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Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z- module homomorphism. Since an abelian group is a \Z-module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwise) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the '' variety of groups''. History Before the nineteenth century, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of ''commutative algebra'', a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul E
Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) * Paul (surname), a list of people People Christianity *Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Christian missionary and writer * Pope Paul (other), multiple Popes of the Roman Catholic Church * Saint Paul (other), multiple other people and locations named "Saint Paul" Roman and Byzantine empire * Lucius Aemilius Paullus Macedonicus (c. 229 BC – 160 BC), Roman general * Julius Paulus Prudentissimus (), Roman jurist * Paulus Catena (died 362), Roman notary *Paulus Alexandrinus (4th century), Hellenistic astrologer * Paul of Aegina or Paulus Aegineta (625–690), Greek surgeon Royals *Paul I of Russia (1754–1801), Tsar of Russia * Paul of Greece (1901–1964), King of Greece Other people *Paul the Deacon or Paulus Diaconus (c. 720 – c. 799), Italian Benedictine monk *Paul (father of Maurice), the father of Mauric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger C
Roger is a given name, usually masculine, and a surname. The given name is derived from the Old French personal names ' and '. These names are of Germanic origin, derived from the elements ', ''χrōþi'' ("fame", "renown", "honour") and ', ' ("spear", "lance") (Hrōþigēraz). The name was introduced into England by the Normans. In Normandy, the Frankish name had been reinforced by the Old Norse cognate '. The name introduced into England replaced the Old English cognate '. ''Roger'' became a very common given name during the Middle Ages. A variant form of the given name ''Roger'' that is closer to the name's origin is ''Rodger''. Slang and other uses Roger is also a short version of the term "Jolly Roger", which refers to a black flag with a white skull and crossbones, formerly used by sea pirates since as early as 1723. From up to , Roger was slang for the word "penis". In ''Under Milk Wood'', Dylan Thomas writes "jolly, rodgered" suggesting both the sexual double enten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Homogeneity
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain and codomain are vector spaces over a field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of functions of several real variables and real v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Over The Rational Numbers
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain and codomain are vector spaces over a field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of functions of several real variables and real v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unital Module
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Homomorphism
In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ''R''-''linear map'' if for any ''x'', ''y'' in ''M'' and ''r'' in ''R'', :f(x + y) = f(x) + f(y), :f(rx) = rf(x). In other words, ''f'' is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If ''M'', ''N'' are right ''R''-modules, then the second condition is replaced with :f(xr) = f(x)r. The preimage of the zero element under ''f'' is called the kernel of ''f''. The set of all module homomorphisms from ''M'' to ''N'' is denoted by \operatorname_R(M, N). It is an abelian group (under pointwise addition) but is not necessarily a module unless ''R'' is commutative. The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain and codomain are vector spaces over a field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of functions of several real variables and rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. Pointwise operations Formal definition A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by Commonly, ''o'' and ''O'' are denoted by the same symbol. A similar definition is used for unary operations ''o'', and for operations of other arity. Examples \begin (f+g)(x) & = f(x)+g(x) & \text \\ (f\cdot g)(x) & = f(x) \cdot g(x) & \text \\ (\lambda \cdot f)(x) & = \lambda \cdot f(x) & \text \end where f, g : X \to R. See also pointwise product, and scala ... [...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] |
