Formulation of the problem of constructing a complete involutive collection of functions
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Panova, E. Vallejo, Kronecker products, characters, partitions, and the tensor square conjectures , arXiv On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers. Exponentiation with integer exponents can be defined in any multiplicative monoid. Exponentiation is defined inductively by:. Monoids include many structures of importance in mathematics, including groups and rings under multiplication , with more specific examples of the latter being matrix rings and fields.
If A is a square matrix, then the product of A with itself n times is called the matrix power. Matrix powers appear often in the context of discrete dynamical systems , where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system. So computing matrix powers is equivalent to solving the evolution of the dynamical system.
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In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors. Apart from matrices, more general linear operators can also be exponentiated. The n -th power of the differentiation operator is the n -th derivative:. These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents.
This is the starting point of the mathematical theory of semigroups. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral , is one of the basic operations of the fractional calculus. A field is an algebraic structure in which multiplication, addition, subtraction, and division are all well-defined and satisfy their familiar properties.
The real numbers, for example, form a field, as do the complex numbers and rational numbers. Unlike these familiar examples of fields, which are all infinite sets , some fields have only finitely many elements. Exponentiation in finite fields has applications in public key cryptography. For example, the Diffie—Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm the inverse of exponentiation is computationally expensive.
This prime number is called the characteristic of the field. This is called the Frobenius automorphism of F. The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield. Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.
Let X be a set with a power-associative binary operation which is written multiplicatively. Then x n is defined for any element x of X and any nonzero natural number n as the product of n copies of x , which is recursively defined by. If the operation has a two-sided identity element 1, then x 0 is defined to be equal to 1 for any x. If the operation also has two-sided inverses and is associative, then the magma is a group. If the multiplication operation is commutative as for instance in abelian groups , then the following holds:. If the binary operation is written additively, as it often is for abelian groups , then "exponentiation is repeated multiplication" can be reinterpreted as " multiplication is repeated addition ".
Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.presnestighpo.ga
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When there are several power-associative binary operations defined on a set, any of which might be iterated, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Superscript notation is also used, especially in group theory , to indicate conjugation.
Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role. If n is a natural number and A is an arbitrary set, the expression A n is often used to denote the set of ordered n -tuples of elements of A.
This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra , it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of. We can again replace the set N with a cardinal number n to get V n , although without choosing a specific standard set with cardinality n , this is defined only up to isomorphism. Taking V to be the field R of real numbers thought of as a vector space over itself and n to be some natural number , we get the vector space that is most commonly studied in linear algebra, the real vector space R n.
If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Since multiple Cartesian products produce an n - tuple , which can be represented by a function on a set of appropriate cardinality, S N becomes simply the set of all functions from N to S in this case:. In a Cartesian closed category , the exponential operation can be used to raise an arbitrary object to the power of another object.
This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 0 0 is isomorphic to any terminal object 1. In set theory , there are exponential operations for cardinal and ordinal numbers. Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration.