group theory

A complex in chemistry is a reversible association of molecules, atoms, or ions through weak non-covalent chemical bonds. Simple salts are usually not considered complexes. Complex chemistry - Metal complexes. A metal complex, also known as coordination compound, is a structure composed of a central metal atom or ion, generally a cation, surrounded by a number of negatively charged ions or neutral molecules possessing lone pairs. Counter ions often surround the metal complex ion, causing the compound to hav ...

Including:

• Complex chemistry - Metal complexes
• Complex chemistry - Naming complexes
• Complex chemistry - Receptor-ligand complexes

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For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Commutator - Group theory. The commutator of two elements g and h of a group G is the element [g, h] = g−1h−1gh Including:

• Commutator - Group theory
• Commutator - Identities
• Commutator - Ring theory
• Commutator - Identities

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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Monoid - Definition. A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms: Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) Identity ...

Including:

• Monoid - Definition
• Monoid - Examples
• Monoid - Properties
• Monoid - Monoid homomorphisms
• Monoid - Relation to category theory

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In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc.) is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way. One of the most common places in mathematics in which the term well-defined is used is in dealing with cosets in group theory. It is as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmet ...

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Commensurability mathematics - Commensurability in general. Generally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable. A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080. Commensurabilit ...

Including:

• Commensurability mathematics - Commensurability in general
• Commensurability mathematics - Commensurability in mathematics

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In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one which is considered equivalent to it. In group theory, for example, we may have a group G acting on a set X, in which case we say that two elements of X are equivalent "up to the group action" if they lie in the same orbit. Up to - Exam ...

Including:

• Up to - Examples

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In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field theory, and gauge theory. 3-manifold theory is a part o ...

Including:

• 3-manifold - Some famous examples of 3-manifolds
• 3-manifold - Some important classes of 3-manifolds
• 3-manifold - Some important structures on 3-manifolds
• 3-manifold - Foundational results
• 3-manifold - Some famous conjectures

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Ultra (sometimes capitalised ULTRA) was the name used by the British for intelligence resulting from decryption of German communications in World War II. The term eventually became the standard designation in both Britain and the United States for all intelligence from high-level cryptanalytic sources. The name arose because the code-breaking success was considered more important than the highest security classification available at the time (Most Secret) and so was regarded as being Ultra Secret. Much of t ...

Including:

• Ultra - Sources and history
• Ultra - Encrypted messages
• Ultra - Breaking the cipher
• Ultra - Use of Ultra
• Ultra - Magic and Purple in Europe
• Ultra - Postwar public disclosure of Ultra
• Ultra - Difficulties with some disclosures
• Ultra - Ultra's strategic consequences
• Ultra - Wartime consequences
• Ultra - Postwar consequences
• Ultra - Notes

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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 (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of a (or na when the notation is additive). That is, we say G is cyclic if G = { an for any integer n }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains a is ...

Including:

• Cyclic group - Properties
• Cyclic group - Examples
• Cyclic group - Representation
• Cyclic group - Subgroups
• Cyclic group - Endomorphisms

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In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans. Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory< ...

Including:

• Axiomatic system - Properties
• Axiomatic system - Models
• Axiomatic system - Axiomatic method

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Core - Science. In archaeology, a core is a distinctive artifact that results from the practice of lithic reduction. In this sense, a core is the scarred nucleus resulting from the detachment of one or more lithic flakes from a lump of source material or tool stone, usually by using a hard hammer percussor such as a hammerstone. In astrophysics, the core of a star is its center where nuclear fusion takes place. In astrophysics and geology, a planetary core is the composite material at the centre of ...

Including:

• Core - Science
• Core - Other

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In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. Constructible polygon - Conditions for constructibility. Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not ...

Including:

• Constructible polygon - Conditions for constructibility
• Constructible polygon - General theory
• Constructible polygon - Detailed results in terms of Fermat primes
• Constructible polygon - Compass-and-straightedge constructions
• Constructible polygon - Other constructions

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In mathematics, the term abelian is used in many different definitions: Abelian - In group theory. An abelian group is a group in which the binary operation is commutative. The category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms. A metabelian group is a group where the commutator subgroup is contained in the center. Any group is "made abelian" by its abelianisation. Abelian - In Galois theoryIncluding:

• Abelian - In group theory
• Abelian - In Galois theory
• Abelian - In real analysis
• Abelian - In functional analysis
• Abelian - In topology and number theory
• Abelian - In category theory

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In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism that is a bijection from an object to itself. In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism, φ(xy) = φ(y)φ(x) for all x,y in X. The map that sends x to x ...

Including:

• Antihomomorphism - Examples

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In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. Involution - General properties. The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other exa ...

Including:

• Involution - General properties
• Involution - Involutions in Euclidean geometry
• Involution - Involutions in differential geometry
• Involution - Involutions in ring theory
• Involution - Involutions in group theory

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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

• Axiom - Etymology
• Axiom - Mathematics
• Axiom - Logical axioms
• Axiom - Non-logical axioms
• Axiom - Role in mathematical logic
• Axiom - Further discussion

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Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Etymology. The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not completely resembling real. The term ‘theoria’ (a noun) was already used by ...

Including:

• Theory - Etymology
• Theory - Science
• Theory - Models
• Theory - Types of theories
• Theory - Further explanation of a scientific theory
• Theory - Characteristics
• Theory - Mathematics
• Theory - Other fields
• Theory - List of famous theories
• Theory - Reference

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Field of sets - Sigma algebras and measure spaces. If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebra and the corresponding field of sets is called a measureable space. The complexes of a measurable space are called measureable sets. A measure space is a triple where is a measurable space and μ is a measure defined on it. If μ is in fact a probability measure w ...

See also:

Field of sets, Field of sets - Fields of sets in the representation theory of Boolean algebras, Field of sets - Stone representation, Field of sets - Separative and compact fields of sets: towards Stone duality, Field of sets - Fields of sets with additional structure, Field of sets - Sigma algebras and measure spaces, Field of sets - Topological fields of sets, Field of sets - Preorder fields, Field of sets - Complex algebras and fields of sets on relational structures

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The practical uses of the Riemann hypothesis include many propositions which are stated to be true under the Riemann hypothesis, and some which can be shown to be equivalent to the Riemann hypothesis. One is the rate of growth in the error term of the prime number theorem given above. Other formulations equivalent to the Riemann hypothesis involve the Möbius function μ. The statement that the equation is valid for every s with real part greater than ½, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this w ...

See also:

Riemann hypothesis, Riemann hypothesis - History, Riemann hypothesis - The Riemann hypothesis and primes, Riemann hypothesis - Other consequences of the Riemann hypothesis, Riemann hypothesis - Attempted proofs of the Riemann hypothesis, Riemann hypothesis - Possible connection with operator theory, Riemann hypothesis - Searching for ζ-function zeroes

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According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A). Given a natural transformation Φ : Hom(A,–) → F the corresponding element of u ∈ F(A) is given by Conversely, given any element u ∈ F(A) we may define a natural transformation Φ
...

See also:

Representable functor, Representable functor - Definition, Representable functor - Universal elements, Representable functor - Uniqueness, Representable functor - Examples, Representable functor - Relation to universal morphisms and adjoints

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