isomorphism - Article Index

Encyclopedia - Bijection: Encyclopedia Ii - Bijection - Composition And Inverses
A function f is bijective if and only if its inverse relation f-1 is a function. In that case, f-1 is a bijection. The composition (mathe...   » Read the article

Encyclopedia - Cartan Connection Applications: Encyclopedia - Cartan Connection Applications
This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. Cartan connection a...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia - Category Mathematics
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories...   » Read the article

Encyclopedia - Algebraic Number: Encyclopedia - Algebraic Number
In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a...   » Read the article

Encyclopedia - Algebra Over A Field: Encyclopedia - Algebra Over A Field
In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication...   » Read the article

Encyclopedia - Category Theory: Encyclopedia - Category Theory
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is...   » Read the article

Encyclopedia - Relativism: Encyclopedia - Relativism
Relativism is the view that the meaning and value of human beliefs and behaviors have no absolute reference. Relativists claim that human...   » Read the article

Encyclopedia - Analogy: Encyclopedia - Analogy
Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particu...   » Read the article

Encyclopedia - Complex Number: Encyclopedia - Complex Number
In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of...   » Read the article

Encyclopedia - Coproduct: Encyclopedia - Coproduct
In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definitio...   » Read the article

Encyclopedia - Cyclic Group: Encyclopedia - Cyclic Group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (calle...   » Read the article

Encyclopedia - Uniform Space: Encyclopedia - Uniform Space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with addi...   » Read the article

Encyclopedia - Coordinate Vector: Encyclopedia - Coordinate Vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of number...   » Read the article

Encyclopedia - Functor: Encyclopedia - Functor
In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of...   » Read the article

Encyclopedia - Monoid: Encyclopedia - Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an ident...   » Read the article

Encyclopedia - Axiom: Encyclopedia - Axiom
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all e...   » Read the article

Encyclopedia - Vector Space: Encyclopedia - Vector Space
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. If one considers geomet...   » Read the article

Encyclopedia - Up To: Encyclopedia - Up To
In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some pur...   » Read the article

Encyclopedia - Canonical: Encyclopedia - Canonical
Canonical is an adjective derived from canon. It essentially means "standard", "generally accepted" or "part of the back-story." basic, c...   » Read the article

Encyclopedia - Completeness: Encyclopedia - Completeness
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise i...   » Read the article

Encyclopedia - Measure-preserving Dynamical System: Encyclopedia Ii - Measure-preserving Dynamical System - Definition
It is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system with the following ...   » Read the article

Encyclopedia - Lp Space: Encyclopedia Ii - Lp Space - Motivation
The simplest Lp space is the Euclidean space Rn. The length of a vector is usually given by but this is by no means the only way of...   » Read the article

Encyclopedia - Joy Programming Language: Encyclopedia Ii - Joy Programming Language - Mathematical Purity
One of the most appealing aspects of Joy is this: the meaning function is a homomorphism from the syntactic monoid onto the semantic mono...   » Read the article

Encyclopedia - Equaliser: Encyclopedia Ii - Equaliser - In Category Theory
Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categ...   » Read the article

Encyclopedia - Imaginary Unit: Encyclopedia Ii - Imaginary Unit - I And −i
The above equation actually has two distinct solutions which are additive inverses. More precisely, once a solution i of the equation has...   » Read the article

Encyclopedia - Complex Number: Encyclopedia Ii - Complex Number - Definitions
Complex number - Notation and operations. The set of all complex numbers is usually denoted by C, or in blackboard bold by . It include...   » Read the article

Encyclopedia - Euclidean Plane Isometry: Encyclopedia Ii - Euclidean Plane Isometry - Isometries As Reflection Group
Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a reflection group. Euclid...   » Read the article

Encyclopedia - Logarithm: Encyclopedia Ii - Logarithm - History
Jaina mathematicians in ancient India first conceived of logarithms between 200 BC and 400 CE. They performed a number of operations usin...   » Read the article

Encyclopedia - Isometry: Encyclopedia Ii - Isometry - Definitions
The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often c...   » Read the article

Encyclopedia - Linear Algebra: Encyclopedia Ii - Linear Algebra - Elementary Introduction
Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector, here, is a directed line segment, c...   » Read the article

Encyclopedia - Simplified Molecular Input Line Entry Specification: Encyclopedia Ii - Simplified Molecular Input Line Entry Specification - Examples
Atoms are represented by the standard abbreviation of the chemical elements, in square brackets, such as [Au] for gold. The hydroxide ani...   » Read the article

Encyclopedia - Quaternion: Encyclopedia Ii - Quaternion - Definition
While the complex numbers are obtained by adding the element i to the real numbers which satisfies i2 = −1, the quaternions are obtaine...   » Read the article

Encyclopedia - Lattice Group: Encyclopedia Ii - Lattice Group - Lattices In Two Dimensions: Detailed Discussion
There are five 2D lattice types. Below the wallpaper group of the lattice is given in parentheses; note that a pattern with this lattice ...   » Read the article

Encyclopedia - Real Number: Encyclopedia Ii - Real Number - Properties
Real number - Completeness. The main reason for introducing the reals is that the reals contain all limits. More technically, the reals...   » Read the article

Encyclopedia - Riemann Sphere: Encyclopedia Ii - Riemann Sphere - Geometric Introduction
Define (i.e. the extended complex plane: the complex numbers joined with the point at infinity). The Riemann sphere is based on the tran...   » Read the article

Encyclopedia - Division Algebra: Encyclopedia Ii - Division Algebra - Associative Division Algebras
The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real...   » Read the article

Encyclopedia - Galois Connection: Encyclopedia Ii - Galois Connection - Definition
Suppose (A, ≤) and (B, <=) are two partially ordered sets. A Galois connection between these posets consists of two monotone functio...   » Read the article

Encyclopedia - Product Category Theory: Encyclopedia Ii - Product Category Theory - Discussion
The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of...   » Read the article

Encyclopedia - Pullback: Encyclopedia Ii - Pullback - Pullback On Tensors
Let be a linear map between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback f * T on V ca...   » Read the article

Encyclopedia - Functional Analysis: Encyclopedia Ii - Functional Analysis - Normed Vector Spaces
In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spac...   » Read the article

Encyclopedia - Gestalt Psychology: Encyclopedia Ii - Gestalt Psychology - Origins
Although Max Wertheimer is credited as the founder of the movement, the concept of Gestalt was first introduced in contemporary philosoph...   » Read the article

Encyclopedia - Polarization: Encyclopedia Ii - Polarization - Theory
Polarization - Basics - plane waves. The simplest manifestation of polarization to visualize is that of a plane wave, which is a good a...   » Read the article

Encyclopedia - Picard Group: Encyclopedia Ii - Picard Group - Examples
The name is in honour of Charles-Emile Picard's theories, in particular of divisors on algebraic surfaces. The Picard group of the spectr...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Homomorphism For Beginners
Homomorphism is one of the fundamental concepts in abstract algebra. Because abstract algebra studies sets with operations that generate ...   » Read the article

Encyclopedia - Monoid: Encyclopedia Ii - Monoid - Properties
Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an e...   » Read the article

Encyclopedia - Orientation Mathematics: Encyclopedia Ii - Orientation Mathematics - Definition
Let V be a real vector space and let b1 and b2 be two ordered bases for V. It is a standard result in linear algebra that there exists a ...   » Read the article

Encyclopedia - Dual Space: Encyclopedia Ii - Dual Space - Algebraic Dual Space
Given any vector space V over some field F, we define the dual space V* to be the set of all linear functionals on V, i.e., scalar-valued...   » Read the article

Encyclopedia - Homeomorphism: Encyclopedia Ii - Homeomorphism - Definition
A function f between two topological spaces X and Y is called a homeomorphism if it has the following properties f is a bijection, f is ...   » Read the article

Encyclopedia - List Of Small Groups: Encyclopedia Ii - List Of Small Groups - Glossary
The notations Zn and Dihn have the advantage that point groups in three dimensions Cn and Dn do not have the same notation. There are mor...   » Read the article

Encyclopedia - Axiom: Encyclopedia Ii - Axiom - Mathematics
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Ax...   » Read the article

Encyclopedia - Algebra Over A Field: Encyclopedia Ii - Algebra Over A Field - Kinds Of Algebras And Examples
A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative. Thes...   » Read the article

Encyclopedia - Fundamental Theorem Of Galois Theory: Encyclopedia Ii - Fundamental Theorem Of Galois Theory - Example
The following is the simplest case where the Galois group is not abelian (that is, not commutative). Consider the splitting field K of th...   » Read the article

Encyclopedia - Natural Transformation: Encyclopedia Ii - Natural Transformation - Examples
Natural transformation - A worked example. Statements like "Every group is naturally isomorphic to its opposite group" abound in mode...   » Read the article

Encyclopedia - Vector Space: Encyclopedia Ii - Vector Space - Formal Definition
A vector space over a field F (such as the field of real or of complex numbers) is a set V together with two operations: vector addition...   » Read the article

Encyclopedia - Groupoid: Encyclopedia Ii - Groupoid - Definitions
From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, ...   » Read the article

Encyclopedia - Natural Logarithm: Encyclopedia Ii - Natural Logarithm - Derivative Taylor Series And Complex Arguments
The derivative of the natural logarithm is given by This leads to the Taylor series One may define ln(z) also for all non-zero comp...   » Read the article

Encyclopedia - Null Space: Encyclopedia Ii - Null Space - Example
Consider the function f(x,y) = x − y defined for x and y real numbers, which is linear as f(x + z,y + w) = (x + z) − (y + w) = f(x,y)...   » Read the article

Encyclopedia - Polarization: Encyclopedia Ii - Polarization - Theory
Polarization - Basics - plane waves. The simplest manifestation of polarization to visualize is that of a plane wave, which is a good a...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia Ii - Category Mathematics - Definition
A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target ...   » Read the article

Encyclopedia - Analogy: Encyclopedia Ii - Analogy - Models And Theories Of Analogy
Analogy - Identity of relation. In ancient Greek the word αναλογια (analogia) originally meant proportionality, in the mathemat...   » Read the article

Encyclopedia - Group Representation: Encyclopedia Ii - Group Representation - Branches Of Representation Theory
Representation theory divides into subtheories depending on the kind of group being represented. The various theories are quite different...   » Read the article

Encyclopedia - Algebraic Number: Encyclopedia Ii - Algebraic Number - Algebraic Integers
An algebraic number which satisfies a polynomial equation of degree n with leading coefficient an = 1 (that is, a monic polynom...   » Read the article

Encyclopedia - Morphism: Encyclopedia Ii - Morphism - Definition
A category C is given by two pieces of data: a class of objects and a class of morphisms. There are two operations defined on every morph...   » Read the article

Encyclopedia - Morphism: Encyclopedia Ii - Morphism - Definition
A category C is given by two pieces of data: a class of objects and a class of morphisms. There are two operations defined on every morph...   » Read the article

Encyclopedia - Fundamental Group: Encyclopedia Ii - Fundamental Group - Intuition And Definition
Before giving a precise definition of the fundamental group, we try to describe the general idea in non-mathematical terms. Take some spa...   » Read the article

Encyclopedia - Relativism: Encyclopedia Ii - Relativism - The Catholic Church And Relativism
The Catholic Church for some time now, especially with Pope Benedict XVI, head of the Congregation of the Doctrine of the Faith when he w...   » Read the article

Encyclopedia - Pullback Category Theory: Encyclopedia Ii - Pullback Category Theory - Examples
In the category of sets the pullback of f and g is the set X ×Z Y = {(x, y) ∈ X × Y | f(x) = g(y)}, together with the restrictions ...   » Read the article

Encyclopedia - Up To: Encyclopedia Ii - Up To - Examples
In the eight queens puzzle, if the eight queens are considered to be distinct, there are 3 709 440 distinct solutions. Normally...   » Read the article

Encyclopedia - Distributive Lattice: Encyclopedia Ii - Distributive Lattice - Representation Theory
The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive iff it is iso...   » Read the article

Encyclopedia - Lie Algebra: Encyclopedia Ii - Lie Algebra - Relation To Lie Groups
Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a ...   » Read the article

Encyclopedia - Möbius Transformation: Encyclopedia Ii - Möbius Transformation - Geometric Interpretation Of The Characteristic Constant
The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transfo...   » Read the article

Encyclopedia - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Definition
Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case,...   » Read the article

Encyclopedia - Initial Object: Encyclopedia Ii - Initial Object - Properties
Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Informal Discussion
Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting ...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Motivation
In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such...   » Read the article

Encyclopedia - Inverse Limit: Encyclopedia Ii - Inverse Limit - Formal Definition
Inverse limit - Algebraic objects. We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a dir...   » Read the article

Encyclopedia - Isomorphism Of Categories: Encyclopedia Ii - Isomorphism Of Categories - Properties
As is true for any notion of isomorphism, we have the following general properties formally similar to an equivalence relation: any cate...   » Read the article

Encyclopedia - Polynomial Interpolation: Encyclopedia Ii - Polynomial Interpolation - Convergence Properties
It is natural to ask, for which classes of functions and for which interpolation nodes the sequence of interpolating polynomials converge...   » Read the article

Encyclopedia - Group Action: Encyclopedia Ii - Group Action - Orbits And Stabilizers
Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of ...   » Read the article

Encyclopedia - Complex Number: Encyclopedia Ii - Complex Number - Definition
Complex number - The complex number field. Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) togethe...   » Read the article

Encyclopedia - Signed Graph: Encyclopedia Ii - Signed Graph - Matroid Theory
There are two matroids associated with a signed graph, called the signed-graphic matroid (or the frame matroid or bias matroid) and the l...   » Read the article

Encyclopedia - Symplectomorphism: Encyclopedia Ii - Symplectomorphism - Formal Definition
Specifically, let (M1, ω1) and (M2, ω2) by symplectic manifolds. A map f : M1 → M2 is a symplectomorphism if it is a diffeomor...   » Read the article

Encyclopedia - Chinese Remainder Theorem: Encyclopedia Ii - Chinese Remainder Theorem - Simultaneous Congruences Of Integers
The original form of the theorem, contained in a third-century book by Chinese mathematician Sun Tzu and later republished in a 1247 book...   » Read the article

Encyclopedia - Boolean Prime Ideal Theorem: Encyclopedia Ii - Boolean Prime Ideal Theorem - Boolean Prime Ideal Theorem
The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is: Let B be a Boolean...   » Read the article

Encyclopedia - Diffeomorphism: Encyclopedia Ii - Diffeomorphism - Local Description
Model example: if U and V are two open subsets of , a differentiable map f from U to V is a diffeomorphism if it is a bijection, its dif...   » Read the article

Encyclopedia - Direct Limit: Encyclopedia Ii - Direct Limit - Formal Definition
Direct limit - Algebraic objects. In this section we will understand objects to be sets with a given algebraic structure such as groups...   » Read the article

Encyclopedia - Automorphism: Encyclopedia Ii - Automorphism - Definition
The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "iso...   » Read the article

Encyclopedia - Monoid: Encyclopedia Ii - Monoid - Properties
Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an e...   » Read the article

Encyclopedia - Coordinate Vector: Encyclopedia Ii - Coordinate Vector - Definition
Let V be a vector space of dimension n over a field F and let be an ordered basis for V. Then for every there is a unique linear comb...   » Read the article

Encyclopedia - Signed Measure: Encyclopedia Ii - Signed Measure - Properties
What follows are two results which will imply that a signed measure is the difference of two nonnegative measures, and as such, that sign...   » Read the article

Encyclopedia - Functor: Encyclopedia Ii - Functor - Examples
Constant functor: A very boring functor C → D is one which maps every object of C to a fixed object X in D and every morphism in C to t...   » Read the article

Encyclopedia - Character Theory: Encyclopedia Ii - Character Theory - Basic Definitions
In mathematics, the character of a group representation of a group G is the function which sends g in G to the trace (the sum of th...   » Read the article

Encyclopedia - Coequalizer: Encyclopedia Ii - Coequalizer - Definition
The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects ...   » Read the article

Encyclopedia - Cyclic Group: Encyclopedia Ii - Cyclic Group - Properties
Every cyclic group is isomorphic to (essentially the same as) the group { 0, 1, 2, ... n - 1 } under addition modulo n, or Z, the additiv...   » Read the article

Encyclopedia - Tensor Product: Encyclopedia Ii - Tensor Product - Tensor Product Of Hilbert Spaces
The tensor product of two Hilbert spaces is another Hilbert space, which is defined as described below. Tensor product - Definition. Le...   » Read the article

Encyclopedia - Vector Space: Encyclopedia Ii - Vector Space - Formal Definition
A vector space over a field F (such as the field of real numbers or the field of complex numbers) is a set V together with two operations...   » Read the article

Encyclopedia - Canonical: Encyclopedia Ii - Canonical - Mathematics
Mathematicians have for perhaps a century or more used the word canonical to refer to concepts that have a kind of uniqueness or naturaln...   » Read the article

Encyclopedia - Category Theory: Encyclopedia Ii - Category Theory - Background
The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider t...   » Read the article

Encyclopedia - Uniform Isomorphism: Encyclopedia Ii - Uniform Isomorphism - Definition
A function f between two uniform spaces X and Y is called uniform isomorphism if it satisfies the following properties f is a bijection ...   » Read the article

Encyclopedia - Tangent Space: Encyclopedia Ii - Tangent Space - Formal Definitions
There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite st...   » Read the article

Encyclopedia - Distinct: Encyclopedia Ii - Distinct - Example
A quadratic equation over the complex numbers always has two roots. The equation x2 − 3x + 2 = 0 factors as (x − 1)(x − 2) = 0 ...   » Read the article