text stringlengths 42 3.65k | source stringclasses 1
value |
|---|---|
Classification theorem Summary Classification_theorem In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Classification theorem Summary Classification_theorem A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it. A computable complete set of invariants (together with which invariants are realizable) solves both the classification ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Classifying topos Summary Classifying_topos In mathematics, a classifying topos for some sort of structure is a topos T such that there is a natural equivalence between geometric morphisms from a cocomplete topos E to T and the category of models for the structure in E. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Clean ring Summary Clean_ring In mathematics, a clean ring is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it is clean and has no idempotents other than 0 and 1. The endomorphism ring of a continuous module is a clean ring. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Clean ring Summary Clean_ring Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact surface Summary Compact_manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boundary parallel Summary Boundary_parallel In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Closure space Summary Closure_operator In mathematics, a closure operator on a set S is a function cl: P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)} from the power set of S to itself that satisfies the following conditions for all sets X , Y ⊆ S {\displaystyle X,Y\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-s-cobordism Summary Semi-s-cobordism In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the inclusion M ↪ W {\displaystyle M\hookrightarrow W} is a simple homotopy equiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cocountability Summary Cocountable_set In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocoun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Terminology and definition Coefficient > Terminology and definition In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial with variables x {\displaystyle x} and y {\displaystyle y} , the first two terms have... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Terminology and definition Coefficient > Terminology and definition In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Terminology and definition Coefficient > Terminology and definition For example, if y is considered a parameter in the above expression, then the coefficient of x would be −3y, and the constant coefficient (with respect to x) would be 1.5 + y. When one writes it is generally assumed that x is the on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Terminology and definition Coefficient > Terminology and definition For the largest i {\displaystyle i} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} (if any), a i {\displaystyle a_{i}} is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial is 4. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Summary Constant_coefficient In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Summary Constant_coefficient When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter.For example, the polynomial 2 x 2 − x + 3 {\displaystyle 2x^{2}-x+3} has coefficients 2, −1, and 3, and the powers of the variable x {\displaystyle x} i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Summary Constant_coefficient The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively. In the context of differential equations, an equation can ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leading coefficient Summary Constant_coefficient In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, the coefficient that is not atta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coercive operator Summary Coercive_bilinear_form In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cofinite set Summary Finite_complement_topology In mathematics, a cofinite subset of a set X {\displaystyle X} is a subset A {\displaystyle A} whose complement in X {\displaystyle X} is a finite set. In other words, A {\displaystyle A} contains all but finitely many elements of X . {\displaystyle X.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cofinite set Summary Finite_complement_topology If the complement is not finite, but is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum. This use of the prefix... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coframe Summary Coframe In mathematics, a coframe or coframe field on a smooth manifold M {\displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M {\displaystyle M} , one has a natural map from v k: ⨁ k T ∗ M → ⋀ k T ∗ M {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coherent topos Summary Coherent_topos In mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomological invariant Summary Cohomological_invariant In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coincidence point Summary Coincidence_point In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image. Formally, given two functions f , g: X → Y {\displaystyle f,g\colon X\rightarrow Y} we say that a point x in X is a coincidence point of f and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coincidence point Summary Coincidence_point Notable among them, in the setting of manifolds, is the Lefschetz coincidence theorem, which is typically known only in its special case formulation for fixed points.Coincidence points, like fixed points, are today studied using many tools from mathematical analysis and topol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Collapsing algebra Summary Collapsing_algebra In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.The collapsing algebra of λω is a complete Boolea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rational dependence Summary Rational_dependence In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationall... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Point finite Summary Point-finite_collection In mathematics, a collection or family U {\displaystyle {\mathcal {U}}} of subsets of a topological space X {\displaystyle X} is said to be point-finite if every point of X {\displaystyle X} lies in only finitely many members of U . {\displaystyle {\mathcal {U}}.} A metacomp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Point finite Summary Point-finite_collection Every locally finite collection of subsets of a topological space is also point-finite. A topological space in which every open cover admits a locally finite open refinement is called a paracompact space. Every paracompact space is therefore metacompact. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Collocation point Summary Orthogonal_collocation In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a cer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Colored matroid Summary Colored_matroid In mathematics, a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first n positive integers, or the sign set {+, −}. The interest in colored matroids is through their invariants... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Colossally abundant number Summary Colossally_abundant_number In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it's defined by a ratio between the sum of an integer's divisors and that integer raised to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination If the set has n elements, the number of k-combinations, denoted by C ( n , k ) {\displaystyle C(n,k)} or C k n {\displaystyle C_{k}^{n}} , is equal to the binomial coefficient which can be written using factorials as n ! k ! ( n − k ) ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination {\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}} whenever k ≤ n {\displaystyle k\leq n} , and which is zero when k > n {\displaystyle k>n} . This formula can be derived from the fact that each k-combination of a set S of n members has k ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination {\displaystyle k!} permutations so P k n = C k n × k ! {\displaystyle P_{k}^{n}=C_{k}^{n}\times k!} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination or C k n = P k n / k ! {\displaystyle C_{k}^{n}=P_{k}^{n}/k!} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination The set of all k-combinations of a set S is often denoted by ( S k ) {\displaystyle \textstyle {\binom {S}{k}}} . A combination is a combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-combination with repetit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combination Summary Combination If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears. Although the set of three fruits was small enough to write a complete list of combinations, this becomes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combinatorial class Summary Combinatorial_class In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combinatorial explosion (communication) Summary Combinatorial_explosion_(communication) In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem. Combinatorial explosion is so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Comma category Summary Comma_category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Comma category Summary Comma_category Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Commutation theory Summary Commutation_theorem_for_traces In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 193... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universally catenary Summary Universally_catenary_ring In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p = p0 ⊂ p1 ⊂ ... ⊂ pn = qof prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universally catenary Summary Universally_catenary_ring The word 'catenary' is derived from the Latin word catena, which means "chain". There is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Commutative rings Summary Commutative_ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braided monoidal category Summary Braided_monoidal_category In mathematics, a commutativity constraint γ {\displaystyle \gamma } on a monoidal category C {\displaystyle {\mathcal {C}}} is a choice of isomorphism γ A , B: A ⊗ B → B ⊗ A {\displaystyle \gamma _{A,B}:A\otimes B\rightarrow B\otimes A} for each pair of objec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braided monoidal category Summary Braided_monoidal_category Partly for this reason, braided monoidal categories and other topics are related in the theory of knot invariants. Alternatively, a braided monoidal category can be seen as a tricategory with one 0-cell and one 1-cell. Braided monoidal categories were introduc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Comodule Summary Comodule In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact group Summary Compact_group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact quantum group Summary Compact_quantum_group In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group. The basic motivation for this theory ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact quantum group Summary Compact_quantum_group On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact semigroup Summary Compact_semigroup In mathematics, a compact semigroup is a semigroup in which the sets of solutions to equations can be described by finite sets of equations. The term "compact" here does not refer to any topology on the semigroup. Let S be a semigroup and X a finite set of letters. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact semigroup Summary Compact_semigroup A system of equations is a subset E of the Cartesian product X∗ × X∗ of the free monoid (finite strings) over X with itself. The system E is satisfiable in S if there is a map f from X to S, which extends to a semigroup morphism f from X+ to S, such that for all (u,v) in E we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compactly generated group Summary Compactly_generated_group In mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should not be confused with the unrelated notion (widely used in algebraic topology) of a compactly generate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete Boolean algebra Summary Complete_Boolean_algebra In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finitely complete category Summary Finitely_complete_category In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F: J → C (where J is small) has a limit in C. Dually, a cocomplete category is one in which all small colimits exist. A bico... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finitely complete category Summary Finitely_complete_category Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete field Summary Complete_field In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete lattices Summary Complete_free_lattice In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a conditionally complete lattice. Specifically, every non-empty fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete lattices Summary Complete_free_lattice Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geodesically complete Summary Geodesic_manifold In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point p, is defined on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete measure Summary Complete_measure_space In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if S ⊆ N ∈ Σ and μ ( N ) = 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete set of invariants Summary Complete_set_of_invariants In mathematics, a complete set of invariants for a classification problem is a collection of maps f i: X → Y i {\displaystyle f_{i}:X\to Y_{i}} (where X {\displaystyle X} is the collection of objects being classified, up to some equivalence relation ∼ {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely metrizable Summary Completely_metrizable_space In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that (X, d) is a complete metric space and d induces the topology T. The term topological... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely regular semigroup Summary Completely_regular_semigroup In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse sem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely regular semigroup Summary Completely_regular_semigroup In the Russian literature, completely regular semigroups are often called "Clifford semigroups". In the English literature, the name "Clifford semigroup" is used synonymously to "inverse Clifford semigroup", and refers to a completely regular inverse sem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Completely regular semigroup Summary Completely_regular_semigroup In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups. Hence completely regular semigroups are also referred to as "unions of groups". Epigroups generalize this notion and their class includes all... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Butson-type Hadamard matrix Summary Butson-type_Hadamard_matrices In mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity, ( H j k ) q = 1 f o r j , k = 1 , 2 , … , N . {\displaystyle (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex Lie algebra Summary Complex_Lie_algebra In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K3 surface Summary K3_manifold In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same cond... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K3 surface Summary K3_manifold Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K3 surface Summary K3_manifold K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifold... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
D-bar operator Summary D-bar_operator In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
D-bar operator Summary D-bar_operator Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex k-form can be decomposed uniquely into a sum of so-called (p, q)-forms: roughly, wedges of p differentials of the holomorphic coord... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex geodesic Summary Complex_geodesic In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex line Summary Complex_line In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex logarithm Summary Complex_logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number z {\displaystyle z} , defined to be any complex nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex logarithm Summary Complex_logarithm These logarithms are equally spaced along a vertical line in the complex plane. A complex-valued function log: U → C {\displaystyle \log \colon U\to \mathbb {C} } , defined on some subset U {\displaystyle U} of the set C ∗ {\displaystyle \mathbb {C} ^{*}} of nonzero complex n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex Numbers Summary Complex_quantity In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} ; every complex number can be expressed in the form a + b i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex Numbers Summary Complex_quantity More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation ( x + 1 ) 2 = − 9 {\displaystyle (x+1)^{2}=-9} has no real solution, since... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex Numbers Summary Complex_quantity Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex Numbers Summary Complex_quantity This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex Numbers Summary Complex_quantity The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with resp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex reflection group Summary Complex_reflection In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex representation Summary Complex_representation In mathematics, a complex representation is a representation of a group (or that of Lie algebra) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Normal matrix Summary Normal_matrix In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*: The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Normal matrix Summary Normal_matrix The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A*A = AA* is diagonalizable. The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear complex structure Summary Linear_complex_structure In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex torus Summary Complex_torus In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles). Here N must be the even number 2n, where n is the complex dimension of M. All such complex ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex torus Summary Complex_torus For n > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conjugate bundle Summary Complex_vector_bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be promoted to a complex vector b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conjugate bundle Summary Complex_vector_bundle The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multiplicative quadratic form Summary Multiplicative_quadratic_form In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)} for all x and y in A. A comp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Composition (combinatorics) Summary Composition_(combinatorics) In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to def... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Composition (combinatorics) Summary Composition_(combinatorics) Each positive integer n has 2n−1 distinct compositions. A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Composition ring Summary Composition_ring In mathematics, a composition ring, introduced in (Adler 1962), is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation ∘: R × R → R {\displaystyle \circ :R\times R\rightarrow R} such that, for any three elements f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Concave function Summary Concave_downward In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.