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.
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.
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.
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.
Projective varietyIn algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
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.
EtherIn organic chemistry, ethers are a class of compounds that contain an ether group—an oxygen atom connected to two alkyl or aryl groups. They have the general formula , where R and R′ represent the alkyl or aryl groups. Ethers can again be classified into two varieties: if the alkyl or aryl groups are the same on both sides of the oxygen atom, then it is a simple or symmetrical ether, whereas if they are different, the ethers are called mixed or unsymmetrical ethers.
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.
Ketone bodiesKetone bodies are water-soluble molecules or compounds that contain the ketone groups produced from fatty acids by the liver (ketogenesis). Ketone bodies are readily transported into tissues outside the liver, where they are converted into acetyl-CoA (acetyl-Coenzyme A)—which then enters the citric acid cycle (Krebs cycle) and is oxidized for energy. These liver-derived ketone groups include acetoacetic acid (acetoacetate), beta-hydroxybutyrate, and acetone, a spontaneous breakdown product of acetoacetate (see graphic).
EnolateIn organic chemistry, enolates are organic anions derived from the deprotonation of carbonyl () compounds. Rarely isolated, they are widely used as reagents in the synthesis of organic compounds. Enolate anions are electronically related to allyl anions. The anionic charge is delocalized over the oxygen and the two carbon sites. Thus they have the character of both an alkoxide and a carbanion. Although they are often drawn as being simple salts, in fact they adopt complicated structures often featuring aggregates.
Sulfur dioxideSulfur dioxide (IUPAC-recommended spelling) or sulphur dioxide (traditional Commonwealth English) is the chemical compound with the formula SO2. It is a toxic gas responsible for the odor of burnt matches. It is released naturally by volcanic activity and is produced as a by-product of copper extraction and the burning of sulfur-bearing fossil fuels. SO2 is a bent molecule with C2v symmetry point group. A valence bond theory approach considering just s and p orbitals would describe the bonding in terms of resonance between two resonance structures.
Sonata formSonata form (also sonata-allegro form or first movement form) is a musical structure generally consisting of three main sections: an exposition, a development, and a recapitulation. It has been used widely since the middle of the 18th century (the early Classical period). While it is typically used in the first movement of multi-movement pieces, it is sometimes used in subsequent movements as well—particularly the final movement.
Sulfur trioxideSulfur trioxide (alternative spelling sulphur trioxide, also known as nisso sulfan) is the chemical compound with the formula SO3. It has been described as "unquestionably the most important economically" sulfur oxide. It is prepared on an industrial scale as a precursor to sulfuric acid. Sulfur trioxide exists in several forms - gaseous monomer, crystalline trimer, and solid polymer. Sulfur trioxide is a solid at just below room temperature with a relatively narrow liquid range. Gaseous SO3 is the primary precursor to acid rain.
Musical formIn music, form refers to the structure of a musical composition or performance. In his book, Worlds of Music, Jeff Todd Titon suggests that a number of organizational elements may determine the formal structure of a piece of music, such as "the arrangement of musical units of rhythm, melody, and/or harmony that show repetition or variation, the arrangement of the instruments (as in the order of solos in a jazz or bluegrass performance), or the way a symphonic piece is orchestrated", among other factors.
Chlorine dioxideChlorine dioxide is a chemical compound with the formula ClO2 that exists as yellowish-green gas above 11 °C, a reddish-brown liquid between 11 °C and −59 °C, and as bright orange crystals below −59 °C. It is usually handled as an aqueous solution. It is commonly used as a bleach. More recent developments have extended its applications in food processing and as a disinfectant. The molecule ClO2 has an odd number of valence electrons, and therefore, it is a paramagnetic radical.
