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\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s) $$

=Rayleigh–Ritz method=

''I propose the following cleanup of the article Rayleigh–Ritz method. --jftsang 05:28, 20 April 2014 (UTC)''

In mathematics, the Rayleigh–Ritz method is a method for finding an approximation to the smallest eigenvalue of a Sturm-Liouville operator. It is named after Walther Ritz and Lord Rayleigh.

Description
The Rayleigh-Ritz method is a direct variational method, in which the minimum of a functional defined on a normed linear space is approximated by a linear combination of elements from that space.

Suppose we are given the eigenvalue equation
 * $$Ly = \lambda w y,$$

where $$L$$ is the Sturm-Liouville operator
 * $$Ly = -\frac{\mathrm{d}}{\mathrm{d}x}\left(p(x) \frac{\mathrm{d}y}{\mathrm{d}x}\right) + q(x)y$$

with appropriate boundary conditions. We seek the smallest eigenvalue of $$L$$. Consider the functionals
 * $$F[y] = \int w y Ly \mathrm{d}x$$

and
 * $$G[y] = \int w y^2 \mathrm{d}x.$$

It can be shown that the problem of finding the smallest eigenvalue of $$L$$ is equivalent to the variational problem of minimising $$F[y]$$, subject to $$G[y]=1$$, which in turn is equivalent to minimising $$F - \lambda G$$. Here $$\lambda$$ is a Lagrange multiplier, the possible values of which being also the eigenvalues of $$L$$. We can minimise $$F - \lambda G$$ by considering a trial function $$\tilde{y}$$, which satisfies the boundary conditions and $$G[\tilde{y}]=1$$. We make an ansatz about the form of $$\tilde{y}$$ and minimise $$F - \lambda G$$ amongst functions of this form. For example, $$\tilde{y}$$ might be taken to be a truncated Taylor series; and we conduct the minimisation of $$F-\lambda G$$ by choosing the coefficients of the series properly. The method is effective if the form that we guess for $$\tilde{y}$$ can approximate the lowest-eigenvalue eigenfunction well.

Example
Consider the eigenvalue problem
 * $$Ly = -y'' + y = \lambda y$$

subject to $$y \rightarrow 0$$ as $$|x| \rightarrow \infty$$. This is the eigenvalue problem for the quantum harmonic oscillator, in suitable units. We seek approximations to the lowest eigenvalue and eigenfunction. Here $$w = 1$$, and
 * $$F-\lambda G = \int (-y y'' + y^2 - \lambda y^2) \mathrm{d}x.$$

Mechanical engineering
The Rayleigh-Ritz method is widely used in applied mathematics and mechanical engineering for the calculation of the natural vibration frequency of a structure in the second or higher order, as it can give a useful approximation even when the true solution may be intractable.

Typically in mechanical engineering it is used for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section. It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.

Other applications
The Rayleigh-Ritz method is also widely used in quantum chemistry.

Between England and the Ming Dynasty (1638—1644)
The first direct contact between the English and the Chinese was on 27 June 1637, when four heavily armed ships under Captain John Wendell arrived at Macau in an attempt to open trade between England and China. They were not backed by the East India Company, but rather by a private group led by Sir William Courteen, including King Charles I's personal interest of £10,000. They were opposed by the Portuguese authorities in Macau (who were required to do so under their agreements with China) and quickly infuriated the Ming authorities. Later in the summer they easily captured one of the Bogue forts, and spent several weeks engaged in low-level fighting and smuggling. After being forced to seek Portuguese help in the release of three hostages, they left the Pearl River on 27 December. It is unclear whether they returned home.

Between the UK and the Qing Dynasty (1644—1911)


The Ming dynasty was conquered by the Manchu-ruled Qing dynasty in 1644. In 1706, England and Scotland signed the Treaty of Union and became the Kingdom of Great Britain. Later, the Acts of Union 1800 would establish the United Kingdom of Great Britain and Ireland.

In 1685, Michael Shen Fu-Tsung visited England and met James II. A century later, George Macartney, 1st Earl Macartney led the Macartney Embassy to Beijing in 1793.

The countries traded throughout the 18th century, especially as China gradually loosened restrictions. During this time, Britain imported much tea and other goods from China, against whom it built up a large trade deficit. To correct the deficit, Britain started exporting opium to China in 1817. In the 1820s, British merchants turned Lintin Island in the Pearl River estuary into a centre of opium trade. Opium use became very prevalent in China and the balance of trade was reversed. The Qing, concerned at both the trade deficit and the social ills caused by opium, began to restrict the import of opium. Britain declared the First Opium War in 1839, after opium carried by British traders was seized and destroyed by a Chinese official.

In 1841, the Convention of Chuenpee was signed. The convention was intended to end the war with Hong Kong Island ceded to Britain, but it was never ratified. The war lasted until the signing of the Treaty of Nanking on 29 August 1842. Under the treaty, Hong Kong Island was ceded the British, and China opened five treaty ports to international trade. The Treaty of Nanking was supplemented by the Treaty of the Bogue (October 1843), which granted extraterritoriality to British subjects in China and most favoured nation status to Britain.

Further British and French demands for the legalisation of opium and the opening of ports were rejected by the Qing, and in 1856 the Second Opium War broke out. The war had a brief interlude in 1858, in the June of which the Treaty of Tientsin was signed by Lord Elgin. The treaty would open more ports to trade, grant more privileges to foreigners in China, and legalise opium in China, but it was at first not ratified by China. Fighting resumed until October 1860, when the victorious British and French troops sacked and destroyed Old Summer Palace. China ratified the Treaty of Tientsin and signed the Convention of Peking, ceding Kowloon Peninsula to Britain.

In accordance with the treaties, a British Legation opened in Beijing (Peking) on 26 March 1861. In the following few years consulates opened throughout the Empire, including Hankou (Wuhan), Takao (Kaohsiung), Tamsui (near Taipei), Shanghai and Xiamen.


 * 1868 - The Yangzhou riot
 * 1877 - A Chinese Legation opens in London under Guo Songtao (Kuo Sung-t'ao)
 * 1888 - War in Sikkim between the British and Tibetans. By the Treaty of Calcutta (1890), China recognizes British suzerainty over northern Sikkim.
 * 1896 - Sun Yat-sen is detained in the Chinese Legation in London. Under pressure from the British public, the Foreign Office secures his release.
 * 9 June 1898 - Convention for the Extension of Hong Kong Territory (Second Convention of Peking): New Territories are leased to Britain for 99 years, and are incorporated in Hong Kong
 * 1898 - The British obtain a lease on Weihai Harbour, Shandong, to run for as long as the Russians lease Port Arthur. (The reference to the Russians was replaced with one to the Japanese after 1905).
 * 1900-1901 - The Boxer Rebellion
 * 1901 - The Boxer Protocol
 * 1906 - Anglo-Chinese Treaty on Tibet, which the UK interprets as limiting China to suzerainty over the region
 * 1909 - The Japanese Government claims foreign consulates in Taiwan; the British consulates at Tamsui and Takoa close the following year.

Between the UK and the Republic of China (1912 - )

 * 1930 - Weihai Harbour returned to China.
 * 17 May 1935 - Following decades of Chinese complaints about the low rank of Western diplomats, the British Legation in Beijing is upgraded to an Embassy.
 * 1936-37 - British Embassy moves to Nanjing (Nanking), following the earlier transfer there of the Chinese capital.
 * 1937-41 - British public and official opinion favours China in its war against Japan, but Britain focuses on defending Singapore and the Empire and can give little help.  It does provide training in India for Chinese infantry divisions, and air bases in India used by the Americans to fly supplies and warplanes to China.
 * 1941-45 - Chinese and British fight side by side against Japan in World War II. The British train Chinese troops in India and use them in the Burma campaign.
 * 6 January 1950 - His Majesty's Government (HMG) removes recognition from the Republic of China. The Nanjing Embassy is then wound down. The Tamsui Consulate is kept open under the guise of liaison with the Taiwan Provincial Government.
 * 13 March 1972 - The Tamsui Consulate is closed.
 * February 1976 - The Anglo Taiwan Trade Committee is formed to promote trade between Britain and Taiwan.
 * 30 June 1980 - Fort San Domingo is seized by the Republic of China authorities in lieu of unpaid rent.
 * 1989 - The Anglo Taiwan Trade Committee begins issuing British visas in Taipei.
 * 1993 - British Trade and Cultural Office opened in Taipei.

Between the UK and the People's Republic of China (1949 - )
The United Kingdom and the anti-Communist Nationalist Chinese government were allies during World War II. Britain sought stability in China after the war to protect its more than £300 million in investments, much more than from the United States. It agreed in the Moscow Agreement of 1945 to not interfere in Chinese affairs but sympathized with the Nationalists, who until 1947 were winning the Chinese Civil War against the Communist Party of China. By August 1948, however, the Communists' victories caused the British government to begin preparing for a Communist takeover of the country. It kept open consulates in Communist-controlled areas and rejected the Nationalists' requests that British citizens assist in the defense of Shanghai. By December, the government concluded that although British property in China would likely be nationalized, British traders would benefit in the long run from a stable, industrializing Communist China. Retaining Hong Kong was especially important; although the Communists promised to not interfere with its rule, Britain reinforced the Hong Kong Garrison during 1949. When the victorious Communist government declared on 1 October 1949 that it would exchange diplomats with any country that ended relations with the Nationalists, Britain&mdash;after discussions with other Commonwealth members and European countries&mdash;formally recognised the People's Republic of China in January 1950.


 * 20 April 1949 - The People's Liberation Army attacks HMS Amethyst (F116) travelling to the British Embassy in Nanjing and forces a successful British rescue mission. Since the Communist Party of China does not recognise the UK or the Unequal Treaties, it argues that the ship has no right to be on the Yangtse.
 * 6 January 1950 - The United Kingdom recognises the PRC as the government of China and posts a chargé d'affaires ad interim in Beijing (Peking). The British expect a rapid exchange of Ambassadors. However, the PRC demands concessions on the Chinese seat at the UN and the foreign assets of the Republic of China.
 * c.1950 - British companies seeking trade with the PRC form the Group of 48 (now China-Britain Business Council).
 * 1950 - British Commonwealth Forces in Korea successfully defend Hill 282 against Chinese and North Korean forces in the Battle of Pakchon, part of the Korean War.
 * 1950 - The Chinese People's Volunteer Army defeat the British at the Battle of Chosin Reservoir, part of the Korean War
 * 1951 - The Chinese defeat the British at the Battle of the Imjin River after numerous casualties in a pyrrhic victory, in the Korean War.
 * 1951 - Chinese forces attacking outnumbered British Commonwealth forces are held back in the Battle of Kapyong.
 * 1951 - British Commonwealth forces successfully capture Hill 317 from Chinese forces in the Battle of Maryang San.
 * 1953 - Outnumbered British forces successfully defend Yong Dong against Chinese forces in the Battle of the Hook.
 * 1954 - The Sino-British Trade Committee formed as semi-official trade body (later merged with the Group of 48).
 * 1954 - A British Labour Party delegation including Clement Attlee visits China at the invitation of then Foreign Minister Zhou Enlai.
 * 17 June 1954 - Following talks at the Geneva Conference, the PRC agrees to station a chargé d'affaires in London. The same talks resulted in an agreement to re-open a British office in Shanghai, and the grant of exit visas to several British businessmen confined to the mainland since 1951.
 * 1961 - The UK begins to vote in the General Assembly for PRC membership of the United Nations. It has abstained on votes since 1950.
 * June 1967 - Red Guards break into the British Legation in Beijing and assault three diplomats and a secretary. The PRC authorities refuse to condemn the action. British officials in Shanghai were attacked in a separate incident, as the PRC authorities attempted to close the office there.
 * June–August 1967 - Hong Kong 1967 riots. The commander of the Guangzhou Military Region, Huang Yongsheng, secretly suggests invading Hong Kong, but  his plan is vetoed by Zhou Enlai.
 * July 1967 - Hong Kong 1967 riots - Chinese People's Liberation Army troops fire on British Hong Kong Police, killing 5 of them.
 * 23 July 1967-25 September 1969 - Anthony Grey, a young Reuters journalist, is kept under house arrest in Beijing, in retaliation for the imprisonment of Communist journalists in Hong Kong.
 * 23 August 1967 - A Red Guard mob sacks the British Legation in Beijing, slightly injuring the chargé d'affaires and other staff, in response to British arrests of Communist agents in Hong Kong. A Reuters correspondent, Anthony Grey, was also imprisoned by the PRC authorities.
 * 29 August 1967 - Armed Chinese diplomats attack British police guarding the Chinese Legation in London.
 * 13 March 1972 - PRC accords full recognition to HMG, permitting the exchange of ambassadors. HMG acknowledges the PRC's position on Taiwan without accepting it.
 * 1982 - During negotiations with Margaret Thatcher about the return of Hong Kong, Deng Xiaoping tells her that China can simply invade Hong Kong. It was revealed later (2007) that such plans indeed existed.
 * 1984 - Sino-British Joint Declaration.
 * 30 June-1 July 1997 - Return of Hong Kong to China.
 * 1997 - China and Britain forge a strategic partnership.
 * 24 August 2008 - Olympic flag was handed over from the Beijing mayor Guo Jinlong to London mayor Boris Johnson, for the 2012 Summer Olympics in London.
 * 21 January 2005 - Sino-UK relations have entered mature period.
 * 29 October 2008 - HMG recognizes Tibet as an integral part of the PRC. It had previously only recognized Chinese "suzerainty" (supremacy over the local ruler) over the region.
 * 29 December 2009 - Sino-British relations strain after the execution of Akmal Shaikh, a British national who was said to have a mental illness, for drug smuggling.
 * 26 June 2010 - President Hu Jintao invites British PM for talks in Beijing at the start of what looks like a fresh start for the two nations.
 * 5 July 2010 - Both countries pledge closer military cooperation.
 * 8 November 2010 - A pledge was made at talks between Chinese Vice Premier Li Keqiang and UK Chancellor of the Exchequer George Osborne for greater cooperation, who will attend the third China-UK Economic and Financial Dialogue in Beijing.
 * 9 November 2010 - Sino-UK ties are "in good momentum".
 * 10 November 2010 - Both countries pledge closer cooperation.
 * 25 November 2010 - senior military officials met in Beijing to discuss military cooperation, including the deputy chief of staff of the People's Liberation Army, and the chief of the general staff of the British army.
 * 9 January 2011 - Sino-UK relations are off to a good start.
 * 12 January 2011 - China and Britain plans to intensify their strategic partnership.
 * 26 June 2011 - Chinese PM visits London in order to plan out trade between the two countries which is worth billions of pounds.
 * 7 May 2013 - China bans David Cameron from visiting Beijing ever again after he meets with the Dalai Lama. As a result, British-Chinese relationships take a turn for the worse and there are threats that China will halt trade with the island as a result of Cameron's actions.
 * October 2013 - Britain's chancellor George Osborne visited China to look at making new trade links. He said that the UK and China have "much in common" in a speech on a visit to China.

=Eigenvalue perturbation=

A matrix eigenvalue problem
Suppose the symmetric matrix $M_{0}$ has known eigenvalues and eigenvectors: in particular, suppose that
 * $$Mx_0 = \lambda_0x_0$$

with $x_{0}&ne;0$, and real $&lambda;$. Assume that this is the only eigenvector (up to scalar multiplication) of $M$ with this eigenvalue. Now consider a second matrix
 * $$M = M_0 + \epsilon M'$$

where $&epsilon;$ is small. For sufficiently small $&epsilon;$, there will be an eigenvalue $&lambda;$ and an eigenvector $x$ of $M$ that 'corresponds' to $&lambda;_{0}$ and $x_{0}$.

We write the new eigenvalue and eigenvector as a perturbation series in $&epsilon;$:
 * $$\lambda = \lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + \cdots$$
 * $$x = x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots.$$

Then the equation
 * $$Mx = \lambda x$$

becomes
 * $$(M_0 + \epsilon M')(x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots) = (\lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + \cdots)(x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots)$$

Multiplying this out and comparing the powers of $&epsilon;$ lets us find the corrections iteratively. For example, comparing the $O(&epsilon;)$ gives
 * $$M_0 x_1 + M' x_0 = \lambda_0 x_1 + \lambda_1 x_0$$
 * $$(M_0 - \lambda_0 I)x_1 = -(M' - \lambda_1 I)x_0$$

Multiply both sides on the left by $x_{0}^{T}$. (If $M$ were not symmetric, then this would not necessarily be the left eigenvector of $M$.) The left hand side vanishes, and so
 * $$0 = x_0^T(M' - \lambda_1 I)x_0.$$

But $x_{0}&ne;0$. Therefore