Athanassios Fokas - Böcker

Modern Mathematical Methods for Scientists and Engineers is a modern introduction to basic topics in mathematics at the undergraduate level, with emphasis on explanations and applications to real-life problems. There is also an "Application" section at the end of each chapter, with topics drawn from a variety of areas, including neural networks, fluid dynamics, and the behavior of "put" and "call" options in financial markets. The book presents several modern important and computationally efficient topics, including feedforward neural networks, wavelets, generalized functions, stochastic optimization methods, and numerical methods.A unique and novel feature of the book is the introduction of a recently developed method for solving partial differential equations (PDEs), called the unified transform. PDEs are the mathematical cornerstone for describing an astonishingly wide range of phenomena, from quantum mechanics to ocean waves, to the diffusion of heat in matter and the behavior of financial markets. Despite the efforts of many famous mathematicians, physicists and engineers, the solution of partial differential equations remains a challenge.The unified transform greatly facilitates this task. For example, two and a half centuries after Jean d'Alembert formulated the wave equation and presented a solution for solving a simple problem for this equation, the unified transform derives in a simple manner a generalization of the d'Alembert solution, valid for general boundary value problems. Moreover, two centuries after Joseph Fourier introduced the classical tool of the Fourier series for solving the heat equation, the unified transform constructs a new solution to this ubiquitous PDE, with important analytical and numerical advantages in comparison to the classical solutions. The authors present the unified transform pedagogically, building all the necessary background, including functions of real and of complex variables and the Fourier transform, illustrating the method with numerous examples.Broad in scope, but pedagogical in style and content, the book is an introduction to powerful mathematical concepts and modern tools for students in science and engineering.

Modern Mathematical Methods for Scientists and Engineers is a modern introduction to basic topics in mathematics at the undergraduate level, with emphasis on explanations and applications to real-life problems. There is also an "Application" section at the end of each chapter, with topics drawn from a variety of areas, including neural networks, fluid dynamics, and the behavior of "put" and "call" options in financial markets. The book presents several modern important and computationally efficient topics, including feedforward neural networks, wavelets, generalized functions, stochastic optimization methods, and numerical methods.A unique and novel feature of the book is the introduction of a recently developed method for solving partial differential equations (PDEs), called the unified transform. PDEs are the mathematical cornerstone for describing an astonishingly wide range of phenomena, from quantum mechanics to ocean waves, to the diffusion of heat in matter and the behavior of financial markets. Despite the efforts of many famous mathematicians, physicists and engineers, the solution of partial differential equations remains a challenge.The unified transform greatly facilitates this task. For example, two and a half centuries after Jean d'Alembert formulated the wave equation and presented a solution for solving a simple problem for this equation, the unified transform derives in a simple manner a generalization of the d'Alembert solution, valid for general boundary value problems. Moreover, two centuries after Joseph Fourier introduced the classical tool of the Fourier series for solving the heat equation, the unified transform constructs a new solution to this ubiquitous PDE, with important analytical and numerical advantages in comparison to the classical solutions. The authors present the unified transform pedagogically, building all the necessary background, including functions of real and of complex variables and the Fourier transform, illustrating the method with numerous examples.Broad in scope, but pedagogical in style and content, the book is an introduction to powerful mathematical concepts and modern tools for students in science and engineering.

To comprehend the world around us, we first have to decipher how our brains work. This book outlines a new approach to knowledge and comprehension based on the elucidation of several basic neuronal mechanisms. It explores the crucial fact that unconscious processes and conscious experiences form a continuum and, importantly, introduces the concept of metarepresentations. This fundamental notion captures the essence of being human, namely, what separates us from our evolutionary predecessors. Important examples of metarepresentations are language, mathematics, technology and the arts. The interdisciplinary, integrative approach of this book demonstrates how different disciplines and notions which appear disconnected are actually closely related. Furthermore, this approach elucidates how we think about topics such as mathematics, physics, painting, music and philosophy. Most significantly, it helps us understand the origin of our thoughts, feelings and actions, bringing deep insight and fulfilment or, more precisely, eudemonia.

To comprehend the world around us, we first have to decipher how our brains work. This book outlines a new approach to knowledge and comprehension based on the elucidation of several basic neuronal mechanisms. It explores the crucial fact that unconscious processes and conscious experiences form a continuum and, importantly, introduces the concept of metarepresentations. This fundamental notion captures the essence of being human, namely, what separates us from our evolutionary predecessors. Important examples of metarepresentations are language, mathematics, technology and the arts. The interdisciplinary, integrative approach of this book demonstrates how different disciplines and notions which appear disconnected are actually closely related. Furthermore, this approach elucidates how we think about topics such as mathematics, physics, painting, music and philosophy. Most significantly, it helps us understand the origin of our thoughts, feelings and actions, bringing deep insight and fulfilment or, more precisely, eudemonia.

The Embodied Mind approaches AI and the modern revolutionary breakthroughs in medicine, biology, and physics in conversation with the role of the humanities. A mathematician, engineer, and physician, Professor Thanasis Fokas is interested in intensive and focused analysis of these fields: the achievements, uses, neurophysiology, and possible problems of AI; advances in immunology, cancer, and gene editing; the striking inventiveness of Einstein and the primacy of experimental physics; and, finally, the fundamental unconscious processes of the brain and the critical importance of sleep. Throughout the book, Fokas emphasises the crucial role of the arts and letters in today's society, which is increasingly driven towards mechanisation, where emotional and affectional aspects are ignored and emphasis is seemingly directed only towards purely algorithmic, machine-driven gadgets.No other book matches this title in its combination of breadth, depth, and interdisciplinary focus. Postgraduate researchers across a wide variety of fields will find cutting-edge discoveries presented with appropriate vigour, while general readers will be able to appreciate the nuances and complexities of the topics covered.What, if any, significant role will be left for humans to play in this anxious, unpredictable world of AI and robotics? Fokas distils the defining aspects of our modern condition to consider, suggest, and advise readers on ways to perceive and engage with the world, avoid catastrophe, and unravel our understandings of AI, medicine, and physics.

The Embodied Mind approaches AI and the modern revolutionary breakthroughs in medicine, biology, and physics in conversation with the role of the humanities. A mathematician, engineer, and physician, Professor Thanasis Fokas is interested in intensive and focused analysis of these fields: the achievements, uses, neurophysiology, and possible problems of AI; advances in immunology, cancer, and gene editing; the striking inventiveness of Einstein and the primacy of experimental physics; and, finally, the fundamental unconscious processes of the brain and the critical importance of sleep. Throughout the book, Fokas emphasises the crucial role of the arts and letters in today's society, which is increasingly driven towards mechanisation, where emotional and affectional aspects are ignored and emphasis is seemingly directed only towards purely algorithmic, machine-driven gadgets.No other book matches this title in its combination of breadth, depth, and interdisciplinary focus. Postgraduate researchers across a wide variety of fields will find cutting-edge discoveries presented with appropriate vigour, while general readers will be able to appreciate the nuances and complexities of the topics covered.What, if any, significant role will be left for humans to play in this anxious, unpredictable world of AI and robotics? Fokas distils the defining aspects of our modern condition to consider, suggest, and advise readers on ways to perceive and engage with the world, avoid catastrophe, and unravel our understandings of AI, medicine, and physics.