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In this project, bridging entropy econometrics, game theory and information theory, a game theoretic approach will be investigated to quantum information, during which new mathematical definitions for quantum relative entropy, quantum mutual information, and quantum channel capacity will be given and monotonicity of entangled quantum relative entropy and additivity of quantum channel capacity will be obtained rigorously in mathematics; also quantum state will be explored in Kelly criterion, during which game theoretic interpretations will be given to classical relative entropy, mutual information, and asymptotical information. In specific, after briefly introducing probability inequalities, C*-algebra, including von Neumann algebra, and quantum probability, we will overview quantum entanglement, quantum relative entropy, quantum mutual information, and entangled quantum channel capacity in the direction of R. L. Stratonovich and V. P. Belavkin, and upon the monotonicity property of quantum mutual information of Araki-Umegaki type and Belavkin-Staszewski type, we will prove the additivity property of entangled quantum channel capacities, extending the results of V. P. Belavkin to products of arbitrary quantum channel to quantum relative entropy of both Araki-Umegaki type and Belavkin-Staszewski type. We will obtain a sufficient condition for minimax theorem in an introduction to strategic game, after which, in the exploration of classical/quantum estimate (here we still use the terminology of quantum estimate in the sense of game theory in accordance to classical estimate, but NOT in the sense of quantum physics or quantum probability), we will find the existence of the minimax value of this game and its minimax strategy, and applying biconjugation in convex analysis, we will arrive at one new approach to quantum relative entropy, quantum mutual entropy, and quantum channel capacity, in the sense, independent on Radon-Nikodym derivative, also the monotonicity of quantum relative entropy and the additivity of quantum communication channel capacity will be obtained. Applying Kelly's criterion, we will give a practical game theoretic interpretation, in the model to identify quantum state, to relative entropy, mutual information, and asymptotical information, during which we will find that the decrement in the doubling rate achieved with true knowledge of the distribution F over that achieved with incorrect knowledge G is bounded by relative entropy R(F;G) of F relative to G; the increment [Delta] in the doubling rate resulting from side information Y is less than or equal to the mutual information I(X,Y); a good sequence to identify the true quantum state leads to asymptotically optimal growth rate of utility; and applying the asymptotic behavior of classical relative entropy, the utility of the Bayes' strategy will be bounded below in terms of the optimal utility. The first two main parts are to extend classical entropy econometrics, in the view of strategic game theory, to non-commutative data, for example, quantum data in physical implementation, while the third main part is to intrinsically and practically give a game theoretic interpretation of classical relative entropy, mutual information, and asymptotical information, in the model to identify quantum state, upon which a pregnant financial stock may be designed, which may be called "quantum" stock, for its physical implementation.

A matching in a graph is a set of pairwise nonadjacent edges, a k-factor is a k-regular spanning subgraph, and a cycle is a closed path. This thesis has two parts. In Part I (by far the larger part) we study sufficient conditions for structures involving matchings, factors and cycles. The three main types of conditions involve: the minimum degree; the degree sum of pairs of nonadjacent vertices (Ore-type conditions); and the neighbourhoods of independent sets of vertices. We show that most of our theorems are best possible by giving appropriate extremal graphs. We study Ore-type conditions for a graph to have a Hamilton cycle or 2-factor containing a given matching or path-system, and for any matching and single vertex to be contained in a cycle. We give Ore-type and neighbourhood conditions for a matching L of l edges to be contained in a matching of k edges (l < k). We generalise two different aspects of this result: the l = 0 case with an Ore-type condition for a heavy matching in an edge-weighted graph; and the conditions for a perfect matching containing L with degree and neighbourhood conditions for a k-factor (k > 2) containing a given set of edges. We also establish neighbourhood conditions for the existence of a cycle of length at least k. A list-edge-colouring of a graph is an assignment of a colour to each edge from its own list of colours. In Part II we study edge colourings of powers of cycles, and prove the List-Edge-Colouring Conjecture for squares of cycles of odd length.

In this thesis two quantum search algorithms on two different graphs, a hypercube and a d-dimensional square lattice, are analysed and some applications of the lattice search are discussed. The approach in this thesis generalises a picture drawn by Shenvi, Kempe and Whaley, which was later adapted by Ambainis, Kempe and Rivosh. It defines a one parameter family of unitary operators U_λ with parameter λ. It will be shown that two eigenvalues of U_λ form an avoided crossing at the λ-value where U_λ is equal to the old search operator. This generalised picture opens the way for a construction of two approximate eigen- vectors at the crossing and gives rise to a 2×2 model Hamiltonian that is used to approximate the operator U_λ near the crossing. The thus defined Hamiltonian can be used to calculate the leading order of search time and success probability for the search. To the best of my knowledge only the scaling of these quantities has been known. For the algorithm searching the regular lattice, a generalisation of the model Hamiltonian for m target vertices is constructed. This algorithm can be used to send a signal from one vertex of the graph to a set of vertices. The signal is transmitted between these vertices exclusively and is localised only on the sender and the receiving vertices while the probability to measure the signal at one of the remaining vertices is significantly smaller. However, this effect can be used to introduce an additional sender to search settings and send a continuous signal to all target vertices where the signal will localise. This effect is an improvement compared to the original search algorithm as it does not need to know the number of target vertices.

A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a graph is outerplanar if it is both K4-minor-free and K2,3-minor-free, then by a near-outerplanar graph we mean a graph that is either K4-minor-free or K2,3-minor-free. Chapter 1 gives an introduction to the area of graph colourings, and includes a review of several results and conjectures in this area. In particular, four important and interesting conjectures in graph theory are the List-Edge-Colouring Conjecture (LECC), the List-Total-Colouring Conjecture (LTCC), the Entire Colouring Conjecture (ECC), and the List-Square-Colouring Conjecture (LSCC), each of which will be discussed in Chapter 1. In Chapter 2 we include a proof of the LECC and LTCC for all near-outerplanar graphs. In Chapter 3 we will study the list-colouring of a near-outerplanar graph in which vertices and faces, edges and faces, or vertices, edges and face are to be coloured. The results for the case when all elements are to be coloured will prove the ECC for all near-outerplanar graphs. In Chapter 4 we will study the list-colouring of the square of a K4-minor-free graph, and in Chapter 5 we will study the list-colouring of the square of a K2,3-minor-free graph. In Chapter 5 we include a proof of the LSCC for all K2,3-minor-free graphs with maximum degree at least six.

This work considers the problem of Equations Over Groups and settles the KL- conjecture for equations of length five. Firstly, the problem of equations over groups is stated and discussed and the results, which were up to now obtained, are presented. Then, by way of contradiction, it is assumed that for the remaining cases of equations of length five a solution does not exist. The methodology adopted uses the combinatorial and topological arguments of relative diagrams. If D is a relative diagram representing the counter example, all types of interior regions of positive curvature are listed for each type of equation of length five. For each interior region of positive curvature, one region of negative curvature is found and the positive curvature is added to it to obtain the total curvature in the interior of diagram D. In the final chapter the curvature of the interior of D is added to the curvature of the boundary regions to obtain the total curvature of the diagram. It is proved that the total curvature of 4pi cannot be achieved, our desired contradiction, and therefore equations of length five have a solution.

In this thesis we study some graph colouring problems which arise from mathematical models of frequency assignment in radiocommunications networks, in particular from models formulated by Hale and by Tesman in the 1980s. The main body of the thesis is divided into four chapters. Chapter 2 is the shortest, and is largely self-contained; it contains some early work on the frequency assignment problem, in which each edge of a graph is assigned a positive integer weight, and an assignment of integer colours to the vertices is sought in which the colours of adjacent vertices differ by at least the weight of the edge joining them. The remaining three chapters focus on problems which combine frequency assignment with list colouring, in which each vertex has a list of integers from which its colour must be chosen. In Chapter 3 we study list colourings where the colours of adjacent vertices must differ by at least a fixed integer s, and in Chapter 4 we add the additional restriction that the lists must be sets of consecutive integers. In both cases we investigate the required size of the lists so that a colouring can always be found. By considering the behaviour of these parameters as s tends to infinity, we formulate continuous analogues of the two problems, considering lists which are real intervals in Chapter 4, and arbitrary closed real sets in Chapter 5. This gives rise to two new graph invariants, the consecutive choosability ratio tau(G) and the choosability ratio sigma(G). We relate these to other known graph invariants, provide general bounds on their values, and determine specific values for various classes of graphs.

The thesis is divided into five sections: (a) Trigonometric sums involving prime numbers and applications, (b) Mean-values and Sign-changes of S(t)-- related to Riemann's Zeta function, (c) Mean-values of strongly additive arithmetical functions, (d) Combinatorial identities and sieves, (e) A Goldbach-type problem. Parts (b) and (c) are related by means of the techniques used but otherwise the sections are disjoint. (a) We consider the question of finding upper bounds for sums like ∑_PSN▒〖e(ap2)〗 and using a method of Vaughan, we get estimates which are much better than those obtained by Vinogradov. We then consider two applications of these, namely, the distribution of the sequence (αp2) modulo one. Of course we could have used the improved results to get improvements in estimates in various other problems involving p[superscript]2 but we do not do so. We also obtain an estimate for the sum ∑_PSN▒〖(ap3)〗 and get improved estimates by the same method. (b) Let N(T) denote the number of zeros of ς(s) - Riemann's Zeta function. It is well known that N(T) = L(T) + S(T), where L(T) = 1/2π Tlog(T/2π)-T⁄(2π+7⁄(8+0 ((1)⁄(T))))but the finer behaviour of S(T) is not known. It is known that S(t) ≪ log t ; ∫_o^t▒〖Slu)du〗 ≪ log t so that S(T) has many changes of sign. In 1942, A. Selberg showed that the number of sign changes of Set) for t ∈ (O,T) exceeds T (log T)1/3 exp(-A loglog T), (1) but stated to Professor Halberstam in 1979 that one can improve the constant 1/3 in (1) to 1 – ∈. It can be shown easily that the upper bound for the number of changes of sign is log T. We give a proof of Selberg's statement in (b), but in the process we do much more. Selberg showed that if k is a positive integer then ∫_T^(T+H)▒〖ls(t)l〖2k〗_dt 〗 = C CkH(loglog T) k ,{1+0( (loglogT)(-1)/2) } (2) where TT 1⁄2< H ≤ T[superscript]2 and C[subscript]k is some explicit constant in k. We have found a simple technique which gives (2) with the constant k replaced by any non-negative real number. Using this type of result, I prove Selberg's statement, with (log T)-∈ replaced by Exp (-A√loglogT (logloglogT) -□(1/2)). (c) I use the" method for finding mean-values above to answer similar questions for a class of strongly additive arithemetical functions. We say that f is strongly additive if (1) f(mn) = f(m) = f(n), if m and n are coprime, (2) f(p[superscript]a) = f(p) for all primes p and positive integer a. (d) This section contains joint work with Professor Halberstam and is still in its infancy. We have found a general identity and a type of convolution which serves to be the starting point of most investigations in Prime Number Theory involving the local and the global sieves. The term global refers to sieve methods of Brun, Selberg, Rosser and many more. The term local refers to things like Selberg's formula in the elementary proof of the prime number theorem, Vaughan's identity and so on. We have shown that both methods stem from the same source and so leads to a unified approach to such research. (e) I considered the question of solving the representation of an integer N in the form N = P_(1^2 )+ P _(2^2 ) +K P[subscript] 3, where the Pi’s are prime numbers. This problem was motivated by Goldbach's Problem and is exceedingly difficult. So I looked into getting partial answers. Let E(x) denote the numbers less than x not representable in the required form. Then there is a computable constant δ > 0 such that E(x) ≪ X i- δ To do this we use a method of Montgomery and Vaughan but the proof is long and technical, and we do not give it here. We show by sieve methods that the following result holds true: N = P_(1^2 ) + P _(2^2 ) +kP3P4P5. We have been unable to replace the product of three primes by two. Note: k is a constant depending on the residue class of N modulo 12.

In this thesis, we shall investigate chromatic polynomials of graphs, and some related polynomials. In Chapter 1, we study the chromatic polynomial written in a modified form, and use these results to characterise the chromatic polynomials of polygon trees. In Chapter 2, we consider the chromatic polynomial written as a sum of the chromatic polynomials of complete graphs; in particular, we determine for which graphs the coefficients are symmetrical, and show that the coefficients exhibit a skewed property. In Chapter 3, we dualise many results about chromatic polynomials to flow polynomials, including the results in Chapter 1, and a result about a zero-free interval. Finally, in Chapter 4, we investigate the zeros of the Tutte Polynomial; in particular their observed proximity to certain hyperbole in the xy-plane.

Discrete breathers are time-periodic and spatially localised exact solutions in translationally invariant nonlinear lattices. They are generic solutions, since only moderate conditions are required for their existence. Closed analytic forms for breather solutions are generally not known. We use asymptotic methods to determine both the properties and the approximate form of discrete breather solutions in various lattices. We find the conditions for which the one-dimensional FPU chain admits breather solutions, generalising a known result for stationary breathers to include moving breathers. These conditions are verified by numerical simulations. We show that the FPU chain with quartic interaction potential supports long-lived waveforms which are combinations of a breather and a kink. The amplitude of classical monotone kinks is shown to have a nonzero minimum, whereas the amplitude of breathing-kinks can be arbitrarily small. We consider a two-dimensional FPU lattice with square rotational symmetry. An analysis to third-order in the wave amplitude is inadequate, since this leads to a partial differential equation which does not admit stable soliton solutions for the breather envelope. We overcome this by extending the analysis to higher-order, obtaining a modified partial differential equation which includes known stabilising terms. From this, we determine regions of parameter space where breather solutions are expected. Our analytic results are supported by extensive numerical simulations, which suggest that the two-dimensional square FPU lattice supports long-lived stationary and moving breather modes. We find no restriction upon the direction in which breathers can travel through the lattice. Asymptotic estimates for the breather energy confirm that there is a minimum threshold energy which must be exceeded for breathers to exist in the two-dimensional lattice. We find similar results for a two-dimensional FPU lattice with hexagonal rotational symmetry.