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In mathematics, a regular matroid is a matroid that can be represented over all fields.
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https://en.wikipedia.org/wiki/Regular_matroid
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In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.
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https://en.wikipedia.org/wiki/Regular_measure
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets (-faces) and regular vertex figures.
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https://en.wikipedia.org/wiki/Regular_(geometry)
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These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form {a, b, c, ..., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}.
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https://en.wikipedia.org/wiki/Regular_(geometry)
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In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a in S there exists an element x in S such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.
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https://en.wikipedia.org/wiki/Regular_semigroup
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In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Nicolas Bourbaki in 1949, in their book "Livre IV: Fonctions d'une variable réelle".
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https://en.wikipedia.org/wiki/Regulated_function
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In mathematics, a relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Each partial order as well as each equivalence relation needs to be transitive.
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https://en.wikipedia.org/wiki/Axiom_of_transitivity
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In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all distinct pairs of elements of the set in one direction or the other while it is called strongly connected if it relates all pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all x ∈ X {\displaystyle x\in X} there is a y ∈ X {\displaystyle y\in X} so that x R y {\displaystyle x\mathrel {R} y} (see serial relation). Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected.
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https://en.wikipedia.org/wiki/Strongly_connected_relation
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Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain).
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https://en.wikipedia.org/wiki/Strongly_connected_relation
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In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform, on an n-dimensional manifold obeys the following equation where that is, the determinant of the Jacobian of the transformation. A scalar density refers to the w = 1 {\displaystyle w=1} case. Relative scalars are an important special case of the more general concept of a relative tensor.
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https://en.wikipedia.org/wiki/Relative_scalar
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In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact.
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https://en.wikipedia.org/wiki/Relatively_compact_subset
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In mathematics, a remarkable cardinal is a certain kind of large cardinal number. A cardinal κ is called remarkable if for all regular cardinals θ > κ, there exist π, M, λ, σ, N and ρ such that π: M → Hθ is an elementary embedding M is countable and transitive π(λ) = κ σ: M → N is an elementary embedding with critical point λ N is countable and transitive ρ = M ∩ Ord is a regular cardinal in N σ(λ) > ρ M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"Equivalently, κ {\displaystyle \kappa } is remarkable if and only if for every λ > κ {\displaystyle \lambda >\kappa } there is λ ¯ < κ {\displaystyle {\bar {\lambda }}<\kappa } such that in some forcing extension V {\displaystyle V} , there is an elementary embedding j: V λ ¯ V → V λ V {\displaystyle j:V_{\bar {\lambda }}^{V}\rightarrow V_{\lambda }^{V}} satisfying j ( crit ( j ) ) = κ {\displaystyle j(\operatorname {crit} (j))=\kappa } . Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in V {\displaystyle V} , not in V {\displaystyle V} .
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https://en.wikipedia.org/wiki/Remarkable_cardinal
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In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yi conform, in some consistent way, to those existing among the corresponding represented objects xi. More specifically, given a set Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphism that preserves Π. The label representation is sometimes also applied to the homomorphism itself (such as group homomorphism in group theory).
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https://en.wikipedia.org/wiki/Representation_(mathematics)
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In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties. Let X be an affine algebraic variety over an algebraically closed field k of characteristic zero with the action of a reductive algebraic group G. G then acts on the coordinate ring k {\displaystyle k} of X as a left regular representation: ( g ⋅ f ) ( x ) = f ( g − 1 x ) {\displaystyle (g\cdot f)(x)=f(g^{-1}x)} . This is a representation of G on the coordinate ring of X. The most basic case is when X is an affine space (that is, X is a finite-dimensional representation of G) and the coordinate ring is a polynomial ring. The most important case is when X is a symmetric variety; i.e., the quotient of G by a fixed-point subgroup of an involution.
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https://en.wikipedia.org/wiki/Representation_on_coordinate_rings
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In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.
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https://en.wikipedia.org/wiki/Representation_theorem
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In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.
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https://en.wikipedia.org/wiki/Residuated_Boolean_algebra
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In mathematics, a residuated lattice is an algebraic structure L = (L, ≤, •, I) such that (i) (L, ≤) is a lattice. (ii) (L, •, I) is a monoid. (iii) For all z there exists for every x a greatest y, and for every y a greatest x, such that x•y ≤ z (the residuation properties).In (iii), the "greatest y", being a function of z and x, is denoted x\z and called the right residual of z by x. Think of it as what remains of z on the right after "dividing" z on the left by x. Dually, the "greatest x" is denoted z/y and called the left residual of z by y. An equivalent, more formal statement of (iii) that uses these operations to name these greatest values is (iii)' for all x, y, z in L, y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y. As suggested by the notation, the residuals are a form of quotient.
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https://en.wikipedia.org/wiki/Residuated_lattice
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More precisely, for a given x in L, the unary operations x• and x\ are respectively the lower and upper adjoints of a Galois connection on L, and dually for the two functions •y and /y. By the same reasoning that applies to any Galois connection, we have yet another definition of the residuals, namely, x•(x\y) ≤ y ≤ x\(x•y), and (y/x)•x ≤ y ≤ (y•x)/x,together with the requirement that x•y be monotone in x and y. (When axiomatized using (iii) or (iii)' monotonicity becomes a theorem and hence not required in the axiomatization.)
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https://en.wikipedia.org/wiki/Residuated_lattice
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These give a sense in which the functions x• and x\ are pseudoinverses or adjoints of each other, and likewise for •x and /x. This last definition is purely in terms of inequalities, noting that monotonicity can be axiomatized as x•y ≤ (x∨z)•y and similarly for the other operations and their arguments. Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \).
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https://en.wikipedia.org/wiki/Residuated_lattice
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When thus organized, residuated lattices form an equational class or variety, whose homomorphisms respect the residuals as well as the lattice and monoid operations. Note that distributivity x•(y ∨ z) = (x•y) ∨ (x•z) and x•0 = 0 are consequences of these axioms and so do not need to be made part of the definition.
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https://en.wikipedia.org/wiki/Residuated_lattice
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This necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a Heyting algebra. Alternative notations for x•y include x◦y, x;y (relation algebra), and x⊗y (linear logic).
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https://en.wikipedia.org/wiki/Residuated_lattice
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Alternatives for I include e and 1'. Alternative notations for the residuals are x → y for x\y and y ← x for y/x, suggested by the similarity between residuation and implication in logic, with the multiplication of the monoid understood as a form of conjunction that need not be commutative. When the monoid is commutative the two residuals coincide. When not commutative, the intuitive meaning of the monoid as conjunction and the residuals as implications can be understood as having a temporal quality: x•y means x and then y, x → y means had x (in the past) then y (now), and y ← x means if-ever x (in the future) then y (at that time), as illustrated by the natural language example at the end of the examples.
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https://en.wikipedia.org/wiki/Residuated_lattice
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In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."
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https://en.wikipedia.org/wiki/Restricted_Lie_algebra
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In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov.
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https://en.wikipedia.org/wiki/Kolmogorov's_characterization_of_reversible_diffusions
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In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category.
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https://en.wikipedia.org/wiki/Strongly_ribbon_category
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In mathematics, a ridge function is any function f: R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that can be written as the composition of a univariate function with an affine transformation, that is: f ( x ) = g ( x ⋅ a ) {\displaystyle f({\boldsymbol {x}})=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})} for some g: R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } and a ∈ R d {\displaystyle {\boldsymbol {a}}\in \mathbb {R} ^{d}} . Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.
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https://en.wikipedia.org/wiki/Ridge_function
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In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
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https://en.wikipedia.org/wiki/Gelfand_triple
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In mathematics, a right group is an algebraic structure consisting of a set together with a binary operation that combines two elements into a third element while obeying the right group axioms. The right group axioms are similar to the group axioms, but while groups can have only one identity and any element can have only one inverse, right groups allow for multiple one-sided identity elements and multiple one-sided inverse elements. It can be proven (theorem 1.27 in ) that a right group is isomorphic to the direct product of a right zero semigroup and a group, while a right abelian group is the direct product of a right zero semigroup and an abelian group. Left group and left abelian group are defined in analogous way, by substituting right for left in the definitions. The rest of this article will be mostly concerned about right groups, but everything applies to left groups by doing the appropriate right/left substitutions.
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https://en.wikipedia.org/wiki/Right_group
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In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness.
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https://en.wikipedia.org/wiki/Rigid-analytic_space
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In mathematics, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect. The above statement does not define a mathematical property; instead, it describes in what sense the adjective "rigid" is typically used in mathematics, by mathematicians.
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https://en.wikipedia.org/wiki/Rigidity_(mathematics)
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In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a proper rigid transformation, or rototranslation.
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https://en.wikipedia.org/wiki/Rigid_Transformation
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Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections. Any object will keep the same shape and size after a proper rigid transformation. All rigid transformations are examples of affine transformations.
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https://en.wikipedia.org/wiki/Rigid_Transformation
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The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformations is called special Euclidean group, denoted SE(n). In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw displacement.
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https://en.wikipedia.org/wiki/Rigid_Transformation
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In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.
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https://en.wikipedia.org/wiki/Ring_class_field
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In mathematics, a ring is said to be a Dedekind-finite ring if ab = 1 implies ba = 1 for any two ring elements a and b. In other words, all one-sided inverses in the ring are two-sided. These rings have also been called directly finite rings and von Neumann finite rings.
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https://en.wikipedia.org/wiki/Dedekind-finite_ring
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In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets.
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https://en.wikipedia.org/wiki/Structure_sheaf
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Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.
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https://en.wikipedia.org/wiki/Structure_sheaf
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In mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The notion of a ringed topos has applications to deformation theory in algebraic geometry (cf. cotangent complex) and the mathematical foundation of quantum mechanics. In the latter subject, a Bohr topos is a ringed topos that plays the role of a quantum phase space.The definition of a topos-version of a "locally ringed space" is not straightforward, as the meaning of "local" in this context is not obvious. One can introduce the notion of a locally ringed topos by introducing a sort of geometric conditions of local rings (see SGA4, Exposé IV, Exercise 13.9), which is equivalent to saying that all the stalks of the structure ring object are local rings when there are enough points.
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https://en.wikipedia.org/wiki/Ringed_topos
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In mathematics, a rod group is a three-dimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some three-dimensional lattice. Table of the 75 rod groups, organized by crystal system or lattice type, and by their point groups: The double entries are for orientation variants of a group relative to the perpendicular-directions lattice. Among these groups, there are 8 enantiomorphic pairs.
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https://en.wikipedia.org/wiki/Rod_group
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In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.
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https://en.wikipedia.org/wiki/Cube_roots_of_unity
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In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.
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https://en.wikipedia.org/wiki/Positive_root
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In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups.
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https://en.wikipedia.org/wiki/Bouquet_of_circles
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In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.
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https://en.wikipedia.org/wiki/Rose_(mathematics)
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In mathematics, a rotating body is commonly represented by a pseudovector along the axis of rotation. The length of the vector gives the speed of rotation and the direction of the axis gives the direction of rotation according to the right-hand rule: right fingers curled in the direction of rotation and the right thumb pointing in the positive direction of the axis. This allows some easy calculations using the vector cross-product.
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https://en.wikipedia.org/wiki/Corkscrew_rule
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No part of the body is moving in the direction of the axis arrow. By coincidence, if the thumb is pointing north, Earth rotates according to the right-hand rule (prograde motion). This causes the Sun, Moon, and stars to appear to revolve westward according to the left-hand rule.
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https://en.wikipedia.org/wiki/Corkscrew_rule
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In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties. Given a vertex v {\displaystyle v} and an edge label i {\displaystyle i} , the rotation map returns the i {\displaystyle i} 'th neighbor of v {\displaystyle v} and the edge label that would lead back to v {\displaystyle v} .
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https://en.wikipedia.org/wiki/Rotation_map
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In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle θ {\displaystyle \theta } . A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle θ {\displaystyle \theta } . A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a linear map and a rigid transformation.
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https://en.wikipedia.org/wiki/Rotation_of_axes_in_two_dimensions
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In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f ( x , y ) = x 2 + y 3 {\displaystyle f(x,y)=x^{2}+y^{3}} has a critical point at ( 0 , 0 ) {\displaystyle (0,0)} that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y {\displaystyle y} -direction.
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https://en.wikipedia.org/wiki/Saddle_point
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The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour map with a pair of lines intersecting at the point. Such intersections are rare in actual ordnance survey maps, as the height of the saddle point is unlikely to coincide with the integer multiples used in such maps.
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https://en.wikipedia.org/wiki/Saddle_point
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Instead, the saddle point appears as a blank space in the middle of four sets of contour lines that approach and veer away from it. For a basic saddle point, these sets occur in pairs, with an opposing high pair and an opposing low pair positioned in orthogonal directions. The critical contour lines generally do not have to intersect orthogonally.
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https://en.wikipedia.org/wiki/Saddle_point
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In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.
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https://en.wikipedia.org/wiki/Sample-continuous_process
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In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X contains a point isolated in A. A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology.
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https://en.wikipedia.org/wiki/Scattered_space
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In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra.
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https://en.wikipedia.org/wiki/Scheme_theory
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Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a ringed space which is locally a spectrum of a commutative ring.
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https://en.wikipedia.org/wiki/Scheme_theory
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The relative point of view is that much of algebraic geometry should be developed for a morphism X → Y of schemes (called a scheme X over Y), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space. For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.
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https://en.wikipedia.org/wiki/Scheme_theory
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In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.
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https://en.wikipedia.org/wiki/Seashell_surface
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In mathematics, a secondary cohomology operation is a functorial correspondence between cohomology groups. More precisely, it is a natural transformation from the kernel of some primary cohomology operation to the cokernel of another primary operation. They were introduced by J. Frank Adams (1960) in his solution to the Hopf invariant problem. Similarly one can define tertiary cohomology operations from the kernel to the cokernel of secondary operations, and continue like this to define higher cohomology operations, as in Maunder (1963).
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https://en.wikipedia.org/wiki/Secondary_cohomology_operation
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Michael Atiyah pointed out in the 1960s that many of the classical applications could be proved more easily using generalized cohomology theories, such as in his reproof of the Hopf invariant one theorem. Despite this, secondary cohomology operations still see modern usage, for example, in the obstruction theory of commutative ring spectra. Examples of secondary and higher cohomology operations include the Massey product, the Toda bracket, and differentials of spectral sequences.
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https://en.wikipedia.org/wiki/Secondary_cohomology_operation
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In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.
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https://en.wikipedia.org/wiki/Selection_principle
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In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
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https://en.wikipedia.org/wiki/Symmetric_operator
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Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} defined by H ^ ψ = − ℏ 2 2 m ∇ 2 ψ + V ψ , {\displaystyle {\hat {H}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi ,} which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators.
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https://en.wikipedia.org/wiki/Symmetric_operator
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The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators.
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https://en.wikipedia.org/wiki/Symmetric_operator
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With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail.
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https://en.wikipedia.org/wiki/Symmetric_operator
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In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice.
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https://en.wikipedia.org/wiki/Self-avoiding_walks
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Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations. In computational physics, a self-avoiding walk is a chain-like path in R2 or R3 with a certain number of nodes, typically a fixed step length and has the property that it doesn't cross itself or another walk. A system of SAWs satisfies the so-called excluded volume condition.
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https://en.wikipedia.org/wiki/Self-avoiding_walks
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In higher dimensions, the SAW is believed to behave much like the ordinary random walk. SAWs and SAPs play a central role in the modeling of the topological and knot-theoretic behavior of thread- and loop-like molecules such as proteins. Indeed, SAWs may have first been introduced by the chemist Paul Flory in order to model the real-life behavior of chain-like entities such as solvents and polymers, whose physical volume prohibits multiple occupation of the same spatial point.
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https://en.wikipedia.org/wiki/Self-avoiding_walks
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SAWs are fractals. For example, in d = 2 the fractal dimension is 4/3, for d = 3 it is close to 5/3 while for d ≥ 4 the fractal dimension is 2. The dimension is called the upper critical dimension above which excluded volume is negligible.
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https://en.wikipedia.org/wiki/Self-avoiding_walks
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A SAW that does not satisfy the excluded volume condition was recently studied to model explicit surface geometry resulting from expansion of a SAW.The properties of SAWs cannot be calculated analytically, so numerical simulations are employed. The pivot algorithm is a common method for Markov chain Monte Carlo simulations for the uniform measure on n-step self-avoiding walks.
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https://en.wikipedia.org/wiki/Self-avoiding_walks
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The pivot algorithm works by taking a self-avoiding walk and randomly choosing a point on this walk, and then applying symmetrical transformations (rotations and reflections) on the walk after the nth step to create a new walk. Calculating the number of self-avoiding walks in any given lattice is a common computational problem. There is currently no known formula, although there are rigorous methods of approximation.
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https://en.wikipedia.org/wiki/Self-avoiding_walks
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In mathematics, a self-descriptive number is an integer m that in a given base b is b digits long in which each digit d at position n (the most significant digit being at position 0 and the least significant at position b−1) counts how many instances of digit n are in m.
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https://en.wikipedia.org/wiki/Self-descriptive_number
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In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole.
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https://en.wikipedia.org/wiki/Self-similarity
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For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales.
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https://en.wikipedia.org/wiki/Self-similarity
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As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity f ( x , t ) {\displaystyle f(x,t)} measured at different times are different but the corresponding dimensionless quantity at given value of x / t z {\displaystyle x/t^{z}} remain invariant. It happens if the quantity f ( x , t ) {\displaystyle f(x,t)} exhibits dynamic scaling.
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https://en.wikipedia.org/wiki/Self-similarity
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The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide. Peitgen et al. explain the concept as such: If parts of a figure are small replicas of the whole, then the figure is called self-similar....A figure is strictly self-similar if the figure can be decomposed into parts which are exact replicas of the whole.
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https://en.wikipedia.org/wiki/Self-similarity
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Any arbitrary part contains an exact replica of the whole figure.Since mathematically, a fractal may show self-similarity under indefinite magnification, it is impossible to recreate this physically. Peitgen et al. suggest studying self-similarity using approximations:In order to give an operational meaning to the property of self-similarity, we are necessarily restricted to dealing with finite approximations of the limit figure. This is done using the method which we will call box self-similarity where measurements are made on finite stages of the figure using grids of various sizes. This vocabulary was introduced by Benoit Mandelbrot in 1964.
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https://en.wikipedia.org/wiki/Self-similarity
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In mathematics, a semi-Hilbert space is a generalization of a Hilbert space in functional analysis, in which, roughly speaking, the inner product is required only to be positive semi-definite rather than positive definite, so that it gives rise to a seminorm rather than a vector space norm. The quotient of this space by the kernel of this seminorm is also required to be a Hilbert space in the usual sense.
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https://en.wikipedia.org/wiki/Semi-Hilbert_space
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In mathematics, a semi-infinite programming (SIP) problem is an optimization problem with a finite number of variables and an infinite number of constraints. The constraints are typically parameterized. In a generalized semi-infinite programming (GSIP) problem, the feasible set of the parameters depends on the variables.
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https://en.wikipedia.org/wiki/Generalized_semi-infinite_programming
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In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7) The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals).
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https://en.wikipedia.org/wiki/Semi-local_ring
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When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals". Some literature refers to a commutative semi-local ring in general as a quasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals. A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.
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https://en.wikipedia.org/wiki/Semi-local_ring
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In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
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https://en.wikipedia.org/wiki/Semifield
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In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality.
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https://en.wikipedia.org/wiki/Completely_simple_semigroup
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Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively.
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https://en.wikipedia.org/wiki/Completely_simple_semigroup
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In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology. As in any algebraic theory, one of the main problems of the theory of semigroups is the classification of all semigroups and a complete description of their structure.
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https://en.wikipedia.org/wiki/Completely_simple_semigroup
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In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example, the structure of the sets of idempotents of regular semigroups is completely known.
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https://en.wikipedia.org/wiki/Completely_simple_semigroup
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Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the group. A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.
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https://en.wikipedia.org/wiki/Completely_simple_semigroup
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In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z) for all x, y and z in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses.
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https://en.wikipedia.org/wiki/Semigroup
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As in the case of groups or magmas, the semigroup operation need not be commutative, so x·y is not necessarily equal to y·x; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called an abelian semigroup. A monoid is an algebraic structure intermediate between semigroups and groups, and is a semigroup having an identity element, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid.
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https://en.wikipedia.org/wiki/Semigroup
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A natural example is strings with concatenation as the binary operation, and the empty string as the identity element. Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid.
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https://en.wikipedia.org/wiki/Semigroup
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A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups, which are a generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups a notion of division.
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https://en.wikipedia.org/wiki/Semigroup
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Division in semigroups (or in monoids) is not possible in general. The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as transformation semigroup, in which arbitrary functions replace the role of bijections from group theory.
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https://en.wikipedia.org/wiki/Semigroup
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A deep result in the classification of finite semigroups is Krohn–Rhodes theory, analogous to the Jordan–Hölder decomposition for finite groups. Some other techniques for studying semigroups, like Green's relations, do not resemble anything in group theory. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid.
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https://en.wikipedia.org/wiki/Semigroup
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In probability theory, semigroups are associated with Markov processes. In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time.
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https://en.wikipedia.org/wiki/Semigroup
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There are numerous special classes of semigroups, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting classes of semigroups that do not contain any groups except the trivial group; examples of the latter kind are bands and their commutative subclass—semilattices, which are also ordered algebraic structures.
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https://en.wikipedia.org/wiki/Semigroup
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In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist on the underlying set of a semigroup being non-empty.
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https://en.wikipedia.org/wiki/Empty_semigroup
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One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from S × S to S. This operation vacuously satisfies the closure and associativity axioms of a semigroup. Not excluding the empty semigroup simplifies certain results on semigroups.
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https://en.wikipedia.org/wiki/Empty_semigroup
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For example, the result that the intersection of two subsemigroups of a semigroup T is a subsemigroup of T becomes valid even when the intersection is empty. When a semigroup is defined to have additional structure, the issue may not arise. For example, the definition of a monoid requires an identity element, which rules out the empty semigroup as a monoid.
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https://en.wikipedia.org/wiki/Empty_semigroup
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In category theory, the empty semigroup is always admitted. It is the unique initial object of the category of semigroups. A semigroup with no elements is an inverse semigroup, since the necessary condition is vacuously satisfied.
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https://en.wikipedia.org/wiki/Empty_semigroup
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In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements: O2, the null semigroup of order two, LO2, the left zero semigroup of order two, RO2, the right zero semigroup of order two, ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra, (Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: the only group of order two.The semigroups LO2 and RO2 are antiisomorphic. O2, ({0,1}, ∧) and (Z2, +2) are commutative, and LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are bands.
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https://en.wikipedia.org/wiki/Semigroup_with_two_elements
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In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise groups. Semigroupoids have applications in the structural theory of semigroups. Formally, a semigroupoid consists of: a set of things called objects.
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https://en.wikipedia.org/wiki/Semigroupoid
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for every two objects A and B a set Mor(A,B) of things called morphisms from A to B. If f is in Mor(A,B), we write f: A → B. for every three objects A, B and C a binary operation Mor(A,B) × Mor(B,C) → Mor(A,C) called composition of morphisms. The composition of f: A → B and g: B → C is written as g ∘ f or gf.
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https://en.wikipedia.org/wiki/Semigroupoid
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