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Elsevier

Books in Mathematics and applied mathematics

This portfolio offers comprehensive coverage of pure and applied mathematics, including algebra, calculus, differential equations, and computational techniques. It supports researchers, educators, and students in advancing theoretical understanding and practical problem-solving across engineering, physics, and data science. Featuring innovative methodologies, real-world case studies, and emerging trends, these resources enable breakthroughs in modelling, simulation, and analytical skills critical for technological progress and scientific discovery.

  • Artificial Intelligence Modeling for Dynamical Problems

    • 1st Edition
    • Snehashish Chakraverty + 2 more
    • English
    Artificial Intelligence Modeling for Dynamical Problems provides a comprehensive exploration of AI-driven methodologies tailored to address the intricate challenges posed by dynamical systems. The chapters in this book delve into cutting-edge techniques, including scientific machine learning, operator learning, fuzzy logic, and optimization algorithms, highlighting their applications in structural dynamics, fluid dynamics, robotics, and wave dynamics. Readers will gain insights into innovative state-of-the-art approaches such as Physics-Informed Neural Networks (PINNs) for precise navigation and control in autonomous vehicles, convolutional neural networks (CNNs) for frequency dynamics recognition, and data-driven models for market sentiment analysis. Additionally, the book explores the role of uncertainty modeling, statistical inference, and hybrid AI techniques, such as Type-2 fuzzy fractional modeling, in advancing the field of dynamical system analysis. Each chapter combines foundational principles with practical applications, including recent investigations, making it a valuable resource for researchers, practitioners, students of STEM seeking to apply theoretical AI models to real-world dynamical problems.

  • Ensuring AI-enabled Zero Trust Security for Smart City and Healthcare Applications Part A

    • 1st Edition
    • Volume 146
    • English
    Ensuring AI-enabled Zero Trust Security for Smart City and Healthcare Applications Part A, published as Volume 146 of the serial Advances in Computers, provides a comprehensive examination of the convergence of artificial intelligence, Zero Trust security frameworks, smart city infrastructures, and healthcare technologies. The volume explores how AI-driven security architectures can safeguard increasingly interconnected urban and healthcare ecosystems, addressing challenges associated with IoT networks, autonomous vehicles, wearable devices, telemedicine platforms, medical imaging systems, and healthcare data interoperability. It highlights emerging approaches that integrate AI, blockchain, edge computing, digital twins, federated learning, threat intelligence, and quantum computing to strengthen cyber resilience, privacy protection, and trust management across critical applications. By combining foundational concepts with practical implementations and future-oriented perspectives, this volume serves as an authoritative resource for researchers, cybersecurity professionals, healthcare technologists, and smart city stakeholders seeking to develop secure, adaptive, and resilient digital environments.

  • Doing Bayesian Data Analysis

    A Tutorial With R, Stan, brms, and the tidyverse
    • 3rd Edition
    • John K. Kruschke + 1 more
    • English
    Doing Bayesian Data Analysis: A Tutorial with R, Stan, brms, and the tidyverse, Third Edition, provides a carefully scaffolded tutorial from beginning concepts to advanced, realistic data analyses. The book uses a proven sequence of topics unique to Doing Bayesian Data Analysis. The first part covers foundational concepts of statistical models, probability, Bayesian reasoning, and computer programming in R and the tidyverse. The second part introduces all the concepts and methods of Bayesian data analysis, including the Stan modeling language, by using the simplest possible data structures and statistical models. The third part covers the generalized linear model, including multilevel (a.k.a. hierarchical) versions of regression and analysis of variance, for a variety of data types (metric, ordinal, nominal, dichotomous, count), using the convenient computer package called brms. Every concept and case is illustrated with detailed examples, and all computer code is available at the book’s website. This book is intended for self-learners or classroom learning, for first-year graduate students, advanced undergraduates, and professionals. The methods apply to any field, including social sciences, biological sciences, and physical sciences, for any setting, including academia, government, business, and industry.

  • Physiology, Medicine, and Health Sciences

    Catalyzed by Biomedical Engineering Modeling
    • 1st Edition
    • Dhanjoo N. Ghista + 1 more
    • English
    Physiology, Medicine, and Health Sciences: Catalyzed by Biomedical Engineering Modeling is a quantitative handbook for mathematical and quantitative analysis of anatomical structures, physiological and organ systems, medical tests and surgical procedures. This book’s purpose is to demonstrate how fundamental mathematical modeling can be incorporated into physiology, medicine, and health sciences research, application, and clinical practice to make these disciplines more quantitative and computational, and hence more explanatory and informative. The book also provides quantitative formulation of medical procedures, towards supporting the growing field of precision medicine. Each chapter includes a robust introduction to the BME and mathematical concepts necessary to understand their application to the indicated medical and biological systems, beginning with Intrinsic Anatomy, showing how anatomical structures are intrinsically optimally designed for their functional performance, such as the intrinsic hyperboloid shape of the spinal vertebral body (as a lightweight, high-strength structure), and the ellipsoidal shape of the left ventricle that enables it to contract efficiently. Other BME and modelling concepts covered include: Physiological Engineering, in quantifying physiological systems and developing indices for assessing their function and dysfunction leading to medical diagnostics; Precision Medicine, by developing biomedical engineering formulation of medical diagnostic and assessment methods and indices, including the new concept of non-dimensional indices in medical assessment, such as cardiac contractility index for risk of heart failure and diabetes index for accurate diagnosis of diabetes; and, Computational Surgery, involving surgical design of prosthetic devices based on the intrinsic design of the fractured structure, such as the design of the spinal vertebral body cage for fractured vertebral body to preserve its intrinsic hyperboloid shape. Each chapter provides in-depth guidance on the engineering and mathematical concepts, methodology, and procedures for applying the formulas and equations to the appropriate biomedical and physiological systems, as well as step-by-step procedures for creating models for a wealth of specific applications.

  • Quantum Computing for Internet of Things

    • 1st Edition
    • Volume 145
    • English
    Quantum Computing for Internet of Things, published as Volume 147 of the serial Advances in Computers, provides a comprehensive examination of the convergence of quantum computing and the Internet of Things (IoT). The volume introduces the fundamental principles of quantum computing, including qubits and quantum mechanics, and explores the architectures, enabling technologies, and intelligent applications that underpin modern IoT ecosystems. It highlights how quantum technologies can enhance IoT performance, security, optimisation, and data processing capabilities across diverse application domains.The volume presents emerging advances in quantum-enabled IoT systems, including distributed quantum computing frameworks, quantum machine learning approaches, quantum simulation tools, and novel optimisation techniques tailored to IoT environments. Case studies and application-focused chapters demonstrate the use of quantum methods in military systems, climate change analysis, and healthcare analytics. The volume concludes by addressing key research challenges, implementation barriers, and future directions, making it a valuable resource for researchers and practitioners working at the intersection of quantum technologies and connected intelligent systems.

  • Mathematical Foundations of Physiology, Medicine, and Health Sciences

    • 1st Edition
    • Dhanjoo N. Ghista + 3 more
    • English
    Mathematics and Biomedical Engineering in Medicine, Physiology and Health Sciences is a visionary book, whose purpose is to demonstrate how fundamental mathematics and biomedical engineering can be incorporated into physiology, medicine, and health sciences teaching, research,and clinical practice to make these disciplines more quantitative and computational, and hence more explanatory and informative. The book also provides more quantitative formulation of medical procedures, towards supporting the growing field of precision medicine.

  • Digital Signal Processing with MATLAB®

    From Fundamentals to AI Applications
    • 1st Edition
    • Samir I. Abood + 1 more
    • English
    Digital Signal Processing with MATLAB®: From Fundamentals to AI Applications provides coverage of discrete-time signals and systems, and lays out the mathematical fundamentals of digital signal processing (DSP) systems. The book contains a solution to large-scale signal processing problems, namely the development of methods and algorithms that increase (a) operational efficiency under different uncertainty conditions and (b) system stability rate. It offers an algebraic approach to digital signal processing, including algorithms related to digital systems, the setting of parameters, plus operational estimation, detection, classification, resolution, and reliability. Digital Signal Processing with MATLAB®: From Fundamentals to AI Applications begins with a thorough explanation of MATLAB® in recognition of the Accreditation Board for Engineering and Technology (ABET) requirements for integrating computer tools into DSP. The principles of DSP are presented using a step-by-step approach; each chapter starts with an introduction that states the chapter objectives, and ends with a discussion of practical applications of DSP, including providing problems that help the reader understand how the concepts are used in real-life situations. Sometimes, problems are solved in two or three ways to facilitate a deeper understanding and comparison of different approaches. Digital Signal Processing with MATLAB®: From Fundamentals to AI Applications teaches the fundamental principles of digital signal processing, and explains how MATLAB® can be used in the development of methods and algorithms for DSP.

  • Advances in Machine Learning for Engineering Applications and Sustainable Development

    • 1st Edition
    • Volume 145
    • English
    Advances in Machine Learning for Engineering Applications and Sustainable Development, published as Volume 145 of the serial Advances in Computers, provides a comprehensive and timely exploration of how artificial intelligence (AI) and machine learning (ML) are transforming engineering practice and advancing sustainability goals. The volume brings together contributions from leading researchers to examine the integration of data-driven methods in concrete technology, digital fabrication, structural health monitoring, materials engineering, and industrial process optimisation. It highlights emerging paradigms such as AI-driven 3D printing, predictive analytics for failure detection, lifecycle and environmental impact assessment, and circular economy strategies.Spanning foundational methods and real-world implementations, the volume emphasises sustainable materials, eco-efficient construction processes, and intelligent system design. It pays particular attention to the use of ML in improving resource efficiency, enabling recycling of industrial by-products, and enhancing the performance of green materials such as biochar-based composites. By combining theoretical insights, mathematical modelling, and application-oriented case studies, this volume serves as a critical reference for advancing resilient, efficient, and environmentally responsible engineering systems.

  • Handbook of MRI Pulse Sequences

    • 2nd Edition
    • Xiaohong Joe Zhou + 4 more
    • English
    Handbook of MRI Pulse Sequences, Second Edition includes 92 self-contained sections, with each section focusing on a single subject. A new section on detailing the advanced pulse sequence techniques covers a variety of basic and advanced image reconstruction methods. The extensive topic coverage and cross-referencing makes this book ideal for beginners learning the building blocks of MRI pulse sequence design, as well as for experienced professionals who are seeking deeper knowledge of a particular technique.This book is among the most important medical imaging techniques available today. Each of these scanners is capable of running many different "pulse sequences." These sequences are governed by physics and engineering principles and implemented by software programs that control the MRI hardware.

  • Mathematical Modeling

    Process and Presentation
    • 1st Edition
    • Eric Marland + 3 more
    • English
    Mathematical Modeling: Process and Presentation offers a comprehensive introduction to the art of mathematical modeling, engaging students through thought-provoking questions and interactive content. Each chapter presents a different canonical modeling type, guiding readers through the essential components of the modeling process, from problem posing and assumptions to reflection and purpose definition. With relatable examples, such as estimating efforts in balloon filling or developing metrics for college roommate matching, students are encouraged to explore and expand on the models presented. This text stands out by revisiting the modeling process throughout, rather than focusing solely on examples, fostering a deeper understanding of the methodology. Accessible to a broad audience, it invites readers to apply their mathematical tools at varying levels of sophistication, ensuring that the focus remains on the core process, which remains relevant as their knowledge grows. Mathematical Modeling: Process and Presentation is an invaluable resource for undergraduate students and early graduate-level mathematics learners seeking to enhance their modeling skills and engage with real-world applications.

  • Machine Learning Solutions for Inverse Problems: Part B

    • 1st Edition
    • Volume 27
    • English
    Machine Learning Solutions for Inverse Problems: Part B, Volume 27 in the Handbook of Numerical Analysis, continues the exploration of emerging approaches at the intersection of machine learning and inverse problem theory. This volume presents a collection of chapters addressing a wide range of contemporary topics, including deep image prior methods for computed tomography, data-consistent learning strategies, and unified frameworks for training and inversion in machine learning-based reconstruction methods. Additional chapters examine learned regularization techniques, generative models for inverse problems, and the integration of deep learning with traditional computational frameworks such as full waveform inversion and PDE-based inverse modeling.The volume also discusses advances in self-supervised learning, data selection strategies, plug-and-play denoising methods, and diffusion models for solving imaging inverse problems. Further contributions explore neural network representations, operator learning, and learned iterative schemes, along with theoretical perspectives on stability, approximation hardness, hallucinations, and trustworthiness in AI-driven inverse problem methodologies.

  • Graph Theory

    Fundamentals and Applications
    • 1st Edition
    • Sovan Samanta + 1 more
    • English
    Graph theory is a rapidly evolving and expanding mathematical discipline, with new discoveries, challenges, and techniques emerging every year. Graph Theory: Fundamentals and Applications provides a fully up-to-date and accessible introduction to graph theory, covering both the classical and the modern topics, as well as algorithms and evolving challenges addressed by discipline. Based on the latest syllabi and research trends worldwide, this book includes practical, solved problems that are user friendly to undergraduate, postgraduate, and PhD students, and acts as a key aid in learning the fundamentals and the frontiers of graph theory, as well as developing independent problem-solving and critical thinking skills. This book includes clear instruction in graph representation, basic graph operations, graph connectivity, trees and forests, matching theory, planar graphs and graph drawing, algebraic graph theory, graph traversals, network flows, topological graph theory, and cryptography, among other topics. Each chapter features key term definitions, proofs and algorithms, summary points, and unique exercises to reinforce learning, as well as open problems and research challenges that present unsolved or conjectural problems in graph theory for discussion. Supporting student and instructor sites offer additional exercises, solutions, examples, and case studies in graph theory applications.

  • Functional Analysis

    Theory and Applications
    • 1st Edition
    • Anita Tomar + 2 more
    • English
    An applied understanding of functional analysis is essential for students pursuing research or careers in pure mathematics, applied mathematics, mathematical physics, and engineering, among other disciplines.Function... Analysis: Theory and Applications offers a comprehensive exploration of functional analysis. Authored by esteemed mathematicians with extensive expertise in the field, this book thoroughly introduces fundamental concepts in functional analysis, including Banach spaces, Hilbert spaces, operator theory, nonlinear analysis, linear operators, and normed spaces, and implements these in real-world problems across various scientific and engineering disciplines.The book's rigorous mathematical treatment is combined with worked examples, exercises and solutions, visual aids, application case studies, and future directions across all chapters to reinforce learning, while appendices offer supplementary materials, proofs of theorems, and tables of important results, among other resources.

  • Lebesgue Measure and Integration Theory

    Foundations and Solved Exercises
    • 1st Edition
    • Alberto Cabada + 1 more
    • English
    Lebesgue Measure and Integration Theory: Foundations and Solved Exercises offers a thorough, engaging introduction to Lebesgue measure and the theory of integration for students of mathematics and physics. This book provides the complete theoretical underpinnings of this theory, with the corresponding proofs, adapted to the level of advanced undergraduate and graduate students in these disciplines. Beginning with a fundamental discussion of measure spaces, the book moves onto measurable and non-measurable sets, approximation of measurable sets, measurable functions, the Lebesgue integral, the relationship between differentiation and integration on R, and product measures, among other topics. Examples and solved exercises are included across chapters to reinforce understanding and application.

  • Introduction to Fuzzy Mathematics

    With Applications to Global Problems
    • 1st Edition
    • John Mordeson + 2 more
    • English
    Delve into the intricate landscape of fuzzy mathematics, where the boundaries of traditional mathematical disciplines— analysis, abstract algebra, geometry, topology, and graph theory—are blurred to address pressing global issues. Through a rigorous examination of fuzzy sets and similarity measures, An Introduction to Fuzzy Mathematics: With Applications to Global Problems lays the groundwork for innovative solutions to complex problems, from medical diagnostics to sustainability, refugee crises, and the fight against human trafficking. Meanwhile, research projects and exercises integrated across chapters reinforce learning and apply fuzzy mathematics to real-world scenarios. Chapters are meticulously organized to guide readers through foundational concepts, including fuzzy sets, evidence theory, and implication operators, before advancing to applications in sustainability and climate change. Further, the book examines refugee dynamics and public health models, culminating in a thorough exploration of fuzzy algebraic structures, geometry, topology, and graph theory. This comprehensive resource not only enhances understanding of fuzzy mathematics but also equips readers—researchers, practitioners, and policymakers alike—with the tools to tackle critical global issues. By integrating mathematical rigor with real-life applications, the book serves as a vital reference for anyone seeking to navigate the complexities of our world through the lens of fuzzy mathematics.

  • Quantum Machine Learning (QML): Platform, Tools and Applications

    • 1st Edition
    • Volume 140
    • English
    Quantum Machine Learning (QML): Platform, Tools and Applications, Volume 140 in the Advances in Computers series, explores the intersection of quantum computing and artificial intelligence, highlighting advances that promise to revolutionize computational science. The book introduces foundational concepts in quantum computing and circuits, building toward the practical implementation of quantum machine learning (QML) algorithms. Chapters address challenges such as the gradient vanishing problem in variational quantum circuits and explore powerful optimization methods enabled by quantum mechanics. The volume also covers advanced applications, including quantum approaches to smart grid management, quantum Monte Carlo simulations, and predictive modeling in numerical solvers using quantum neural networks.Real-world relevance is underscored through discussions of transformative quantum algorithms and their potential to reshape machine learning, enabling unprecedented performance in data analysis, optimization, and beyond.

  • Trigonometry

    From Theory to Application
    • 1st Edition
    • João Ricardo Branco + 1 more
    • English
    Trigonometry: From Theory to Application introduces the basics of trigonometry and key areas of practice, fully considering in straightforward, pragmatic terms the characterization of triangles, coordinate transport, and coordinate systems, with emphasis on interpreting key concepts and applying them. The book begins with a review of basic trigonometric functions and their properties, and then moves on to trigonometric identities, equations, and practical applications. Special sections dedicated to real-world and technological problems demonstrate how trigonometry is applied in everyday situations, inspiring and motivating students to see the relevance of trigonometry beyond the classroom, across fields ranging from engineering to physics and beyond.Each chapter is interconnected, ensuring that students build a solid and cumulative understanding of concepts, with detailed examples, varied exercises, and full-color illustrations.

  • Quantitative Biology

    Mathematical Modeling and Computation
    • 1st Edition
    • Alonso Ogueda-Oliva + 1 more
    • English
    Quantitative Biology provides quantitative and data-driven approaches for analyzing biological and bio-inspired systems, covering the foundations of mathematical modeling, analysis, and computation. The book presents a practical mix of both theory and computation for a variety of biological applications, with tied-in, engaging project activities, instruction, programming language, and technological tools. Modeling approaches combine mathematical foundations, statistical reasoning, and computational thinking, with applications in compartmental, agent-based, bio image, biological interaction, and neural network modeling, as well as machine learning, parameter identification, and applications across societal challenges.Each chapter includes exposure to models and modeling, a foundational instructional framework, benchmark applications, and numerical simulations with a literate programming guided style that helps readers go beyond replication models and into prediction and data-driven discovery. A companion website also features interactive code to accompany projects across each chapter.

  • An Introduction to Writing Mathematical Proofs

    Shifting Gears from Calculus to Advanced Mathematics
    • 1st Edition
    • Thomas Bieske
    • English
    An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics addresses a critical gap in mathematics education, particularly for students transitioning from calculus to more advanced coursework. It provides a structured and supportive approach, guiding students through the intricacies of writing proofs while building a solid foundation in essential mathematical concepts. Sections introduce elementary proof methods, beginning with fundamental topics such as sets and mathematical logic, systematically develop the properties of real numbers and geometry from a proof-writing perspective, and delve into advanced proof methods, introducing quantifiers and techniques such as proof by induction, counterexamples, contraposition, and contradiction. Finally, the book applies these techniques to a variety of mathematical topics, including functions, equivalence relations, countability, and a variety of algebraic activities, allowing students to synthesize their learning in meaningful ways. It not only equips students with essential proof-writing skills but also fosters a deeper understanding of mathematical reasoning. Each chapter features clearly defined objectives, fully worked examples, and a diverse array of exercises designed to encourage exploration and independent learning. Supplemented by an Instructors' Resources guide hosted online, this text is an invaluable companion for undergraduate students eager to master the art of writing mathematical proofs.

  • Boundary Value Problems and Partial Differential Equations

    • 7th Edition
    • David L. Powers + 3 more
    • English
    For over fifty years, Boundary Value Problems and Partial Differential Equations, Seventh Edition has provided advanced students an accessible and practical introduction to deriving, solving, and interpreting explicit solutions involving partial differential equations with boundary and initial conditions. Fully revised and now in its Seventh Edition, this valued text aims to be comprehensive without affecting the accessibility and convenience of the original. The resource’s main tool is Fourier analysis, but the work covers other techniques, including Laplace transform, Fourier transform, numerical methods, characteristics, and separation of variables, as well, to provide well-rounded coverage. Mathematical modeling techniques are illustrated in derivations, which are widely used in engineering and science. In particular, this includes the modeling of heat distribution, a vibrating string or beam under various boundary conditions and constraints. New to this edition, the text also now uniquely discusses the beam equation. Throughout the text, examples and exercises have been included, pulled from the literature based on popular problems from engineering and science. These include some "outside-the-box" exercises at the end of each chapter, which provide challenging and thought-provoking practice that can also be used to promote classroom discussion. Chapters also include Projects, problems that synthesize or dig more deeply into the material that are slightly more involved than standard book exercises, and which are intended to support team solutions. Additional materials, exercises, animations, and more are also accessible to students via links and in-text QR codes to support practice and subject mastery.

  • Measure and Integration

    Concepts, Examples, and Applications
    • 1st Edition
    • Ahmed Ghatasheh + 2 more
    • English
    Measure and Integration: Examples, Concepts, and Applications instructs on core proofs, theorems, and approaches of real analysis as illustrated via compelling exercises and carefully crafted, practical examples. Following early chapters on core concepts and approaches of real analysis, the authors apply real analysis across integration on product spaces, radon functionals, bounded variation and lebesgue-stieltjes measures, convolutions, probability, and differential equations, among other topics. From chapter one onward, students are asked to apply concepts to reinforce understanding and gain applied experience in real analysis. In particular, exercises challenge students to use key proofs of major real analysis theorems to encourage independent thinking, problem-solving, and new areas of research powered by real analysis.

  • An Introduction to Point-Set Topology

    A Hybrid Texas Style Approach
    • 1st Edition
    • Shelby J. Kilmer
    • English
    An Introduction to Point-Set Topology is intended for use in a beginning topology course for undergraduates or as an elective course for graduate students. The book’s style can be thought of as a hybrid between the Texas style (Moore method) of teaching topology and the more traditional styles. In the Texas style the students are given the definitions and the statements of the theorems and then they present their proofs to the class. This type of participation builds students’ confidence and provides them with a deeper understanding of the subject that they will retain longer. This text offers some of the theorems with their proofs and leaves others for the students to prove and present. Those theorems chosen to have their proofs presented in the text keep the course moving forward under the instructors’ guidance and increase student comprehension. An Introduction to Point-Set Topology covers a broad range of topological concepts, including but not limited to, metric spaces, topological spaces, homeomorphisms, connected sets, compact sets, product spaces, Hausdorff spaces, sequences, limits, weak topologies, the axiom of choice, Zorn’s lemma, and Nets. Incorporating both historical references and color graphics, the material keeps readers engaged. The book’s goals include increasing student participation, thus promoting a deeper knowledge through an intuitive understanding of how and why topology was developed in the way that it was. This “instructor-friendly... accessible text is also accompanied by a detailed solutions manual to support both experienced topologists and other mathematicians who would like to teach topology.

  • Machine Learning Solutions for Inverse Problems: Part A

    • 1st Edition
    • Volume 26
    • English
    Machine Learning Solutions for Inverse Problems: Part A, Volume 26 in the Handbook of Numerical Analysis, highlights new advances in the field, with this new volume presenting interesting chapters on a variety of timely topics, including Data-Driven Approaches for Generalized Lasso Problems, Implicit Regularization of the Deep Inverse Prior via (Inertial) Gradient Flow, Generalized Hardness of Approximation, Hallucinations, and Trustworthiness in Machine Learning for Inverse Problems, Energy-Based Models for Inverse Imaging Problems, Regularization Theory of Stochastic Iterative Methods for Solving Inverse Problems, and more.Other sections cover Advances in Identifying Differential Equations from Noisy Data Observations, The Complete Electrode Model for Electrical Impedance Tomography: A Comparative Study of Deep Learning and Analytical Methods, Learned Iterative Schemes: Neural Network Architectures for Operator Learning, Jacobian-Free Backpropagation for Unfolded Schemes with Convergence Guarantees, and Operator Learning Meets Inverse Problems: A Probabilistic Perspective

  • Recent Developments in Theory and Applications of Fractional Order Systems

    • 1st Edition
    • Mehmet Yavuz + 2 more
    • English
    Recent Developments in Theory and Applications of Fractional Order Systems presents a rigorous and thorough analysis of various aspects of Fractional Calculus. The book provides readers with a thorough understanding of fundamental concepts and methods of applied mathematics utilized in a variety of scientific and engineering disciplines. The authors present each computational modeling concept with a definition, methods, theorems, and observations followed by typical application problems and step-by-step solutions. Each topic is covered in detail, followed typically by several meticulously worked out examples and a problem set containing many additional related problems.In addition, the book discusses recent developments and the latest research on Fractional Calculus and its applications, demonstrating important applications in Engineering, Computer Science, Management, Social Science, and the Humanities.

  • Quaternion Generalized Inverses

    Foundations, Theory, Problems, and Solutions
    • 1st Edition
    • Ivan I. Kyrchei
    • English
    A cornerstone of linear algebra, the determinant's utility in real and complex fields is undeniable, though traditionally limited to invertibility, rank, and solving linear systems. Quaternion Generalized Inverses: Foundations, Theory, Problems, and Solutions ventures into uncharted territory: extending these concepts to linear algebra over the noncommutative quaternion skew field. The author's groundbreaking theory of "noncommutative" row–column determinants is central to this exploration, a significant advancement beyond the Moore determinant. This seven-chapter work thoroughly introduces the history of noncommutative determinants before delving into the author's theory and its application to inverse matrix computation and Cramer's rule for quaternion systems. The main portion of this work is dedicated to a comprehensive examination of quaternion generalized inverses, spanning the well-established Moore–Penrose and Drazin inverses to more recent developments such as core-EP and composite inverses. The book provides their definitions, properties, and, uniquely, their determinantal representations based on the author's noncommutative determinants. It culminates in demonstrating their powerful applications in solving a wide range of quaternion matrix equations, including Sylvester-type and constrained equations, as well as differential matrix equations.

  • Numerical Linear Algebra with Applications

    Using MATLAB and Octave
    • 2nd Edition
    • William Ford + 1 more
    • English
    Numerical Linear Algebra with Applications: Using MATLAB and Octave, Second Edition provides practical knowledge on modern computational techniques for the numerical solution of linear algebra problems. The book offers a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions. Useful to readers regardless of background, the text begins with six introductory courses to provide background for those who haven’t taken applied or theoretical linear algebra. This approach offers a thorough explanation of the issues and methods for practical computing using MATLAB as the vehicle for computation.Appropri... for advanced undergraduate and early graduate courses on numerical linear algebra, this useful textbook explores numerous applications to engineering and science.

  • An Introduction to Discrete Mathematics

    • 1st Edition
    • Vidyadhar Kulkarni
    • English
    An Introduction to Discrete Mathematics offers an engaging and accessible introduction to discrete mathematics for beginning undergraduate students across a wide range of application areas, from mathematics to statistics, operations research, business, engineering, and the sciences. It provides solid foundation in precise proof writing methods, with early chapters introducing set theory and logic that are followed by deductive and inductive proof techniques, number theory, counting principles, permutations and combinations, probability of events, random variables, graphs, and weighted graphs.The book illustrates fundamental concepts in discrete mathematics with clear and precise definitions that are paired with examples and counter-examples as applied in combinatorics, discrete probability, and graph theory. Chapters include student exercises to enhance learning, and a solutions manual and example questions are available for instructors on a companion website.

  • Mathematical Modeling in Bioscience

    Theory and Applications
    • 1st Edition
    • Hemen Dutta
    • English
    Mathematical Modeling in Bioscience: Theory and Applications provides readers with tools and techniques for mathematical modeling in bioscience through a wide range of novel and intriguing topics. The book concentrates on larger elements of mathematical modeling in bioscience, including topics such as modeling of the Topp--Leone new power generalized Weibull-G distribution family, vector-borne disease modeling, transmission modeling of SARS-COV-2 among other infectious diseases, pattern formulation models, compartmental models for HIV/AIDS transmission, population models, irrigation scheduling models, and predator--prey models. The readers will discover a variety of new methods, approaches, and techniques, as well as a wide range of applications demonstrating key concepts in bioscience modeling. This book provides a leading-edge resource for researchers in a variety of scientific fields who are interested in mathematical modeling, including mathematics, statistics, biology, biomedical engineering, computer science, and applied sciences.

  • Generalized Quantum Calculus with Applications

    • 1st Edition
    • Svetlin G. Georgiev + 1 more
    • English
    Generalized Quantum Calculus with Applications is devoted to the qualitative theory of general quantum calculus and its applications to general quantum differential equations and inequalities. The book is aimed at upper-level undergraduate students and beginning graduate students in a range of interdisciplinary courses including physical sciences and engineering, from quantum mechanics to differential equations, with pedagogically organized chapters that each concludes with a section of practical problems. Generalized quantum calculus includes a generalization of the q-quantum calculus and the time scale calculus. There are many open problems and difficulties in q-quantum calculus and time-scale calculus, and this book explores how to use the generalized quantum operators to solve difficulties arising in q-quantum calculus and time-scale calculus, including but not limited to generalized quantum integration, generalized quantum chain rules, and generalized quantum Taylor formula.Since generalized quantum calculus includes the q-quantum and time-scale calculus, this book can be utilized by a wide audience of researchers and students. This text is one of few foundational books on generalized quantum calculus and can be used for future discoveries in the area of integral transforms, variational calculus, integral equations, and inequalities in the language of generalized quantum calculus. This book also offers detailed proofs, exercises, and examples to aid instructors, researchers, and users in their studies.

  • Approximation Theory and Applications

    Piecewise Linear and Generalized Functions
    • 1st Edition
    • Sergei Aliukov
    • English
    Approximation Theory and Applications: Piecewise Linear and Generalized Functions presents the main provisions of approximation theory, and considers existing and new methods for approximating piecewise linear and generalized functions, widely used to solve problems related to mathematical modeling of systems, processes, and phenomena in fields ranging from engineering to economics. The widespread use of piecewise linear and generalized functions is explained by the simplicity of their structure. However, challenges often arise when constructing solutions over the entire domain of these functions, requiring the use special mathematical methods to put theory into practice. This book first offers a first full foundation in approximation theory as it relates to piecewise linear and generalized functions, followed by staged methods to resolve common problems in practice, with applications examined across structural mechanics, medicine, quantum theory, signal theory, semiconductor theory, mechanical engineering, heat engineering, and other fields. Later chapters consider numerical verification of approximation methods, and approximation theory as the basis for new macroeconomic theory with impulse and jump characteristics. Each chapter includes questions for review and sample problems, accompanied by a separate Solutions Manual hosted for instructor access.

  • Irreducible Tensorial Sets

    • 1st Edition
    • U. Fano + 1 more
    • H. S. W. Massey + 1 more
    • English
    Irreducible Tensorial Sets discusses mathematical methods originating from the theory of coupling and recoupling of angular momenta in atomic physics that constitute an extension of vector and tensor algebra. The book presents a unified treatment with a compact system of notations from different approaches, such as group theory, algebra, and quantum mechanical transformation theory. It discusses irreducible tensorial sets that cover different sets of quantities such as tensor components and states of atomic systems. It also explains quantum mechanical applications, coupling and recoupling of atomic and nuclear states, the Wigner-Eckart theorem, and the products of tensorial sets of operators. The text shows how to calculate the interaction energy between atomic systems couple with one another with a constant total angular momentum. The book also explains the correlations which are functions of the Euler angles between the frame of reference in which a radiation is observed and a frame of reference attached to the orienting radiation or field. It then cites sample problems related to the angular distribution of radiations. The book will prove useful for physicists, for mathematicians, or for readers with some knowledge in theoretical physics, particularly on theory of groups and quantum mechanics.

  • Mathematical Methods in Medical and Biological Sciences

    • 1st Edition
    • Harendra Singh + 1 more
    • English
    Mathematical Methods in Medical and Biological Sciences presents mathematical methods for computational models arising in the medical and biological sciences. The book presents several real-life medical and biological models, such as infectious and non-infectious diseases that can be modeled mathematically to accomplish　profound research in virtual environments when the cost of laboratory expenses is relatively high. It focuses on mathematical techniques that provide global solutions for models arising in medical and biological sciences by considering their long-term benefits.In addition, the book provides leading-edge developments and insights for a range of applications, including epidemiological modeling of pandemic dynamics, viral infection developments, cancer developments, blood oxygen dynamics, HIV infection spread, reaction-diffusion models, polio infection spread, and chaos modeling with fractional order derivatives.

  • Encyclopedia of Mathematical Physics

    • 2nd Edition
    • Richard Szabo + 1 more
    • English
    Encyclopedia of Mathematical Physics, Second Edition provides a complete resource for researchers, students, and lecturers with an interest in mathematical physics. The book enables readers to access basic information on topics peripheral to their own areas by providing a repository of core information that can be used to refresh even the experienced researcher’s memory and aid teachers in directing students to entries relevant to their course-work. The impressive amount of information in this work - approximately 270 chapters - has been distilled, organized into 10 distinct sections and presented as a complete reference tool to the userThe book is a stimulus for new researchers working in mathematical physics—or in areas using the methods originating from work in mathematical physics—providing them with focused, high-quality background information.

  • Difference Equations and Applications

    • 1st Edition
    • Youssef N. Raffoul
    • English
    Difference Equations and Applications provides unique coverage of high-level topics in the application of difference equations and dynamical systems. The book begins with extensive coverage of the calculus of difference equations, including contemporary topics on l_p stability, exponential stability, and parameters that can be used to qualitatively study solutions to non-linear difference equations, including variations of parameters and equations with constant coefficients, before moving on to the Z-Transform and its various functions, scalings, and applications. It covers systems, Lyapunov functions, and stability, a subject rarely covered in competitor titles, before concluding with a comprehensive section on new variations of parameters.Exercises are provided after each section, ranging from an easy to medium level of difficulty. When finished, students are set up to conduct meaningful research in discrete dynamical systems. In summary, this book is a comprehensive resource that delves into the mathematical theory of difference equations while highlighting their practical applications in various dynamic systems. It is highly likely to be of interest to students, researchers, and professionals in fields where discrete modeling and analysis are essential.

  • Fuzzy Mathematics, Graphs, and Similarity Measures

    Analysis and Application Across Global Challenges
    • 1st Edition
    • John Mordeson + 1 more
    • English
    Fuzzy Mathematics, Graphs, and Similarity Measures provides a solid foundation in core analytical tracks of mathematics of uncertainty, from fuzzy mathematics to graphs and similarity measures with applications in a range of timely cases studies and world challenges. Following a full grounding in fuzzy graph indices, connectivity in fuzzy graph structures, lattice isomorphisms, and similarity measures, the book applies these models in analyzing world challenges, from human trafficking to modern slavery, global poverty, global hunger, homelessness, biodiversity, extinction, terrorism and bioterrorism, pandemics, and climate change.Connections and constructive steps forward are tied throughout to UN Sustainable Development Goals (SDGs). The authors demonstrate and instruct readers in applying techniques from mathematics of uncertainty in examining issues where accurate data is impossible to obtain. In addition to a diverse range of cases studies, exercises reinforce key concepts in each chapter, and an online instructor’s manual supports teaching across a range of course contexts.

  • Computer and Information Security Handbook (2-Volume Set)

    • 4th Edition
    • John Vacca
    • English
    Computer and Information Security Handbook, Fourth Edition offers deep coverage of an extremely wide range of issues in computer and cybersecurity theory, along with applications and best practices, offering the latest insights into established and emerging technologies and advancements. With new parts devoted to such current topics as Cyber Security for the Smart City and Smart Homes, Cyber Security of Connected and Automated Vehicles, and Future Cyber Security Trends and Directions, the book now has 104 chapters in 2 Volumes written by leading experts in their fields, as well as 8 updated appendices and an expanded glossary.Chapters new to this edition include such timely topics as Threat Landscape and Good Practices for Internet Infrastructure, Cyber Attacks Against the Grid Infrastructure, Threat Landscape and Good Practices for the Smart Grid Infrastructure, Energy Infrastructure Cyber Security, Smart Cities Cyber Security Concerns, Community Preparedness Action Groups for Smart City Cyber Security, Smart City Disaster Preparedness and Resilience, Cyber Security in Smart Homes, Threat Landscape and Good Practices for Smart Homes and Converged Media, Future Trends for Cyber Security for Smart Cities and Smart Homes, Cyber Attacks and Defenses on Intelligent Connected Vehicles, Cyber Security Issues in VANETs, Use of AI in Cyber Security, New Cyber Security Vulnerabilities and Trends Facing Aerospace and Defense Systems, and much more.

  • Mechanism Design, Behavioral Science and Artificial Intelligence in International Relations

    • 1st Edition
    • Tshilidzi Marwala
    • English
    Recent advances in AI and Mechanism Design provide a vital tool for solving collective action problems, common in international relations. By using AI to optimize mechanisms for cooperation and coordination, we can better address issues such as climate change, trade, and security. Mechanism Design, Behavioral Science and Artificial Intelligence in International Relations shows readers how the intersection of Mechanism Design and Artificial Intelligence is revolutionizing the way we approach international relations. By using AI to optimize mechanisms, we can design better institutions, policies, and agreements that are more effective and efficient. Dr. Tshilidzi Marwala, United Nations University Rector and UN Under-Secretary General, presents the essential technologies used in Game Theory, Mechanism Design and AI and applies these to significant global issues such as interstate conflict, cybersecurity, and energy. International relations are a complex field, with many different actors and interests in play. By incorporating AI into our analysis and decision-making processes, we can better understand and predict the behavior of multiple actors and design mechanisms that take these behaviors into account, thereby producing more desirable and creative interdisciplinary approaches. The book presents real-world applications of these rapidly evolving technologies in crucial research fields such as Interstate Conflict, International Trade, Climate Change, Water management, Energy, cybersecurity, and global finance.

  • Information Modeling and Relational Databases

    • 3rd Edition
    • Terry Halpin + 1 more
    • English
    Information Modeling and Relational Databases, Third Edition, provides thorough coverage of information modeling approaches, including object-role modeling (ORM), entity-relationship (ER) modeling, and the unified modeling language (UML). It shows how to map models developed with those approaches to a variety of relational and nonrelational database systems, including document databases, column-oriented databases, graph databases, and deductive databases. Process and state modeling, ontological modeling, and metamodeling are also covered. For this new edition, the coverage of ORM, ER, UML, SQL, OWL, and BPMN has been thoroughly updated to include their latest versions. A significant amount of new material has been added. Various data file formats such as CSV, XML, JSON, YAML, and some other markup languages are now covered, and a more thorough treatment is provided for nonrelational databases, especially NoSQL. One of the major features of the book is its large number of exercises, which have been thoroughly class-tested. This book is intended for anyone with a stake in the accuracy and efficacy of databases such as systems analysts, information modelers, database designers and administrators, and programmers.

  • Fractional Calculus

    Bridging Theory with Computational and Contemporary Advances
    • 1st Edition
    • Behzad Ghanbari
    • English
    Fractional Calculus: Bridging Theory with Computational and Contemporary Advances is an authoritative and comprehensive guide that delves into the world of fractional calculus, offering a unique blend of theoretical foundations, numerical algorithms, practical applications, and innovative perspectives. This book explores the mathematical framework of fractional calculus and its relevance across various disciplines, providing readers with a deep understanding of this rapidly growing field. The author presents a rigorous yet accessible approach to fractional calculus, making it suitable for mathematicians, researchers, academics, graduate students, and professionals in engineering and applied sciences. The book covers a wide range of topics, including numerical methods for fractional calculus equations, fractional differential equations, fractal dynamics, and fractional control systems. It also explores applications in areas such as physics, engineering, signal processing, and data analysis. Fractional Calculus: Bridging Theory with Computational and Contemporary Advances equips readers with the necessary tools to tackle challenging problems involving fractional calculus, empowering them to apply these techniques in their research, professional work, or academic pursuits. The book provides a comprehensive introduction to the fundamentals of fractional calculus, explaining the theoretical concepts and key definitions in a clear and accessible manner. This helps readers build a strong foundation in the subject. The book then covers a range of numerical algorithms specifically designed for fractional calculus problems, explaining the underlying principles, step-by-step implementation, and computational aspects of these algorithms. This enables readers to apply numerical techniques to solve fractional calculus problems effectively. The book also provides examples that illustrate how fractional calculus is applied to solve real-world problems, providing readers with insights into the wide-ranging applications of the subject.

  • Advanced Calculus for Mathematical Modeling in Engineering and Physics

    With Discrete and Numerical Analogies
    • 1st Edition
    • David Stapleton
    • English
    Advanced Calculus for Mathematical Modeling in Engineering and Physics: With Discrete and Numerical Analogies introduces the principles and methods of advanced calculus for mathematical modeling through a balance of theory and application using a state space approach with elementary functional analysis. This framework facilitates a deeper understanding of the nature of mathematical models, and of the behavior of their solutions. The work provides a variety of advanced calculus models for mathematical, physical science, and engineering audiences, with discussions on how calculus-based models and their discrete analogies are generated. This valuable textbook offers scientific computations driven by Octave/MATLAB script.

  • Numerical Analysis meets Machine Learning

    • 1st Edition
    • Volume 25
    • English
    Numerical Analysis Meets Machine Learning series, highlights new advances in the field, with this new volume presenting interesting chapters. Each chapter is written by an international board of authors.

  • Fractional Differential Equations

    Theoretical Aspects and Applications
    • 1st Edition
    • Praveen Agarwal + 2 more
    • English
    Fractional Differential Equations: Theoretical Aspects and Applications presents the latest mathematical and conceptual developments in the field of Fractional Calculus and explores the scope of applications in research science and computational modeling. The book delves into these methods and applied computational modelling techniques, including analysis of equations involving fractional derivatives, fractional derivatives and the wave equation, analysis of FDE on groups, direct and inverse problems, functional inequalities, and computational methods for FDEs in physics and engineering. Other modeling techniques and applications explored include general fractional derivatives involving the special functions in analysis and fractional derivatives with respect to other functions. Fractional Calculus, the field of mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modelling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economics, finance, biology, and chemistry, among others.

  • Applications of Nature-Inspired Computing and Optimization Techniques

    • 1st Edition
    • Volume 135
    • English
    Advances in Computers, Volume 135 highlights advances in the field, with this new volume, Applications of Nature-inspired Computing and Optimization Techniques presenting interesting chapters on a variety of timely topics, including A Brief Introduction to Nature-inspired Computing, Optimization and Applications, Overview of Non-linear Interval Optimization Problems, Solving the Aircraft Landing Problem using the Bee Colony Optimization (BCO) Algorithm, Situation-based Genetic Network Programming to Solve Agent Control Problems, Small Signal Stability Enhancement of Large Interconnected Power System using Grasshopper Optimization Algorithm Tuned Power System Stabilizer, Air Quality Modelling for Smart Cities of India by Nature Inspired AI – A Sustainable Approach, and much more.Other sections cover Genetic Algorithm for the Optimization of Infectiological Parameter Values under Different Nutritional Status, A Novel Influencer Mutation Strategy for Nature-inspired Optimization Algorithms to Solve Electricity Price Forecasting Problem, Recent Trends in Human and Bio Inspired Computing: Use Case Study from Retail Perspective, Domain Knowledge Enriched Summarization of Legal Judgment Documents via Grey Wolf Optimization, and a host of other topics.

  • Computation and Modeling for Fractional Order Systems

    • 1st Edition
    • Snehashish Chakraverty + 1 more
    • English
    Computation and Modeling for Fractional Order Systems provides readers with problem-solving techniques for obtaining exact and/or approximate solutions of governing equations arising in fractional dynamical systems presented using various analytical, semi-analytical, and numerical methods. Various analytical/semi-anal... methods are applied for solving real-life fractional order problems. The comprehensive descriptions of different recently developed fractional singular, non-singular, fractal-fractional, and discrete fractional operators, along with computationally efficient methods, are included for the reader to understand how these may be applied to real-world systems, and a wide variety of dynamical systems such as deterministic, stochastic, continuous, and discrete are addressed.Fractional calculus has gained increasing popularity and relevance over the last few decades, due to its well-established applications in various fields of science and engineering. It deals with the differential and integral operators with non-integral powers. Fractional differential equations are the pillar of various systems occurring in a wide range of science and engineering disciplines, namely physics, chemical engineering, mathematical biology, financial mathematics, structural mechanics, control theory, circuit analysis, and biomechanics, among others.

  • Actuaries' Survival Guide

    Navigating the Exams as Applications of Data Science
    • 3rd Edition
    • Ping Wang + 1 more
    • English
    Actuaries' Survival Guide: Navigating the Exams as Applications of Data Science, Third Edition explains what actuaries are, what they do, and where they do it. The book describes exciting combinations of ideas, techniques, and skills involved in their day-to-day work. This edition has been updated to reflect the rise of social networking and the internet, the progress toward a global knowledge-based economy, and the global expansion of the actuarial field that has occurred since the prior edition.

  • Applying Computational Intelligence for Social Good

    Track, Understand and Build a Better world
    • 1st Edition
    • Volume 132
    • English
    Applying Computational Intelligence for Social Good: Track, Understand and Build a Better World, Volume 132 presents views on how Computational Intelligent and ICT technologies can be applied to ease or solve social problems by sharing examples of research results from studies of social anxiety, environmental issues, mobility of the disabled, and problems in social safety. Sample chapters in this release include Why is implementing Computational Intelligence for social good so challenging? Principles and its Application, Smart crisis management system for road accidents using Geo-Spacial Machine Learning Techniques, Residential Energy Management System (REMS) Using Machine Learning, Text-Based Personality Prediction using XLNet, and much more.

  • Handbook of Truly Concurrent Process Algebra

    • 1st Edition
    • Yong Wang
    • English
    Handbook of Truly Concurrent Process Algebra provides readers with a detailed and in-depth explanation of the algebra used for concurrent computing. This complete handbook is divided into five Parts: Algebraic Theory for Reversible Computing, Probabilistic Process Algebra for True Concurrency, Actors – A Process Algebra-Based Approach, Secure Process Algebra, and Verification of Patterns. The author demonstrates actor models which are captured using the following characteristics: Concurrency, Asynchrony, Uniqueness, Concentration, Communication Dependency, Abstraction, and Persistence. Every pattern is detailed according to a regular format to be understood and utilized easily, which includes introduction to a pattern and its verifications.Patter... of the vertical domains are also provided, including the domains of networked objects and resource management. To help readers develop and implement the software patterns scientifically, the pattern languages are also presented.

  • Fundamentals of Data Science

    Theory and Practice
    • 1st Edition
    • Jugal K. Kalita + 2 more
    • English
    Fundamentals of Data Science: Theory and Practice presents basic and advanced concepts in data science along with real-life applications. The book provides students, researchers and professionals at different levels a good understanding of the concepts of data science, machine learning, data mining and analytics. Users will find the authors’ research experiences and achievements in data science applications, along with in-depth discussions on topics that are essential for data science projects, including pre-processing, that is carried out before applying predictive and descriptive data analysis tasks and proximity measures for numeric, categorical and mixed-type data. The book's authors include a systematic presentation of many predictive and descriptive learning algorithms, including recent developments that have successfully handled large datasets with high accuracy. In addition, a number of descriptive learning tasks are included.

  • Optical Communications from a Fourier Perspective

    Fourier Theory and Optical Fiber Devices and Systems
    • 1st Edition
    • Palle Jeppesen + 1 more
    • English
    Optical Communications from a Fourier Perspective: Fourier Theory and Optical Fiber Devices and Systems covers Fourier theory and signal analysis over photonic components, including time lenses in optical communication. Sections cover wave propagation in optical waveguides based on Maxwell equations and the nonlinear Schrödinger equation. Optical Fourier transform in the form of time lens is covered, for example in modulation format conversion and spectrum magnification, and couplers and their use for optical discrete Fourier transformation are also discussed. Other important subjects are discussed such as shot noise, thermal noise and also the basics of four wave mixing in relation to time lenses. Detailed derivations and a deeper background for the chapters are provided in appendices where appropriate. Some of the theory is more generally applicable beyond optical communication and is of relevance also for communications engineering. The Fourier theory dimension of the book presents the relationship between Fourier series and Fourier integrals and also the related Laplace transform.

  • Picture Fuzzy Logic and Its Applications in Decision Making Problems

    • 1st Edition
    • Chiranjibe Jana + 3 more
    • English
    Picture Fuzzy Logic and Its Applications in Decision Making Problems provides methodological frameworks and the latest empirical research findings in the field of picture fuzzy operators and their applications in scientific research and real-world engineering problems. In this book, picture fuzzy sets are investigated, and different types of operators are defined to solve a number of important decision-making and optimization problems. The hybrid operator on picture fuzzy set based on the combination of picture fuzzy weighted averaging operators and picture fuzzy weighted geometric operators is developed and named Hybrid Picture Fuzzy Weighted Averaging Geometric (H-PFWAG) operator. In addition, another operator is developed for interval-valued picture fuzzy environment, which is named Hybrid Interval-Valued Picture Fuzzy Weighted Averaging Geometric (H-IVPFWAG) operator. These two operators are then demonstrated as solutions to Multiple-Attribute Decision-Making (MADM) problems. The picture fuzzy soft weighted aggregation operators (averaging and geometric) are defined, and these are applied to develop a multi-criteria group decision making system.

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