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ggplot2 is an open-source data visualization package for the statistical programming language R.Created by Hadley Wickham in 2005, ggplot2 is an implementation of Leland Wilkinson's Grammar of Graphics—a general scheme for data visualization which breaks up graphs into semantic components such as scales and layers. ggplot2 can serve as a replacement for the base graphics in R and contains a ...
Flush right is often used when formatting tables of data. It is used to align text to the right margin; in this case, the left ends will be unequal. The term "right alignment" is frequently used when the right side of text is aligned along a visible or invisible vertical line which may or may not coincide with the right margin.
Margins also play an important role in digital word-processing and can be changed using the page setup menu. The default margins for Microsoft Word from version 2007 onward have been 1 inch (25.4 mm) all around; in Word 2003, the default top and bottom margins were 1 inch (25.4 mm), but 1.25 inches (31.7 mm) were given at the left and the right.
The arithmetic–geometric mean of two numbers, a 0 and b 0, is found by calculating the limit of the sequences + = +, + =, which both converge to the same limit. If = and = then the limit is () where () is the complete elliptic integral of the first kind
The Gauss-Legendre methods are implicit, so in general they cannot be applied exactly. Instead one makes an educated guess of , and then uses Newton's method to converge arbitrarily close to the true solution.
The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for R the field of residues modulo a prime number p, and χ the Legendre symbol.In this case Gauss proved that G(χ) = p 1 ⁄ 2 or ip 1 ⁄ 2 for p congruent to 1 or 3 modulo 4 respectively (the quadratic Gauss sum can also be evaluated by Fourier analysis as well as by contour integration).
This interval is called the confidence interval, and the radius (half the interval) is called the margin of error, corresponding to a 95% confidence level. Generally, at a confidence level γ {\displaystyle \gamma } , a sample sized n {\displaystyle n} of a population having expected standard deviation σ {\displaystyle \sigma } has a margin of ...
The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...