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This thesis concerns the study of the Lefschetz properties of artinian monomial algebras. An artinian algebra is said to satisfy the strong Lefschetz property if multiplication by all powers of a general linear form has maximal rank in every degree. If it holds for the first power it is said to have the weak Lefschetz property (WLP). In the first paper, we study the Lefschetz properties of monomial algebras by studying their minimal free resolutions. In particular, we give an afirmative answer to an specific case of a conjecture by Eisenbud, Huneke and Ulrich for algebras having almost linear resolutions. Since many algebras are expected to have the Lefschetz properties, studying algebras failing the Lefschetz properties is of a great interest. In the second paper, we provide sharp lower bounds for the number of generators of monomial ideals failing the WLP extending a result by Mezzetti and Miró-Roig which provides upper bounds for such ideals. In the second paper, we also study the WLP of ideals generated by forms of a certain degree invariant under an action of a cyclic group. We give a complete classication of such ideals satisfying the WLP in terms of the representation of the group generalizing a result by Mezzetti and Miró-Roig. / <p>QC 20180220</p>

The study of stochastic queueing networks is quite important due to the many applications including transportation, telecommunication, and manufacturing industries. Since there is often no explicit solution to these types of control problems, numerical methods are needed. Following the method of Boué-Dupuis, we use a Dynamic Programming approach of optimization on a controlled Markov Chain that simulates the behavior of a fluid limit of the original process. The search for an optimal control in this case involves a Skorokhod problem to describe the dynamics on the boundary of closed, convex domain. Using relaxed stochastic controls we show that the approximating numerical solution converges to the actual solution as the size of the mesh in the discretized state space goes to zero, and illustrate with an example. / Ph. D.

This dissertation is intended to provide a statistical foundation for the IV models and shed lights on a number of issues related to the IV method. The first chapter shows that the theoretical Instrumental Variable model can be derived by reparameterization of a well-specified statistical model defined on the joint distribution of the involved random variables as the actual (local) data generation process. This reveals the covariance structure of the error terms of the usual theory-driven instrumental variable model. The revealed covariance structure of the IV model have important implications, particularly, for designing simulation studies. Monte Carlo simulations are used to reexamine the Nelson and Startz (1990a) findings regarding the performance of IV estimators when the instruments are weak. The results from the simulation exercises indicate that the sampling distribution of ^Î _IV is concentrated around ^Î _OLS. The second chapter considers the underlying joint distribution function of the instrumental variable (IV) model and presents an alternative definition for the exogenous and relevant instruments. The paper extracts a system of independent and orthogonal equations that covers up a non-orthogonal structural model and argues that the estimated IV regression is well-specified if the underlying system of equations is well-specified. It proposes a new instrument relevancy measure that does not suffer from the first-stage <i>R²</i> deficiencies. Third chapter argues the application of the IV method in estimation of models with omitted variable. The paper considers the implicit parametrization of statistical models and presents five conditions for an appropriate instruments. Two of them are empirically measurable and can be tested. This improves the literature by adding one more objective criterion for the selection of instruments. This chapter applies the IV method to estimate the rate of return to education in Iran. It argues that the education of two cohorts of Iranians was delayed or cut short by the Cultural Revolution. Therefore, the Cultural Revolution, as an exogenous shock to the supply of education, establishes the year of birth as the exogenous and relevant instrument for education. Using the standard Mincerian earnings function with control for experience, ethnicity, location of residence and sector of employment, the instrumental variable estimate of the return to schooling is equal to 5.6&#37;. The estimation results indicate that the Iranian labor market values degrees more than years of schooling. / Ph. D.