Fractional Inequalities In Banach Algebras

This book presents generalized Caputo fractional Ostrowski and Grüss-type inequalities involving several Banach algebra valued functions. Furthermore, the author gives generalized Canavati fractional Ostrowski, Opial, Grüss, and Hilbert-Pachpatte-type inequalities for multiple Banach algebra valued functions. By applying the p-Schatten norms over the von Neumann–Schatten classes, the author produces the analogous refined and interesting inequalities. The author provides many applications. This book’s results are expected to find applications in many areas of pure and applied mathematics, especially in fractional inequalities and fractional differential equations. Other interesting applications are in applied sciences like geophysics, physics, chemistry, economics, and engineering. This book is appropriate for researchers, graduate students, practitioners, and seminars of the above disciplines, also to be in all science and engineering libraries.

In this book, we introduce the parametrized, deformed and general activation function of neural networks. The parametrized activation function kills much less neurons than the original one. The asymmetry of the brain is best expressed by deformed activation functions. Along with a great variety of activation functions, general activation functions are also engaged. Thus, in this book, all presented is original work by the author given at a very general level to cover a maximum number of different kinds of neural networks: giving ordinary, fractional, fuzzy and stochastic approximations. It presents here univariate, fractional and multivariate approximations. Iterated sequential multi-layer approximations are also studied. The functions under approximation and neural networks are Banach space valued.

"Volume 205, number 966 (end of volume)."

In these notes the abstract theory of analytic one-parameter semigroups in Banach algebras is discussed, with the Gaussian, Poisson and fractional integral semigroups in convolution Banach algebras serving as motivating examples. Such semigroups are constructed in a Banach algebra with a bounded approximate identity. Growth restrictions on the semigroup are linked to the structure of the underlying Banach algebra. The Hille-Yosida Theorem and a result of J. Esterle's on the nilpotency of semigroups are proved in detail. The lecture notes are an expanded version of lectures given by the author at the University of Edinburgh in 1980 and can be used as a text for a graduate course in functional analysis.

This compact book focuses on self-adjoint operators’ well-known named inequalities and Korovkin approximation theory, both in a Hilbert space environment. It is the first book to study these aspects, and all chapters are self-contained and can be read independently. Further, each chapter includes an extensive list of references for further reading. The book’s results are expected to find applications in many areas of pure and applied mathematics. Given its concise format, it is especially suitable for use in related graduate classes and research projects. As such, the book offers a valuable resource for researchers and graduate students alike, as well as a key addition to all science and engineering libraries.

The first part of this book offers a comprehensive overview of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The second part of the book explores several applications of this theory.

This paper studies derivations, endomorphisms, automorphisms, and various related questions about certain Banach algebras, B, which are continuously embedded in the space of complex formal power series in the indeterminate z.