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Full-Text Articles in Special Functions
Making Mathematical Art Via The Harmonic And Perturbed Harmonic Functions, Mehmet Pakdemirli
Making Mathematical Art Via The Harmonic And Perturbed Harmonic Functions, Mehmet Pakdemirli
Journal of Humanistic Mathematics
Starting from the definition of a harmonic function series, we define a new series which we call the perturbed harmonic function series. We explore the relationship of the new series with the Weierstrass and Riemann fractal functions as well as its convergence and differentiability properties. We then illustrate the potential of the harmonic functions in generating complex figures that sometimes resemble natural objects, with explicit numerical examples.
Uniform Convergence Of Deep Neural Networks With Lipschitz Continuous Activation Functions And Variable Widths, Yuesheng Xu, Haizhang Zhang
Uniform Convergence Of Deep Neural Networks With Lipschitz Continuous Activation Functions And Variable Widths, Yuesheng Xu, Haizhang Zhang
Mathematics & Statistics Faculty Publications
We consider deep neural networks (DNNs) with a Lipschitz continuous activation function and with weight matrices of variable widths. We establish a uniform convergence analysis framework in which sufficient conditions on weight matrices and bias vectors together with the Lipschitz constant are provided to ensure uniform convergence of DNNs to a meaningful function as the number of their layers tends to infinity. In the framework, special results on uniform convergence of DNNs with a fixed width, bounded widths and unbounded widths are presented. In particular, as convolutional neural networks are special DNNs with weight matrices of increasing widths, we put …