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where the term n = 0 is equal to 1 and where is the time-ordering operator. It is a higher-order operation that ensures the exponential is time-ordered, so that any product of a(t) that occurs in the expansion of the exponential is ordered such that the value of t is increasing from right to left of the product. For example:
Given any random variables X 1, X 2, ..., X n, the order statistics X (1), X (2), ..., X (n) are also random variables, defined by sorting the values (realizations) of X 1, ..., X n in increasing order. When the random variables X 1, X 2, ..., X n form a sample they are independent and identically distributed. This is the case treated below. In ...
43 as: The shortest X-ray laser pulse [8] 53 as: The shortest electron laser pulse [9] [10] 10 −15: femtosecond: fs One quadrillionth of one second 1 fs: The cycle time for ultraviolet light with a wavelength of 300 nanometres; The time it takes light to travel a distance of 0.3 micrometres (μm). 7.58fs: The period of vibration of a hydrogen ...
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
Total stopping time is on the x axis, frequency on the y axis. Histogram of total stopping times for the numbers 1 to 10 9. Total stopping time is on the x axis, frequency on the y axis. Iteration time for inputs of 2 to 10 7. Total stopping time of numbers up to 250, 1000, 4000, 20000, 100000, 500000
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If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a tuple, a.k.a. string