Preface

Calculus is the elementary subject of applied analysis and its study includes a rich variety of functions and their behavior. This book brings together a range of different concepts from across the wide spectrum of the concept of calculus.

The book is in two sections. The first deals with advances in analysis and the second with the application of some results of functional calculus to applied problems.

The first section opens with an analysis of logarithmic potential transform. The singular values of this transform are discussed on the Poincaré disk. This potential can be used to illustrate some of the important features of field theory such as dimensional regularization and renormalization. Although most recent textbooks do not discuss this potential in detail, the calculations to demonstrate some of its unique features are quite simple. The bound state energy of this logarithmic potential is obtained through the uncertainty principle, phase space quantization and the Hellmann‒Feynman theorem.

In the second chapter of the first section, the authors define and prove new Tauberian theorems under triple statistically Norlund-Cesaro summability. Some theorems, lemmas and corollaries can be defined and proved similarly by using the (1, 0, 0), (0, 1, 0) and (0, 0, 1) summability method. Although Tauberian theorems for single sequences of a single variable are well established, they remain in their infancy for triple sequences.

The final chapter of the first section is devoted to the study of the Calderon operator, which is the sum of the Hardy averaging operator and its adjoint and plays an important role in the theory of real interpolation. On the other hand, the Hilbert operator arises from the continuous version of Hilbert's inequality. Both operators appear in different contexts and have numerous applications within the harmonic analysis. In this chapter, the authors briefly review the Calderon and Hilbert operators, showing some of the most relevant results within the functional analysis and presenting recent results on these operators within Fourier analysis.

The second section of the book collects some results from applied analysis The first chapter deals with the study of heat transfer development of titanium oxide nanofluid of platelet-shaped nanoparticles over a vertical stretching cylinder. A set of nonlinear equations is obtained using suitable transformation on the governing equations which are then solved with numerical scheme BVP4C. The results obtained are interpreted graphically and numerically. The effects of Prandtl, Eckert and unsteadiness parameters on temperature distribution are depicted, and skin friction and Nusselt number are also computed. In the second chapter, a nonlinear response of the follower motion is simulated at different cam speeds, different coefficients of restitution and different internal distances of the follower guide from inside. The nonlinear response of the follower is employed to investigate the chaotic phenomenon in the cam follower system in the presence of follower offset. The numerical results are achieved using Solid Works software. The chaos phenomenon is detected using Poincaré maps with phase-plane portraits, the largest Lyapunov exponent parameter, and a bifurcation diagram. The largest Lyapunov exponent has its maximum value when the follower offsets to the right, and its minimum value when the follower offsets to the left. The chaotic phenomenon in cam follower systems when the follower offsets to the left is greater than the chaotic phenomenon when the follower offsets to the right. The final chapter investigates the decision fusion problem for large-scale sensor networks associated with the Internet of Things and artificial intelligence. The sensor networks discussed are those with unavoidable transmission channel interference and non-ideal channels that are prone to errors. A generalized algorithm is proposed that enables decision fusion rules to be designed for large-scale sensor networks and can at the same time search for the optimal sensor rules and the optimal fusion rule. Finally, numerical examples show the effectiveness of the new algorithms for large-scale sensor networks with non-ideal channels.

> **Hammad Khalil** Department of Mathematics, University of Education, Lahore, Pakistan

Section 1

Advances in Theory
