isomorphism

It is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system with the following structure: X is a set, is a σ-algebra over X, is a probability measure, so that μ(X) = 1, and is a measurable transformation which preserves the measure μ, i. e. each measurable satisfies μ(T < ...

See also:

Measure-preserving dynamical system, Measure-preserving dynamical system - Definition, Measure-preserving dynamical system - Examples, Measure-preserving dynamical system - Discussion, Measure-preserving dynamical system - Homomorphisms, Measure-preserving dynamical system - Generic points, Measure-preserving dynamical system - Symbolic names and generators, Measure-preserving dynamical system - Operations on partitions, Measure-preserving dynamical system - Measure-theoretic Entropy

Read more here: » Measure-preserving dynamical system: Encyclopedia II - Measure-preserving dynamical system - Definition

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 defining length. If p is a real number, p≥1, define for any vector . It turns out that this definition indeed satisfies the properties of a length function (or norm), which are that only the length of the zero vector is zero, the length of the vector scales proport ...

See also:

Lp space, Lp space - Motivation, Lp space - l^p spaces, Lp space - Properties of l^p spaces, Lp space - L^p spaces, Lp space - Special cases, Lp space - Relation to l^p spaces, Lp space - Properties of L^p spaces

Read more here: » Lp space: Encyclopedia II - Lp space - Motivation

One of the most appealing aspects of Joy is this: the meaning function is a homomorphism from the syntactic monoid onto the semantic monoid. That is, the syntactic relation of concatenation of symbols maps directly onto the semantic relation of composition of functions. It is a homomorphism instead of an isomorphism because it is onto but not one-to-one, that is, some sequences of symbols have the same meaning ( ...

See also:

Joy programming language, Joy programming language - Mathematical purity, Joy programming language - External link

Read more here: » Joy programming language: Encyclopedia II - Joy programming language - Mathematical purity

Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categories. In the general context, X and Y are objects, while f and g are morphisms from X to Y. These objects and morphisms form a diagram in the category in question, and the equaliser is simply the limit of that diagram. In more explicit terms, the equaliser consists of an object E and a morphism eq : E → X satisfying ...

See also:

Equaliser, Equaliser - Definitions, Equaliser - Difference kernels, Equaliser - In category theory

Read more here: » Equaliser: Encyclopedia II - Equaliser - In category theory

The above equation actually has two distinct solutions which are additive inverses. More precisely, once a solution i of the equation has been fixed, −i (≠ i) is also a solution. Since the equation is the only definition of i, it appears that the definition is ambiguous (more precisely, not well-defined). However, no ambiguity results, as long as we choose a solution and fix it forever as "positive i". The issue is a subtle one. The most precise explanation is to say that although the comp ...

See also:

Imaginary unit, Imaginary unit - Definition, Imaginary unit - i and −i, Imaginary unit - Warning, Imaginary unit - Powers of i, Imaginary unit - i and Euler's formula, Imaginary unit - Alternate notation

Read more here: » Imaginary unit: Encyclopedia II - Imaginary unit - i and −i

Complex number - Notation and operations. The set of all complex numbers is usually denoted by C, or in blackboard bold by . It includes the real numbers because every real number can be regarded as complex: a = a + 0i. Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1: (a + bi) + (c + diSee also:

Complex number, Complex number - Definitions, Complex number - Notation and operations, Complex number - The complex number field, Complex number - The complex plane, Complex number - Absolute value conjugation and distance, Complex number - Complex number division, Complex number - Matrix representation of complex numbers, Complex number - Geometric interpretation of the operations on complex numbers, Complex number - Some properties, Complex number - Real vector space, Complex number - Solutions of polynomial equations, Complex number - Algebraic characterization, Complex number - Characterization as a topological field, Complex number - Complex analysis, Complex number - Applications, Complex number - Control theory, Complex number - Signal analysis, Complex number - Improper integrals, Complex number - Quantum mechanics, Complex number - Relativity, Complex number - Applied mathematics, Complex number - Fluid dynamics, Complex number - Fractals, Complex number - History

Read more here: » Complex number: Encyclopedia II - Complex number - Definitions

Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a reflection group. Euclidean plane isometry - Mirror combinations. In the Euclidean plane, we have the following possibilities. [d  ] Identity Two reflections in the same mirror restore each point to its original position. All points are left fixed. Any pair of identical mirrors has the same effect. See also:

Euclidean plane isometry, Euclidean plane isometry - Informal discussion, Euclidean plane isometry - Formal definition, Euclidean plane isometry - Classification of Euclidean plane isometries, Euclidean plane isometry - Isometries as reflection group, Euclidean plane isometry - Mirror combinations, Euclidean plane isometry - Three mirrors suffice, Euclidean plane isometry - Recognition, Euclidean plane isometry - Group structure, Euclidean plane isometry - Composition, Euclidean plane isometry - Translation rotation and orthogonal subgroups, Euclidean plane isometry - Nested group construction, Euclidean plane isometry - Discrete subgroups, Euclidean plane isometry - Isometries in the complex plane

Read more here: » Euclidean plane isometry: Encyclopedia II - Euclidean plane isometry - Isometries as reflection group

Jaina mathematicians in ancient India first conceived of logarithms between 200 BC and 400 CE. They performed a number of operations using logarithmic functions to base-2. From the 13th century, logarithmic tables were produced by Muslim mathematicians. Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first discovered logarithms as a computational tool; however he did not publish his discovery until 1620. The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithm ...

See also:

Logarithm, Logarithm - Bases, Logarithm - Other notations, Logarithm - Change of base, Logarithm - Uses of logarithms, Logarithm - Science and engineering, Logarithm - Exponential functions, Logarithm - Easier computations, Logarithm - Calculus, Logarithm - Generalizations, Logarithm - History, Logarithm - Tables of logarithms, Logarithm - Trivia, Logarithm - Unicode glyph, Logarithm - Graphical interpretation, Logarithm - Irrationality, Logarithm - Relationships between binary natural and common logarithms

Read more here: » Logarithm: Encyclopedia II - Logarithm - History

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should guess from context which one is intended. Let X and Y be metric spaces with metrics dX and dY. A map is called distance preserving if for any one has < ...

See also:

Isometry, Isometry - Definitions, Isometry - Examples, Isometry - Generalizations

Read more here: » Isometry: Encyclopedia II - Isometry - Definitions

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, characterized by both its magnitude represented by length, and its direction. Vectors can be used to represent physical entities such as forces, and they can be added and multiplied with scalars, thus forming the first example of a real vector space. Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. A vector space of dimension n is called an n ...

See also:

Linear algebra, Linear algebra - History, Linear algebra - Elementary introduction, Linear algebra - Some useful theorems, Linear algebra - Generalization and related topics

Read more here: » Linear algebra: Encyclopedia II - Linear algebra - Elementary introduction

Atoms are represented by the standard abbreviation of the chemical elements, in square brackets, such as [Au] for gold. The hydroxide anion is [OH-]. Brackets can be omitted for the "organic subset" of C, N, O, P, S, Br, Cl, I. All other elements must be enclosed in brackets. If the brackets are omitted, the proper number of implicit hydrogen atoms is assumed; for instance the SMILES for water is simply O and that for ethanol is CCO. The double-bonded carbo ...

See also:

Simplified molecular input line entry specification, Simplified molecular input line entry specification - Canonical SMILES and Isomeric SMILES, Simplified molecular input line entry specification - Graph-based definition, Simplified molecular input line entry specification - Examples, Simplified molecular input line entry specification - Isomeric SMILES, Simplified molecular input line entry specification - Extensions, Simplified molecular input line entry specification - Conversion

Read more here: » Simplified molecular input line entry specification: Encyclopedia II - Simplified molecular input line entry specification - Examples

While the complex numbers are obtained by adding the element i to the real numbers which satisfies i2 = −1, the quaternions are obtained by adding the elements i, j and k to the real numbers which satisfy the following relations. If the multiplication is assumed to be associative (as indeed it is), the following relations follow directly: (these are derived in detail below). Every quaternion is a real linear combination of the basis qua ...

See also:

Quaternion, Quaternion - Definition, Quaternion - Example, Quaternion - Arithmetic, Quaternion - Fundamental formula, Quaternion - Profile, Quaternion - Rotation group, Quaternion - Representing quaternions by matrices, Quaternion - Quaternion operations, Quaternion - Addition and products, Quaternion - Functions of a quaternion variable, Quaternion - Exponentials and logarithms, Quaternion - Trigonometry, Quaternion - Hyperbolic, Quaternion - Inverse hyperbolic functions, Quaternion - Inverse trigonometric functions, Quaternion - Construction of quaternions from complex numbers, Quaternion - Generalizations, Quaternion - History, Quaternion - Use controversy, Quaternion - Recent years, Quaternion - Quotes about quaternions, Quaternion - External articles and resources

Read more here: » Quaternion: Encyclopedia II - Quaternion - Definition

There are five 2D lattice types. Below the wallpaper group of the lattice is given in parentheses; note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. If the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. a rhombic lattice, also called centered rectangular lattice or isosceles triangular lattice (cmm), with evenly ...

See also:

Lattice group, Lattice group - Symmetry considerations and examples, Lattice group - Dividing space according to a lattice, Lattice group - Lattice points in convex sets, Lattice group - Computing with lattices, Lattice group - Lattices in two dimensions: detailed discussion, Lattice group - Lattices in three dimensions, Lattice group - Lattices in complex space, Lattice group - In Lie groups, Lattice group - Lattices over general vector-spaces

Read more here: » Lattice group: Encyclopedia II - Lattice group - Lattices in two dimensions: detailed discussion

Real number - Completeness. The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such t ...

See also:

Real number, Real number - History, Real number - Definition, Real number - Construction from the rational numbers, Real number - Axiomatic approach, Real number - Properties, Real number - Completeness, Real number - The complete ordered field, Real number - Advanced properties, Real number - Generalizations and extensions

Read more here: » Real number: Encyclopedia II - Real number - Properties

Define (i.e. the extended complex plane: the complex numbers joined with the point at infinity). The Riemann sphere is based on the transformation from to in the form , where and . We visualize the Riemman sphere as a sphere in 3-space, i.e. in . Every point on the sphere has both a z value and w value, related by the above transformation. That is, f(z) transforms the sphere onto itself. See also:

Riemann sphere, Riemann sphere - Geometric introduction, Riemann sphere - Stereographic projection, Riemann sphere - Möbius transformations, Riemann sphere - Complex structure, Riemann sphere - The complex projective line, Riemann sphere - Properties

Read more here: » Riemann sphere: Encyclopedia II - Riemann sphere - Geometric introduction

The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-dimensional as a vector space over the reals). Up to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4). This was proved by Frobenius in 1877. Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensiona ...

See also:

Division algebra, Division algebra - Definitions, Division algebra - Associative division algebras, Division algebra - Not necessarily associative division algebras

Read more here: » Division algebra: Encyclopedia II - Division algebra - Associative division algebras

Suppose (A, ≤) and (B, <=) are two partially ordered sets. A Galois connection between these posets consists of two monotone functions: F : A → B and G : B → A, such that for all a in A and b in B, we have F(a) <= b if and ...

See also:

Galois connection, Galois connection - Definition, Galois connection - Alternative definition, Galois connection - Examples, Galois connection - Properties, Galois connection - Closure operators and Galois connections, Galois connection - Existence and uniqueness of Galois connections, Galois connection - Galois connections as morphisms, Galois connection - Connection to category theory, Galois connection - Applications in the theory of programming

Read more here: » Galois connection: Encyclopedia II - Galois connection - Definition

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 any discrete subcategory in C. Not every family {Xi} needs to have a product, but if it does, then the product is unique in a strong sense: if πi : X → Xi and π’i : X’ → Xi are two products of the family {Xi}, then (by ...

See also:

Product category theory, Product category theory - Definition, Product category theory - Examples, Product category theory - Discussion

Read more here: » Product category theory: Encyclopedia II - Product category theory - Discussion

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 can be defined. That is, given a tensor and a set of vectors one then defines the pullback as . The result f * T is again a tensor, so that f *See also:

Pullback, Pullback - Pullback on tensors, Pullback - Pullback of cotangent bundles, Pullback - Pullback on tensor bundles, Pullback - Pullback of diffeomorphisms

Read more here: » Pullback: Encyclopedia II - Pullback - Pullback on tensors

In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in the mathematical formulation of quantum mechanics. More generally, functional analysis includes the study of Fréchet spaces ...

See also:

Functional analysis, Functional analysis - Normed vector spaces, Functional analysis - Hilbert spaces, Functional analysis - Banach spaces, Functional analysis - Major and foundational results, Functional analysis - Foundations of mathematics considerations, Functional analysis - Points of view

Read more here: » Functional analysis: Encyclopedia II - Functional analysis - Normed vector spaces

Although Max Wertheimer is credited as the founder of the movement, the concept of Gestalt was first introduced in contemporary philosophy and psychology by Christian von Ehrenfels (a member of the School of Brentano). The idea of Gestalt has its roots in theories by Johann Wolfgang von Goethe, Immanuel Kant, and Ernst Mach. Both von Ehrenfels and Edmund Husserl seem to have been inspired by Mach's work Beiträge zur Analyse der Empfindungen (Contributions to the Analysis of the Sensations, 1886), in formulating their very similar concepts of Gestal ...

See also:

Gestalt psychology, Gestalt psychology - Origins, Gestalt psychology - Theoretical framework and methodology, Gestalt psychology - Prägnanz, Gestalt psychology - Relationship to gestalt therapy

Read more here: » Gestalt psychology: Encyclopedia II - Gestalt psychology - Origins