{"fact":"Cats can be right-pawed or left-pawed.","length":38}
{"slip": { "id": 91, "advice": "Drink a glass of water before meals."}}
{"fact":"Contrary to popular belief, the cat is a social animal. A pet cat will respond and answer to speech , and seems to enjoy human companionship.","length":141}
{"type":"standard","title":"Tannaka–Krein duality","displaytitle":"Tannaka–Krein duality","namespace":{"id":0,"text":""},"wikibase_item":"Q7683608","titles":{"canonical":"Tannaka–Krein_duality","normalized":"Tannaka–Krein duality","display":"Tannaka–Krein duality"},"pageid":2588624,"lang":"en","dir":"ltr","revision":"1110714679","tid":"c8012960-3636-11ed-86bb-f57d10cc1b18","timestamp":"2022-09-17T03:14:03Z","description":"Duality between a group and its representations","description_source":"local","content_urls":{"desktop":{"page":"https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality","revisions":"https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality?action=history","edit":"https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality?action=edit","talk":"https://en.wikipedia.org/wiki/Talk:Tannaka%E2%80%93Krein_duality"},"mobile":{"page":"https://en.m.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality","revisions":"https://en.m.wikipedia.org/wiki/Special:History/Tannaka%E2%80%93Krein_duality","edit":"https://en.m.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality?action=edit","talk":"https://en.m.wikipedia.org/wiki/Talk:Tannaka%E2%80%93Krein_duality"}},"extract":"In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named after Tadao Tannaka and Mark Grigorievich Krein. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G.","extract_html":"
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named after Tadao Tannaka and Mark Grigorievich Krein. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G.
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{"type":"standard","title":"Saint-Oyen, Aosta Valley","displaytitle":"Saint-Oyen, Aosta Valley","namespace":{"id":0,"text":""},"wikibase_item":"Q35411","titles":{"canonical":"Saint-Oyen,_Aosta_Valley","normalized":"Saint-Oyen, Aosta Valley","display":"Saint-Oyen, Aosta Valley"},"pageid":6210522,"thumbnail":{"source":"https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/St_oyen.jpg/330px-St_oyen.jpg","width":320,"height":240},"originalimage":{"source":"https://upload.wikimedia.org/wikipedia/commons/b/b7/St_oyen.jpg","width":640,"height":480},"lang":"en","dir":"ltr","revision":"1273888563","tid":"f31370dd-e2ee-11ef-8266-10e67fd5488b","timestamp":"2025-02-04T11:55:38Z","description":"Comune in Aosta Valley, Italy","description_source":"local","coordinates":{"lat":45.81666667,"lon":7.21666667},"content_urls":{"desktop":{"page":"https://en.wikipedia.org/wiki/Saint-Oyen%2C_Aosta_Valley","revisions":"https://en.wikipedia.org/wiki/Saint-Oyen%2C_Aosta_Valley?action=history","edit":"https://en.wikipedia.org/wiki/Saint-Oyen%2C_Aosta_Valley?action=edit","talk":"https://en.wikipedia.org/wiki/Talk:Saint-Oyen%2C_Aosta_Valley"},"mobile":{"page":"https://en.m.wikipedia.org/wiki/Saint-Oyen%2C_Aosta_Valley","revisions":"https://en.m.wikipedia.org/wiki/Special:History/Saint-Oyen%2C_Aosta_Valley","edit":"https://en.m.wikipedia.org/wiki/Saint-Oyen%2C_Aosta_Valley?action=edit","talk":"https://en.m.wikipedia.org/wiki/Talk:Saint-Oyen%2C_Aosta_Valley"}},"extract":"Saint-Oyen is a village and comune in the Aosta Valley region of north-western Italy.","extract_html":"
Saint-Oyen is a village and comune in the Aosta Valley region of north-western Italy.
"}{"slip": { "id": 192, "advice": "Don't take it personally."}}
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