In this paper, we prove that for any Kähler metrics ω and χ on M, there exists a Kähler metric ω_φ = ω_0 + √-1 ∂ \bar∂ φ > 0 satisfying the J-equation tr_{ω_φ} χ = c if and only if (M, [ω0], [χ]) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kähler metrics with c_1 < 0. Using the same method, we also prove a similar result for the supercritical deformed Hermitian–Yang–Mills equation.

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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.

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It has been an open question whether the family of merit functions $\psi _p\(p&gt; 1) $, the generalized Fischer-Burmeister (FB) merit function, associated to the second-order cone is smooth or not. In this paper we answer it partly, and show that \psi _p is smooth for \psi _p , and we provide the condition for its coerciveness. Numerical results are reported to illustrate the influence of \psi _p on the performance of the merit function method based on \psi _p .