Multivariate normal distributionIn probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.
Complex normal distributionIn probability theory, the family of complex normal distributions, denoted or , characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix , and the relation matrix . The standard complex normal is the univariate distribution with , , and . An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: and .
Linear equationIn mathematics, a linear equation is an equation that may be put in the form where are the variables (or unknowns), and are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients are required to not all be zero. Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken.
Differentiable functionIn mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists.
Inflection pointIn differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.
Schrödinger equationThe Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. The equation is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics.
Partial differential equationIn mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.
Linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that where is the remainder term.
Diophantine equationIn mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations.
OrthogonalityIn mathematics, orthogonality is the generalization of the geometric notion of perpendicularity. Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. The word comes from the Ancient Greek ὀρθός (), meaning "upright", and γωνία (), meaning "angle". The Ancient Greek ὀρθογώνιον () and Classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle.
CurvatureIn mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point.
TetrahydrofuranTetrahydrofuran (THF), or oxolane, is an organic compound with the formula (CH2)4O. The compound is classified as heterocyclic compound, specifically a cyclic ether. It is a colorless, water-miscible organic liquid with low viscosity. It is mainly used as a precursor to polymers. Being polar and having a wide liquid range, THF is a versatile solvent. About 200,000 tonnes of tetrahydrofuran are produced annually. The most widely used industrial process involves the acid-catalyzed dehydration of 1,4-butanediol.
