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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glagolitic Script
The Glagolitic script (, , ''glagolitsa'') is the oldest known Slavic alphabet. It is generally agreed to have been created in the 9th century by Saint Cyril, a monk from Thessalonica. He and his brother Saint Methodius were sent by the Byzantine Emperor Michael III in 863 to Great Moravia to spread Christianity among the West Slavs in the area. The brothers decided to translate liturgical books into the contemporary Slavic language understandable to the general population (now known as Old Church Slavonic). As the words of that language could not be easily written by using either the Greek or Latin alphabets, Cyril decided to invent a new script, Glagolitic, which he based on the local dialect of the Slavic tribes from the Byzantine theme of Thessalonica. After the deaths of Cyril and Methodius, the Glagolitic alphabet ceased to be used in Moravia for political or religious needs. In 885, Pope Stephen V issued a papal bull to restrict spreading and reading Christian servi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hebrew Alphabet
The Hebrew alphabet ( he, אָלֶף־בֵּית עִבְרִי, ), known variously by scholars as the Ktav Ashuri, Jewish script, square script and block script, is an abjad script used in the writing of the Hebrew language and other Jewish languages, most notably Yiddish, Ladino, Judeo-Arabic, and Judeo-Persian. It is also used informally in Israel to write Levantine Arabic, especially among Druze. It is an offshoot of the Imperial Aramaic alphabet, which flourished during the Achaemenid Empire and which itself derives from the Phoenician alphabet. Historically, two separate abjad scripts have been used to write Hebrew. The original, old Hebrew script, known as the paleo-Hebrew alphabet, has been largely preserved in a variant form as the Samaritan alphabet. The present "Jewish script" or "square script", on the contrary, is a stylized form of the Aramaic alphabet and was technically known by Jewish sages as Ashurit (lit. "Assyrian script"), since its origins we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glagolitic
The Glagolitic script (, , ''glagolitsa'') is the oldest known Slavic alphabet. It is generally agreed to have been created in the 9th century by Saint Cyril, a monk from Thessalonica. He and his brother Saint Methodius were sent by the Byzantine Emperor Michael III in 863 to Great Moravia to spread Christianity among the West Slavs in the area. The brothers decided to translate liturgical books into the contemporary Slavic language understandable to the general population (now known as Old Church Slavonic). As the words of that language could not be easily written by using either the Greek or Latin alphabets, Cyril decided to invent a new script, Glagolitic, which he based on the local dialect of the Slavic tribes from the Byzantine theme of Thessalonica. After the deaths of Cyril and Methodius, the Glagolitic alphabet ceased to be used in Moravia for political or religious needs. In 885, Pope Stephen V issued a papal bull to restrict spreading and reading Christian servic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and the sum extends over all integers ''k.'' The Dirac delta function \delta and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the \deltas as the comb's ''teeth''), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol \operatorname\,\,(t), where the period is omitted, represents a Dirac comb of unit period. This implies \operatorname_(t) \ = \frac\operatorname\ \!\!\!\left(\frac\right). Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel: \operatorname_(t) = \frac\sum_^ e^. The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were (controversially) described as anti-semitic. Mathematics From his early years, Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups. Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate–Shafarevich Group
In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of (i.e. the -adic fields obtained from , as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as :\bigcap_v\mathrm\left(H^1\left(G_K,A\right)\rightarrow H^1\left(G_,A_v\right)\right). This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation , where is the Cyrillic letter " Sha", for Shafarevich, replacing the older notation or . Elements of the Tate–Shafarevich group Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of that have -rational points for every place of , but no -rational point. Thus, the group measures the extent to which the Hasse principle fails to ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shcha
Shcha (Щ щ; italics: ), Shta or Sha with descender is a letter of the Cyrillic script. In Russian, it represents the voiceless alveolo-palatal fricative , similar to the pronunciation of in ''sheep'' (but longer). In Ukrainian and Rusyn, it represents the consonant cluster . In Bulgarian, it represents the consonant cluster . Other non-Slavic languages written in Cyrillic use this letter to spell the few loanwords that use it or foreign names; it is usually pronounced and is often omitted when teaching those languages. In English, Shcha is romanized as or (with háčeks) or occasionally as , all reflecting the historical Russian pronunciation of the letter (as a combined ''Ш'' and ''Ч''). English-speaking learners of Russian are often instructed to pronounce it in this way although it is no longer the standard pronunciation in Russian (it still is in Ukrainian and Rusyn, as above). The letter Щ in Russian and Ukrainian corresponds to ШЧ in related words in Bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coptic Alphabet
The Coptic alphabet is the script used for writing the Coptic language. The repertoire of glyphs is based on the Greek alphabet augmented by letters borrowed from the Egyptian Demotic and is the first alphabetic script used for the Egyptian language. There are several Coptic alphabets, as the Coptic writing system may vary greatly among the various dialects and subdialects of the Coptic language. History The Coptic alphabet has a long history, going back to the Hellenistic period, when the Greek alphabet was used to transcribe Demotic texts, with the aim of recording the correct pronunciation of Demotic. During the first two centuries of the Common Era, an entire series of spiritual texts were written in what scholars term Old Coptic, Egyptian language texts written in the Greek alphabet. A number of letters, however, were derived from Demotic, and many of these (though not all) are used in "true" form of Coptic writing. With the spread of Christianity in Egypt, by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
