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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 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 tex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 newspaper was controlled by Tony O'Reilly's Irish Independent News & Media from 1997 until it was sold to the Russian oligarch and former KGB Officer Alexander Lebedev in 2010. In 2017, Sultan Muhammad Abuljadayel bought a 30% stake in it. The daily edition was named National Newspaper of the Year at the 2004 British Press Awards. The website and mobile app had a combined monthly reach of 19,826,000 in 2021. History 1986 to 1990 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 produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, Walter Rudin, and Harold S. Shapiro, who independently 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, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Of Hypergraphs
Discrepancy of hypergraphs is an area of discrepancy theory. 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 sequences is nearly sharp", page 319-325. Combinatorica, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 between bitstreams and 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals. Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance. Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pulse Compression
Pulse compression is a signal processing technique commonly used by radar, sonar and echography to increase the range resolution as well as the signal to noise ratio. This is achieved by modulating the transmitted pulse and then correlating the received signal with the transmitted pulse. Simple pulse Signal description The simplest signal a pulse radar can transmit is a sinusoidal-amplitude pulse, A and carrier frequency, f_0, truncated by a rectangular function of width, T. The pulse is transmitted periodically, but that is not the main topic of this article; we will consider only a single pulse, s. If we assume the pulse to start at time t=0, the signal can be written the following way, using the complex notation: :s(t) = \begin A e^ &\text \; 0 \leq t where it reaches its maximum 1, and it decreases linearly on ,\frac{1}{2}/math> until it reaches 0 again. Figures at the end of this paragraph show the shape of the intercorrelation for a sample signal (in red), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct-sequence Spread Spectrum
In telecommunications, direct-sequence spread spectrum (DSSS) is a spread-spectrum modulation technique primarily used to reduce overall signal interference. The direct-sequence modulation makes the transmitted signal wider in bandwidth than the information bandwidth. After the despreading or removal of the direct-sequence modulation in the receiver, the information bandwidth is restored, while the unintentional and intentional interference is substantially reduced. The first known scheme for this technique was introduced by a Swiss inventor, Gustav Guanella. With DSSS, the message bits are modulated by a pseudorandom bit sequence known as a spreading sequence. Each spreading-sequence bit, which is known as a chip, has a much shorter duration (larger bandwidth) than the original message bits. The modulation of the message bits scrambles and spreads the pieces of data, and thereby results in a bandwidth size nearly identical to that of the spreading sequence. The smaller the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Analysis
''Discrete Analysis'' is a mathematics journal covering the applications of analysis to discrete structures. ''Discrete Analysis'' is an arXiv overlay journal, meaning the journal's content is hosted on the arXiv. History ''Discrete Analysis'' was created by Timothy Gowers to demonstrate that a high-quality mathematics journal could be inexpensively produced outside of the traditional academic publishing industry. The journal is open access, and submissions are free for authors. The journal's 2018 MCQ is 1.21.''Discrete Analysis'', MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ..., 2019. Accessed 2019-09-02. References * * External links *{{Official, https://discreteanalysisjournal.com/ Open access journals Mathematics journals Publications established in 2016 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
