Orthogonal basisIn mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. Any orthogonal basis can be used to define a system of orthogonal coordinates Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.
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.
Basis set (chemistry)In theoretical and computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate resolution of the identity: the orbitals are expanded within the basis set as a linear combination of the basis functions , where the expansion coefficients are given by .
Pericyclic reactionIn organic chemistry, a pericyclic reaction is the type of organic reaction wherein the transition state of the molecule has a cyclic geometry, the reaction progresses in a concerted fashion, and the bond orbitals involved in the reaction overlap in a continuous cycle at the transition state. Pericyclic reactions stand in contrast to linear reactions, encompassing most organic transformations and proceeding through an acyclic transition state, on the one hand and coarctate reactions, which proceed through a doubly cyclic, concerted transition state on the other hand.
Organosulfur chemistryOrganosulfur chemistry is the study of the properties and synthesis of organosulfur compounds, which are organic compounds that contain sulfur. They are often associated with foul odors, but many of the sweetest compounds known are organosulfur derivatives, e.g., saccharin. Nature abounds with organosulfur compounds—sulfur is vital for life. Of the 20 common amino acids, two (cysteine and methionine) are organosulfur compounds, and the antibiotics penicillin and sulfa drugs both contain sulfur.
Metal carbonylMetal carbonyls are coordination complexes of transition metals with carbon monoxide ligands. Metal carbonyls are useful in organic synthesis and as catalysts or catalyst precursors in homogeneous catalysis, such as hydroformylation and Reppe chemistry. In the Mond process, nickel tetracarbonyl is used to produce pure nickel. In organometallic chemistry, metal carbonyls serve as precursors for the preparation of other organometallic complexes.
Ethanol fermentationEthanol fermentation, also called alcoholic fermentation, is a biological process which converts sugars such as glucose, fructose, and sucrose into cellular energy, producing ethanol and carbon dioxide as by-products. Because yeasts perform this conversion in the absence of oxygen, alcoholic fermentation is considered an anaerobic process. It also takes place in some species of fish (including goldfish and carp) where (along with lactic acid fermentation) it provides energy when oxygen is scarce.
Standard basisIn mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors Similarly, the standard basis for the three-dimensional space is formed by vectors Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction.
NitrosoIn organic chemistry, nitroso refers to a functional group in which the nitric oxide () group is attached to an organic moiety. As such, various nitroso groups can be categorized as C-nitroso compounds (e.g., nitrosoalkanes; ), S-nitroso compounds (nitrosothiols; ), N-nitroso compounds (e.g., nitrosamines, ), and O-nitroso compounds (alkyl nitrites; ). Nitroso compounds can be prepared by the reduction of nitro compounds or by the oxidation of hydroxylamines. Ortho-nitrosophenols may be produced by the Baudisch reaction.
Gröbner basisIn mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K[x1, ..., xn] over a field K. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite.
