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https://en.wikipedia.org/wiki/Injective_module
Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules
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/Variable_(mathematics)
[14] So for example, what would be written in modern notation as: x 3 − 2 x 2 + 10 x − 1 , {\displaystyle x^{3}-2x^{2}+10x-1,} would be written in Diophantus's syncopated notation as:
K υ α ¯ ζ ι ¯ ⋔ Δ υ β ¯ M α ¯ {\displaystyle \mathrm {K} ^{\upsilon }{\overline {\al
https://en.wikipedia.org/wiki/Geometrical_optics
of optics that describes light propagation in terms of rays
https://en.wikipedia.org/wiki/Mathematical_proof
In most mathematical literature, proofs are written in terms of rigorous informal logic
https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms;[1] often the variance is related to the squares of the previous innovations
https://en.wikipedia.org/wiki/Apocalypse
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https://en.wikipedia.org/wiki/Russell's_paradox
[6][7] At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction (to Cantor's theorem)[8], as he told Hilbert and Richard Dedekind by letter
https://en.wikipedia.org/wiki/Hippopotamus
en |access-date=19 November 2021}}</ref>"},"status2":{"wt":"CITES_A2"},"status2_system":{"wt":"CITES"},"status2_ref":{"wt":"<ref name=iucn/>"},"genus":{"wt":"Hippopotamus"},"species":{"wt":"amphibius"},"authority":{"wt":"[[Carl Linnaeus|Linnaeus]], [[10th edition of Systema Naturae|1758]]<ref name=Linnæus>{{cite book
https://en.wikipedia.org/wiki/Common_hippo
en |access-date=19 November 2021}}</ref>"},"status2":{"wt":"CITES_A2"},"status2_system":{"wt":"CITES"},"status2_ref":{"wt":"<ref name=iucn/>"},"genus":{"wt":"Hippopotamus"},"species":{"wt":"amphibius"},"authority":{"wt":"[[Carl Linnaeus|Linnaeus]], [[10th edition of Systema Naturae|1758]]<ref name=Linnæus>{{cite book
https://en.wikipedia.org/wiki/Hodge_star_operator
n}V} is defined in terms of an oriented orthonormal basis { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} of V as:
ω := ± e 1 ∧ ⋯ ∧ e n {\displaystyle \omega :=\pm e_{1}\wedge \cdots \wedge e_{n}} ,
where the sign is free to be chosen and fixed as plus or minus
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
https://en.wikipedia.org/wiki/Autoregressive_model
Then by the AR equation for X 2 {\displaystyle X_{2}} in terms of X 1 {\displaystyle X_{1}} , this affects X 2 {\displaystyle X_{2}} by the amount φ 1 ε 1 {\displaystyle \varphi _{1}\varepsilon _{1}}
https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field
In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking
https://en.wikipedia.org/wiki/Excess_kurtosis
Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis
https://en.wikipedia.org/wiki/Affine_connection
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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/Mathematical_logic
Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory
https://en.wikipedia.org/wiki/Hypothesis_test
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https://en.wikipedia.org/wiki/Hypothesis_testing
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https://en.wikipedia.org/wiki/Outline_of_finance
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https://en.wikipedia.org/wiki/Group_(mathematics)
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https://en.wikipedia.org/wiki/Child_development
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https://en.wikipedia.org/wiki/Frederick_Scherger
Faced with the frightened woman and the enraged husband crying that he would "shoot the bitch", Scherger knocked the man down with a
https://en.wikipedia.org/wiki/Spanish_America
Of equal importance was the Spanish conquest of the Inca Empire by Francisco Pizarro, which would become the Viceroyalty of Peru
https://en.wikipedia.org/wiki/Roman_aqueduct
The water of the River Tiber was close at hand, but would have been polluted by water-borne disease
https://en.wikipedia.org/wiki/Admiralty_Islands_campaign
[6] By July 1943, the Joint Chiefs were considering the possibility of neutralising and bypassing Rabaul, but the navy would still need a forward fleet base
https://en.wikipedia.org/wiki/Gladiatorial_games
Other groups and tribes would join the cast list as Roman territories expanded
https://en.wikipedia.org/wiki/Renewable_energy_development
(February 2019)
This article may require cleanup to meet Wikipedia's quality standards
https://en.wikipedia.org/wiki/Spanish_colonization_of_the_Americas
The process of Spanish settlement, now called "colonization" and the "colonial era" are terms contested by scholars of Latin America[2][3][4] and more generally