Algebraic groupIn mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.
Classical groupIn mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups.
Chemical synthesisIn chemistry, chemical synthesis (chemical combination) is the artificial execution of chemical reactions to obtain one or several products. This occurs by physical and chemical manipulations usually involving one or more reactions. In modern laboratory uses, the process is reproducible and reliable. A chemical synthesis involves one or more compounds (known as reagents or reactants) that will experience a transformation when subjected to certain conditions. Various reaction types can be applied to formulate a desired product.
Newton's methodIn numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then is a better approximation of the root than x0.
Succinic acidSuccinic acid (səkˈsɪnᵻk) is a dicarboxylic acid with the chemical formula (CH2)2(CO2H)2. In living organisms, succinic acid takes the form of an anion, succinate, which has multiple biological roles as a metabolic intermediate being converted into fumarate by the enzyme succinate dehydrogenase in complex 2 of the electron transport chain which is involved in making ATP, and as a signaling molecule reflecting the cellular metabolic state. Succinate is generated in mitochondria via the tricarboxylic acid cycle (TCA).
Group of Lie typeIn mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
PyrrolePyrrole is a heterocyclic, aromatic, organic compound, a five-membered ring with the formula . It is a colorless volatile liquid that darkens readily upon exposure to air. Substituted derivatives are also called pyrroles, e.g., N-methylpyrrole, . Porphobilinogen, a trisubstituted pyrrole, is the biosynthetic precursor to many natural products such as heme. Pyrroles are components of more complex macrocycles, including the porphyrinogens and products derived therefrom, including porphyrins of heme, the chlorins, bacteriochlorins, and chlorophylls.
Propionic acidPropionic acid (proʊpiˈɒnɪk, from the Greek words πρῶτος : prōtos, meaning "first", and πίων : píōn, meaning "fat"; also known as propanoic acid) is a naturally occurring carboxylic acid with chemical formula CH3CH2CO2H. It is a liquid with a pungent and unpleasant smell somewhat resembling body odor. The anion CH3CH2CO2- as well as the salts and esters of propionic acid are known as propionates or propanoates. Propionic acid was first described in 1844 by Johann Gottlieb, who found it among the degradation products of sugar.
Total synthesisTotal synthesis is the complete chemical synthesis of a complex molecule, often a natural product, from simple, commercially-available precursors. It usually refers to a process not involving the aid of biological processes, which distinguishes it from semisynthesis. Syntheses may sometimes conclude at a precursor with further known synthetic pathways to a target molecule, in which case it is known as a formal synthesis. Total synthesis target molecules can be natural products, medicinally-important active ingredients, known intermediates, or molecules of theoretical interest.
Α-Ketoglutaric acidα-Ketoglutaric acid (2-oxoglutaric acid) is one of two ketone derivatives of glutaric acid. The term "ketoglutaric acid," when not further qualified, almost always refers to the alpha variant. β-Ketoglutaric acid varies only by the position of the ketone functional group, and is much less common. Its carboxylate, α-ketoglutarate (also called 2-oxoglutarate), is an important biological compound. It is the keto acid produced by deamination of glutamate, and is an intermediate in the Krebs cycle.
