Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.

Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas.

This thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.

New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs near-optimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly efficient. Explicit convergence rates are shown for the singular values of differentiable and separately analytic functions, and examples are given to demonstrate some paradoxical features of low rank approximation theory. Analogues of QR, LU, and Cholesky factorizations are introduced for matrices that are continuous in one or both directions, deriving a continuous linear algebra. New notions of triangular structures are proposed and the convergence of the infinite series associated with these factorizations is proved under certain smoothness assumptions. A robust numerical bivariate rootfinder is developed for computing the common zeros of two smooth functions via a resultant method. Using several specialized techniques the algorithm can accurately find the simple common zeros of two functions with polynomial approximants of high degree (≥ 1,000). Lastly, low rank ideas are extended to linear partial differential equations (PDEs) with variable coefficients defined on rectangles. When these ideas are used in conjunction with a new one-dimensional spectral method the resulting solver is spectrally accurate and efficient, requiring O(n²) operations for rank $1$ partial differential operators, O(n³) for rank 2, and O(n⁴) for rank &geq,3 to compute an n x n matrix of bivariate Chebyshev expansion coefficients for the PDE solution. The algorithms in this thesis are realized in a software package called Chebfun2, which is an integrated two-dimensional component of Chebfun.

Integrable systems arise in nonlinear processes and, both in their classical and quantum version, have many applications in various fields of mathematics and physics, which makes them a very active research area. In this thesis, the problem of integrability of multidimensional equations, especially in three dimensions (3D), is explored. We investigate systems of differential, differential-difference and discrete equations, which are studied via a novel approach that was developed over the last few years. This approach, is essentially a perturbation technique based on the so called method of dispersive deformations of hydrodynamic reductions . This method is used to classify a variety of differential equations, including soliton equations and scalar higher-order quasilinear PDEs. As part of this research, the method is extended to differential-difference equations and consequently to purely discrete equations. The passage to discrete equations is important, since, in the case of multidimensional systems, there exist very few integrability criteria. Complete lists of various classes of integrable equations in three dimensions are provided, as well as partial results related to the theory of dispersive shock waves. A new definition of integrability, based on hydrodynamic reductions, is used throughout, which is a natural analogue of the generalized hodograph transform in higher dimensions. The definition is also justified by the fact that Lax pairs the most well-known integrability criteria are given for all classification results obtained.

No description available.

This thesis is about double product integrals with pseudo rotational generator, and aims to exhibit them as unitary implementors of Bogolubov transformations. We further introduce these concepts in this abstract and describe their roles in the thesis's chapters. The notion of product integral, (simple product integral, not double) is not a new one, but is unfamiliar to many a mathematician. Product integrals were first investigated by Volterra in the nineteenth century. Though often regarded as merely a notation for solutions of differential equations, they provide a priori a multiplicative analogue of the additive integration theories of Riemann, Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject. Extensions of the theory of product integrals to multiplicative versions of Ito and especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy in the 1980's, Hudson et al. [1982]. The first developments of double product integrals was a theory of an algebraic kind developed by Hudson and Pulmannova motivated by the study of the solution of the quantum Yang-Baxter equation by the construction of quantum groups, see Hudson and Pulmaanova [2005]. This was a purely algebraic theory based on formal power series in a formal parameter. However, there also exists a developing analytic theory of double product integral. This thesis contributes to this analytic theory. The first papers in that direction are Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum extension of Girsanov's theorem and hence a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics. From a different direction "causal" double products, (see Hudson [2005b]), have become of interest in connection with quantum versions of the Levy area, and in particular quantum Levy area formula (Hudson [2011] and Chen and Hudson [2013]) for its characteristic function. There is a close association of causal double products with the double products of rectangular type (Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward" rectangular double products. In the first chapter we give our notation which will be used in the following chapters and we introduce some simple double products and show heuristically that they are the solution of two different quantum stochastic differential equations. For each example the order in which the products are taken is shown to be unimportant; either calculation gives the same answer. This is in fact a consequence of the so called multiplicative Fubini Theorem Hudson and Pulmaanova [2005]. In Chapter two we formally introduce the notion of product integral as a solution of two particular quantum stochastic differential equations. In Chapter three we introduce the Fock representation of the canonical commutation relations, and discuss the Stone-von Neumann uniqueness theorem. We define the notion of Bogolubov transformation (often called a symplectic automorphism, see Parthasarathy [1992] for example), implementation of these transformations by an implementor (a unitary operator) and introduce Shale's theorem which will be relevant to the following chapters. For an alternative coverage of Shale's Theorem, symplectic automorphism and their implementors see Derezinski [2003]. In Chapter four we study double product integrals of the pseudo rotational type. This is in contrast to double product integrals of the rotational type that have been studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the product integral is suggestive of a natural discretisation scheme where the infinitesimals are replaced by discrete increments i.e. discretised creation and annihilation operators of quantum mechanics. Because of a weak commutativity condition, between the discretised creation and annihilation operators corresponding on different subintervals of R, the order of the factors of the product are unimportant (Hudson [2005a]), and hence the discrete product is well defined; we call this result the discrete multiplicative Fubini Theorem. It is also the case that the order in which the products are taken in the continuous (non-discretised case) does not matter (Hudson [2005a], Hudson and Jones [2012]). The resulting discrete double product is shown to be the implementor (a unitary operator) of a Bogolubov transformation acting on discretised creation and annihilation operators (Bogolubov transformations are invertible real linear operators on a Hilbert space that preserve the imaginary part of the inner product, but here we may regard them equivalently as liner transformations acting directly on creation and annihilations operators but preserving adjointness and commutation relations). Unitary operators on the same Hilbert space are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov transformations are used to construct new canonical pairs from old ones (In the literature Bogolubov transformations are often called symplectic automorphisms). The aforementioned Bogolubov transformation (acting on the discretised creation and annihilation operators) can be embedded into the space L2(R+) L2(R+) and limits can be taken resulting in a limiting Bogolubov transformation in the space L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov transformation has three important properties, namely bi-evolution, shift covariance and time-reversal covariance, see (Hudson [2007]) for a detailed description of these properties. Subsequently we show rigorously that this transformation really is a Bogolubov transformation. We remark that these transformations are Hilbert-Schmidt perturbations of the identity map and satisfy a criterion specified by Shale's theorem. By Shale's theorem we then know that each Bogolubov transformation is implemented in the Fock representation of the CCR. We also compute the constituent kernels of the integral operators making up the Hilbert-Schmidt operators involved in the Bogolubov transformations, and show that the order in which the approximating discrete products are taken has no bearing on the final Bogolubov transformation got by the limiting procedure, as would be expected from the multiplicative Fubini Theorem. In Chapter five we generalise the canonical form of the double product studied in Chapter four by the use of gauge transformations. We show that all the theory of Chapter four carries over to these generalised double product integrals. This is because there is unitary equivalence between the Bogolubov transformation got from the generalised canonical form of the double product and the corresponding original one. In Chapter six we make progress towards showing that a system of implementors of this family of Bogolubov transformations can be found which inherits properties of the original family such as being a bi-evolution and being covariant under shifts. We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More specifically, Shale's theorem ensures that each Bogolubov transformation of our system is implemented by a unitary operator which is unique to with multiplicaiton by a scalar of modulus 1. We expect that there is a unique system of implementors, which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which inherits the properties of the corresponding system of Bogolubov transformation). This is partly on-going research. We also expect the implementor of the Bogolubov transformation to be the original double product. In Evans [1988], Evan's showed that the the implementor of a Bogolubov transformation in the simple product case is indeed the simple product. If given more time it might be possible to adapt Evan's result to the double product case.

There are many problems that lead to analysis of dynamical systems with phase variables of two types, slow and fast ones. Such systems are called slow-fast systems. The dynamics of such systems is usually described by means of different versions of perturbation theory. Many questions about accuracy of this description are still open. The difficulties are related to presence of resonances. The goal of the proposed thesis is to establish some estimates of the accuracy of the perturbation theory for slow-fast systems in the presence of resonances. We consider slow-fast Hamiltonian systems and study an accuracy of one of the methods of perturbation theory: the averaging method. In this thesis, we start with the case of slow-fast Hamiltonian systems with two degrees of freedom. One degree of freedom corresponds to fast variables, and the other degree of freedom corresponds to slow variables. Action variable of fast sub-system is an adiabatic invariant of the problem. Let this adiabatic invariant have limiting values along trajectories as time tends to plus and minus infinity. The difference of these two limits for a trajectory is known to be exponentially small in analytic systems. We obtain an exponent in this estimate. To this end, by means of iso-energetic reduction and canonical transformations in complexified phase space, we reduce the problem to the case of one and a half degrees of freedom, where the exponent is known. We then consider a quasi-linear Hamiltonian system with one and a half degrees of freedom. The Hamiltonian of this system differs by a small, ~ε, perturbing term from the Hamiltonian of a linear oscillatory system. We consider passage through a resonance: the frequency of the latter system slowly changes with time and passes through 0. The speed of this passage is of order of ε. We provide asymptotic formulas that describe effects of passage through a resonance with an improved accuracy O(ε3/2). A numerical verification is also provided. The problem under consideration is a model problem that describes passage through an isolated resonance in multi-frequency quasi-linear Hamiltonian systems. We also discuss a resonant phenomenon of scattering on resonances associated with discretisation arising in a numerical solving of systems with one rotating phase. Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. For arbitrarily small time step of a numerical method, discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances, that is absent in the original system.

The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ is the origin, this is just the classical Funk–Radon transform. We find two mappings from the sphere to itself that enable us to represent the generalized Radon transform in terms of the Funk–Radon transform. This representation is utilized to characterize the nullspace and range as well as to prove an inversion formula of the generalized Radon transform.

Το όρος Γκιώνα είναι το υψηλότερο βουνό της Στερεάς Ελλάδας και το πέμπτο μεγαλύτερο της Ελλάδας. Έχει έκταση περίπου 300.000 στρεμμάτων, χωρίζει κάθετα τη Στερεά Ελλάδα σε δυτική και ανατολική και αριθμεί εικοσιτέσσερις (24) κορυφές με υψόμετρο μεγαλύτερο των 2.000 μ. Αρχικά, η παρούσα διατριβή ερευνά τη χλωρίδα της εξεταζόμενης περιοχής, η οποία αποτελείται από 1.273 taxa, 572 εκ των οποίων, αναφέρονται για πρώτη φορά. Η γεωγραφική θέση του συγκεκριμένου ορεινού όγκου στη Στερεά Ελλάδα, ο οποίος γειτνιάζει με τα υπόλοιπα ψηλά βουνά αυτής της περιφέρειας (Βαρδούσια, Οίτη, Παρνασσό), αποτελεί σημείο συνάντησης φυτών με διαφορετική γεωγραφική εξάπλωση. Αυτό το δεδομένο αποκτά ακόμη μεγαλύτερη αναλυτική σημασία στην περίπτωση των ενδημικών στοιχείων, καθώς το ορεινό συγκρότημα της Γκιώνας με το επιβλητικό τοπίο, τα μεγάλα υψόμετρα, την εκτεταμένη αλπική ζώνη και τη συνολικά μεγάλη του έκταση υποστηρίζει ένα πολύ υψηλό ποσοστό ενδημισμού (11,88%). Αξιοσημείωτο είναι ότι μεταξύ των φυτικών στοιχείων που εξετάσαμε, περιλαμβάνονται και 171 taxa, τα οποία είναι σπάνια ή προστατευόμενα. Επιπρόσθετα, διερευνήσαμε τις φυτογεωγραφικές σχέσεις μεταξύ των βουνών της Στερεάς Ελλάδας και της Πελοποννήσου, όπου διαφαίνεται η υψηλή χλωριδική συγγένεια του όρους Γκιώνα με τα όρη Παρνασσός και Βαρδούσια. Στη συνέχεια, μελετήσαμε τη βλάστηση της εξεταζόμενης περιοχής, όπου εντοπίστηκαν έντεκα (11) ενότητες βλάστησης. Από την επεξεργασία των δειγματοληψιών, διακρίθηκαν δεκαεννέα (19) τύποι οικοτόπων, με οκτώ (8) φυτοκοινότητες να αναφέρονται για πρώτη φορά. Για την αξιολόγηση των ανθρώπινων επιδράσεων που λαμβάνουν χώρα στο όρος Γκιώνα εφαρμόστηκε το πλαίσιο DPSIR, το οποίο εφαρμόζεται για πρώτη φορά σε ορεινό οικοσύστημα στην Ελλάδα. Στην παρούσα μελέτη αναλύονται πενήντα πέντε (55) δείκτες του προαναφερθέντος πλαισίου αξιολόγησης, ενώ ιδιαίτερη βαρύτητα δίνεται στις πιέσεις που ασκούνται από τα μεταλλεία βωξίτη στην περιοχή. Τα διαχειριστικά μέτρα, τα οποία προτείνονται, λαμβάνουν υπόψη όλες τις εξεταζόμενες παραμέτρους αξιολόγησης και αποτελούν ένα χρήσιμο εργαλείο για τους παράγοντες λήψης αποφάσεων. Όσα προαναφέρθηκαν συνεπικουρούνται από μία σειρά θεματικών χαρτών, τους οποίους κατασκευάσαμε με τη χρήση GIS και απεικονίζουν την παρούσα κατάσταση της περιοχής μελέτης. / Mount Giona is the highest mountain of the Sterea Ellas region and Greece’s fifth biggest mountain. It covers 30,000 hectares, divides Sterea Ellas into west and east and includes twenty-four (24) peaks above 2,000 m. Initially, the present thesis investigates the flora of the examined area, which comprises 1,273 taxa, 572 of which are reported here for the first time. The geographical position of Mount Giona in Sterea Ellas, which neighbors with the other high mountains of that region (namely: Vardousia, Oiti, Parnassos), constitutes a meeting point for plants with different geographical distributions. This fact is extremely important in the case of the endemic elements, which constitute a large percentage of the total flora (11.88%), due to the highly diversified landscape, the high altitudes, the extended alpine vegetation zone and the vast area covered by the mountain. The fact that 171 plant taxa of the total flora are rare or under some protection status, is also notable. Furthermore, we have examined the phytogeographical relation among the mountains of Sterea Ellas and Peloponnese, where the high floristic affinity of Mount Giona with the mountains Parnassos and Vardousia was revealed. Furthermore, we have studied the vegetation of the examined area, where we have recognized eleven (11) vegetation groups. The elaboration of vegetation samplings via vegetation analysis techniques showed nineteen (19) habitat types, with eight (8) plant communities being reported here for the first time. In order to evaluate the human impacts taking place on Mount Giona, we implemented the DPSIR causal framework, which is applied for the first time in a mountainous ecosystem in Greece. Fifty-five (55) indicators of the above-mentioned framework are analyzed in the present study, with special attention to the pressures caused by the bauxite mines on the area. The management measures, proposed here, have taken under consideration all the examined evaluation parameters and aim to be a valuable tool in the hands of decision makers. Finally, a series of thematic maps, created by means of GIS, has been designed in order to support our studies and reflect the present state of Mount Giona’s ecosystems.