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  • Optimization and Approximation
    Optimization and Approximation

    This book provides a basic, initial resource, introducing science and engineering students to the field of optimization.It covers three main areas: mathematical programming, calculus of variations and optimal control, highlighting the ideas and concepts and offering insights into the importance of optimality conditions in each area.It also systematically presents affordable approximation methods.Exercises at various levels have been included to support the learning process.

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  • Practical Applied Mathematics : Modelling, Analysis, Approximation
    Practical Applied Mathematics : Modelling, Analysis, Approximation

    Drawing from a wide variety of mathematical subjects, this book aims to show how mathematics is realised in practice in the everyday world.Dozens of applications are used to show that applied mathematics is much more than a series of academic calculations.Mathematical topics covered include distributions, ordinary and partial differential equations, and asymptotic methods as well as basics of modelling.The range of applications is similarly varied, from the modelling of hair to piano tuning, egg incubation and traffic flow.The style is informal but not superficial. In addition, the text is supplemented by a large number of exercises and sideline discussions, assisting the reader's grasp of the material.Used either in the classroom by upper-undergraduate students, or as extra reading for any applied mathematician, this book illustrates how the reader's knowledge can be used to describe the world around them.

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  • Statistical Modeling Using Local Gaussian Approximation
    Statistical Modeling Using Local Gaussian Approximation

    Statistical Modeling using Local Gaussian Approximation extends powerful characteristics of the Gaussian distribution, perhaps, the most well-known and most used distribution in statistics, to a large class of non-Gaussian and nonlinear situations through local approximation.This extension enables the reader to follow new methods in assessing dependence and conditional dependence, in estimating probability and spectral density functions, and in discrimination.Chapters in this release cover Parametric, nonparametric, locally parametric, Dependence, Local Gaussian correlation and dependence, Local Gaussian correlation and the copula, Applications in finance, and more. Additional chapters explores Measuring dependence and testing for independence, Time series dependence and spectral analysis, Multivariate density estimation, Conditional density estimation, The local Gaussian partial correlation, Regression and conditional regression quantiles, and a A local Gaussian Fisher discriminant.

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  • Wave Phenomena : Mathematical Analysis and Numerical Approximation
    Wave Phenomena : Mathematical Analysis and Numerical Approximation

    This book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach. The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering.Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing. The goal of this book is to present a comprehensive introduction to the research on wave phenomena.Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered.The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.

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  • What is an approximation in mathematics?

    An approximation in mathematics is a value that is close to the true value of a number or quantity, but not necessarily exact. It is used when the exact value is difficult to calculate or when only an estimate is needed. Approximations are often used in real-life situations where precise values are not necessary, such as in engineering, physics, or finance. Common methods of approximation include rounding, truncating, or using simplified formulas.

  • What is the small angle approximation?

    The small angle approximation is a method used in mathematics and physics to simplify trigonometric functions when dealing with small angles. It states that for small angles, the sine and tangent of the angle can be approximated by the angle itself, and the cosine of the angle can be approximated by 1. This approximation is useful because it makes calculations easier and more manageable when dealing with small angles.

  • What is the approximation for 3n5x?

    The approximation for 3n5x is 15x. This is because when we multiply 3 and 5, we get 15, and the variable x remains unchanged. Therefore, the approximation for 3n5x is 15x.

  • How do you calculate an exponential approximation?

    To calculate an exponential approximation, you can use the formula for the Taylor series expansion of the exponential function: e^x ≈ 1 + x + x^2/2! + x^3/3! + ... + x^n/n!. By choosing an appropriate value for n, you can determine how many terms of the series you want to include in your approximation. The more terms you include, the more accurate your approximation will be. Finally, plug in the value of x into the formula to calculate the exponential approximation.

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  • Sparse Polynomial Approximation of High-Dimensional Functions
    Sparse Polynomial Approximation of High-Dimensional Functions

    Over seventy years ago, Richard Bellman coined the term "the curse of dimensionality" to describe phenomena and computational challenges that arise in high dimensions.These challenges, in tandem with the ubiquity of high-dimensional functions in real-world applications, have led to a lengthy, focused research effort on high-dimensional approximation—that is, the development of methods for approximating functions of many variables accurately and efficiently from data.This book provides an in-depth treatment of one of the latest installments in this long and ongoing story: sparse polynomial approximation methods.These methods have emerged as useful tools for various high-dimensional approximation tasks arising in a range of applications in computational science and engineering.It begins with a comprehensive overview of best s-term polynomial approximation theory for holomorphic, high-dimensional functions, as well as a detailed survey of applications to parametric differential equations.It then describes methods for computing sparse polynomial approximations, focusing on least squares and compressed sensing techniques. Sparse Polynomial Approximation of High-Dimensional Functions presents the first comprehensive and unified treatment of polynomial approximation techniques that can mitigate the curse of dimensionality in high-dimensional approximation, including least squares and compressed sensing.It develops main concepts in a mathematically rigorous manner, with full proofs given wherever possible, and it contains many numerical examples, each accompanied by downloadable code.The authors provide an extensive bibliography of over 350 relevant references, with an additional annotated bibliography available on the book's companion website (www.sparse-hd-book.com). This text is aimed at graduate students, postdoctoral fellows, and researchers in mathematics, computer science, and engineering who are interested in high-dimensional polynomial approximation techniques.

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  • Ethical and Responsible Tourism : Managing Sustainability in Local Tourism Destinations
    Ethical and Responsible Tourism : Managing Sustainability in Local Tourism Destinations

    Ethical and Responsible Tourism explains the methods and practices used to manage the environmental impact of tourism on local communities and destinations.This new edition takes into account recent global events such as the Covid-19 health crisis, the impacts of the war in Ukraine on tourism in neighbouring regions and the consequences of the energy and cost of living crisis.The three core themes of the book – destination management, environmental and social aspects of ethical sustainable development and business impacts – are discussed across both topic and case study chapters, alongside explanatory editorial analysis with all chapters clearly signposted and interlinked.The case studies address specific and practical examples from a global range of examples including sites in Australasia, Central America, Europe, Asia, North America and South America.In this new edition, further case studies are included from the USA and Japan, as well as new examples from Brazil, Croatia and Malta.Used as a core textbook, the linking of theory in the topic chapters, and practice gained through case studies, alongside further reading and editorial commentary, Ethical and Responsible Tourism provides a detailed and comprehensive learning experience.Specific case studies can be used as standalone examples as part of a case teaching approach, and the editorial and discussion elements are designed to be suitable for those simply seeking a concise overview, such as tourism professionals or potential investors in sustainable tourism projects.This revised edition continues to be essential reading for students, researchers and practitioners of tourism, environmental and sustainability studies.

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  • Approximation Theory and Applications : Piecewise Linear and Generalized Functions
    Approximation Theory and Applications : Piecewise Linear and Generalized Functions

    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.

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  • Biodiversity : Threats and Conservation
    Biodiversity : Threats and Conservation

    The term "biodiversity" or "biological diversity" describes the variety of living beings on Earth encompassing microorganisms, plants, animals, and ecosystems, such as coral reefs, forests, and deserts.In fact, it also represents a wealth of biological resources available to us.Today, instead of exploring and preserving its unmapped biodiversity, mankind has gone away from mother nature, conquering the domain through relentless exploitation.This has resulted in an ecological imbalance and thereby has posed serious threat to biodiversity.Declining biodiversity is, therefore, a concern for countless reasons.Biodiversity’s wealth of genetic information can be conserved by storing biological material from endangered species.Cell lines, gametes, and embryos are preserved so that in case a species becomes extinct or there is a need to increase the population of the species, then cloning and artificial reproductive techniques can be used to revive and reproduce the said species.The present volume discusses the biodiversity of specific animal groups, such as cnidarians and fish, as well as potential risks to and methods for conserving some of them.The following important aspects are addressed: 1. The importance of biodiversity in maintaining ecosystem balance for sustainability 2.The emerging role of biodiversity as a source of important materials, pharmaceuticals, food, and so on as such or through genetic manipulation to meet the contemporary and future challenges 3.The impact of environment degradation on biodiversity and conservation in a changing environment 4.Conventional and emerging biotechnological techniques to conserve biodiversity The book is intended for academics, scientists, and naturalists working in the fields of biology, biochemistry, biophysics, and biomedical sciences.

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  • What is the approximation method for differentiation?

    The approximation method for differentiation involves using the concept of limits to estimate the derivative of a function at a specific point. One common approach is to use the difference quotient, which calculates the average rate of change of the function over a small interval. By taking the limit of this average rate of change as the interval approaches zero, we can approximate the derivative at a particular point. This method is useful when the function is not easily differentiable or when an exact derivative is difficult to calculate.

  • Is the small angle approximation not correct?

    The small angle approximation is a useful approximation in physics and engineering when dealing with small angles, typically less than 10 degrees. However, it is not always correct, especially when dealing with very precise measurements or high accuracy requirements. In some cases, the small angle approximation may introduce significant errors, and it is important to carefully consider the specific application and the level of accuracy needed before using this approximation.

  • Can you help me with approximation in math?

    Yes, I can help you with approximation in math. Approximation is the process of finding an estimate or close value of a number or quantity. I can show you different methods and techniques to approximate numbers, such as rounding, truncating, or using significant figures. Just let me know what specific concept or problem you need help with, and I'll be happy to assist you.

  • How does the Newton's method approximation technique work?

    Newton's method is an iterative technique used to find the roots of a real-valued function. It starts with an initial guess and then refines this guess by using the function's derivative to find a better approximation. The process is repeated until a sufficiently accurate solution is found. This method is based on linear approximation and can converge quickly to the root of the function if the initial guess is close enough.

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