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±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] |
Erdős Discrepancy Problem
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 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 than the entire text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Crowdsourcing
Crowdsourcing involves a large group of dispersed participants contributing or producing goods or services—including ideas, votes, micro-tasks, and finances—for payment or as volunteers. Contemporary crowdsourcing often involves digital platforms to attract and divide work between participants to achieve a cumulative result. Crowdsourcing is not limited to online activity, however, and there are various historical examples of crowdsourcing. The word crowdsourcing is a portmanteau of "crowd" and "outsourcing". In contrast to outsourcing, crowdsourcing usually involves less specific and more public groups of participants. Advantages of using crowdsourcing include lowered costs, improved speed, improved quality, increased flexibility, and/or increased scalability of the work, as well as promoting diversity. Crowdsourcing methods include competitions, virtual labor markets, open online collaboration and data donation. Some forms of crowdsourcing, such as in "idea competiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
The Independent
''The Independent'' is a British online newspaper. It was established in 1986 as a national morning printed paper. Nicknamed the ''Indy'', it began as a broadsheet and changed to tabloid format in 2003. The last printed edition was published on Saturday 26 March 2016, leaving only the online edition. The daily edition was named National Newspaper of the Year at the 2004 British Press Awards. ''The Independent'' won the Brand of the Year Award in The Drum Awards for Online Media 2023. History 1980s Launched in 1986, the first issue of ''The Independent'' was published on 7 October in broadsheet format.Dennis Griffiths (ed.) ''The Encyclopedia of the British Press, 1422–1992'', London & Basingstoke: Macmillan, 1992, p. 330. It was produced by Newspaper Publishing plc and created by Andreas Whittam Smith, Stephen Glover and Matthew Symonds. All three partners were former journalists at ''The Daily Telegraph'' who had left the paper towards the end of Lord Hartwell' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rudin–Shapiro Sequence
In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2- automatic sequence named after Marcel Golay, Harold S. Shapiro, and Walter Rudin, who investigated its properties. Definition Each term of the Rudin–Shapiro sequence is either 1 or -1. If the binary expansion of n is given by : n = \sum_ \epsilon_k(n) 2^k, then let : u_n = \sum_ \epsilon_k(n)\epsilon_(n). (So u_n is the number of times the block 11 appears in the binary expansion of n.) The Rudin–Shapiro sequence (r_n)_ is then defined by : r_n = (-1)^. Thus r_n = 1 if u_n is even and r_n = -1 if u_n is odd. The sequence u_n is known as the complete Rudin–Shapiro sequence, and starting at n = 0, its first few terms are: : 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... and the corresponding terms r_n of the Rudin–Shapiro sequence are: : +1, +1, +1, −1, +1, +1, −1, +1, +1, +1, +1, −1, −1, −1, +1, −1, ... For exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discrepancy Of Hypergraphs
Discrepancy of hypergraphs is an area of discrepancy theory that studies the discrepancy of general set systems. Definitions In the classical setting, we aim at partitioning the vertices of a hypergraph \mathcal=(V, \mathcal) into two classes in such a way that ideally each hyperedge contains the same number of vertices in both classes. A partition into two classes can be represented by a coloring \chi \colon V \rightarrow \. We call −1 and +1 ''colors''. The color-classes \chi^(-1) and \chi^(+1) form the corresponding partition. For a hyperedge E \in \mathcal, set :\chi(E) := \sum_ \chi(v). The ''discrepancy of \mathcal with respect to \chi'' and the ''discrepancy of \mathcal'' are defined by :\operatorname(\mathcal,\chi) := \; \max_ , \chi(E), , :\operatorname(\mathcal) := \min_ \operatorname(\mathcal, \chi). These notions as well as the term 'discrepancy' seem to have appeared for the first time in a paper of Beck.J. Beck: "Roth's estimate of the discrepancy of integer seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Binary Sequence
A bitstream (or bit stream), also known as binary sequence, is a sequence of bits. A bytestream is a sequence of bytes. Typically, each byte is an 8-bit quantity, and so the term octet stream is sometimes used interchangeably. An octet may be encoded as a sequence of 8 bits in multiple different ways (see bit numbering) so there is no unique and direct translation between bytestreams and bitstreams. Bitstreams and bytestreams are used extensively in telecommunications and computing. For example, synchronous bitstreams are carried by SONET, and Transmission Control Protocol transports an asynchronous bytestream. Relationship to bytestreams In practice, bitstreams are not used directly to encode bytestreams; a communication channel may use a signalling method that does not directly translate to bits (for instance, by transmitting signals of multiple frequencies) and typically also encodes other information such as framing and error correction together with its data. Exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
