Mutation (algebra)

In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original.

Definitions

Let A be an algebra over a field F with multiplication (not assumed to be associative) denoted by juxtaposition. For an element a of A, define the left a-homotope to be the algebra with multiplication

Similarly define the left (a,b) mutation

Right homotope and mutation are defined analogously. Since the right (p,q) mutation of A is the left (−q, −p) mutation of the opposite algebra to A, it suffices to study left mutations.

If A is a unital algebra and a is invertible, we refer to the isotope by a.

Properties

Mutation (disambiguation)

A mutation is a change in the sequence of an organism's genetic material.

Mutation may also refer to:

Literature

Music

Sciences

Consonant mutation

Consonant mutation is change in a consonant in a word according to its morphological or syntactic environment.

Mutation phenomena occur in languages around the world. A prototypical example of consonant mutation is the initial consonant mutation of all modern Celtic languages. Initial consonant mutation is also found in Indonesian or Malay, in Southern Paiute and in several West African languages such as Fula. The Nilotic language Dholuo, spoken in Kenya, shows mutation of stem-final consonants, as does English to a small extent. Mutation of initial, medial, and final consonants is found in Modern Hebrew. Japanese exhibits word medial consonant mutation involving voicing, rendaku, in many compounds.

Similar sound changes

Initial consonant mutation must not be confused with sandhi, which can refer to word-initial alternations triggered by their phonological environment, unlike mutations, which are triggered by their morphosyntactic environment. Some examples of word-initial sandhi are listed below.

Mutation (genetic algorithm)

Mutation is a genetic operator used to maintain genetic diversity from one generation of a population of genetic algorithm chromosomes to the next. It is analogous to biological mutation. Mutation alters one or more gene values in a chromosome from its initial state. In mutation, the solution may change entirely from the previous solution. Hence GA can come to better solution by using mutation. Mutation occurs during evolution according to a user-definable mutation probability. This probability should be set low. If it is set too high, the search will turn into a primitive random search.

The classic example of a mutation operator involves a probability that an arbitrary bit in a genetic sequence will be changed from its original state. A common method of implementing the mutation operator involves generating a random variable for each bit in a sequence. This random variable tells whether or not a particular bit will be modified. This mutation procedure, based on the biological point mutation, is called single point mutation. Other types are inversion and floating point mutation. When the gene encoding is restrictive as in permutation problems, mutations are swaps, inversions, and scrambles.

*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:

for all x, y in A.

This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.

Algebra

Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).

†-algebra

A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.

References