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Technical math for dummies
Are you a vocational student or a trade professional? This is your one-stop, hands-on guide to mastering the math you'll encounter on the job or while working toward your degree or certification.
Book
Handbook of mathematical formulas and integrals
The updated Handbook is an essential reference for researchers and students in applied mathematics, engineering, and physics. It provides quick access to important formulas, relations, and methods from algebra, trigonometric and exponential functions, combinatorics, probability, matrix theory, calculus and vector calculus, ordinary and partial differential equations, Fourier series, orthogonal polynomials, and Laplace transforms. Many of the entries are based upon the updated sixth edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and other important reference works.The Third Edition has new chapters covering solutions of elliptic, parabolic and hyperbolic equations and qualitative properties of the heat and Laplace equation. Key Features: * Comprehensive coverage of frequently used integrals, functions and fundamental mathematical results * Contents selected and organized to suit the needs of students, scientists, and engineers * Contains tables of Laplace and Fourier transform pairs * New section on numerical approximation * New section on the z-transform * Easy reference system
eBook
Patently Mathematical
Uncovers the surprising ways math shapes our lives—from whom we date to what we learn.
How do dating sites match compatible partners? What do cell phones and sea coasts have in common? And why do computer scientists keep ant colonies? Jeff Suzuki answers these questions and more in Patently Mathematical, which explores the mathematics behind some of the key inventions that have changed our world.
In recent years, patents based on mathematics have been issued by the thousands—from search engines and image recognition technology to educational software and LEGO designs. Suzuki delves into the details of cutting-edge devices, programs, and products to show how even the simplest mathematical principles can be turned into patentable ideas worth billions of dollars. Readers will discover
• whether secure credit cards are really secure
• how improved data compression made streaming video services like Netflix a hit
• the mathematics behind self-correcting golf balls
• why Google is such an effective and popular search engine
• how eHarmony and Match.com find the perfect partner for those seeking a mate
• and much more!
A gifted writer who combines quirky historical anecdotes with relatable, everyday examples, Suzuki makes math interesting for everyone who likes to ponder the world of numerical relationships.
Praise for Jeff Suzuki's Constitutional Calculus
\"Presents an entertaining and insightful approach to the mathematics that underlies the American system of government. The book is neatly organized, breaking down the United States Constitution by article, section, and amendment. Within each piece, Suzuki reviews the mathematical principles that went into the underlying framework.\"— Mathematical Reviews
\"A breath of fresh air... A reaffirmation that mathematics should be used more often to make general public policy.\"— MAA Reviews
eBook
Thermodynamic Unification of Optimal Transport: Thermodynamic Uncertainty Relation, Minimum Dissipation, and Thermodynamic Speed Limits
Thermodynamics serves as a universal means for studying physical systems from an energy perspective. In recent years, with the establishment of the field of stochastic and quantum thermodynamics, the ideas of thermodynamics have been generalized to small fluctuating systems. Independently developed in mathematics and statistics, the optimal transport theory concerns the means by which one can optimally transport a source distribution to a target distribution, deriving a useful metric between probability distributions, called the Wasserstein distance. Despite their seemingly unrelated nature, an intimate connection between these fields has been unveiled in the context of continuous-state Langevin dynamics, providing several important implications for nonequilibrium systems. In this study, we elucidate an analogous connection for discrete cases by developing a thermodynamic framework for discrete optimal transport. We first introduce a novel quantity called dynamical state mobility, which significantly improves the thermodynamic uncertainty relation and provides insights into the precision of currents in nonequilibrium Markov jump processes. We then derive variational formulas that connect the discrete Wasserstein distances to stochastic and quantum thermodynamics of discrete Markovian dynamics described by master equations. Specifically, we rigorously prove that the Wasserstein distance equals the minimum product of irreversible entropy production and dynamical state mobility over all admissible Markovian dynamics. These formulas not only unify the relationship between thermodynamics and the optimal transport theory for discrete and continuous cases, but also generalize it to the quantum case. In addition, we demonstrate that the obtained variational formulas lead to remarkable applications in stochastic and quantum thermodynamics, such as stringent thermodynamic speed limits and the finite-time Landauer principle. These bounds are tight and can be saturated for arbitrary temperatures, even in the zero-temperature limit. Notably, the finite-time Landauer principle can explain finite dissipation even at extremely low temperatures, which cannot be explained by the conventional Landauer principle.
Journal Article
Estimation of particle size using the Debye equation and the Scherrer formula for polyphasic TiO2 powder
There are two methods to estimate the particle size from X-ray diffraction data: the Debye equation and the Scherrer formula. The main goal of this study is to describe the methodology of particle size estimation on the base of two these methods and to apply it to TiO2 powder to determine the diameters and the mass content of anatase and brookite components. The studied nano-dispersed TiO2 powder was synthesized by the sol-gel method. The proposed method of particle size estimation consists of several steps: 1. Approximation of diffraction peaks by Gaussians and calculation of initial values of particle size with the use of the Scherrer formula; 2. Iterations with the use of the Debye equation to obtain more accurate particle size values; 3. Calculation of the mass content of different components corresponding to the minimum R-factor.
Journal Article
Formulas and calculations for drilling operations
\"Presented in an easy-to-use format, this second edition of Formulas and Calculations for Drilling Operations is a quick reference for day-to-day work out on the rig; It also serves as a handy study guide for drilling and well control certification courses; Virtually all the mathematics required on a drilling rig is here in one convenient source, including formulas for pressure gradient, specific gravity, pump, output, annular velocity, buoyancy factor, and many other topics; Whether open on your desk, on the hood of your truck at the well, or on an offshore platform, this is the only book available that covers the gamut of the formulas and calculations for petroleum engineers that have been compiled over decades; Some of these formulas and calculations have been used for decades, while others are meant to help guide the engineer through some of the more recent breakthroughs in the industry's technology, such as hydraulic fracturing and enhanced oil recovery. There is no other source for these useful formulas and calculations that is this thorough; An instant classic when the first edition was published, the much-improved revision is even better, offering new information not available in the first edition, making it as up-to-date as possible in book form; Truly a state-of-the-art masterpiece for the oil and gas industry, if there is only one book you buy to help you do your job, this is it!\"-- Provided by publisher.
Book
Algebraic multigrid methods
This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother
$R$
for a matrix
$A$
, such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension
$n_{c}$
is the span of the eigenvectors corresponding to the first
$n_{c}$
eigenvectors
$\\bar{R}A$
(with
$\\bar{R}=R+R^{T}-R^{T}AR$
). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with
$\\bar{R}A$
, and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.
Journal Article