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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
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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
https://en.wikipedia.org/wiki/Wedderburn%E2%80%93Artin_theorem
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Here we used right modules because R ≅ E n d ( R R ) {\displaystyle R\cong \mathrm {End} (R_{R})} ; if we used left modules R {\displaystyle R} would be isomorphic to the opposite algebra of E n d ( R R ) {\displaystyle \mathrm {End} ({}_{R}R)} , but the proof would still go through
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
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
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
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
https://en.wikipedia.org/wiki/Double_centralizer_theorem
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https://en.wikipedia.org/wiki/Conjugate_diameters
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https://en.wikipedia.org/wiki/Ideal_norm
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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
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/Module_homomorphism
Note: the above verification would "fail" if one used the left R-action in place of the right S-action
https://en.wikipedia.org/wiki/Korean_verbs
If speaking both to and of one's elders, one would use both the formal and the honorific suffixes
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/Bali_myna
[1]
In 2018, there were three locations on Bali where the birds existed in the wild: the West Bali National Park; Bali's small island of Nusa Penida and Begawan Foundation's breeding and release site at Melinggih Kelod, Payangan
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/Variable_(mathematics)
The values a variable can take are usually of the same kind, often numbers
https://en.wikipedia.org/wiki/Theorem
[1]
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven
https://en.wikipedia.org/wiki/Russell's_paradox
If R were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal
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/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/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/Grammatical_gender
Because of this, languages with gender distinction can often use pronouns where in English a noun would have to be repeated in order to avoid confusion
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
https://en.wikipedia.org/wiki/Outline_of_category_theory
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https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
that ideals in a Boolean algebra can be extended to prime ideals
https://en.wikipedia.org/wiki/Krull's_theorem
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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/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/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]