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A sequential time is one in which the numbers form a normal sequence, such as 1:02:03 4/5/06 (two minutes and three seconds past 1 am on 4 May 2006 (or April 5, 2006 in the United States) or the same time and date in the "06" year of any other century). Short sequential times such as 1:23:45 or 12:34:56 appear every day.
In set theory, a normal sequence is one that is continuous and strictly increasing. In probability theory, a normal number is a number whose representation is a normal sequence in all bases, i.e. regardless of which base is chosen (e.g. base 2, base 8, base 10, etc.) the sequence of digits contains every finite subsequence with equal probability.
We say that x is simply normal in base b if the sequence S x, b is simply normal [5] and that x is normal in base b if the sequence S x, b is normal. [6] The number x is called a normal number (or sometimes an absolutely normal number) if it is normal in base b for every integer b greater than 1. [7] [8]
Recamán's sequence: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence. A005132: Look-and ...
Counter-intuitively, the most likely sequence is often not a member of the typical set. For example, suppose that X is an i.i.d Bernoulli random variable with p(0)=0.1 and p(1)=0.9. In n independent trials, since p(1)>p(0), the most likely sequence of outcome is the sequence
If Y is a metric space, then the compact-open topology is equivalent to the topology of compact convergence, [1] and we obtain a definition which is closer to the classical one: A collection F of continuous functions is called a normal family if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets ...
The definition of the Champernowne constant immediately gives rise to an infinite series representation involving a double sum, = = = (+), where () = = is the number of digits between the decimal point and the first contribution from an n-digit base-10 number; these expressions generalize to an arbitrary base b by replacing 10 and 9 with b and b − 1 respectively.
The question of what groups are extensions of by is called the extension problem, and has been studied heavily since the late nineteenth century.As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {}, where each {+} is an extension of {} by some simple group.