Abstract. Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the Fisher market game in which buyers strategize by posing different utility functions. We show that existence of a conflict-free allocation is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs.
We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property preservation under substructures modulo bounded cores, and present a syntactic characterization via Σ 0 2 sentences for properties of arbitrary structures definable by FO sentences. As a sharper characterization, we further show that the count of existential quantifiers in the Σ 0 2 sentence equals the size of the smallest bounded core. We also present our results on the sharper characterization for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the sharper characterization fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.
Chairman Date:Place: DeclarationI declare that this written submission represents my ideas in my own words and where others' ideas or words have been included, I have adequately cited and referenced the original sources.I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falsified any idea/data/fact/source in my submission. I understand that any violation of the above will be cause for disciplinary action by the Institute and can also evoke penal action from the sources which have thus not been properly cited or from whom proper permission has not been taken when needed.(Signature of student) and preservation under extensions, when k equals 0. As a consequence, we get a parameterized generalization of the Łoś-Tarski theorem for sentences, in both its substructural and extensional forms. We call our characterizations collectively the generalized Łoś-Tarski theorem for sentences at level k, abbreviated GLT(k). To the best of our knowledge, GLT(k) is the first to relate counts of quantifiers appearing in the sentences of the Σ In summary, the properties introduced in this thesis are interesting in both the classical and finite model theory contexts, and yield in both these contexts, a new parameterized generalization of the Łoś-Tarski preservation theorem.viii
Given two irreducible representations μ, ν of the symmetric group S d , the Kronecker problem is to find an explicit rule, giving the multiplicity of an irreducible representation, λ, of S d , in the tensor product of μ and ν. We propose a geometric approach to investigate this problem. We demonstrate its effectiveness by obtaining explicit formulas for the tensor product multiplicities, when the irreducible representations are parameterized by partitions with at most two rows.
In this work, we propose a structured computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a "local analysis", which covers parametrizations and singularities, as well as a "global theory" which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the "funnel" which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface-surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as decomposable and nondecomposable. Further, we construct an invariant function θ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between θ , local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach of testing points, θ is a computationally robust global function. It is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included.
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