Datasets:
thewordsmiths
/
wiki_stem

Dataset card Data Studio Files
xet
Community
content_id
stringlengths
14
14
page_title
stringlengths
1
250
section_title
stringlengths
1
1.26k
⌀
breadcrumb
stringlengths
1
1.39k
text
stringlengths
9
3.55k
c_pv7tf88ek5al
Theory of
Mathematical
Theoretical_framework > Mathematical
In mathematics the use of the term theory is different, necessarily so, since mathematics contains no explanations of natural phenomena, per se, even though it may help provide insight into natural systems or be inspired by them. In the general sense, a mathematical theory is a branch of or topic in mathematics, such as Set theory, Number theory, Group theory, Probability theory, Game theory, Control theory, Perturbation theory, etc., such as might be appropriate for a single textbook. In the same sense, but more specifically, the word theory is an extensive, structured collection of theorems, organized so that the proof of each theorem only requires the theorems and axioms that preceded it (no circular proofs), occurs as early as feasible in sequence (no postponed proofs), and the whole is as succinct as possible (no redundant proofs). Ideally, the sequence in which the theorems are presented is as easy to understand as possible, although illuminating a branch of mathematics is the purpose of textbooks, rather than the mathematical theory they might be written to cover.
c_54t75mh5lbcd
Map coloring
Mathematics
Map_coloring > Mathematics
In mathematics there is a close link between map coloring and graph coloring, since every map showing different areas has a corresponding graph. By far the most famous result in this area is the four color theorem, which states that any planar map can be colored with at most four colors.
c_ahdoz27k1rpp
No-go theorem
Proof of impossibility
No-go_theorem > Proof of impossibility
In mathematics there is the concept of proof of impossibility referring to problems impossible to solve. The difference between this impossibility and that of the no-go theorems is: a proof of impossibility states a category of logical proposition that may never be true; a no-go theorem instead presents a sequence of events that may never occur.
c_bk139jogh1cc
Angled bracket
Curly brackets in mathematics
Curly_brackets > Curly brackets > Uses of { } > Curly brackets in mathematics
In mathematics they delimit sets and are often also used to denote the Poisson bracket between two quantities. In ring theory, braces denote the anticommutator where {a, b} is defined as a b + b a .
c_j6tat5zvkfdm
Parthasarathy's theorem
Summary
Parthasarathy's_theorem
In mathematics – and in particular the study of games on the unit square – Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem. It states that a particular class of games has a mixed value, provided that at least one of the players has a strategy that is restricted to absolutely continuous distributions with respect to the Lebesgue measure (in other words, one of the players is forbidden to use a pure strategy). The theorem is attributed to the Indian mathematician Thiruvenkatachari Parthasarathy.
c_uosmcyut8cov
Stechkin's lemma
Summary
Stechkin's_lemma
In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the ℓq norm of the tail of a sequence, when the whole sequence is known to have finite ℓp norm. Here, the term "tail" means those terms in the sequence that are not among the N largest terms, for an arbitrary natural number N. Stechkin's lemma is often useful when analysing best-N-term approximations to functions in a given basis of a function space. The result was originally proved by Stechkin in the case q = 2 {\displaystyle q=2} .
c_b3gr9nvgop0d
Bochner measurable function
Summary
Bochner_measurable_function
In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e., f ( t ) = lim n → ∞ f n ( t ) for almost every t , {\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,} where the functions f n {\displaystyle f_{n}} each have a countable range and for which the pre-image f n − 1 ( { x } ) {\displaystyle f_{n}^{-1}(\{x\})} is measurable for each element x. The concept is named after Salomon Bochner. Bochner-measurable functions are sometimes called strongly measurable, μ {\displaystyle \mu } -measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).
c_v70xczh0jaw4
Densely defined
Summary
Densely_defined
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".
c_ti47wi8bxm5x
Itō diffusion
Summary
Itō_diffusion
In mathematics – specifically, in stochastic analysis – an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.
c_du7bxp0nw831
Doob's martingale convergence theorems
Summary
Lévy's_zero–one_law
In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.
c_qdid3qd9hmlc
Bochner identity
Summary
Bochner_identity
In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner.
c_aej9azz7ffh7
Geodesically convex
Summary
Geodesically_convex
In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function.
c_krnvcqlbxxwi
Berezin transform
Summary
Berezin_transform
In mathematics — specifically, in complex analysis — the Berezin transform is an integral operator acting on functions defined on the open unit disk D of the complex plane C. Formally, for a function ƒ: D → C, the Berezin transform of ƒ is a new function Bƒ: D → C defined at a point z ∈ D by ( B f ) ( z ) = ∫ D ( 1 − | z | 2 ) 2 | 1 − z w ¯ | 4 f ( w ) d A ( w ) , {\displaystyle (Bf)(z)=\int _{D}{\frac {(1-|z|^{2})^{2}}{|1-z{\bar {w}}|^{4}}}f(w)\,\mathrm {d} A(w),} where w denotes the complex conjugate of w and d A {\displaystyle \mathrm {d} A} is the area measure. It is named after Felix Alexandrovich Berezin.
c_hgh5ftj03rqa
Maximising measure
Summary
Maximising_measure
In mathematics — specifically, in ergodic theory — a maximising measure is a particular kind of probability measure. Informally, a probability measure μ is a maximising measure for some function f if the integral of f with respect to μ is "as big as it can be". The theory of maximising measures is relatively young and quite little is known about their general structure and properties.
c_j72ops06bimx
Assouad dimension
Summary
Assouad_dimension
In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by Patrice Assouad in his 1977 PhD thesis and later published in 1979, although the same notion had been studied in 1928 by Georges Bouligand. As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.
c_cqzl018hrr6z
Sphere measure
Summary
Sphere_measure
In mathematics — specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1.
c_1h5csbgen0uj
Rate function
Summary
Rate_function
In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation principle. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities. A rate function is also called a Cramér function, after the Swedish probabilist Harald Cramér.
c_716je27skptr
Contraction principle (large deviations theory)
Summary
Contraction_principle_(large_deviations_theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space via a continuous function.
c_hoyycdxyrbpa
Tilted large deviation principle
Summary
Tilted_large_deviation_principle
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i.e. integration against an exponential functional. It can be seen as an alternative formulation of Varadhan's lemma.
c_erqzkomy8hx0
Cylindrical σ-algebra
Summary
Cylindrical_σ-algebra
In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces. For a product space, the cylinder σ-algebra is the one that is generated by cylinder sets. In the context of a Banach space X , {\displaystyle X,} the cylindrical σ-algebra Cyl ⁡ ( X ) {\displaystyle \operatorname {Cyl} (X)} is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on X {\displaystyle X} is a measurable function. In general, Cyl ⁡ ( X ) {\displaystyle \operatorname {Cyl} (X)} is not the same as the Borel σ-algebra on X , {\displaystyle X,} which is the coarsest σ-algebra that contains all open subsets of X . {\displaystyle X.}
c_1nxqi9qcs458
Malliavin's absolute continuity lemma
Summary
Malliavin's_absolute_continuity_lemma
In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus. Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.
c_7cypfbyf54c7
Perfect measure
Summary
Perfect_measure
In mathematics — specifically, in measure theory — a perfect measure (or, more accurately, a perfect measure space) is one that is "well-behaved" in some sense. Intuitively, a perfect measure μ is one for which, if we consider the pushforward measure on the real line R, then every measurable set is "μ-approximately a Borel set". The notion of perfectness is closely related to tightness of measures: indeed, in metric spaces, tight measures are always perfect.
c_gofdew96pu0h
Concentration dimension
Summary
Concentration_dimension
In mathematics — specifically, in probability theory — the concentration dimension of a Banach space-valued random variable is a numerical measure of how "spread out" the random variable is compared to the norm on the space.
c_1ubwqg3j47y2
Dynkin's formula
Summary
Dynkin's_formula
In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin.
c_wisvpzu2c525
Green measure
Summary
Green_measure
In mathematics — specifically, in stochastic analysis — the Green measure is a measure associated to an Itō diffusion. There is an associated Green formula representing suitably smooth functions in terms of the Green measure and first exit times of the diffusion. The concepts are named after the British mathematician George Green and are generalizations of the classical Green's function and Green formula to the stochastic case using Dynkin's formula.
c_y73q3i66jrig
Infinitesimal generator (stochastic processes)
Summary
Infinitesimal_generator_(stochastic_processes)
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation, which describes the evolution of statistics of the process; its L2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density functions of the process. The Kolmogorov forward equation in the notation is just ∂ t ρ = A ∗ ρ {\displaystyle \partial _{t}\rho ={\mathcal {A}}^{*}\rho } , where ρ {\displaystyle \rho } is the probability density function, and A ∗ {\displaystyle {\mathcal {A}}^{*}} is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that.
c_0o4jtlxumv93
Besov measure
Summary
Besov_measure
In mathematics — specifically, in the fields of probability theory and inverse problems — Besov measures and associated Besov-distributed random variables are generalisations of the notions of Gaussian measures and random variables, Laplace distributions, and other classical distributions. They are particularly useful in the study of inverse problems on function spaces for which a Gaussian Bayesian prior is an inappropriate model. The construction of a Besov measure is similar to the construction of a Besov space, hence the nomenclature.
c_422ei4yzid04
Semi-elliptic operator
Summary
Semi-elliptic_operator
In mathematics — specifically, in the theory of partial differential equations — a semi-elliptic operator is a partial differential operator satisfying a positivity condition slightly weaker than that of being an elliptic operator. Every elliptic operator is also semi-elliptic, and semi-elliptic operators share many of the nice properties of elliptic operators: for example, much of the same existence and uniqueness theory is applicable, and semi-elliptic Dirichlet problems can be solved using the methods of stochastic analysis.
c_1p6ji1j4f1bn
Question Mark
Mathematics and formal logic
Question_Mark > Mathematics and formal logic
In mathematics, "?" commonly denotes Minkowski's question mark function. In equations, it can mean "questioned" as opposed to "defined". U+225F ≟ QUESTIONED EQUAL TO U+2A7B ⩻ LESS-THAN WITH QUESTION MARK ABOVE U+2A7C ⩼ GREATER-THAN WITH QUESTION MARK ABOVEIn linear logic, the question mark denotes one of the exponential modalities that control weakening and contraction.
c_7s8xe4d78bit
0.999...
Summary
0.999...
In mathematics, 0.999... (also written as 0.9 or 0..9) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence. This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1 – rather, "0.999..." and "1" represent exactly the same number.
c_vj884chuujob
0.999...
Summary
0.999...
There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined.
c_z1qv9llt3ofp
0.999...
Summary
0.999...
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.
c_cs5q7lriphtp
1 + 1 + 1 + 1 + ⋯
Summary
1_+_1_+_1_+_1_+_⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑ n = 1 ∞ n 0 {\displaystyle \sum _{n=1}^{\infty }n^{0}} , ∑ n = 1 ∞ 1 n {\displaystyle \sum _{n=1}^{\infty }1^{n}} , or simply ∑ n = 1 ∞ 1 {\displaystyle \sum _{n=1}^{\infty }1} , is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1n can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line ∑ n = 1 ∞ 1 = + ∞ , {\displaystyle \sum _{n=1}^{\infty }1=+\infty \,,} since its sequence of partial sums increases monotonically without bound. Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function: ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 1 − 2 1 − s ∑ n = 1 ∞ ( − 1 ) n + 1 n s .
c_36bh0kgjw8tx
1 + 1 + 1 + 1 + ⋯
Summary
1_+_1_+_1_+_1_+_⋯
{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1-2^{1-s}}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}\,.} The two formulas given above are not valid at zero however, but the analytic continuation is. ζ ( s ) = 2 s π s − 1 sin ⁡ ( π s 2 ) Γ ( 1 − s ) ζ ( 1 − s ) , {\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\!,} Using this one gets (given that Γ(1) = 1), ζ ( 0 ) = 1 π lim s → 0 sin ⁡ ( π s 2 ) ζ ( 1 − s ) = 1 π lim s → 0 ( π s 2 − π 3 s 3 48 + .
c_1w3phydxq834
1 + 1 + 1 + 1 + ⋯
Summary
1_+_1_+_1_+_1_+_⋯
( − 1 s + . . . )
c_wueu3vdwfkhv
1 + 1 + 1 + 1 + ⋯
Summary
1_+_1_+_1_+_1_+_⋯
= − 1 2 {\displaystyle \zeta (0)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \sin \left({\frac {\pi s}{2}}\right)\ \zeta (1-s)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \left({\frac {\pi s}{2}}-{\frac {\pi ^{3}s^{3}}{48}}+...\right)\ \left(-{\frac {1}{s}}+...\right)=-{\frac {1}{2}}} where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + ⋯ = ζ(0) = −1/2. Emilio Elizalde presents a comment from others about the series: In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + ⋯ = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.
c_m5fnciv0ataw
1 − 2 + 3 − 4 + ⋯
Summary
1_−_2_+_3_−_4_+_⋯
In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation: A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts.
c_cs1vayk61yc9
1 − 2 + 3 − 4 + ⋯
Summary
1_−_2_+_3_−_4_+_⋯
Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the few methods that do not sum 1 − 2 + 3 − 4 + ..., so the series is an example where a slightly stronger method, such as Abel summation, is required. The series 1 − 2 + 3 − 4 + ... is closely related to Grandi's series 1 − 1 + 1 − 1 + .... Euler treated these two as special cases of the more general sequence 1 − 2n + 3n − 4n + ..., where n = 1 and n = 0 respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function.
c_qhpy05564vtr
2E6 (mathematics)
Summary
2E6_(mathematics)
In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L) (thinking of the group as a subgroup of E6(L) fixed by an outer involution). Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).
c_1w11d01kri4o
6-sphere coordinates
Summary
6-sphere_coordinates
In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} . They are so named because the loci where one coordinate is constant form spheres tangent to the origin from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative). They have nothing whatsoever to do with the 6-sphere, which is an object of considerable interest in its own right.
c_ilgndgvy4t0x
6-sphere coordinates
Summary
6-sphere_coordinates
The three coordinates are u = x x 2 + y 2 + z 2 , v = y x 2 + y 2 + z 2 , w = z x 2 + y 2 + z 2 . {\displaystyle u={\frac {x}{x^{2}+y^{2}+z^{2}}},\quad v={\frac {y}{x^{2}+y^{2}+z^{2}}},\quad w={\frac {z}{x^{2}+y^{2}+z^{2}}}.} Since inversion is its own inverse, the equations for x, y, and z in terms of u, v, and w are similar: x = u u 2 + v 2 + w 2 , y = v u 2 + v 2 + w 2 , z = w u 2 + v 2 + w 2 . {\displaystyle x={\frac {u}{u^{2}+v^{2}+w^{2}}},\quad y={\frac {v}{u^{2}+v^{2}+w^{2}}},\quad z={\frac {w}{u^{2}+v^{2}+w^{2}}}.} This coordinate system is R {\displaystyle R} -separable for the 3-variable Laplace equation.
c_x5pxubu0gcn6
A-equivalence
Summary
A-equivalence
In mathematics, A {\displaystyle {\mathcal {A}}} -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let M {\displaystyle M} and N {\displaystyle N} be two manifolds, and let f , g: ( M , x ) → ( N , y ) {\displaystyle f,g:(M,x)\to (N,y)} be two smooth map germs. We say that f {\displaystyle f} and g {\displaystyle g} are A {\displaystyle {\mathcal {A}}} -equivalent if there exist diffeomorphism germs ϕ: ( M , x ) → ( M , x ) {\displaystyle \phi :(M,x)\to (M,x)} and ψ: ( N , y ) → ( N , y ) {\displaystyle \psi :(N,y)\to (N,y)} such that ψ ∘ f = g ∘ ϕ . {\displaystyle \psi \circ f=g\circ \phi .}
c_anb4uq6ip3sk
A-equivalence
Summary
A-equivalence
In other words, two map germs are A {\displaystyle {\mathcal {A}}} -equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. M {\displaystyle M} ) and the target (i.e. N {\displaystyle N} ). Let Ω ( M x , N y ) {\displaystyle \Omega (M_{x},N_{y})} denote the space of smooth map germs ( M , x ) → ( N , y ) . {\displaystyle (M,x)\to (N,y).}
c_leri8j5axpr6
A-equivalence
Summary
A-equivalence
Let diff ( M x ) {\displaystyle {\mbox{diff}}(M_{x})} be the group of diffeomorphism germs ( M , x ) → ( M , x ) {\displaystyle (M,x)\to (M,x)} and diff ( N y ) {\displaystyle {\mbox{diff}}(N_{y})} be the group of diffeomorphism germs ( N , y ) → ( N , y ) . {\displaystyle (N,y)\to (N,y).} The group G := diff ( M x ) × diff ( N y ) {\displaystyle G:={\mbox{diff}}(M_{x})\times {\mbox{diff}}(N_{y})} acts on Ω ( M x , N y ) {\displaystyle \Omega (M_{x},N_{y})} in the natural way: ( ϕ , ψ ) ⋅ f = ψ − 1 ∘ f ∘ ϕ .
c_cvnaaufbs5yw
A-equivalence
Summary
A-equivalence
{\displaystyle (\phi ,\psi )\cdot f=\psi ^{-1}\circ f\circ \phi .} Under this action we see that the map germs f , g: ( M , x ) → ( N , y ) {\displaystyle f,g:(M,x)\to (N,y)} are A {\displaystyle {\mathcal {A}}} -equivalent if, and only if, g {\displaystyle g} lies in the orbit of f {\displaystyle f} , i.e. g ∈ orb G ( f ) {\displaystyle g\in {\mbox{orb}}_{G}(f)} (or vice versa). A map germ is called stable if its orbit under the action of G := diff ( M x ) × diff ( N y ) {\displaystyle G:={\mbox{diff}}(M_{x})\times {\mbox{diff}}(N_{y})} is open relative to the Whitney topology.
c_twn46xnsi076
A-equivalence
Summary
A-equivalence
Since Ω ( M x , N y ) {\displaystyle \Omega (M_{x},N_{y})} is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking k {\displaystyle k} -jets for every k {\displaystyle k} and taking open neighbourhoods in the ordinary Euclidean sense.
c_si0s4hf4xx5t
A-equivalence
Summary
A-equivalence
Open sets in the topology are then unions of these base sets. Consider the orbit of some map germ o r b G ( f ) .
c_7dnptv24yizh
A-equivalence
Summary
A-equivalence
{\displaystyle orb_{G}(f).} The map germ f {\displaystyle f} is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs ( R n , 0 ) → ( R , 0 ) {\displaystyle (\mathbb {R} ^{n},0)\to (\mathbb {R} ,0)} for 1 ≤ n ≤ 3 {\displaystyle 1\leq n\leq 3} are the infinite sequence A k {\displaystyle A_{k}} ( k ∈ N {\displaystyle k\in \mathbb {N} } ), the infinite sequence D 4 + k {\displaystyle D_{4+k}} ( k ∈ N {\displaystyle k\in \mathbb {N} } ), E 6 , {\displaystyle E_{6},} E 7 , {\displaystyle E_{7},} and E 8 . {\displaystyle E_{8}.}
c_9bmz4q2ksxdt
Abel's formula
Summary
Abel's_formula
In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be generalised to nth-order linear ordinary differential equations. The identity is named after the Norwegian mathematician Niels Henrik Abel. Since Abel's identity relates the different linearly independent solutions of the differential equation, it can be used to find one solution from the other.
c_basqtfni77vn
Abel's formula
Summary
Abel's_formula
It provides useful identities relating the solutions, and is also useful as a part of other techniques such as the method of variation of parameters. It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly. A generalisation to first-order systems of homogeneous linear differential equations is given by Liouville's formula.
c_zcygpaxidzta
Abel's inequality
Summary
Abel's_inequality
In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case.
c_i5yd488su5np
Abel's irreducibility theorem
Summary
Abel's_irreducibility_theorem
In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.Corollaries of the theorem include: If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has 2 {\displaystyle {\sqrt {2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2 {\displaystyle {\sqrt {2}}} as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x). If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.
c_f9j8x168hp47
Abel's summation formula
Summary
Abel's_summation_formula
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.
c_st2kfo5cy8k7
Abel's test
Summary
Abel's_test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.
c_uemqygnncl1r
Abel's Theorem
Summary
Abel's_Theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
c_sr53zvmox462
Tauberian theorem
Summary
Abelian_theorem
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods.
c_ndktcb56z4rl
Tauberian theorem
Summary
Abelian_theorem
There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In the theory of integral transforms, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.
c_2su8g5vvpvxi
Abel–Goncharov interpolation
Summary
Abel–Goncharov_interpolation
In mathematics, Abel–Goncharov interpolation determines a polynomial such that various higher derivatives are the same as those of a given function at given points. It was introduced by Whittaker (1935) and rediscovered by Goncharov (1954).
c_6of3fxbts927
Abhyankar's lemma
Summary
Abhyankar's_lemma
In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified extension of A.
c_6du13piuvtvd
Al-Salam–Carlitz polynomials
Summary
Al-Salam–Carlitz_polynomials
In mathematics, Al-Salam–Carlitz polynomials U(a)n(x;q) and V(a)n(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Waleed Al-Salam and Leonard Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14.24, 14.25) give a detailed list of their properties.
c_whpv3vtlns5m
Alcuin's sequence
Summary
Alcuin's_sequence
In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: x 3 ( 1 − x 2 ) ( 1 − x 3 ) ( 1 − x 4 ) = x 3 + x 5 + x 6 + 2 x 7 + x 8 + 3 x 9 + ⋯ . {\displaystyle {\frac {x^{3}}{(1-x^{2})(1-x^{3})(1-x^{4})}}=x^{3}+x^{5}+x^{6}+2x^{7}+x^{8}+3x^{9}+\cdots .} The sequence begins with these integers: 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21 (sequence A005044 in the OEIS)The nth term is the number of triangles with integer sides and perimeter n. It is also the number of triangles with distinct integer sides and perimeter n + 6, i.e. number of triples (a, b, c) such that 1 ≤ a < b < c < a + b, a + b + c = n + 6. If one deletes the three leading zeros, then it is the number of ways in which n empty casks, n casks half-full of wine and n full casks can be distributed to three persons in such a way that each one gets the same number of casks and the same amount of wine.
c_sijmcv8aqj8o
Alcuin's sequence
Summary
Alcuin's_sequence
This is the generalization of problem 12 appearing in Propositiones ad Acuendos Juvenes ("Problems to Sharpen the Young") usually attributed to Alcuin. That problem is given as, Problem 12: A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil, another 10 were half full, while another 10 were empty. Divide the oil and flasks so that an equal share of the commodities should equally come down to the three sons, both of oil and glass.The term "Alcuin's sequence" may be traced back to D. Olivastro's 1993 book on mathematical games, Ancient Puzzle: Classical Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries (Bantam, New York).The sequence with the three leading zeros deleted is obtained as the sequence of coefficients of the power-series expansion of 1 ( 1 − x 2 ) ( 1 − x 3 ) ( 1 − x 4 ) = 1 + x 2 + x 3 + 2 x 4 + x 5 + 3 x 6 + ⋯ .
c_oy1jz91vpvlq
Alcuin's sequence
Summary
Alcuin's_sequence
{\displaystyle {\frac {1}{(1-x^{2})(1-x^{3})(1-x^{4})}}=1+x^{2}+x^{3}+2x^{4}+x^{5}+3x^{6}+\cdots .} This sequence has also been called Alcuin's sequence by some authors. == References ==
c_mskghmhy93pa
Aleksandrov–Clark measure
Summary
Aleksandrov–Clark_measure
In mathematics, Aleksandrov–Clark (AC) measures are specially constructed measures named after the two mathematicians, A. B. Aleksandrov and Douglas Clark, who discovered some of their deepest properties. The measures are also called either Aleksandrov measures, Clark measures, or occasionally spectral measures. AC measures are used to extract information about self-maps of the unit disc, and have applications in a number of areas of complex analysis, most notably those related to operator theory. Systems of AC measures have also been constructed for higher dimensions, and for the half-plane.
c_flr3e2tnhgpa
AB5 category
Summary
AB5_category
In mathematics, Alexander Grothendieck (1957) in his "Tôhoku paper" introduced a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according to the axiom (AB2). AB3 categories are abelian categories possessing arbitrary coproducts (hence, by the existence of quotients in abelian categories, also all colimits). AB5 categories are the AB3 categories in which filtered colimits of exact sequences are exact. Grothendieck categories are the AB5 categories with a generator.
c_hp4mx5choabm
Alexander duality
Summary
Alexander_duality
In mathematics, Alexander duality refers to a duality theory initiated by a result of J. W. Alexander in 1915, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other manifold. It is generalized by Spanier–Whitehead duality.
c_20x0nmg1v9xq
Ibn al-Haytham
Mathematical works
Ibn_al-Haytham > Mathematical works
In mathematics, Alhazen built on the mathematical works of Euclid and Thabit ibn Qurra and worked on "the beginnings of the link between algebra and geometry".He developed a formula for summing the first 100 natural numbers, using a geometric proof to prove the formula.
c_u7wpim2a0f07
Anderson acceleration
Summary
Anderson_acceleration
In mathematics, Anderson acceleration, also called Anderson mixing, is a method for the acceleration of the convergence rate of fixed-point iterations. Introduced by Donald G. Anderson, this technique can be used to find the solution to fixed point equations f ( x ) = x {\displaystyle f(x)=x} often arising in the field of computational science.
c_50yfwmq0mbgb
Anderson's theorem
Summary
Anderson's_theorem
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function f over an n-dimensional convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph of f can be thought of as a hill with a single peak over the origin; however, for n ≥ 2, the proof is not entirely obvious, as there may be points x of the body K where the value f(x) is larger than at the corresponding translate of x. Anderson's theorem, named after Theodore Wilbur Anderson, also has an interesting application to probability theory.
c_gv3pfaawwojh
Appell series
Summary
Appell_series
In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series 2F1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.
c_73vqitp8g1yf
Apéry's constant
Summary
Apéry's_constant
In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = lim n → ∞ ( 1 1 3 + 1 2 3 + ⋯ + 1 n 3 ) , {\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right),\end{aligned}}} where ζ is the Riemann zeta function. It has an approximate value of ζ(3) = 1.202056903159594285399738161511449990764986292… (sequence A002117 in the OEIS).The constant is named after Roger Apéry. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient, which appear occasionally in physics, for instance, when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.
c_vqult7pt0gxi
Apéry's theorem
Summary
Apéry's_theorem
In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + ⋯ = 1.2020569 … {\displaystyle \zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.2020569\ldots } cannot be written as a fraction p / q {\displaystyle p/q} where p and q are integers. The theorem is named after Roger Apéry. The special values of the Riemann zeta function at even integers 2 n {\displaystyle 2n} ( n > 0 {\displaystyle n>0} ) can be shown in terms of Bernoulli numbers to be irrational, while it remains open whether the function's values are in general rational or not at the odd integers 2 n + 1 {\displaystyle 2n+1} ( n > 1 {\displaystyle n>1} ) (though they are conjectured to be irrational).
c_hnfnzcxrlk9x
Arithmetic scheme
Summary
Arakelov_geometry
In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
c_t4ooow2yrvfu
Arakelyan's theorem
Summary
Arakelyan's_theorem
In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.
c_r5osbdanem15
Arf semigroup
Summary
Arf_semigroup
In mathematics, Arf semigroups are certain subsets of the non-negative integers closed under addition, that were studied by Cahit Arf (1948). They appeared as the semigroups of values of Arf rings. A subset of the integers forms a monoid if it includes zero, and if every two elements in the subset have a sum that also belongs to the subset. In this case, it is called a "numerical semigroup". A numerical semigroup is called an Arf semigroup if, for every three elements x, y, and z with z = min(x, y, and z), the semigroup also contains the element x + y − z. For instance, the set containing zero and all even numbers greater than 10 is an Arf semigroup.
c_pxxnjwi523k9
Arnold's cat map
Summary
Arnold's_cat_map
In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name.Thinking of the torus T 2 {\displaystyle \mathbb {T} ^{2}} as the quotient space R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} , Arnold's cat map is the transformation Γ: T 2 → T 2 {\displaystyle \Gamma :\mathbb {T} ^{2}\to \mathbb {T} ^{2}} given by the formula Γ ( x , y ) = ( 2 x + y , x + y ) mod 1 . {\displaystyle \Gamma (x,y)=(2x+y,x+y){\bmod {1}}.} Equivalently, in matrix notation, this is Γ ( ) = mod 1 = mod 1 . {\displaystyle \Gamma \left({\begin{bmatrix}x\\y\end{bmatrix}}\right)={\begin{bmatrix}2&1\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}{\bmod {1}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}{\bmod {1}}.} That is, with a unit equal to the width of the square image, the image is sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the square.
c_9cnppssflty0
Arnold's spectral sequence
Summary
Arnold's_spectral_sequence
In mathematics, Arnold's spectral sequence (also spelled Arnol'd) is a spectral sequence used in singularity theory and normal form theory as an efficient computational tool for reducing a function to canonical form near critical points. It was introduced by Vladimir Arnold in 1975.
c_nj0xsp2wsue4
Artin's criterion
Summary
Artin's_criterion
In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces or as Algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of pointed curves.
c_kl9by6qz4zam
Artin-Verdier duality
Summary
Artin–Verdier_duality
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Michael Artin and Jean-Louis Verdier (1964), that generalizes Tate duality. It shows that, as far as etale (or flat) cohomology is concerned, the ring of integers in a number field behaves like a 3-dimensional mathematical object.
c_hfv95rfezmyb
Atkin–Lehner involution
Summary
Atkin–Lehner_involution
In mathematics, Atkin–Lehner theory is part of the theory of modular forms describing when they arise at a given integer level N in such a way that the theory of Hecke operators can be extended to higher levels. Atkin–Lehner theory is based on the concept of a newform, which is a cusp form 'new' at a given level N, where the levels are the nested congruence subgroups: Γ 0 ( N ) = { ( a b c d ) ∈ SL ( 2 , Z ): c ≡ 0 ( mod N ) } {\displaystyle \Gamma _{0}(N)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv 0{\pmod {N}}\right\}} of the modular group, with N ordered by divisibility. That is, if M divides N, Γ0(N) is a subgroup of Γ0(M). The oldforms for Γ0(N) are those modular forms f(τ) of level N of the form g(d τ) for modular forms g of level M with M a proper divisor of N, where d divides N/M.
c_vbegoftktmv1
Atkin–Lehner involution
Summary
Atkin–Lehner_involution
The newforms are defined as a vector subspace of the modular forms of level N, complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product. The Hecke operators, which act on the space of all cusp forms, preserve the subspace of newforms and are self-adjoint and commuting operators (with respect to the Petersson inner product) when restricted to this subspace. Therefore, the algebra of operators on newforms they generate is a finite-dimensional C*-algebra that is commutative; and by the spectral theory of such operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra.
c_9sa72akktzvr
Auerbach's lemma
Summary
Auerbach's_lemma
In mathematics, Auerbach's lemma, named after Herman Auerbach, is a theorem in functional analysis which asserts that a certain property of Euclidean spaces holds for general finite-dimensional normed vector spaces.
c_fu3jtuw5fyam
BCK algebra
Summary
BCK_algebra
In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics.
c_9nfrddemcsrq
BF-algebra
Summary
BF-algebra
In mathematics, BF algebras are a class of algebraic structures arising out of a symmetric "Yin Yang" concept for Bipolar Fuzzy logic, the name was introduced by Andrzej Walendziak in 2007. The name covers discrete versions, but a canonical example arises in the BF space x of pairs of (false-ness, truth-ness).
c_682oky1a77ru
Baire function
Summary
Baire_function
In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function. (There are other similar, but inequivalent definitions of Baire sets.)
c_ox19zd53ooau
Banach algebra cohomology
Summary
Banach_algebra_cohomology
In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is a cohomology theory defined in a similar way to Hochschild cohomology of an abstract algebra, except that one takes the topology into account so that all cochains and so on are continuous.
c_t96gog3squw9
Beez's theorem
Summary
Beez's_theorem
In mathematics, Beez's theorem, introduced by Richard Beez in 1875, implies that if n > 3 then in general an (n – 1)-dimensional hypersurface immersed in Rn cannot be deformed. == References ==
c_cpgtnsqy60fy
Belyi's theorem
Summary
Belyi's_theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data.
c_0ef4mpxdsd10
Bender–Dunne polynomials
Summary
Bender–Dunne_polynomials
In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Carl M. Bender and Gerald Dunne (1988, 1996). They may be defined by the recursion: P 0 ( x ) = 1 {\displaystyle P_{0}(x)=1} , P 1 ( x ) = x {\displaystyle P_{1}(x)=x} ,and for n > 1 {\displaystyle n>1}: P n ( x ) = x P n − 1 ( x ) + 16 ( n − 1 ) ( n − J − 1 ) ( n + 2 s − 2 ) P n − 2 ( x ) {\displaystyle P_{n}(x)=xP_{n-1}(x)+16(n-1)(n-J-1)(n+2s-2)P_{n-2}(x)} where J {\displaystyle J} and s {\displaystyle s} are arbitrary parameters.
c_qyuhbg4mceqd
Bendixson's inequality
Summary
Bendixson's_inequality
In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices. A special case of this inequality leads to the result that characteristic roots of a real symmetric matrix are always real. The inequality relating to the imaginary parts of characteristic roots of real matrices (Theorem I in ) is stated as: Let A = ( a i j ) {\displaystyle A=\left(a_{ij}\right)} be a real n × n {\displaystyle n\times n} matrix and α = max 1 ≤ i , j ≤ n 1 2 | a i j − a j i | {\displaystyle \alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|} .
c_2n82gwhpbabg
Bendixson's inequality
Summary
Bendixson's_inequality
If λ {\displaystyle \lambda } is any characteristic root of A {\displaystyle A} , then | Im ⁡ ( λ ) | ≤ α n ( n − 1 ) 2 . {\displaystyle \left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}} If A {\displaystyle A} is symmetric then α = 0 {\displaystyle \alpha =0} and consequently the inequality implies that λ {\displaystyle \lambda } must be real. The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in ) is stated as: Let m {\displaystyle m} and M {\displaystyle M} be the smallest and largest characteristic roots of A + A H 2 {\displaystyle {\tfrac {A+A^{H}}{2}}} , then m ≤ Re ⁡ ( λ ) ≤ M {\displaystyle m\leq \operatorname {Re} (\lambda )\leq M} .
c_y0oelw7cnc2v
Berger's isoembolic inequality
Summary
Berger's_isoembolic_inequality
In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the m-dimensional sphere with its usual "round" metric. The theorem is named after the mathematician Marcel Berger, who derived it from an inequality proved by Jerry Kazdan.
c_jg216rxtfw4n
Bernoulli's inequality
Summary
Bernoulli's_inequality
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x {\displaystyle 1+x} . It is often employed in real analysis. It has several useful variants:
c_vxcvbeqommu5
Proof of Bertrand's postulate
Summary
Proof_of_Bertrand's_postulate
In mathematics, Bertrand's postulate (actually now a theorem) states that for each n ≥ 2 {\displaystyle n\geq 2} there is a prime p {\displaystyle p} such that n < p < 2 n {\displaystyle n
c_2wrepabjr0sf
Bhargava factorial
Summary
Bhargava_factorial
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his thesis in Harvard University in 1996. The Bhargava factorial has the property that many number-theoretic results involving the ordinary factorials remain true even when the factorials are replaced by the Bhargava factorials. Using an arbitrary infinite subset S of the set Z {\displaystyle \mathbb {Z} } of integers, Bhargava associated a positive integer with every positive integer k, which he denoted by k !S, with the property that if one takes S = Z {\displaystyle \mathbb {Z} } itself, then the integer associated with k, that is k ! Z {\displaystyle \mathbb {Z} } , would turn out to be the ordinary factorial of k.
c_swhct5y7g4dl
Bhāskara I's sine approximation formula
Summary
Bhāskara_I's_sine_approximation_formula
In mathematics, Bhāskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhāskara I (c. 600 – c. 680), a seventh-century Indian mathematician.
c_rhkd9c3zybnm
Bhāskara I's sine approximation formula
Summary
Bhāskara_I's_sine_approximation_formula
This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhāskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhāskara might have used to arrive at his formula. The formula is elegant and simple, and it enables the computation of reasonably accurate values of trigonometric sines without the use of geometry.
c_dmtbk74x4sa5
Birch's theorem
Summary
Birch's_theorem
In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.
c_bl5y6sxwjajo
Birkhoff factorization
Summary
Birkhoff_factorization
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by George David Birkhoff (1909), is the factorization of an invertible matrix M with coefficients that are Laurent polynomials in z into a product M = M+M0M−, where M+ has entries that are polynomials in z, M0 is diagonal, and M− has entries that are polynomials in z−1. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to Alexander Grothendieck (1957). Birkhoff factorization implies the Birkhoff–Grothendieck theorem of Grothendieck (1957) that vector bundles over the projective line are sums of line bundles. Birkhoff factorization follows from the Bruhat decomposition for affine Kac–Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine general linear group follows from Birkhoff factorization together with the Bruhat decomposition for the ordinary general linear group.
c_uc8j4fecnvlp
Birkhoff interpolation
Summary
Birkhoff_interpolation
In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial p of degree d such that certain derivatives have specified values at specified points: p ( n i ) ( x i ) = y i for i = 1 , … , d , {\displaystyle p^{(n_{i})}(x_{i})=y_{i}\qquad {\mbox{for }}i=1,\ldots ,d,} where the data points ( x i , y i ) {\displaystyle (x_{i},y_{i})} and the nonnegative integers n i {\displaystyle n_{i}} are given. It differs from Hermite interpolation in that it is possible to specify derivatives of p at some points without specifying the lower derivatives or the polynomial itself. The name refers to George David Birkhoff, who first studied the problem.