Computational Methods for Deep Learning
The first edition of this textbook was published in 2021. Over the past two years, we have invested in enhancing all aspects of deep learning methods to ensure the book is comprehensive and impeccable. Taking into account feedback from our readers and audience, the author has diligently updated this book. The second edition of this textbook presents control theory, transformer models, and graph neural networks (GNN) in deep learning. We have incorporated the latest algorithmic advances and large-scale deep learning models, such as GPTs, to align with the current research trends. Through the second edition, this book showcases how computational methods in deep learning serve as a dynamic driving force in this era of artificial intelligence (AI). This book is intended for research students, engineers, as well as computer scientists with interest in computational methods in deep learning. Furthermore, it is also well-suited for researchers exploring topics such as machine intelligence, robotic control, and related areas.
Analyticity and Sparsity in Uncertainty Quantification for Pdes with Gaussian Random Field Inputs
The present book develops the mathematical and numerical analysis of linear, elliptic and parabolic partial differential equations (PDEs) with coefficients whose logarithms are modelled as Gaussian random fields (GRFs), in polygonal and polyhedral physical domains. Both, forward and Bayesian inverse PDE problems subject to GRF priors are considered.Adopting a pathwise, affine-parametric representation of the GRFs, turns the random PDEs into equivalent, countably-parametric, deterministic PDEs, with nonuniform ellipticity constants. A detailed sparsity analysis of Wiener-Hermite polynomial chaos expansions of the corresponding parametric PDE solution families by analytic continuation into the complex domain is developed, in corner- and edge-weighted function spaces on the physical domain.The presented Algorithms and results are relevant for the mathematical analysis of many approximation methods for PDEs with GRF inputs, such as model order reduction, neural network and tensor-formatted surrogates of parametric solution families. They are expected to impact computational uncertainty quantification subject to GRF models of uncertainty in PDEs, and are of interest for researchers and graduate students in both, applied and computational mathematics, as well as in computational science and engineering.
Numerical Methods for Unsteady Compressible Flow Problems
This book is written to give both mathematicians and engineers an overview of the state of the art in the field, as well as of new developments. The focus is on methods for the compressible Navier-Stokes equations, the solutions of which can exhibit shocks, boundary layers and turbulence.
Probability
This textbook offers a complete one-semester course in probability, covering the essential topics necessary for further study in the areas of probability and statistics. The book begins with a review of the fundamentals of measure theory and integration. Probability measures, random variables, and their laws are introduced next, along with the main analytic tools for their investigation, accompanied by some applications to statistics. Questions of convergence lead to classical results such as the law of large numbers and the central limit theorem with their applications also to statistical analysis and more. Conditioning is the next main topic, followed by a thorough introduction to discrete time martingales. Some attention is given to computer simulation. Through the text, over 150 exercises with full solutions not only reinforce the concepts presented, but also provide students with opportunities to develop their problem-solving skills, and make this textbook suitable forguided self-study. Based on years of teaching experience, the author's expertise will be evident in the clear presentation of material and the carefully chosen exercises. Assuming familiarity with measure and integration theory as well as elementary notions of probability, the book is specifically designed for teaching in parallel with a first course in measure theory. An invaluable resource for both instructors and students alike, it offers ideal preparation for further courses in statistics or probability, such as stochastic calculus, as covered in the author's book on the topic.
Smooth-Automorphic Forms and Smooth-Automorphic Representations
This book provides a conceptual introduction into the representation theory of local and global groups, with final emphasis on automorphic representations of reductive groups G over number fields F.Our approach to automorphic representations differs from the usual literature: We do not consider 'K-finite' automorphic forms, but we allow a richer class of smooth functions of uniform moderate growth. Contrasting the usual approach, our space of 'smooth-automorphic forms' is intrinsic to the group scheme G/F.This setup also covers the advantage that a perfect representation-theoretical symmetry between the archimedean and non-archimedean places of the number field F is regained, by making the bigger space of smooth-automorphic forms into a proper, continuous representation of the full group of adelic points of G.Graduate students and researchers will find the covered topics appear for the first time in a book, where the theory of smooth-automorphic representations is robustly developed and presented in great detail.
Representation Theory and Algebraic Geometry
The chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference "Interactions Between Representation Theory and Algebraic Geometry", held at the University of Chicago, August 21-25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes: Groups, algebras, categories, and representation theoryD-modules and perverse sheavesAnalogous varieties defined by quivers Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.
A new perspective on the determination of the prime numbers
Academic Paper from the year 2023 in the subject Mathematics - Number Theory, grade: 2.00, language: English, abstract: A procedure is developed for determining the primality of a number, N, but does not examine the number, rather a function of that number. The function has several properties, by means of which many of the non-primes may be identified and discarded without further investigation. Using the concept of cage numbers (defined in the text) a matrix of all of the counting numbers is produced, where, with the exception of the prime numbers 2 and 3, all of the prime numbers are embedded in one column of the matrix. Strings of consecutive numbers are extracted from the matrix, and it is seen that the only places within the whole of the range of the counting numbers where prime numbers can exist is at the positions of the second and penultimate numbers within a string By means of the function of N, all of the odd numbers may be generated and examined for primality.
Mathematicians Playing Games
This book explores a wide variety of popular mathematical games, including their historical beginnings and the mathematical theories that underpin them. Its academic level is suitable for high school students and higher, but people of any age or level will find something to entertain them, and something new to learn.
Mathematicians Playing Games
This book explores a wide variety of popular mathematical games, including their historical beginnings and the mathematical theories that underpin them. Its academic level is suitable for high school students and higher, but people of any age or level will find something to entertain them, and something new to learn.
The Navier-Stokes Problem in the 21st Century
This book provides a self-contained guide to the role of harmonic analysis in the PDEs of fluid mechanics, now revised to include fresh examples, theorems, results, and references that have become relevant since the first edition published in 2016.
The Geometry of Cubic Hypersurfaces
Cubic hypersurfaces are described by almost the simplest possible polynomial equations, yet their behaviour is rich enough to demonstrate many of the central challenges in algebraic geometry. With exercises and detailed references to the wider literature, this thorough text introduces cubic hypersurfaces and all the techniques needed to study them. The book starts by laying the foundations for the study of cubic hypersurfaces and of many other algebraic varieties, covering cohomology and Hodge theory of hypersurfaces, moduli spaces of those and Fano varieties of linear subspaces contained in hypersurfaces. The next three chapters examine the general machinery applied to cubic hypersurfaces of dimension two, three, and four. Finally, the author looks at cubic hypersurfaces from a categorical point of view and describes motivic features. Based on the author's lecture courses, this is an ideal text for graduate students as well as an invaluable reference for researchers in algebraic geometry.
Time Series for Data Scientists
Learn by doing with this user-friendly introduction to time series data analysis in R. This book explores the intricacies of managing and cleaning time series data of different sizes, scales and granularity, data preparation for analysis and visualization, and different approaches to classical and machine learning time series modeling and forecasting. A range of pedagogical features support students, including end-of-chapter exercises, problems, quizzes and case studies. The case studies are designed to stretch the learner, introducing larger data sets, enhanced data management skills, and R packages and functions appropriate for real-world data analysis. On top of providing commented R programs and data sets, the book's companion website offers extra case studies, lecture slides, videos and exercise solutions. Accessible to those with a basic background in statistics and probability, this is an ideal hands-on text for undergraduate and graduate students, as well as researchers in data-rich disciplines
Asymptotic and Stationary Preserving Schemes for Kinetic and Hyperbolic Partial Differential Equations
In this thesis, we are interested in numerically preserving stationary solutions of balance laws. We start by developing finite volume well-balanced schemes for the system of Euler equations and the system of Magnetohydrodynamics (MHD) equations with gravitational source term. Since fluid models and kinetic models are related, this leads us to investigate Asymptotic Preserving (AP) schemes for kinetic equations and their ability to preserve stationary solutions. In an attempt to mimic our result for kinetic equations in the context of fluid models, for the isentropic Euler equations we developed an AP scheme in the limit of the Mach number going to zero. The properties of the schemes we developed and its criteria are validated numerically by various test cases from the literature.
Outlier Detection Using Power Mean
Outliers have been regarded as the noisy data in statistics which have now turned out to be an important problem and are now been researched in diverse fields and application domains. Outlier detection has been in core interest of not only the statisticians but all the professionals who are working on a particular data set. Many outlier detection techniques have been developed specific to certain application domains, while some techniques are more generic. This work has added one more technique to the bucket list of all those professionals. Power mean which has been used as a general method to calculate various means like arithmetic mean (power mean with power 1), geometric mean (power mean with power 0), Lorentz mean (power mean with power 1/3) etc. can also be used to detect the sensitivity of the data towards being the outlier. This work studies various powers of power mean for outlier detection. It contains two different data sets, one containing fractions and the other integers. The results have been verified by the existing standard techniques of outlier detection. Thus this book contains description of detecting outliers using power mean with different types of data sets, graphs and figures for better understanding. The idea is to check the efficacy of the method using a data set in which the outlier or the anomaly is already known and then testing the same method for a data set in which the outliers are not known to us. The open research issues and challenges at the end will provide researchers a clear path for the future of outlier detection methods. The book would be useful for practitioners of applied statistics and data analysts.
The Square Root of Two to One Million Digits
A conundrum faced ancient mathematicians. Which number to use for the square root of 2? Ancient Babylonians used the fraction 305470/216000. This was accurate to about six decimal places. Ancient Indians used the fraction 577/408. This fraction is accurate to five decimal places. Today, mathematicians understand that the square root of 2 is irrational, its digits go on forever without repeating. It is calculated using an infinite series. This book lists the first million digits of the square root of 2.
Mathematical Topics on Modelling Complex Systems
This book explores recent developments in theoretical research and mathematical modelling of real-world complex systems, organized in four parts. The first part of the book is devoted to the mathematical tools for the design and analysis in engineering and social science study cases. We discuss the periodic evolutions in nonlinear chemical processes, vibro-compact systems and their behaviour, different types of metal-semiconductor self-assembled samples, made of silver nanowires and zinc oxide nanorods. The second part of the book is devoted to mathematical description and modelling of the critical events, climate change and robust emergency scales. In three chapters, we consider a climate-economy model with endogenous carbon intensity and the behaviour of Tehran Stock Exchange market under international sanctions. The third part of the book is devoted to fractional dynamic and fractional control problems. We discuss the novel operational matrix technique for variable-order fractional optimal control problems, the nonlinear variable-order time fractional convection-diffusion equation with generalized polynomials The fourth part of the book concerns solvability and inverse problems in differential and integro-differential equations. The book facilitates a better understanding of the mechanisms and phenomena in nonlinear dynamics and develops the corresponding mathematical theory to apply nonlinear design to practical engineering. It can be read by mathematicians, physicists, complex systems scientists, IT specialists, civil engineers, data scientists and urban planners.
Art and IR Theory
This book examines the correspondence between international relations (IR) theories of structural realism and constructivism and paintings, notably the artwork of Mark Rothko and Jackson Pollock, in a game theory setting. This interdisciplinary approach, through the lens of game theory and semiotics, permits different, enriched interpretations of structural realism and constructivism. These theories constitute an axis of debate between social and systemic approaches to international politics, as well as an axis of differentiation between scientific realism and positivism as philosophies of science. As such, the interpretations explored in this book contribute to what we know about international relations, how semiotics intersect with strategic uncertainty, and explains these interactions in the proposed games model.The book's use of game theory and semiotics generate 'visual semiotic games' (VSGs) that shed light on the debate axes through strategic uncertainty, interactions, and players' interactive belief systems. VSGs will contribute to literature on experimental semiotics in the sense of players' coordination behavior, beliefs, and artistic evaluations. The equilibria, interpreted through branches of philosophy of mind and theories of explanation, will reveal possibilities of agreement among players about which artwork representing the theory at hand is the best, opening innovative research perspectives for the discipline of IR theory.
Modeling, Simulation and Optimization in the Health- And Energy-Sector
This volume is addressed to people who are interested in modern mathematical solutions for real life applications. In particular, mathematical modeling, simulation and optimization is nowadays successfully used in various fields of application, like the energy- or health-sector. Here, mathematics is often the driving force for new innovations and most relevant for the success of many interdisciplinary projects. The presented chapters demonstrate the power of this emerging research field and show how society can benefit from applied mathematics.
Topics in Global Real Analytic Geometry
In the first two chapters we review the theory developped by Cartan, Whitney and Tognoli. Then Nullstellensatz is proved both for Stein algebras and for the algebra of real analytic functions on a C-analytic space. Here we find a relation between real Nullstellensatz and seventeenth Hilbert's problem for positive semidefinite analytic functions. Namely, a positive answer to Hilbert's problem implies a solution for the real Nullstellensatz more similar to the one for real polinomials. A chapter is devoted to the state of the art on this problem that is far from a complete answer. In the last chapter we deal with inequalities. We describe a class of semianalytic sets defined by countably many global real analytic functions that is stable under topological properties and under proper holomorphic maps between Stein spaces, that is, verifies a direct image theorem. A smaller class admits also a decomposition into irreducible components as it happens for semialgebraic sets. Duringthe redaction some proofs have been simplified with respect to the original ones.
Sequences and Series in Calculus
The book Sequences and Series in Calculus is designed as the first college/university calculus course for students who take and do well on the AP AB exam in high school and who are interested in a more proof-oriented treatment of calculus. The text begins with an ε-ℕ treatment of sequence convergence, then builds on this to discuss convergence of series--first series of real numbers, then series of functions. The difference between uniform and pointwise convergence is discussed in some detail. This is followed by a discussion of calculus on power series and Taylor series. Finally improper integrals, integration by parts and partial fractions integration all are introduced. This book is designed both to teach calculus, and to give the readers and students a taste of analysis to help them determine if they wish to study this material even more deeply. It might be used by colleges and universities who teach special versions of calculus courses for their most mathematically advanced entering first-year students, as might its older sibling text Multivariable and Vector Calculus which appeared in 2020 and is intended for students who take and do well on the AP BC exam.
The fast guide to statistical testing with JASP
FeaturesWorked examples from the social sciencesClearly written, without mathematical formulaeWell-structured, for beginnersComprehensive chapter on categorical analysis - not just 'Chi squared'Effect sizes and confidence intervalsIntroduces Bayesian statisticsReproduces a well-received short chapter on making presentationsSummary chapter on intermediate statistical testsData sets and case studies on the website
The fast guide to statistical testing with JASP
FeaturesWorked examples from the social sciencesClearly written, without mathematical formulaeWell-structured, for beginnersComprehensive chapter on categorical analysis - not just 'Chi squared'Effect sizes and confidence intervalsIntroduces Bayesian statisticsReproduces a well-received short chapter on making presentationsSummary chapter on intermediate statistical testsData sets and case studies on the website
Metric Spaces and Related Analysis
This book offers the comprehensive study of one of the foundational topics in Mathematics, known as Metric Spaces. The book delivers the concepts in an appropriate and concise manner, at the same time rich in illustrations and exercise problems. Special focus has been laid on important theorems like Baire's Category theorem, Heine-Borel theorem, Ascoli-Arzela Theorem, etc, which play a crucial role in the study of metric spaces.The additional chapter on Cofinal completeness, UC spaces and finite chainability makes the text unique of its kind. This helps the students in: Readers will also find brief discussions on various subtleties of continuity like subcontinuity, upper semi-continuity, lower semi-continuity, etc. The interested readers will be motivated to explore the special classes of functions between metric spaces to further extent.Consequently, the book becomes a complete package: it makes the foundational pillars strong and develops the interest of students to pursue research in metric spaces. The book is useful for third and fourth year undergraduate students and it is also helpful for graduate students and researchers.
Introduction to Proofs and Proof Strategies
Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof. The material revolves around possible strategies to approaching a problem without classifying 'types of proofs' or providing proof templates. Instead, it helps students develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow. Beginning by discussing familiar and fundamental topics from a more theoretical perspective, the book moves on to inequalities, induction, relations, cardinality, and elementary number theory. The final supplementary chapters allow students to apply these strategies to the topics they will learn in future courses. With its focus on 'doing mathematics' through 200 worked examples, over 370 problems, illustrations, discussions, and minimal prerequisites, this course will be indispensable to first- and second-year students in mathematics, statistics, and computer science. Instructor resources include solutions to select problems.
R Programming
R is an open-source statistical environment and programming language that has grown in popularity for data management and analysis in various industries. "R" Programming teaches you all the R you'll ever need in a rapid and painless manner. This accessible tutorial taught you your way around a list with no previous programming expertise and loads of practical examples, step-by-step exercises, and sample code. This book covers the most significant modeling and prediction methods, as well as their applications.Learn how to use R to transform raw data into knowledge, understanding, and insight. This book introduces you to R, RStudio, and the tidyverse, a set of R tools that work together to make data research simple, fluent, and enjoyable. This book is meant to get you practicing data science as fast as possible, even if you have no prior programming expertise. You'll get a comprehensive grasp of the data science cycle and the fundamental tools you'll need to handle the details.R is becoming more well-known by the day, as large institutions embrace it as a standard. Its popularity stems partly from the fact that it is a free tool replacing expensive statistical software products that may take an undue amount of time to master. Furthermore, R allows a user to do complicated statistical analyses with only a few keystrokes, making advanced studies accessible and clear to a broad audience.Learn how to import data, construct and dismantle data objects, traverse R's environment system, develop your own functions, and utilize all of R's programming tools with this book. This book will not only teach you how to program but also how to use R for more than simply displaying and analyzing data.Most of the chapters are written for you to understand statistical data, so if you are a student, this book can guarantee to teach you some basic statistics that will help you get good grades.The first two chapters are based on beginner's level of R programming, mainly teaching how to successfully install and run the program on your device and introducing Rstudio to you. You will get complete guidance.Chapters three and four will teach you some basic and simple lines of code and how you can successfully execute them on your device. There will be some simple maths included. After that, you will start chapter five, which is completely based on statistics. Chapters six and seven will take your attention to more IT. If you already know basic HTML you will get more information in these chapters as they include similar topics as HTML.Finally, you arrived at chapters eight, nine, and ten, which will enable you to create and run your data. These chapters will complete your journey by allowing you to understand how data frames work and how you can work with different types of data. You will also see how you can install premade data packages that can be helpful when you want to practice more. After that, you will know how to visualize your data in different forms of charts, graphs, etc.The book is highly informative with lots of code, and also I added tons of pictures of how the code will look in r studio. Following the book, you will be able to start Rstudio and use the program smoothly.
The Indian Number System
This book presents well-established modern methods of representation and quantitative evaluation for a variety of entities, constants, functions, vectors, random quantities, fuzzy sets, and abstract entities such as relationships, cooperation, dependence, etc. and reveals striking similarities between these modern concepts and the Hindu-Arabic Decimal Number System. We regularly work with them without realizing these similarities perhaps because the centuries old Indian Number System became so commonplace today that these connections are not evidenced either by citations in the literature. It is hoped that this book will serve to throw some new light on the INS which is held in high esteem by great minds. The size of this book, its presentation in simple terms devoid of mystifying jargon, and its capture of the basic elements of the vast subject of mathematical concepts, especially in the field of systems and signals, are expected to be attractive to both teachers and students.
Mathematics of Multilevel Systems: Data, Scaling, Images, Signals, and Fractals
This book presents the mathematics of wavelet theory and its applications in a broader sense, comprising entropy encoding, lifting scheme, matrix factorization, and fractals. It also encompasses image compression examples using wavelet transform and includes the principal component analysis which is a hot topic on data dimension reduction in machine learning.Readers will find equal coverage on the following three themes: The book entails a varied choice of diverse interdisciplinary themes. While the topics can be found in various parts of the pure and applied literature, this book fulfills the need for an accessible presentation which cuts across the fields.As the target audience is wide-ranging, a detailed and systematic discussion of issues involving infinite dimensions and Hilbert space is presented in later chapters on wavelets, transform theory and, entropy encoding and probability. For the problems addressed there, the case of infinite dimension will be more natural, and well-motivated.
Mathematics for Human Flourishing in the Time of Covid-19 and Post Covid-19
The International Chair in Mathematical Physics and Applications (ICMPA - UNESCO chair), University of Abomey-Calavi, Benin, and the Center for Applied Mathematics of the Faculty of Mechanical Engineering Nis, CAM-FMEN, organized a webinar on Mathematics for human flourishing in the time of COVID-19 and post COVID-19, 21 October 2020, supported by the City of Nis. The objectives of the webinar were to give precise information about the work that scientists do to cure the disease, to push forward technology, to understand our society and create new expressions of humanity, and to question the role of mathematics in the responses to this pandemic.
An Introduction to Optimization with Applications in Machine Learning and Data Analytics
The text introduces students to numerous methods in solving a variety of Optimization problems. Also, the narrow focus of most math textbooks is completely dedicated to nonlinear programming, linear programming, combinatorial or convex optimization.
Bayesian Filtering and Smoothing
Now in its second edition, this accessible text presents a unified Bayesian treatment of state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models. The book focuses on discrete-time state space models and carefully introduces fundamental aspects related to optimal filtering and smoothing. In particular, it covers a range of efficient non-linear Gaussian filtering and smoothing algorithms, as well as Monte Carlo-based algorithms. This updated edition features new chapters on constructing state space models of practical systems, the discretization of continuous-time state space models, Gaussian filtering by enabling approximations, posterior linearization filtering, and the corresponding smoothers. Coverage of key topics is expanded, including extended Kalman filtering and smoothing, and parameter estimation. The book's practical, algorithmic approach assumes only modest mathematical prerequisites, suitable for graduate and advanced undergraduate students. Many examples are included, with Matlab and Python code available online, enabling readers to implement algorithms in their own projects.
Ordinary Differential Equations
The textbook presents a rather unique combination of topics in ODEs, examples and presentation style. The primary intended audience is undergraduate (2nd, 3rd, or 4th year) students in engineering and science (physics, biology, economics). The needed pre-requisite is a mastery of single-variable calculus. A wealth of included topics allows using the textbook in up to three sequential, one-semester ODE courses. Presentation emphasizes the development of practical solution skills by including a very large number of in-text examples and end-of-section exercises. All in-text examples, be they of a mathematical nature or a real-world examples, are fully solved, and the solution logic and flow are explained. Even advanced topics are presented in the same undergraduate-friendly style as the rest of the textbook. Completely optional interactive laboratory-type software is included with the textbook. Email Mikhail.Khenner@wku.edu with proof of textbook purchase to request access to optional software download.
Computational Mathematics
This textbook is a comprehensive introduction to computational mathematics and scientific computing suitable for undergraduate and postgraduate courses. It presents both practical and theoretical aspects of the subject, as well as advantages and pitfalls of classical numerical methods alongside with computer code and experiments in Python. Each chapter closes with modern applications in physics, engineering, and computer science.Features: No previous experience in Python is required. Includes simplified computer code for fast-paced learning and transferable skills development. Includes practical problems ideal for project assignments and distance learning. Presents both intuitive and rigorous faces of modern scientific computing. Provides an introduction to neural networks and machine learning.
Artificial Intelligence for Financial Markets
This book introduces the novel artificial intelligence technique of polymodels and applies it to the prediction of stock returns. The idea of polymodels is to describe a system by its sensitivities to an environment, and to monitor it, imitating what a natural brain does spontaneously. In practice this involves running a collection of non-linear univariate models. This very powerful standalone technique has several advantages over traditional multivariate regressions. With its easy to interpret results, this method provides an ideal preliminary step towards the traditional neural network approach. The first two chapters compare the technique with other regression alternatives and introduces an estimation method which regularizes a polynomial regression using cross-validation. The rest of the book applies these ideas to financial markets. Certain equity return components are predicted using polymodels in very different ways, and a genetic algorithm is describedwhich combines these different predictions into a single portfolio, aiming to optimize the portfolio returns net of transaction costs. Addressed to investors at all levels of experience this book will also be of interest to both seasoned and non-seasoned statisticians.
An Introduction to Smooth Manifolds
Targeted to graduate students of mathematics, this book discusses major topics like the Lie group in the study of smooth manifolds. It is said that mathematics can be learned by solving problems and not only by just reading it. To serve this purpose, this book contains a sufficient number of examples and exercises after each section in every chapter. Some of the exercises are routine ones for the general understanding of topics. The book also contains hints to difficult exercises. Answers to all exercises are given at the end of each section. It also provides proofs of all theorems in a lucid manner. The only pre-requisites are good working knowledge of point-set topology and linear algebra.
Numerical Methods Fundamentals
The book is designed to cover all major aspects of applied numerical methods, including numerical computations, solution of algebraic and transcendental equations, finite differences and interpolation, curve fitting, correlation and regression, numerical differentiation and integration, matrices and linear system of equations, numerical solution of ordinary differential equations, and numerical solution of partial differential equations. It uses a numerical problem-solving orientation with numerous examples, figures, and end of chapter exercises. Presentations are limited to very basic topics to serve as an introduction to more advanced topics.
Oblique Derivative Problems for Elliptic Equations in Conical Domains
The aim of our book is the investigation of the behavior of strong and weak solutions to the regular oblique derivative problems for second order elliptic equations, linear and quasi-linear, in the neighborhood of the boundary singularities. The main goal is to establish the precise exponent of the solution decrease rate and under the best possible conditions. The question on the behavior of solutions of elliptic boundary value problems near boundary singularities is of great importance for its many applications, e.g., in hydrodynamics, aerodynamics, fracture mechanics, in the geodesy etc. Only few works are devoted to the regular oblique derivative problems for second order elliptic equations in non-smooth domains. All results are given with complete proofs. The monograph will be of interest to graduate students and specialists in elliptic boundary value problems and their applications.
Statistical Inference Based on Kernel Distribution Function Estimators
This book presents a study of statistical inferences based on the kernel-type estimators of distribution functions. The inferences involve matters such as quantile estimation, nonparametric tests, and mean residual life expectation, to name just some. Convergence rates for the kernel estimators of density functions are slower than ordinary parametric estimators, which have root-n consistency. If the appropriate kernel function is used, the kernel estimators of the distribution functions recover the root-n consistency, and the inferences based on kernel distribution estimators have root-n consistency. Further, the kernel-type estimator produces smooth estimation results. The estimators based on the empirical distribution function have discrete distribution, and the normal approximation cannot be improved--that is, the validity of the Edgeworth expansion cannot be proved. If the support of the population density function is bounded, there is a boundary problem, namely the estimator does not have consistency near the boundary. The book also contains a study of the mean squared errors of the estimators and the Edgeworth expansion for quantile estimators.
Nonautonomous Bifurcation Theory
Bifurcation theory is a major topic in dynamical systems theory with profound applications. However, in contrast to autonomous dynamical systems, it is not clear what a bifurcation of a nonautonomous dynamical system actually is, and so far, various different approaches to describe qualitative changes have been suggested in the literature. The aim of this book is to provide a concise survey of the area and equip the reader with suitable tools to tackle nonautonomous problems. A review, discussion and comparison of several concepts of bifurcation is provided, and these are formulated in a unified notation and illustrated by means of comprehensible examples. Additionally, certain relevant tools needed in a corresponding analysis are presented.
Why Does Math Work ... If It's Not Real?
According to G. H. Hardy, the 'real' mathematics of the greats like Fermat and Euler is 'useless, ' and thus the work of mathematicians should not be judged on its applicability to real-world problems. Yet, mysteriously, much of mathematics used in modern science and technology was derived from this 'useless' mathematics. Mobile phone technology is based on trig functions, which were invented centuries ago. Newton observed that the Earth's orbit is an ellipse, a curve discovered by ancient Greeks in their futile attempt to double the cube. It is like some magic hand had guided the ancient mathematicians so their formulas were perfectly fitted for the sophisticated technology of today. Using anecdotes and witty storytelling, this book explores that mystery. Through a series of fascinating stories of mathematical effectiveness, including Planck's discovery of quanta, mathematically curious readers will get a sense of how mathematicians develop their concepts.
Measurement Uncertainty
This reprint focuses on a very important topic in metrology, which is represent by measurement uncertainty. Any good metrologist or scientist in engineering knows that no measurement makes sense without an associated uncertainty value: without an uncertainty value, no decision can be taken; no comparisons can be made; no conformity can be assessed. Any decision, comparison or conformity assessment made without considering the measurement uncertainty affecting the measurement value is completely useless and meaningless. Stated that, it becomes very clear that uncertainty in measurement plays indeed a very important rule in our everyday life. This is the reason why there is a great need to have a fruitful academic and scientific discussion on this topic.We have been speaking about measurement uncertainty for less than 30 years, since the concept of "measurement uncertainty" has been introduced in 1995 by the "Guide to the expression of uncertainty in measurement" (GUM). Thirty years seems to be many, but still the concept of measurement uncertainty has not been spread worldwide and the GUM is a document that is not known everywhere. On the other hand, this document should be considered not only in academic scenario, but also in any technical and industrial scenario, where it is pivotal to know the meaning of measurement uncertainty, identify the uncertainty contributions and know how these contributions affect the final measurement result.
Why Does Math Work ... If It's Not Real?
According to G. H. Hardy, the 'real' mathematics of the greats like Fermat and Euler is 'useless, ' and thus the work of mathematicians should not be judged on its applicability to real-world problems. Yet, mysteriously, much of mathematics used in modern science and technology was derived from this 'useless' mathematics. Mobile phone technology is based on trig functions, which were invented centuries ago. Newton observed that the Earth's orbit is an ellipse, a curve discovered by ancient Greeks in their futile attempt to double the cube. It is like some magic hand had guided the ancient mathematicians so their formulas were perfectly fitted for the sophisticated technology of today. Using anecdotes and witty storytelling, this book explores that mystery. Through a series of fascinating stories of mathematical effectiveness, including Planck's discovery of quanta, mathematically curious readers will get a sense of how mathematicians develop their concepts.
An Introduction to Numerical Methods
An Introduction to Numerical Methods: A MATLAB(R) Approach, Fifth Edition continues to offer readers an accessible and practical introduction to numerical analysis. It presents a wide range of algorithms for scientific and engineering applications, using MATLAB to illustrate each numerical method.
Multiplicative Differential Equations
Multiplicative Differential Equations: Volume I is the first part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics.
An Introduction to R and Python for Data Analysis
An Introduction to R and Python for Data Analysis helps teach students to code in both R and Python simultaneously. As both R and Python can be used in similar manners, it is useful and efficient to learn both at the same time, helping lecturers and students to teach and learn more and save time.
Multiplicative Differential Equations
Multiplicative Differential Equations: Volume 2 is the second part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics.
