|
Min-plus Semiring
In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical semiring has various applications (see tropical analysis), and forms the basis of tropical geometry. The name ''tropical'' is a reference to the Hungarian-born computer scientist Imre Simon, so named because he lived and worked in Brazil. Definition The ' (or or ) is the semiring (\mathbb \cup \, \oplus, \otimes), with the operations: : x \oplus y = \min\, : x \otimes y = x + y. The operations \oplus and \otimes are referred to as ''tropical addition'' and ''tropical multiplication'' respectively. The identity element for \oplus is +\infty, and the identity element for \otimes is 0. Similarly, the ' (or or or ) is the semiring (\mathbb \cup \, \oplus, \otimes), with operations: : x \oplus y = \max\, : x \otimes y = x + y. The ident ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent Analysis
In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A \oplus A = A. References * {{mathanalysis-stub Idempotent analysis, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotence
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Puiseux Series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + x^ + \cdots \end is a Puiseux series in the indeterminate . Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.Puiseux (1850, 1851) The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator , a Puiseux series becomes a Laurent series in an th root of the indeterminate. For example, the example above is a Laurent series in x^. Because a complex number has th roots, a convergent Puiseux series typically defines functions in a neighborhood of . Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation P(x,y)=0 with complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Laurent Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Valuation
In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides . It is denoted \nu_p(n). Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of n. The P-adic number, -adic valuation is a Valuation (algebra), valuation and gives rise to an analogue of the usual absolute value. Whereas the Complete metric space, completion of the rational numbers with respect to the usual absolute value results in the real numbers \mathbb, the completion of the rational numbers with respect to the p-adic absolute value results in the p-adic number, numbers \mathbb_p. Definition and properties Let be a prime number. Integers The -adic valuation of an integer n is defined to be : \nu_p(n)= \begin \mathrm\ & \text n \neq 0\\ \infty & \text n=0, \end where \mathbb_0 denotes the set of natural numbers (including zero) and m \mid n denotes divisor, divisibility of n by m. In particular, \nu_p is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valued Field
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : that satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (algebra)
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : that satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a module or vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a commutative ring ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggestin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log Semiring
In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base for the exponent and logarithm ( is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base is equivalent to using a negative sign and using the inverse . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggestin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
