Mathematics for Digital Science 2 E-Book

Gerard-Michel Cochard

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: ISTE Invoiced
• Sprache: Englisch

Seitenzahl: 330

Veröffentlichungsjahr: 2025

Beschreibung

Over the past century, advancements in computer science have consistently resulted from extensive mathematical work. Even today, innovations in the digital domain continue to be grounded in a strong mathematical foundation. To succeed in this profession, both today's students and tomorrow’s computer engineers need a solid mathematical background.

The goal of this book series is to offer a solid foundation of the knowledge essential to working in the digital sector. Across three volumes, it explores fundamental principles, digital information, data analysis, and optimization. Whether the reader is pursuing initial training or looking to deepen their expertise, the Mathematics for Digital Science series revisits familiar concepts, helping them refresh and expand their knowledge while also introducing equally essential, newer topics.

Leseprobe

SCIENCES

Computer Science,Field Director – Jean-Charles Pomerol

Operational Research and Decision,Subject Head – Patrick Siarry

Mathematics for Digital Science 2

Digital Information

Gérard-Michel CochardMhand Hifi

First published 2025 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the under mentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2025The rights of Gérard-Michel Cochard and Mhand Hifi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2025930227

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78945-195-5

ERC code:PE1 MathematicsPE1_17 Mathematical aspects of computer sciencePE1_20 Control theory, optimisation and operational research

Preface

The term “digital” relates to information and communication sciences and technologies, covering areas such as computing, digital electronics and telecommunications. Over the last 50 years, advances in these fields, through discoveries, developments and applications, have grown exponentially. Such progress has profoundly transformed human activities, making the transition to an “all digital” world a significant economic and political issue. Concurrently, professions related to digital technology are constantly evolving. It is noteworthy that these technological advances are grounded in a substantial mathematical foundation. Thus, any engineer or researcher aiming to drive innovation must possess a strong knowledge of mathematics.

Many students opt for computer science-related courses early on without first acquiring the essential mathematical foundations they might need in the future. While they may excel as technicians, they might not be as well prepared to become effective engineers or researchers.

It should be recognized that computer science cannot be fully understood without a solid mathematical foundation. Advances in digital technologies have been closely linked to mathematical support. The pioneers of computing were primarily mathematicians: Alan Turing, Claude Shannon, John von Neumann and Charles Babbage, often referred to as the “grandfather of computing”. They could never have designed computers without their mathematical expertise. Other historical figures such as Euclid, who formulated the first algorithm, George Boole, the father of binary algebra, Ada Lovelace, a pioneer in creation the first computer programs, Grace Hopper, who developed the first language compiler, and Margaret Hamilton, a key figure in software engineering, were also mathematicians. Today, areas such as artificial intelligence, managing large datasets, and information security are at the center of computer science research. Once again, it is evident that without a robust mathematical foundation, innovation in these areas would be challenging.

The digital mathematics courses offered are designed to provide the fundamental mathematical knowledge essential for mastering and advancing digital technologies. While the book primarily targets university and engineering students, it also serves as a resource for IT professionals seeking to enhance their mathematical skills as part of their professional development in the field.

The three-volume work compiles the lessons taught to numerous generations of students in the first two university cycles, specifically in bachelor’s/master’s degrees in Computer Science or in Computer Methods Applied to Business Management. The first volume covers the essential mathematical foundations for approaching digital technologies. The second – and present – volume focuses on digital information, while the third volume is devoted to data analysis and optimization.

Chapters 1 and 2 of this second volume address the representation of information (such as numbers, texts, images, sounds, videos, labels) in the form of combinations of 0s and 1s. This binary representation facilitates the processing, analysis and transmission of information using computer resources.

Chapter 3 addresses the classification of communication signals, both analog and digital, and introduces Fourier series analysis for periodic signals.

Chapter 4 presents mathematical tools for signal analysis, including z-transforms, Fourier transforms and Laplace transforms.

Chapter 5 is dedicated to the digitalization of analog signals, while Chapters 6 and 7 focus on signal processing, specifically modulation and filtering.

The following four chapters are devoted to digital images. Chapter 8 presents the characteristics of digital images. Chapter 9 describes the main techniques of 2D computer graphics. Chapter 10 covers the fundamental elements of image processing and analysis, while Chapter 11 explains the essential concepts of image compression.

Finally, Chapter 12 describes the main algorithms of numerical analysis used to find approximate solutions to problems that are difficult to solve.

To achieve the goal of mastering the application of mathematical results, demonstrations are presented whenever they are accessible to the reader, who is assumed to be familiar with basic knowledge. In some cases, the validity of the results is accepted without detailed proof, with references to more advanced works where such demonstrations can be found.

Each chapter includes numerous examples to illustrate the concepts presented. These examples are generally elaborated in detail to ensure a better understanding for the reader.

Finally, each topic is presented exhaustively, starting with the initial definitions and hypotheses. While many readers may already be familiar with these basic concepts, this thorough approach is intended to assist those with potential knowledge gaps, reducing the need to consult additional sources.

January 2025