Linear mapIn mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .
Inner product spaceIn mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates.
Acyl chlorideIn organic chemistry, an acyl chloride (or acid chloride) is an organic compound with the functional group . Their formula is usually written , where R is a side chain. They are reactive derivatives of carboxylic acids (). A specific example of an acyl chloride is acetyl chloride, . Acyl chlorides are the most important subset of acyl halides. Where the acyl chloride moiety takes priority, acyl chlorides are named by taking the name of the parent carboxylic acid, and substituting -yl chloride for -ic acid.
Catalytic cycleIn chemistry, a catalytic cycle is a multistep reaction mechanism that involves a catalyst. The catalytic cycle is the main method for describing the role of catalysts in biochemistry, organometallic chemistry, bioinorganic chemistry, materials science, etc. Since catalysts are regenerated, catalytic cycles are usually written as a sequence of chemical reactions in the form of a loop. In such loops, the initial step entails binding of one or more reactants by the catalyst, and the final step is the release of the product and regeneration of the catalyst.
Water resourcesWater resources are natural resources of water that are potentially useful for humans, for example as a source of drinking water supply or irrigation water. 97% of the water on Earth is salt water and only three percent is fresh water; slightly over two-thirds of this is frozen in glaciers and polar ice caps. The remaining unfrozen freshwater is found mainly as groundwater, with only a small fraction present above ground or in the air. Natural sources of fresh water include surface water, under river flow, groundwater and frozen water.
Cross-coupling reactionIn organic chemistry, a cross-coupling reaction is a reaction where two different fragments are joined. Cross-couplings are a subset of the more general coupling reactions. Often cross-coupling reactions require metal catalysts. One important reaction type is this: (R, R' = organic fragments, usually aryle; M = main group center such as Li or MgX; X = halide) These reactions are used to form carbon–carbon bonds but also carbon-heteroatom bonds. Cross-coupling reaction are a subset of coupling reactions.
Water scarcityWater scarcity (closely related to water stress or water crisis) is the lack of fresh water resources to meet the standard water demand. There are two types of water scarcity namely physical and economic water scarcity. Physical water scarcity is where there is not enough water to meet all demands, including that needed for ecosystems to function. Arid areas for example Central and West Asia, and North Africa often experience physical water scarcity.
Heck reactionThe Heck reaction (also called the Mizoroki–Heck reaction) is the chemical reaction of an unsaturated halide (or triflate) with an alkene in the presence of a base and a palladium catalyst to form a substituted alkene. It is named after Tsutomu Mizoroki and Richard F. Heck. Heck was awarded the 2010 Nobel Prize in Chemistry, which he shared with Ei-ichi Negishi and Akira Suzuki, for the discovery and development of this reaction.
Linear independenceIn the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.
Catalytic reformingCatalytic reforming is a chemical process used to convert petroleum refinery naphthas distilled from crude oil (typically having low octane ratings) into high-octane liquid products called reformates, which are premium blending stocks for high-octane gasoline. The process converts low-octane linear hydrocarbons (paraffins) into branched alkanes (isoparaffins) and cyclic naphthenes, which are then partially dehydrogenated to produce high-octane aromatic hydrocarbons.
