In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).} They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.
One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces.[1][2] The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariant of number fields such as the discriminant. Arithmetic groups can be thought of as a vast generalisation of the unit groups of number fields to a noncommutative setting.
The same groups also appeared in analytic number theory as the study of classical modular forms and their generalisations developed. Of course the two topics were related, as can be seen for example in Langlands' computation of the volume of certain fundamental domains using analytic methods.[3] This classical theory culminated with the work of Siegel, who showed the finiteness of the volume of a fundamental domain in many cases.
For the modern theory to begin foundational work was needed, and was provided by the work of Armand Borel, André Weil, Jacques Tits and others on algebraic groups.[4][5] Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra.[6] Meanwhile, there was progress on the general theory of lattices in Lie groups by Atle Selberg, Grigori Margulis, David Kazhdan, M. S. Raghunathan and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972.[7]
In the seventies Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group.[8] Some limited results in this direction had been obtained earlier by Selberg, but Margulis' methods (the use of ergodic-theoretical tools for actions on homogeneous spaces) were completely new in this context and were to be extremely influential on later developments, effectively renewing the old subject of geometry of numbers and allowing Margulis himself to prove the Oppenheim conjecture; stronger results (Ratner's theorems) were later obtained by Marina Ratner.
In another direction the classical topic of modular forms has blossomed into the modern theory of automorphic forms. The driving force behind this effort is mainly the Langlands program initiated by Robert Langlands. One of the main tool used there is the trace formula originating in Selberg's work[9] and developed in the most general setting by James Arthur.[10]
Finally arithmetic groups are often used to construct interesting examples of locally symmetric Riemannian manifolds. A particularly active research topic has been arithmetic hyperbolic 3-manifolds, which as William Thurston wrote,[11] "...often seem to have special beauty."
If G {\displaystyle \mathrm {G} } is an algebraic subgroup of G L n ( Q ) {\displaystyle \mathrm {GL} _{n}(\mathbb {Q} )} for some n {\displaystyle n} then we can define an arithmetic subgroup of G ( Q ) {\displaystyle \mathrm {G} (\mathbb {Q} )} as the group of integer points Γ Γ --> = G L n ( Z ) ∩ ∩ --> G ( Q ) . {\displaystyle \Gamma =\mathrm {GL} _{n}(\mathbb {Z} )\cap \mathrm {G} (\mathbb {Q} ).} In general it is not so obvious how to make precise sense of the notion of "integer points" of a Q {\displaystyle \mathbb {Q} } -group, and the subgroup defined above can change when we take different embeddings G → → --> G L n ( Q ) . {\displaystyle \mathrm {G} \to \mathrm {GL} _{n}(\mathbb {Q} ).}
Thus a better notion is to take for definition of an arithmetic subgroup of G ( Q ) {\displaystyle \mathrm {G} (\mathbb {Q} )} any group Λ Λ --> {\displaystyle \Lambda } which is commensurable (this means that both Γ Γ --> / ( Γ Γ --> ∩ ∩ --> Λ Λ --> ) {\displaystyle \Gamma /(\Gamma \cap \Lambda )} and Λ Λ --> / ( Γ Γ --> ∩ ∩ --> Λ Λ --> ) {\displaystyle \Lambda /(\Gamma \cap \Lambda )} are finite sets) with a group Γ Γ --> {\displaystyle \Gamma } defined as above (with respect to any embedding into G L n {\displaystyle \mathrm {GL} _{n}} ). With this definition, to the algebraic group G {\displaystyle \mathrm {G} } is associated a collection of "discrete" subgroups all commensurable to each other.
A natural generalisation of the construction above is as follows: let F {\displaystyle F} be a number field with ring of integers O {\displaystyle O} and G {\displaystyle \mathrm {G} } an algebraic group over F {\displaystyle F} . If we are given an embedding ρ ρ --> : G → → --> G L n {\displaystyle \rho :\mathrm {G} \to \mathrm {GL} _{n}} defined over F {\displaystyle F} then the subgroup ρ ρ --> − − --> 1 ( G L n ( O ) ) ⊂ ⊂ --> G ( F ) {\displaystyle \rho ^{-1}(\mathrm {GL} _{n}(O))\subset \mathrm {G} (F)} can legitimately be called an arithmetic group.
On the other hand, the class of groups thus obtained is not larger than the class of arithmetic groups as defined above. Indeed, if we consider the algebraic group G ′ {\displaystyle \mathrm {G} '} over Q {\displaystyle \mathbb {Q} } obtained by restricting scalars from F {\displaystyle F} to Q {\displaystyle \mathbb {Q} } and the Q {\displaystyle \mathbb {Q} } -embedding ρ ρ --> ′ : G ′ → → --> G L d n {\displaystyle \rho ':\mathrm {G} '\to \mathrm {GL} _{dn}} induced by ρ ρ --> {\displaystyle \rho } (where d = [ F : Q ] {\displaystyle d=[F:\mathbb {Q} ]} ) then the group constructed above is equal to ( ρ ρ --> ′ ) − − --> 1 ( G L n d ( Z ) ) {\displaystyle (\rho ')^{-1}(\mathrm {GL} _{nd}(\mathbb {Z} ))} .
The classical example of an arithmetic group is S L n ( Z ) {\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )} , or the closely related groups P S L n ( Z ) {\displaystyle \mathrm {PSL} _{n}(\mathbb {Z} )} , G L n ( Z ) {\displaystyle \mathrm {GL} _{n}(\mathbb {Z} )} and P G L n ( Z ) {\displaystyle \mathrm {PGL} _{n}(\mathbb {Z} )} . For n = 2 {\displaystyle n=2} the group P S L 2 ( Z ) {\displaystyle \mathrm {PSL} _{2}(\mathbb {Z} )} , or sometimes S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} , is called the modular group as it is related to the modular curve. Similar examples are the Siegel modular groups S p 2 g ( Z ) {\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )} .
Other well-known and studied examples include the Bianchi groups S L 2 ( O − − --> m ) , {\displaystyle \mathrm {SL} _{2}(O_{-m}),} where m > 0 {\displaystyle m>0} is a square-free integer and O − − --> m {\displaystyle O_{-m}} is the ring of integers in the field Q ( − − --> m ) , {\displaystyle \mathbb {Q} ({\sqrt {-m}}),} and the Hilbert–Blumenthal modular groups S L 2 ( O m ) {\displaystyle \mathrm {SL} _{2}(O_{m})} .
Another classical example is given by the integral elements in the orthogonal group of a quadratic form defined over a number field, for example S O ( n , 1 ) ( Z ) {\displaystyle \mathrm {SO} (n,1)(\mathbb {Z} )} . A related construction is by taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitz quaternion order). Similar constructions can be performed with unitary groups of hermitian forms, a well-known example is the Picard modular group.
When G {\displaystyle G} is a Lie group one can define an arithmetic lattice in G {\displaystyle G} as follows: for any algebraic group G {\displaystyle \mathrm {G} } defined over Q {\displaystyle \mathbb {Q} } such that there is a morphism G ( R ) → → --> G {\displaystyle \mathrm {G} (\mathbb {R} )\to G} with compact kernel, the image of an arithmetic subgroup in G ( Q ) {\displaystyle \mathrm {G} (\mathbb {Q} )} is an arithmetic lattice in G {\displaystyle G} . Thus, for example, if G = G ( R ) {\displaystyle G=\mathrm {G} (\mathbb {R} )} and G {\displaystyle G} is a subgroup of G L n {\displaystyle \mathrm {GL} _{n}} then G ∩ ∩ --> G L n ( Z ) {\displaystyle G\cap \mathrm {GL} _{n}(\mathbb {Z} )} is an arithmetic lattice in G {\displaystyle G} (but there are many more, corresponding to other embeddings); for instance, S L n ( Z ) {\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )} is an arithmetic lattice in S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbb {R} )} .
A lattice in a Lie group is usually defined as a discrete subgroup with finite covolume. The terminology introduced above is coherent with this, as a theorem due to Borel and Harish-Chandra states that an arithmetic subgroup in a semisimple Lie group is of finite covolume (the discreteness is obvious).
The theorem is more precise: it says that the arithmetic lattice is cocompact if and only if the "form" of G {\displaystyle G} used to define it (i.e. the Q {\displaystyle \mathbb {Q} } -group G {\displaystyle \mathrm {G} } ) is anisotropic. For example, the arithmetic lattice associated to a quadratic form in n {\displaystyle n} variables over Q {\displaystyle \mathbb {Q} } will be co-compact in the associated orthogonal group if and only if the quadratic form does not vanish at any point in Q n ∖ ∖ --> { 0 } {\displaystyle \mathbb {Q} ^{n}\setminus \{0\}} .
The spectacular result that Margulis obtained is a partial converse to the Borel—Harish-Chandra theorem: for certain Lie groups any lattice is arithmetic. This result is true for all irreducible lattice in semisimple Lie groups of real rank larger than two.[12][13] For example, all lattices in S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbb {R} )} are arithmetic when n ≥ ≥ --> 3 {\displaystyle n\geq 3} . The main new ingredient that Margulis used to prove his theorem was the superrigidity of lattices in higher-rank groups that he proved for this purpose.
Irreducibility only plays a role when G {\displaystyle G} has a factor of real rank one (otherwise the theorem always holds) and is not simple: it means that for any product decomposition G = G 1 × × --> G 2 {\displaystyle G=G_{1}\times G_{2}} the lattice is not commensurable to a product of lattices in each of the factors G i {\displaystyle G_{i}} . For example, the lattice S L 2 ( Z [ 2 ] ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} [{\sqrt {2}}])} in S L 2 ( R ) × × --> S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {R} )} is irreducible, while S L 2 ( Z ) × × --> S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\times \mathrm {SL} _{2}(\mathbb {Z} )} is not.
The Margulis arithmeticity (and superrigidity) theorem holds for certain rank 1 Lie groups, namely S p ( n , 1 ) {\displaystyle \mathrm {Sp} (n,1)} for n ⩾ ⩾ --> 1 {\displaystyle n\geqslant 1} and the exceptional group F 4 − − --> 20 {\displaystyle F_{4}^{-20}} .[14][15] It is known not to hold in all groups S O ( n , 1 ) {\displaystyle \mathrm {SO} (n,1)} for n ⩾ ⩾ --> 2 {\displaystyle n\geqslant 2} (ref to GPS) and for S U ( n , 1 ) {\displaystyle \mathrm {SU} (n,1)} when n = 1 , 2 , 3 {\displaystyle n=1,2,3} . There are no known non-arithmetic lattices in the groups S U ( n , 1 ) {\displaystyle \mathrm {SU} (n,1)} when n ⩾ ⩾ --> 4 {\displaystyle n\geqslant 4} .
An arithmetic Fuchsian group is constructed from the following data: a totally real number field F {\displaystyle F} , a quaternion algebra A {\displaystyle A} over F {\displaystyle F} and an order O {\displaystyle {\mathcal {O}}} in A {\displaystyle A} . It is asked that for one embedding σ σ --> : F → → --> R {\displaystyle \sigma :F\to \mathbb {R} } the algebra A σ σ --> ⊗ ⊗ --> F R {\displaystyle A^{\sigma }\otimes _{F}\mathbb {R} } be isomorphic to the matrix algebra M 2 ( R ) {\displaystyle M_{2}(\mathbb {R} )} and for all others to the Hamilton quaternions. Then the group of units O 1 {\displaystyle {\mathcal {O}}^{1}} is a lattice in ( A σ σ --> ⊗ ⊗ --> F R ) 1 {\displaystyle (A^{\sigma }\otimes _{F}\mathbb {R} )^{1}} which is isomorphic to S L 2 ( R ) , {\displaystyle \mathrm {SL} _{2}(\mathbb {R} ),} and it is co-compact in all cases except when A {\displaystyle A} is the matrix algebra over Q . {\displaystyle \mathbb {Q} .} All arithmetic lattices in S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} are obtained in this way (up to commensurability).
Arithmetic Kleinian groups are constructed similarly except that F {\displaystyle F} is required to have exactly one complex place and A {\displaystyle A} to be the Hamilton quaternions at all real places. They exhaust all arithmetic commensurability classes in S L 2 ( C ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {C} ).}
For every semisimple Lie group G {\displaystyle G} it is in theory possible to classify (up to commensurability) all arithmetic lattices in G {\displaystyle G} , in a manner similar to the cases G = S L 2 ( R ) , S L 2 ( C ) {\displaystyle G=\mathrm {SL} _{2}(\mathbb {R} ),\mathrm {SL} _{2}(\mathbb {C} )} explained above. This amounts to classifying the algebraic groups whose real points are isomorphic up to a compact factor to G {\displaystyle G} .[16]
A congruence subgroup is (roughly) a subgroup of an arithmetic group defined by taking all matrices satisfying certain equations modulo an integer, for example the group of 2 by 2 integer matrices with diagonal (respectively off-diagonal) coefficients congruent to 1 (respectively 0) modulo a positive integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture (usually attributed to Jean-Pierre Serre) is that this is true for (irreducible) arithmetic lattices in higher-rank groups and false in rank-one groups. It is still open in this generality but there are many results establishing it for specific lattices (in both its positive and negative cases).
Instead of taking integral points in the definition of an arithmetic lattice one can take points which are only integral away from a finite number of primes. This leads to the notion of an S {\displaystyle S} -arithmetic lattice (where S {\displaystyle S} stands for the set of primes inverted). The prototypical example is S L 2 ( Z [ 1 p ] ) {\displaystyle \mathrm {SL} _{2}\left(\mathbb {Z} \left[{\tfrac {1}{p}}\right]\right)} . They are also naturally lattices in certain topological groups, for example S L 2 ( Z [ 1 p ] ) {\displaystyle \mathrm {SL} _{2}\left(\mathbb {Z} \left[{\tfrac {1}{p}}\right]\right)} is a lattice in S L 2 ( R ) × × --> S L 2 ( Q p ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {Q} _{p}).}
The formal definition of an S {\displaystyle S} -arithmetic group for S {\displaystyle S} a finite set of prime numbers is the same as for arithmetic groups with G L n ( Z ) {\displaystyle \mathrm {GL} _{n}(\mathbb {Z} )} replaced by G L n ( Z [ 1 N ] ) {\displaystyle \mathrm {GL} _{n}\left(\mathbb {Z} \left[{\tfrac {1}{N}}\right]\right)} where N {\displaystyle N} is the product of the primes in S {\displaystyle S} .
The Borel–Harish-Chandra theorem generalizes to S {\displaystyle S} -arithmetic groups as follows: if Γ Γ --> {\displaystyle \Gamma } is an S {\displaystyle S} -arithmetic group in a Q {\displaystyle \mathbb {Q} } -algebraic group G {\displaystyle \mathrm {G} } then Γ Γ --> {\displaystyle \Gamma } is a lattice in the locally compact group
Arithmetic groups with Kazhdan's property (T) or the weaker property ( τ τ --> {\displaystyle \tau } ) of Lubotzky and Zimmer can be used to construct expander graphs (Margulis), or even Ramanujan graphs (Lubotzky-Phillips-Sarnak[17][18]). Such graphs are known to exist in abundance by probabilistic results but the explicit nature of these constructions makes them interesting.
Congruence covers of arithmetic surfaces are known to give rise to surfaces with large injectivity radius.[19] Likewise the Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak have large girth. It is in fact known that the Ramanujan property itself implies that the local girths of the graph are almost always large.[20]
Arithmetic groups can be used to construct isospectral manifolds. This was first realised by Marie-France Vignéras[21] and numerous variations on her construction have appeared since. The isospectrality problem is in fact particularly amenable to study in the restricted setting of arithmetic manifolds.[22]
A fake projective plane[23] is a complex surface which has the same Betti numbers as the projective plane P 2 ( C ) {\displaystyle \mathbb {P} ^{2}(\mathbb {C} )} but is not biholomorphic to it; the first example was discovered by Mumford. By work of Klingler (also proved independently by Yeung) all such are quotients of the 2-ball by arithmetic lattices in P U ( 2 , 1 ) {\displaystyle \mathrm {PU} (2,1)} . The possible lattices have been classified by Prasad and Yeung and the classification was completed by Cartwright and Steger who determined, by computer assisted computations, all the fake projective planes in each Prasad-Yeung class.
{{citation}}
رائد الفضاء سيرجي فولكوف يعمل خارج محطة الفضاء الدولية في 3 أغسطس 2011. ستيفن روبينسون يركب ذراعا روبوتية خلال مهمة إس تي سي-114، ويقوم بأول عملية إصلاح لمكوك فضائي أثناء الطيران. نشاط خارج المركبة (بالإنجليزية: Extravehicular activity وإختصارا EVA) هو أي نشاط يقوم به رائد الفضاء خارج المرك
Para otros usos de este término, véase Prometeo (desambiguación). Óleo en lienzo de Heinrich Friedrich Füger: Prometeo lleva el fuego a la humanidad (Prometheus bringt der Menschheit das Feuer, ca. 1817). En la mitología griega, Prometeo (en griego antiguo Προμηθεύς, ‘previsión’, ‘prospección’) es el titán amigo de los mortales, conocido principalmente por desafiar a los dioses robándoles el fuego a ellos en el tallo de una cañaheja, darlo a los hombres para su uso y …
Lukaskirche mit dem neuen Kirchturm Alter Turm von 1901 Die Lukaskirche ist eine evangelisch-lutherische Kirche in Hannover-Vahrenwald. Inhaltsverzeichnis 1 Geschichte 2 Orgel 3 Literatur 4 Einzelnachweise 5 Weblinks Geschichte Die Lukas-Kirchengemeinde wurde Ende des 19. Jahrhunderts durch Abtrennung des Pfarrbezirks von der Apostelkirche gegründet. Das neugotische Kirchengebäude wurde von 1899 bis 1901 durch Karl Börgemann erbaut. Eingeweiht wurde die Kirche 1901. 1931 bis 1933 schuf der Ki…
Darcy Ribeiro Información personalNacimiento 26 de octubre de 1922 Montes Claros (Brasil) Fallecimiento 17 de febrero de 1997 (74 años)Brasilia (Brasil) Causa de muerte Cáncer Nacionalidad BrasileñaReligión Catolicismo Lengua materna Portugués FamiliaCónyuge Berta Gleizer Ribeiro (1948-1975) EducaciónEducado en Universidad de São Paulo Información profesionalOcupación Antropólogo, político, escritor, sociólogo y novelista Cargos ocupados Minister of Education of Brazil (1…
Übersichtskarte der Autobahnen in Deutschland A 20 bei Langsdorf A 11 Uckermark, Brandenburg Die Autobahnen in Deutschland sind allein den Kraftfahrzeugen vorbehaltene Fernstraßen, die frei von Anbauten und plangleichen Kreuzungen sind, getrennte Fahrbahnen für den Richtungsverkehr mit jeweils mehreren Fahrstreifen haben und mit besonderen Anschlussstellen für die Zu- und Ausfahrt ausgestattet sind. Das Autobahnnetz Deutschlands ist mit über 13.192 Kilometern (Stand 2021)[1 …
フランス革命 > フランス革命の年表 フランス革命の年表(フランスかくめいのねんぴょう、仏: Chronologie de la Révolution française)は、フランス革命(1789年〜1799年)とそれに関連する出来事、さらにその原因にまで遡って取りまとめた年表である。 革命以前 啓蒙思想の世紀 ジュネーヴのルソー像 モンテスキュー像 1709年:厳冬により西ヨーロッパ全域で大飢饉[…
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: The Wilderness Years Nick Lowe album – news · newspapers · books · scholar · JSTOR (May 2019) (Learn how and when to remove this template message) 1991 compilation album by Nick LoweThe Wilderness YearsCompilation album by Nick LoweReleased1991Recorded1974–1977GenreR…
United States historic placePioneer Oil Company Filling StationU.S. National Register of Historic Places Show map of IowaShow map of the United StatesLocation831 West St.Grinnell, IowaCoordinates41°44′35″N 92°43′39″W / 41.74306°N 92.72750°W / 41.74306; -92.72750Built1937Built byL.0. WilsonArchitectural styleModern MovementNRHP reference No.13000070[1]Added to NRHPMarch 13, 2013 The Pioneer Oil Company Filling Station is a historic buildi…
Car marque owned by Jaguar Land Rover and former British car company Jaguar (marque)Product typeCarsOwnerJaguar Land Rover (since 2013)[1]Produced byJaguar Land RoverCountryEnglandIntroducedSeptember 1935; 88 years ago (1935-09)Related brandsLand RoverMarketsWorldwidePrevious ownersJaguar Cars LimitedTaglineThe Art of PerformanceWebsitejaguar.com Jaguar Cars LimitedFormerlySS Cars(1933–1945)TypePrivate (1933–1966)Subsidiary (1966–present)[2]Indus…
2010 studio album by Richard PoonI'll Be Seeing YouStudio album by Richard PoonReleasedOctober 2010Recorded2010GenrePop, jazzLanguageEnglish, TagalogLabelMCAProducerRicky R. IlacadRichard Poon chronology For You(2009) I'll Be Seeing You(2010) Singles from I'll Be Seeing You PanalanginReleased: October 2010 I'll Be Seeing You is the third studio album by Filipino pop-jazz singer Richard Poon, released in the Philippines in October 2010 by MCA Records. The album consists mostly of cover ve…
Location of Cuba Cuba is a country comprising the island of Cuba as well as Isla de la Juventud and several minor archipelagos. The Cuban state claims to adhere to socialist principles in organizing its largely state-controlled planned economy. Most of the means of production are owned and run by the government and most of the labor force is employed by the state. Recent years have seen a trend toward more private sector employment. By 2006, public sector employment was 78% and private sector 22…
ドミニク・ミステリオプロフィールリングネーム ドミニク・ミステリオ本名 ドミニク・グティエレスニックネーム ダーティ身長 185cm体重 91kg誕生日 (1997-04-05) 1997年4月5日(26歳)出身地 アメリカ合衆国・カリフォルニア州サンディエゴ所属 WWEトレーナー レイ・ミステリオジェイ・リーサルコナンランス・ストームデビュー 2020年8月23日テンプレートを表示 ドミニク・グ…
Lot and His DaughtersArtistSimon VouetYear1633Mediumoil painting on canvasMovementBaroqueHistory paintingSubjectLot and his daughtersDimensions160 cm × 130 cm (63 in × 51 in)[1]LocationMusée des Beaux-Arts, StrasbourgAccession1937 Lot and His Daughters is a 1633 oil-on-canvas painting of Lot and his daughters by the French artist Simon Vouet, now in the Musée des Beaux-Arts de Strasbourg. It depicts the Book of Genesis story in which, after t…
2002 book by Carolyn Meyer This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Doomed Queen Anne – news · newspapers · books · scholar · JSTOR (November 2008) (Learn how and when to remove this template message) Doomed Queen Anne AuthorCarolyn MeyerCountry United StatesLanguageEnglishSeriesYoung RoyalsGenreYou…
Former British bank This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Northern Rock – news · newspapers · books · scholar · JSTOR (September 2022) (Learn how and when to remove this template message) This article is about the former consumer bank. For other uses, see Northern Rock (disambiguation). Not to be conf…
Ukiran Flammarion (1888) yang menggambarkan seorang penjelajah yang sampai ke ujung Bumi datar, dan menyembulkan kepalanya ke luar kubah langit. Model Bumi datar adalah sebuah konsepsi arkais dari bentuk Bumi sebagai bidang atau cakram. Banyak dari kebudayaan kuno menganut kosmografi bumi datar, yang meliputi Yunani sampai zaman klasik, peradaban Zaman Perunggu dan Zaman Besi dari Timur Dekat sampai periode Helenistik, India sampai zaman Gupta (awal abad-abad Masehi), dan Tiongkok sampai abad ke…
ОТ-54 «Объект 481» (прототип ОТ-54) на испытаниях в мае 1952 года ОТ-54 Классификация Огнемётный танк Боевая масса, т 36,5 Компоновочная схема классическая Экипаж, чел. 4 История Разработчик Завод №112/СКБ-1 Производитель Годы разработки с 1946 по 1955 Годы производства с 1948 по 1959 Годы эксп…
American comedy Flash-animated series Homestar RunnerHomestar Runner logoGenreAnimationSurreal humorParodySatireCartoon seriesCreated byMike ChapmanMatt ChapmanCraig ZobelWritten byMatt ChapmanMike ChapmanVoices ofMatt ChapmanMissy PalmerMike ChapmanCountry of originUnited StatesOriginal languageEnglishProductionAnimatorsMike ChapmanMatt ChapmanOriginal releaseRelease2000 (2000) –present Homestar Runner is an American comedy animated web series and website created by Mike and Matt Ch…
Nepal Bhasa poet (1867 - 1929) Postage stamp of Mahaju issued in 1980. Pamphlet advertising drama performance about Siddhidas Mahaju to celebrate his birth centenary in 1967. Siddhidas Mahaju (Nepali: सिद्धिदास महाजु) (alternative name: Siddhidas Amatya) (15 October 1867 – 29 December 1929) was a Nepalese poet and one of the Four Pillars of Nepal Bhasa. He was at the forefront in the endeavour to revive literature in Nepal Bhasa that had become stagnant as a res…
SMA Negeri 11 BandungInformasiDidirikan1968JenisSekolah NegeriAkreditasiA[1]Nomor Statistik Sekolah30.1.02.60.14.101Nomor Pokok Sekolah Nasional20219243[2]Kepala SekolahDrs. H. Suparman, M.M.Pd.( Desember 2021-sekarang)Jumlah kelas34 Kelas 12 Kelas X 10 Kelas XI 12 Kelas XIIJurusan atau peminatanIPA, IPS, dan BahasaRentang kelasX IPA, X IPS, X IBB, XI IPA, XI IPS, XI IBB, XII IPA, XII IPS, XII IBBKurikulumKurikulum 2013Jumlah siswa418 murid kelas X 414 murid kelas …
Lokasi Pengunjung: 3.142.133.226