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In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on L p ([ - 1 , 1 ] s) for integers s ≥ 1 and 1 ≤ p < ∞ . However, their theoretical properties are largely unknown beyond universality of approximation or the existing analysis does not apply to the rectified linear unit (ReLU) activation function. To fill in this void, we investigate here the approximation power of functional deep neural networks associated with the ReLU activation function by constructing a continuous piecewise linear interpolation under a simple triangulation. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions. Finally, our study may also shed some light on the understanding of functional data learning algorithms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give some sequences that converge quickly to the Ioachimescu constant by a multiple-correction method. [ABSTRACT FROM AUTHOR]

In this paper, we provide some new continued fraction approximation and inequalities of the Somos quadratic recurrence constant, using its relation with the generalized Euler constant. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some new continued fraction inequalities for the Euler-Mascheroni constant by multiple-correction method. [ABSTRACT FROM AUTHOR]

Support vector machines regression (SVMR) is a regularized learning algorithm in reproducing kernel Hilbert spaces with a loss function called the ε-insensitive loss function. Compared with the well-understood least square regression, the study of SVMR is not satisfactory, especially the quantitative estimates of the convergence of this algorithm. This paper provides an error analysis for SVMR, and introduces some recently developed methods for analysis of classification algorithms such as the projection operator and the iteration technique. The main result is an explicit learning rate for the SVMR algorithm under some assumptions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new sequence converging to the Euler-Mascheroni constant. Finally, we establish some inequalities for the Euler-Mascheroni constant by the new sequence. [ABSTRACT FROM AUTHOR]