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No description available.

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No description available.

This thesis gives some order estimates and asymptotic formulae associated with general classes of non-negative multiplicative functions as well as some particular multiplicative functions such as the divisor functions dk(n). In Chapter One we give a lower estimate for the number of distinct values assumed by the divisor function d(n) in 1 <n <x .We also identify the smallest positive integer which is a product of triangular numbers and not equal to d3(n) for 1 <n <x . In Chapter Two we show that if f(n) satisfies some conditions and if M=max {f(2a)}1/a, if a> or = 1 then the maximum value of f(n) in 1<n< x is about log x / Mloglog x. We also show that a function which has a finite mean value cannot be large too often. In Chapter Three we give an upper estimate to the average value of f(n) as n runs through a short interval in an arithmetic progression with a large modulus . As an application of our general theorem we show, for example, that if f(n) is the characteristic function of the set of integers which are the sum of two squares, then as x -> infinity. We call a positive integer n a k-full integer if pk divides n whenever p is a prime divisor of n, and in Chapter Four we give an asymptotic formula for the number of k-full integers not exceeding x. In Chapter Five we give an asymptotic formula for the number of 2-full integers in an interval. We also study the problem of the distribution of the perfect squares among the sequence of 2-full integers. The materials in the first three chapters have been accepted for publications and will appear [31], [22], [33] and [32].

Algebraic equations on complex numbers and functional equations on generating functions are often used to solve combinatorial problems. But the introduction of common arithmetic operators such as subtraction and division always causes panic in the world of objects which are generated from constants by applying products and coproducts. Over the years, researchers have been endeavouring to interpretate some absurd calculations on objects which lead to meaningful combinatorial results. This thesis investigates connections between algebraic equations on complex numbers and isomorphisms of recursively defined objects. We are attempting to work out conditions under which isomorphisms between recursively defined objects can be decided by equalities between polynomials on multi-variables with integers as coefficients.

This thesis studies topological and algebraic aspects of higher dimensional local fields and relations to other neighbouring research areas such as nonarchimedean functional analysis and higher dimensional arithmetic geometry. We establish how a higher local field can be described as a locally convex space once an embedding of a local field into it has been fixed. We study the resulting spaces from a functional analytic point of view: in particular we introduce and study bounded, c-compact and compactoid submodules of characteristic zero higher local fields. We show how these spaces are isomorphic to their appropriately topologized duals and study the implications of this fact in terms of polarity. We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields and their rings of integers and include higher local fields. Our results extend the constructions of Weil over (one-dimensional) local fields. We establish the existence of an appropriate topology on the set of rational points of schemes of finite type over the rings considered, study the functoriality of this construction and deduce several properties.

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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.