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Abstract. Sensitivity analysis in chemical transport models quantifies the response of output variables to changes in input parameters. This information is valuable for researchers engaged in data assimilation and model development. Additionally, environmental decision-makers depend upon these expected responses of concentrations to emissions when designing and justifying air pollution control strategies. Existing sensitivity analysis methods include the finite-difference method, the direct decoupled method (DDM), the complex variable method, and the adjoint method. These methods are either prone to significant numerical errors when applied to nonlinear models with complex components (e.g. finite difference and complex step methods) or difficult to maintain when the original model is updated (e.g. direct decoupled and adjoint methods). Here, we present the implementation of the hyperdual-step method in the Community Multiscale Air Quality Model (CMAQ) version 5.3.2 as CMAQ-hyd. CMAQ-hyd can be applied to compute numerically exact first- and second-order sensitivities of species concentrations with respect to emissions or concentrations. Compared to CMAQ-DDM and CMAQ-adjoint, CMAQ-hyd is more straightforward to update and maintain, while it remains free of subtractive cancellation and truncation errors, just as those augmented models do. To evaluate the accuracy of the implementation, the sensitivities computed by CMAQ-hyd are compared with those calculated with other traditional methods or a hybrid of the traditional and advanced methods. We demonstrate the capability of CMAQ-hyd with the newly implemented gas-phase chemistry and biogenic aerosol formation mechanism in CMAQ. We also explore the cross-sensitivity of monoterpene nitrate aerosol formation to its anthropogenic and biogenic precursors to show the additional sensitivity information computed by CMAQ-hyd. Compared with the traditional finite difference method, CMAQ-hyd consumes fewer computational resources when the same sensitivity coefficients are calculated. This novel method implemented in CMAQ is also computationally competitive with other existing methods and could be further optimized to reduce memory and computational time overheads. 

Abstract We study the Yamabe flow starting from an asymptotically flat manifold $(M^n,g_0)$(M^{n},g_{0}).We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if $Y(M,\left[g_0\right])>0$Y(M,[g_{0}])>0, and show that the flow does not converge otherwise.If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow. 

Abstract In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset $\mathrm{\Omega }$\Omegaof the CR manifold ${\mathbb{S}}^3${{\mathbb{S}}}^{3}bounded by the Clifford torus $\mathrm{\Sigma }$\Sigmais discussed. The Yamabe-type problem of finding a contact form on $\mathrm{\Omega }$\Omegawhich has zero Tanaka-Webster scalar curvature and for which $\mathrm{\Sigma }$\Sigmahas a constant $p$p-mean curvature is also discussed. 

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.