isomorphism

Polarization - Basics - plane waves. The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation to most light waves. A plane wave propagates everywhere in the same direction, and like all electromagnetic waves has the electric and magnetic fields perpendicular to the propagation direction. Either vector at a point in space can be decomposed into two orthogonal components in the plane perpendicular to the direction of propagation. Conventionally, when considering ...

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Polarization, Polarization - Theory, Polarization - Basics - plane waves, Polarization - Incoherent radiation, Polarization - Parameterizing polarization, Polarization - Propagation reflection and scattering, Polarization - Polarization in nature science and technology, Polarization - Observing polarization effects in everyday life, Polarization - Biology, Polarization - Geology, Polarization - Chemistry, Polarization - Astronomy, Polarization - Technology

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The name is in honour of Charles-Emile Picard's theories, in particular of divisors on algebraic surfaces. The Picard group of the spectrum of a Dedekind domain is its ideal class group. The invertible sheaves on projective space , for k a field, are the twisting sheaves , so the Picard group of is isomorphic to . The Picard group of the affine line with two origins over See also:

Picard group, Picard group - Examples, Picard group - Picard scheme

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Homomorphism is one of the fundamental concepts in abstract algebra. Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operation. For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). Note that f(x) = 3x is a homomorphism, ...

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Homomorphism, Homomorphism - Homomorphism for beginners, Homomorphism - Homomorphism for mathematicians, Homomorphism - Types of homomorphisms, Homomorphism - Kernel of a homomorphism

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Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an element x is called invertible if there exists an element y such x*y = e and y*x = e. The element y is called the inverse of x and is usually written x−1. Associativity guarantees that inverses, if they exist, are unique. The set of all invertible elements in a monoid M, together with the operation *, forms a group. In that ...

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

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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 unique linear transformation A : V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation (or be consistently oriented) if A has positive determinant; otherwise they have opposite orientations. The property of h ...

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Orientation mathematics, Orientation mathematics - Definition, Orientation mathematics - Zero-dimensional case, Orientation mathematics - Alternate viewpoints, Orientation mathematics - Orientation on manifolds

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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 linear transformations on V (in this context, a "scalar" is a member of the base-field F). V* itself becomes a vector space over F under the following definition of addition and scalar multiplication: for all φ, ψ in V*, a in F and x in V. In the language of tensors, elements of V are sometimes called covariant vectors, and elements of V*, contravariant vectors, covectors or one-forms.< ...

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Dual space, Dual space - Algebraic dual space, Dual space - Examples, Dual space - Transpose of a linear map, Dual space - Bilinear products and dual spaces, Dual space - Injection into the double-dual, Dual space - Continuous dual space, Dual space - Examples, Dual space - Further properties

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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 continuous, the inverse function f −1 is continuous. If such a function exists we say X and Y are homeomorphic. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence clas ...

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Homeomorphism, Homeomorphism - Definition, Homeomorphism - Examples, Homeomorphism - Notes, Homeomorphism - Properties, Homeomorphism - Informal discussion

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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 more isometry groups than these two, of the same abstract group type. The notation G × H stands for the direct product of the two groups. Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Zn, where n is p ...

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List of small groups, List of small groups - Glossary, List of small groups - List of small non-abelian groups, List of small groups - Combined list, List of small groups - Small groups library

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In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Axiom - Logical axioms. These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function . More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all ...

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

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A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative. These include the most familiar kinds of algebras. Associative algebras: the algebra of all n-by-n matrices over the field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication. Group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication the commutative ...

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Algebra over a field, Algebra over a field - Definitions, Algebra over a field - Properties, Algebra over a field - Kinds of algebras and examples, Algebra over a field - Index-free notation, Algebra over a field - K-algebra morphism

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The following is the simplest case where the Galois group is not abelian (that is, not commutative). Consider the splitting field K of the polynomial x2−2 over Q; that is, where θ is a cube root of 2, and ω is a cube root of 1 (but not 1 itself). For example, if we imagine K to be inside the field of complex numbers, we may take θ to be the real cube root of 2, and ω to be See also:

Fundamental theorem of Galois theory, Fundamental theorem of Galois theory - Proof, Fundamental theorem of Galois theory - Explicit description of the correspondence, Fundamental theorem of Galois theory - Properties of the correspondence, Fundamental theorem of Galois theory - Example, Fundamental theorem of Galois theory - Example, Fundamental theorem of Galois theory - Applications, Fundamental theorem of Galois theory - Infinite case

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Natural transformation - A worked example. Statements like "Every group is naturally isomorphic to its opposite group" abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms. If (G,*) is a group, we define its opposite group (Gop,*op) as follows: Gop is the same set as G, and the operation *< ...

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Natural transformation, Natural transformation - Definition, Natural transformation - Examples, Natural transformation - A worked example, Natural transformation - Further examples, Natural transformation - Operations with natural transformations, Natural transformation - Functor categories, Natural transformation - Yoneda lemma, Natural transformation - Historical notes

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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: defined on the Cartesian product V × V with values in V and denoted v + w, where v, w ∈ V, and scalar multiplication: defined on the Cartesian product F × V with values in V and denoted a v, where a< ...

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Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures

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From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, a groupoid G is: A set G0 of objects; For each pair of objects x and y in G0, a set G(x,y) of morphisms (or arrows) from x to y — we write f : x → y to indicate that f is an elemen ...

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Groupoid, Groupoid - Definitions, Groupoid - Examples, Groupoid - Relation to groups, Groupoid - Covariance in special relativity, Groupoid - Lie groupoids and Lie algebroids

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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 complex numbers z, but it is usually denoted log(z) for mostly two reasons: To distinguish the function from the usual ln, and because no base other than the natural base is used in the complex domain, making it the only Log function for complex numbers. The above Taylor expansion remains valid for all complex numbers x with ab ...

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Natural logarithm, Natural logarithm - Notational conventions, Natural logarithm - Ln is the inverse of the natural exponential function, Natural logarithm - Reason for being natural, Natural logarithm - Definitions, Natural logarithm - Derivative Taylor series and complex arguments, Natural logarithm - Numerical value, Natural logarithm - The natural logarithm in integration

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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) + f(z,w). Its null space consists of vectors whose first and second coordinates coincide, that is the set {(x,x): x is a re ...

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Null space, Null space - Example, Null space - Properties

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Polarization - Basics - plane waves. The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation to most light waves. A plane wave is one where the direction of the magnetic and electric fields are confined to a plane perpendicular to the propagation direction. Simply because the plane is two-dimensional, the electric vector in the plane at a point in space can be decomposed into two orthogonal components. Call these the x and y components (followin ...

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Polarization, Polarization - Theory, Polarization - Basics - plane waves, Polarization - Incoherent radiation, Polarization - Parameterizing polarization, Polarization - Propagation reflection and scattering, Polarization - Polarization in nature science and technology, Polarization - Observing polarization effects in everyday life, Polarization - Biology, Polarization - Geology, Polarization - Chemistry, Polarization - Astronomy, Polarization - Technology

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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 object b where a and b are in ob(C). We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or homC(a, b)) to denote the hom-class of all morphisms from a to b. (Some a ...

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Category mathematics, Category mathematics - Definition, Category mathematics - Examples, Category mathematics - Types of morphisms, Category mathematics - Types of categories

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Analogy - Identity of relation. In ancient Greek the word αναλογια (analogia) originally meant proportionality, in the mathematical sense, and it was indeed sometimes translated to Latin as proportio. From there analogy was understood as identity of relation between any two ordered pairs, whether of mathematical nature or not. Kant's Critique of Judgment held to this notion. Kant argued that there can be exactly the same relation between two completely different objects. ...

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Analogy, Analogy - Models and theories of analogy, Analogy - Identity of relation, Analogy - Shared abstraction, Analogy - Special case of induction, Analogy - Hidden deduction, Analogy - Shared structure, Analogy - High-level perception, Analogy - Applications and types of analogy, Analogy - Linguistics, Analogy - Mathematics, Analogy - Artificial intelligence, Analogy - Anatomy, Analogy - Law, Analogy - Engineering

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Representation theory divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry. If the field of scalars of the vector space has characteristic p, and if p divides ...

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Group representation, Group representation - Branches of representation theory, Group representation - Basic definitions, Group representation - Simple example, Group representation - Equivalence of representations, Group representation - Reducibility, Group representation - Character theory, Group representation - Generalizations, Group representation - Set-theoretical representations, Group representation - Representations in other categories

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An algebraic number which satisfies a polynomial equation of degree n with leading coefficient an = 1 (that is, a monic polynomial) and all other other coefficients ai belonging to the set Z of integers, is called an algebraic integer. Examples of algebraic integers are 3√2 + 5 and 6i - 2. The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. Th ...

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Algebraic number, Algebraic number - The field of algebraic numbers, Algebraic number - Numbers defined by radicals, Algebraic number - Algebraic integers, Algebraic number - Special classes of algebraic number

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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 morphism, the domain (or source) and the codomain (or target). Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. The set of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y). (Some authors write MorC(X ...

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Morphism, Morphism - Definition, Morphism - Types of morphisms, Morphism - Examples

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