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Since its formation the ROC Marine Corps has received training from the United States Marine Corps, from 1979 to 2020 that training was conducted secretly however in 2020 the annual month long training exercise held with trainers from the USMC's Marine Raider Regiment was conducted publicly. |
The Legislative Assembly had a total of 137 seats, 50 of which were elected and 87 of which were appointed. |
The top twelve teams will qualify for the tournament. Teams will be seeded by record within the conference, with a tiebreaker system to seed teams with identical conference records. |
This Cumberland County, New Jersey state location article is a stub. You can help Wikipedia by expanding it. |
This article on a moth of the family Yponomeutidae is a stub. You can help Wikipedia by expanding it. |
Nimislyarovo is located 18 km southwest of Krasnaya Gorka (the district's administrative centre) by road. Novokulevo is the nearest rural locality. |
Dean was educated at the University of York, where she was awarded a Bachelor of Arts degree in Biology in 1978 and a PhD. in Biology in 1982. |
FLC regulation involves an antisense-mediated chromatin mechanism that coordinately influences transcription initiation and elongation. As plants overwinter FLC expression is then epigenetically silenced through a cold-induced, cis-based, Polycomb switching mechanism. The group are mechanistically dissecting these cons... |
She uses Arabidopsis as a reference to establish the regulatory hierarchy and then use this information to translate into other species. She was a pioneer in Arabidopsis becoming a key model organism in plant science. |
Gabriela M. Mosquera (born January 3, 1977) is an American Democratic Party politician, who has served in the New Jersey General Assembly since March 5, 2012, where she represents the 4th Legislative District. |
Each of the forty districts in the New Jersey Legislature has one representative in the New Jersey Senate and two members in the New Jersey General Assembly. The other representatives from the 4th District for the 218th Legislature are: |
He made his debut in the Russian Second Division for FC Podolye Podolsky district on 11 October 2012 in a game against FC Metallurg-Oskol Stary Oskol. |
He made his Russian Premier League debut for FC Anzhi Makhachkala on 18 October 2015 in a game against FC Krasnodar. |
This biographical article related to a Russian association football defender born in 1993 is a stub. You can help Wikipedia by expanding it. |
Oakle Street railway station served the village of Oakle Street, Gloucestershire, England from 1851 to 1964 on the Gloucester-Newport line. |
The station opened on 19 September 1851 by the South Wales Railway. It closed on 31 March 1856 but reopened on 2 July 1870, before closing permanently on 2 November 1964. |
This article about a railway station in South West England is a stub. You can help Wikipedia by expanding it. |
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 ... |
This may be summarized by saying: a field has two operations, called addition and multiplication; it is an abelian group under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition. |
The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows: |
The real numbers R, with the usual operations of addition and multiplication, also form a field. The complex numbers C consist of expressions |
Not all real numbers are constructible. It can be shown that 2 3 {\displaystyle {\sqrt[{3}]{2}}} is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a cube with volume 2, another problem posed by the ancient Greeks. |
In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A, and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms above), and I ... |
The axioms of a field F imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by (F, +) when denoting it simply as F could be confusing. |
then F is said to have characteristic 0. For example, the field of rational numbers Q has characteristic 0 since no positive integer n is zero. Otherwise, if there is a positive integer n satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by p and the f... |
is compatible with the addition in F (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic p quite different from fields of characteristic 0. |
A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field F contains a prime field. If the characteristic of F is p (a prime number), the prime field is isomorphic to the finite field Fp introduced below. Otherwise the prime field is isomorphic to Q. |
The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. For a fixed positive integer n, arithmetic "modulo n" means to work with the numbers |
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry. A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros x1, x2, x3 of a cubic polyno... |
By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system. |
In the hierarchy of algebraic structures fields can be characterized as the commutative rings R in which every nonzero element is a unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct ideals, (0) and R. Fields are also precisely the commutative rings i... |
Given a commutative ring R, there are two ways to construct a field related to R, i.e., two ways of modifying R such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of Z is Q, the rationals, while the residue fields of Z are the finite fiel... |
It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field. |
The field F(x) of the rational fractions over a field (or an integral domain) F is the field of fractions of the polynomial ring F[x]. The field F((x)) of Laurent series |
In addition to the field of fractions, which embeds R injectively into a field, a field can be obtained from a commutative ring R by means of a surjective map onto a field F. Any field obtained in this way is a quotient R / m, where m is a maximal ideal of R. If R has only one maximal ideal m, this field is called the ... |
The ideal generated by a single polynomial f in the polynomial ring R = E[X] (over a field E) is maximal if and only if f is irreducible in E, i.e., if f cannot be expressed as the product of two polynomials in E[X] of smaller degree. This yields a field |
The compositum of two subfields E and E' of some field F is the smallest subfield of F containing both E and E'. The compositum can be used to construct the biggest subfield of F satisfying a certain property, for example the biggest subfield of F, which is, in the language introduced below, algebraic over E.[nb 3] |
Extensions whose degree is finite are referred to as finite extensions. The extensions C / R and F4 / F2 are of degree 2, whereas R / Q is an infinite extension. |
A field extension in which every element of F is algebraic over E is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula. |
The subfield E(x) generated by an element x, as above, is an algebraic extension of E if and only if x is an algebraic element. That is to say, if x is algebraic, all other elements of E(x) are necessarily algebraic as well. Moreover, the degree of the extension E(x) / E, i.e., the dimension of E(x) as an E-vector spac... |
The above-mentioned field of rational fractions E(X), where X is an indeterminate, is not an algebraic extension of E since there is no polynomial equation with coefficients in E whose zero is X. Elements, such as X, which are not algebraic are called transcendental. Informally speaking, the indeterminate X and its pow... |
Once again, the field extension E(x) / E discussed above is a key example: if x is not algebraic (i.e., x is not a root of a polynomial with coefficients in E), then E(x) is isomorphic to E(X). This isomorphism is obtained by substituting x to X in rational fractions. |
A subset S of a field F is a transcendence basis if it is algebraically independent (don't satisfy any polynomial relations) over E and if F is an algebraic extension of E(S). Any field extension F / E has a transcendence basis. Thus, field extensions can be split into ones of the form E(S) / E (purely transcendental e... |
A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation |
does not have any rational or real solution. A field containing F is called an algebraic closure of F if it is algebraic over F (roughly speaking, not too big compared to F) and is algebraically closed (big enough to contain solutions of all polynomial equations). |
By the above, C is an algebraic closure of R. The situation that the algebraic closure is a finite extension of the field F is quite special: by the Artin-Schreier theorem, the degree of this extension is necessarily 2, and F is elementarily equivalent to R. Such fields are also known as real closed fields. |
Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas. |
An Archimedean field is an ordered field such that for each element there exists a finite expression |
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of R. |
An ordered field is Dedekind-complete if all upper bounds, lower bounds (see Dedekind cut) and limits, which should exist, do exist. More formally, each bounded subset of F is required to have a least upper bound. Any complete field is necessarily Archimedean, since in any non-Archimedean field there is neither a great... |
Since every proper subfield of the reals also contains such gaps, R is the unique complete ordered field, up to isomorphism. Several foundational results in calculus follow directly from this characterization of the reals. |
The hyperreals R* form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis. |
The field Qp is used in number theory and p-adic analysis. The algebraic closure Qp carries a unique norm extending the one on Qp, but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of com... |
Differential fields are fields equipped with a derivation, i.e., allow to take derivatives of elements in the field. For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory, a variant of Galois theory dealing wi... |
Galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. An important notion in this area is that of finite Galois extensions F / E, which are, by definition, those that are separable and normal. The primitive element theorem shows that ... |
where f is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of f are contained in F and that f has only simple zeros. The latter condition is always satisfied if E has characteristic 0. |
The tensor product of fields is not usually a field. For example, a finite extension F / E of degree n is a Galois extension if and only if there is an isomorphism of F-algebras |
This fact is the beginning of Grothendieck's Galois theory, a far-reaching extension of Galois theory applicable to algebro-geometric objects. |
In model theory, a branch of mathematical logic, two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. The mathematical statements in question are required to be first-order sentences (involving 0, 1, the addition and multiplication).... |
If U is an ultrafilter on a set I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect to U is a field. It is denoted by |
A description in terms of generators and relations is also known for the Galois groups of p-adic number fields (finite extensions of Qp). |
Representations of Galois groups and of related groups such as the Weil group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using Galois cohomology. For example, the Brauer group, which is classically defined as the group of centra... |
The norm residue isomorphism theorem, proved around 2000 by Vladimir Voevodsky, relates this to Galois cohomology by means of an isomorphism |
The theory of modules (the analogue of vector spaces over rings instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular systems of linear equations over a ring are much more difficult to solve than in the case of fields, even in the specially simple case o... |
in a (large) finite field Fq can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution n to an equation |
In elliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form |
Functions on a suitable topological space X into a field k can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain: |
For having a field of functions, one must consider algebras of functions that are integral domains. In this case the ratios of two functions, i.e., expressions of the form |
This occurs in two main cases. When X is a complex manifold X. In this case, one considers the algebra of holomorphic functions, i.e., complex differentiable functions. Their ratios form the field of meromorphic functions on X. |
There are also proper classes with field structure, which are sometimes called Fields, with a capital F. The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The nimbers, a concept from game theory, form such a Field as well. |
Pain out of proportion or pain out of proportion to physical examination is a medical sign where apparent pain in the individual does not correspond to other signs. It is found in a number of conditions: |
Marta Zlatic is a Croatian neuroscientist who is group leader at the MRC Laboratory of Molecular Biology in Cambridge, UK. Her research investigates how neural circuits generate behaviour. |
Zlatic is interested the complex functions of the human brain, including language and communication. She has studied the brains of different species; including Drosophila and maggots. She made use of electron microscopy to map the entire Drosophila connectome, and studies the strengths of the connections between neuron... |
Cullom was drafted in the 24th round of the 1949 NFL Draft and again in the seventeenth round of the 1950 NFL Draft by the Washington Redskins. After one season playing for the New York Yanks, he was recalled by the Marines and was wounded in the Korean War. Cullom later returned to Cal where he worked as an assistan... |
This biographical article relating to an American football player, coach, or other figure is a stub. You can help Wikipedia by expanding it. |
Erskine recounts these events to the narrator, who is so struck by the Willie Hughes theory that he begins his own research and further fleshes out Graham's findings until he is without a doubt convinced that Willie Hughes was real and was the subject of the sonnets. He presents the evidence to Erskine but then finds h... |
He assumes Erskine committed suicide like Graham, but the doctor tells him the real cause was a lingering illness that Erskine had known about for some months; he had come to Cannes specifically to die. He left his friend the portrait of Mr. W. H. The portrait now hangs in his home, where many comment on it, but he doe... |
It is not known whether or not Wilde himself subscribed to the theory presented in the story. His lover Lord Alfred Douglas stated that he did believe it. Samuel Butler accepted some aspects of it, regarding the name "Will Hughes" as a "plausible conjecture". |
Wilde's story may have been an influence on John Masefield, whose book Shakespeare and Spiritual Life (1924) suggests that the Fair Youth was an actor who was delicate and small enough to play parts such as the boy-servant Moth in Love's Labours Lost and the sprite Ariel in The Tempest. He believed that he may even hav... |
In G. S. Viereck's novel My First Two Thousand Years, the protagonist, the Wandering Jew, watches a performance given by Willie Hughes, who is "positively enchanting as Juliet". He learns that Shakespeare had dedicated his sonnets to the boy-actor, but when he meets him he discovers that the boy is actually a girl in d... |
Oberto Della Torre (Genoa, 1617 - Genoa, 1698) was the 130th Doge of the Republic of Genoa and king of Corsica. |
The biosphere reserve is divided into a core zone (Kernzone) and a managed zone (Pflegezone), which include six nature reserves |
as well as six Natura 2000 areas. Whilst, in the core zone, human intervention is banned, in the managed zone, some land usage is possible, as long as it conforms to the nature reserve regulations and does not affect the conservation and care of the eco-system. |
The "development zone" covers the rest of the area of the biosphere reserve, including parts of the protected landscape of "Harz and Southern Harz Foreland". Here, human management, including its use as a recreational area and the needs of nature, are considered together. |
The Karst Trail runs through the eastern part of the biosphere reserve. In November 2010, over 20 nature and landscape guides were trained. This community project by the biosphere reserve's authorities and the county college of further education was enabled local nature lovers to be qualified to look after tourists. |
Runner3 is a rhythm platform game developed by Choice Provisions. A sequel to Bit.Trip Presents... Runner2: Future Legend of Rhythm Alien, Runner3 is part of the Bit.Trip series, starring the character CommanderVideo. The game was released on May 22, 2018 on Microsoft Windows, macOS, and Nintendo Switch, and was releas... |
Runner3 is a rhythm platform game in which players take control of CommanderVideo, the protagonist of the Bit.Trip series. Similar to the previous two Runner games, CommanderVideo runs forward automatically and the player controls actions such as jumping, sliding, and kicking to overcome obstacles and collect items. Ma... |
There are 27 main levels (each containing many branching paths, collectibles, and secrets), and three main worlds (Foodland, Spookyland, and Machineland) with differently themed scenery between each world. |
Runner3 was developed by video game development studio Choice Provisions. One focal point of Runner3's design was creating a rewarding and enjoyable experience for playing on any difficulty. Choice Provisions' co-founder Alex Neuse emphasized that many games rewarded only hardcore players and offered a lesser experienc... |
Runner3 was announced in September 2016. In February 2017, Nintendo revealed that Runner3 would launch on the Nintendo Switch console in the latter half of 2017. However, in August 2017, Choice Provisions announced that the game would not release until 2018. A physical version of Runner3 was announced by Nicalis on Dec... |
Following the release, Choice Provisions had announced upgrades to the game, which will include such features as enemy density, more checkpoints, stair assist, bonk counter and gold bars and gems. |
The video game aggregator site Metacritic gave the Nintendo Switch version a 73/100, while the PC version got a 78/100. |
Seth Macy of IGN criticized the game's "outdated" graphics and repetitiveness of levels, giving it a 7 out of 10. |
The original name for this community was New Sour Lake. The first significant landmark discovery of this area was the sulfur-smelling spring by J. F. Cotton in the 1850s. As late as 1865 he tried to establish an oil well on the site, but failed due to inadequate machinery. |
In the 1880s a man named P. S. Watts wanted to profit from the spring using the unique "medicinal" properties of the water (a popular trend at the time). To draw would-be visitors to the site, Watts changed the name of the site to Saratoga to replicate the famous resort at Saratoga Springs, New York. He built a hotel a... |
Mehrabian was born in 1970 in Isfahan, Iran. His family is from Isfahan. Mehrabi studied at Shiraz University and Tehran university. He obtained his master degree in economic science from Tehran University . |
As a minister, Mehrabian travelled the world promoting Iran's commercial interests in friendly countries such as Qatar, Belarus, Egypt and Venezuela. He involved in the discussions of a potential World Oil Bank involving Russia and Venezuela, and agreed on a cooperation protocol with the Democratic Republic of the Cong... |
He expressed deep concern over the non-adherence of industrial states to their commitments to control the emission of greenhouse gases. |
He implemented policies to encourage production of cars powered by compressed natural gas (CNG). Because Iran suffers from a lack of refining capacity, gasoline is rationed. Hundreds of thousands of gasoline-powered vehicles are also being converted to use a bi-fuel system allowing CNG as an alternative fuel. He encou... |
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