Change of basisIn mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.
Rotations and reflections in two dimensionsIn Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another. A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1. Then reflect P′ to its image P′′ on the other side of line L2. If lines L1 and L2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the intersection of L1 and L2. I.e.
Invertible matrixIn linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
Rotation of axes in two dimensionsIn mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle . A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle .
Hessian matrixIn mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".
Power inverterA power inverter, inverter or invertor is a power electronic device or circuitry that changes direct current (DC) to alternating current (AC). The resulting AC frequency obtained depends on the particular device employed. Inverters do the opposite of rectifiers which were originally large electromechanical devices converting AC to DC. The input voltage, output voltage and frequency, and overall power handling depend on the design of the specific device or circuitry. The inverter does not produce any power; the power is provided by the DC source.
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.
Mortal sinA mortal sin (peccatum mortale), in Catholic theology, is a gravely sinful act which can lead to damnation if a person does not repent of the sin before death. It is alternatively called deadly, grave, and serious. A sin is considered to be "mortal" when its quality is such that it leads to a separation of that person from God's saving grace. Three conditions must together be met for a sin to be mortal: "Mortal sin is sin whose object is grave matter and which is also committed with full knowledge and deliberate consent.
Christian views on sinIn Christianity, 'sin' is an immoral act considered to be a transgression of divine law. The doctrine of sin is central to the Christian faith, since its basic message is about redemption in Christ. Hamartiology, a branch of Christian theology which is the study of sin, describes sin as an act of offence against God by despising his persons and Christian biblical law, and by injuring others. Christian hamartiology is closely related to concepts of natural law, moral theology and Christian ethics.
Eternal sinIn Christian hamartiology, eternal sin, the unforgivable sin, unpardonable sin, or ultimate sin is the sin which will not be forgiven by God. One eternal or unforgivable sin (blasphemy against the Holy Spirit), also known as the sin unto death, is specified in several passages of the Synoptic Gospels, including Mark 3:28–29, Matthew 12:31–32, and Luke 12:10, as well as other New Testament passages including Hebrews 6:4–6, Hebrews 10:26–31, and 1 John 5:16.
Jyā, koti-jyā and utkrama-jyāJyā, koṭi-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematicians and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and koti-jyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to the Sanskrit words jyā and koti-jyā.
Holomorphic functionIn mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.
