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.
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 .
Chow varietyIn mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety is the fine moduli variety parametrizing all effective algebraic cycles of dimension and degree in . The Chow variety may be constructed via a Chow embedding into a sufficiently large projective space.
Organic synthesisOrganic synthesis is a special branch of chemical synthesis and is concerned with the intentional construction of organic compounds. Organic molecules are often more complex than inorganic compounds, and their synthesis has developed into one of the most important branches of organic chemistry. There are several main areas of research within the general area of organic synthesis: total synthesis, semisynthesis, and methodology.
Generalized flag varietyIn mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F.
Medicinal chemistryMedicinal or pharmaceutical chemistry is a scientific discipline at the intersection of chemistry and pharmacy involved with designing and developing pharmaceutical drugs. Medicinal chemistry involves the identification, synthesis and development of new chemical entities suitable for therapeutic use. It also includes the study of existing drugs, their biological properties, and their quantitative structure-activity relationships (QSAR).
Danishefsky Taxol total synthesisThe Danishefsky Taxol total synthesis in organic chemistry is an important third Taxol synthesis published by the group of Samuel Danishefsky in 1996 two years after the first two efforts described in the Holton Taxol total synthesis and the Nicolaou Taxol total synthesis. Combined they provide a good insight in the application of organic chemistry in total synthesis. Danishefsky's route to Taxol has many similarities with that of Nicolaou. Both are examples of convergent synthesis with a coupling of the A and the C ring from two precursors.
Nicolaou Taxol total synthesisThe Nicolaou Taxol total synthesis, published by K. C. Nicolaou and his group in 1994 concerns the total synthesis of taxol. Taxol is an important drug in the treatment of cancer but also expensive because the compound is harvested from a scarce resource, namely the pacific yew. This synthetic route to taxol is one of several; other groups have presented their own solutions, notably the group of Holton with a linear synthesis starting from borneol, the Samuel Danishefsky group starting from the Wieland-Miescher ketone and the Wender group from pinene.
Functional groupIn organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the rest of the molecule's composition. This enables systematic prediction of chemical reactions and behavior of chemical compounds and the design of chemical synthesis. The reactivity of a functional group can be modified by other functional groups nearby.
Affine varietyIn algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime. Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).
Ring (chemistry)In chemistry, a ring is an ambiguous term referring either to a simple cycle of atoms and bonds in a molecule or to a connected set of atoms and bonds in which every atom and bond is a member of a cycle (also called a ring system). A ring system that is a simple cycle is called a monocycle or simple ring, and one that is not a simple cycle is called a polycycle or polycyclic ring system. A simple ring contains the same number of sigma bonds as atoms, and a polycyclic ring system contains more sigma bonds than atoms.
Blowing upIn mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup.
Cyclic compoundA cyclic compound (or ring compound) is a term for a compound in the field of chemistry in which one or more series of atoms in the compound is connected to form a ring. Rings may vary in size from three to many atoms, and include examples where all the atoms are carbon (i.e., are carbocycles), none of the atoms are carbon (inorganic cyclic compounds), or where both carbon and non-carbon atoms are present (heterocyclic compounds with rings containing both carbon and non-carbon).
Yield (chemistry)In chemistry, yield, also referred to as reaction yield, is a measure of the quantity of moles of a product formed in relation to the reactant consumed, obtained in a chemical reaction, usually expressed as a percentage. Yield is one of the primary factors that scientists must consider in organic and inorganic chemical synthesis processes. In chemical reaction engineering, "yield", "conversion" and "selectivity" are terms used to describe ratios of how much of a reactant was consumed (conversion), how much desired product was formed (yield) in relation to the undesired product (selectivity), represented as X, Y, and S.
CarbeneIn organic chemistry, a carbene is a molecule containing a neutral carbon atom with a valence of two and two unshared valence electrons. The general formula is or where the R represents substituents or hydrogen atoms. The term "carbene" may also refer to the specific compound , also called methylene, the parent hydride from which all other carbene compounds are formally derived. Carbenes are classified as either singlets or triplets, depending upon their electronic structure.
CyclopropanationIn organic chemistry, cyclopropanation refers to any chemical process which generates cyclopropane () rings. It is an important process in modern chemistry as many useful compounds bear this motif; for example pyrethroid insecticides and a number of quinolone antibiotics (ciprofloxacin, sparfloxacin, etc.). However, the high ring strain present in cyclopropanes makes them challenging to produce and generally requires the use of highly reactive species, such as carbenes, ylids and carbanions.
