|
±1-sequence
In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence (1, −1, 1, −1, ...). Such sequences are commonly studied in discrepancy theory. ErdÅ‘s discrepancy problem Around 1932, mathematician Paul ErdÅ‘s conjectured that for any infinite ±1-sequence (x_1, x_2, \ldots) and any integer ''C'', there exist integers ''k'' and ''d'' such that : \left, \sum_^k x_ \ > C. The ErdÅ‘s discrepancy problem asks for a mathematical proof, proof or disproof of this conjecture. In February 2014, Alexei Lisitsa and Boris Konev of the University of Liverpool showed that every sequence of 1161 or more elements satisfies the conjecture in the special case ''C'' = 2, which proves the conjecture for ''C'' ≤ 2. This was the best such bound available at the time. Their proof relied on a SAT-solver computer algorithm whose output takes up 13 gigabytes of data, more tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
