|
Shuffle Algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product ''X'' ⧢ ''Y'' of two words ''X'', ''Y'': the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative. Shuffle product The shuffle product of words of lengths ''m'' and ''n'' is a sum over the ways of interleaving the two words, as shown in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Property
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replacement for well-formed formula, expressions in Formal proof, logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the Operation (mathematics), operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds ... [...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] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zinbiel Algebra
In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: :(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b). Zinbiel algebras were introduced by . The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra. In any Zinbiel algebra, the symmetrised product :a \star b = a \circ b + b \circ a is associative. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over ''V'' is the tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ... with product :(x_0 \otimes \cdots \otimes x_p) \circ (x_ \otimes \cdots \otimes x_) = x_0 \sum_ (x_1,\ldots,x_), where the sum is over all (p,q) shuffles. Referenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra Of Permutations
In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the finite symmetric groups ''S''''n'', and is a non-commutative analogue of the Hopf algebra of symmetric functions. It is both free as an algebra and graded- cofree as a graded coalgebra, so is in some sense as far as possible from being either commutative or cocommutative. It was introduced by and studied by . Definition The underlying free abelian group of the MPR algebra has a basis consisting of the disjoint union of the symmetric groups ''S''''n'' for ''n'' = 0, 1, 2, .... , which can be thought of as permutations. The identity 1 is the empty permutation, and the counit takes the empty permutation to 1 and the others to 0. The product of two permutations (''a''1,...,''a''''m'') and (''b''1,...,''b''''n'') in MPR is given by the shuffle product (''a''1,...,''a''''m'') ''ш'' (''m'' + ''b''1,...,''m'' +&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sha (Cyrillic)
Sha, alternatively transliterated Ša (Ш ш; italics: ) is a letter of the Glagolitic and Cyrillic scripts. It commonly represents the voiceless postalveolar fricative , like the pronunciation of sh in "ship". More precisely, the sound in Russian denoted by ш is often falsely transcribed as a palatoalveolar fricative, but is actually a voiceless retroflex fricative . It is used in every variation of the Cyrillic alphabet for Slavic and non-Slavic languages. In English, Sha is romanized as sh or as š, the latter being the equivalent letter in the Latin alphabets of Czech, Slovak, Slovene, Serbo-Croatian, Latvian and Lithuanian. History Sha has its earliest origins in Phoenician Shin and is possibly linked closely to Shin's Greek equivalent: Sigma (Σ, σ, ς). (The similar form of the modern Hebrew Shin (ש), which is probably where the Cyrillic letter was actually derived from, derives from the same Proto-Canaanite source). Sha already possessed its current ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyrillic
The Cyrillic script ( ) is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking countries in Southeastern Europe, Eastern Europe, the Caucasus, Central Asia, North Asia, and East Asia, and used by many other minority languages. , around 250 million people in Eurasia use Cyrillic as the official script for their national languages, with Russia accounting for about half of them. With the accession of Bulgaria to the European Union on 1 January 2007, Cyrillic became the third official script of the European Union, following the Latin and Greek alphabets. The Early Cyrillic alphabet was developed during the 9th century AD at the Preslav Literary School in the First Bulgarian Empire during the reign of Tsar Simeon I the Great, probably by the disciples of the two Byzantine brothers Cyril and Methodius, who had previously created the Gl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicode
Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Character (computing), characters and 168 script (Unicode), scripts used in various ordinary, literary, academic, and technical contexts. Unicode has largely supplanted the previous environment of a myriad of incompatible character sets used within different locales and on different computer architectures. The entire repertoire of these sets, plus many additional characters, were merged into the single Unicode set. Unicode is used to encode the vast majority of text on the Internet, including most web pages, and relevant Unicode support has become a common consideration in contemporary software development. Unicode is ultimately capable of encoding more than 1.1 million characters. The Unicode character repertoire is synchronized with Univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied. Definition A binary operation * on a set ''S'' is ''commutative'' if x * y = y * x for all x,y \in S. An operation that is not commutative is said to be ''noncommutative''. One says ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riffle Shuffle Permutation
In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into two packets and then the two packets are interleaved (e.g. by moving cards one at a time from the bottom of one or the other of the packets to the top of the sorted deck). Beginning with an ordered set (1 rising sequence), mathematically a riffle shuffle is defined as a permutation on this set containing 1 or 2 rising sequences. The permutations with 1 rising sequence are the identity permutations. As a special case of this, a (p,q)-shuffle, for numbers p and q with p+q=n, is a riffle in which the first packet has p cards and the second packet has q cards.Weibel, Charles (1994). ''An Introduction to Homological Algebra'', p. 181. Cambridge University Press, Cambridge. Combinatorial enumeration Since a (p,q)-shuffle is completely determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riffle Shuffle
Shuffling is a technique used to randomize a deck of playing cards, introducing an element of chance into card games. Various shuffling methods exist, each with its own characteristics and potential for manipulation. One of the simplest shuffling techniques is the overhand shuffle, where small packets of cards are transferred from one hand to the other. This method is easy to perform but can be manipulated to control the order of cards. Another common technique is the riffle shuffle, where the deck is split into two halves and interleaved. This method is more complex but minimizes the risk of exposing cards. The Gilbert–Shannon–Reeds model suggests that seven riffle shuffles are sufficient to thoroughly randomize a deck, although some studies indicate that six shuffles may be enough. Other shuffling methods include the Hindu shuffle, commonly used in Asia, and the pile shuffle, where cards are dealt into piles and then stacked. The Mongean shuffle involves a specific sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
