|
Puiseux Expansion
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 a 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 coeffici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Formal Power 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, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrametric Space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. Formal definition An ultrametric on a set is a real-valued function :d\colon M \times M \rightarrow \mathbb (where denote the real numbers), such that for all : # ; # (''symmetry''); # ; # if then ; # (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a metric). If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on . In the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation Group
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of 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 : which satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(''a'') � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the ''additive group'' of the integers, of a vector space and of a ring. This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...s. References {{DEFAULTSORT:Additive group Algebraic structures Group theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valued Field
Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western sources * Social imaginary is the set of values, institutions, laws, and symbols common to a particular social group Economics * Value (economics), a measure of the benefit that may be gained from goods or service ** Theory of value (economics), the study of the concept of economic value ** Value (marketing), the difference between a customer's evaluation of benefits and costs ** Value investing, an investment paradigm * Values (heritage), the measure by which the cultural significance of heritage items is assessed * Present value * Present value of benefits Business * Business value * Customer value proposition * Employee value proposition * Value (marketing) * Value proposition Other uses ... [...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 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 : which satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(''a'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, ''x'' is unique up to ''R'' means that all objects ''x'' under consideration are in the same equivalence class with respect to the relation ''R''. Moreover, the equivalence relation ''R'' is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation ''R'' that relates two lists if one can be obtained by reordering (permutation) from the other. As another example, the sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct System
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from a small filtered category ''I'' to ''C''. The dual concept is the pro-completion, Pro(''C''). Definitions Filtered categories Direct systems depend on the notion of ''filtered categories''. For example, the category N, whose objects are natural numbers, and with exactly one morphism from ''n'' to ''m'' whenever n \le m, is a filtered category. Direct systems A ''direct system'' or an ''ind-object'' in a category ''C'' is defined to be a functor :F : I \to C from a small filtered category ''I'' to ''C''. For example, if ''I'' is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in ''C'' together with morphisms as displayed. The ind-completion Ind-objects in ''C'' form a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Homomorphism
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ring (''F'',+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field ''F'' form an abelian group under multiplication; this group is typically denoted by ''F''×; The ring of polynomials in the variable ''x'' with coefficients in ''F'' is denoted by ''F'' 'x'' Basic definitions ; Characteristic : The ''characteristic'' of the field ''F'' is the smallest positive integer ''n'' such that ''n''·1 = 0; here ''n''·1 stands for ''n'' summands 1 + 1 + 1 + ... + 1. If no such ''n'' exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the ratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms ( group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.) Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are also a special case of limits in category theory. Formal definition We will first give the definition for algebraic structures like groups and modules, and then the general defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
