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Commutator - Wikipedia

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)

Ideal (order theory) - Wikipedia

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

Cohen–Macaulay ring - Wikipedia

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

Discrete space - Wikipedia

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

Embedding - Wikipedia

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

Projective module - Wikipedia

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)

Riemann–Stieltjes integral - Wikipedia

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 

Monotonic function - Wikipedia

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\

Monotonic function - Wikipedia

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\

Noncommutative ring - Wikipedia

https://en.wikipedia.org/wiki/Noncommutative_ring
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Complex analysis - Wikipedia

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

Non-associative algebra - Wikipedia

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

Isomorphism - Wikipedia

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} ⁠

Moment generating function - Wikipedia

https://en.wikipedia.org/wiki/Moment_generating_function
If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist

Bijection - Wikipedia

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

Injective module - Wikipedia

https://en.wikipedia.org/wiki/Injective_module
Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q

Poincaré recurrence theorem - Wikipedia

https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem
[3][4] In simple terms, it states that if a system has a finite volume (or total probability measure) and preserves that volume over time, then almost every initial state will eventually return arbitrarily close to its starting position, and it will do so infinitely many times

Lehmann–Scheffé theorem - Wikipedia

https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem
This is where the Lehmann–Scheffé theorem comes in, if T {\displaystyle T} is also complete, that is, for any real-valued measurable function φ {\displaystyle \varphi } it holds E θ [ φ ( T ( X ) ) ] = 0 ⟹ φ ( T ( X ) ) = 0 P θ -a

Slavery in ancient Egypt - Wikipedia

https://en.wikipedia.org/wiki/Slavery_in_ancient_Egypt
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Truth table - Wikipedia

https://en.wikipedia.org/wiki/Truth_tables
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White noise - Wikipedia

https://en.wikipedia.org/wiki/White_noise
[3] In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise

Mathematical proof - Wikipedia

https://en.wikipedia.org/wiki/Mathematical_proof
A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work

Interval (mathematics) - Wikipedia

https://en.wikipedia.org/wiki/Interval_(mathematics)
[1] The endpoints of an interval are its supremum (least upper bound), and its infimum (greatest lower bound), if they exist as real numbers

Semigroup - Wikipedia

https://en.wikipedia.org/wiki/Semigroup
If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called an abelian semigroup

Ideal (ring theory) - Wikipedia

https://en.wikipedia.org/wiki/Ideal_(ring_theory)
If the ring is commutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of an ideal

Autoregressive conditional heteroskedasticity - Wikipedia

https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model

Projective variety - Wikipedia

https://en.wikipedia.org/wiki/Projective_varieties
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial

Axiomatic system - Wikipedia

https://en.wikipedia.org/wiki/Axiomatic_system
It is not, however, evident whether the existence of inverse elements was axiomatic: it would follow from the semantic assumption that groups were permutation groups (permutations being invertible by definition) or geometric transformations with the same property

Autoregressive model - Wikipedia

https://en.wikipedia.org/wiki/Autoregressive_model
Continuing this process shows that the effect of ε 1 {\displaystyle \varepsilon _{1}} never ends, although if the process is stationary then the effect diminishes toward zero in the limit

Mathematical descriptions of the electromagnetic field - Wikipedia

https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field
If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field