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Search engine - Wikipedia

https://en.wikipedia.org/wiki/Search_engine
For a search provider, its engine is part of a distributed computing system that can encompass many data centers throughout the world

Search engine - Wikipedia

https://en.wikipedia.org/wiki/Web_search_engine
For a search provider, its engine is part of a distributed computing system that can encompass many data centers throughout the world

Double centralizer theorem - Wikipedia

https://en.wikipedia.org/wiki/Double_centralizer_theorem
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Up to - Wikipedia

https://en.wikipedia.org/wiki/Up_to
For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation R that relates two lists if one can be obtained by reordering (permuting) the other

Conjugate diameters - Wikipedia

https://en.wikipedia.org/wiki/Conjugate_diameters
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Ideal norm - Wikipedia

https://en.wikipedia.org/wiki/Ideal_norm
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Wedderburn–Artin theorem - Wikipedia

https://en.wikipedia.org/wiki/Wedderburn%E2%80%93Artin_theorem
If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra over k

Invariant basis number - Wikipedia

https://en.wikipedia.org/wiki/Invariant_basis_number
Assuming the ultrafilter lemma (a strictly weaker form of the axiom of choice), this result is actually equivalent to the definition given here, and can be taken as an alternative definition

Lithium–sulfur battery - Wikipedia

https://en.wikipedia.org/wiki/Lithium%E2%80%93sulfur_battery
These constraints are interdependent: increasing sulfur loading or lowering E/S improves projected energy density, but can also increase polarization and lower reversible capacity if ion/electron transport and interfacial stability are not maintained

Rule of inference - Wikipedia

https://en.wikipedia.org/wiki/Inference_rule
If an argument with true premises follows a rule of inference then the conclusion cannot be false

Grammatical gender

https://en.wikipedia.org/wiki/Grammatical_gender
com/blog/culture-conscious/201209/masculine-or-feminine-and-why-it-matters |work=Psychology Today |access-date=2 July 2015}}</ref>"}},"i":0}}]}'>[note 1][4] The values present in a given language, of which there are usually two or three, are called the genders of that language

Nakayama's lemma - Wikipedia

https://en.wikipedia.org/wiki/Nakayama's_lemma
Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle in some sense

Euclidean division - Wikipedia

https://en.wikipedia.org/wiki/Euclidean_division
This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total

Collusion - Wikipedia

https://en.wikipedia.org/wiki/Collusion
If the practice of the industry causes more complicated pricing, which is hard for the consumer to understand (such as risk-based pricing, hidden taxes and fees in the wireless industry, negotiable pricing), this can cause competition based on price to be meaningless (because it would be too complicated to explain to

Outline of category theory - Wikipedia

https://en.wikipedia.org/wiki/Outline_of_category_theory
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Boolean prime ideal theorem - Wikipedia

https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
that ideals in a Boolean algebra can be extended to prime ideals

Krull's theorem - Wikipedia

https://en.wikipedia.org/wiki/Krull's_theorem
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Prime ideal - Wikipedia

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

Basis (linear algebra) - Wikipedia

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

Annihilator (ring theory) - Wikipedia

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

Integral element - Wikipedia

https://en.wikipedia.org/wiki/Integral_extension
over a subring A of B if b is a root of some monic polynomial over A

Measurable function - Wikipedia

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

Local ring - Wikipedia

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

Stephen Smale - Wikipedia

https://en.wikipedia.org/wiki/Stephen_Smale
]]<br>[[Morse–Smale system]]<br>[[Morse–Smale diffeomorphism]]<br>[[Palais–Smale compactness condition]]<br>[[Blum–Shub–Smale machine]]<br>[[Smale–Williams attractor]]<br>[[Morse–Palais lemma]]<br>[[Regular homotopy]]<br>[[Sard's theorem]]<br>[[Sphere eversion]]<br>[[Structural stability]]<br>[[Whitehead

Consistent estimator - Wikipedia

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

Division ring - Wikipedia

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

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
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} \

Noether's second theorem

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

Detection theory - Wikipedia

https://en.wikipedia.org/wiki/Signal_recovery
Bias can be desirable if false alarms and misses lead to different costs