Abelian varietyIn 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.
Algebraic varietyAlgebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly.
Silyl enol etherSilyl enol ethers in organic chemistry are a class of organic compounds that share a common functional group composed of an enolate bonded through its oxygen end to an organosilicon group. They are important intermediates in organic synthesis. Silyl enol ethers are generally prepared by reacting an enolizable carbonyl compound with a silyl electrophile and a base, or just reacting an enolate with a silyl electrophile. Since silyl electrophiles are hard and silicon-oxygen bonds are very strong, the oxygen (of the carbonyl compound or enolate) acts as the nucleophile to form a Si-O single bond.
KetoneIn organic chemistry, a ketone ˈkiːtoʊn is a functional group with the structure , where R and R' can be a variety of carbon-containing substituents. Ketones contain a carbonyl group (which contains a carbon-oxygen double bond C=O). The simplest ketone is acetone (where R and R' is methyl), with the formula . Many ketones are of great importance in biology and in industry. Examples include many sugars (ketoses), many steroids (e.g., testosterone), and the solvent acetone.
EsterIn chemistry, an ester is a compound derived from an acid (organic or inorganic) in which the hydrogen atom (H) of at least one acidic hydroxyl group () of that acid is replaced by an organyl group (). Analogues derived from oxygen replaced by other chalcogens belong to the ester category as well. According to some authors, organyl derivatives of acidic hydrogen of other acids are esters as well (e.g. amides), but not according to the IUPAC.
Rational varietyIn mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set of indeterminates, where d is the dimension of the variety. Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in . If V is rational, then there are n + 1 polynomials g0, ..., gn in such that In order words, we have a of the variety.
Complete varietyIn mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the morphism is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products. The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete.
EnolIn organic chemistry, alkenols (shortened to enols) are a type of reactive structure or intermediate in organic chemistry that is represented as an alkene (olefin) with a hydroxyl group attached to one end of the alkene double bond (). The terms enol and alkenol are portmanteaus deriving from "-ene"/"alkene" and the "-ol" suffix indicating the hydroxyl group of alcohols, dropping the terminal "-e" of the first term. Generation of enols often involves deprotonation at the α position to the carbonyl group—i.
Quasi-projective varietyIn mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subscheme of some projective space. An affine space is a Zariski-open subset of a projective space, and since any closed affine subset can be expressed as an intersection of the projective completion and the affine space embedded in the projective space, this implies that any affine variety is quasiprojective.
Α-Halo ketoneIn organic chemistry, an α-halo ketone is a functional group consisting of a ketone group or more generally a carbonyl group with an α-halogen substituent. α-Halo ketones are alkylating agents. Prominent α-halo ketones include phenacyl bromide and chloroacetone. The general structure is RR′C(X)C(=O)R where R is an alkyl or aryl residue and X any one of the halogens. The preferred conformation of a halo ketone is that of a cisoid with the halogen and carbonyl sharing the same plane as the steric hindrance with the carbonyl alkyl group is generally larger.
