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Primitive Polynomial (field Theory)
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field . This means that a polynomial of degree with coefficients in is a ''primitive polynomial'' if it has a root in such that is the entire field . This implies that is a primitive ()-root of unity in . Properties * Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible. * A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by ''x''. Over GF(2), is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by (it has 1 as a root). * An irreducible polynomial ''F''(''x'') of degree ''m'' over GF(''p''), where ''p'' is prime, is a primitive polynomial if the smallest positive integer ''n'' such that ''F''(''x'') divides is . * Over GF(''p''''m'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Theory (mathematics) , a sociological theory concerning the relationship between social actors and local social orders
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Field theory may refer to: Science * Field (mathematics), the theory of the algebraic concept of field * Field theory (physics), a physical theory which employs fields in the physical sense, consisting of three types: ** Classical field theory, the theory and dynamics of classical fields ** Quantum field theory, the theory of quantum mechanical fields ** Statistical field theory, the theory of critical phase transitions **Grand unified theory Social science * Field theory (psychology), a psychological theory which examines patterns of interaction between the individual and his or her environment * Field theory (sociology) In sociology, field theory examines how individuals construct social fields, and how they are affected by such fields. Social fields are environments in which competition between individuals and between groups takes place, such as markets, academi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow at the Australian National University. His research interests include number theory (in particular factorisation), random number generators, computer architecture, and analysis of algorithms. In 1973, he published a root-finding algorithm (an algorithm for solving equations numerically) which is now known as Brent's method. In 1975 he and Eugene Salamin independently conceived the Salamin–Brent algorithm, used in high-precision calculation of \pi. At the same time, he showed that all the elementary functions (such as log(''x''), sin(''x'') etc.) can be evaluated to high precision in the same time as \pi (apart from a small constant factor) using the arithmetic-geometric mean of Carl Friedrich Gauss. In 1979 he showed that the first 75 million complex zeros of the Riema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Twister
The Mersenne Twister is a general-purpose pseudorandom number generator (PRNG) developed in 1997 by and . Its name derives from the fact that its period length is chosen to be a Mersenne prime. The Mersenne Twister was designed specifically to rectify most of the flaws found in older PRNGs. The most commonly used version of the Mersenne Twister algorithm is based on the Mersenne prime 2^-1. The standard implementation of that, MT19937, uses a 32-bit word length. There is another implementation (with five variants) that uses a 64-bit word length, MT19937-64; it generates a different sequence. Application Software The Mersenne Twister is used as default PRNG by the following software: * Programming languages: Dyalog APL, IDL, R, Ruby, Free Pascal, PHP, Python (also available in NumPy, however the default was changed to PCG64 instead as of version 1.17),, CMU Common Lisp, Embeddable Common Lisp, Steel Bank Common Lisp, Julia (up to Julia 1.6 LTS, still available i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of CRC
The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF(2) (the integers modulo 2), that is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the coefficients of a message polynomial of this sort, and to find the CRC, we multiply the message polynomial by x^n and then find the remainder when dividing by the degree-n generator polynomial. The coefficients of the remainder polynomial are the bits of the CRC. Maths In general, computation of CRC corresponds to Euclidean division of polynomials over GF(2): :M(x) \cdot x^n = Q(x) \cdot G(x) + R(x). Here M(x) is the original message polynomial and G(x) is the degree-n generator polynomial. The bits of M(x) \cdot x^n are the original message with n zeroes added at the end. The CRC 'checksum' is formed by the coefficients of the remainder polynomial R(x) whose degree is strictly less t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Redundancy Check
A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get a short ''check value'' attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption. CRCs can be used for error correction (see bitfilters). CRCs are so called because the ''check'' (data verification) value is a ''redundancy'' (it expands the message without adding information) and the algorithm is based on ''cyclic'' codes. CRCs are popular because they are simple to implement in binary hardware, easy to analyze mathematically, and particularly good at detecting common errors caused by noise in transmission channels. Because the check value has a fixed length, the function that generates it is occasionally used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear-feedback Shift Register
In computing, a linear-feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. The most commonly used linear function of single bits is exclusive-or (XOR). Thus, an LFSR is most often a shift register whose input bit is driven by the XOR of some bits of the overall shift register value. The initial value of the LFSR is called the seed, and because the operation of the register is deterministic, the stream of values produced by the register is completely determined by its current (or previous) state. Likewise, because the register has a finite number of possible states, it must eventually enter a repeating cycle. However, an LFSR with a well-chosen feedback function can produce a sequence of bits that appears random and has a very long cycle. Applications of LFSRs include generating pseudo-random numbers, pseudo-noise sequences, fast digital counters, and whitening sequences. Both hardware and software implementation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Number Generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's ''seed'' (which may include truly random values). Although sequences that are closer to truly random can be generated using hardware random number generators, ''pseudorandom number generators'' are important in practice for their speed in number generation and their reproducibility. PRNGs are central in applications such as simulations (e.g. for the Monte Carlo method), electronic games (e.g. for procedural generation), and cryptography. Cryptographic applications require the output not to be predictable from earlier outputs, and more elaborate algorithms, which do not inherit the linearity of simpler PRNGs, are needed. Good statis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These programs enable computers to perform a wide range of tasks. A computer system is a nominally complete computer that includes the hardware, operating system (main software), and peripheral equipment needed and used for full operation. This term may also refer to a group of computers that are linked and function together, such as a computer network or computer cluster. A broad range of industrial and consumer products use computers as control systems. Simple special-purpose devices like microwave ovens and remote controls are included, as are factory devices like industrial robots and computer-aided design, as well as general-purpose devices like personal computers and mobile devices like smartphones. Computers power the Internet ... [...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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (group Theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
