Permutation groupIn mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by Sn, and may be called the symmetric group on n letters.
Cyclic permutationIn mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has k elements, it may be called a k-cycle. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.
Parity of a permutationIn mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd.
PermutationIn mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1, 2, 3}, namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).
Invariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
Eigenvalue algorithmIn numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors and Generalized eigenvector Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.
Dissociation (chemistry)Dissociation in chemistry is a general process in which molecules (or ionic compounds such as salts, or complexes) separate or split into other things such as atoms, ions, or radicals, usually in a reversible manner. For instance, when an acid dissolves in water, a covalent bond between an electronegative atom and a hydrogen atom is broken by heterolytic fission, which gives a proton (H+) and a negative ion. Dissociation is the opposite of association or recombination.
Eigenvalues and eigenvectorsIn linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Permutation matrixIn mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A.
Acid dissociation constantIn chemistry, an acid dissociation constant (also known as acidity constant, or acid-ionization constant; denoted K_a) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction HA A^- + H^+ known as dissociation in the context of acid–base reactions. The chemical species HA is an acid that dissociates into , the conjugate base of the acid and a hydrogen ion, .
Inverse iterationIn numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics.
Dissociation constantIn chemistry, biochemistry, and pharmacology, a dissociation constant () is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. The dissociation constant is the inverse of the association constant. In the special case of salts, the dissociation constant can also be called an ionization constant.
Coordinate systemIn geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring.
Order and disorderIn physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are: the melting of ice: solid-liquid transition, loss of crystalline order; the demagnetization of iron by heating above the Curie temperature: ferromagnetic-paramagnetic transition, loss of magnetic order.
Power iterationIn mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix , the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, . The algorithm is also known as the Von Mises iteration. Power iteration is a very simple algorithm, but it may converge slowly.
IsomerIn chemistry, isomers are molecules or polyatomic ions with identical molecular formula – that is, same number of atoms of each element – but distinct arrangements of atoms in space. Isomerism refers to the existence or possibility of isomers. Isomers do not necessarily share similar chemical or physical properties. Two main forms of isomerism are structural or constitutional isomerism, in which bonds between the atoms differ; and stereoisomerism or spatial isomerism, in which the bonds are the same but the relative positions of the atoms differ.
Bond-dissociation energyThe bond-dissociation energy (BDE, D0, or DH°) is one measure of the strength of a chemical bond . It can be defined as the standard enthalpy change when is cleaved by homolysis to give fragments A and B, which are usually radical species. The enthalpy change is temperature-dependent, and the bond-dissociation energy is often defined to be the enthalpy change of the homolysis at 0 K (absolute zero), although the enthalpy change at 298 K (standard conditions) is also a frequently encountered parameter.
Conformational isomerismIn chemistry, conformational isomerism is a form of stereoisomerism in which the isomers can be interconverted just by rotations about formally single bonds (refer to figure on single bond rotation). While any two arrangements of atoms in a molecule that differ by rotation about single bonds can be referred to as different conformations, conformations that correspond to local minima on the potential energy surface are specifically called conformational isomers or conformers.
Acid–base reactionAn acid–base reaction is a chemical reaction that occurs between an acid and a base. It can be used to determine pH via titration. Several theoretical frameworks provide alternative conceptions of the reaction mechanisms and their application in solving related problems; these are called the acid–base theories, for example, Brønsted–Lowry acid–base theory. Their importance becomes apparent in analyzing acid–base reactions for gaseous or liquid species, or when acid or base character may be somewhat less apparent.
Chemical bondA chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules, crystals, and other structures. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of electrons as in covalent bonds. The strength of chemical bonds varies considerably; there are "strong bonds" or "primary bonds" such as covalent, ionic and metallic bonds, and "weak bonds" or "secondary bonds" such as dipole–dipole interactions, the London dispersion force, and hydrogen bonding.
