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Krohn–Rhodes Theory
In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner (called a " wreath product" or "cascade"). Krohn and Rhodes found a general decomposition for finite automata. The authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups. Definitions and description of the Krohn–Rhodes theorem Let ''T'' be a semigroup. A semigroup ''S'' that is a homomorphic image of a subsemigroup of ''T'' is said to be a divisor of ''T''. The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup ''S'' is a divisor of a finite alternating wreath product of finite simple groups, each a ... [...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] |
Semigroup With Three Elements
In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplication. Multiplication of integers is associative, and the product of any two of these three integers is again one of these three integers. There are 18 inequivalent ways to define an associative operation on three elements: while there are, altogether, a total of 39 = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are isomorphic or antiisomorphic so that there are essentially only 18 possibilities.Andreas DistlerClassification and enumeration of finite semigroups, PhD thesis, University of St. Andrews One of these is C3, the cyclic group with three elements. The others all have a semigroup with two elements as subsemigroups. In the example above, the set under multiplication contain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massachusetts Institute Of Technology
The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and science. In response to the increasing Technological and industrial history of the United States, industrialization of the United States, William Barton Rogers organized a school in Boston to create "useful knowledge." Initially funded by a land-grant universities, federal land grant, the institute adopted a Polytechnic, polytechnic model that stressed laboratory instruction in applied science and engineering. MIT moved from Boston to Cambridge in 1916 and grew rapidly through collaboration with private industry, military branches, and new federal basic research agencies, the formation of which was influenced by MIT faculty like Vannevar Bush. In the late twentieth century, MIT became a leading center for research in compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University
Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyman John Harvard (clergyman), John Harvard, it is the oldest institution of higher learning in the United States. Its influence, wealth, and rankings have made it one of the most prestigious universities in the world. Harvard was founded and authorized by the Massachusetts General Court, the governing legislature of Colonial history of the United States, colonial-era Massachusetts Bay Colony. While never formally affiliated with any Religious denomination, denomination, Harvard trained Congregationalism in the United States, Congregational clergy until its curriculum and student body were gradually secularized in the 18th century. By the 19th century, Harvard emerged as the most prominent academic and cultural institution among the Boston B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Finite Automaton
In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state automaton (DFSA)—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by the string. ''Deterministic'' refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943. The figure illustrates a deterministic finite automaton using a state diagram. In this example automaton, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenneth Krohn
Kenneth is a given name of Gaelic origin. The name is an Anglicised form of two entirely different Gaelic personal names: ''Cainnech'' and '' Cináed''. The modern Gaelic form of ''Cainnech'' is ''Coinneach''; the name was derived from a byname meaning "handsome", "comely". Etymology The second part of the name ''Cinaed'' is derived either from the Celtic ''*aidhu'', meaning "fire", or else Brittonic ''jʉ:ð'' meaning "lord". People Fictional characters * Kenneth Widmerpool, character in Anthony Powell's novel sequence ''A Dance to the Music of Time'' *Kenneth Parcell from 30 Rock Places In the United States: * Kenneth, Minnesota * Kenneth City, Florida In Scotland: * Inch Kenneth, an island off the west coast of the Isle of Mull Other * " What's the Frequency, Kenneth?", a song by R.E.M. * Hurricane Kenneth * Cyclone Kenneth Intense Tropical Cyclone Kenneth was the strongest tropical cyclone to make landfall in Mozambique since modern records began. The cyclone ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Japan
Japan is an island country in East Asia. Located in the Pacific Ocean off the northeast coast of the Asia, Asian mainland, it is bordered on the west by the Sea of Japan and extends from the Sea of Okhotsk in the north to the East China Sea in the south. The Japanese archipelago consists of four major islands—Hokkaido, Honshu, Shikoku, and Kyushu—and List of islands of Japan, thousands of smaller islands, covering . Japan has a population of over 123 million as of 2025, making it the List of countries and dependencies by population, eleventh-most populous country. The capital of Japan and List of cities in Japan, its largest city is Tokyo; the Greater Tokyo Area is the List of largest cities, largest metropolitan area in the world, with more than 37 million inhabitants as of 2024. Japan is divided into 47 Prefectures of Japan, administrative prefectures and List of regions of Japan, eight traditional regions. About three-quarters of Geography of Japan, the countr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aizu
is the westernmost of the three regions of Fukushima Prefecture, Japan, the other two regions being Nakadōri in the central area of the prefecture and Hamadōri in the east. As of October 1, 2010, it had a population of 291,838. The principal city of the area is Aizuwakamatsu. It was part of Mutsu Province; the area once was part of Iwase Province created during the reign of Empress Genshō.Meyners d'Estrey, Guillaume Henry Jean (1884). ; excerpt, '' Genshō crée sept provinces : Idzumi, Noto, Atoa, Iwaki, Iwase, Suwa et Sado en empiétant sur celles de Kawachi, Echizen, Etchū, Kazusa, Mutsu and Shinano'' The ''Yōrō Ritsuryo'' established the Iwase Province in 718 through the division of the Michinoku Province ( Mutsu Province). It was composed of five districts of Shirakawa (白河), Iwase (石背), Aizu (会津), Asaka (安積) and Shinobu (信夫). The area encompassed by the province reverted to Mutsu some time between 722 and 724. During the Edo p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication Table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication binary operation, operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9. History Pre-modern times The oldest known multiplication tables were used by the Babylonian mathematics, Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese mathematics, Chinese Tsinghua Bamboo Slips#Decimal multiplication table, decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the matrix multiplication, product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minor (linear algebra), minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower trian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
