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https://en.wikipedia.org/wiki/Mashta%20Elshalahmeh | Mashta Elshalahmeh () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Mashta Elshalahmeh had a population of 326 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Nabe%20Eltib | Nabe Eltib () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Nabe Eltib had a population of 1521 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Oweina | Oweina () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Oweina had a population of 1832 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Saidiyeh%2C%20Hama | Saidiyeh, Hama () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Saidiyeh, Hama had a population of 324 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Salba%2C%20Hama | Salba, Hama () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Salba, Hama had a population of 695 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Bibliography
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Saqiyet%20Nijm | Saqiyet Nijm () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Saqiyet Nijm had a population of 1932 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Shahta | Shahta () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Shahta had a population of 454 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Shajar | Al-Shajar () is a Syrian village located in the Suqaylabiyah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Shajar had a population of 941 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Tahunet%20Elhalawa | Tahunet Elhalawa () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Tahunet Elhalawa had a population of 1049 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Tal%20Eltiten | Tal Eltiten () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Tal Eltiten had a population of 435 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Tal%20Kombatri | Tal Kombatri () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Tal Kombatri had a population of 850 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Hurra%2C%20Syria | Al-Hurra () is a village in northern Syria located in the Suqaylabiyah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Hurra had a population of 843 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Elrihana | Elrihana () is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Elrihana had a population of 692 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Rudat%20al-Tar | Rudat al-Tar ( rūḍat aṭ-ṭār) is a Syrian village located in Al-Suqaylabiyah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 744 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Abu%20Faraj%2C%20Hama | Abu Faraj, Hama () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Abu Faraj, Hama had a population of 834 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Asharinah | Al-Asharinah () is a Syrian village located in the Tell Salhab Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Asharinah had a population of 6,347 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Bibliography
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Biret%20Eljabal | Biret Eljabal () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Biret Eljabal had a population of 543 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Ein%20Eljern | Ein Eljern () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ein Eljern had a population of 728 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Hir%20Elmosil | Hir Elmosil () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Hir Elmosil had a population of 579 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Kanayes | Kanayes () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Kanayes had a population of 481 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Khareb | Khareb (; also spelled Kharayeb) is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Khareb had a population of 241 in the 2004 census.
In 2018, archaeologists revealed a Byzantine mosaic painting of a church dates back to the fifth century AD. The painting, which was decorated with geometric shapes and inscriptions in Latin, was unearthed,
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Latma | Latmana () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Latmana had a population of 406 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Mazhal | Mazhal () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Mazhal had a population of 2077 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Hawayeq | Hawayeq () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Hawayeq had a population of 747 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Rasm%20Eljern | Rasm Eljern () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Rasm Eljern had a population of 314 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Tamaza | Tamaza () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Tamaza had a population of 101 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Tobeh | Tobeh () is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Tobeh had a population of 1,045 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Qashati | Al-Qashati ( al-qashāṭī) is a Syrian village located in Tell Salhab Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 969 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Ankawi | Al-Ankawi () is a Syrian town located in the Ziyarah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Ankawi had a population of 2,298 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Ein%20Elhamam | Ein Elhamam () is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ein Elhamam had a population of 734 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Closed-subgroup%20theorem | In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group , then is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding.
One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.
Overview
Let be a Lie group with Lie algebra . Now let be an arbitrary closed subgroup of . It is necessary to show that is a smooth embedded submanifold of . The first step is to identify something that could be the Lie algebra of , that is, the tangent space of at the identity. The challenge is that is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space. To proceed, define the "Lie algebra" of by the formula
It is not difficult to show that is a Lie subalgebra of . In particular, is a subspace of , which one might hope to be the tangent space of at the identity. For this idea to work, however, must be big enough to capture some interesting information about . If, for example, were some large subgroup of but turned out to be zero, would not be helpful.
The key step, then, is to show that actually captures all the elements of that are sufficiently close to the identity. That is to say, it is necessary to prove the following critical lemma:
Once this has been established, one can use exponential coordinates on , that is, writing each (not necessarily in ) as for . In these coordinates, the lemma says that corresponds to a point in precisely if belongs to . That is to say, in exponential coordinates near the identity, looks like . Since is just a subspace of , this means that is just like , with and . Thus, we have exhibited a "slice coordinate system" in which looks locally like , which is the condition for an embedded submanifold.
It is worth noting that Rossmann shows that for any subgroup of (not necessarily closed), the Lie algebra of is a Lie subalgebra of . Rossmann then goes on to introduce coordinates on that make the identity component of into a Lie group. It is important to note, however, that the topology on coming from these coordinates is not the subset topology. That it so say, the identity component of is an immersed submanifold of but not an embedded submanifold.
In particular, the lemma stated above does not hold if is not closed.
Example of a non-closed subgroup
For an example of a subgroup that is not an embedded Lie subgroup, consider the torus and an "irrational winding of the torus".
and its subgroup
with irrational. Then is dense in and hence not closed. In the relative topology, a small open subset of is composed of infinitely many almost parallel line segments on the surface of the torus. This means that is not locally path |
https://en.wikipedia.org/wiki/2014%E2%80%9315%20FK%20Partizan%20season | The 2014–15 season is FK Partizan's 9th season in Serbian SuperLiga. This article shows player statistics and all matches (official and friendly) that the club have and will play during the 2014–15 season.
Season overview
In summer 2014, Partizan has qualified to the UEFA Europa League where they have won two points in a group with Tottenham, Beşiktaş and Asteras.
Partizan scored one goal in 2014–15 UEFA Europa League. That one goal scored Saša Marković vs Beşiktaş on Atatürk Olympic Stadium in Istanbul.
The first game in Europe was held on 15 July 2014 against HB Tórshavn on Partizan Stadium in UEFA Champions League Second qualifying round. That night Partizan won 3–0, in a rematch also won 3–1, and passed in Third qualifying round. In the third qualifying round Partizan's opponent was Ludogorets Razgrad. In Razgrad the match was played without goals, but in Belgrade it was 2-2. Ludogorets won on away goals. In Play-off round UEFA Europa League Partizan played against Neftchi Baku. First game in Belgrade Partizan won 3–2,in that game 2 own goals were witnessed, but in Baku Partizan also won n 2–1. On 18 September Partizan played against Tottenham in Belgrade. In this game there was no winner and the result was 0-0.Partizan was attacking and had more shots from Tottenham. On 23 October Partizan played against Beşiktaş and lost 0–4. In this match Škuletić was played first time in UEFA Europa League after the suspension of 2 game. On 25 March manager Marko Nikolić replaced Zoran Milinković as the head coach at the club.
Marko Nikolic was sacked because of poor performance in the spring part although he had a contract until the end of the season. On 4 April Milinkovic debuted on the Partizan bench against Radnički 1923. In that match Partizan won n 2–1. Same day Partizans biggest rival Red Star lost and so Partizan went from +2 to +5. On 13 May Partizan won against Napredak Kruševac and won the title of Serbian champion. Then Partizan won 26 title and equalized with the biggest rival (Red Star) by number of champion titles.
In Serbian Cup, Partizan came to the finals by beating FK Bežanija 1–0, 3-0 FK Sloga Petrovac na Mlavi and Rad 1–0.In the semi-finals Partizan won against Jagodina Total score 5–0. To the finals Partizan won all his matches without conceding a goal. On 20 May in the final Partizan lost to Čukarički 1–0.And he didn't won the double crown.
Transfers
In
Out
For recent transfers, see List of Serbian football transfers winter 2014–15. For summer transfers, see List of Serbian football transfers summer 2014.
Players
Squad
Competitions
Overview
Serbian SuperLiga
League table
Results and positions by round
Matches
Serbian Cup
Serbian Cup
UEFA Champions League
Second qualifying round
Third qualifying round
UEFA Europa League
Play-off round
Group stage
Friendlies
Statistics
Goalscorers
This includes all competitive matches. The list is sorted by shirt number when total goals are equal.
Last Updated: 4 June |
https://en.wikipedia.org/wiki/1976%E2%80%9377%20VfL%20Bochum%20season | The 1976–77 VfL Bochum season was the 39th season in club history.
Review and events
Matches
Legend
Bundesliga
DFB-Pokal
Squad
Squad and statistics
Squad, appearances and goals scored
Transfers
Summer
In:
Out:
Winter
In:
Out:
Sources
External links
1976–77 VfL Bochum season at Weltfussball.de
1976–77 VfL Bochum season at kicker.de
1976–77 VfL Bochum season at Fussballdaten.de
Bochum
VfL Bochum seasons |
https://en.wikipedia.org/wiki/Al-Huwayz%20al-Shamali | Huweiz Elshmali () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Huweiz Elshmali had a population of 771 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Hamra%2C%20al-Suqaylabiyah | Al-Hamra () is a Syrian village located in the Qalaat al-Madiq Subdistrict of the Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Hamra had a population of 932 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Hawash%2C%20Hama | Al-Hawash () is a Syrian village located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Hawash had a population of 3,306 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Humayrat | Hmeirat () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Hmeirat had a population of 658 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Hurriyah%2C%20Syria | Al-Hurriyah () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Hurriyah had a population of 2,525 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Karim%2C%20Syria | Al-Karim () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Karim had a population of 2,879 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Muhajirin%2C%20Hama%20Governorate | Al-Muhajirin ( al-muhājirīn, meaning "the emigrants) is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 158 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Qabr%20Fidda | Qabr Fiddah () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Qabr Fiddah had a population of 1,490 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Qahirah%2C%20Hama | Al-Qahirah, Hama ( al-qāhirah) is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 708 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Qiratah | Qirata () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Qirata had a population of 267 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Bibliography
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Ramlah%2C%20Syria | Ramleh, Hama () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ramleh, Hama had a population of 2,359 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Rasif | Al-Rasif () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Rasif had a population of 1,689 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Salihiyah%2C%20Hama | Salhiyeh () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Salhiyeh had a population of 556 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Sahariyah | Sehriyeh () is a Syrian village in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics, Sehriyeh had a population of 959 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Bibliography
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Shahranaz | Shahranaz () is a Syrian town located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Shahranaz had a population of 1,646 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Populated places in al-Suqaylabiyah District
Populated places in Jabal Zawiya |
https://en.wikipedia.org/wiki/Tell%20Huwash | Tal Hawash () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Tal Hawash had a population of 2,498 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Bani | Al-Bani () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Bani had a population of 1,328 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Zitiyah | Elzitia () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Elzitia had a population of 305 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Sha%27irah | Al-Sha'irah () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Sha'irah had a population of 529 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Kurkat | Elkrkat () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Elkrkat had a population of 847 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in Jabal Zawiya |
https://en.wikipedia.org/wiki/Hawijah%20Sayyad | Hweijeh Elsiad () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Hweijeh Elsiad had a population of 846 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Ojt%20Eltoba | Ojt Eltoba () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ojt Eltoba had a population of 540 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Kawri | Kaori, Hama () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Kaori, Hama had a population of 671 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Mastarihat%20Afamiyah | Mstriht Afamia () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Mstriht Afamia had a population of 826 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Midan%20Ghazal | Midan Ghazal () is a Syrian village located in the Qalaat al-Madiq Subdistrict in the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Midan Ghazal had a population of 631 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Populated places in al-Suqaylabiyah District
Populated places in Jabal Zawiya |
https://en.wikipedia.org/wiki/Fawru | Fawru () is a village in Syria located in the Subdistrict of the al-Suqaylabiyah District in the Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Fawru had a population of 475 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Khirbat%20al-Naqus | Khirbat al-Naqus ( khirbat al-nāqūs) is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 1186 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Mansura%2C%20Hama | Mansura (), also spelt Mansoura, is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), it had a population of 770 in the 2004 census.
Syrian Civil War
During the Syrian Civil War Mansura came under the control of Hayat Tahrir al-Sham. The Syrian Arab Army began shelling the village on August 28, 2018 in preparation for their operation against the last remaining areas under rebel control.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Marana%2C%20Hama | Marana, Hama () is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Marana, Hama had a population of 308 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Qalidin | Qalidin () is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Qalidin had a population of 3636 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Al-Safsafah%2C%20Hama | Al-Safsafah () is a Syrian village located in the Ziyarah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Safsafah had a population of 676 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Sindiyana%2C%20Suqaylabiyah | Sindyana () is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Sindyana had a population of 117 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Tal%20Wassit | Tal Wassit (), also spelt Tal Wasat and Tall Waset, is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), it had a population of 1254 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Zayzun | Zayzun () is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Zayzun had a population of 1944 in the 2004 census.
See also
Zeyzoun Dam
References
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Elzaqum | Elzaqum () is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Elzaqum had a population of 913 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-%27Amrah | Al-'Amrah ( al-‘āmrah) is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 575 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Qastal%20al-Burj | Qastal al-Burj ( qasṭal al-burj) is a Syrian village located in Al-Ziyarah Nahiyah in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 36 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Mashik | Mashik (, also spelled Mshik or Msheek) is a Syrian village located in the Ziyarah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Mashik had a population of 311 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Ein%20Jorjin | Ein Jorjin () is a Syrian village located in Shathah Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ein Jorjin had a population of 988 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Haydariyeh | Haydariyeh () is a Syrian village located in Shathah Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics, Haydariyeh had a population of 2554 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Jubb%20al-Ghar | Jubb al-Ghar ( jubb al-ghār) is a Syrian village located in Shathah Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 821 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Jurin | Jurin () is a village in northern Syria, located in the Shathah Subdistrict of the Suqaylabiyah District in the Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Jurin had a population of 2,326 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Maradash | Maradash (, also spelled Mirdash) is a Syrian village located in Shathah Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Maradesh had a population of 1899 in the 2004 census. It had a population of 600 in the early 1960s. Its inhabitants are predominantly Alawites.
References
Bibliography
Populated places in al-Suqaylabiyah District
Alawite communities in Syria |
https://en.wikipedia.org/wiki/Naur%20Jurin | Na'ur Jurin () is a village in northern Syria located in the Shathah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Na'ur Jurin had a population of 1,439 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Qatrat%20al-Rihan | Qatrat al-Rihan () is a village in northern Syria located in the Shathah Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Qatrat al-Rihan had a population of 659 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Rihana%2C%20Hama | Rihana, Hama () is a Syrian village located in Shathah Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Rihana, Hama had a population of 985 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Mashta%20Mahfuz | Mashta Mahfoz () is a Syrian village located in Shathah Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Mashta Mahfoz had a population of 648 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Ashrafiyah%2C%20Hama | Ashrafiyah () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ashrafiyah had a population of 492 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Thawrah%2C%20Hama%20Governorate | Al-Thawrah ( ath-thawrah, meaning "the revolution") is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), the village had a population of 632 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Al-Aziziyah%2C%20al-Suqaylabiyah | Al-Aziziyah () is a village in northern Syria located in the Qalaat al-Madiq Subdistrict of the al-Suqaylabiyah District in Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), al-Aziziyah had a population of 1,828 in the 2004 census. Its inhabitants are predominantly Alawites.
References
Alawite communities in Syria
Populated places in al-Suqaylabiyah District
Populated places in al-Ghab Plain |
https://en.wikipedia.org/wiki/Bab%20al-Taqa | Bab Eltaqa () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Bab Eltaqa had a population of 1000 in the 2004 census.
References
Populated places in al-Suqaylabiyah District |
https://en.wikipedia.org/wiki/Deir%20Sunbul | Deir Sunbul () is a Syrian village located in Qalaat al-Madiq Subdistrict in Al-Suqaylabiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Deir Sunbul had a population of 1037 in the 2004 census.
References
Populated places in al-Suqaylabiyah District
Populated places in Jabal Zawiya |
https://en.wikipedia.org/wiki/Gamma%20number | In mathematics a gamma number may be:
A value of the gamma function
An additively indecomposable ordinal
An ordinal Γα that is a fixed point of the Veblen hierarchy |
https://en.wikipedia.org/wiki/Gene%20drive | A gene drive is a natural process and technology of genetic engineering that propagates a particular suite of genes throughout a population by altering the probability that a specific allele will be transmitted to offspring (instead of the Mendelian 50% probability). Gene drives can arise through a variety of mechanisms. They have been proposed to provide an effective means of genetically modifying specific populations and entire species.
The technique can employ adding, deleting, disrupting, or modifying genes.
Proposed applications include exterminating insects that carry pathogens (notably mosquitoes that transmit malaria, dengue, and zika pathogens), controlling invasive species, or eliminating herbicide or pesticide resistance.
As with any potentially powerful technique, gene drives can be misused in a variety of ways or induce unintended consequences. For example, a gene drive intended to affect only a local population might spread across an entire species. Gene drives that eradicate populations of invasive species in their non-native habitats may have consequences for the population of the species as a whole, even in its native habitat. Any accidental return of individuals of the species to its original habitats, through natural migration, environmental disruption (storms, floods, etc.), accidental human transportation, or purposeful relocation, could unintentionally drive the species to extinction if the relocated individuals carried harmful gene drives.
Gene drives can be built from many naturally occurring selfish genetic elements that use a variety of molecular mechanisms. These naturally occurring mechanisms induce similar segregation distortion in the wild, arising when alleles evolve molecular mechanisms that give them a transmission chance greater than the normal 50%.
Most gene drives have been developed in insects, notably mosquitoes, as a way to control insect-borne pathogens. Recent developments designed gene drives directly in viruses, notably herpesviruses. These viral gene drives can propagate a modification into the population of viruses, and aim to reduce the infectivity of the virus.
Mechanism
In sexually-reproducing species, most genes are present in two copies (which can be the same or different alleles), either one of which has a 50% chance of passing to a descendant. By biasing the inheritance of particular altered genes, synthetic gene drives could spread alterations through a population.
Molecular mechanisms
At the molecular level, an endonuclease gene drive works by cutting a chromosome at a specific site that does not encode the drive, inducing the cell to repair the damage by copying the drive sequence onto the damaged chromosome. The cell then has two copies of the drive sequence. The method derives from genome editing techniques and relies on the fact that double strand breaks are most frequently repaired by homologous recombination, (in the presence of a template), rather than non-homologous end joini |
https://en.wikipedia.org/wiki/2014%E2%80%9315%20Rayo%20Vallecano%20season | The 2014–15 Rayo Vallecano season is the club's 90th season in its history and its 16th in La Liga.
Squad statistics
Appearances and goals
Updated as of 30 May 2015.
|-
! colspan=14 style=background:#dcdcdc; text-align:center| Players who have made an appearance or had a squad number this season but have been loaned out or transferred
|}
Transfers
In:
Out:
Friendlies
Competitions
Overall
Primera División
League table
Matches
Kickoff times are in CET and CEST
Copa del Rey
Round of 32
References
Rayo Vallecano seasons
Rayo |
https://en.wikipedia.org/wiki/Milnor%E2%80%93Moore%20theorem | In algebra, the Milnor–Moore theorem, introduced by classifies an important class of Hopf algebras, of the sort that often show up as cohomology rings in algebraic topology.
The theorem states: given a connected, graded, cocommutative Hopf algebra A over a field of characteristic zero with for all n, the natural Hopf algebra homomorphism
from the universal enveloping algebra of the graded Lie algebra of primitive elements of A to A is an isomorphism. Here we say A is connected if is the field and for negative n. The universal enveloping algebra of a graded Lie algebra L is the quotient of the tensor algebra of L by the two-sided ideal generated by all elements of the form .
In algebraic topology, the term usually refers to the corollary of the aforementioned result, that for a pointed, simply connected space X, the following isomorphism holds:
where denotes the loop space of X,
compare with Theorem 21.5 from . This work may also be compared with that of .
References
External links
Theorems about algebras
Hopf algebras |
https://en.wikipedia.org/wiki/Primitive%20element%20%28co-algebra%29 | In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies
where is the co-multiplication and g is an element of C that maps to the multiplicative identity 1 of the base field under the co-unit (g is called group-like).
If C is a bi-algebra, i.e., a co-algebra that is also an algebra (with certain compatibility conditions satisfied), then one usually takes g to be 1, the multiplicative identity of C. The bi-algebra C is said to be primitively generated if it is generated by primitive elements (as an algebra).
If C is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket (graded commutator if C is graded).
If A is a connected graded cocommutative Hopf algebra over a field of characteristic zero, then the Milnor–Moore theorem states the universal enveloping algebra of the graded Lie algebra of primitive elements of A is isomorphic to A. (This also holds under slightly weaker requirements.)
References
http://www.encyclopediaofmath.org/index.php/Primitive_element_in_a_co-algebra
Coalgebras |
https://en.wikipedia.org/wiki/Augmentation%20%28algebra%29 | In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism , typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided ideal called the augmentation ideal of A.
For example, if is the group algebra of a finite group G, then
is an augmentation.
If A is a graded algebra which is connected, i.e. , then the homomorphism which maps an element to its homogeneous component of degree 0 is an augmentation. For example,
is an augmentation on the polynomial ring .
References
Algebras |
https://en.wikipedia.org/wiki/Laure%20Saint-Raymond | Laure Saint-Raymond (born 1975) is a French mathematician, and a professor of mathematics at Institut des Hautes Études Scientifiques (IHES). She was previously a professor at École Normale Supérieure de Lyon. She is known for her work in partial differential equations, and in particular for her contributions to the mathematically rigorous study of the connections between interacting particle systems, the Boltzmann equation, and fluid mechanics. In 2008 she was awarded the European Mathematical Society Prize, with her citation reading:
Biography
Laure Saint-Raymond studied in Paris, entering École Normale Supérieure in 1994. In 1996, she received a Master's degrees in plasma physics from Versailles Saint-Quentin-en-Yvelines University and in applied mathematics from Pierre and Marie Curie University. In 2000 she finished her Ph.D. in applied mathematics at Paris Diderot University, under the supervision of François Golse.
She worked for two years for the French National Centre for Scientific Research, and was named in 2002 full professor of mathematics at the Pierre and Marie Curie University at the age of 27. In 2007, she moved to the École Normale Supérieure, and is now professor at the École Normale Supérieure de Lyon. In 2021, she joined the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France as permanent professor, the first woman to hold that position in the history of the Institute.
She is the mother of six children.
Awards and honors
Her work has been recognized by many prestigious international awards, notably the Fermat Prize and the Bôcher Memorial Prize. In 2013, she was elected to the French Academy of Sciences, and in 2014 was an invited speaker at the International Congress of Mathematicians. A partial list of her awards and honors include:
2003 : Louis Armand Prize of the French Academy of Sciences
2004 : Claude-Antoine Peccot Award of Collège de France
2004 : Pius XI Medal of the Pontifical Academy of Sciences
2006 : The Best Paper prize of the SIAM Activity Group on Analysis of PDE for:
(with François Golse) "The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels." Invent. Math. 155 (2004), no. 1, 81–161.
2006 : Prize of the City of Paris for Young Scientists (joint with Isabelle Gallagher)
2008 : Prize of the European Mathematical Society
2009 : Ruth Lyttle Satter Prize in Mathematics of the American Mathematical Society for:
(with François Golse) "The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels." Invent. Math. 155 (2004), no. 1, 81–161.
"Convergence of solutions to the Boltzmann equation in the incompressible Euler limit." Arch. Ration. Mech. Anal. 166 (2003), no. 1, 47–80.
2011 : Irène Joliot-Curie Prize of the "Young Scientist Woman"
2013 : Elected to the French Academy of Sciences (on 10 December 2013)
2015 : Fermat Prize for:
the development of asymptotic theories of partial differential equations, including the fluid limits of rarefied flow |
https://en.wikipedia.org/wiki/PAOK%20FC%20in%20European%20football | PAOK FC history and statistics in UEFA competitions.
Match table
European Cup / UEFA Champions League
1: The first leg finished 2–1 to Maccabi Tel Aviv but was awarded 3–0 against PAOK for fielding the suspended player Liasos Louka.
2: Metalist Kharkiv were removed from the Champions League following their win over PAOK, after UEFA found them guilty of match fixing in the Ukrainian league in contravention of UEFA regulations. As a result, PAOK were reinstated in the Champions League.
Inter-Cities Fairs Cup / UEFA Cup / UEFA Europa League
1: The game was 2–0 to PSG in half-time. Due to episodes in Toumba Stadium between PAOK fans and police, Dutch referee John Blankenstein decided to stop game. In 4 of October UEFA awarded 3–0 to PSG.
UEFA Europa Conference League
European Cup Winners' Cup
Scorers
Opponents per country and per season
Last Update: 1 September 2023
Team statistics
By competition
Last updated: 26 October 2023
Biggest home win: 2015–16 Europa League, 23 July 2015, PAOK 6–0 Lokomotiva Zagreb
Biggest away win: 1999–00 UEFA Cup, 16 September 1999, Locomotive Tbilisi 0–7 PAOK
Biggest home defeat: 2018–19 UEFA Champions League, 29 August 2018, PAOK 1–4 Benfica
Biggest away defeat: 1965–66 Inter-Cities Fairs Cup, 29 September 1965, Wiener 6–0 PAOK
Highest scoring games: 1997–98 UEFA Cup, 12 August 1997, PAOK 5–3 Spartak Trnava & 1997–98 UEFA Cup, 4 November 1997, PAOK 4–4 Atlético Madrid
By country
Last updated: 26 October 2023Source: Statistics at UEFA.com
Player and Coach statistics
Last updated: 26 October 2023
Players who scored for and against PAOK in European competitions
The youngest players who scored for PAOK in European competitions
Players in bold are currently active for PAOK
Last updated: 26 October 2023 Source: Statistics in Europe at paokfc.gr Source: Statistics at
Match statistics
European matches on penalty shoot-out
Attendance record
Top attendance records at home and away European matches (not available for all the matches; the rank could be different).
Penalties in European matches
Last updated: 26 October 2023
Red cards in European matches
UEFA club ranking
{|
|-
| style="vertical-align:top; width:33%;"|
Current ranking
| style="vertical-align:top; width:75%;" |
All time ranking
Last updated: 20 June 2023
Season 2023–24 in progress
Source: Club coefficients at kassiesa.net
Notable wins
Notable wins
Biggest wins
References
European football
Greek football clubs in international competitions |
https://en.wikipedia.org/wiki/Primitive%20recursive%20set%20function | In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by .
Definition
A primitive recursive set function is a function from sets to sets that can be obtained from the following basic functions by repeatedly applying the following rules of substitution and recursion:
The basic functions are:
Projection: Pn,m(x1, ..., xn) = xm for 0 ≤ m ≤ n
Zero: F(x) = 0
Adjoining an element to a set: F(x, y) = x ∪ {y}
Testing membership: C(x, y, u, v) = x if u ∈ v, and C(x, y, u, v) = y otherwise.
The rules for generating new functions by substitution are
F(x, y) = G(x, H(x), y)
F(x, y) = G(H(x), y)
where x and y are finite sequences of variables.
The rule for generating new functions by recursion is
F(z, x) = G(∪u∈z F(u, x), z, x)
A primitive recursive ordinal function is defined in the same way, except that the initial function F(x, y) = x ∪ {y} is replaced by F(x) = x ∪ {x} (the successor of x). The primitive recursive ordinal functions are the same as the primitive recursive set functions that map ordinals to ordinals.
Examples of primitive recursive set functions:
TC, the function assigning to a set its transitive closure.
Given hereditarily finite , the constant function .
Extensions
One can also add more initial functions to obtain a larger class of functions. For example, the ordinal function is not primitive recursive, because the constant function with value ω (or any other infinite set) is not primitive recursive, so one might want to add this constant function to the initial functions.
The notion of a set function being primitive recursive in ω has the same definition as that of primitive recursion, except with ω as a parameter kept fixed, not altered by the primitive recursion schemata.
Examples of functions primitive recursive in ω: pp.28--29
.
The function assigning to the th level of Godel's constructible hierarchy.
Primitive recursive closure
Let be the function , and for all , and . Let Lα denote the αth stage of Godel's constructible universe. Lα is closed under primitive recursive set functions iff α is closed under each for all .
References
Inline
Computability theory
Theory of computation
Functions and mappings
Recursion
Set theory
Ordinal numbers |
https://en.wikipedia.org/wiki/Basic%20Elementary%20Skills%20Test | The Basic Elementary Skills Test (BEST) is a test used to verify the ability of children in mathematics and English (writing, reading, and spelling). Currently the test is available in Cambodian, Chinese, Persian, Vietnamese, Arabic, and Spanish.
Different sub-tests
The reading test requires the child to read a list of words as quickly as possible. The writing test examines the ability of the child to write sentences or paragraphs based on their age. The mathematics sub-test involves identifying numbers and solving simple problems. In the spelling test the child's ability to spell words correctly is measured.
Scoring marks
Marks are awarded for each correct answer, except for writing test where points are taken away for making mistakes and more points are earned for longer sentences. The raw marks are converted to approximate grade levels which are then used to determine if the child needs intervention classes.
References
Achievement tests
Mathematics tests |
https://en.wikipedia.org/wiki/Principal%20orbit%20type%20theorem | In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type.
Definitions
Suppose G is a compact Lie group acting smoothly on a connected differentiable manifold M.
An isotropy group is the subgroup of G fixing some point of M.
An isotropy type is a conjugacy class of isotropy groups.
The principal orbit type theorem states that there is a unique isotropy type such that the set of points of M with isotropy groups in this isotropy type is open and dense.
The principal orbit type is the space G/H, where H is a subgroup in the isotropy type above.
References
Lie groups
Group actions (mathematics) |
https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler%20theorem | The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematicians Euclid and Leonhard Euler, who respectively proved the "if" and "only if" aspects of the theorem.
It has been conjectured that there are infinitely many Mersenne primes. Although the truth of this conjecture remains unknown, it is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. However, it is also unknown whether there exists even a single odd perfect number.
Statement and examples
A perfect number is a natural number that equals the sum of its proper divisors, the numbers that are less than it and divide it evenly (with remainder zero). For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect.
A Mersenne prime is a prime number of the form , one less than a power of two. For a number of this form to be prime, itself must also be prime, but not all primes give rise to Mersenne primes in this way. For instance, is a Mersenne prime, but is not.
The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form , where is a Mersenne prime. The perfect number 6 comes from in this way, as , and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
History
Euclid proved that is an even perfect number whenever is prime. This is the final result on number theory in Euclid's Elements; the later books in the Elements instead concern irrational numbers, solid geometry, and the golden ratio. Euclid expresses the result by stating that if a finite geometric series beginning at 1 with ratio 2 has a prime sum , then this sum multiplied by the last term in the series is perfect. Expressed in these terms, the sum of the finite series is the Mersenne prime and the last term in the series is the power of two . Euclid proves that is perfect by observing that the geometric series with ratio 2 starting at , with the same number of terms, is proportional to the original series; therefore, since the original series sums to , the second series sums to , and both series together add to , two times the supposed perfect number. However, these two series are disjoint from each other and (by the primality of ) exhaust all the divisors of , so has divisors that sum to , showing that it is perfect.
Over a millennium after Euclid, Alhazen conjectured that even perfect number is of the form where is prime, but he was not able to prove this result. It was not until the 18th century, over 2000 years after Euclid, that Leonhard Euler proved that the formula will yield all the even perfect numbers. Thus, there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. After Euler |
https://en.wikipedia.org/wiki/Ali%20Kasun | Ali Kasun () is a Syrian village located in Salamiyah Subdistrict in Salamiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Ali Kasun had a population of 1336 in the 2004 census.
References
Populated places in Salamiyah District |
https://en.wikipedia.org/wiki/Bardunah | Berdoneh () is a Syrian village located in Salamiyah Subdistrict in Salamiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Berdoneh had a population of 862 in the 2004 census.
References
Populated places in Salamiyah District |
https://en.wikipedia.org/wiki/Al-Buwaydah | Buwayda () is a Syrian village located in Salamiyah Subdistrict in Salamiyah District, Hama. According to the Syria Central Bureau of Statistics (CBS), Buwayda had a population of 174 in the 2004 census.
References
Populated places in Salamiyah District |
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