text:
Sparse grid

Sparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician Sergey A. Smolyak, a student of Lazar Lyusternik, and are based on a sparse tensor product construction. Computer algorithms for efficient implementations of such grids were later developed by Michael Griebel and Christoph Zenger.

Curse of dimensionality
The standard way of representing multidimensional functions are tensor or full grids. The number of basis functions or nodes (grid points) that have to be stored and processed depend exponentially on the number of dimensions.

The curse of dimensionality is expressed in the order of the integration error that is made by a quadrature of level $$l$$, with $$N_{l}$$ points. The function has regularity $$r$$, i.e. is $$r$$ times differentiable. The number of dimensions is $$d$$.

$$|E_l| = O(N_l^{-\frac{r}{d}})$$

Smolyak's quadrature rule
Smolyak found a computationally more efficient method of integrating multidimensional functions based on a univariate quadrature rule $$Q^{(1)}$$. The $$d$$-dimensional Smolyak integral $$Q^{(d)}$$ of a function $$f$$ can be written as a recursion formula with the tensor product.

$$Q_l^{(d)} f = \left(\sum_{i=1}^l \left(Q_i^{(1)}-Q_{i-1}^{(1)}\right)\otimes Q_{l-i+1}^{(d-1)}\right)f$$

The index to $$Q$$ is the level of the discretization. If a 1-dimension integration on level $$i$$ is computed by the evaluation of $$O(2^{i})$$ points, the error estimate for a function of regularity $$r$$ will be $$|E_l| = O\left(N_l^{-r}\left(\log N_l\right)^{(d-1)(r+1)}\right)$$

title: Sparse grid


text:
Kingborough Council

Kingborough Council is a local government body in Tasmania, and one of the five municipalities that constitutes the Greater Hobart Area. Kingborough is classified as an urban local government area and has a population of 37,734, it covers the transition from the southern urban areas of Hobart through Kingston, as well as encompassing Bruny Island.

Etymology
The origin of Kingborough Council is a simple derivation from the name of the main town. The name Kingston was suggested by Mr Lucas in 1851, and the area had been known as Brown's River before then. Why he suggested this name is unknown. Mr Lucas' parents had been raised in England near New Kingston, they had come from Norfolk Island where the capital was Kingston or it might have been named after the Governor of New Norfolk Philip Gidley King.

History
Europeans settled in the Kingborough Council’s district in 1808 at Brown’s River (Promenalinah), named after Robert Brown, botanist in 1804. The town and district were both known as Brown’s River during the earliest periods of settlement. The Lucas family settled the area and built The Red House, a prominent mansion which is now Kingston Golf Club. Early Kingston developed primarily around The Red House. Development of Kingborough was slow and the first post office opened only in the 1840s. The first road to Hobart, Proctors Road, was opened in 1835.

Kingston was proclaimed a town in 1851. The town’s name was suggested by the then Police Magistrate, a member of the Lucas family. By 1890 Kingston had a population of 249. When proclaimed on 20 August 1907, the Kingborough Municipality comprised 3 wards: Kingston, Margate and Longley. On 2 April 1993 Kingborough absorbed the former municipality of Bruny.

In 1877 Keen's Curry was invented in the municipality.

The area developed as a holiday area, especially the suburbs of Kingston Beach and Blackmans Bay. Margate and Kettering operated fishing fleets and Taroona had a manufacturing industry, primarily through Taroona Shot Tower. Sandfly became a centre for small fruits and apple orcharding. The decline of the apple industry progressed during the 1960s, but the industry collapsed entirely following the 1967 Tasmanian fires, when much of the municipality was devastated and most orchards burnt down. A memorial to the 62 fatalities of the fire was constructed in Snug following the fire, where 80 of the towns 120 buildings burned and a permanent exhibition is present at the Channel Heritage Centre.

Demographics
Localities and smaller towns of the region include Taroona, Kettering, Margate, Snug, Blackmans Bay and Woodbridge. Kingston is classified as urban, fringe and medium (UFM) under the Australian Classification of Local Governments.

Hobart suburbs
•	•	•	•	•	•	•

Other localities
•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	Pelverata •	•	•	•	•	•	•

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

title: Kingborough Council


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Mitral stenosis

Mitral stenosis is a valvular heart disease characterized by the narrowing of the opening of the mitral valve of the heart. It is almost always caused by rheumatic valvular heart disease. Normally, the mitral valve is about 5 cm2 during diastole. Any decrease in area below 2 cm2 causes mitral stenosis. Early diagnosis of mitral stenosis in pregnancy is very important as the heart cannot tolerate increased cardiac output demand as in the case of exercise and pregnancy. Atrial fibrillation is a common complication of resulting left atrial enlargement, which can lead to systemic thromboembolic complications such as stroke.

Signs and symptoms
Signs and symptoms of mitral stenosis include the following: Fatigue and weakness increase with exercise and pregnancy.
 * Heart failure symptoms, such as dyspnea on exertion, orthopnea and paroxysmal nocturnal dyspnea (PND)
 * Palpitations
 * Chest pain
 * Hemoptysis
 * Thromboembolism in later stages when the left atrial volume is increased (i.e., dilation). The latter leads to increase risk of atrial fibrillation, which increases the risk of blood stasis (motionless). This increases the risk of coagulation.
 * Ascites and edema and hepatomegaly (if right-side heart failure develops)

Natural history
The natural history of mitral stenosis secondary to rheumatic fever (the most common cause) is an asymptomatic latent phase following the initial episode of rheumatic fever. This latent period lasts an average of 16.3 ± 5.2 years. Once symptoms of mitral stenosis begin to develop, progression to severe disability takes 9.2 ± 4.3 years.

In individuals having been offered mitral valve surgery but refused, survival with medical therapy alone was 44 ± 6% at 5 years, and 32 ± 8% at 10 years after they were offered correction.

Cause
Almost all cases of mitral stenosis are due to disease in the heart secondary to rheumatic fever and the consequent rheumatic heart disease. Uncommon causes of mitral stenosis are calcification of the mitral valve leaflets, and as a form of congenital heart disease. It is the most common valvular heart disease in pregnancy.

Other causes include infective endocarditis where the vegetations may favor increase risk of stenosis. Other rare causes include mitral annular calcification, endomyocardial fibroelastosis, malignant carcinoid syndrome, systemic lupus erythematosus, whipple disease, fabry disease, and rheumatoid arthritis. hurler' disease, hunter's disease, amyloidosis.

Pathophysiology
The normal area of the mitral valve orifice is about 4 to 6 cm2. In normal cardiac physiology, the mitral valve opens during left ventricular diastole, to allow blood to flow from the left atrium to the left ventricle. A normal mitral valve will not impede the flow of blood from the left atrium to the left ventricle during (ventricular) diastole, and the pressures in the left atrium and the left ventricle during ventricular diastole will be equal. The result is that the left ventricle gets filled with blood during early ventricular diastole, with only a small portion of extra blood contributed by contraction of the left atrium (the "atrial kick") during late ventricular diastole.

When the mitral valve area goes below 2 cm2, the valve causes an impediment to the flow of blood into the left ventricle, creating a pressure gradient across the mitral valve. This gradient may be increased by increases in the heart rate or cardiac output. As the gradient across the mitral valve increases, the amount of time necessary to fill the left ventricle with blood increases. Eventually, the left ventricle requires the atrial kick to fill with blood. As the heart rate increases, the amount of time that the ventricle is in diastole and can fill up with blood (called the diastolic filling period) decreases. When the heart rate goes above a certain point, the diastolic filling period is insufficient to fill the ventricle with blood and pressure builds up in the left atrium, leading to pulmonary congestion.

When the mitral valve area goes less than 1 cm2, there will be an increase in the left atrial pressures (required to push blood through the stenotic valve). Since the normal left ventricular diastolic pressures is about 5 mmHg, a pressure gradient across the mitral valve of 20 mmHg due to severe mitral stenosis will cause a left atrial pressure of about 25 mmHg. This left atrial pressure is transmitted to the pulmonary vasculature and causes pulmonary hypertension. Pulmonary capillary pressures in this level cause an imbalance between the hydrostatic pressure and the oncotic pressure, leading to extravasation of fluid from the vascular tree and pooling of fluid in the lungs (congestive heart failure causing pulmonary edema).

The constant pressure overload of the left atrium will cause the left atrium to increase in size. As the left atrium increases in size, it becomes more prone to develop atrial fibrillation (AF).

In individuals with severe mitral stenosis, the left ventricular filling is dependent on the atrial kick. The loss of the atrial kick due to atrial fibrillation (i.e., blood cannot flow into the left ventricle thus accumulating in the left atrium ) can cause a precipitous decrease in cardiac output and sudden congestive heart failure.

Patients with mitral stenosis prompts a series of hemodynamic changes that frequently cause deterioration of the patient's clinical status. A reduction in cardiac output, associated with acceleration of heart rate and shortening of the diastolic time, frequently leads to congestive heart failure. In addition, when AF sets in, systemic embolization becomes a real danger.

Mitral stenosis typically progresses slowly (over decades) from the initial signs of mitral stenosis to NYHA functional class II symptoms to the development of atrial fibrillation to the development of NYHA functional class III or IV symptoms. Once an individual develops NYHA class III or IV symptoms, the progression of the disease accelerates and the patient's condition deteriorates.

Physical examination
Upon auscultation of an individual with mitral stenosis, the first heart sound is usually loud and may be palpable (tapping apex beat) because of increased force in closing the mitral valve. The first heart sound is made by the mitral and tricuspid heart valves closing. These are normally synchronous, and the sounds are termed M1 and T1, respectively. M1 becomes louder in mitral stenosis. It may be the most prominent sign.

If pulmonary hypertension secondary to mitral stenosis is severe, the P2 (pulmonic) component of the second heart sound (S2) will become loud.

An opening snap that is a high-pitch additional sound may be heard after the A2 (aortic) component of the second heart sound (S2), which correlates to the forceful opening of the mitral valve. The mitral valve opens when the pressure in the left atrium is greater than the pressure in the left ventricle. This happens in ventricular diastole (after closure of the aortic valve), when the pressure in the ventricle precipitously drops. In individuals with mitral stenosis, the pressure in the left atrium correlates with the severity of the mitral stenosis. As the severity of the mitral stenosis increases, the pressure in the left atrium increases, and the mitral valve opens earlier in ventricular diastole.

A mid-diastolic rumbling murmur with presystolic accentuation will be heard after the opening snap. The murmur is best heard at the apical region and is not radiated. Since it is a low-pitch sound, it is heard best with the bell of the stethoscope. Its duration increases with worsening disease.

Advanced disease may present with signs of right-sided heart failure such as parasternal heave, jugular venous distension, hepatomegaly, ascites and/or pulmonary hypertension, the latter often presenting with a loud P2.

Almost all signs increase with exercise and pregnancy.

Other peripheral signs include:
 * Malar flush - due to back pressure and buildup of carbon dioxide . is a natural vasodilator.
 * Atrial fibrillation - irregular pulse and loss of 'a' wave in jugular venous pressure
 * Left parasternal heave - presence of right ventricular hypertrophy due to pulmonary hypertension
 * Tapping apex beat that is not displaced

Medical signs of atrial fibrillation include:

Heart rate is about 100-150/min. Irregularly irregular pulse with a pulse deficit>10. Varying first heart sound intensity. Opening snap is not heard sometimes. Absent a waves in the neck veins. Presystolic accentuation of diastolic murmur disappears. Embolic manifestations may appear.

Associated lesions
With severe pulmonary hypertension, a pansystolic murmur produced by functional tricuspid regurgitation may be audible along the left sternal border. This murmur is usually louder during inspiration and diminishes during forced expiration (Carvallo's sign). When the cardiac output is markedly reduced in MS, the typical auscultatory findings, including the diastolic rumbling murmur, may not be detectable (silent MS), but they may reappear as compensation is restored. The Graham Steell murmur of pulmonary regurgitation, a high-pitched, diastolic, decrescendo blowing murmur along the left sternal border, results from dilation of the pulmonary valve ring and occurs in patients with mitral valve disease and severe pulmonary hypertension. This murmur may be indistinguishable from the more common murmur produced by aortic regurgitation (AR), although it may increase in intensity with inspiration and is accompanied by a loud and often palpable P2.

Echocardiography
In most cases, the diagnosis of mitral stenosis is most easily made by echocardiography, which shows left atrial enlargement, thick and calcified mitral valve with narrow and "fish-mouth"-shaped orifice and signs of right ventricular failure in advanced disease. It can also show decreased opening of the mitral valve leaflets, and increased blood flow velocity during diastole. The trans-mitral gradient as measured by Doppler echocardiography is the gold standard in the evaluation of the severity of mitral stenosis.

Cardiac chamber catheterization
Another method of measuring the severity of mitral stenosis is the simultaneous left and right heart chamber catheterization. The right heart catheterization (commonly known as Swan-Ganz catheterization) gives the physician the mean pulmonary capillary wedge pressure, which is a reflection of the left atrial pressure. The left heart catheterization, on the other hand, gives the pressure in the left ventricle. By simultaneously taking these pressures, it is possible to determine the gradient between the left atrium and left ventricle during ventricular diastole, which is a marker for the severity of mitral stenosis. This method of evaluating mitral stenosis tends to overestimate the degree of mitral stenosis, however, because of the time lag in the pressure tracings seen on the right-heart catheterization and the slow Y descent seen on the wedge tracings. If a trans-septal puncture is made during right heart catheterization, however, the pressure gradient can accurately quantify the severity of mitral stenosis.

Other techniques
Chest X-ray may also assist in diagnosis, showing left atrial enlargement.

Electrocardiography may show P mitrale, that is, broad, notched P waves in several or many leads with a prominent late negative component to the P wave in lead V1, and may also be seen in mitral regurgitation, and, potentially, any cause of overload of the left atrium. Thus, P-sinistrocardiale may be a more appropriate term.

Treatment
Treatment is not necessary in asymptomatic patients.

The treatment options for mitral stenosis include mitral valve replacement by surgery, and percutaneous mitral valvuloplasty by balloon catheter.

The indication for invasive treatment with either a mitral valve replacement or valvuloplasty is NYHA functional class III or IV symptoms.

Another option is balloon dilatation. To determine which patients would benefit from percutaneous balloon mitral valvuloplasty, a scoring system has been developed. Scoring is based on 4 echocardiographic criteria: leaflet mobility, leaflet thickening, subvalvular thickening, and calcification. Individuals with a score of ≥ 8 tended to have suboptimal results. Superb results with valvotomy are seen in individuals with a crisp opening snap, score < 8, and no calcium in the commissures.

Treatment also focuses on concomitant conditions often seen in mitral stenosis:
 * Any angina is treated with short-acting nitrovasodilators, beta-blockers and/or calcium blockers
 * Any hypertension is treated aggressively, but caution must be taken in administering beta-blockers
 * Any heart failure is treated with digoxin, diuretics, nitrovasodilators and, if not contraindicated, cautious inpatient administration of ACE inhibitors

Mitral valvuloplasty
Mitral valvuloplasty is a minimally invasive therapeutic procedure to correct an uncomplicated mitral stenosis by dilating the valve using a balloon. Under local anaesthetic, a catheter with a special balloon is passed from the right femoral vein, up the inferior vena cava and into the right atrium. The interatrial septum is punctured and the catheter passed into the left atrium using a "trans-septal technique." The balloon is sub-divided into 3 segments and is dilated in 3 stages. First, the distal portion (lying in the left ventricle) is inflated and pulled against the valve cusps. Second, the proximal portion is dilated, in order to fix the centre segment at the valve orifice. Finally, the central section is inflated, this should take no longer than 30 seconds, since full inflation obstructs the valve and causes congestion, leading to circulatory arrest and flash pulmonary edema.

With careful patient pre-selection, percutaneous balloon mitral valvuloplasty (PBMV) is associated with good success rates and a low rate of complications. By far the most serious adverse event is the occurrence of acute severe mitral regurgitation. Severe mitral regurgitation usually results from a tear in one of the valve leaflets or the subvalvular apparatus. It can lead to pulmonary edema and hemodynamic compromise, necessitating urgent surgical mitral valve replacement.

Other serious complications with PBMV usually relate to the technique of trans-septal puncture (TSP). The ideal site for TSP is the region of the fossa ovalis in the inter-atrial septum. Occasionally, however, the sharp needle used for TSP may inadvertently traumatize other cardiac structures, leading to cardiac tamponade or serious blood loss.

Although the immediate results of PBMV are often quite gratifying, the procedure does not provide permanent relief from mitral stenosis. Regular follow-up is mandatory, to detect restenosis. Long-term follow-up data from patients undergoing PBMV indicates that up to 70–75% individuals can be free of restenosis 10 years following the procedure. The number falls to about 40% 15 years post-PBMV.

title: Mitral stenosis


text:
Asia Agcaoili

Junelyn Alexis "Asia" Unana Agcaoili (born September 20, 1977 ) is a Filipina actress, model, entrepreneur, television and radio host, and former columnist.

Entertainment career
Agcaoili was already a model when she joined the ABC reality show SINGLE in 2002. In four episodes, she bested the two other contestants to become the "Single Girl". The show subsequently featured her for 13 episodes.

In 2003, she joined the second batch of the Viva Hot Babes, a girl group known for double entendre lyrics, sensual performances, as well as appearances in men's magazines and films. The group released three albums before disbanding in 2008. With her fellow Hot Babes, Agcaoili appeared in the film First Time (2003) and then starred in the sexually-informative direct-to-video films Sex Guru (2004) and Erotica: Lessons of the Flesh (2005).

In March 2004, Agcaoili was suspended for 20 days by ABS-CBN after her left breast was exposed during a Viva Hot Babes dance performance for the live variety show MTB. She explained that the incident was unintentional. In July that year, she appeared on the cover of FHM Philippines and became the magazine's sex columnist, writing monthly until 2009. She also appeared several times on the magazine's 100 Sexiest Women list, with her highest rank being 13th place in 2005 and 2006.

She then joined an episode on the GMA Network reality game show Extra Challenge, where she displayed her intelligence and confidence while facing conservative moralists.

In radio, she was a disc jockey on the radio station Magic 89.9 until January 14, 2007. Agcaoíli is known for being very open in talking about sex.

She was also a regular on the ABS-CBN show Bida si Mister, Bida si Misis.

Agcaoíli had the role of a villain in Till Death Do Us Part on ABS-CBN.

Her first television appearance in New Zealand was in a 2020 COVID-19 public service announcement.

Personal life
Agcaoili graduated from the University of the Philippines Diliman with a bachelor's degree in clothing technology.

She moved to Amsterdam, Netherlands in 2007 to join her boyfriend, Dutch computer programmer Bram van der Kolk. She gave birth to their son, Xander, in April 2009. The couple then married on September 9 that year. During her time in the Netherlands, she took Dutch integration courses at the University of Amsterdam. She also met with the self-exiled communist leader Jose Maria Sison.

Agcaoili, along with her husband and son, later moved to Auckland, New Zealand. There, Van der Kolk worked as the chief programmer of Megaupload. He, along with his colleagues, were charged with copyright infringement, racketeering, and money laundering when the company was shut down in 2012. When the company was re-launched as Mega in 2013, Van der Kolk was retained as chief programmer while Agcaoili became a shareholder.

She is bisexual.

Later shows
Agcaoili spearheaded the Pinoy Big Brother show for the viewers of Studio 23. Her show, called Pinoy Big Brother: Si Kuya, KaBarkada Mo, featured snippets from the primetime telecast the night before, but also featured opinion polls both from the man on the street and those sending SMS, spoof segments (such as predictions from "prominent" fortune tellers and insider news from the "resident" deep penetration agent, a cockroach), and live feeds from inside the house.

She was also one of the hosts of Breakfast on Studio 23 every Thursday and was part of the teleserye Vietnam Rose on ABS-CBN, where she teamed up with her Bida si Mister, Bida si Misis and the Philippines' Diamond Star, Maricel Soriano.

She also hosted the Sunday afternoon TV show iPBA on ABC, a show that featured the ins and outs of the PBA which included interviews with league personalities.

title: Asia Agcaoili


text: User talk:62.232.66.118

title: User talk:62.232.66.118


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Talk:National League North

Finances
Are any teams fully professional in Conference North? Are any players or teams amateur? Ffda (talk) 01:03, 15 January 2008 (UTC)
 * They are mostly semi-professional.--EchetusXe (talk) 15:57, 5 October 2008 (UTC)

Location of page
Shouldn't this page really be at Football Conference North?

 * Apparently not: English football league system

Fair use rationale for Image:Football Conference.png
Image:Football Conference.png is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale. Using one of the templates at Fair use rationale guideline is an easy way to insure that your image is in compliance with Wikipedia policy, but remember that you must complete the template. Do not simply insert a blank template on an image page.

If there is other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images lacking such an explanation can be deleted one week after being tagged, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.

BetacommandBot (talk) 06:25, 2 January 2008 (UTC)

Done. Chrislintott (talk) 10:03, 2 January 2008 (UTC)

Map is incorrect
It says teams from the counties in red or maybe purple go into the Conference North, yet Lowestoft Toen in Suffolk (yellow) are in the Conference North — Preceding unsigned comment added by 212.159.177.110 (talk) 15:50, 13 September 2014 (UTC)

What happens if both teams relegated from the Conference Premier are in Conference North or South ?
Explanation is needed of what happens if both relegated teams are clearly in Conference North or South region. E.g. if both relegated teams are relegated to the Conference North region, would the southernmost team in the North area have to be moved to the South area for next season to keep the same number of teams in each region ? This would imply that teams on the border between North and South can find themselves moving North and South fairly often. Rcbutcher (talk) 23:07, 8 May 2015 (UTC)

When all the promotion and relegation issues are sorted Conf North and Conf South are regarded as one big pool of clubs, then divided equally between the 22 northern most clubs and the 22 southern most clubs. The FA.L.C. advise their recommendations on allocations, but then nothing is settled until the Football Conference AGM approves the constitution, usually around the beginning of June. Which is why I find it strange that someone has already amended the member clubs for 2015/16 when they haven't actually been decided yet... DJhinckley (talk) 19:49, 13 May 2015 (UTC)

Requested move 31 May 2015

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Moved. EdJohnston (talk) 23:46, 11 June 2015 (UTC)

Conference North → National League North – The league is to be renamed the National League North due to a re-branding – Source.  Del ♉ sion 23  (talk)  00:32, 31 May 2015 (UTC)
 * Note: This discussion has been included in WikiProject Football's list of association football-related page moves.  Del ♉ sion 23  (talk)  00:39, 31 May 2015 (UTC)

 * Support. Because this is not a sponsor rename (Vanarama had no say in the decision), a move to the new name makes sense. ONR (talk) 20:25, 1 June 2015 (UTC)
 * Support However, I do wonder what the point in having separate articles for the three divisions is. There is barely any prose in there, quite a lot of content could be moved to the season articles (or duplicates them); the remainder would easily fit in a single article. Number   5  7  10:47, 2 June 2015 (UTC)
 * Support - no-brainer. The official (i.e. unsponsored) name of the league has changed, so the article title should reflect this fact -- ChrisTheDude (talk) 15:52, 2 June 2015 (UTC)
 * Support per nom. Mattythewhite (talk) 15:57, 2 June 2015 (UTC)
 * Support - to match change in name. GiantSnowman 17:35, 2 June 2015 (UTC)

 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

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title: Talk:National League North


text:
Henry FitzAlan-Howard, 2nd Viscount FitzAlan of Derwent

Henry Edmund FitzAlan-Howard, 2nd Viscount FitzAlan of Derwent OBE (30 October 1883 – 17 May 1962), was a British peer.

Biography
FitzAlan-Howard was the only son of Edmund FitzAlan-Howard, 1st Viscount FitzAlan of Derwent, second infancy-surviving son of Henry Fitzalan-Howard, 14th Duke of Norfolk, and Lady Mary Bertie (1859–1938), daughter of Montagu Bertie, 7th Earl of Abingdon.

He served as a captain in the First World War and was wounded. After the war he was assistant private secretary from 1921 to 1922 to his father, the last Lord Lieutenant of Ireland. He succeeded to the viscountcy on his father's death in 1947, after which he became heir to the Dukedom of Norfolk, which he was until his own death.

Marriage and children
FitzAlan was married on 9 May 1922 to Joyce Elizabeth Mary Langdale (born 25 April 1898, died 1995), eldest daughter and heiress of Lt Col Philip Joseph Langdale, OBE, JP, DL of Houghton Hall, East Riding of Yorkshire. They divorced in 1955, having had two daughters:
 * Hon Alathea Gwendoline Alys Mary Fitzalan-Howard (born 24 November 1923, died 5 March 2001), married Hon Edward Frederick Ward, a younger son of William Ward, 2nd Earl of Dudley.
 * Hon Elizabeth Anne Marie Gabrielle Fitzalan-Howard (born 26 January 1934, died 20 March 1997), married Sir Vivyan Edward Naylor-Leyland, 3rd Baronet.

Alathea's wartime diaries were published in 2020 as The Windsor Diaries.

Death
Lord FitzAlan of Derwent died on 17 May 1962, aged 78, at which time the family home was 12 Montagu Square, his taxable estate probated at GB£81307 1962. The Viscountcy became extinct on his death.


text:
Miss World 2003

Miss World 2003 was the 53rd edition of the Miss World pageant, held at the Crown of Beauty Theatre in Sanya, China on 6 December 2003. It was the first time that the Miss World pageant held in China.

At the end of the event, Azra Akın of Turkey crowned Rosanna Davison of Ireland as Miss World 2003. It is Ireland's first victory in the history of the pageant.

Contestants 106 contestants competed in this year's pageant, surpassing the previous record of 95 contestants in 2000. The pageant was hosted by Phil Keoghan, Amanda Byram, and Angela Chow. Luis Fonsi and Bryan Ferry performed in this year's pageant.

Contestants

106 contestants competed for the title.

Judges
Miss World 2003 had nine judges.
 * Candace Bushnell
 * Jackie Chan
 * Agbani Darego – Miss World 2001 from Nigeria
 * Bruce Forsyth†
 * Gustavo Gianetti
 * Julia Morley – Chairwoman of the Miss World Organisation
 * Krish Naidoo
 * Clive Robertson
 * Dick Zimmermann

Debuts

 * 🇦🇩 Andorra
 * 🇪🇹 Ethiopia
 *  Georgia
 * 🇬🇵 Guadeloupe
 *  Northern Marianas

Returns

 * Last competed in 1981:
 * 🇱🇸 Lesotho
 * Last competed in 1998:
 * 🇲🇺 Mauritius
 * Last competed in 1999:
 * 🇿🇲 Zambia
 * Last competed in 2000:
 * 🇧🇾 Belarus
 * 🇩🇰 Denmark
 * 🇬🇹 Guatemala
 * 🇲🇩 Moldova
 * 🇳🇵 Nepal
 * 🇵🇾 Paraguay
 * 🇱🇰 Sri Lanka
 * Last competed in 2001:
 * 🇰🇾 Cayman Islands
 * 🇨🇷 Costa Rica
 * 🇩🇴 Dominican Republic
 * 🇮🇸 Iceland
 * 🇰🇷 South Korea
 * 🇵🇹 Portugal
 * 🇨🇭 Switzerland

Withdrawals

 * 🇦🇲 Armenia – Lusine Tovmasyan - She withdrew due to financial problems.
 * 🇫🇯 Fiji – Aishwarya Sukhdeo - She withdrew at the last minute for unknown reasons. however, she competed in Miss World a year later.

No Shows

 * 🇩🇿 Algeria – Mounia Achlaf
 * 🇦🇹 Austria – Miss Austria 2003, Tanja Duhovich did not compete due to not meeting the age requirements, she was just 17. Then the Miss Austria corporation decided to appoint the Miss Austria 2003 second runner up, Bianca Zudrell at the last minute with no time to prepare her visa.
 *  US Virgin Islands – Alexandrya Evans. She competed at Miss Universe 2011 8 years later.
 *  British Virgin Islands – No contest
 * 🇭🇹 Haiti - No Contest
 * 🇲🇼 Malawi – Mable Pulu
 *  Tahiti - Heitiare Tribondeau
 *  Turks and Caicos- No contest
 * 🇬🇭 Ghana – Did not compete because of a shift in the Miss Ghana calendar. Organizers held the national final, Miss Ghana 2003, on the same day Miss World 2003 was held. This is the reason Ghanaian delegates to Miss World have their titles dating back by a year.

Replacements

 * 🇱🇻 Latvia – Agnese Eiduka
 * 🇸🇪 Sweden – Isabelle Jonsson - She was the first runner up of Fröken Sverige 2003, but the organization just lost the MW licence that year to the newly Miss World Sweden contest.
 * 🇻🇪 Venezuela – Amara Barroeta Seijas.

Country Changes

 * Yugoslavia changed its name to Serbia and Montenegro.

Other notes

 * Northern Marianas was originally crowned for the Miss Universe pageant but was sent to Miss World instead.

title: Miss World 2003


text:
Mark Little (Australian actor)

Mark Little (born 20 October 1959) is an Australian actor, television presenter, comedian and screen/stage writer. He is known for portraying the role of Joe Mangel from 1988 to 1991, 2005 and 2022 on the Australian soap opera Neighbours.

Career
Little appeared in a string of Australian films and TV series during the 1980s, including Short Changed (1986), written by Aboriginal playwright Bob Merritt and directed by George Ogilvie. He was nominated for Best Supporting Actor in the AFI Awards for this role.

He also performed his own comedy in Melbourne's comedy clubs throughout the eighties, while his longest-running television role was as Joe Mangel in the soap opera Neighbours, from 1988 to 1991. Owing to the show's popularity in the UK, he became known in the country and subsequently moved there. In 1990, he co-hosted with Tania Lacy on Countdown Revolution, a music show that was on ABC each week night. He and Lacy were fired from the show in June after their infamous protest about acts having to mime and the hosts having to pretend to like an artist, even if they didn't.

In 1994, he replaced Chris Evans as the presenter of The Big Breakfast. He has also narrated cult TV series The Villa on Sky TV.

He has worked on films such as Blackball and in the West End with his one-man show Defending the Caveman, which also won the Laurence Olivier Award in 2000. He returned to Neighbours as Joe for a brief period in 2005.

Little appeared as Roy in a production of Louis Nowra's play Così at The White Bear Theatre in Kennington, London, in 2008. From the following year, he started making regular appearances on the discussion show The Wright Stuff.

As well as extended runs of Defending the Caveman at London's Leicester Square Theatre, he toured the play around the UK in 2010.

In 2010, he made his directorial debut starring in, designing and directing Jack Hibberd's A Stretch of the Imagination at The Cock Tavern Theatre in Kilburn, London.

Little has also written comedy films and poetry. He appeared in a documentary special celebrating Neighbours' 30th anniversary titled Neighbours 30th: The Stars Reunite, which aired in Australia and the UK in March 2015.

In 2017, Little played the character of "Fleshcreep" in a pantomime production of Jack and the Beanstalk at the Palace Theatre, Mansfield.

In 2019, Little took part in the eleventh series of Dancing on Ice, partnered with Brianne Delcourt. They become the first couple to leave after the judges chose to save ex-cricketer Ryan Sidebottom and his skating partner Brandee Malto in the skate-off.

He has also taken many self-devised shows to the Edinburgh Festival Fringe and toured them nationally throughout the UK.

In 2021, he appeared at the Middlesbrough Little Theatre in a pantomime production of Peter Pan, playing the part of Captain Hook.

In 2022, he reprised his role as Joe Mangel for the ending of Neighbours.

Awards

 * 1985 Australian Television Society Award, Best Actor for The Flying Doctors
 * 1997 Paramount Comedy Award, Edinburgh Festival Fringe for Psychobubble
 * 2000 Laurence Olivier Award for Best Entertainment, Defending the Caveman
 * 2008 Contributing Artist: Brighton Festival of Artists Open Houses, Best Open House: Cath Farr's The Art of Fun
 * Australian Film Institute: Nominations for Best Supporting Actor: 1985 An Indecent Obsession (Ben Maynard) and 1987 Short Changed (Curly)
 * Logie Awards: Nomination for Best Supporting Actor 1984 Waterfront (Alan)
 * Australian Television Society: Nomination for Best Actor 1990 Neighbours (Joe Mangel)

title: Mark Little (Australian actor)


text:
Pamir (ship)

Pamir was a four-masted barque built for the German shipping company F. Laeisz. One of their famous Flying P-Liners, she was the last commercial sailing ship to round Cape Horn, in 1949. By 1957, she had been outmoded by modern bulk carriers and could not operate at a profit. Her shipping consortium's inability to finance much-needed repairs or to recruit sufficient sail-trained officers caused severe technical difficulties. On 21 September 1957, she was caught in Hurricane Carrie and sank off the Azores, with only six survivors rescued after an extensive search.

Early days and World War I
She was built at the Blohm & Voss shipyards in Hamburg, launched on 29 July 1905. She had a steel hull and tonnage of 3,020 GRT (2,777 net). She had an overall length of 114.5 m (375 ft), a beam of about 14 m (46 ft) and a draught of 7.25 m (23.5 ft). Three masts stood 51.2 m (168 ft) above deck and the main yard was 28 m (92 ft) wide. She carried 3,800 m² (40,900 ft²) of sails and could reach a top speed of 16 kn. Her regular cruise speed was around 8-9 knots.

She was the fifth of ten near-sister ships. She was commissioned on 18 October 1905 and used by the Laeisz company in the South American nitrate trade. By 1914, she had made eight voyages to Chile, taking between 64 and about 70 days for a one-way trip from Hamburg to Valparaíso or Iquique, the foremost Chilean nitrate ports at the time. From October 1914, she stayed in Santa Cruz de la Palma port in La Palma Island, Canary Islands. Due to post war conditions, she did not return from Santa Cruz de la Palma to Hamburg until 17 March 1920.

In the same year, she was handed over to Italy as war reparation. On 15 July 1920, she left Hamburg via Rotterdam to Naples towed by tugs. The Italian government was unable to find a deep-water sailing ship crew, so she was laid up near Castellamare in the Gulf of Naples.

In 1924, the F. Laeisz Company bought her back for £7,000 and put her into service in the nitrate trade again. Laeisz sold her in 1931 to the Finnish shipping company of Gustaf Erikson, which used her in the Australian wheat trade.

World War II and beyond
During World War II, Pamir was seized as a prize of war by the New Zealand government on 3 August 1941 while in port at Wellington. Ten commercial voyages were made under the New Zealand flag: five to San Francisco, three to Vancouver, one to Sydney and her last voyage across the Tasman from Sydney to Wellington carrying 2,700 tons of cement and 400 tons of nail wire. Weathering a storm during the last Tasman voyage is described in detail by one of the mates, Andrew Keyworth, in a letter never posted.

She escaped the war unscathed despite a close call in 1943 when a Japanese submarine was spotted. Evidently as a fast-moving barque under a strong and fair wind, she did not interest the submarine's commander. After the war, she made one voyage from Wellington via Cape Horn to London, then Antwerp to Auckland and Wellington in 1948.

She was returned to the Erikson Line on 12 November 1948 at Wellington and sailed to Port Victoria on Spencer Gulf to load Australian grain. On her 128-day journey to Falmouth, she was the last windjammer carrying a commercial load around Cape Horn, on 11 July 1949.

Gustaf Erikson had died in 1947. His son Edgar found he could no longer operate her (or Passat) at a profit, owing primarily to changing regulations and union contracts governing employment aboard ships; the standard two-watch system on sailing ships was replaced by the three-watch system in use on motor-ships, requiring more crew.

In March 1951, Belgian shipbreakers paid £40,000 for her and Passat. As she was being towed to Antwerp, German shipowner Heinz Schliewen, who had sailed on her in the late 1920s, bought her (and Passat, thus often erroneously referred to as a sister ship). The ships were modernized with refurbished quarters to accommodate merchant marine trainees and fitted with an auxiliary engine, a refrigeration system for the galleys (precluding the need to carry live animals for fresh meat), modern communications equipment and water ballast tanks. Her first trip was to Brazil in 1952 with cement, to return to Germany with iron ore. Early in the outbound voyage the propeller dropped off, "much to the satisfaction of the sail-favoring crew if not the owner." The enterprise went briefly bankrupt but was bought by a new consortium of 40 German shipowners. For the next five years, the ships continued to sail between Europe and the east coast of South America, but not around Cape Horn. They were used as cargo-carrying school ships, primarily to Argentina. Although the German public supported the concept as maritime symbols and sources of national pride, the economic realities of the post-war years placed restraints on the operation. The ships were no longer profitable as freight haulers, and Pamir had increasing technical problems such as leaking decks and serious corrosion. The consortium was unable to get sufficient increased funding from German governments or contributions from shipping companies or public donations, and thus let both vessels deteriorate.

Last voyage
Due to ill health, her regular captain, Hermann Eggers, had been replaced by Captain Johannes Diebitsch, who had sailed on her as a young seaman and had commanded sail-training ships, but had little experience as master of cargo-carrying sailing ships. His first officer, Rolf Köhler, was only 29 at the time, and wrote that he was "getting thin with anger" over the state of the ship and that he was intending to quit the ship's company after arriving home from the next voyage. Captain Diebitsch was criticized for being a harsh and inflexible officer.

On 10 August 1957, she left Buenos Aires for Hamburg with a crew of 86, including 52 cadets. Her cargo of 3,780 tons of barley was stored loose in the holds and ballast tanks, secured by 255 tons in sacks on top of the loose grain. Records indicate that this was one of the major mistakes implicated in the sinking – she had been held up by a dockworkers' strike, and Diebitsch, under severe pressure to sail, decided to let the trimming (the correct storage of loose cargo so that it does not shift in the hold) be done by his own untrained crew. It was later found that he also had the ballast tank filled with barley. Even though testing of the roll period (the time the ship took to right itself after load transfers) showed that she was dangerously unstable, Diebitsch decided to sail.

On the morning of 21 September 1957, she was caught in Hurricane Carrie before shortening sails. It was later considered that because the radio officer had also been given substantial administrative tasks (to save the money required for another officer's position), he had likely not received any of the radio storm warnings. She had also not responded to radio hails by ships that had sighted her earlier in the voyage. She soon listed severely to port in the sudden storm. As hatchways and other openings were not closed at once, they probably allowed considerable amounts of water to enter, as found by the commission which examined the probable causes of the sinking. The shipping company's lawyer at the investigation claimed that the water entered her due to a leak. According to the commission, the water caused her to list further and the grain to shift, which aggravated the list.

The captain did not order the flooding of her grain-filled ballast tanks, which would have helped her to right herself. Once she listed severely, the lifeboats could not be deployed because her port side was underwater and her starboard side was raised to an angle that did not allow use of the boats.

She sent distress signals before capsizing at 13:03 local time, and sinking after drifting keel-up for 30 minutes in the middle of the Atlantic 600 nmi west-southwest of the Azores at position 35.95°N, -40.33333°W. Three damaged lifeboats that had come loose before or during the capsizing and the only lifeboat that had been deployed were drifting nearby. None contained any provisions or working distress signal rockets. Many sharks were later seen near the position.

A nine-day search for survivors was organized by the United States Coast Guard Cutter USCGC Absecon (WAVP-374), but only four crewmen and two cadets were rescued alive, from two of the lifeboats. It was reported that many of the 86 men aboard had managed to reach the boats, but most died in the next three days. As none of the officers nor the captain survived, the reasons for the capsizing remained uncertain.

The sinking made headlines around the world; it was a national tragedy in Germany.

False coverage
The sinking received extensive but not always accurate press coverage. For example, The New Zealand Herald reported the following fabrication mixing the real event with imagined details, supposedly directly based on the survivor "Gunter Hasselback" (his real name was Günter Haselbach):

 * ‘The Loss of the Pamir’
 * Last of the ‘P’ Line

 * “Overwhelmed in a hurricane off the Azores on September 21, 1957, - complement of about 80 crew and training cadets – 5 survivors picked up on Tuesday, 24th. Survivors tell of how terror struck into the hearts of the naval cadets in the Pamir when huge waves tossed her around like a shuttle cock.  Her cargo of wheat shifted and she took on a 45 degree list.  Her crew fought to right her and to calm the cadets who were making their first sea voyage, but hysteria gripped them.  The captain (Diebitsch) led the cadets in prayer but it was impossible to calm them.  He ordered them away in three boats with three experienced hands in each.  As the boats were launched they were caught by the mountainous waves and sent hurtling hundreds of feet away from the ship. The boats had hardly been launched when the masts snapped and her sails were blown away.  The pounding of the seas and the drag of the masts and rigging over the side, heeled the ship over further and further.  It was now impossible to keep the Pamir’s bows head into the wind – she was lying broadside on.  There was no time to send another SOS (aerials were down) – the end was here.  In the trough of a giant wave she rolled right over and we last saw her was her bottom up and going down by the bow like a submarine slowly diving.  The few men who were still on board when she capsized were struggling in the water.  I don’t know how we got away but it seemed to me that our lifeboat was the only one successfully launched.  We had no flares or smoke signals that worked.  I could see nothing of the three boats in which the cadets had been put out from the ship.  Seventeen of the men in my boat were washed overboard in the hurricane while rescue aircraft were flying overhead above the storm.  Three others, screaming like demons, jumped overboard into the sea on Monday afternoon.  I was too weak to stop them.  If you had not found me on Tuesday, I would have done the same thing myself."

 * Gunter Hasselback

The facts as reported by the survivors and established by the Seeamt Lübeck, the German authority that investigated the sinking, are that Haselbach was the only survivor rescued from the second lifeboat, not one of the five survivors found together as the article suggests; Pamir had a cargo of barley, not wheat. It is not certain when the cargo shifted; the official opinion of the investigation indicated it was early in the storm, but others have suggested it shifted only at the very end; the cadets had already gone from Hamburg to Buenos Aires before they started the return trip.

The survivors reported that crew and cadets stayed very calm until close to the loss of the ship because the ship was not believed to be in difficulties – cadets were still taking photographs, and supposedly some complained when ordered to put on life jackets. Even at the very end, there was no panic. She was not going head into the wind at any time, and her engine was not used.

She was mounted more and more to the wind, with waves, more from abaft the beam (behind), hitting roughly from the side. Radio contact was maintained until the end. She sent her last audible SOS call at 12:54 and an indecipherable one at 13:03; she was capsizing around this time. At least one lifeboat broke free before the capsize; others detached briefly before or during the capsizing and sinking. Nobody boarded a lifeboat before she capsized, and nobody jumped overboard: when she capsized, all 86 men were still on board.

Her masts did not break nor did any yards or anything else fall down, so nothing was dragging to the side. Some sails were left until they were blown out, but others were shortened or cut off by the crew; the headsail had to be cut with knives before it would blow out. When she sank, she still had set about a third of the mizzen sail and some tarpaulin in the shrouds of the mizzen mast.

The boat that Haselbach reached was badly damaged (as were the other two that were salvaged) and almost entirely submerged when he was rescued. Nothing indicates that four boats were manned - the existence of a third manned boat is only assumed, mostly on grounds of survivors' reports of seeing flares one night. Of the 20 or 22 men who were on Haselbach's boat, ten were still on board 24 hours before he was rescued, after the hurricane dissipated. The rescue aircraft could be deployed only after the storm had calmed down. The official documents, including a report by Haselbach during the hours after he was found, say nothing about people screaming when they left the lifeboats.

The Pamir was not last of the ‘P’ Line, as the newspaper's subtitle suggests. The Passat was still in service, and other ‘P’ Liners still existed but not under sail, including the Kruzenshtern (ex Padua, the only ‘P’ Liner still under sail), the Peking and the Pommern.

Insurance
Pamir's last voyage was the only one in her school ship career during which she made a profit, as the insurance sum of about 2.2 million Deutsche Mark was sufficient to cover the company losses for that year. While there was no indication that this was the intention of the consortium, which was never legally blamed for the sinking, some later researchers have considered that through its neglect it was at least strongly implicated in the loss. But the Germanischer Lloyd had recently certified her as seaworthy.

Captains

 * 1905–1908 Carl Martin Prützmann (DE)
 * 1908–1911 Heinrich Horn (DE)
 * 1911–1912 Robert Miethe (DE)
 * 1912–1913 Gustav A. H. H. Becker (DE)
 * 1913–1914 Wilhem Johann Ehlert (DE)
 * 1914–1920 Jürgen Jürs (DE)
 * 1920–1921 C. Ambrogi (IT)
 * 1924–1925 Jochim Hans Hinrich Nissen (DE)
 * 1925–1926 Heinrich Oellrich (DE)
 * 1926–1929 Carl Martin Brockhöft (DE)
 * 1929–1930 Robert Clauß (DE)
 * 1930–1931 Walter Schaer (DE)
 * 1931–1932 Karl Gerhard Sjögren (FI)
 * 1933–1936 Mauritz Mattson (FI)
 * 1936–1937 Uno Mörn (FI)
 * 1937–1937 Linus Lindvall (FI)
 * 1937–1941 Verner Björkfelt (FI)
 * 1942–1943 Christopher Stanich (NZ)
 * 1943–1944 David McLeish (NZ)
 * 1944–1945 Roy Champion (NZ)
 * 1946–1946 Desmond Champion (NZ)
 * 1946–1948 Horace Stanley Collier (Two-Gun Pete) (NZ)
 * 1948–1949 Verner Björkfelt (FI)
 * 1951–1952 Paul Greiff (DE)
 * 1955–1957 Hermann Eggers (DE)
 * 1957 Johannes Diebitsch (DE)

title: Pamir (ship)


text:
K3 surface

In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface
 * $$x^4+y^4+z^4+w^4=0$$

in complex projective 3-space.

Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory.

It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations.

Definition
There are several equivalent ways to define K3 surfaces. The only compact complex surfaces with trivial canonical bundle are K3 surfaces and compact complex tori, and so one can add any condition excluding the latter to define K3 surfaces. For example, it is equivalent to define a complex analytic K3 surface as a simply connected compact complex manifold of dimension 2 with a nowhere-vanishing holomorphic 2-form. (The latter condition says exactly that the canonical bundle is trivial.)

There are also some variants of the definition. Over the complex numbers, some authors consider only the algebraic K3 surfaces. (An algebraic K3 surface is automatically projective. ) Or one may allow K3 surfaces to have du Val singularities (the canonical singularities of dimension 2), rather than being smooth.

Calculation of the Betti numbers
The Betti numbers of a complex analytic K3 surface are computed as follows. (A similar argument gives the same answer for the Betti numbers of an algebraic K3 surface over any field, defined using l-adic cohomology.) By definition, the canonical bundle $$ K_X = \Omega^2_X$$ is trivial, and the irregularity q(X) (the dimension $$h^1(X,O_X)$$ of the coherent sheaf cohomology group $$H^1(X,O_X)$$) is zero. By Serre duality,
 * $$h^2(X,\mathcal{O}_X)=h^0(X,K_X)=1.$$

As a result, the arithmetic genus (or holomorphic Euler characteristic) of X is:
 * $$\chi(X,\mathcal{O}_X):=\sum_i (-1)^i h^i(X,\mathcal{O}_X)=1-0+1=2.$$

On the other hand, the Riemann–Roch theorem (Noether's formula) says:
 * $$\chi(X,\mathcal{O}_X) = \frac{1}{12} \left(c_1(X)^2+c_2(X)\right),$$

where $$c_i(X)$$ is the i-th Chern class of the tangent bundle. Since $$K_X$$ is trivial, its first Chern class $$c_1(K_X)=-c_1(X)$$ is zero, and so $$c_2(X)=24$$.

Next, the exponential sequence $$0\to \Z_X\to O_X\to O_X^*\to 0$$ gives an exact sequence of cohomology groups $$0\to H^1(X,\Z) \to H^1(X,O_X)$$, and so $$H^1(X,\Z)=0$$. Thus the Betti number $$b_1(X)$$ is zero, and by Poincaré duality, $$b_3(X)$$ is also zero. Finally, $$c_2(X)=24$$ is equal to the topological Euler characteristic
 * $$\chi(X)=\sum_i (-1)^ib_i(X).$$

Since $$b_0(X)=b_4(X)=1$$ and $$b_1(X)=b_3(X)=0$$, it follows that $$b_2(X)=22$$.

Properties

 * Any two complex analytic K3 surfaces are diffeomorphic as smooth 4-manifolds, by Kunihiko Kodaira.
 * Every complex analytic K3 surface has a Kähler metric, by Yum-Tong Siu. (Analogously, but much easier: every algebraic K3 surface over a field is projective.) By Shing-Tung Yau's solution to the Calabi conjecture, it follows that every complex analytic K3 surface has a Ricci-flat Kähler metric.
 * The Hodge numbers of any K3 surface are listed in the Hodge diamond:
 * One way to show this is to calculate the Jacobian ideal of a specific K3 surface, and then using a variation of Hodge structure on the moduli of algebraic K3 surfaces to show that all such K3 surfaces have the same Hodge numbers. A more low-brow calculation can be done using the calculation of the Betti numbers along with the parts of the Hodge structure computed on $$H^2(X;\Z) $$ for an arbitrary K3 surface. In this case, Hodge symmetry forces $$H^0(X;\Omega_X^2)\cong \mathbb{C}$$, hence $$H^1(X,\Omega_X) \cong \mathbb{C}^{20}$$. For K3 surfaces in characteristic p > 0, this was first shown by Alexey Rudakov and Igor Shafarevich.
 * For a complex analytic K3 surface X, the intersection form (or cup product) on $$H^2(X,\Z)\cong\Z^{22}$$ is a symmetric bilinear form with values in the integers, known as the K3 lattice. This is isomorphic to the even unimodular lattice $$\operatorname{II}_{3,19}$$, or equivalently $$E_8(-1)^{\oplus 2}\oplus U^{\oplus 3}$$, where U is the hyperbolic lattice of rank 2 and $$E_8$$ is the E8 lattice.
 * Yukio Matsumoto's 11/8 conjecture predicts that every smooth oriented 4-manifold X with even intersection form has second Betti number at least 11/8 times the absolute value of the signature. This would be optimal if true, since equality holds for a complex K3 surface, which has signature 3−19 = −16. The conjecture would imply that every simply connected smooth 4-manifold with even intersection form is homeomorphic to a connected sum of copies of the K3 surface and of $$S^2\times S^2$$.
 * Every complex surface that is diffeomorphic to a K3 surface is a K3 surface, by Robert Friedman and John Morgan. On the other hand, there are smooth complex surfaces (some of them projective) that are homeomorphic but not diffeomorphic to a K3 surface, by Kodaira and Michael Freedman. These "homotopy K3 surfaces" all have Kodaira dimension 1.
 * Every complex surface that is diffeomorphic to a K3 surface is a K3 surface, by Robert Friedman and John Morgan. On the other hand, there are smooth complex surfaces (some of them projective) that are homeomorphic but not diffeomorphic to a K3 surface, by Kodaira and Michael Freedman. These "homotopy K3 surfaces" all have Kodaira dimension 1.

Examples

 * The double cover X of the projective plane branched along a smooth sextic (degree 6) curve is a K3 surface of genus 2 (that is, degree 2g−2 = 2). (This terminology means that the inverse image in X of a general hyperplane in $$\mathbf{P}^2$$ is a smooth curve of genus 2.)
 * A smooth quartic (degree 4) surface in $$\mathbf{P}^3$$ is a K3 surface of genus 3 (that is, degree 4).
 * A Kummer surface is the quotient of a two-dimensional abelian variety A by the action $$a\mapsto -a$$. This results in 16 singularities, at the 2-torsion points of A. The minimal resolution of this singular surface may also be called a Kummer surface; that resolution is a K3 surface. When A is the Jacobian of a curve of genus 2, Kummer showed that the quotient $$A/(\pm 1)$$ can be embedded into $$\mathbf{P}^3$$ as a quartic surface with 16 nodes.
 * More generally: for any quartic surface Y with du Val singularities, the minimal resolution of Y is an algebraic K3 surface.
 * The intersection of a quadric and a cubic in $$\mathbf{P}^4$$ is a K3 surface of genus 4 (that is, degree 6).
 * The intersection of three quadrics in $$\mathbf{P}^5$$ is a K3 surface of genus 5 (that is, degree 8).
 * There are several databases of K3 surfaces with du Val singularities in weighted projective spaces.

The Picard lattice
The Picard group Pic(X) of a complex analytic K3 surface X means the abelian group of complex analytic line bundles on X. For an algebraic K3 surface, Pic(X) means the group of algebraic line bundles on X. The two definitions agree for a complex algebraic K3 surface, by Jean-Pierre Serre's GAGA theorem.

The Picard group of a K3 surface X is always a finitely generated free abelian group; its rank is called the Picard number $$\rho$$. In the complex case, Pic(X) is a subgroup of $$H^2(X,\Z)\cong\Z^{22}$$. It is an important feature of K3 surfaces that many different Picard numbers can occur. For X a complex algebraic K3 surface, $$\rho$$ can be any integer between 1 and 20. In the complex analytic case, $$\rho$$ may also be zero. (In that case, X contains no closed complex curves at all. By contrast, an algebraic surface always contains many continuous families of curves.) Over an algebraically closed field of characteristic p > 0, there is a special class of K3 surfaces, supersingular K3 surfaces, with Picard number 22.

The Picard lattice of a K3 surface means the abelian group Pic(X) together with its intersection form, a symmetric bilinear form with values in the integers. (Over $$\Complex$$, the intersection form means the restriction of the intersection form on $$H^2(X,\Z)$$. Over a general field, the intersection form can be defined using the intersection theory of curves on a surface, by identifying the Picard group with the divisor class group.) The Picard lattice of a K3 surface is always even, meaning that the integer $$u^2$$ is even for each $$u\in\operatorname{Pic}(X)$$.

The Hodge index theorem implies that the Picard lattice of an algebraic K3 surface has signature $$(1,\rho-1)$$. Many properties of a K3 surface are determined by its Picard lattice, as a symmetric bilinear form over the integers. This leads to a strong connection between the theory of K3 surfaces and the arithmetic of symmetric bilinear forms. As a first example of this connection: a complex analytic K3 surface is algebraic if and only if there is an element $$u\in\operatorname{Pic}(X)$$ with $$u^2>0$$.

Roughly speaking, the space of all complex analytic K3 surfaces has complex dimension 20, while the space of K3 surfaces with Picard number $$\rho$$ has dimension $$20-\rho$$ (excluding the supersingular case). In particular, algebraic K3 surfaces occur in 19-dimensional families. More details about moduli spaces of K3 surfaces are given below.

The precise description of which lattices can occur as Picard lattices of K3 surfaces is complicated. One clear statement, due to Viacheslav Nikulin and David Morrison, is that every even lattice of signature $$(1,\rho-1)$$ with $$\rho\leq 11$$ is the Picard lattice of some complex projective K3 surface. The space of such surfaces has dimension $$20-\rho$$.

Elliptic K3 surfaces
An important subclass of K3 surfaces, easier to analyze than the general case, consists of the K3 surfaces with an elliptic fibration $$X\to\mathbf{P}^1$$. "Elliptic" means that all but finitely many fibers of this morphism are smooth curves of genus 1. The singular fibers are unions of rational curves, with the possible types of singular fibers classified by Kodaira. There are always some singular fibers, since the sum of the topological Euler characteristics of the singular fibers is $$\chi(X)=24$$. A general elliptic K3 surface has exactly 24 singular fibers, each of type $$I_1$$ (a nodal cubic curve).

Whether a K3 surface is elliptic can be read from its Picard lattice. Namely, in characteristic not 2 or 3, a K3 surface X has an elliptic fibration if and only if there is a nonzero element $$u\in\operatorname{Pic}(X)$$ with $$u^2=0$$. (In characteristic 2 or 3, the latter condition may also correspond to a quasi-elliptic fibration.) It follows that having an elliptic fibration is a codimension-1 condition on a K3 surface. So there are 19-dimensional families of complex analytic K3 surfaces with an elliptic fibration, and 18-dimensional moduli spaces of projective K3 surfaces with an elliptic fibration.

Example: Every smooth quartic surface X in $$\mathbf{P}^3$$ that contains a line L has an elliptic fibration $$X\to \mathbf{P}^1$$, given by projecting away from L. The moduli space of all smooth quartic surfaces (up to isomorphism) has dimension 19, while the subspace of quartic surfaces containing a line has dimension 18.

Rational curves on K3 surfaces
In contrast to positively curved varieties such as del Pezzo surfaces, a complex algebraic K3 surface X is not uniruled; that is, it is not covered by a continuous family of rational curves. On the other hand, in contrast to negatively curved varieties such as surfaces of general type, X contains a large discrete set of rational curves (possibly singular). In particular, Fedor Bogomolov and David Mumford showed that every curve on X is linearly equivalent to a positive linear combination of rational curves.

Another contrast to negatively curved varieties is that the Kobayashi metric on a complex analytic K3 surface X is identically zero. The proof uses that an algebraic K3 surface X is always covered by a continuous family of images of elliptic curves. (These curves are singular in X, unless X happens to be an elliptic K3 surface.) A stronger question that remains open is whether every complex K3 surface admits a nondegenerate holomorphic map from $$\C^2$$ (where "nondegenerate" means that the derivative of the map is an isomorphism at some point).

The period map
Define a marking of a complex analytic K3 surface X to be an isomorphism of lattices from $$H^2(X,\Z)$$ to the K3 lattice $$\Lambda=E_8(-1)^{\oplus 2}\oplus U^{\oplus 3}$$. The space N of marked complex K3 surfaces is a non-Hausdorff complex manifold of dimension 20. The set of isomorphism classes of complex analytic K3 surfaces is the quotient of N by the orthogonal group $$O(\Lambda)$$, but this quotient is not a geometrically meaningful moduli space, because the action of $$O(\Lambda)$$ is far from being properly discontinuous. (For example, the space of smooth quartic surfaces is irreducible of dimension 19, and yet every complex analytic K3 surface in the 20-dimensional family N has arbitrarily small deformations which are isomorphic to smooth quartics. ) For the same reason, there is not a meaningful moduli space of compact complex tori of dimension at least 2.

The period mapping sends a K3 surface to its Hodge structure. When stated carefully, the Torelli theorem holds: a K3 surface is determined by its Hodge structure. The period domain is defined as the 20-dimensional complex manifold
 * $$D=\{u\in P(\Lambda\otimes\Complex): u^2=0,\, u\cdot\overline{u} > 0\}.$$

The period mapping $$N\to D$$ sends a marked K3 surface X to the complex line $$H^0(X,\Omega^2)\subset H^2(X,\Complex)\cong \Lambda\otimes\Complex$$. This is surjective, and a local isomorphism, but not an isomorphism (in particular because D is Hausdorff and N is not). However, the global Torelli theorem for K3 surfaces says that the quotient map of sets
 * $$N/O(\Lambda)\to D/O(\Lambda)$$

is bijective. It follows that two complex analytic K3 surfaces X and Y are isomorphic if and only if there is a Hodge isometry from $$H^2(X,\Z)$$ to $$H^2(Y,\Z)$$, that is, an isomorphism of abelian groups that preserves the intersection form and sends $$H^0(X,\Omega^2)\subset H^2(X,\Complex)$$ to $$H^0(Y,\Omega^2)$$.

Moduli spaces of projective K3 surfaces
A polarized K3 surface X of genus g is defined to be a projective K3 surface together with an ample line bundle L such that L is primitive (that is, not 2 or more times another line bundle) and $$c_1(L)^2=2g-2$$. This is also called a polarized K3 surface of degree 2g−2.

Under these assumptions, L is basepoint-free. In characteristic zero, Bertini's theorem implies that there is a smooth curve C in the linear system |L|. All such curves have genus g, which explains why (X,L) is said to have genus g.

The vector space of sections of L has dimension g + 1, and so L gives a morphism from X to projective space $$\mathbf{P}^g$$. In most cases, this morphism is an embedding, so that X is isomorphic to a surface of degree 2g−2 in $$\mathbf{P}^g$$.

There is an irreducible coarse moduli space $$\mathcal{F}_g$$ of polarized complex K3 surfaces of genus g for each $$g\geq 2$$; it can be viewed as a Zariski open subset of a Shimura variety for the group SO(2,19). For each g, $$\mathcal{F}_g$$ is a quasi-projective complex variety of dimension 19. Shigeru Mukai showed that this moduli space is unirational if $$g\leq 13$$ or $$g=18,20$$. In contrast, Valery Gritsenko, Klaus Hulek and Gregory Sankaran showed that $$\mathcal{F}_g$$ is of general type if $$g\geq 63$$ or $$g=47,51,55,58,59,61$$. A survey of this area was given by.

The different 19-dimensional moduli spaces $$\mathcal{F}_g$$ overlap in an intricate way. Indeed, there is a countably infinite set of codimension-1 subvarieties of each $$\mathcal{F}_g$$ corresponding to K3 surfaces of Picard number at least 2. Those K3 surfaces have polarizations of infinitely many different degrees, not just 2g–2. So one can say that infinitely many of the other moduli spaces $$\mathcal{F}_h$$ meet $$\mathcal{F}_g$$. This is imprecise, since there is not a well-behaved space containing all the moduli spaces $$\mathcal{F}_g$$. However, a concrete version of this idea is the fact that any two complex algebraic K3 surfaces are deformation-equivalent through algebraic K3 surfaces.

More generally, a quasi-polarized K3 surface of genus g means a projective K3 surface with a primitive nef and big line bundle L such that $$c_1(L)^2=2g-2$$. Such a line bundle still gives a morphism to $$\mathbf{P}^g$$, but now it may contract finitely many (−2)-curves, so that the image Y of X is singular. (A (−2)-curve on a surface means a curve isomorphic to $$\mathbf{P}^1$$ with self-intersection −2.) The moduli space of quasi-polarized K3 surfaces of genus g is still irreducible of dimension 19 (containing the previous moduli space as an open subset). Formally, it works better to view this as a moduli space of K3 surfaces Y with du Val singularities.

The ample cone and the cone of curves
A remarkable feature of algebraic K3 surfaces is that the Picard lattice determines many geometric properties of the surface, including the convex cone of ample divisors (up to automorphisms of the Picard lattice). The ample cone is determined by the Picard lattice as follows. By the Hodge index theorem, the intersection form on the real vector space $$N^1(X):=\operatorname{Pic}(X)\otimes\R$$ has signature $$(1,\rho-1)$$. It follows that the set of elements of $$N^1(X)$$ with positive self-intersection has two connected components. Call the positive cone the component that contains any ample divisor on X.

Case 1: There is no element u of Pic(X) with $$u^2=-2$$. Then the ample cone is equal to the positive cone. Thus it is the standard round cone.

Case 2: Otherwise, let $$\Delta=\{u\in\operatorname{Pic}(X):u^2=-2\}$$, the set of roots of the Picard lattice. The orthogonal complements of the roots form a set of hyperplanes which all go through the positive cone. Then the ample cone is a connected component of the complement of these hyperplanes in the positive cone. Any two such components are isomorphic via the orthogonal group of the lattice Pic(X), since that contains the reflection across each root hyperplane. In this sense, the Picard lattice determines the ample cone up to isomorphism.

A related statement, due to Sándor Kovács, is that knowing one ample divisor A in Pic(X) determines the whole cone of curves of X. Namely, suppose that X has Picard number $$\rho\geq 3$$. If the set of roots $$\Delta$$ is empty, then the closed cone of curves is the closure of the positive cone. Otherwise, the closed cone of curves is the closed convex cone spanned by all elements $$u\in\Delta$$ with $$A\cdot u>0$$. In the first case, X contains no (−2)-curves; in the second case, the closed cone of curves is the closed convex cone spanned by all (−2)-curves. (If $$\rho=2$$, there is one other possibility: the cone of curves may be spanned by one (−2)-curve and one curve with self-intersection 0.) So the cone of curves is either the standard round cone, or else it has "sharp corners" (because every (−2)-curve spans an isolated extremal ray of the cone of curves).

Automorphism group
K3 surfaces are somewhat unusual among algebraic varieties in that their automorphism groups may be infinite, discrete, and highly nonabelian. By a version of the Torelli theorem, the Picard lattice of a complex algebraic K3 surface X determines the automorphism group of X up to commensurability. Namely, let the Weyl group W be the subgroup of the orthogonal group O(Pic(X)) generated by reflections in the set of roots $$\Delta$$. Then W is a normal subgroup of O(Pic(X)), and the automorphism group of X is commensurable with the quotient group O(Pic(X))/W. A related statement, due to Hans Sterk, is that Aut(X) acts on the nef cone of X with a rational polyhedral fundamental domain.

Relation to string duality
K3 surfaces appear almost ubiquitously in string duality and provide an important tool for the understanding of it. String compactifications on these surfaces are not trivial, yet they are simple enough to analyze most of their properties in detail. The type IIA string, the type IIB string, the E8×E8 heterotic string, the Spin(32)/Z2 heterotic string, and M-theory are related by compactification on a K3 surface. For example, the Type IIA string compactified on a K3 surface is equivalent to the heterotic string compactified on a 4-torus.

History
Quartic surfaces in $$\mathbf{P}^3$$ were studied by Ernst Kummer, Arthur Cayley, Friedrich Schur and other 19th-century geometers. More generally, Federigo Enriques observed in 1893 that for various numbers g, there are surfaces of degree 2g−2 in $$\mathbf{P}^g$$ with trivial canonical bundle and irregularity zero. In 1909, Enriques showed that such surfaces exist for all $$g\geq 3$$, and Francesco Severi showed that the moduli space of such surfaces has dimension 19 for each g.

André gave K3 surfaces their name (see the quotation above) and made several influential conjectures about their classification. Kunihiko Kodaira completed the basic theory around 1960, in particular making the first systematic study of complex analytic K3 surfaces which are not algebraic. He showed that any two complex analytic K3 surfaces are deformation-equivalent and hence diffeomorphic, which was new even for algebraic K3 surfaces. An important later advance was the proof of the Torelli theorem for complex algebraic K3 surfaces by Ilya Piatetski-Shapiro and Igor Shafarevich (1971), extended to complex analytic K3 surfaces by Daniel Burns and Michael Rapoport (1975).

title: K3 surface


text:
Talk:Pamir (ship)

Untitled
I have not found any non-copyrighted picture of the Pamir on the web. However, having one or two images in this article would be very nice. Lupo 13:14, 10 May 2004 (UTC)
 * Got you a stamp from the Falkland Islands. I think as a stamp it is not copyrighted. Another stamp from the cook islands was here, but smaller. The australian government has another photo here , for no-commercial use according to their copyright licenses, so I am not sure if we can use that. Hope this helps -- Chris 73 | Talk 14:20, 10 May 2004 (UTC)

 * Well, thanks. Actually, I posted this on requested pictures in hope of an image of the ship, not of a stamp, but it's better than nothing... At least one can discern the four masts. And as you also noted, most other pictures available on the web have a (sometimes very) unclear copyright status. Lupo 14:59, 10 May 2004 (UTC)

In Tall Ships Down Daniel Parrott gives the LAO of the Pamir as only 316ft. Anyone know whether which figure is correct, or whether the "overall length" in this article is actually referring to something other than the technical LAO? BruceRD 10:43, 18 Sep 2004 (UTC)

 * 316 ft would be the deck length. See . Lupo 08:02, September 9, 2005 (UTC)

Sigh, it is
I have again removed a problem sentence. The article had become less POV but the inference that the captain was to blame for not flooding the ballast tanks was a problem. Even if true, it needs citing. If it isn't true, then it is an attack by Wikipedia. The sentence removed said (1) " For unknown reasons, the captain did not have the ballast tanks flooded, which would have helped the ship to right herself again" and was also a problem. You could equally say, in part, (2) "as the ballast tanks were not flooded, the ship rolled over". The primary effect the lack of balance from ballasted tanks had on the disaster was to capsize Pamir. Not being able to provide self-righting was a secondary effect, but as the vessel was already capsized in the middle of a Category 1 Hurricane, it was already history. OK, I know my last two sentences are my opinion, but I would to see evidence why (1) is more accurate than (2). Moriori 01:09, 4 December 2006 (UTC)

 * Moriori, I'm glad we can have the discussion here... so: 1) Whether or not someone said something is not POV. It's a fact. It may be hard to track down (as in the case of a ship where most people died), but it remains a fact. In this case (the captain ordering or not ordering): Nobody (including the most outspoken supporters of the captain, esp. the shipping company's former lawyer Willner) claims that Diebitsch [the captain] actually gave the order. That's not surprising: On large sailing vessels as the Pamir, and especially in those times, orders are given hierarchically; and the person who would be giving orders with consequences as far-reaching as flooding a tank which was filled with grain (thereby destroying part of the cargo) would be the captain. Due to that very hierarchy, such orders would be followed unless a) they are (physically) impossible to be followed, or b) you have a mutiny. There was no mutiny, and there has not been the slightest hint that the orders could not be followed. Consequently, as the tanks were not flooded, there has not been any reason to assume that the captain did give the orders. Consequently, as of my knowledge, nobody has assumed that Diebitsch gave those orders. (Unfortunately it's logically impossible to "prove" or "cite evidence" that something did not happen. So if you disagree and believe that Diebitsch did in fact give those orders, I would ask you to show me who argues it. Anyway, I think the last version of the article did not even claim he did, it just said he "did not have the tanks flooded" = merely outcome-based.)

 * 2) Whether or not someone should be blamed for saying or not saying something is an entirely different matter. The article has never blamed the captain for not having said anything (and thus did not infer anything about that matter), it merely stated that he didn't. Just pointing out that someone did not do something (i.e. did not have the tanks flooded) is no attack, it's a fact. If you think that it sounds nonetheless like blame: Feel free to rephrase.

 * 3) I don't know what exact difference you see between capsizing and self-righting. As capsizing is simply further down the line, I would prefer to state simply that not flooding the tanks impacted the ships ability to right herself. But maybe that's a vocabulary problem on my part (see your discussion page), then it shouldn't be a problem for you as a native speaker to take care of it. As for the content rather than the vocabulary: The problem of the Pamir was generally that the ship listed more and more, then "fell over" to lie on her side, then capsized and finally sank. These were not independent events, but each was a consequence of the preceding event: The ship "fell over" because it listed too much, it capsized because it had been laying on its side, etc. Flooding the ballast tanks would have counteracted that spiral: It would have righted the ship, thereby preventing the "falling over" part and all the subsequent events. (Whether or not flooding the tanks would have been enough, is mere speculation. Saying that it "would have helped" should be okay, but again, feel free to rephrase.) - BTW, saying "as (!) the tanks weren't flooded, the Pamir capsized" would IMHO be a POV. And to my knowledge, nobody argues that anyway. Not flooding the tanks may have contributed, but it probably did not cause the shipwreck.

 * As a result, I still believe that there's nothing wrong whatsoever with stating that the captain did not have the tanks flooded, which would have helped to right the ship again. --Ibn Battuta 02:43, 4 December 2006 (UTC)
 * Please do not use "it's a fact....it may be hard to track down" to try to justify inclusion of any content in any article. If contested content isn't cited, it doesn't merit inclusion. Moriori 04:50, 4 December 2006 (UTC)
 * I did not use "it's a fact....it may be hard to track down" to "try" to justify anything. Citing out of context doesn't get you anywhere. I used "it's a fact....it may be hard to track down", followed by a pretty long paragraph arguing why there is no way and no need to "prove" (and hence cite) why it is the captain who does or does not give orders. I can explain it to you again: The ship capsized with the starboard tank not having been flooded. The command to flood tanks is given by a ship's captain. And this is in fact so agreed-upon that you don't even find sources arguing back and forth about it. I don't know if you've ever gone to sea or served in the military or anywhere else where hierarchy matters. If you haven't, please feel free to consult with someone who has. Flooding tanks in the midst of a hurricane and thereby destroying part of your cargo is not something that a cadet will order. And there is simply no literature on this topic because nobody but you seems to contest it. Is that so hard to understand?

 * You, on the contrary, could maybe elaborate on your opinion. Which part of the sentence is an "inference that the captain was to blame", and where does the "POV" come into play? I've tried to address anything that I thought you could object to. If you aren't satisfied, why don't you try to formulate a bit more precisely what you actually object to? --Ibn Battuta 07:25, 4 December 2006 (UTC)

Queensbury
I've unintentionally reverted an edit about the Queensbury survey (no idea why I didn't get a message that someone had worked at the article in the meantime):
 * Subsequently the UK maritime authorities did a very detailed survey of the Motor Vessel Queensbury, before it docked in Liverpool. The "Queensbury had also loaded grain in Buenos Aires and used the same method of loading as the Pamir. She sailed from Monte Video only twelve hours before the Pamir and went through the same storm that sunk the Pamir. The Queensbury had no movement of her cargo and the officers of the Queensbury were convinced that open or broken hatch covers were the most likely cause of this terrible accident.

I'm hesitant to put it back in, though. I've never heard of this survey before, and I don't buy the argument without further information:
 * Motor vessels should not list along a passage (only in severe storm); they also move differently with the regular swell.
 * Pamir's bulkhead "along the ship's length" was suspected to "leak" (because this was discovered to happen on board the Passat). Motor vessels usually don't have such a bulkhead because excessive listing is not expected in the first place; the consequences of both the bulkhead and the "leaking" would not be experienced on board the Queensbury.
 * What does "same method of loading" mean? Loose grain with some sacks? How many and where? Did they fill the compartments the same way the Pamir did (not filling all of them up to the hatches)? It is suspected that the grain was not stowed sufficiently into the corners of the cargo hold of the Pamir - does that also apply to the Queensbury method? How much did the Queensbury cargo settle down over the course of the voyage? Is that known?
 * It is not clear where Queensbury was at the time of the loss of the Pamir: The Pamir took the trade wind route for the sailing vessels which took her close to the route of the hurricane. Queensbury is likely to have traveled further east, which would have spared her the proximity of the hurricane. Which side of the hurricane was she on? (There's evidence suggesting that the navigable quarter of Carrie had actually higher wind velocities than the others.) Apart from that, it's unlikely that both ships traveled at the same speed. Generally, Queensbury seems to have been nowhere close to Pamir at the time of her loss, otherwise they would have had radio contact (which to my knowledge they didn't); and Pamir's radio message was received by vessels up to some 600 nm away if I remember correctly (sorry, I can't look it up at the moment).
 * But most of all, what exactly does it tell us if the officers of the Queensbury were convinced of anything?
 * How about the UK maritime authorities? What did they find?
 * Why does at least some of the commons literature on the Pamir not mention this survey? (I can't tell about all, of course, but I haven't seen it anywhere yet.)

Again, I don't rule out that this information is very interesting and relevant, but I'd ask for more information and/or (preferably) a valid source before putting it back in. Thanks, Ibn Battuta 20:44, 25 May 2007 (UTC)

New Zealand newspaper article
What I had originally wanted to ask: Who knows the source of this dubious newspaper article from New Zealand? It was added by an IP, so I can't follow-up on that. I don't like citing it without even knowing where it's from. It could be from the equivalence of the Sun for all we know. Which might go some way in explaining its quality... Apart from that, I doubt its relevancy given that it's getting it all wrong anyways. --Ibn Battuta 01:27, 26 May 2007 (UTC)

The New Zealand Herald of Auckland is the largest newspaper in New Zealand, and respected; though the article may be a correct quote - but wrong! But the quote/issue should be dated - will try and find out! Hugo999 (talk) 21:57, 6 December 2008 (UTC)

Coord problem
This shows up on Google maps in the Pacific at 24.487149,-146.777344. It looks like the coord templates in the article are correct. Does anyone know what is causing this?  Randall Bart    Talk   19:41, 21 June 2012 (UTC)

The 'false coverage' paragraph...
Is quite unusual in first quoting a report known to be false, at length, and then going through most of its statements to refute them one by one. I think it would be better to rewrite this in such a way that we do not reiterate misleading information before correcting it. -- Theoprakt (talk) 20:41, 6 November 2019 (UTC)

needs something on last sailing vessels
clearly, along with https://en.wikipedia.org/wiki/Omega_(barque) pamir was one of the last working sail vessels some discussion here https://www.shippingwondersoftheworld.com/last_days-sail.html — Preceding unsigned comment added by 2601:197:D00:3CA0:1992:A295:68DF:1781 (talk) 14:48, 1 July 2023 (UTC)

title: Talk:Pamir (ship)


text:
Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold $$(M, g)$$ endowed with three integrable almost complex structures $$I, J, K$$ that are Kähler with respect to the Riemannian metric $$g$$ and satisfy the quaternionic relations $$I^2=J^2=K^2=IJK=-1$$. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.

Hyperkähler manifolds were defined by Eugenio Calabi in 1979.

Early history
Marcel Berger's 1955 paper on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1).Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan and Kraines who have independently proven that any such manifold admits a parallel 4-form $$\Omega$$.The long awaited analog of strong Lefschetz theorem was published in 1982 :    $$ \Omega^{n-k}\wedge\bigwedge^{2k}T^*M=\bigwedge^{4n-2k}T^*M.$$

Equivalent definition in terms of holonomy
Equivalently, a hyperkähler manifold is a Riemannian manifold $$(M, g)$$ of dimension $$4n$$ whose holonomy group is contained in the compact symplectic group $Sp(n)$.

Indeed, if $$(M, g, I, J, K)$$ is a hyperkähler manifold, then the tangent space $SU(2n)$ is a quaternionic vector space for each point $Sp(n)$ of $T_{x}M$, i.e. it is isomorphic to $$\mathbb{H}^n$$ for some integer $$n$$, where $$\mathbb{H}$$ is the algebra of quaternions. The compact symplectic group $x$ can be considered as the group of orthogonal transformations of $$\mathbb{H}^n$$ which are linear with respect to $M$, $Sp(n)$ and $I$. From this, it follows that the holonomy group of the Riemannian manifold $$(M, g)$$ is contained in $J$. Conversely, if the holonomy group of a Riemannian manifold $$(M, g)$$ of dimension $$4n$$ is contained in $K$, choose complex structures $Sp(n)$, $Sp(n)$ and $I_{x}$ on $J_{x}$ which make $K_{x}$ into a quaternionic vector space. Parallel transport of these complex structures gives the required complex structures $$I, J, K$$ on $T_{x}M$ making $$(M, g, I, J, K)$$ into a hyperkähler manifold.

Two-sphere of complex structures
Every hyperkähler manifold $$(M, g, I, J, K)$$ has a 2-sphere of complex structures with respect to which the metric $$g$$ is Kähler. Indeed, for any real numbers $$a, b, c$$ such that

 * $$a^2 + b^2 + c^2 = 1 \, $$

the linear combination

 * $$aI + bJ + cK \, $$

is a complex structures that is Kähler with respect to $$g$$. If $$\omega_I, \omega_J, \omega_K$$ denotes the Kähler forms of $$(g, I), (g, J), (g, K)$$, respectively, then the Kähler form of $$aI + bJ + cK$$ is
 * $$a \omega_I + b \omega_J + c \omega_K.$$

Holomorphic symplectic form
A hyperkähler manifold $$(M, g, I, J, K)$$, considered as a complex manifold $$(M, I)$$, is holomorphically symplectic (equipped with a holomorphic, non-degenerate, closed 2-form). More precisely, if $$\omega_I, \omega_J, \omega_K$$ denotes the Kähler forms of $$(g, I), (g, J), (g, K)$$, respectively, then

 * $$\Omega := \omega_J + i\omega_K$$

is holomorphic symplectic with respect to $$I$$.

Conversely, Shing-Tung Yau's proof of the Calabi conjecture implies that a compact, Kähler, holomorphically symplectic manifold $$(M,I,\Omega)$$ is always equipped with a compatible hyperkähler metric. Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under the name holomorphically symplectic manifolds. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic manifold of complex dimension $$2n$$ with $$H^{2,0}(M)=1$$ is exactly $T_{x}M$; and if the simply connected Calabi–Yau manifold instead has $$H^{2,0}(M)\geq 2$$, it is just the Riemannian product of lower-dimensional hyperkähler manifolds. This fact immediately follows from the Bochner formula for holomorphic forms on a Kähler manifold, together the Berger classification of holonomy groups; ironically, it is often attributed to Bogomolov, who incorrectly went on to claim in the same paper that compact hyperkähler manifolds actually do not exist!

Examples
For any integer $$n \ge 1$$, the space $$\mathbb{H}^n$$ of $$n$$-tuples of quaternions endowed with the flat Euclidean metric is a hyperkähler manifold. The first non-trivial example discovered is the Eguchi–Hanson metric on the cotangent bundle $$T^*S^2$$ of the two-sphere. It was also independently discovered by Eugenio Calabi, who showed the more general statement that cotangent bundle $$T^*\mathbb{CP}^n$$ of any complex projective space has a complete hyperkähler metric. More generally, Birte Feix and Dmitry Kaledin showed that the cotangent bundle of any Kähler manifold has a hyperkähler structure on a neighbourhood of its zero section, although it is generally incomplete.

Due to Kunihiko Kodaira's classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus $$T^4$$. (Every Calabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because $M$ is isomorphic to $Sp(n)$.)

As was discovered by Beauville, the Hilbert scheme of $SU(2)$ points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension $Sp(1)$. This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties.

Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to $k$, where $4k$ denotes the quaternions and $H/G$ is a finite subgroup of $H$, are known as asymptotically locally Euclidean, or ALE, spaces. These spaces, and various generalizations involving different asymptotic behaviors, are studied in physics under the name gravitational instantons. The Gibbons–Hawking ansatz gives examples invariant under a circle action.

Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to certain gauge theory equations which arise from the dimensional reduction of the anti-self dual Yang–Mills equations: instanton moduli spaces, monopole moduli spaces, spaces of solutions to Nigel Hitchin's self-duality equations on Riemann surfaces, space of solutions to Nahm equations. Another class of examples are the Nakajima quiver varieties, which are of great importance in representation theory.

Cohomology
show that the cohomology of any compact hyperkähler manifold embeds into the cohomology of a torus, in a way that preserves the Hodge structure.

title: Hyperkähler manifold