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Abstract: This thesis aims to characterize the alternating group \(A_n\), for \(n \geq 8\), by its fusion system over a Sylow 2-subgroup. The results here will contribute to Aschbacher's programme which is aimed at providing a simplified proof of the theorem of classification of finite simple groups.

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Abstract: This study brings together a connection among the Sheffer polynomial sequences, the Riordan arrays and the Stirling numbers. Besides the algebraic view, a differential equation that gives rise to Sheffer sequences is discussed; serving as a differential approach to the study of Sheffer polynomials. Numbers generated by the function \(e^{e^{t}-1} := e^{Gt}\) have already been shown to satisfy the congruence relation \(G_{n+p}\equiv G_{n} + G_{n+1} \; (\mathrm{mod} p)\) for any \(p\) prime. We derive a congruence relation for the Touchard polynomial sequence and the Stirling numbers of the second kind analogous to the known congruence relation for the \(G_{n}\). Also, some Riordan arrays with generating functions related to \(e^{e^{t}-1}\) for polynomial sequences such as the Touchard polynomial, Toscano polynomial, Charlier polynomial and the Poisson - Charlier polynomial are discussed, and their corresponding inverses constructed.

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Abstract: Stochastic calculus has been applied to the problems of pricing financial derivatives since 1973 when Black and Scholes published their famous paper "The pricing of options and corporate liabilities" in the journal of political economy. In this work, we introduce basic concepts of probability theory which gives a better understanding in the study of stochastic processes, such as Markov process, Martingale and Brownian motion. We then construct the It^o's integral under stochastic calculus and it was used to study stochastic differential equations. The lognormal model was used to model asset prices showing its usefulness in financial mathematics. Finally, we show how the famous Black-Scholes model for option pricing was obtained from the lognormal asset model.

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Abstract: In this work, we review 3-dimensional gravity by canonically analyzing the Hilbert-Palatini action. We apply a time gauge used in 4-dimensional loop quantum gravity [14] to the simpler case of the 3-dimensional loop quantum gravity. By time gauge _fixing this Hilbert-Palatini action it leads to the Gauss constraints, the spatial diffeomorphism constraints, the Hamiltonian constraint and a new constraint C. The gauge symmetries generated by this new constraints are spacetime diffeomorphism and \(\mathfrak{S}\)\(\mathfrak{O}\)(2) gauge transformations. We solve the dynamics of the theory by providing a regularization of the generalized projector operator in terms of the Hamiltonian constraint. We provide the denition of the physical scalar product which can be represented in terms of a sum over finite spinfoam amplitudes. Then we establish a clear-cut link between the canonical quantization of the new theory to the spinfoam model (Ponzano-Regge model) defined in terms of the \(\mathfrak{S}\)\(\mathfrak{U}\)(2) spin networks.

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Abstract: We obtain the operator square root (􀀀D)1=2 of the Laplacian, called the fractional Laplacian, from the harmonic extension problem to the upper half space. It turns out that this operator maps the Dirichlet boundary condition to the Neumann condition. In this thesis, we extend the work of [2] by establishing the fractional Laplacian using semi-group methods and also providing proofs to certain claims and propositions in [2]. We also study some properties of the fractional Laplacian and relate it to an extension problem.

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Abstract: Chaos poses technical challenges to constrained Hamiltonian systems. This is an important topic for discussion, because general relativity in its Hamiltonian formulation is a constrained system, and there is strong evidence that it exhibits chaotic features. We review concepts in gauge systems and their association with Hamiltonian constraints, relational Dirac observables as gauge invariant encodings of physical information, and chaos in unconstrained Hamiltonian systems. We then construct a non-integrable, ergodic toy model, and with it explicitly illustrate the non-existence of a maximal set of Dirac observables, and a solution space which fails to be a manifold. The potential consequences of these qualitative features of a chaotic constrained Hamiltonian system for general relativity and the quest for its quantum theory are deliberated.

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Abstract: The G-graph \(\mathfrak{T}\) (G, S) is a graph from the group G generated by S \leqslant G, where the vertices are the right cosets of the cyclic subgroups s, s \in S with k-edges between two distinct cosets if there is an intersection of k elements. In this thesis, after presenting some important properties of G-graphs, we show how the G-graph depends on the generating set of the group. We give the G-graphs of the symmetric group, alternating group and the semi-dihedral group with respect to various generating sets. We give a characterisation of finite G-graphs; in the general case and a bipartite case. Using these characterisations, we give several classes of graphs that are G-graphs. For instance, we consider the Turán graphs, the platonic graphs and biregular graphs such as the Levi graphs of geometric configurations. We emphasis the structural properties of G-graphs and their relations to the group G and the generating set S. As preliminary results for further studies, we give the adjacency matrix and spectrum of various finite G-graphs. As an application, we compute the energy of these graphs. We also present some preliminary results on infinite G-graphs where we consider the G-graphs of the infinite group \text{SL}_2\(\mathfrak{Z}\) and an infinite non-Abelian matrix group.

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Abstract: This thesis investigates the relationship between copy number variations and neuro-image features of Glioblastoma patients. Canonical correlation analysis was employed to elicit these relationships. This thesis highlights some of the concepts of the technique which enabled us to obtain our main results. We found three pairs of significant canonical variates with correlations of 0.6704, 0.6347 and 0.5552 respectively, which was used to identify genes and neuro-image features related to Glioblastoma.

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Abstract: Linear Quadratic Control Problems are control problems with a quadratic cost function and linear dynamic system and a linear terminal constraint. This work looks at linear quadratic control problems without terminal conditions. We will first look at the controllability and observability of linear dynamical systems and then establish the necessary conditons for the variation in the cost criterion to be non-negative for strong perturbations in the control. These conditions are the first order necessary conditions for optimality. We shall also consider the necessary and sufficient conditon for positivity of the quadratic cost criterion. Moreover, necessary and sufficient conditions for strong positivity are derived and we shall show that these conditions are based on the existence of solution to a Riccati matrix differential equation. The symplectic property of the Hamiltonian system helps us to derive the Riccati matrix differential equation. We also look at some of the properties of the Riccati variable. Using these ideas, as an illustration, we consider an application in the control of disease, by considering a variant of the SIR model.

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Abstract: We show that the right derived functors of the limits of the Khovanov presheaf describes the Khovanov homology. We also look at the cellular cohomology of a poset \(\mathfrak{P}\) with coe cients in a presheaf F and show by example that the Khovanov homology can be computed cellular.

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Abstract: The properties of graphs can be studied via the algebraic characteristics of its adjacency or Laplacian matrix. The second eigenvector of the graph Laplacian is one very useful tool which provides information as to how to partition a graph. In this thesis, we study spectral clustering and how to apply it in solving the image segmentation problem in computer vision.

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Abstract: We show that the normal Appell subgroup of the Riordan group is a pseudo ring under a multiplication given by the componentwise composition. We develop formulae for calculating the degree of the root in generating trees and we establish isomorphisms between the four groups : the hitting time, Bell, associated and the derivative which are all subgroups of the Riordan group. We have found the average number of trees with left branch length in the class of ordered trees and the Motzkin trees. In the last chapter we examine the uplift principle and some known examples. We generalise some of the examples and we show that the average portion of protected points in the hex trees approaches 76/125 as \[\lim_{n \ to \infty} frac{1}{n} = 0 \]

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Abstract: This paper looks at the study of a derivation of a known inequality for spectral functions of products of exponentials using the Baker - Campbell - Hausdor formula. This known inequality is called the Golden-Thompson inequality. The Kalman and the Gramian matrices, Lyapunov equation and matrix exponential, Hadamard's lemma and Duhamel formula as well as the trace inequality due to Araki-Lieb-Thirring are all considered in this work.

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Abstract: This Research, studies the biological control of pest or invasive species given the use of additional food. We model this using the Beddington-DeAngelis functional response. We add an additional equation to the existing model to practically study the e ect that this additional food will have on the predator -prey dynamics. We observe that at high predator mutual interference there is a stability in the system, even though both quantity and quality continue to increase beyond a certain threshold. Also, at low values of mutual interferences, the system exhibits some interesting Hopf-Bifurcations which progress from stability to instability. Biological control is successful at low predator interactions, high quality of low quantity additional food.

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Abstract: The Pentagram map is a well notable integrable system that is de_ned on the moduli space of polygons. In 2005, Richard Evan Schwartz introduced certain polynomials called pentagram integrals (Monodromy invariants) of the pentagram map and defined certain associated integrals, the analogous first integrals. Schwartz further studied in 2011 with S. Tabachnikov on how these integrals behave on inscribed polygons. They discovered that the integrals are equal for every given weight of polygons inscribed in non-degenerate conics. However, the proof of their outcome was combinatorial which appeared to be more involving hence there was a need for quite a simple proof. Anton Izosimov in 2016 gave quite a simple conceptual geometric proof of these invariants of polygons inscribed in non-degenerate conics. In this thesis, we seek to analyse the geometry of these invariants by reviewing Anton's work. Our core analyses is that for any polygon inscribed in a non-degenerate conic, the analogous monodromy should satisfy a certain self- duality relation.

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Abstract: In steady flows, the notion of boundaries separating dynamically distinct regions is not ambiguous. This is because the invariant manifolds of time-independent flows and the critical points of time-periodic flows provide adequate information to determine the behaviour of the solutions of these systems. However, for time dependent systems, it is strenuous to determine the nature of their solutions due to their dependence on time. Nevertheless, it was observed that just like steady flows, most time-dependent systems have boundaries that prevent cross-mixing of dynamically distinct regions. They are known as Lagrangian Coherent Structures(LCSs) and they are embedded in time-dependent flows as robust structures that determine the flow pattern of fluid particles. This project investigates LCSs and also employs a numerical method to compute the Finite Time Lyapunov Exponent to detect these structures. Initially, the coherent structures are defined as hyperbolic material lines that separate dynamically distinct regions in an unsteady flow. Then, the LCSs are classified into attracting and repelling structures based on their in_uence on the time-dependent flow. Subsequently, the LCSs are also defined as a second derivative ridges of the FTLE fields. This definition is perceptible from the numerical computations of the double-gyre model where the coherent structures are extracted as ridges of the computed FTLE fields. Furthermore, we employ the Finite time Lyapunov Exponent model to carry out numerical simulations on satellite observed surface velocities along the coast of Ghana. The aim of this realistic application is to determine the Lagrangian Coherent Structures that are formed in geophysical flows. Finally, based on these results, we hypothesize the implications of a crude oil spill along the coast of Ghana. It was realized that in the event of a spill, the oil is likely to be confined to the coast temporarily due to the concentration of repelling LCSs. Also, for a longer time interval the oil spill is likely to be adverted from the coastline.

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Abstract: Given a Lie algebra \(\mathfrak{g}\) and its complexification \(\mathfrak{g} \Bbb{C}\); the representations of \(\mathfrak{g} \Bbb{C}\) are isomorphic to those of \(\mathfrak{g}\). Moreover, if \(\mathfrak{g}\) is the corresponding Lie algebra of a connected and simply connected Lie group \(G\) then the representations of the Lie group in question are isomorphic to those of \(\mathfrak{g} \Bbb{C}\) . This thesis explains the basic concepts of Lie groups and Lie algebras. Further, the basic representation theory of Lie groups and Lie algebras, particularly those of semisimple Lie algebras is discussed. In addition, an exposition of a method of constructing induced representations, with the particular case of the Poincar\'e group and an application in Physics is given. Finally, some physical applications of Lie groups and Lie algebras are outlined and discussed.

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Abstract: We address the single container packing problem of a company that has to serve its customers by first placing the products in boxes and then loading the boxes into a container. We approach the problem by developing and solving mixed-integer linear models. Our models consider geometric constraints which feature non-overlappping constraints, box orientation constraints, dimensionality constraints, relative packing position constraints and linearity constraints. We also consider extension of the models by integrating load balance as well as deviation of the center of gravity. The models have been tested on a large set of real instances involving up to 41 boxes and obtaining optimal solutions in most cases and very small gaps when optimality could not be proven.

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Abstract: In this study, we provide two analytic solutions to a one dimensional Advection-Diffusion Equation (ADE). The ADE is solved using a constant and an exponentially decaying inlet boundary condition, together with a Dirichlet and Neumann outlet conditions. The analytic solutions are shown to be simple if a combination of the initial concentration and the transformed boundary condition result in a non-zero singularity pole of inverse Laplace transform. The differences between the two analytic solutions are elucidated. Additionally, to verify the analytic solutions provided, comparisons are done with a solution to a mathematically related problem in Kim (2020a) and an error that appears in the paper is corrected. Moreover, the analytical solutions are compared to some observational data from the Fena river in the Ashanti region of Ghana and the difference between the two (analytical and observational) are clarified.

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Abstract: It is known that certain polynomials of degree one, with integer coefficients, admit infinitely-many primes. In this thesis, we provide an alternative proof of Dirichlets theorem concerning primes in arithmetic progressions, without applying methods involving Dirichlet characters or the Riemann Zeta function. A more general result concerning multiples of primes in short-intervals is also provided. This thesis also considers problems concerning the existence of odd perfect numbers. The main contribution is a good upper-bound on the largest prime divisor of an odd perfect number. In addition, we show how new results concerning odd perfect numbers or k - perfect numbers can be obtained by applying a property of completely-multiplicative functions.

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Abstract: This thesis presents a detailed description and analysis of Bregman’s iterative method for convex programming with linear constraints. Row and block action methods for large scale problems are adopted for convex feasibility problems. This motivates Bregman type methods for optimization. A new simultaneous version of the Bregman’s method for the optimization of Bregman function subject to linear constraints is presented and an extension of the method and its application to solving convex optimization problems is also made. Closed-form formulae are known for Bregman’s method for the particular cases of entropy maximization like Shannon and Burg’s entropies. The algorithms such as the Multiplicative Algebraic Reconstruction Technique (MART) and the related methods use closed-form formulae in their iterations. We present a generalization of these closed-form formulae of Bregman’s method when the objective function variables are separated and analyze its convergence. We also analyze the algorithm MART when the problem is inconsistent and give some convergence results.

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Abstract:  In this work, we introduce the new spaces, \(H^{s}_{p,q\),  \(H^{S}_{p,q}\), \(H^{*}_{p,q}\), \(\mathcal{Q}_{p,q}\), \mathcal{P}_{p,q}, called the Martingale Hardy-Amalgam spaces. We study some of the properties of these newly introduced spaces; two definitions of atoms are given and hence two atomic decompositions are given, dualities of these spaces are characterized and the Martingale inequalities and embeddings of these spaces are also discussed. It is proved that the dual of \(H^{s}_{p,q\), \((0<p\leq q \eq 1)\), is a Campanato-type space and the dual of \(H^{s}_{p,q\), \((1<p\leq q < \infty)\), is \(H^{s}_{p',q'\) where \((p, p'), (q, q')\) are conjugate pairs. The variation integrable space  \(\mathcal{G}_{p,q}\) is also introduced and it is established that the jump bounded space  \(\mathcal{B}\mathcal{D}_{p,q}\) is the dual of  \(\mathcal{G}_{p,q}\). To be able to characterize this duality, a larger space, which we denote by \(\mathcal{K}\left(L_{p,q},l_{r}\right)\), is introduced, such that \(\mathcal{G}_{p,q}\) can be embedded into. The classical Doob's Martingale inequality is also extended from the classical Martingale Hardy spaces to the newly introduced Martingale Hardy-Amalgam spaces. The Burkholder-Davis-Gundy inequality is also extended from the classical Martingale Hardy spaces to the Martingale Hardy-Amalgam spaces as well as the convexity inequality and the concavity inequalities involving measurable functions. The classical Martingale Hardy space embeddings are also extended to the Martingale Hardy-Amalgam spaces. The Davis decompositions of Martingales in the classical Martingale Hardy spaces are also extended to the Martingale Hardy-Amalgam spaces. As an application of the Davis decomposition and the Garsia space, a duality theorem for \(H^{*}_{p,q} \left(1\leq p,q \leq 2\right)\) is provided. Finally, the boundedness of martingale transforms between the Martingale Hardy-Amalgam spaces are also discussed. No data was collected for this study as the methodology used is purely theoretical in nature.

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Abstract: This thesis presents the analysis of upwelling events in the Gulf of Guinea and the Equatorial Atlantic region. The upwelling events in the West African subregion are usually experienced in the Gulf of Guinea, impacting the West African coastlines. During upwelling events, the wind blows across the water surface; a vacuum is created and filled with cold water and nutrient from the deep part of the ocean’s surface. This process leads to heat distribution in the ocean, preserving the ocean’s temperature. Classically, the upwelling events in the Gulf of Guinea were assumed to be generally caused by wind, Coriolis force, and Ekman transport. However, this research demonstrates that the wind that blows along the coast is usually in the north-northeastern direction. This means the correlation between the local wind stress and sea surface temperature (SST) anomaly in the Gulf of Guinea is considerably smaller. 

Early research on the upwelling events in the eastern and tropical Atlantic region used trends and patterns from observations and mathematical numerical models to examine the cause of upwelling in the Atlantic region. [13] and [3] hypothesized that upwelling in the Gulf of Guinea is associated with Kelvin waves propagating eastward from the Brazillian coast along the equator. The Kelvin waves are trapped along the coast once they reach Equatorial Guinea as coastally-trapped Kelvin waves. A major goal of this study is to test this hypothesis using satellite observational data and output from the state-of-the-art Estimating the Circulation and Climate of the
Ocean (ECCO) model.

We first investigate classical mathematical theory to understand the dynamics of upwelling. Exploring the classical theory of Ekman transport, Ekman pumping and suction will aid in explaining the reasons for the upwelling in the Gulf of Guinea. Based on [3], some mathematical derivations of the Kelvin waves theory will be used to find the features of the eastward propagating Kelvin waves on the ocean’s surface.

Data collected from the Prediction and Research Moored Array in the Atlantic (PIRATA) will aid the understanding of the interactions between the ocean and atmosphere in the tropical Atlantic region. Temperature as a function of depth and time is collected from four different locations to understand how the ocean’s temperature varies with depth. The rise of the thermal structure along the Atlantic region makes the thermocline shallower, leading to coastal upwelling, as discussed by [2] and [3]. June, July, August, and September usually are associated with low sea surface temperature (SST). However, the low SST is not enough evidence of what causes upwelling in the Gulf of Guinea, even though SST is important.

Sea surface height (SSH) signals from observations (satellite) and model data (ECCO) are composed of various waves whose characteristics and structures are different in terms of their period, wavelength, frequency, amplitude, and phase speeds. The Kelvin waves extracted from satellite data are examined and compared with the
Kelvin waves extracted from model data. The phase speed of the Kelvin waves from observations and model data is about 1.8 m/s which is consistent with the result of [7]. The lag correlation between Kelvin waves from observation and ECCO for some specific years is good. The question is, how does the observational data compare with the model data, and can the model data be useful for future predictions? It is observed from analyzing the results that the error between Kelvin waves from observations and ECCO was getting smaller in recent years. This result shows a marginal improvement in the model.

Some selected parameters in a few areas of relevance in the Atlantic region, such as wind stress from the Brazilian region and SST from the Guinea-west region, were considered to examine the upwelling analysis used by [6], to explain the remote influence of upwelling in the Gulf of Guinea region. The results obtained by [6] were
reproduced using recent observational data. In addition to the work by [6], model data from ECCO is used to produce similar results, supporting the hypothesis of [13].

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Abstract: Arterial stenosis is a major cause of death in cardiovascular-related diseases. Mathematical models can be used to describe the flow of blood through such diseased arteries via the Navier-Stokes (N-S) equation. An understanding of the dynamics of blood flow will result in a better understanding of the relationship between key parameters (velocity and pressure) and the stenosed artery size. Gaining insights into blood flow dynamics has useful applications in measuring pressure and velocity in a dialysis tube, for early detection/prediction of a heart attack or stroke and the design of diagnostic tools and/or equipment.

In this study, we use the Navier-Stokes equation to model the flow of blood in a stenosed artery. We show that the N-S equation has unique solutions within a bounded domain of the hyperboloid of one sheet. The model is then solved numerically using the finite elements method. We use the hyperboloid of one sheet as a geometry of the stenosed artery in our modelling framework since it gives a better geometrical representation of a stenosed artery as compared to the cylindrical geometry. Numerical solutions are presented and flow parameters; pressure and velocity are analysed using the COMSOL Multiphysics software. Our results showed an inverse relationship between pressure and the size of the stenosis. There is also an inverse relationship between the shear rate and arc length of the stenosis. The numerical results agree with Bernoulli’s principle.

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Abstract: In the context of Poisson jumps, substantial research has been done on the stochastic optimal control of jump-diffusion models. However, it is well known that jumps do occur quickly after one another, which causes clustering effects and, in systems of interest, contagion phenomena. Hawkes jump processes oer a more realistic representation of such events. There is a gap in the literature when it comes to stochastic optimal control problems in which the state dynamics follow Hawkes jump-diffusion models (contagion models). In this thesis, we solve stochastic optimal control problems under contagion and we look at their financial applications, namely mean-variance portfolio selection and risk- indifference pricing. Four main projects were considered. In the first project, we formulated a generalised mean-variance portfolio selection problem and derived the sufficient maximum principle for the problem. The result was applied to various special cases to obtain the optimal control (investment strategy). In the more general case, the optimal control obtained was in a semi-closed form subject to solutions to some partial integro differential equations (PIDEs). The results obtained extend and generalise the results obtained in the classical jump-diffusion case. The second project was on risk-indifference pricing under contagion. The problem was formulated as a stochastic differential game for which we obtained a sufficient maximum principle whilst considering a generic convex risk measure. The result was applied to the case of entropic risk measure to obtain explicit formulae for the buyer's and seller's risk-indifference prices. The third and fourth projects aimed at mean-eld singular control problems. The case of no contagion was considered in the third project and the case of contagion was considered in the fourth project. In both cases, we derived necessary and sufficient principles for the associated optimal control problems and applied the results to mean-variance portfolio selection with transaction cost. In both cases, we obtained an expressions for the optimal control in semi-closed form.

NB: The methodology used in the thesis is theoretical in nature and does not require data
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Abstract: This dissertation presents several results on existence, uniqueness, regularity and smoothness properties of probability densities of solutions to a class of forward-backward stochastic differential equations (FBSDEs, for short) for which the driver involves this type of nonlinearity f (y)|z|2 and the drift does not satisfy the standard Lipschitz continuous assumption. Since the groundbreaking works of Bismut ([3]), Bensoussan ([2]), Pardoux & Peng ([4], [5]), and Antonelli ([1]), the theory of FBSDEs has remained a very dynamic area of research, despite the fact that the fundamental concepts relating to the well-posedness of FBSDEs are now well understood. This is heavily motivated by the fact that the theory has produced a number of intriguing applications in both theoretical and applied mathematics, such as the theory of partial differential equations (PDEs) via the Feynman-Kac type representation or a probabilistic interpretation of the parabolic PDE, stochastic optimal control by utilizing the Pontryagin’s stochastic maximum principle, in quantitative finance, physics, and biology. Properties and mainly sensitivity analysis of solutions for such equations have been investigated so far only for locally Lipschitz coefficients. This thesis addresses precisely this question of lack of regularity of the coefficients in general and of the drift in particular. Indeed, we scrutinize the differentiability with respect to the initial condition of the forward system and in the variational sense of Malliavin of these equations when the drift term of the forward equation is either Hölder continuous, or Dini continuous or merely bounded and measurable in the spatial variable. In addition, we revisit several results regarding path regularity of quadratic FBSDEs without any assumptions on the boundedness of the control variable and particularly we extend the celebrate Zhang’s path regularity theorem to a more large class of quadratic FBSDE with non smooth coefficients. Furthermore, we examine some conditions under which each component of the solution to such equations has a continuous marginal law with respect to the Lebesgue measure. We also derive Gaussians-type bounds and tail estimates of the densities, while at least the globally Lipschitz condition is a crucial requirement in most of the scholars. At last, we examine if the solution of a more general class of quadratic FBSDEs has a structure of regularity when the terminal value involved is allowed to depend on the whole path of the diffusion process and the drift only belongs to the class of slowly varying function at zero. These type of equations are well known as Path-dependent FBSDEs.

Keywords: Quadratic FBSDEs; Parabolic PDE; Malliavin calculus; Local time; Path regularity; Rough drifts, Path-dependent terminal value

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Abstract: Given a group, the uniqueness of the duplicial set structure on the nerve of the group is discussed and proved. Furthermore it is shown that with nontrivial coefficients, all these duplicial sets arise from a construction due to Bohm and Stefan. Next, the self-duality of the paracyclic category is extended to a certain class of homotopy categories of (2,1)-categories. These generalise the orbit category of a group and are associated to certain self-dual preorders equipped with a presheaf of groups and a cosieve. Slice 2-categories of equidimensional submanifolds of a compact manifold without oundary form a particular case, and for $S^1$, one recovers cyclic duality.

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