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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/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
The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals, where two ideals I1, I2 are called coprime if 1</sub> + ''I''<sub>2</sub>"}},"i":0}}]}'>R = I1 + I2
https://en.wikipedia.org/wiki/List_of_differential_geometry_topics
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https://en.wikipedia.org/wiki/Consistent_estimator
If the sequence of estimates can be mathematically shown to converge in probability to the true value θ0, it is called a consistent estimator; otherwise the estimator is said to be inconsistent
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/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
But we can expand the definition for a more general Noetherian ring: If R {\displaystyle R} is a commutative Noetherian ring, then an R-module M is called Cohen–Macaulay module if M m {\displaystyle M_{\mathrm {m} }} is a Cohen–Macaulay module for all maximal ideals m ∈ S u p p ( M ) {\displaystyle \mathrm {m} \
https://en.wikipedia.org/wiki/Noether's_second_theorem
The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least action
https://en.wikipedia.org/wiki/Signal_recovery
Bias can be desirable if false alarms and misses lead to different costs
https://en.wikipedia.org/wiki/Module_homomorphism
Explicitly, if M and N are left modules over a ring R, then a function f : M → N {\displaystyle f:M\to N} is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,
f ( x + y ) = f ( x ) + f ( y ) , {\displaystyle f(x+y)=f(x)+f(y),} f ( r x ) = r f ( x )
https://en.wikipedia.org/wiki/Discrete_space
The discrete topology is the finest topology that can be given on a set
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/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/Korean_verbs
Some verbs can be either stative or active, depending on meaning
https://en.wikipedia.org/wiki/Canonical_isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping
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/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
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
x n ⟩ / ( x i x j − q i j x j x i ) {\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})} , called a quantum plane, where q i j ∈ C {\displaystyle q_{ij}\in \mathbb {C} }
Any Clifford algebra can be described explicitly using an algebra presentation: given an F {\
https://en.wikipedia.org/wiki/Moment_generating_function
As its name implies, the moment generating function can be used to compute a distribution’s moments: the n-th moment about 0 is the n-th derivative of the moment generating function, evaluated at 0
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/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/Truth_tables
[1] In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid
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
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
https://en.wikipedia.org/wiki/Russell's_paradox
If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves
https://en.wikipedia.org/wiki/White_noise
Even a binary signal which can only take on the values 1 or -1 will be white if the sequence is statistically uncorrelated
https://en.wikipedia.org/wiki/Variable_(mathematics)
The values a variable can take are usually of the same kind, often numbers