Category of modulesIn algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).
Tetrakis(triphenylphosphine)palladium(0)Tetrakis(triphenylphosphine)palladium(0) (sometimes called quatrotriphenylphosphine palladium) is the chemical compound [Pd(P(C6H5)3)4], often abbreviated Pd(PPh3)4, or rarely PdP4. It is a bright yellow crystalline solid that becomes brown upon decomposition in air. The four phosphorus atoms are at the corners of a tetrahedron surrounding the palladium(0) center. This structure is typical for four-coordinate 18 e− complexes. The corresponding complexes Ni(PPh3)4 and Pt(PPh3)4 are also well known.
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.
Compact closed categoryIn , a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the having finite-dimensional vector spaces as s and linear maps as s, with tensor product as the structure. Another example is , the category having sets as objects and relations as morphisms, with .
Hermitian matrixIn mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: or in matrix form: Hermitian matrices can be understood as the complex extension of real symmetric matrices.
Organopalladium chemistryOrganopalladium chemistry is a branch of organometallic chemistry that deals with organic palladium compounds and their reactions. Palladium is often used as a catalyst in the reduction of alkenes and alkynes with hydrogen. This process involves the formation of a palladium-carbon covalent bond. Palladium is also prominent in carbon-carbon coupling reactions, as demonstrated in tandem reactions. 1873 - A. N. Zaitsev reports reduction of benzophenone over palladium with hydrogen.
Closed monoidal categoryIn mathematics, especially in , a closed monoidal category (or a monoidal closed category) is a that is both a and a in such a way that the structures are compatible. A classic example is the , Set, where the monoidal product of sets and is the usual cartesian product , and the internal Hom is the set of functions from to . A non- example is the , K-Vect, over a field . Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another.
Monoid (category theory)In , a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a (C, ⊗, I) is an M together with two morphisms μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the Cop.
Schur decompositionIn the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. The Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as where Q is a unitary matrix (so that its inverse Q−1 is also the conjugate transpose Q* of Q), and U is an upper triangular matrix, which is called a Schur form of A.
Euclidean vectorIn mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".
Spectrum (functional analysis)In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, is the identity operator. By the closed graph theorem, is in the spectrum if and only if the bounded operator is non-bijective on .
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
