Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of O(<i>h</i>²_<i>n</i>). In this note, a method of correcting the bias in the kernel density estimates is provided, which reduces the bias to a smaller order. Effectively, this method produces a higher order kernel based on a second order kernel. For a kernel function <i>K</i>, the functions W_k(x)=^k-1₁₌₀(^k_l+1)x^lK^(l)(x)/l! and [1/<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> d <i>x</i>]<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> are kernels of order <i>k</i>, under some mild conditions.

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In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

An orthogonal discrete auditory transform (ODAT) from sound signal to spectrum is constructed by combining the auditory spreading matrix of Schroeder et. al. and the time one map of a discrete nonlocal Schrodinger equation. Thanks to the dispersive smoothing property of the Schrodinger evolution, ODAT spectrum is a smoother than that of the discrete Fourier transform (DFT) consistent with human audition. ODAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. ODAT outperforms DFT in signal to noise ratio (SNR) when the noise level is relatively high.

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