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https://en.wikipedia.org/wiki/Theory_of_knots
Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself
https://en.wikipedia.org/wiki/Formal_power_series_ring
} If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1 by the Cauchy–Hadamard theorem
https://en.wikipedia.org/wiki/Saul_Kripke
|edition=Fall 2016 |language=en}}</ref> [[direct reference theory]], criticism of the [[Frege–Russell view]])<br />[[Admissible ordinal]]<br />[[Kripke structure]]<br />[[Rigid designator|Rigid]] vs
https://en.wikipedia.org/wiki/Tragedy_of_the_commons
He postulated that if a herder put more than his allotted number of cattle on the common, overgrazing could result
https://en.wikipedia.org/wiki/Nakayama's_lemma
Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field
https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
A variation of this statement for filters on sets is known as the ultrafilter lemma
https://en.wikipedia.org/wiki/Basis_(linear_algebra)
: bases) if every element of V can be written in a unique way as a finite linear combination of elements of B
https://en.wikipedia.org/wiki/Prime_ideal
This generalizes the following property of prime numbers, known as Euclid's lemma: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b
https://en.wikipedia.org/wiki/Jerzy_Neyman
[6] Another noted contribution is the Neyman–Pearson lemma, the basis of hypothesis testing
https://en.wikipedia.org/wiki/Elliott_H._Lieb
entropy principle]]<br>[[AKLT model|Affleck-Kennedy-Lieb-Tasaki model]]<br>[[Borell–Brascamp–Lieb inequality]]<br>[[Brascamp–Lieb inequality]]<br>[[Brezis–Lieb lemma]]<br>[[Strong subadditivity of quantum entropy#Carlen-Lieb Extension|Carlen–Lieb extension]]<br>[[Temperley–Lieb algebra]]<br>[[Dicke model]
https://en.wikipedia.org/wiki/Integral_extension
over a subring A of B if b is a root of some monic polynomial over A
https://en.wikipedia.org/wiki/Skew-field
In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;[6] every division ring arises in this fashion from some simple module
https://en.wikipedia.org/wiki/Noether's_second_theorem
Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations
https://en.wikipedia.org/wiki/Measurable_function
} A function f : X → Y {\displaystyle f:X\to Y} is said to be measurable if for every E ∈ T {\displaystyle E\in \mathrm {T} } the pre-image of E {\displaystyle E} under f {\displaystyle f} is in Σ {\displaystyle \Sigma } ; that is, for all E ∈ T {\displaystyle E\in \mathrm {T} } f − 1 ( E ) := { x ∈ X ∣ f
https://en.wikipedia.org/wiki/Local_ring
1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit
https://en.wikipedia.org/wiki/Faithful_module
For example, if R {\displaystyle R} is a commutative ring and I {\displaystyle I} is an ideal of R {\displaystyle R} , we can consider the quotient ring R / I {\displaystyle R/I} to be an R {\displaystyle R} -module
https://en.wikipedia.org/wiki/Commutator_(ring_theory)
[1]
This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg)
https://en.wikipedia.org/wiki/Ideal_(order_theory)
In this case, the following equivalent definition can be given: a subset I of a lattice ( P , ≤ ) {\displaystyle (P,\leq )} is an ideal if and only if it is a lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element x ∨ y {\displaystyle x\vee y} of P is also
https://en.wikipedia.org/wiki/Unmixed_ideal
finitely generated) R-module M ≠ 0 {\displaystyle M\neq 0} is a Cohen–Macaulay module if d e p t h ( M ) = d i m ( M ) {\displaystyle \mathrm {depth} (M)=\mathrm {dim} (M)} (in general we have: d e p t h ( M ) ≤ d i m ( M ) {\displaystyle \mathrm {depth} (M)\leq \mathrm {dim} (M)} , see Auslander–Buchsbaum
https://en.wikipedia.org/wiki/Discrete_space
X {\displaystyle X} is defined by letting every superset of the diagonal { ( x , x ) : x ∈ X } {\displaystyle \{(x,x):x\in X\}} in X × X {\displaystyle X\times X} be an entourage, and X {\displaystyle X} is a discrete uniform space if it is equipped with its discrete uniformity
https://en.wikipedia.org/wiki/Embedding
[3] More explicitly, an injective continuous map f : X → Y {\displaystyle f:X\to Y} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is a topological embedding if f {\displaystyle f} yields a homeomorphism between X {\displaystyle X} and f ( X ) {\displaystyle f(X)} (where f ( X ) {\displaystyle
https://en.wikipedia.org/wiki/Projective_dimension
However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem)
https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
[2]
On the other hand, a classical result[3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with 1"}},"i":0}}]}'>α + β > 1
https://en.wikipedia.org/wiki/Monotonic
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\
https://en.wikipedia.org/wiki/Monotonic_function
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\
https://en.wikipedia.org/wiki/Noncommutative_ring
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https://en.wikipedia.org/wiki/Complex_analysis
}
A complex function is continuous if and only if its associated vector-valued function of two variables is also continuous
https://en.wikipedia.org/wiki/Non-associative_ring
That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative
https://en.wikipedia.org/wiki/Canonical_isomorphism
Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as A ≅ B {\displaystyle A\cong B}
https://en.wikipedia.org/wiki/Bijective
A function is bijective if and only if it is invertible; that is, a function f : X → Y {\displaystyle f:X\to Y} is bijective if and only if there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f, such that each of the two ways for composing the two functions produces an identity function: g ( f