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https://en.wikipedia.org/wiki/Group_(mathematics)
In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory
https://en.wikipedia.org/wiki/Glossary_of_commutative_algebra
The Artin-Rees lemma establishes a certain stability of filtration by an ideal
https://en.wikipedia.org/wiki/Shing-Tung_Yau
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https://en.wikipedia.org/wiki/Nicolaus_Copernicus
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https://en.wikipedia.org/wiki/AI
[16] These models can produce incorrect outputs or "hallucinations," unlike symbolic reasoning systems
https://en.wikipedia.org/wiki/Artificial_intelligence
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https://en.wikipedia.org/wiki/Survivor:_Marquesas
com/shows/survivor/episodes/20143 Recap]\n"},"7":{"wt":" OriginalAirDate = {{Start date|2002|2|21}}\n"},"8":{"wt":" Aux4 = 9
https://en.wikipedia.org/wiki/Tragedy_of_the_commons
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https://en.wikipedia.org/wiki/Skype
com/en-us/store/p/skype-preview/9wzdncrfj364 |title=Skype |publisher=Microsoft Store}}</ref>"},"genre":{"wt":"[[Videoconferencing]], [[VoIP]] and [[Instant messaging]]"},"license":{"wt":"[[Proprietary software]]"},"website":{"wt":"{{URL| https://web
https://en.wikipedia.org/wiki/Brooklyn,_New_York
com/codes/newyorkcity/latest/NYCadmin/0-0-0-89 2–105: Official flag; borough of Brooklyn], New York City Administrative Code
https://en.wikipedia.org/wiki/Module_homomorphism
}
The preimage of the zero element under f is called the kernel of f
https://en.wikipedia.org/wiki/Financial_economics
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https://en.wikipedia.org/wiki/Flat_spacetime
In Euclidean space, the distance between any two distinct points is greater than zero, but in Minkowski spacetime, the interval between two distinct events can be zero
https://en.wikipedia.org/wiki/Historicity_of_Jesus
[note 1] However, scholars distinguish between the 'Christ of faith' as presented in the New Testament and the subsequent Christian theology, and a minimal 'Jesus of history', about whom almost nothing can be known
https://en.wikipedia.org/wiki/The_Buddha
Founder of Buddhism
Not to be confused with Budai.
''Śhākyamuni'' ({{literal_translation|Sage of the [[Shakya]]s}})"},"image":{"wt":"Buddha in Sarnath Museum (Dhammajak Mutra).jpg"},"caption":{"wt":"[[Buddha Preaching his First Sermon
https://en.wikipedia.org/wiki/Operator_algebra
From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator
https://en.wikipedia.org/wiki/Embedding
However, this does not apply to immersions; for instance, R P 2 {\displaystyle \mathbb {R} \mathrm {P} ^{2}} can be immersed in R 3 {\displaystyle \mathbb {R} ^{3}} as is explicitly shown by Boy's surface—which has self-intersections
https://en.wikipedia.org/wiki/Ideal_(ring_theory)
An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group
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/Semigroup
A semigroup without an identity element can be easily turned into a monoid by just adding an identity element
https://en.wikipedia.org/wiki/Gardariki
2: \"Their common ancestor can be reconstructed fairly certainly as a proto-Germanic adjective ''*rīkja-'', showing a root ''*rīk-'' and a suffix ''*-ja-'' which forms adjectives
https://en.wikipedia.org/wiki/Dirac_delta_function
[2] Thus it can be represented heuristically as δ ( x ) = { 0 , x ≠ 0 ∞ , x = 0 {\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}} such that ∫ − ∞ ∞ δ ( x ) d x = 1
https://en.wikipedia.org/wiki/List_of_abstract_algebra_topics
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https://en.wikipedia.org/wiki/Glossary_of_ring_theory
"}},"i":2}},"\n\n",{"template":{"target":{"wt":"term","href":
https://en.wikipedia.org/wiki/Neopets
com led in site "stickiness" in May and June, with the average user spending 117 minutes a week
https://en.wikipedia.org/wiki/Gentoo_Linux
-- Weekly minimal install builds are available on Gentoo's site above the old 2008 releases -->"},"latest release date":{"wt":
https://en.wikipedia.org/wiki/Outline_of_category_theory
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https://en.wikipedia.org/wiki/Double_centralizer_theorem
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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/Up_to
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