Let λ * > 0 denote the largest possible value of λ such thathas a solution, where B is the unit ball in R n centered at the origin, p > n+4 n−4 and n is the exterior unit normal vector. We show that for λ = λ * this problem possesses a unique weak solution u * , called the extremal solution. We prove that u * is singular when n ≥ 13 for p large enough, in which case u * (x) ≤ r − 4 p−1 − 1 on the unit ball and actually solve part of the open problem which [9] left.
Decarburization refers to a phenomenon that the content of carbon in metal is reduced, and this could be advantageous or detrimental depending on the usage scenario. In this paper, a new feature, called Barkhausen frequency scale cepstrum coefficients (BFCCs), which was based on the cepstrum was proposed to evaluate the depth of the decarburized layer. The Genetic algorithm was used to optimize the parameters of the BFCCs feature. In the experiment part, the regression precision was compared under BFCCs and MBN envelope features using multiple linear regression (MLR) and BP neural network (BP). The regression experiment result was measured by mean absolute error (MAE) and shows the good representative ability of BFCCs for the depth of the decarburized layer.
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