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In this paper, we present a new approximation of the Wallis ratio. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation over other classical ones.

Primary squamous cell carcinoma (SCC) of the ovary in humans is rare. Most cases represent a malignant transformation of ovarian teratoma, Brenner tumor, or endometriosis. The etiology of this cancer remains largely unknown. Human papillomavirus (HPV) infection is a critical factor that induces tumor formation, particularly cervical cancer. Therefore, this study aimed to evaluate the association of HPV with malignant transformation of mature cystic teratoma (MCT) into SCC of the ovary.

Minimizing the nuclear norm is recently considered as the convex relaxation of the rank minimization problem and arises in many applications as Netflix challenge. A closest nonconvex relaxation - Schatten norm minimization has been proposed to replace the NP hard rank minimization. In this paper, an algorithm based on Majorization Minimization has be proposed to solve Schatten norm minimization. The numerical experiments show that Schatten norm with recovers low rank matrix from fewer measurements than nuclear norm minimization. The numerical results also indicate that our algorithm give a more accurate reconstruction

This paper proposes a least square regularized regression algorithm in sum space of reproducing kernel Hilbert spaces (RKHSs) for nonflat function approximation, and obtains the solution of the algorithm by solving a system of linear equations. This algorithm can approximate the low- and high-frequency component of the target function with large and small scale kernels, respectively. The convergence and learning rate are analyzed. We measure the complexity of the sum space by its covering number and demonstrate that the covering number can be bounded by the product of the covering numbers of basic RKHSs. For sum space of RKHSs with Gaussian kernels, by choosing appropriate parameters, we tradeoff the sample error and regularization error, and obtain a polynomial learning rate, which is better than that in any single RKHS. The utility of this method is illustrated with two simulated data sets and five real-life databases.

This article discusses the local regularity of refinable function vectors associated with a dilation matrix M. Suppose that D is a complete set of representatives of Zs/MZs. Under the assumptions that the self-affine tile T (M,D), associated with dilation matrix M and digit set D, has measure 1 and that the corresponding refinable function vector φ ∈ L∞, we prove that there is a set H ⫅ Rsof full measure such that the restriction φ|Hof φ on H has a positive Hölder exponent α(x) at every x ∈ H. Similar result holds for the derivatives of refinable function vectors provided that the dilation matrix is diagonalizable.

This note studies Bessel sequences and frames of shift-invariant spaces generated by a countable set of periodic functions. We give characterizations under which the set of translations of the countable set is a Bessel sequence or a frame in terms of spectral decompositions of some self-adjoint operators.

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms.

Subdivision operators play an important role in wavelet analysis. This paper studies the algebraic properties of subdivision operators with matrix mask, especially their action on polynomial sequences and on some of their invariant subspaces. As an application, we characterize, under a mild condition, the approximation order provided by refinable vectors in terms of the eigenvalues and eigenvectors of polynomial sequences of the associated subdivision operator. Moreover, some necessary conditions, in terms of nondegeneracy and simplicity of eigenvalues of a matrix related to the subdivision operator for the refinable vector to be smooth are given. The main results are new even in the scalar case