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O2 (brand)

O2 (typeset as O2) is a global brand name owned by the Spanish telecommunications company Telefónica. The company uses the O2 brand for its subsidiaries in the United Kingdom and Germany. Since 2018, it is also used as an online-only flanker brand in Spain.

Formerly, all European branches of Telefónica were managed by Telefónica Europe plc, a broadband and telecommunications company. The company originated as a collection of worldwide telecommunications companies, known in the later half of the 1990s as BT Wireless, and a global mobile data business known then as Genie Internet, both subsidiaries of British Telecommunications (British Telecom or BT). As part of a reorganisation of Telefónica in 2014, Telefónica Europe's holdings and operations became direct subsidiaries of its parent: Telefónica S.A.

Telefónica Europe provided mobile telephony operations in the United Kingdom and Germany, and licensed its brand to former operations in the Republic of Ireland, the Czech Republic, and Slovakia. O2 has additionally established a joint venture with Tesco Mobile in the United Kingdom, Republic of Ireland and in Slovakia, and the Tchibo Mobilfunk network in Germany.

United Kingdom
Cellnet was launched in January 1985 as a subsidiary of Telecom Securicor Cellular Radio Limited, a 60:40 venture between British Telecommunications (British Telecom or BT) and Securicor respectively.

In 1999, BT purchased Securicor's shares in Cellnet for £3.15 billion. The company was later rebranded as BT Cellnet, and it became a part of BT Wireless, a group of subsidiary companies owned by BT. BT invested £2 billion in BT Cellnet, and in 2000, BT Cellnet had over 6 million customers.

In October 2001, at a general meeting held in Birmingham, 4.297 billion BT shares voted in favour of BT Cellnet's demerger from BT, with 0.67 million shares voting against. In 2001, BT Cellnet demerged from BT as mmO2 plc, and was relaunched on 1 May 2002 using the O2 brand. The company changed its name from mmO2 to O2 plc in March 2005. In 2021 it merged with Virgin Media to form Virgin Media O2.

Republic of Ireland
Digifone was created in 2000, as the demerged mobile business of the prior Esat Digifone which had just been purchased by BT. Esat Digifone was originally a joint venture between Esat Telecom and Telenor. After the acquisition of Esat Telecom by BT, the Telenor share in Esat Digifone was also acquired by BT. Subsequently, the fixed operations, Esat Telecom later became BT Ireland and the demerged mobile operations, rebranded without the Esat name as Digifone, became a part of the newly formed separate company, mmO2 plc. Digifone became O2 Ireland in 2001 as a subsidiary of mmO2 plc following the demerger from BT. O2 Ireland became a subsidiary of Telefónica in 2006, after its parent company O2 in the United Kingdom was purchased. In June 2013, Hutchison Whampoa announced it would acquire the Republic of Ireland arm of O2 for €780m. O2 was merged into Hutchison Whampoa's subsidiary Three Ireland in March 2015.

Germany
Viag Interkom was created in 1995 as a joint venture of the German power supply firm VIAG (45%), British BT Group (then called British Telecommunications) (45%) and Norwegian Telenor (10%) in order to compete for official licence to provide services in the rapidly liberalising German (fixed and mobile) telecommunications market at that time. In 2001 BT acquired VIAG's share (owned by German energy company E.ON) for €11.4 billion following the German UMTS licence auction the previous year. Viag Interkom's mobile operations were transferred into the BT Wireless family and the fixed business became BT (Germany) GmbH & CO. Viag Interkom became O2 Germany as a subsidiary of mmO2 plc in 2001, following the demerger from BT.

The Netherlands
Telfort was created in March 1997 as a 50:50 joint venture between BT and Nederlandse Spoorwegen (the Dutch national railways operator) and headquartered in Amsterdam. Telfort was awarded a licence by the government of the Netherlands to provide fixed telecommunications services in November of that year, and a mobile licence was awarded the following year. In 2000, the mobile operations of Telfort became a part of the BT Wireless family. Telfort became O2 Netherlands as a subsidiary of mmO2 plc in 2001, following the demerger from BT.

In 2003, O2 sold the company to Greenfield Capital Partners which reverted the company back to the Telfort name. Telfort eventually became a wholly owned subsidiary of KPN.

Isle of Man
Manx Telecom was demerged from BT as part of the mmO2 flotation in 2001. Unlike some of the company's counterparts, the fixed and mobile arms of Manx Telecom were not split prior to the demerger, which meant that Manx Telecom was the mmO2 group's only fixed operation at the time.

After the demerger from BT in 2001, Manx Telecom continued to trade under its brand as a subsidiary of mmO2 plc.

On 4 June 2010, Manx Telecom was sold to UK private equity investor HG Capital (who were buying the majority stake), alongside telecoms management company CPS Partners. HG Capital indicated that the enterprise value of the deal was £158.8 million ($232.5 million).

Genie Internet and Genie Asia
Genie Internet Ltd and Genie Asia were created as autonomous but wholly owned subsidiaries of BT in 2000. By the time it became a part of the BT Wireless family of companies in 2001, Genie had mobile portal operations in the UK, Germany, Spain, Italy, the Netherlands, Hong Kong and Japan, and an Internet MVNO operation in UK called Genie Mobile. Prior to the demerger of O2 from BT, Genie scaled back its global operations in order to align with the mobile businesses demerging with it; namely UK, Germany, Ireland and the Netherlands. The Genie Asia business continued portal operations in Asia.

After demerging from BT in 2001, the European Genie business became the basis of a central products and services division called 'Products O2' and the Genie Mobile business was rebranded to 'O2 Online' which continues in the UK as a mobile communications service provider tied to the O2 UK network. The Genie portal properties were rebranded O2 Active in UK, Germany, Ireland and the Netherlands, and Genie Asia became O2 Asia. O2 Asia became a subsidiary of O2 UK for governance purposes.

2001–2005: mmO2 plc era
Following the demerger of British Telecommunications plc in 2001, mmO2 plc was created as the holding company for BT's former mobile network assets and was split from the rest of the company, now named BT Group plc. During this period, O2 Netherlands was sold to Dutch investment group Greenfield Capital Partners in 2003. mmO2 plc was later renamed O2 plc before being taken over by Telefónica.

2006–present: Telefónica subsidiary
On 31 October 2005, O2 plc agreed to be taken over by Telefónica, a Spanish telecommunications company, with a cash offer of £17.7 billion, or £2 per share. According to the merger announcement, O2 retained its name and continued to be based in the United Kingdom, keeping both the brand and the management team. The merger became unconditional on 23 January 2006.

Following the completion of the acquisition in February 2006, Telefónica undertook a corporate organisational change that saw the merging of its fixed and mobile businesses in Spain, and the transfer of Telefónica's non-Spanish European telecommunications properties into the O2 brand. Thus, the Český Telecom and Eurotel operations in the Czech Republic as well as the Telefónica Deutschland business in Germany were brought into the governance of O2, which retained its UK registered public company status with its own board of directors and corporate governance structures and processes.

Telefónica chose to keep their existing mobile phone operations in the rest of the world under the brand Movistar. This name is used in Spain and in most Latin America countries, operated by a separate management team.

Telefónica sold its O2 Asia subsidiary in a management buyout in 2007. O2 Asia, headquartered in Singapore, operating in Far East, South Asia, Middle East, and Australasian countries, for a short time developed and marketed a range of wirelessly connected PDA and smartphone products branded Xda for both the Asian and European markets, and further such products under the "MWg" brand, short for Mobile & Wireless Group.

In 2015 there were talks for Li Ka-shing, owner of the rival UK network Three, to buy the company. The following year, the European Commission blocked the takeover.

Operations
Telefónica operates its businesses under the O2 brand through its subsidiaries Telefónica UK and Telefónica Germany. Telefónica Spain uses Movistar as its flagship brand, while O2 is a secondary low-cost brand.

Telefónica owns the Giffgaff MVNO in the United Kingdom, which uses the O2 network.

In May 2020, it was reported that Telefónica and Liberty Global plan to merge the UK operations of O2 with Virgin Media, a pay-TV and broadband provider in that country, owned by Liberty Global.

Ireland
Telefónica Ireland was providing broadband and telecommunications in Ireland under the O2 brand. O2 Ireland was previously called Esat Digifone when it was owned by Esat Telecommunications (and Telenor) from 1997 to 2006. O2 Ireland became part of Telefónica in 2006 as part of its purchase of O2 in the UK.

In June 2013, Hutchison Whampoa announced it would acquire the Irish arm of O2 for €780m. In March 2015, the company was merged into Hutchison Whampoa's subsidiary Three Ireland.

Czech Republic
Český Telecom was the new name given to a company previously known as SPT Telecom which was formed from the splitting off of the Czech Ministry of Posts and Telecommunications into postal and telecom sides in 1993. The company was renamed to Český Telecom in 2000 and remained a government-controlled company until its complete privatisation in 2004. Along the way, rapid modernisation of the network occurred with the help of funding and expertise of the Dutch and Swiss consortium called TelSource.

In parallel to the evolution of Český Telecom, the Eurotel mobile phone business was established in 1990 in a joint venture between what was then known as the Czechoslovak telecom ministry, Bell Atlantic and US West. In 1996, GSM services were introduced, and in 2003 Český Telecom acquired full control of Eurotel.

The Czech Republic government sold its remaining stake in Český Telecom (along with its Eurotel subsidiary) to Telefónica in 2005 which renamed it to Telefónica O2 Czech Republic and since 2006 started to use the O2 brand for both fixed and mobile services. In 2011 company was renamed to Telefónica Czech Republic.

In 2013 was announced that Telefónica would sell its stake in the company to PPF. Under the terms of the sale, the company would continue to use the O2 brand for a maximum of four years. The agreement was later extended, and the company continues under the O2 brand as of January 2024.

Slovakia
In 2002 Český Telecom founded Czech Telecom Slovakia, its subsidiary in Slovakia. The company was providing services for large corporations only. In 2006 the company was renamed Telefónica O2 Slovakia and won a tender for the 3rd mobile phone operator in Slovakia. On 2 February 2007, Telefónica O2 Slovakia started providing services in Slovakia for first customers "O2 Jednotky". On 28 February O2 launched classic services and opened first shop in Slovakia (in Bratislava on Obchodná street). In 2013 it was announced that it would be sold with its parent company to PPF.

Branding
The BT Cellnet consumer brand was renamed O2 – the chemical symbol for unbound oxygen – as were all the group's other businesses (other than Manx Telecom). The re-branding was designed by the Lambie-Nairn design agency, which developed the idea of the company supplying services that were essential, much the same as oxygen is essential for life. With this, the company logo and associated graphics were designed, using air bubbles to present this.

Sponsorship
O2 currently sponsor the England rugby team, and in 2003 launched a mobile video service allowing customers to download or stream video content related to the 2003 Rugby World Cup. In 2005, Telefónica O2 extended their involvement in rugby union, signing a new deal with the England rugby team and the Rugby Football Union, as well as O2 rugbyclass and Premier Rugby Ltd for the English Premiership.

Telefónica O2 also had a long-standing relationship with Arsenal F.C., being their shirt sponsor from 2002 to 2006. In 2005, a three-year deal was signed that saw Telefónica O2 become the team's exclusive Mobile Communications Partner.

From 2005 to 2008, O2 sponsored the Wireless Festival, a music festival held annually in London (and near Leeds in 2006 and 2007).

Telefónica O2 was the sponsor of the Cork GAA Football and Hurling teams and the Irish rugby team. The partnership with Irish rugby went interactive in 2006, when Ireland fans were offered access to daily updates from head coach Eddie O'Sullivan. In 2009, O2 launched their 'BeTheDifference' integrated advertising campaign where fans had the chance to have their names included on the O2 Rugby posters and on the Irish Rugby players' jerseys. The integrated campaign was created by O2's two Irish advertising agencies: McConnells (above-the-line) and Brando (below-the-line). The interactive campaign won an award at the 2009 Kinsale Sharks.

Naming rights
On 31 May 2005, Telefónica O2 acquired the naming rights for the redeveloped Millennium Dome in London from Anschutz Entertainment Group (AEG). The Dome is therefore now officially known as 'The O2'. The company extended their relationship with music venues in 2008 when all Academy Music venues became known as 'O2 Academies', replacing Carling as venue sponsors.

For short time, the naming rights arrangements with AEG also apply in Germany, with large sport and concert venues known as O2 World in Berlin and Hamburg.

title: O2 (brand)


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Talk:Eqbal Ahmad

Comments
isnt it spelled Iqbal?
 * Google results for "Iqbal Ahmad" : 1,820 and google results for "Eqbal Ahmad": 4,510. Perl 21:45, 23 Feb 2004 (UTC)

He published under the name "Eqbal Ahmad". -- Rbellin 01:28, 25 Feb 2004 (UTC)

As a young man he used to spell his first name, somewhat idiosyncratically, Egbal (with a G), but for most of his life it was Eqbal -- always with an E, never with an I.

Link
I have removed this link: which was not working anymore. --Lionni 14:53, 16 September 2006 (UTC)
 * A tribute to Eqbal Ahmad, by Edward W. Said

I have inserted the following link:
 * Eqbal Ahmad, by Edward W. Said which may or may not be the same as the removed link.

Can someone provide family details
The page is missing details on his family of origin and of marriage. All I have been able to find is: "Eqbal Ahmad was born in the state of Bihar, India in 1932. When he was 4, his father was killed in a land dispute; in 1948, during the partition of India, he and his family (who were Muslims) moved to Lahore, Pakistan. He attended Foreman Christian College in Lahore, then came to Occidental College in California on a Rotary scholarship in the mid-1950s. He entered Princeton in 1958, where he obtained his doctorate in 1965. He taught at Cornell in the 1960s, was appointed a fellow at the Adlai Stevenson Institute of International Affairs, at the University of Chicago, and later joined the Washington Institute for Policy Studies. He was the first directory of the Transnational Institute in Amsterdam. He married Julie Diamond in 1969, and had one daughter, Dohra.""Source: http://asteria.fivecolleges.edu/findaids/hampshire/mah7_bioghist.html"

Can someone who knows more provide details? Thanks.

Year of birth
The article says he was born in 1933 or 1934; the LC authority file says he was born either in 1932 or 1933; is there a way to get a more definitive answer?; LC got their information from the publisher, so I figure the publisher had spoken with the author.--FeanorStar7 (talk) 10:44, 1 February 2009 (UTC)
 * The answer is no, and the current date of 1933 in the article is likely inaccurate as the subject said he thought it might be 1934, but didn’t know for sure. So this needs to be corrected in some way or another. Viriditas (talk) 01:34, 31 December 2022 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Eqbal Ahmad


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Talk:Battle of Suomussalmi

New Analysis Dot
I added a small bit of information to the analysis, only to give a more full understanding of the various factors leading to Finnish success. Let me know if you have any problems.

-Jclingerman (4/25/06) — Preceding unsigned comment added by Jclingerman (talk • contribs) 07:34, 25 April 2006 (UTC)

Casualties
Hi,

Just some notices. Soviets lost more than 40 000 troops in this battle. Only 5000 out of 48 000 men of the 9th Army made it back to Soviet Union. Two divisions (163rd and 44th) and one ski brigade were totally destroyed.

The battle was a great humiliation to the mighty Red Army which had not gone trough this kind of defied before. And as much it was humiliation for RA it was a glorious victory for Finnish Defence Forces.

Best wishes,

OJ from Finland — Preceding unsigned comment added by 64.241.194.70 (talk) 21:39, 23 February 2004 (UTC)

 * Ski brigade?? Tank brigade is right, I think... Cinik 62.209.237.4 08:25, 13 Apr 2005 (UTC)

Please provide sources for those numbers. - Fire For Effect  - 18:59, 2 February 2008 (UTC)

The number of Soviet losses has no relation to reality and is a propaganda myth. Real soviet casualties - 2303 dead and 3868 missing. http://slon-76.livejournal.com/97558.html 178.70.151.125 (talk) 13:44, 18 November 2015 (UTC)

Clean up
I just cleaned up the grammer a bit in the Battle section. I do not believe I changed anything of what the author intended to say. I think that the word 'trace' is not what was intended, but could not figure out what should be there. Also, there is mention of the Finns destroying a village. Is this Raat? It is not clear from the text. PerlKnitter 15:10, 6 October 2005 (UTC)

Trouble with casualty numbers
Can anyone find verifiable sources for the casualties? The amount of soviet dead was just changed from 16000-35000 to c. 22500 - which seems somewhat suspicious, but I can find even fewer sources for the former numbers than for the latter (though one refers to it as a conservative estimate). Additionally, the article currently cites Finnish losses as 350 KIA, which seems a bit low, and some reports I've found speak of about 900 with over a thousand wounded. It's true enough that specific figures for the Soviet side are not available, due to the circumstances. --Kizor 00:52, 5 January 2006 (UTC)
 * This link http://www.winterwar.com/Battles/Suomussalmi2.htm gives 5000 killed or wounded for the Soviets not 27K and gives this for the Finns 350 KIA, 600 wounded and 70 missing —The preceding unsigned comment was added by 83.249.112.176 (talk) 02:36, 13 March 2007 (UTC).

 * Sorry, but link http://www.winterwar.com/Battles/Suomussalmi2.htm gives not this! He gives 5000 bodies, that was found in area Suomussalmi-Alassalmi 1)this area is full deep forrest - many bodies was not found 2) this is not all area of battle 3) this is only tail battle - without battle on Raate road, diferent from this article. On Raateroad was totaly destroyed 44th division and 21th tank brigade. Cinik 09:25, 13 March 2007 (UTC)

Here's a quotation from the book "A Frozen Hell: The Russo-Finnish War of 1939-1940" - "Daylight revealed a scene that staggered the senses of even the most battle-wise correspondents who were brought in to see and photograph it. From the high-water mark of the 163rd Division, up at Piispajarvim to the last burned out truck between Raate village and the Soviet border, were scattered the stone-stiff bodies of 27,500 Russian soldiers. Forty-three tanks and 270 other vehicles, trucks, tractors, and prime moviers clotted the narrow road or lay, windscreens spiderwebbed with bullet holes and turrets blackened by fire, in the snowy morass of roadside ditches. The victors acquired substantial booty: 4 dozen pieces of artillery, 600 working rifles, 300 functional machine guns, a few mortars and salvageable tanks, and a motley but welcome assortment of trucks and armoured cars. Finnish losses amounted to 900 dead and 1770 wounded.

I think this book is more accurate than the website that has been linked to above. Therefore, I am going to change the casualties to (note: although the book does mention captured Soviets, it doesn't mention how many. I am putting 2,100 because it seems like a reasonable amount):

Soviet: 27,500 dead, 2,100 captured, 43 tanks destroyed Finnish: 900 dead, 1770 wounded

If anyone has any problems with this please let me know.

One last thing, I have to go now, so I'll edit the figures when I get back. - Fire For Effect  - 16:45, 25 October 2007 (UTC) Again, if you have a problem with these numbers and you have a credible source, post it here. Do not edit the numbers without first posting here. - Fire For Effect  - 19:20, 2 February 2008 (UTC)

 * So we decided to abandon Russian sources for Russian losses? Also, the 43 "tanks" includes even Komsomolets tractors. 69.165.233.6 (talk) 17:16, 30 November 2023 (UTC)

WW II
Winter War is often acepted as part of WWII. Soviet Union was ally of Germany in this time an his expansion is part of WWII. And factor, that somebody do not agree with this, is not relevant reason for displacing this article from category, who somebody was this article search. Cinik 14:50, 10 February 2006 (UTC)

 * I need sources. Winter War currently stands as a contemporary war to World War II.  Until that changes, we can't change any of the battles. Oberiko 16:37, 10 February 2006 (UTC)

Change
I wrote "Soviet army being poorly equipped for winter warfare" because they were better equipped in almost every other aspect than winter clothing. —Preceding unsigned comment added by Irtehprwn (talk • contribs) 18:49, 31 March 2008 (UTC)

And the sources of course are from your ass, as always Shipseggsbasket (talk) 18:51, 31 March 2008 (UTC)
 * Leave the oroginal (Shipseggsbasket's version) because your citation hasn't references. MOJSKA   666  (msg) 19:20, 31 March 2008 (UTC)

I think Finnish were too. —Preceding unsigned comment added by 87.94.243.68 (talk) 23:41, 2 November 2010 (UTC)

I have change the link Raate to Battle of raate road. It was linked to some Indian movie. Sebastian — Preceding unsigned comment added by 88.115.22.117 (talk) 12:27, 14 February 2016 (UTC)

Soviet casualties poor source
Are we seriously not using Russian sources for Russian casualties? Estimates from the opposing force (not like those are known for notorious exaggerations, right?) made in 1940-1970s are somehow superior to contemporary research, based on the opening of Soviet archives in the 1990s, because we have lots of books supporting each other? Contemporary Russian historiography gives much smaller numbers for Soviet casualties in the battle. For example, the above quoted Oleg Kiselev claims 6171 killed and missing (this number includes prisoners of war), 4001 wounded and around 2800 non-combat casualties (mostly from the elements) - some 12 000 overall, with about half of them dead. I suggest editing the infobox accordingly. Mudriy zmei (talk) 01:25, 25 August 2018 (UTC)

Soviet units. Strength and casualties.
1. Soviet units. Number of men in divisions and casualties. Source: Article by Oleg Kiselev. http://raatteentie.heninen.net/sotatapahtumat/klio.htm

1.1 Number of men in divisions (including engineers, medics, etc.)

Strength of the 163rd Division is taken as the sum of the strength of the 163rd Division as of 30 November (12,762) and 1 battalion of the 305 Regiment of the 44th Division (800), which had been attached to the 163rd Division since mid-December. The 47th Corps units strength is taken as the difference between the total strength of 163 Division and Corps units (16,000) and that of 163 Division (12,762). The strength of the 44th Division is taken as the difference between the number of the 44th Division personnel as of 7th December (16,716) and the number of personnel of 2/305th regiment and its reconnaissance battalion (1,000) assigned to the Reboly sector, as well as the number of units that did not participate in the battle on the Raate road from 1st January to 7th January (3,500) and 1/305/44th Division (800), which returned to the 44th Division only on 9th January. The strength of the 3d NKVD regiment: There is probably a typo in the text of the article. " It is known that by the end of January 9, 850 of the regiment's 1,500 soldiers and commanders were on the Soviet side. Thus, the maximum number of irrecoverable losses of the regiment was up to 750 men. " I think it should have been written 1,600, not 1,500.

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В результате передачи численность 163-й СД уменьшилась до 12762 человек. В этом составе дивизия и перешла границу 30 ноября 1939 года. [..] К ним также следует добавить 1-й батальон 305-го стрелкового полка (около 800 человек), который в середине декабря был направлен на усиление 163-й СД и действовал в её составе. Тем не менее, этот батальон вернулся в состав 44-й СД уже 9-10 января и принял участие в заключительных боях на дороге Раате. [..] Таким образом, общая численность 163-й СД, корпусных частей и подразделений усиления составляла не 23 тысячи человек, а максимум 15,5-16 тысяч. [..] 44-я СД по состоянию на 7 декабря имела в своем составе 16716 человек. [..] Однако следует учитывать, что разведывательная рота и 2-й батальон 305-го стрелкового полка (всего около 1000 человек) были выделены из состава дивизии и направлены на ребольское направление. [..] Общую численность подразделений, не участвовавших в боях на дороге Раате 1-7 января или участвовавших в них незначительными силами, можно оценить приблизительно в 3,5 тысячи человек. [..]

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Сведения о потерях советских войск по Суомуссалми были бы не полными, если бы в статистику не были включены потери 3-го полка НКВД. К сожалению, подробных сведений о его потерях в распоряжении автора не имеется. Известно, что к исходу 9 января из 1,5 тыс. бойцов и командиров полка на советской стороне находилось 850 человек. Таким образом, максимальные безвозвратные потери полка составили до 750 человек. Если рассматривать боевые потери полка в тех же пропорциях, что и потери 44-й дивизии, то они могли составлять примерно 1100-1150 человек (из расчета, что безвозвратные потери составляли 66% боевых потерь). Здесь необходимо учитывать, что данные о не вернувшихся к 9 января могли быть завышенными (люди могли возвращаться и после этой даты). Кроме того, не совсем ясно, 850 вышедших человек уже включают раненных, больных и обмороженных, или нет. Поэтому указанные потери являются максимально возможными.

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1.2 Casualties

See above: "The information on Soviet losses at Suomussalmi would not be complete if the statistics did not include losses of the 3rd NKVD Regiment. Unfortunately, the author has no detailed information on its losses at his disposal." (2014)

[https://www.youtube.com/watch?v=KUdoR9ipbtU Battle on the Raate road. Oleg Kiselev] (2019), time - 1:11:33

3rd NKVD Regiment: 14 killed, 137 wounded, 100** missing, 251 total (** - Estimates)

3-й полк НКВД: Убито 14, Ранено 137, Без вести 100**, Всего 251 (** - Данные расчетные)

The data is the same as in Table 11, in the 2014 article, except for the 3rd NKVD Regiment. With corrections, total losses 12,073. The number of killed is taken as the sum of the number of killed and the number of missing, minus those taken prisoner of war (1,800*). Frostbite - "category of non-combat casualties - sick and partially frostbitten" (2800). * - "According to Finnish figures, about 500 men were taken prisoner during the battle for Suomussalmi in December, and another 1,300 were taken prisoner on the Raate road." ("Согласно финским данным, в боях за Суомуссалми в декабре было взято в плен около 500 человек, и еще 1300 человек было взято в плен на дороге Раате.") CypressL (talk) 21:41, 7 January 2022 (UTC)

title: Talk:Battle of Suomussalmi


text:
Hairy ball theorem

The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0).

The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer.

The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut".

Counting zeros
Every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. Therefore, there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.

Application to computer graphics
A common problem in computer graphics is to generate a non-zero vector in R3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that finding a non-zero vector orthogonal to the given one is equivalent to finding a non-zero vector that is tangent to the surface of that sphere where it touches the radius. However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector).

Lefschetz connection
There is a closely related argument from algebraic topology, using the Lefschetz fixed-point theorem. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology) of the identity mapping is 2. By integrating a vector field we get (at least a small part of) a one-parameter group of diffeomorphisms on the sphere; and all of the mappings in it are homotopic to the identity. Therefore, they all have Lefschetz number 2, also. Hence they have fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general Poincaré-Hopf index theorem.

Corollary
A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional sphere into itself has either a fixed point or a point that maps onto its own antipodal point. This can be seen by transforming the function into a tangential vector field as follows.

Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed. For each point p, construct the stereographic projection of s(p) with p as the point of tangency. Then v(p) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v(p) = 0, so that s(p) = p.

This argument breaks down only if there exists a point p for which s(p) is the antipodal point of p, since such a point is the only one that cannot be stereographically projected onto the tangent plane of p.

A further corollary is that any even-dimensional projective space has the fixed-point property. This follows from the previous result by lifting continuous functions of $$\mathbb{RP}^{2n}$$ into itself to functions of $$S^{2n}$$ into itself.

Higher dimensions
The connection with the Euler characteristic χ suggests the correct generalisation: the 2n-sphere has no non-vanishing vector field for n ≥ 1. The difference between even and odd dimensions is that, because the only nonzero Betti numbers of the m-sphere are b0 and bm, their alternating sum χ is 2 for m even, and 0 for m odd.

Indeed it is easy to see that an odd-dimensional sphere admits a non-vanishing tangent vector field through a simple process of considering coordinates of the ambient even-dimensional Euclidean space $$\mathbb{R}^{2n}$$ in pairs. Namely, one may define a tangent vector field to $$S^{2n-1}$$ by specifying a vector field $$v: \mathbb{R}^{2n} \to \mathbb{R}^{2n}$$ given by
 * $$ v(x_1,\dots,x_{2n}) = (x_2, -x_1,\dots,x_{2n},-x_{2n-1}).$$

In order for this vector field to restrict to a tangent vector field to the unit sphere $$S^{2n-1}\subset \mathbb{R}^{2n}$$ it is enough to verify that the dot product with a unit vector of the form $$x=(x_1,\dots,x_{2n})$$ satisfying $$\|x\|=1$$ vanishes. Due to the pairing of coordinates, one sees
 * $$ v(x_1,\dots,x_{2n}) \bullet (x_1,\dots,x_{2n}) = (x_2 x_1 - x_1 x_2) + \cdots + (x_{2n} x_{2n-1} - x_{2n-1} x_{2n}) = 0.$$

For a 2n-sphere, the ambient Euclidean space is $$\mathbb{R}^{2n+1}$$ which is odd-dimensional, and so this simple process of pairing coordinates is not possible. Whilst this does not preclude the possibility that there may still exist a tangent vector field to the even-dimensional sphere which does not vanish, the hairy ball theorem demonstrates that in fact there is no way of constructing such a vector field.

Physical exemplifications
The hairy ball theorem has numerous physical exemplifications. For example, rotation of a rigid ball around its fixed axis gives rise to a continuous tangential vector field of velocities of the points located on its surface. This field has two zero-velocity points, which disappear after drilling the ball completely through its center, thereby converting the ball into the topological equivalent of a torus, a body to which the “hairy ball” theorem does not apply. The hairy ball theorem may be successfully applied for the analysis of the propagation of electromagnetic waves, in the case when the wave-front forms a surface, topologically equivalent to a sphere (the surface possessing the Euler characteristic χ = 2). At least one point on the surface at which vectors of electric and magnetic fields equal zero will necessarily appear. On certain 2-spheres of parameter space for electromagnetic waves in plasmas (or other complex media), these type of "cowlicks" or "bald points" also appear, which indicates that there exists topological excitation, i.e., robust waves that are immune to scattering and reflections, in the systems. If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. (What is true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a non-zero 3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.

title: Hairy ball theorem


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