Symmetric polynomials are polynomials that are invariant under the action of the symmetric group, and they play an integral role in mathematics. The space of quasiin- variant polynomials, polynomials that are invariant under the action of the symmetric group to a certain order, were introduced by Feigin and Veselov. These spaces are modules over the ring of symmetric polynomials, and their Hilbert series in elds of characteristic 0 were also computed by Feigin and Veselov. In this paper, we study the Hilbert series of these spaces in elds of positive char- acteristic. Braverman, Etingof, and Finkelberg recently introduced spaces of twisted quasiinvariant polynomials, a generalization of quasiinvariant polynomials in which the space is twisted by a monomial. We extend some of their results to spaces twisted by a product of smooth functions and compute the Hilbert series of the space in certain cases.

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This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.

In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new %Akizuki-Kodaira-Nakano type vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

This paper studies the eect of risk-aversion in the competitive newsvendor game. Multiple newsvendors with risk-averse preferences face a random demand and the demand is allocated proportionally to their inventory levels. Each newsvendor aims to maximize his expected utility instead of his expected prot. Assuming a general form of risk-averse utility function, we prove that there exists a pure Nash equilibrium in this game, and it is also unique under certain conditions. We nd that the order quantity of each newsvendor is decreasing in the degree of risk-aversion and increasing in the initial wealth. Newsvendors with moderate preferences of risk-aversion make more prots compared with the risk-neutral situation. We also discuss the joint eect of risk-aversion and competition. If the eect of risk-aversion is strong enough to dominate the eect of competition, the total inventory level under competition will be lower than that under centralized decision-making.