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Gaussian processes are widely employed as versatile modelling and predictive tools in spa- tial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Mat´ern covariogram and heat kernel, the behaviour of asymptotic inference on the param- eters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Build- ing upon a recently introduced Mat´ern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Mat´ern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum like- lihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory 

The unstretched laminar flame speed (LFS) plays a key role in engine models and predictions of flame propagation. It is also an essential parameter in the study of turbulent combustion and can be directly used in many turbulent combustion models. Therefore, it is important to predict the laminar flame speed accurately and efficiently. Two improved correlations for the unstretched laminar flame speed, namely improved power law and improved Arrhenius form correlations, are proposed for iso-octane/air mixtures in this study, using simulated results for typical operating conditions for spark-ignition engines: unburned temperatures of 300-950 K, pressures of 1-120 bar, and equivalence ratios of 0.6-1.5. The original data points used to develop the new correlations were obtained using the detailed combustion kinetics for iso-octane from Lawrence Livermore National Laboratory (LLNL). The three coefficients in the improved power law correlation were determined using a methodology different from previous approaches. The improved Arrhenius form correlation employs a function of unburned gas temperature to replace the flame temperature, making the expression briefer and making the coefficients easier to calculate. The improved Arrhenius method is able to predict the trends and the values of laminar flame speed with improved accuracy over a larger range of operating conditions. The improved power law method also works well but for a relatively narrow range of predictions. The improved Arrhenius method is recommended, considering its overall fitting error was only half of that using the improved power law correlation and it was closer to the experimental measurements. Even though ϕm, the equivalence ratio at which the laminar flame speed reaches its maximum, is not monotonic with pressure, this dependence is still included, since it produces least-rich best torque (LBT). The comparisons between the improved correlations in this study and the experimental measurements and the other correlations from various researchers are shown as well.