In this paper we prove a new matrix Li-Yau-Hamilton (LYH) estimate for K¨ahler-Ricci flow on manifolds with nonnegative bi- sectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated in [Ch] by Chow. This new inequality is shown to be connected with Perelman’s entropy formula through a family of differential equalities. In the rest of the paper, we show several applications of this new estimate and its corresponding estimate for linear heat equation. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian [LT], a manifold version of Stoll’s theorem [St] on the characterization of ‘algebraic divisors’, and a localized monotonicity formula for analytic subvari- eties, which sharpens the Bishop volume comparison theorem. Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman [P], we prove a sharp lower bound of the fundamental solution to the forward conjugate heat equation, which in a certain sense dual to Perelman’s monotonicity of the reduced volume. As an application of this new monotonicity formula, we show that the blow-down limit of a certain type of long-time solution is a gradient expanding soliton, generalizing an earlier result of Cao. We also illustrate the connection between the new LYH estimate and the Hessian comparison theorem of [FIN] on the forward reduced distance. Localized monotonicity formulae on entropy and forward reduced volume are also derived.

This paper focuses on pricing and vertical cooperative advertising decisions in a two-tier supply chain. Using a Stackelberg game model where the manufacturer acts as the game leader and the retailer acts as the game follower, we obtain closed-form equilibrium solution and explicitly show how pricing and advertising decisions are made. When market demand decreases exponentially with respect to the retail price and increases with respect to national and local advertising expenditures in an additive way, the manufacturer benefits from providing percentage reimbursement for the retailer’s local advertising expenditure when demand price elasticity is large enough. Whether the manufacturer benefits from cooperative advertising is also closely related to supply chain member’s relative advertising efficiency. In the decision for adopting coop advertising strategy, it is critical for the manufacturer to identify how market demand depends on national and local advertisements. The findings from this research can enhance our understanding of cooperative advertising decisions in a two-tier supply chain with price-dependent demand.

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In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.

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