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Regular matroid Summary Regular_matroid In mathematics, a regular matroid is a matroid that can be represented over all fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular measure Summary Regular_measure 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular (geometry) Summary Regular_(geometry) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular (geometry) Summary Regular_(geometry) 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 reg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular semigroup Summary Regular_semigroup 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 am... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regulated function Summary Regulated_function 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom of transitivity Summary Axiom_of_transitivity 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strongly connected relation Summary Semi-connex_relation 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 descri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strongly connected relation Summary Semi-connex_relation 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relative scalar Summary Scalar_density 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relatively compact subset Summary Relatively_compact_subset 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Remarkable cardinal Summary Remarkable_cardinal 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:... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation (mathematics) Summary Representation_(mathematics) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation on coordinate rings Summary Representation_on_coordinate_rings 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theorem Summary Representation_theorem 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated Boolean algebra Summary Residuated_Boolean_algebra 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 unde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated lattice Definition Residuated_semilattice > Definition 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 res... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated lattice Definition Residuated_semilattice > Definition 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated lattice Definition Residuated_semilattice > Definition 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 simila... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated lattice Definition Residuated_semilattice > Definition 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated lattice Definition Residuated_semilattice > Definition 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 speci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated lattice Definition Residuated_semilattice > Definition 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 conj... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Restricted Lie algebra Summary Restricted_Lie_algebra In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation." | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kolmogorov's characterization of reversible diffusions Summary Kolmogorov's_characterization_of_reversible_diffusions 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 Nikolaevic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strongly ribbon category Summary Strongly_ribbon_category In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ridge function Summary Ridge_function 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfand triple Summary Gelfand_triple 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. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Right group Summary Right_group 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 id... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rigid-analytic space Summary Rigid_analytic_space 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 multiplicativ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rigidity (mathematics) Summary Rigidity_(mathematics) 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rigid Transformation Summary Rigid_Transformation 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, transla... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rigid Transformation Summary Rigid_Transformation 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rigid Transformation Summary Rigid_Transformation 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, prope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ring class field Summary Ring_class_field 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dedekind-finite ring Summary Dedekind-finite_ring 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Structure sheaf Summary Ringed_space 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Structure sheaf Summary Ringed_space 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ringed topos Summary Ringed_topos 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rod group Summary Rod_group 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 lat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cube roots of unity Summary Primitive_root_of_unity 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 theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive root Summary Root_systems 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. Sinc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bouquet of circles Summary Figure-eight_space 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 relate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rose (mathematics) Summary Rose_curve 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corkscrew rule A rotating body Right-hand_rule > Rotations > A rotating body 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-han... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corkscrew rule A rotating body Right-hand_rule > Rotations > A rotating body 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 westw... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rotation map Summary Rotation_map 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rotation of axes in two dimensions Summary Rotation_of_axes 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Saddle point Summary Saddle_point 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Saddle point Summary Saddle_point 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Saddle point Summary Saddle_point 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sample-continuous process Summary Sample-continuous_process In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scattered space Summary Scattered_space 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 su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scheme theory Summary Category_of_schemes 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scheme theory Summary Category_of_schemes 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 spectr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scheme theory Summary Category_of_schemes 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 su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Seashell surface Summary Seashell_surface 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Secondary cohomology operation Summary Secondary_cohomology_operation 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 operatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Secondary cohomology operation Summary Secondary_cohomology_operation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Selection principle Summary Selection_principle 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 mathemati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric operator Summary Self-adjoint_operator 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) t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric operator Summary Self-adjoint_operator 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 rep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric operator Summary Self-adjoint_operator 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric operator Summary Self-adjoint_operator 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-avoiding walks Summary Self-avoiding_walks 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 wal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-avoiding walks Summary Self-avoiding_walks 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-avoiding walks Summary Self-avoiding_walks 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-avoiding walks Summary Self-avoiding_walks 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-avoiding walks Summary Self-avoiding_walks 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-avoiding walks Summary Self-avoiding_walks 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 walk... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-descriptive number Summary Self-descriptive_number 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 ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-similarity Summary Self_similarity 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 statisti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-similarity Summary Self_similarity 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-similarity Summary Self_similarity 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-similarity Summary Self_similarity 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 concep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self-similarity Summary Self_similarity 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-Hilbert space Summary Semi-Hilbert_space 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 v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized semi-infinite programming Summary Generalized_semi-infinite_programming 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 pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-local ring Summary Semilocal_ring 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 max... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-local ring Summary Semilocal_ring 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semifield Summary Semifield 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely simple semigroup Summary Special_classes_of_semigroups 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely simple semigroup Summary Special_classes_of_semigroups 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 intensivel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely simple semigroup Summary Special_classes_of_semigroups 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 cardinal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely simple semigroup Summary Special_classes_of_semigroups 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 sem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely simple semigroup Summary Special_classes_of_semigroups 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 defi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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·... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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 semigrou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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 grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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 b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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 semigroup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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 inde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup Summary Monoid_theory 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, orth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Empty semigroup Summary Empty_semigroup 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Empty semigroup Summary Empty_semigroup 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 simp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Empty semigroup Summary Empty_semigroup 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Empty semigroup Summary Empty_semigroup 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroup with two elements Summary Semigroup_with_two_elements 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroupoid Summary Semigroupoid 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 wa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semigroupoid Summary Semigroupoid 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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