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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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Innovation in Music: Technology and Creativity
Innovation in Music: Technology and Creativity is a groundbreaking collection bringing together contributions from instructors, researchers, and professionals.Split into two sections, covering composition and performance, and technology and innovation, this volume offers truly international perspectives on ever-evolving practices. Including chapters on audience interaction, dynamic music methods, AI, and live electronic performances, this is recommended reading for professionals, students, and researchers looking for global insights into the fields of music production, music business, and music technology.
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Fashion and Environmental Sustainability : Entrepreneurship, Innovation and Technology
The wide range of topics that the book covers are organised into sections reflecting a cradle to grave view of how entrepreneurial, innovative, and tech-savvy approaches can advance environmental sustainability in the fashion sector.These sections include: sustainable materials; innovation in design, range planning and product development; sustainable innovations in fashion supply chains; sustainable innovations in fashion retail and marketing; sustainable alternatives for end-of-life and circular economy initiatives; and more sustainable alternative fashion business models.
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People Planet Profit : How to Embrace Sustainability for Innovation and Business Growth
Social and environmental issues are more important than ever and consumers are committed to supporting change. 'Doing good' is no longer a peripheral activity but fundamental to every aspect of how we do business, every day, for everyone. People, Planet, Profit is the first book to truly address business growth in the context of social and environmental concerns.It's a practical guide to new business opportunity, operational improvement and competitive advantage.Full of inspiring case studies, it looks at the challenges faced by key players such as Google, Microsoft, Apple, Nokia, Nike, Amazon, M&S and Walmart.With plenty of comments from industry insiders, it's essential reading for CEOs and business managers who are searching for new ways to create value, to make sense of business in a rapidly shifting landscape, and to deliver profitable growth whilst also doing "the right thing".
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How do you prove that exponential growth beats polynomial growth?
Exponential growth beats polynomial growth by showing that the rate of increase in exponential growth is greater than that of polynomial growth. This can be demonstrated by comparing the growth rates of functions representing exponential and polynomial growth over a large number of iterations or time periods. Additionally, one can analyze the behavior of the functions as the input size or time approaches infinity, where exponential growth will eventually surpass polynomial growth. Finally, mathematical proofs and calculations can be used to show that the exponential function will always grow faster than any polynomial function for sufficiently large inputs or time periods.
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How can one prove that exponential growth beats polynomial growth?
One way to prove that exponential growth beats polynomial growth is by comparing the rates at which the functions grow as the input size increases. Exponential growth, such as 2^n, grows at a much faster rate than polynomial growth, such as n^2, as the input size n becomes large. This can be demonstrated by plotting the two functions on a graph and observing how the exponential function quickly surpasses the polynomial function. Additionally, one can calculate the limit of the ratio between the two functions as n approaches infinity, which will show that the exponential function grows faster.
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What is the difference between a polynomial and a polynomial function?
A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division or roots. A polynomial function, on the other hand, is a specific type of function that can be defined by a polynomial expression. In other words, a polynomial function is a function that can be expressed as a polynomial. So, while a polynomial is simply an algebraic expression, a polynomial function is a specific type of mathematical function.
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What are polynomial functions?
Polynomial functions are mathematical functions that can be expressed as a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. These functions can have multiple terms, each with a different power of the variable. Polynomial functions are continuous and smooth, and they can be used to model a wide range of real-world phenomena. They are commonly used in algebra, calculus, and other branches of mathematics to analyze and solve various problems.
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The Pepper Effect : Tap into the Magic of Creativity, Collaboration, and Innovation
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Learn proven methods for unleashing creativity in any business, awaken the long dormant creativity in yourself-and every employeeIn a world that is dominated by analytical thinking, The Creator Mindset activates a long dormant part of the brain: creativity. This is the unexpected missing ingredient between where you are today and why you are not yet an industry leader of tomorrow. In his groundbreaking new book, innovation guru Nir Bashan shows you how to use creativity as a tool, in much the same way we use Excel spreadsheets and data analysis.He provides the knowledge, insight, and guidance for inspiring and training your company and employees into making creativity a part of everything they do. Organized into four sections-What Is the Creator Mindset?, Why the Creator Mindset and Why Now?, Using the Creator Mindset, and Sustaining Your Creator Mindset-The Creator Mindset helps you create an organizational culture where people overcome self-doubt, approach creativity from a "process" standpoint, and use creativity to solve problems.
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While global challenges such as a future pandemics and global warming seem insurmountable, innovation and cumulative small changes can help towards managing such disruptive events.Innovation can encompass a new way of doing things, new products and services, and new solutions; in organizations where innovation can flourish, progress and resilience can be achieved. This edited collection draws together a number of chapters, organized into two parts – developing social responsibility and developing sustainability – both of which are interlinked and interdependent.Topics presented range from: mandatory CSR in the banking industry to the professional integration of displaced persons to knowledge for and about sustainability, and many more.The diversity of the chapters gift readers an interdisciplinary examination of innovation, social responsibility and sustainability. Developments in Corporate Governance and Responsibility offers the latest research on topical issues by international experts and has practical relevance to business managers.
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The world has gone through profound change since 2019, which has impacted economies, organisations, societies, and ways of working.Now, more than ever, businesses need to be prepared and resilient to large-scale changes.Written by experts, the chapters collected here address various issues such as climate change and the pandemic, suggesting ways in which future crises can be managed successfully and sharing best practice from what we have learned from recent crises. The globally diverse authorship in Corporate Resilience brings together a range of perspectives on corporate resilience and crisis management from varying industries to explore this topic in great depth.Areas studied range from building global resilience through sustainable development and social responsibility, to corporate resilience, environmental investment, internet financial reporting and reporting on human rights. Developments in Corporate Governance and Responsibility offers the latest research on topical issues international experts and has practical relevance to business managers.
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What is double polynomial division?
Double polynomial division is a method used to divide one polynomial by another polynomial. It involves dividing the leading term of the dividend by the leading term of the divisor to determine the first term of the quotient. This process is repeated for each subsequent term until the entire dividend is divided by the divisor. The result is a quotient and a remainder, if any.
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What is a polynomial space?
A polynomial space is a vector space whose elements are polynomials. In other words, it is a set of all polynomials of a certain degree, along with the operations of addition and scalar multiplication. The dimension of a polynomial space is determined by the highest degree of the polynomials in the space. Polynomial spaces are commonly used in mathematics and engineering to represent and manipulate functions and data.
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Is 2x a polynomial function?
Yes, 2x is a polynomial function. A polynomial function is a function that can be expressed as a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. In the case of 2x, it can be written as 2x^1, which fits the definition of a polynomial function.
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"Is my polynomial function correct?"
To determine if your polynomial function is correct, you should first check if it satisfies the given conditions or constraints. Then, you can verify if the function produces the expected output for a range of input values. Additionally, you can compare your function with other known correct polynomial functions to see if they match. If your function meets all these criteria, it is likely correct. However, it's always a good idea to double-check your work and seek feedback from others to ensure accuracy.
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