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High Quality Content by WIKIPEDIA articles! In probability theory, a probability density function (abbreviated as pdf, or just density) of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point in the observation space. The probability of a random variable falling within a given set is given by the integral of its density over the set.On rare occasions the term ?probability distribution function? is used to denote the probability density function. However special care should be taken around this term, since in other sources the ?probability distribution function? may refer to either the probability distribution function, or the cumulative distribution function, or may be a probability mass function rather than a density.

High Quality Content by WIKIPEDIA articles! n Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane. Apollonius of Perga (ca. 262 BC ? ca. 190 BC) posed and solved this famous problem in his work ?????? (Epaphai, "Tangencies"); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2) and each solution circle encloses or excludes the three given circles in a different way.

George E. P. Box

Praise for the First Edition "This book . . . is a significant addition to the literature on statistical practice . . . should be of considerable interest to those interested in these topics."—International Journal of Forecasting Recent research has shown that monitoring techniques alone are inadequate for modern Statistical Process Control (SPC), and there exists a need for these techniques to be augmented by methods that indicate when occasional process adjustment is necessary. Statistical Control by Monitoring and Adjustment, Second Edition presents the relationship among these concepts and elementary ideas from Engineering Process Control (EPC), demonstrating how the powerful synergistic association between SPC and EPC can solve numerous problems that are frequently encountered in process monitoring and adjustment. The book begins with a discussion of SPC as it was originally conceived by Dr. Walter A. Shewhart and Dr. W. Edwards Deming. Subsequent chapters outline...

High Quality Content by WIKIPEDIA articles! Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain partial differential equations (PDEs), often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential convergence" being the fastest possible. PDEs describe a wide array of physical processes such as heat conduction, fluid flow, and sound propagation. In many such equations, there are underlying "basic waves" that can be used to give efficient algorithms for computing solutions to these PDEs. In a typical case, spectral methods take advantage of this fact by writing the solution as its Fourier series, substituting this series into the PDE to get a system of ordinary differential equations (ODEs) in the time-dependent coefficients of the trigonometric terms in the series (written in complex exponential form), and using a...

William L. Briggs, Lyle Cochran

Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher’s voice beyond the classroom. That voice–evident in the narrative, the figures, and the questions interspersed in the narrative–is a master teacher leading readers to deeper levels of understanding. The authors appeal to readers’ geometric intuition to introduce fundamental concepts and lay the foundation for the more rigorous development that follows. Comprehensive exercise sets have received praise for their creativity, quality, and scope. Functions; Limits; Derivatives; Applications of the Derivative; Integration; Applications of Integration; Integration Techniques; Sequences and Infinite Series; Power Series; Parametric and Polar Curves. For all readers interested in single variable and multivariable calculus for mathematics, engineering, and science.

High Quality Content by WIKIPEDIA articles! The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of equations: 1. Conservation of momentum: Consisting of a form of the Navier-Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere 2. A Thermal energy equation: Relating the overall temperature of the system to heat sources and sinks 3. A Continuity equation: Representing the conservation of mass. The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined.

In geometry, a dodecagon is any polygon with twelve sides and twelve angles. In a regular dodecagon, all sides have equal length and all angles have measure 150°. Its Schlafli symbol is {12}.

High Quality Content by WIKIPEDIA articles! In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function around a given point by polynomials (the Taylor polynomials of that function) whose coefficients depend only on the derivatives of the function at that point. The theorem also gives precise estimates on the size of the error in the approximation. The theorem is named after the mathematician Brook Taylor, who stated it in 1712, though the result was first discovered 41 years earlier in 1671 by James Gregory.

High Quality Content by WIKIPEDIA articles! A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, like a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil. If the generating ellipse is a circle, the result is a sphere.

Cedric Pralet

In the first part of this book, the authors introduce a generic algebraic framework that encompasses and unifies a large number of such formalisms. This formalism, called the Plausibility–Feasibility–Utility (PFU) framework, is based on algebraic structures, graphical models, and sequences of quantifications.