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The following symbols are reserved characters that either have a special meaning under LaTeX or are unavailable in all the fonts. ... x _ 2 ^ 3 {x _ 2}^ 3: Super ...
The Knuth–Plass algorithm is a line-breaking algorithm designed for use in Donald Knuth's typesetting program TeX.It integrates the problems of text justification and hyphenation into a single algorithm by using a discrete dynamic programming method to minimize a loss function that attempts to quantify the aesthetic qualities desired in the finished output.
2. Denotes the additive inverse and is read as minus, the negative of, or the opposite of; for example, –2. 3. Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory. × (multiplication sign) 1. In elementary arithmetic, denotes multiplication, and is read as times; for example, 3 × 2. 2.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
In the case of our word, 11 such patterns can be matched, namely 1 c 4 l 4, 1 cy, 1 d 4 i 3 a, 4 edi, e 3 dia, 2 i 1 a, ope 5 d, 2 p 2 ed, 3 pedi, pedia 4, y 1 c. For each position in the word, TeX will calculate the maximum value obtained among all matching patterns, yielding en 1 cy 1 c 4 l 4 o 3 p 4 e 5 d 4 i 3 a 4.
Consider the vectors (polynomials) p 1 := 1, p 2 := x + 1, and p 3 := x 2 + x + 1. Is the polynomial x 2 − 1 a linear combination of p 1, p 2, and p 3? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x 2 − 1. Picking arbitrary coefficients a 1, a 2, and a 3, we want
Thus the lexicographic successor of the initial state is permuted: [1, 2, 4, 3]. Following this algorithm, the next lexicographic permutation will be [1, 3, 2, 4], and the 24th permutation will be [4, 3, 2, 1] at which point a[k] < a[k + 1] does not exist, indicating that this is the last permutation.
Prefix sums are trivial to compute in sequential models of computation, by using the formula y i = y i − 1 + x i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.