Search results
Results from the WOW.Com Content Network
Given a sequence of length (r − 1)(s − 1) + 1, label each number n i in the sequence with the pair (a i, b i), where a i is the length of the longest monotonically increasing subsequence ending with n i and b i is the length of the longest monotonically decreasing subsequence ending with n i.
The analogue of Fekete's lemma holds for superadditive sequences as well, that is: + +. (The limit then may be positive infinity: consider the sequence = !.) There are extensions of Fekete's lemma that do not require the inequality a n + m ≤ a n + a m {\displaystyle a_{n+m}\leq a_{n}+a_{m}} to hold for all m and n , but only for m and n ...
The set C = {0, 1} ∞ of all infinite binary sequences is sometimes called the Cantor space. An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language.
[1] [2] When the process is performed on a sequence of samples of a signal or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or density, as in the case of a photograph). Decimation is a term that historically means the removal of every tenth one.
The definition of a type 1 and type 2 sequence was first introduced by Vail et al. (1984). [4] Since they were hard to recognize, they were redefined in 1990 by Van Wagoner et al.. However even with this new definition, type 2 sequence boundaries were hard to recognize in the field due to their lack of subaerial exposure.
For example, the sequence ,, is a subsequence of ,,,,, obtained after removal of elements ,, and . The relation of one sequence being the subsequence of another is a partial order . Subsequences can contain consecutive elements which were not consecutive in the original sequence.
Erdős (1962) showed that for every sum-free sequence there exists an unbounded sequence of numbers for which () = where is the golden ratio, and he exhibited a sum-free sequence for which, for all values of , () = (/), [1] subsequently improved to () = (/) by Deshouillers, Erdős and Melfi in 1999 [3] and to () = (/) by Luczak and Schoen in ...
For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is