We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones are cut-off polynomials, which consist of a single polynomial piece on [0,1] and vanish on [1,~). More precisely, for any given dimension n and prescribed C k smoothness, there is a function in Ck(R"), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degree B-splines.

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We experimentally optimize mixing of a turbulent round jet using machine learning control (MLC) following Li et al (2017). The jet is manipulated with one unsteady minijet blowing in wall-normal direction close to the nozzle exit. The flow is monitored with two hotwire sensors. The first sensor is positioned on the centerline 5 jet diameters downstream of the nozzle exit, i.e. the end of the potential core, while the second is located 3 jet diameters downstream and displaced towards the shear-layer. The mixing performance is monitored with mean velocity at the first sensor. A reduction of this velocity correlates with increased entrainment near the potential core. Machine Learning Control (MLC) is employed to optimize sensor feedback, a general open-loop broadband frequency actuation and combinations of both. MLC has identified the optimal periodic forcing with small duty cycle as the best control policy employing only 400 actuation measurements, each lasting for 5 seconds. This learning rate is comparable if not faster than typical optimization of periodic forcing with two free parameters (frequency and duty cycle). In addition, MLC re-sults indicate that neither new frequencies nor sensor feedback improves mixing further-contrary to many of other turbulence control experiments. The optimality of pure periodic actuation may be attributed to the simple jet flapping mechanism in the minijet plane. The performance of sensor feedback is shown to face a challenge for small duty cycles. The jet mixing results demonstrate the untapped potential of MLC in quickly learning optimal general control policies, even deciding between open-and closed-loop control.

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