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A novel morphing activation function is proposed, motivated by the wavelet theory and the use of wavelets as activation functions. Morphing refers to the gradual change of shape to mimic several apparently unrelated activation functions. The shape is controlled by the fractional order derivative, which is a trainable parameter to be optimized in the neural network learning process. Given the morphing activation function, and taking only integer-order derivatives, efficient piecewise polynomial versions of several existing activation functions are obtained. Experiments show that the performance of polynomial versions PolySigmoid, PolySoftplus, PolyGeLU, PolySwish, and PolyMish is similar or better than their counterparts Sigmoid, Softplus, GeLU, Swish, and Mish. Furthermore, it is possible to learn the best shape from the data by optimizing the fractional-order derivative with gradient descent algorithms, leading to the study of a more general formula based on fractional calculus to build and adapt activation functions with properties useful in machine learning. [ABSTRACT FROM AUTHOR]

The continuous wavelet transform for functions f in Lp (ℝn) where 1 ≤ p < ∞ is defined with respect to a radially symmetric admissible function so that the singularities of f are the singularities of the continuous wavelet transform. [ABSTRACT FROM AUTHOR]

The continuous wavelet transform for functions f in Lp(Rn) where 1 < p < ∞ is defined with respect to a radially symmetric admissible function so that the singularities of f are the singularities of the continuous wavelet transform. [ABSTRACT FROM AUTHOR]

Given two abstract, locally compact topological groups A and B, where A is Abelian, the abstract wavelet transform for a function f ∈ L²(A) with respect to an admissible function h ∈ L²(A) is defined so that the left and right uniform continuity of the wavelet transform holds because of the left and right uniform continuity of f. [ABSTRACT FROM AUTHOR]

In Medicine, a large number of investigations require a morphometric analysis of an organ, tissue or cell and/or the study of their internal composition. Stereological methods are the result of mathematical reasoning explained in applicable formulas designed to determine parameters such as volume density, surface density, numerical density and length density, from bidimensional images. Changes in cellular function, for example, are frequently related to variations in the size of cells and nuclei, or to changes in the number of specific cellular organelles involved in the process. A brief review of the physical-mathematical principles that gave rise to the classic stereological methods is presented. We comment upon a series of conceptions on stereological principles over the past few decades in order to prevent under or overestimations in the appraisal of the results that the method allows. The stereological methods more frequently used are also examined. Currently software development simplifies the application of morphometric studies, whose main advantage is that they allow quantifications in a short time. [ABSTRACT FROM AUTHOR]

A new asymptotic method is proposed for calculation of spectral series and corresponding asymptotic eigenfunctions for the spinless N particle Schrödinger equation when N tends to infinitum but the physical parameters and the volume are fixed (dense matter case). [ABSTRACT FROM AUTHOR]