Young Hee Geum

We study the complex dynamics by analyzing the dynamical planes associated with the family of third order multiple root finders and the parameter spaces related to the free critical points. The conjugacy maps with the theoretical results of dynamical analysis for the iterative schemes are investigated. In addition, the various experiments are implemented to draw the dynamics of the cubic-order schemes as well as existing methods.

Integral Transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily. In this research area, our main objective of this special issue is to address some of the new and interesting engineering and applied science research problems. Although the theory of Integral Transform is not new, we think it is still worthy of further research in an application point of view, such as convolution in convolutional neural networks (CNN) or medical diagnosis image processing. Computed tomography or magnetic resonance imaging can be viewed as successful applications of the typical Integral Transform.

The aim of this research paper is to provide as many as forty-five attractive Laplace transforms of convolution type related to the product of generalized hypergeometric functions. These are achieved by employing summation theorems for the series _pF_p−1 (for , and 5) available in the literature. The obtained research result is to provide an easier method than the existing method.

In this paper, we consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional neural network. Of course, convolution is a main process of deep learning, the learning method of deep neural networks, as a core technology. In addition to this, the generalized continuous form of convolution has been expressed as a new variant of Laplace-type transform that, encompasses almost all existing integral transforms. Finally, we would, in this paper, like to describe the theoretical contents as detailed as possible so that the paper may be self-contained.

An analysis based on an elementary theory of plane curves is presented to locate bifurcation points from a main component in the parameter space of a family of optimal fourth-order multiple-root finders. We explore the basic dynamics of the iterative multiple-root finders under the Möbius conjugacy map on the Riemann sphere. A linear stability theory on local bifurcations is developed from the viewpoint of an arbitrarily small perturbation about the fixed point of the iterative map with a control parameter. Invariant conjugacy properties are established for the fixed point and its multiplier. The parameter spaces and dynamical planes are investigated to analyze the underlying dynamics behind the iterative map. Numerical experiments support the theory of locating bifurcation points of satellite and primitive components in the parameter space.