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wiki_400_chunk_0 | Staircase paradox | It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.For any smooth curve, polygonal chains with segment lengths decreasing to zero, connecting consecutive vertices along the curve, always converge to the arc length. The failure of the staircase curves to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_401_chunk_0 | Chebyshev norm | The name "uniform norm" derives from the fact that a sequence of functions { f n } {\displaystyle \left\{f_{n}\right\}} converges to f {\displaystyle f} under the metric derived from the uniform norm if and only if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly.If f {\displaystyle f} is a continuo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_402_chunk_0 | Universal chord theorem | In mathematical analysis, the universal chord theorem states that if a function f is continuous on and satisfies f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} , then for every natural number n {\displaystyle n} , there exists some x ∈ {\displaystyle x\in } such that f ( x ) = f ( x + b − a n ) {\displaystyle f(x)=f\lef... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_403_chunk_0 | Multimedia | In mathematical and scientific research, multimedia is mainly used for modeling and simulation. For example, a scientist can look at a molecular model of a particular substance and manipulate it to arrive at a new substance. Representative research can be found in journals such as the Journal of Multimedia. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_404_chunk_0 | Community matrix | In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point. For example, the Lotka–Volterra predator–prey model is d x d t = x ( α − β y ) d y d t = − y ( γ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_405_chunk_0 | Community matrix | By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point (x*, y*), which has the form = A , {\displaystyle {\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},} where u = x −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_406_chunk_0 | Pauline van den Driessche | In mathematical biology, van den Driessche's contributions include important work on delay differential equations and on Hopf bifurcations, and the effects of changing population size and immigration on epidemics.She has also done more fundamental research in linear algebra, motivated by applications in mathematical bi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_407_chunk_0 | Transylvania lottery | In mathematical combinatorics, the Transylvania lottery is a lottery where players selected three numbers from 1-14 for each ticket, and then three numbers are chosen randomly. A ticket wins if two of the numbers match the random ones. The problem asks how many tickets the player must buy in order to be certain of winn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_408_chunk_0 | Radó's theorem (Riemann surfaces) | In mathematical complex analysis, Radó's theorem, proved by Tibor Radó (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_409_chunk_0 | Schottky theorem | In mathematical complex analysis, Schottky's theorem, introduced by Schottky (1904) is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1, the value of |f(z)| can be bounded in terms of z and f(0). Schottky's original theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_410_chunk_0 | Quasiconformal map | In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism betwee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_411_chunk_0 | Geometric function theory | In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism betwee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_412_chunk_0 | Topkis's Theorem | In mathematical economics, Topkis's theorem is a result that is useful for establishing comparative statics. The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes. The result states that if f is supermodular in (x,θ), and D is a lattic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_413_chunk_0 | Isoelastic function | In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_414_chunk_0 | Applied general equilibrium | In mathematical economics, applied general equilibrium (AGE) models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow-up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow–Debreu model of general equilibrium theory with empirical data, to provide "“a g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_415_chunk_0 | Applied general equilibrium | Scarf never built an AGE model, but hinted that “these novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environment” (Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price vector could be solve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_416_chunk_0 | Solomon Mikhlin | In mathematical elasticity theory, Mikhlin was concerned by three themes: the plane problem (mainly from 1932 to 1935), the theory of shells (from 1954) and the Cosserat spectrum (from 1967 to 1973). Dealing with the plane elasticity problem, he proposed two methods for its solution in multiply connected domains. The f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_417_chunk_0 | Solomon Mikhlin | He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called purely rotational state of stress. As a result of his study of this problem, Mikhlin also gave a new (invariant) form of the basic equations of the theory. He also proved a theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_418_chunk_0 | Solomon Mikhlin | Mikhlin studied also the spectrum of the operator pencil of the classical linear elastostatic operator or Navier–Cauchy operator A ( ω ) u = Δ 2 u + ω ∇ ( ∇ ⋅ u ) {\displaystyle {\boldsymbol {\mathcal {A}}}(\omega ){\boldsymbol {u}}=\Delta _{2}{\boldsymbol {u}}+\omega \nabla \left(\nabla \cdot {\boldsymbol {u}}\right)}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_419_chunk_0 | Littlewood–Offord problem | In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if V is a vector space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_420_chunk_0 | Littlewood–Offord problem | By subtracting 1 2 ∑ i = 1 n v i {\displaystyle {\frac {1}{2}}\sum _{i=1}^{n}v_{i}} from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form ∑ i = 1 n ε i v i {\displaystyle \su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_421_chunk_0 | Pseudoreal representation | In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map j: V → V {\displaystyle j\colon V\to V} which satisfies j 2 = − 1. {\displaystyle j^{2}=-1.} Together with the imagi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_422_chunk_0 | Pseudoreal representation | From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the ide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_423_chunk_0 | Symplectic representation | In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form ω: V × V → F {\displaystyle \omega \colon V\times V\to \mathbb {F}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_424_chunk_0 | Smoluchowski equation | In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient σ ( X t , t ) {\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_425_chunk_0 | Margrabe's formula | In mathematical finance, Margrabe's formula is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (PhD Chicago) in 1978. Margrabe's paper has been cited by over 2000 subsequent articles. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_426_chunk_0 | Replicating portfolio | In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties (especially cash flows). This is meant in two distinct senses: static replication, where the portfolio has the same cash flows as the reference asset (and no changes need to be ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_427_chunk_0 | Equivalent Martingale Measure | In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives du... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_428_chunk_0 | Convexity (finance) | In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function. Geometrically, the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_429_chunk_0 | Kelly criterion | If portfolio weights are largely a function of estimation errors, then Ex-post performance of a growth-optimal portfolio may differ fantastically from the ex-ante prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_430_chunk_0 | Black-Scholes equation | In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_431_chunk_0 | Constant elasticity of variance model | In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was developed by J... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_432_chunk_0 | Cox–Ingersoll–Ross model | In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" (short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_433_chunk_0 | Doob decomposition theorem | In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option. Let X = (X0, X1, . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_434_chunk_0 | Doob decomposition theorem | , XN) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F0, F1, . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_435_chunk_0 | Doob decomposition theorem | , FN), and let Q {\displaystyle \mathbb {Q} } denote an equivalent martingale measure. Let U = (U0, U1, . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_436_chunk_0 | Doob decomposition theorem | . , UN) denote the Snell envelope of X with respect to Q {\displaystyle \mathbb {Q} } . The Snell envelope is the smallest Q {\displaystyle \mathbb {Q} } -supermartingale dominating X and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_437_chunk_0 | Doob decomposition theorem | Let U = M + A denote the Doob decomposition with respect to Q {\displaystyle \mathbb {Q} } of the Snell envelope U into a martingale M = (M0, M1, . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_438_chunk_0 | Doob decomposition theorem | , MN) and a decreasing predictable process A = (A0, A1, . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_439_chunk_0 | Doob decomposition theorem | , AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way is τ max := { N if A N = 0 , min { n ∈ { 0 , … , N − 1 } ∣ A n + 1 < 0 } if A N < 0. {\displaystyle \tau _{\text{max}}:={\begin{cases}N&{\text{if }}A_{N}=0,\\\min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}&{\text{if }}A_{N}<... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_440_chunk_0 | Doob decomposition theorem | . . , N − 1}, hence τmax is indeed a stopping time. It gives the last moment before the discounted value of the American option will drop in expectation; up to time τmax the discounted value process U is a martingale with respect to Q {\displaystyle \mathbb {Q} } . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_441_chunk_0 | SABR volatility model | In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_442_chunk_0 | Local volatility | In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form d S t = ( r t − d t ) S t d t + σ t S t d W t {\displaystyle dS_{t}=(r_{t}-d_{t})S_{t}\,dt+\sigma _{t}S_{t}\,dW_{t}} ,under the risk neutral measure, where r t {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_443_chunk_0 | Local volatility | And when such volatility is merely a function of the current underlying asset level St and of time t, we have a local volatility model. The local volatility model is a useful simplification of the stochastic volatility model. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_444_chunk_0 | Thompson uniqueness theorem | In mathematical finite group theory, Thompson's original uniqueness theorem (Feit & Thompson 1963, theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1970) gave a shorter proof of the uniq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_445_chunk_0 | Group of symplectic type | In mathematical finite group theory, a p-group of symplectic type is a p-group such that all characteristic abelian subgroups are cyclic. According to Thompson (1968, p.386), the p-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an ext... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_446_chunk_0 | Quadratic pair | In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by Thompson (1971), is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimal polynomial (x − 1)2. Thomp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_447_chunk_0 | Baer–Suzuki theorem | In mathematical finite group theory, the Baer–Suzuki theorem, proved by Baer (1957) and Suzuki (1965), states that if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements of the conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_448_chunk_0 | Brauer–Fowler theorem | In mathematical finite group theory, the Brauer–Fowler theorem, proved by Brauer & Fowler (1955), states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another resul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_449_chunk_0 | Dade isometry | In mathematical finite group theory, the Dade isometry is an isometry from class function on a subgroup H with support on a subset K of H to class functions on a group G (Collins 1990, 6.1). It was introduced by Dade (1964) as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_450_chunk_0 | Dempwolff group | In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} of G L 5 ( F 2 ) {\displaystyle \mathrm {GL} _{5}(\mathbb {F} _{2})} by its natural module... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_451_chunk_0 | Gorenstein–Harada theorem | In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada (1973, 1974) in a 464-page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part of the classification of finite simple groups.Finite simple groups of section 2 that rank at least 5, have S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_452_chunk_0 | L-balance theorem | In mathematical finite group theory, the L-balance theorem was proved by Gorenstein & Walter (1975). The letter L stands for the layer of a group, and "balance" refers to the property discussed below. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_453_chunk_0 | Thompson order formula | In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson (Held 1969, p.279), gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Brauer & Fowler (1955). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_454_chunk_0 | Thompson transitivity theorem | In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_455_chunk_0 | Classical involution theorem | In mathematical finite group theory, the classical involution theorem of Aschbacher (1977a, 1977b, 1980) classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic. Berkman (2001) extended the classical in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_456_chunk_0 | Uniqueness case | In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p who... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_457_chunk_0 | No free lunch theorem | In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready, alludes to the saying "no such thing as a free lunch", that is, there are no easy shortcuts to success. It appeared in the 1997 "No Free Lunch Theorems for Optimization". Wolpert had previously deri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_458_chunk_0 | Kimeme | In mathematical folklore, the no free lunch theorem (sometimes pluralized) of David Wolpert and William G. Macready appears in the 1997 "No Free Lunch Theorems for Optimization. "This mathematical result states the need for a specific effort in the design of a new algorithm, tailored to the specific problem to be optim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_459_chunk_0 | Minus-plus sign | In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.If x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_460_chunk_0 | Minus-plus sign | In older texts one occasionally finds (−)n, which means the same. When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_461_chunk_0 | Penrose inequality | In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case. Specifically, if (M, g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_462_chunk_0 | Penrose inequality | This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvatur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_463_chunk_0 | Penrose inequality | This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray gave the first complete pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_464_chunk_0 | Bernstein algebra | In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weight... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_465_chunk_0 | Higman-Sims graph | In mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors. It was first... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_466_chunk_0 | Hall–Higman theorem | In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman (1956, Theorem B), describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_467_chunk_0 | Balance theorem | In mathematical group theory, the balance theorem states that if G is a group with no core then G either has disconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type (Gorenstein 1983, p. 7). The significance of this theorem is that it splits the classification of finite simple groups ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_468_chunk_0 | Green's function number | In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing solutions easier to identify, store, and retrieve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_469_chunk_0 | Perpetuant | Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography. MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weight w is the coefficient of xw of x 2 n − 1 − 1 ( 1 − x 2 ) ( 1 − x 3 ) ⋯ ( 1 − x n ) {\displaystyle {\frac {x^{2^{n-1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_470_chunk_0 | Equisatisfiability | In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_471_chunk_0 | Equisatisfiability | Whereas within equisatisfiable formulae, only the primitive proposition the formula imposes is valued. Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable. Examples of translations invo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_472_chunk_0 | Valuation (logic) | In this context, a valuation begins with an assignment of a truth value to each propositional variable. This assignment can be uniquely extended to an assignment of truth values to all propositional formulas. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_473_chunk_0 | Valuation (logic) | Formulas are built out of atomic formulas using logical connectives and quantifiers. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. Corresponding to each structure is a unique t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_474_chunk_0 | Resolution inference | In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_475_chunk_0 | Gabbay's separation theorem | In mathematical logic and computer science, Gabbay's separation theorem, named after Dov Gabbay, states that any arbitrary temporal logic formula can be rewritten in a logically equivalent "past → future" form. I.e. the future becomes what must be satisfied. This form can be used as execution rules; a MetateM program i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_476_chunk_0 | Mu-recursive function | In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_477_chunk_0 | Mu-recursive function | However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions that can be computed by Markov algorithms. The subset of all total recursive functions with values... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_478_chunk_0 | Univalence axiom | In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_479_chunk_0 | Calculus of Constructions | In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_480_chunk_0 | Lambda-mu calculus | In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_481_chunk_0 | Lambda-mu calculus | According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law. Semantically these operat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_482_chunk_0 | Two-variable logic with counting | In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulae can be written using only two different variables. This fragment is usually studied without function symbols. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_483_chunk_0 | Analytical hierarchy | In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, N {\displaystyle \mathbb {N} } , a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_484_chunk_0 | Implication graph | In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph G = (V, E) composed of vertex set V and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edge from vertex u to vertex v represents the material implication "If the li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_485_chunk_0 | Universal Horn theory | In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_486_chunk_0 | Completeness (logic) | In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_487_chunk_0 | Skolem paradox | In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absolutene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_488_chunk_0 | Skolem paradox | 295). Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_489_chunk_0 | Collapsing function | In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_490_chunk_0 | Collapsing function | An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e., defining a system of notations up to the Bachmann–Howard ordinal). The use and definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinal analysis, s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_491_chunk_0 | Ordinal notation | In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_492_chunk_0 | Ordinal notation | A recursive ordinal notation must satisfy the following two additional properties: the subset of natural numbers is a recursive set the induced well-ordering on the subset of natural numbers is a recursive relationThere are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_493_chunk_0 | Ordinal notation | In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_494_chunk_0 | Minsky machine | In mathematical logic and theoretical computer science, a register machine is a generic class of abstract machines used in a manner similar to a Turing machine. All the models are Turing equivalent. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_495_chunk_0 | Convergent term rewriting system | In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") toge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_496_chunk_0 | Löwenheim number | In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds. They are named after Leopold Löwenheim, who proved that these exist for a very broad class of logics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_497_chunk_0 | Craig interpolation | In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable symbol in common, then there is a formula ρ, called an interpolant, such tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_498_chunk_0 | Craig's theorem | In mathematical logic, Craig's theorem (also known as Craig's trick) states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_499_chunk_0 | Diaconescu theorem | In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle or restricted forms of it. The theorem was discovered in 1975 by Radu Diaconescu and later by Goodman and Myhill. Already in 1967, Errett Bishop posed t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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