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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.
The sum a + b can be interpreted as a binary operation that combines a and b, in an algebraic sense, or it can be interpreted as the addition of b more units to a. Under the latter interpretation, the parts of a sum a + b play asymmetric roles, and the operation a + b is viewed as applying the unary operation +b to a. [20]
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
The assumption m ≥ n in the algorithm statement is necessary, as otherwise the matrix is not invertible and the normal equations cannot be solved (at least uniquely).. The Gauss–Newton algorithm can be derived by linearly approximating the vector of functions r i.
The numerator of the CH index is the between-cluster separation (BCSS) divided by its degrees of freedom. The number of degrees of freedom of BCSS is k - 1, since fixing the centroids of k - 1 clusters also determines the k th centroid, as its value makes the weighted sum of all centroids match the overall data centroid.
Computing the silhouette coefficient needs all () pairwise distances, making this evaluation much more costly than clustering with k-means. For a clustering with centers for each cluster , we can use the following simplified Silhouette for each point instead, which can be computed using only () distances:
One can normalize input scores by assuming that the sum is zero (subtract the average: where =), and then the softmax takes the hyperplane of points that sum to zero, =, to the open simplex of positive values that sum to 1 =, analogously to how the exponent takes 0 to 1, = and is positive.