Inverse elementIn mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers. Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element such that x * e = x and e * y = y for all x and y for which the left-hand sides are defined.
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.
Smooth schemeIn algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n.
Solar inverterA solar inverter or photovoltaic (PV) inverter is a type of power inverter which converts the variable direct current (DC) output of a photovoltaic solar panel into a utility frequency alternating current (AC) that can be fed into a commercial electrical grid or used by a local, off-grid electrical network. It is a critical balance of system (BOS)–component in a photovoltaic system, allowing the use of ordinary AC-powered equipment.
Continuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
Parallelizable manifoldIn mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectors provide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .
Dual spaceIn mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.
Matrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.
Local diffeomorphismIn mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Let and be differentiable manifolds. A function is a local diffeomorphism, if for each point there exists an open set containing such that is open in and is a diffeomorphism.
H-droppingH-dropping or aitch-dropping is the deletion of the voiceless glottal fricative or "H-sound", [h]. The phenomenon is common in many dialects of English, and is also found in certain other languages, either as a purely historical development or as a contemporary difference between dialects. Although common in most regions of England and in some other English-speaking countries, and linguistically speaking a neutral evolution in languages, H-dropping is often stigmatized as a sign of careless or uneducated speech.
Kähler manifoldIn mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.
Matrix ringIn abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication . The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring Mn(R) is an associative algebra over R, and may be called a matrix algebra.
