-transforming the kernel from the Pieri rule of Schur functions h when taking a random word [O'C03a]. This is followed by the extension to a q-weighted version that has a branching structure, which can be alternatively viewed as a randomisation of the classical algorithm. The q-weighted RS algorithm is related to the q-Whittaker functions in the same way as the classical algorithm is to the Schur functions. That is, when taking a random word, the algorithm Doob h-transforms the Hamiltonian of the quantum Toda lattice where h are the q-Whittaker functions. Moreover, it can also be applied to model the q-totally asymmetric simple exclusion process introduced in [SW98]. Furthermore, the q-RS algorithm also enjoys a symmetry property analogous to that of the symmetry property of the classical algorithm. This is proved by extending Fomin's growth diagram technique [Fom79, Fom88, Fom94, Fom95], which covers a family of the so-called branching insertion algorithms, including the row insertion proposed in [BP13]. In the second part (Chapters 4, 5) we work with quantum stochastic analysis. First we introduce the basic elements in quantum stochastic analysis, including the quantum probability space, the momentum and position Brownian motions [CH77], and the relation between rotations and angular momenta via the second quantisation, which is generalised to a family of rotation-like operators [HP15a]. Then we discuss a family of unitary quantum causal stochastic double product integrals E, which are expected to be the second quantisation of the continuous limit W of a discrete double product of aforementioned rotation-like operators. In one special case, the operator E is related to the quantum Levy stochastic area, while in another case it is related to the quantum 2-d Bessel process. The explicit formula for the kernel of W is obtained by enumerating linear extensions of partial orderings related to a path model, and the combinatorial aspect is closely related to generalisations of the Catalan numbers and the Dyck paths. Furthermore W is shown to be unitary using integrals of the Bessel functions.

The main results in this thesis are the minimax theorems for operators in Schatten ideals of compact operators acting on separable Hilbert spaces, generalized Clarkson-McCarthy inequalities for vector lq-spaces lq (Sp) of operators from Schatten ideals Sp, inequalities for partitioned operators and for Cartesian decomposition of operators. All Clarkson-McCarthy type inequalities are in fact some estimates on the norms of operators acting on the spaces lq (Sp) or from one such space into another.

In this work we present some uniqueness and cloaking results for a related pair of inverse problems. The first concerns recovering the parameter q in a Bessel-type operator pencil, over L^2(0, 1; rdr) from (a generalisation of) the Weyl–-Titchmarsh boundary m-function. We assume that both coefficients, w and q, are singular at 0. We prove q is uniquely determined by the sequence m(-n^2) (n = 1, 2, 3, ...), using asymptotic and spectral analysis and m-function interpolation results. For corollary we find, in a halfdisc with a singular “Dirichlet-point” boundary condition on the straight edge, a singular radial Schroedinger potential is uniquely determined by Dirichlet-to- Neumann boundary measurements on the semi-circular edge. The second result concerns recovery of three things—a Schroedinger potential in a planar domain, a Dirichlet-point boundary condition on part of the boundary, and a self-adjointness-imposing condition—from Dirichlet-to-Neumann measurements on the remaining boundary. With modern approaches to the inverse conductivity problem and a solution-space density argument we show the boundary condition cloaks the potential and vice versa. Appealing to negative eigen-value asymptotics we find the full-frequency problem has full uniqueness.

This dissertation focuses on the study of generalized fractional differential equations involving a general class of non-local operators which are referred to as the generalized fractional derivatives of Caputo and Riemann-Liouville (RL) type. These operators were introduced recently as a probabilistic extension of the classical fractional Caputo and Riemann-Liouville derivatives of order β ε (0,1) (when acting on regular enough functions). The main contribution of this work lies in displaying the use of stochastic analysis as a valuable approach for the study of fractional differential equations and their generalizations. The stochastic representations presented here also lead to many interesting potential applications, e.g., by providing new numerical approaches to approximate solutions to equations for which an explicit solution is not available.