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Calculus all-in-one (+ chapter quizzes online)
Make calculus more manageable with simplified instruction and tons of practice. This book pairs no-nonsense explanations of calculus content with practical examples and practice problems, so you can untangle the difficult concepts and improve your score in any calculus class. Plus, this book comes with access to chapter quizzes online.
BOOK
Topics in Quaternion Linear Algebra
Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.
eBook
Vector : a surprising story of space, time, and mathematical transformation
\"The stars of the latest book by award-winning science writer and mathematician Robyn Arianrhod are unlikely celebrities--vectors and tensors. If you took a high school physics course, the word \"vector\" might remind you of the mathematics needed to determine forces on an amusement park ride, say; or of cross products, a special kind of multiplication using a bespoke table and a right-hand rule. You might also remember the introductory definition of a vector as a quantity that has magnitude and (this is the key) direction. Velocity--for example, 25 miles per hour northwest--is a vector; speed, such as 25 miles per hour, is not. Put another way, a velocity vector in space contains not one number, but three-a measurement of speed along each of three dimensions. It sounds simple, in hindsight--yet, as Arianrhod shows in this intriguing story, the idea of a single symbol expressing several things at once is a sophisticated one, millennia in the making. Vectors are examples of an even more sophisticated idea, the tensor. And it's not just space that vectors and tensors can represent, but information, too. Which means that whenever you use a search engine, say, or AI bot, computer graphics, or a host of other digital applications, vectors and tensors are there somewhere in the software. As for physics, there's much more to it than velocities and simple forces! Arianrhod shows how the discovery of vectors and tensors enabled physicists and mathematicians to think brand new thoughts-such as Maxwell did when he ushered in the wireless electromagnetic age, and Einstein when he predicted the curving of four-dimensional space-time and the existence of gravitational waves. Quantum theory, too, makes fine use of these ideas. In other words, vectors and tensors have been critical not only to the way we see our universe, but also to the invention of Wi-Fi, GPS, micro-technology, and so much else that we take for granted today. In exploring the history and significance of vectors and tensors-and introducing the fascinating people who gave them to us--Arianrhod takes readers on an extraordinary, five-thousand-year journey through the human imagination. A celebration of an idea, Vector shows the genius required to imagine the world in new dimensions-and how a clever mathematical construct can direct the future of discovery\"-- Provided by publisher.
BOOK
When Computers Were Human
Before Palm Pilots and iPods, PCs and laptops, the term \"computer\" referred to the people who did scientific calculations by hand. These workers were neither calculating geniuses nor idiot savants but knowledgeable people who, in other circumstances, might have become scientists in their own right. When Computers Were Human represents the first in-depth account of this little-known, 200-year epoch in the history of science and technology. Beginning with the story of his own grandmother, who was trained as a human computer, David Alan Grier provides a poignant introduction to the wider world of women and men who did the hard computational labor of science. His grandmother's casual remark, \"I wish I'd used my calculus,\" hinted at a career deferred and an education forgotten, a secret life unappreciated; like many highly educated women of her generation, she studied to become a human computer because nothing else would offer her a place in the scientific world. The book begins with the return of Halley's comet in 1758 and the effort of three French astronomers to compute its orbit. It ends four cycles later, with a UNIVAC electronic computer projecting the 1986 orbit. In between, Grier tells us about the surveyors of the French Revolution, describes the calculating machines of Charles Babbage, and guides the reader through the Great Depression to marvel at the giant computing room of the Works Progress Administration. When Computers Were Human is the sad but lyrical story of workers who gladly did the hard labor of research calculation in the hope that they might be part of the scientific community. In the end, they were rewarded by a new electronic machine that took the place and the name of those who were, once, the computers.
eBook
On the differential structure of metric measure spaces and applications
The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the
duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of
analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To
employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a
meaning to an expression like
eBook
Tempered and Hadamard-Type Fractional Calculus with Respect to Functions
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann–Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.
Journal Article
Tracking the transition to agriculture in Southern Europe through ancient DNA analysis of dental calculus
Archaeological dental calculus, or mineralized plaque, is a key tool to track the evolution of oral microbiota across time in response to processes that impacted our culture and biology, such as the rise of farming during the Neolithic. However, the extent to which the human oral flora changed from prehistory until present has remained elusive due to the scarcity of data on the microbiomes of prehistoric humans. Here, we present our reconstruction of oral microbiomes via shotgun metagenomics of dental calculus in 44 ancient foragers and farmers from two regions playing a pivotal role in the spread of farming across Europe—the Balkans and the Italian Peninsula. We show that the introduction of farming in Southern Europe did not alter significantly the oral microbiomes of local forager groups, and it was in particular associated with a higher abundance of the species Olsenella sp. oral taxon 807. The human oral environment in prehistory was dominated by a microbial species, Anaerolineaceae bacterium oral taxon 439, that diversified geographically. A Near Eastern lineage of this bacterial commensal dispersed with Neolithic farmers and replaced the variant present in the local foragers. Our findings also illustrate that major taxonomic shifts in human oral microbiome composition occurred after the Neolithic and that the functional profile of modern humans evolved in recent times to develop peculiar mechanisms of antibiotic resistance that were previously absent.
Journal Article
Local fractional integral transforms and their applications
Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors.
eBook
A Practical Guide to Prabhakar Fractional Calculus
The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.
Journal Article