Beta functionIn mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral for complex number inputs such that . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta. The beta function is symmetric, meaning that for all inputs and .
Alpha-olefinIn organic chemistry, alpha-olefins (or α-olefins) are a family of organic compounds which are alkenes (also known as olefins) with a chemical formula , distinguished by having a double bond at the primary or alpha (α) position. This location of a double bond enhances the reactivity of the compound and makes it useful for a number of applications. There are two types of alpha-olefins, branched and linear (or normal).
DehydrogenationIn chemistry, dehydrogenation is a chemical reaction that involves the removal of hydrogen, usually from an organic molecule. It is the reverse of hydrogenation. Dehydrogenation is important, both as a useful reaction and a serious problem. At its simplest, it is useful way of converting alkanes, which are relatively inert and thus low-valued, to olefins, which are reactive and thus more valuable. Alkenes are precursors to aldehydes (), alcohols (), polymers, and aromatics.
AlkyneAcetylene Propyne 1-Butyne In organic chemistry, an alkyne is an unsaturated hydrocarbon containing at least one carbon—carbon triple bond. The simplest acyclic alkynes with only one triple bond and no other functional groups form a homologous series with the general chemical formula . Alkynes are traditionally known as acetylenes, although the name acetylene also refers specifically to , known formally as ethyne using IUPAC nomenclature. Like other hydrocarbons, alkynes are generally hydrophobic.
Organosilicon chemistryOrganosilicon chemistry is the study of organometallic compounds containing carbon–silicon bonds, to which they are called organosilicon compounds. Most organosilicon compounds are similar to the ordinary organic compounds, being colourless, flammable, hydrophobic, and stable to air. Silicon carbide is an inorganic compound. Organometallic chemistry In 1863 Charles Friedel and James Crafts made the first organochlorosilane compound. The same year they also described a «polysilicic acid ether» in the preparation of ethyl- and methyl-o-silicic acid.
Ene reactionIn organic chemistry, the ene reaction (also known as the Alder-ene reaction by its discoverer Kurt Alder in 1943) is a chemical reaction between an alkene with an allylic hydrogen (the ene) and a compound containing a multiple bond (the enophile), in order to form a new σ-bond with migration of the ene double bond and 1,5 hydrogen shift. The product is a substituted alkene with the double bond shifted to the allylic position. This transformation is a group transfer pericyclic reaction, and therefore, usually requires highly activated substrates and/or high temperatures.
CycloadditionIn organic chemistry, a cycloaddition is a chemical reaction in which "two or more unsaturated molecules (or parts of the same molecule) combine with the formation of a cyclic adduct in which there is a net reduction of the bond multiplicity". The resulting reaction is a cyclization reaction. Many but not all cycloadditions are concerted and thus pericyclic. Nonconcerted cycloadditions are not pericyclic. As a class of addition reaction, cycloadditions permit carbon–carbon bond formation without the use of a nucleophile or electrophile.
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.
Beta distributionIn probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
Free-radical additionIn organic chemistry, free-radical addition is an addition reaction which involves free radicals. The addition may occur between a radical and a non-radical, or between two radicals. The basic steps with examples of the free-radical addition (also known as radical chain mechanism) are: Initiation by a radical initiator: A radical is created from a non-radical precursor.
