George A. Anastassiou Livres

Focusing on the integration of probability theory, finance, and numerical analysis, this volume delves into the burgeoning field of numerical methods in finance. It addresses the limitations of analytical methods in solving complex financial problems, offering insights into areas like risk assessment, asset management, and portfolio optimization. The research contributions explore various methodologies, including Genetic Algorithms, Neural Networks, and Monte-Carlo methods, highlighting both theoretical and practical aspects yet to be fully examined.

Focusing on approximation theory, this monograph compiles the author's research over the past five years. Each chapter is self-contained, allowing for independent reading, and covers a range of diverse topics. The book serves as a resource for advanced courses, making it suitable for both students and educators in the field.

The book explores innovative activation functions in neural networks, emphasizing parametrized and deformed functions that retain more neurons compared to traditional methods. It highlights the brain's asymmetry through these deformed functions and introduces a wide range of general activation functions. The author's original work covers various neural network types, including ordinary, fractional, fuzzy, and stochastic approximations, as well as univariate and multivariate methods. Additionally, it examines iterated sequential multi-layer approximations within the context of Banach space-valued functions.

Focusing on discrete and fractional analysis, this monograph presents innovative concepts in right delta and right nabla fractional calculus on time scales. It explores representation formulas, various inequalities such as Ostrowski and Grüss types, and their implications in time scales. The work delves into integral operator inequalities using function convexity, examines s-convexity, and discusses the general case with multiple functions. Additionally, it introduces fractional Hermite-Hadamard type inequalities and the reduction method in fractional calculus, highlighting its relationship with fractional Ostrowski inequalities.

The book explores advanced concepts in fractional calculus, focusing on generalized methods of Hilfer, Prabhakar, and their combinations. It presents unifying fractional integral inequalities across various types, including Iyengar and Hardy, applicable in both univariate and multivariate contexts. The findings are poised for use in pure and applied mathematics, particularly in fractional inequalities and differential equations, with potential applications in fields like geophysics, chemistry, and engineering. It serves as a valuable resource for researchers, graduate students, and academic libraries.

The book explores various fractional differentiation inequalities, including those by Opial, Poincaré, Sobolev, Hilbert, and Ostrowski, using Canavati, Riemann-Liouville, and Caputo fractional derivatives. It covers both univariate and multivariate cases, with each chapter designed to be self-contained. The systematic presentation of theory is complemented by practical applications, notably in the field of information theory, making it a comprehensive resource for understanding these mathematical concepts.

The book delves into approximation theory, emphasizing both ordinary and fractional smoothness in univariate and multivariate contexts. It introduces innovative concepts such as approximations under convexity and the use of sublinear operators, particularly for max-product operators, which offer rapid and adaptable solutions. The findings have significant implications across various fields of pure and applied mathematics, making it an essential resource for researchers, graduate students, and academic seminars in approximation theory and numerical analysis.

The monograph explores the computational aspects of moduli of continuity for various functions, emphasizing convexity properties and presenting a comprehensive calculus of smoothness. It uniquely avoids the K-functional method to provide explicit error values in approximation theory. Part II delves into the Global Smoothness Preservation Property (GSPP) across numerous linear approximation operators, offering a general theory and diverse applications in mathematics and computer-aided geometric design. This work is notable for its thorough examination of GSPP and its integration with moduli of smoothness.

The book explores the innovative impact of new trigonometric and hyperbolic Taylor's formulas with integral remainders, offering a comprehensive collection of approximations. It delves into perturbed neural network approximations, their links to Brownian motion, and various analytical inequalities. Covering both univariate and multivariate cases, it addresses Korovkin theory and singular integrals. The findings have broad applications across mathematics, computer science, engineering, and artificial intelligence, making it a vital resource for researchers and students in related fields.