Conjugate gradient methodIn mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.
Higher-order functionIn mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), returns a function as its result. All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function.
Fatty acidIn chemistry, particularly in biochemistry, a fatty acid is a carboxylic acid with an aliphatic chain, which is either saturated or unsaturated. Most naturally occurring fatty acids have an unbranched chain of an even number of carbon atoms, from 4 to 28. Fatty acids are a major component of the lipids (up to 70% by weight) in some species such as microalgae but in some other organisms are not found in their standalone form, but instead exist as three main classes of esters: triglycerides, phospholipids, and cholesteryl esters.
Hamiltonian mechanicsHamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
Absorbing elementIn mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid.
Identity elementIn mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Plane curveIn mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form for some specific function f.
Variation of parametersIn mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.
Solution (chemistry)In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent and solute particles are greater than the attractive forces holding the solute particles together, the solvent particles pull the solute particles apart and surround them. These surrounded solute particles then move away from the solid solute and out into the solution.
Four-gradientIn differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors. This article uses the (+ − − −) metric signature. SR and GR are abbreviations for special relativity and general relativity respectively. indicates the speed of light in vacuum. is the flat spacetime metric of SR.
