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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. Erdős discrepancy problem Around 1932, mathematician Paul Erdős 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 Erdős 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 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] |
Polymath Project
The Polymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The project began in January 2009 on Timothy Gowers's blog when he posted a problem and asked his readers to post partial ideas and partial progress toward a solution. This experiment resulted in a new answer to a difficult problem, and since then the Polymath Project has grown to describe a particular crowdsourcing process of using an online collaboration to solve any math problem. Origin In January 2009, Gowers chose to start a social experiment on his blog by choosing an important unsolved mathematical problem and issuing an invitation for other people to help solve it collaboratively in the comments section of his blog. Along with the math problem itself, Gowers asked a question which was included in the title of his blog post, "is massively collaborative ... [...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 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] |
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] |
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] |
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] |
Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at different points in time. The analysis of autocorrelation is a mathematical tool for identifying repeating patterns or hidden periodicities within a signal obscured by noise. Autocorrelation is widely used in signal processing, time domain and time series analysis to understand the behavior of data over time. 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. Various time series models incorporate autocorrelation, such as unit root processes, trend-stationary processes, autoregressive processes, and moving average processes. Autocorrelation of stochastic processes In statistics, the autocorrelation of a real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pulse Compression
Pulse compression is a signal processing technique commonly used by radar, sonar and Ultrasound, echography to either increase the range angular resolution, resolution when pulse length is constrained or increase the Signal-to-noise ratio, signal to noise ratio when the peak power and the Bandwidth_(signal_processing), bandwidth (or equivalently range resolution) of the transmitted signal are constrained. This is achieved by modulation, modulating the transmitted pulse and then Cross-correlation, correlating the received signal with the transmitted pulse. Simple pulse Signal description The ideal model for the simplest, and historically first type of signals a pulse radar or sonar can transmit is a truncated sinusoidal pulse (also called a CW --carrier wave-- pulse), of amplitude 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 ... [...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. Swiss inventor, Gustav Guanella proposed a "means for and method of secret signals". With DSSS, the message symbols are modulated by a sequence of complex values known as ''spreading sequence''. Each element of the spreading sequence, a so-called ''chip'', has a shorter duration than the original message symbols. The modulation of the message symbols scrambles and spreads the signal in the spectrum, and thereby results in a bandwidth of the spreading sequence. The smaller the chip duration, the larger the bandw ... [...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 Mathematical analysis journals Academic journals est ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
USA Today
''USA Today'' (often stylized in all caps) is an American daily middle-market newspaper and news broadcasting company. Founded by Al Neuharth in 1980 and launched on September 14, 1982, the newspaper operates from Gannett's corporate headquarters in New York City. Its newspaper is printed at 37 sites across the United States and at five additional sites internationally. The paper's dynamic design influenced the style of local, regional, and national newspapers worldwide through its use of concise reports, colorized images, informational graphics, and inclusion of popular culture stories, among other distinct features. As of 2023, ''USA Today'' has the fifth largest print circulation in the United States, with 132,640 print subscribers. It has two million digital subscribers, the fourth-largest online circulation of any U.S. newspaper. ''USA Today'' is distributed in all 50 states, Washington, D.C., and Puerto Rico, and an international edition is distributed in Asia, ... [...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] |
