We derive necessary and sufficient conditions for an infinite sequence of Radon measures to be realized by, or to be the sequence of moment functions of, a finite measure concentrated on a pre-given basic semi-algebraic subset of the space of generalized functions on Rd A set of such a kind is given by (not necessarily countable many) polynomial constraints. We get realizability conditions of semi definite type that can be more easily and efficiently verified, via semi definite programming, than the well-known Riesz-Haviland type condition. As a consequence, we characterize the support of the realizing measure in terms of its moments functions. As concrete examples of basic semi-algebraic sets of generalized functions, we present the set of all Radon measures, the set of all bounded Radon measures with Radon-Nikodym density w.r.t. the Lebesgue measure, the set of all probabilities, the set of all sub-probabilities and the set of all point configurations. These examples are considered in numerous areas of applications dealing with the description of large complex system. Our approach is based on a combination of classical results about the moment problem on nuclear spaces and of techniques developed to solve the moment problem on basic semi-algebraic sets of IRd For this reason, we provide a unified exposition of some aspects of the classical real moment problem which have inspired our main result. Particular importance is given to criteria for existence and uniqueness of the realizing measure on IRd via the multivariate Carleman condition and the operator-theoretical approach. We also give a formulation of the moment problem on general finite dimensional spaces in duality which makes clear the analogies with the infinite dimensional moment problem on nuclear spaces.

In this thesis, we look at problems in Number Theory, specifically Diophantine Equations. We investigate Fermat Quartic curves, by presenting a set of methods to determine the existence of rational points on them. We also consider a method of resolving bielliptic curves of genus 2. We show that the method cycles for an infinite family of curves, and find an example where the method fails, however often it is repeatedly applied.

The first problem we consider is a variation of the Piatetski-Shapiro Prime Number Theorem. Consider a function g(y), growing faster than linearly. We ask how often is the integer part of a function g(y) no less than some distance j from a prime number Using Huxley's method of exponential sums the investigation shows how the rate at which g(y) increases is dependent on the size of j. The faster g(y) increases, the larger the value of j. The second problem investigates primes of arithmetic progressions, a mod g, in short intervals of the form (x, x+xe), where x is sufficiently large in terms of q, cp < x for some 77 > 0. Such a result was proved by Fogels, for some 6 < 1. We explicitly determine the relationship between 6 and 77 to establish admissible values for both. Lastly we use our version of Fogels' theorem and a variation of Vaughan's treatment of the ctp problem to investigate the following problem. Given a real number a in the interval (0,1) how many Farey fractions of the Farey sequence of order Q do we have to pass to go from a to a Farey fraction with prime denominator

Selberg's upper bound method provides rather good results in certain circumstances. We wish to apply ideas from this upper bound method to that of the lower bound sifting problem. The sum G(x) arises in Selberg's method and in this account we study the related sum Hz(x). We provide an asymptotic estimate for the sum Hz(x) by investigating the residual sum Iz(x) = Hz(oo) Hz(x) and transferring back to Hz(x). We obtain a lower bound for the sum which counts the number of a G A with the logarithmic weight log pj log z attached to the smallest prime factor of the number a subject to the condition v(D, A) < R combining ideas from Selberg's A2A" method with Richert's weights. v(D, A) counts the number of prime factors p of a number a according to multiplicity when p > D but counting each p at most once when p < D.

This thesis is an accumulation of work regarding a class of constrained Euclidean Distance Matrix (EDM) based optimization models and corresponding numerical approaches. EDM-based optimization is powerful for processing distance information which appears in diverse applications arising from a wide range of fields, from which the motivation for this work comes. Those problems usually involve minimizing the error of distance measurements as well as satisfying some Euclidean distance constraints, which may present enormous challenge to the existing algorithms. In this thesis, we focus on problems with two different types of constraints. The first one consists of spherical constraints which comes from spherical data representation and the other one has a large number of bound constraints which comes from wireless sensor network localization. For spherical data representation, we reformulate the problem as an Euclidean dis-tance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at its each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the exibility of the algorithm in incorporating various constraints. For wireless sensor network localization, we set up a convex optimization model using EDM which integrates connectivity information as lower and upper bounds on the elements of EDM, resulting in an EDM-based localization scheme that possesses both effciency and robustness in dealing with flip ambiguity under the presence of high level of noises in distance measurements and irregular topology of the concerning network of moderate size. To localize a large-scale network effciently, we propose a patching-stitching localization scheme which divides the network into several sub-networks, localizes each sub-network separately and stitching all the sub-networks together to get the recovered network. Mechanism for separating the network is discussed. EDM-based optimization model can be extended to add more constraints, resulting in a exible localization scheme for various kinds of applications. Numerical results show that the proposed algorithm is promising.

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We consider the unicity of types for various classes of supercuspidal representations of reductive p-adic groups, with a view towards establishing instances of the inertial Langlands correspondence. We introduce the notion of an archetype, which we define to be a conjugacy class of typical representations of maximal compact subgroups. In the case of supercuspidal representations of a special linear group, we generalize the functorial results of Bushnell and Kutzko relating simple types in GLN(F) and SLN(F) to cover all archetypes; from this we deduce that any archetype for a supercuspidal representation of SLN(F) is induced from a maximal simple type. We then provide an explicit description of the number of archetypes contained in a given supercuspidal representation of SLN(F). We next consider depth zero supercuspidal representations of an arbitrary group, where we are able to show that theonly archetypes are the depth zero types constructed by Morris. We end by showing that there exists a unique inertial Langlands correspondence from the set of archetypes contained in regular supercuspidal representations to the set of regular inertial types. In the cases of SLN(F) and depth zero supercuspidals of arbitrary groups, we describe completely the fibres of this inertial correspondence; in general, we formulate a conjecture on how these fibres should look for all regular inertial types.

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No description available.