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Let S be any finite set, f be any endofunction from S to itself, and x 0 be any element of S.For any i > 0, let x i = f(x i − 1).Let μ be the smallest index such that the value x μ reappears infinitely often within the sequence of values x i, and let λ (the loop length) be the smallest positive integer such that x μ = x λ + μ.
Given a function that accepts an array, a range query (,) on an array = [,..,] takes two indices and and returns the result of when applied to the subarray [, …,].For example, for a function that returns the sum of all values in an array, the range query (,) returns the sum of all values in the range [,].
The approaches range from subjective reviews to objective statistical tests. One approach that is commonly used is to have the model builders determine validity of the model through a series of tests. [3] Naylor and Finger [1967] formulated a three-step approach to model validation that has been widely followed: [1] Step 1.
For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour −1)×(1 nanosecond) ≈ 6 × 10 −13 (using the unit conversion 3.6 × 10 12 nanoseconds = 1 hour). There is a probability density function f with f(5 hours) = 2 hour −1. The integral of f over any window of time (not only ...
He developed MATLAB's initial linear algebra programming in 1967 with his one-time thesis advisor, George Forsythe. [21] This was followed by Fortran code for linear equations in 1971. [21] Before version 1.0, MATLAB "was not a programming language; it was a simple interactive matrix calculator. There were no programs, no toolboxes, no graphics.
The P versus NP problem is a major unsolved problem in theoretical computer science.Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at x = 0 if 0 < k < 1, infinite positive slope at x = 0 if 1 < k < 2 and null slope at x = 0 if k > 2. For k = 1 the density has a finite negative slope at x = 0.
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.