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Fibonacci Coding
In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no other instances of "11" before the end. The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system that uses Zeckendorf's theorem and has the property that no number has a representation with consecutive 1s. The Fibonacci code word for a particular integer is exactly the integer's Zeckendorf representation with the order of its digits reversed and an additional "1" appended to the end. Definition For a number N\!, if d(0),d(1),\ldots,d(k-1),d(k)\! represent the digits of the code word representing N\! then we have: : N = \sum_^ d(i) F(i+2),\textd(k-1)=d(k)=1.\! where is the th Fibonacci number, and so is the th distinct Fibonacci number starting with 1,2,3,5,8,13,\ldots. The last bit d(k) i ... [...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] |
Entropy Coder
In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared by Shannon's source coding theorem, which states that any lossless data compression method must have an expected code length greater than or equal to the entropy of the source. More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies \operatorname E_ell(d(x))\geq \operatorname E_ \log_b(P(x))/math>, where \ell is the function specifying the number of symbols in a code word, d is the coding function, b is the number of symbols used to make output codes and P is the probability of the source symbol. An entropy coding attempts to approach this lower bound. Two of the most common entropy coding techniques are Huffman coding and arithmetic coding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lossless Compression Algorithms
Lossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. Lossless compression is possible because most real-world data exhibits statistical redundancy. By contrast, lossy compression permits reconstruction only of an approximation of the original data, though usually with greatly improved compression rates (and therefore reduced media sizes). By operation of the pigeonhole principle, no lossless compression algorithm can shrink the size of all possible data: Some data will get longer by at least one symbol or bit. Compression algorithms are usually effective for human- and machine-readable documents and cannot shrink the size of random data that contain no redundancy. Different algorithms exist that are designed either with a specific type of input data in mind or with specific assumptions about what kinds of redundancy the uncompressed data are likely to contain. Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-standard Positional Numeral Systems
Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems: :In a standard positional numeral system, the base ''b'' is a positive integer, and ''b'' different numerals are used to represent all non-negative integers. The standard set of numerals contains the ''b'' values 0, 1, 2, etc., up to ''b'' − 1, but the value is weighted according to the position of the digit in a number. The value of a digit string like ''pqrs'' in base ''b'' is given by the polynomial form ::p\times b^3+q\times b^2+r\times b+s. :The numbers written in superscript represent the powers of the base used. :For instance, in hexadecimal (''b'' = 16), using the numerals A for 10, B for 11 etc., the digit string 7A3F means ::7\times16^3+10\times16^2+3\times16+15, :which written in our normal decimal notation is 31295. :Upon in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World Scientific Publishing
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, with more than 170 journals in various fields. In 1995, World Scientific co-founded the London-based Imperial College Press together with the Imperial College of Science, Technology and Medicine. Company structure The company head office is in Singapore. The Chairman and Editor-in-Chief is Dr Phua Kok Khoo, while the Managing Director is Doreen Liu. The company was co-founded by them in 1981. Imperial College Press In 1995 the company co-founded Imperial College Press, specializing in engineering, medicine and information technology, with Imperial College London Imperial College London, also known as Imperial, is a Public university, public research university in London, England. Its history began with Prince Albert of Saxe-Coburg and Gotha, Prince Albert, h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Entropy Random Walk
A maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy. While a standard random walk samples for every vertex a uniform probability distribution of outgoing edges, locally maximizing entropy rate, MERW maximizes it globally (average entropy production) by sampling a uniform probability distribution among all paths in a given graph. MERW is used in various fields of science. A direct application is choosing probabilities to maximize transmission rate through a constrained channel, analogously to Fibonacci coding. Its properties also made it useful for example in analysis of complex networks, like link prediction, community detection, robust transport over networks and centrality measures. It is also used in image analysis, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Varicode
Varicode is a self-synchronizing code for use in PSK31. It supports all ASCII characters, but the characters used most frequently in English have shorter codes. The space between characters is indicated by a 00 sequence, an implementation of Fibonacci coding. Originally created for speeding up real-time keyboard-to-keyboard exchanges over low bandwidth links, Varicode is freely available. Limitations * Varicode provides somewhat weaker compression in languages other than English that use same characters as in English. Varicode table Control characters Printable characters Character lengths Beginning with the single-bit code "1", valid varicode values may be formed by prefixing a "1" or "10" to a shorter code. Thus, the number of codes of length ''n'' is equal to the Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ostrowski Numeration
In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers. Fix a positive irrational number ''α'' with continued fraction expansion 'a''0; ''a''1, ''a''2, ... Let (''q''''n'') be the sequence of denominators of the convergents ''p''''n''/''q''''n'' to α: so ''q''''n'' = ''a''''n''''q''''n''−1 + ''q''''n''−2. Let ''α''''n'' denote ''T''''n''(''α'') where ''T'' is the Gauss map ''T''(''x'') = , and write ''β''''n'' = (−1)''n''+1 ''α''0 ''α''1 ... ''α''''n'': we have ''β''''n'' = ''a''''n''''β''''n''−1 + ''β''''n''−2. Real number representations Every positive real ''x'' can be written as : x = \sum_^\infty b_n \beta_n \ where the integer coefficients 0 ≤ ''b''''n'' ≤ ''a''''n'' and if ''b''''n'' = ''a''''n'' then ''b''''n''−1 = 0. Integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NegaFibonacci Coding
In mathematics, negafibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end. Encoding method The following steps describe how to encode a nonzero integer x . Note that f denotes the Negafibonacci sequence. # If x is positive, compute the greatest odd negative integer n such that the sum of the odd negative terms of the Negafibonacci sequence from -1 to n with a step of -2, is greater than or equal to x : n \in \ , \quad \sum_^ f(i) x \geq \sum_^ f(i) # Add a 1 at the , n, ^ bit of the binary word. Subtract f(n) from x . # Repeat the process from step 1 with the new value of ''x'', until it reaches 0. # Add a 1 on the left of the resulting binary word to finish the encoding. To decode an encoded binary word, remove the leftmost 1 from the binary word, since it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio Base
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number \frac ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" – this is called a ''standard form''. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic properties of the base φ — most notably that φ + φ = φ. For instance, 11φ = 100φ. Despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating (finite) base-φ expansion. The set of numbers which possess a finite base-φ representation is the ring Z \frac">math display=inline>\frac it p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Entropy Random Walk
A maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy. While a standard random walk samples for every vertex a uniform probability distribution of outgoing edges, locally maximizing entropy rate, MERW maximizes it globally (average entropy production) by sampling a uniform probability distribution among all paths in a given graph. MERW is used in various fields of science. A direct application is choosing probabilities to maximize transmission rate through a constrained channel, analogously to Fibonacci coding. Its properties also made it useful for example in analysis of complex networks, like link prediction, community detection, robust transport over networks and centrality measures. It is also used in image analysis, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
