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Shuffle Product
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 the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the 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 true when performing addition and multiplication on any real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...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''. The n ... [...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 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 Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome. __TOC__ Techniques Overha .... References ... [...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'' +&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sha (Cyrillic)
Sha or Shu (Ш ш; italics: ) is a letter of the Glagolitic and Cyrillic scripts. It commonly represents the voiceless postalveolar fricative . More precisely, the sound in Russian denoted by ш is commonly 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, Macedonian, 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 form in Saints Cyril and Methodius's Glagolitic alphabet. Most Cyri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyrillic
, bg, кирилица , mk, кирилица , russian: кириллица , sr, ћирилица, uk, кирилиця , fam1 = Egyptian hieroglyphs , fam2 = Proto-Sinaitic , fam3 = Phoenician , fam4 = Greek script augmented by Glagolitic , sisters = , children = Old Permic script , unicode = , iso15924 = Cyrl , iso15924 note = Cyrs (Old Church Slavonic variant) , sample = Romanian Traditional Cyrillic - Lord's Prayer text.png , caption = 1780s Romanian text (Lord's Prayer), written with the Cyrillic script The Cyrillic script ( ), Slavonic script or the Slavic 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. , around 250 million people in Eurasia use Cyrillic a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's writing systems. The standard, which is maintained by the Unicode Consortium, defines as of the current version (15.0) 149,186 characters covering 161 modern and historic script (Unicode), scripts, as well as symbols, emoji (including in colors), and non-visual control and formatting codes. Unicode's success at unifying character sets has led to its widespread and predominant use in the internationalization and localization of computer software. The standard has been implemented in many recent technologies, including modern operating systems, XML, and most modern programming languages. The Unicode character repertoire is synchronized with Universal Coded Character Set, ISO/IEC 10646, each being code-for-code id ... [...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. Most familiar as the name of the property that says something like 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 years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symme ... [...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 determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riffle Shuffle
Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome. __TOC__ Techniques Overhand One of the easiest shuffles to accomplish after a little practice is the overhand shuffle. Johan Jonasson wrote, "The overhand shuffle... is the shuffling technique where you gradually transfer the deck from, say, your right hand to your left hand by sliding off small packets from the top of the deck with your thumb." In detail as normally performed, with the pack initially held in the left hand (say), most of the cards are grasped as a group from the bottom of the pack between the thumb and fingers of the right hand and lifted clear of the small group that remains in the left hand. Small packets are then released from the right hand a packet at a time so that they drop on the top of the pack accumulating in the left hand. The process ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
