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wiki_4500_chunk_0 | Group homomorphisms | In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that h ( u ∗ v ) = h ( u ) ⋅ h ( v ) {\displaystyle h(u*v)=h(u)\cdot h(v)} where the group operation on the left side of the equation is that of G and on the rig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4501_chunk_0 | Product measure | In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4502_chunk_0 | Product measure | {\displaystyle B_{2}\in \Sigma _{2}.} This sigma algebra is called the tensor-product σ-algebra on the product space. A product measure μ 1 × μ 2 {\displaystyle \mu _{1}\times \mu _{2}} (also denoted by μ 1 ⊗ μ 2 {\displaystyle \mu _{1}\otimes \mu _{2}} by many authors) is defined to be a measure on the measurable spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4503_chunk_0 | Product measure | (In multiplying measures, some of which are infinite, we define the product to be zero if any factor is zero.) In fact, when the spaces are σ {\displaystyle \sigma } -finite, the product measure is uniquely defined, and for every measurable set E, ( μ 1 × μ 2 ) ( E ) = ∫ X 2 μ 1 ( E y ) d μ 2 ( y ) = ∫ X 1 μ 2 ( E x ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4504_chunk_0 | Scott continuous | In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4505_chunk_0 | Whitney disk | In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that the two marked points go to p and q, one boundary arc of D goes to A and the other to B.Their existence and embeddedness... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4506_chunk_0 | Global analysis | In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold theory and topological spaces of mappings to classify behaviors of differe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4507_chunk_0 | Gradient descent | In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4508_chunk_0 | Gradient descent | Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4509_chunk_0 | Algorithmic graph theory | In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4510_chunk_0 | Algorithmic graph theory | In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4511_chunk_0 | Polynomial functional calculus | In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which naturally extends the function f from complex argument to operator argument. More ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4512_chunk_0 | Homogeneous coordinates | In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4513_chunk_0 | Homological conjectures in commutative algebra | In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4514_chunk_0 | Homological conjectures in commutative algebra | The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension and r ∈ R {\displaystyle r\in R} is not a zero divisor on M {\displaystyle M} , then r {\displaystyle r} is not a zero divisor on R {\displaystyle R} . Bass's Question. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4515_chunk_0 | Homological conjectures in commutative algebra | If M ≠ 0 {\displaystyle M\neq 0} has a finite injective resolution then R {\displaystyle R} is a Cohen–Macaulay ring. The Intersection Theorem. If M ⊗ R N ≠ 0 {\displaystyle M\otimes _{R}N\neq 0} has finite length, then the Krull dimension of N (i.e., the dimension of R modulo the annihilator of N) is at most the proje... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4516_chunk_0 | Homological conjectures in commutative algebra | Let 0 → G n → ⋯ → G 0 → 0 {\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0} denote a finite complex of free R-modules such that ⨁ i H i ( G ∙ ) {\displaystyle \bigoplus \nolimits _{i}H_{i}(G_{\bullet })} has finite length but is not 0. Then the (Krull dimension) dim R ≤ n {\displaystyle \dim R\leq n} . The Improved... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4517_chunk_0 | Homological conjectures in commutative algebra | Let 0 → G n → ⋯ → G 0 → 0 {\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0} denote a finite complex of free R-modules such that H i ( G ∙ ) {\displaystyle H_{i}(G_{\bullet })} has finite length for i > 0 {\displaystyle i>0} and H 0 ( G ∙ ) {\displaystyle H_{0}(G_{\bullet })} has a minimal generator that is killed by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4518_chunk_0 | Homological conjectures in commutative algebra | The conjecture was proven by Yves André using a theory of perfectoid spaces. The Canonical Element Conjecture. Let x 1 , … , x d {\displaystyle x_{1},\ldots ,x_{d}} be a system of parameters for R, let F ∙ {\displaystyle F_{\bullet }} be a free R-resolution of the residue field of R with F 0 = R {\displaystyle F_{0}=R}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4519_chunk_0 | Homological conjectures in commutative algebra | Lift the identity map R = K 0 → F 0 = R {\displaystyle R=K_{0}\to F_{0}=R} to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from R = K d → F d {\displaystyle R=K_{d}\to F_{d}} is not 0. Existence of Balanced Big Cohen–Macaulay Modules Conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4520_chunk_0 | Homological conjectures in commutative algebra | There exists a (not necessarily finitely generated) R-module W such that mRW ≠ W and every system of parameters for R is a regular sequence on W. Cohen-Macaulayness of Direct Summands Conjecture. If R is a direct summand of a regular ring S as an R-module, then R is Cohen–Macaulay (R need not be local, but the result r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4521_chunk_0 | Homological conjectures in commutative algebra | Let A ⊆ R → S {\displaystyle A\subseteq R\to S} be homomorphisms where R is not necessarily local (one can reduce to that case however), with A, S regular and R finitely generated as an A-module. Let W be any A-module. Then the map Tor i A ( W , R ) → Tor i A ( W , S ) {\displaystyle \operatorname {Tor} _{i}^{A}(W,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4522_chunk_0 | Homological conjectures in commutative algebra | The Strong Direct Summand Conjecture. Let R ⊆ S {\displaystyle R\subseteq S} be a map of complete local domains, and let Q be a height one prime ideal of S lying over x R {\displaystyle xR} , where R and R / x R {\displaystyle R/xR} are both regular. Then x R {\displaystyle xR} is a direct summand of Q considered as R-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4523_chunk_0 | Homological conjectures in commutative algebra | Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let R → S {\displaystyle R\to S} be a local homomorphism of complete local domains. Then there exists an R-algebra BR that is a balanced big Cohen–Macaulay algebra for R, an S-algebra B S {\displaystyle B_{S}} that is a balanced big Cohen-Macaulay a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4524_chunk_0 | Homological conjectures in commutative algebra | Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that M ⊗ R N {\displaystyle M\otimes _{R}N} has finite length. Then χ ( M , N ) {\displaystyle \chi (M,N)} , defined as the alternating sum of the lengths of the modules Tor i R ( M , N ) {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4525_chunk_0 | Homological stability | In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups G 1 ⊂ G 2 ⊂ ⋯ {\displaystyle G_{1}\subset G_{2}\subset \cdots } is stable, i.e., H i ( G n ) {\displaystyle H_{i}(G_{n})} is independent of n when n is large enough (depending on i). The smallest... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4526_chunk_0 | Homology (mathematics) | In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4527_chunk_0 | Homotopic algebra | In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases. The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4528_chunk_0 | Relative homotopy group | In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space. Intuitively, homotopy groups record information about the basi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4529_chunk_0 | Relative homotopy group | Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, π n ( X ) , {\displaystyle \pi _{n}(X),} of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4530_chunk_0 | Abstract homotopy theory | In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is learned as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4531_chunk_0 | Dehn filling | In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions. Such an operation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4532_chunk_0 | Dehn filling | So M = M ( ∞ , … , ∞ ) {\displaystyle M(\infty ,\dots ,\infty )} . We equip the space H of finite volume hyperbolic 3-manifolds with the geometric topology. Thurston's hyperbolic Dehn surgery theorem states: M ( u 1 , u 2 , … , u n ) {\displaystyle M(u_{1},u_{2},\dots ,u_{n})} is hyperbolic as long as a finite set of e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4533_chunk_0 | Dehn filling | This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem. Another important result by Thurston is that v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4534_chunk_0 | Dehn filling | In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the Gromov norm. Jørgensen also showed that the volume function on this space is a continuous, proper function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4535_chunk_0 | Dehn filling | This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov. The figure-eight knot and the (-2, 3, 7) pretzel knot are the only two knots whose complements are known to have more than 6 exceptional surgeries; they have 10 and 7, respectively. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4536_chunk_0 | Dehn filling | Cameron Gordon conjectured that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot complement. This was proved by Marc Lackenby and Rob Meyerhoff, who show that the number of exceptional slopes is 10 for any compact orientable 3-manifold with boundary a torus and interior finite-volume hy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4537_chunk_0 | Hyperbolic coordinates | In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane { ( x , y ): x > 0 , y > 0 } = Q {\displaystyle \{(x,y)\ :\ x>0,\ y>0\ \}=Q} .Hyperbolic coordinates take values in the hyperbolic plane defined as: H P = { ( u , v ): u ∈ R , v > 0 } {\displaystyle HP=\{(u,v):u\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4538_chunk_0 | Hyperbolic cotangent | In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the deriva... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4539_chunk_0 | Hyperbolic cotangent | They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4540_chunk_0 | Hyperbolic cotangent | By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4541_chunk_0 | Hyperbolic plane | In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4542_chunk_0 | Hyperbolic plane | Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4543_chunk_0 | Hyperbolic plane | When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4544_chunk_0 | Hyperbolic space | It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, quaternionic hyperbolic spaces and the octononic hyperb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4545_chunk_0 | Hypercomplex analysis | In mathematics, hypercomplex analysis is the extension of complex analysis to the hypercomplex numbers. The first instance is functions of a quaternion variable, where the argument is a quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis). A second instance involves function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4546_chunk_0 | Hypercomplex analysis | The study of functions with arguments from a Clifford algebra is called Clifford analysis. A matrix may be considered a hypercomplex number. For example, the study of functions of 2 × 2 real matrices shows that the topology of the space of hypercomplex numbers determines the function theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4547_chunk_0 | Hypercomplex analysis | Functions such as square root of a matrix, matrix exponential, and logarithm of a matrix are basic examples of hypercomplex analysis. The function theory of diagonalizable matrices is particularly transparent since they have eigendecompositions. Suppose T = ∑ i = 1 N λ i E i {\displaystyle \textstyle T=\sum _{i=1}^{N}\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4548_chunk_0 | Hypercomplex analysis | The modern terminology for a "system of hypercomplex numbers" is an algebra over the real numbers, and the algebras used in applications are often Banach algebras since Cauchy sequences can be taken to be convergent. Then the function theory is enriched by sequences and series. In this context the extension of holomorp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4549_chunk_0 | Hypercomplex numbers | In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4550_chunk_0 | Hyperfunction | In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese, (1959, 1960 in English), building upon earlier work ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4551_chunk_0 | Hypergeometric identities | In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in solutions to combinatorial problems, and also in the analysis of algorithms. These identities were traditionally found 'by han... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4552_chunk_0 | Inclusion mapping | In mathematics, if A {\displaystyle A} is a subset of B , {\displaystyle B,} then the inclusion map (also inclusion function, insertion, or canonical injection) is the function ι {\displaystyle \iota } that sends each element x {\displaystyle x} of A {\displaystyle A} to x , {\displaystyle x,} treated as an element of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4553_chunk_0 | Complex conjugate representation | In mathematics, if G is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate vector space V as follows: Π(g) is the conjugate of Π(g) for all g in G.Π is also a representation, as one may check explicitly. If g is a re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4554_chunk_0 | Complex conjugate representation | See spinor for some examples associated with spinor representations of the spin groups Spin(p + q) and Spin(p, q). If g {\displaystyle {\mathfrak {g}}} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π(X) is the conjugate of −π(X*) for all X in gFor a finite-dimen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4555_chunk_0 | Contragredient representation | In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation ρ* is defined over the dual vector space V* as follows: ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all g ∈ G.The dual representation is also known as the contragredient represen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4556_chunk_0 | Particle diffusion | In the physics and engineering literature, it is common to use ∇2 to denote the Laplacian, rather than ∆. In mathematics as well as in physics and engineering, it is common to use Newton's notation for time derivatives, so that u ˙ {\displaystyle {\dot {u}}} is used to denote ∂u/∂t, so the equation can be written Note ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4557_chunk_0 | Particle diffusion | In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4558_chunk_0 | Particle diffusion | This is not a major difference, for the following reason. Let u be a function with ∂ u ∂ t = α Δ u . {\displaystyle {\frac {\partial u}{\partial t}}=\alpha \Delta u.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4559_chunk_0 | Particle diffusion | Define a new function v ( t , x ) = u ( t / α , x ) {\displaystyle v(t,x)=u(t/\alpha ,x)} . Then, according to the chain rule, one has Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of α and solutions of the heat equation with α = 1. As such, for the sake... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4560_chunk_0 | Conductor of an abelian variety | In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4561_chunk_0 | Harmonic polynomial | In mathematics, in abstract algebra, a multivariate polynomial p over a field such that the Laplacian of p is zero is termed a harmonic polynomial.The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic pol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4562_chunk_0 | Semipermutable subgroup | In mathematics, in algebra, in the realm of group theory, a subgroup H {\displaystyle H} of a finite group G {\displaystyle G} is said to be semipermutable if H {\displaystyle H} commutes with every subgroup K {\displaystyle K} whose order is relatively prime to that of H {\displaystyle H} . Clearly, every permutable s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4563_chunk_0 | Li Shanlan identity | In mathematics, in combinatorics, the Li Shanlan identity (also called Li Shanlan's summation formula) is a certain combinatorial identity attributed to the nineteenth century Chinese mathematician Li Shanlan. Since Li Shanlan is also known as Li Renshu (his courtesy name), this identity is also referred to as the Li R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4564_chunk_0 | Li Shanlan identity | A Czech mathematician Josef Kaucky published an elementary proof of the identity along with a history of the identity in 1964. Kaucky attributed the identity to a certain Li Jen-Shu. From the account of the history of the identity, it has been ascertained that Li Jen-Shu is in fact Li Shanlan. Western scholars had been... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4565_chunk_0 | Poisson wavelet | In mathematics, in functional analysis, several different wavelets are known by the name Poisson wavelet. In one context, the term "Poisson wavelet" is used to denote a family of wavelets labeled by the set of positive integers, the members of which are associated with the Poisson probability distribution. These wavele... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4566_chunk_0 | Seidel adjacency matrix | In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices. It is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4567_chunk_0 | Seidel adjacency matrix | The Seidel matrix of G is also the adjacency matrix of a signed complete graph KG in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and KG. The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4568_chunk_0 | Cyclic decomposition theorem | In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4569_chunk_0 | Weyr canonical form | In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix which (in some sense) induces "nice" properties with matrices it commutes with. It also has a particularly simple structure and the conditions for possessing a Weyr form are fairly weak, making it a suitable tool ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4570_chunk_0 | Weyr canonical form | This form has been variously called as modified Jordan form, reordered Jordan form, second Jordan form, and H-form. The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999.Recently several applications have been found for the Weyr matrix. Of par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4571_chunk_0 | Gauss composition law | In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae, a textbook on number theory published in 1801, in Articles 234 - 244. Gauss com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4572_chunk_0 | Bhargava cube | In mathematics, in number theory, a Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensively used by Manjul Bhargava, a Canadian-American Fields Medal winning mathematician, to study the composition laws of bina... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4573_chunk_0 | Differential graded Lie algebra | In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applications in deformation theory and rational homotopy theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4574_chunk_0 | Homotopy Lie algebra | In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differential graded Lie alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4575_chunk_0 | Graded module | In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i {\displaystyle R_{i}} such that R i R j ⊆ R i + j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}} . The index set is usually the set of nonnegative integers or the set of int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4576_chunk_0 | Graded module | It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z {\displaystyle \mathbb {Z} } -algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4577_chunk_0 | Moduli functor | In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can sho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4578_chunk_0 | Group of fractional ideals | In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where frac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4579_chunk_0 | Singular values | In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator T: X → Y {\displaystyle T:X\rightarrow Y} acting between Hilbert spaces X {\displaystyle X} and Y {\displaystyle Y} , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4580_chunk_0 | Singular values | The singular values are the absolute values of the eigenvalues of a normal matrix A, because the spectral theorem can be applied to obtain unitary diagonalization of A {\displaystyle A} as A = U Λ U ∗ {\displaystyle A=U\Lambda U^{*}} . Therefore, A ∗ A = U Λ ∗ Λ U ∗ = U | Λ | U ∗ {\textstyle {\sqrt {A^{*}A}}={\sqrt {U\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4581_chunk_0 | Singular values | For example, the Ky Fan-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the pth root of the sum of the pth powers of the singular values. Note that each norm is defined only on a special class of operators, hence s-numbers are useful in classifyin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4582_chunk_0 | Polarization formula | In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4583_chunk_0 | Dolbeault cohomology | In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H p , q ( M , C ) {\displaystyle H^{p,q}(M,\mathbb {C} )} depend ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4584_chunk_0 | Completeness of projective varieties | In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y {\displaystyle X\times Y\to Y} is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4585_chunk_0 | Stiefel–Whitney class | In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4586_chunk_0 | Stiefel–Whitney class | A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, S 1 × R {\displaystyle S^{1}\time... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4587_chunk_0 | Total Chern class | In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, kno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4588_chunk_0 | Berlekamp–Zassenhaus algorithm | In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus. As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4589_chunk_0 | Weitzenböck identity | In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4590_chunk_0 | Minimal volume | In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This diffeomorphism invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold (M, g), one may consider its volume vol(M, g) and sectional curvature Kg. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4591_chunk_0 | Minimal volume | By contrast, a positive lower bound for the minimal volume of M amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on M in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if M is a closed and connected two-dimensional manifold, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4592_chunk_0 | Minimal volume | More generally, according to the Chern-Gauss-Bonnet formula, if M is a closed and connected manifold then MinVol ( M ) ≥ c ( n ) | χ ( M ) | . {\displaystyle \operatorname {MinVol} (M)\geq c(n){\big |}\chi (M){\big |}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4593_chunk_0 | Minimal volume | Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality MinVol ( M ) ≥ ‖ M ‖ ( n − 1 ) n n ! . {\displaystyle \operatorname {MinVol} (M)\geq {\frac {\|M\|}{(n-1)^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4594_chunk_0 | Formally real field | In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4595_chunk_0 | Differentiation in Fréchet spaces | In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, is significantly weaker than the derivative in a Banach space, even between g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4596_chunk_0 | DG Algebra | In mathematics, in particular in homological algebra, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4597_chunk_0 | Homotopy lifting property | In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4598_chunk_0 | Whitney extension theorem | In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4599_chunk_0 | Content (measure theory) | In mathematics, in particular in measure theory, a content μ {\displaystyle \mu } is a real-valued function defined on a collection of subsets A {\displaystyle {\mathcal {A}}} such that μ ( A ) ∈ whenever A ∈ A . {\displaystyle \mu (A)\in \ {\text{ whenever }}A\in {\mathcal {A}}.} μ ( ∅ ) = 0. {\displaystyle \mu (\var... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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