The president of the Republic of Singapore is the head of state of Singapore. The president represents the country in official diplomatic functions and possesses certain executive powers over the Government of Singapore, including the control of the national reserves and the ability to revoke and appoint public service appointments. The president also holds the prerogative to grant pardons.

After Singapore achieved full internal self-governance from the British Empire in 1959, the ceremonial office of the Yang di-Pertuan Negara ( "Lord of the State") was created. The office was later succeeded by the President of Singapore after Singapore's independence in 1965. The initial role of the president was largely ceremonial and symbolic, carrying residual powers, but the role was later vested with some executive power which includes the reserve power to veto certain bills, most notably in relation to the country's reserves as a check and balance process, as well as to revoke and appoint public service appointments among other powers listed in the Constitution.

Prior to 1991, the president was appointed by Parliament. A constitutional amendment was made that year to allow for the president to be directly elected by a popular vote, which was subsequently first held in 1993. Singapore follows a non-executive model of the Westminster parliamentary system whereby the president is not the head of government but rather the head of state. These powers are instead vested in the Cabinet, led by the prime minister. Another constitutional amendment was made in 2016, that allows for a presidential election to be reserved for an ethnic community in Singapore if no one from that community had been president for the previous five presidential terms.

The president has far-reaching formal obligations and duty to act above party politics. Under the Constitution, the president must be a Singaporean citizen, non-partisan and elected by a popular vote.

The incumbent president is Tharman Shanmugaratnam, who took office on 14 September 2023 after being duly elected in the 2023 presidential election with 70.41% of the vote.

History
The office of the President of the Republic of Singapore was created on 9 August 1965 when Singapore achieved independence from Malaysia. It succeeded the office of Yang di-Pertuan Negara ( "Lord of the State") which was created when Singapore attained self-governance from the British Empire in 1959. The last Yang di-Pertuan Negara, Yusof Ishak, became the first president of Singapore. After his death in 1971, he was succeeded by Benjamin Sheares who was appointed by Parliament and served until his death in 1981.

Sheares was succeeded by Devan Nair, who was the Member of Parliament (MP) for Anson SMC from 1979 to 1981, prior to his presidency, who then resigned in 1985. Whilst the apparent reason for Nair's resignation was to seek recovery from his alcoholism, Nair disputes the account, claiming that he was forced out of office by former Prime Minister Goh Chok Tong. Nair was succeeded by Wee Kim Wee, who served as the Ambassador of South Korea and Japan from 1980 to 1984 prior to his presidency until 1993 and was the first president to exercise custodial powers pursuant to the constitutional amendments in 1991.

In the early 1980s, the People's Action Party (PAP), who at the time occupied all the parliamentary seats in Singapore, suffered their first parliamentary loss in 15 years to the J. B. Jeyaretnam of the Worker's Party (WP). As such, then-Prime Minister Lee Kuan Yew became worried that there would one day be a "freak election result" where the PAP would no longer have control over the Parliament and opposition parties would have unfettered access to the government's reserves. As such, in January 1991, a constitutional amendment was passed by Parliament to redefine the role of the president.

The amendment provided that the president would be elected by a popular vote, subject to strict eligibility requirements. The president was also empowered by the amendment to veto the use of the country's reserves and revoke or appoint public service appointments. The president can also examine the government's usage of the Internal Security Act and Maintenance of Religious Harmony Act, and concur with the director of the Corrupt Practices Investigation Bureau (CPIB) to investigate persons for corrupt practices, even if the prime minister refuses consent.

The first elected president was Ong Teng Cheong, who served as Deputy Prime Minister from 1985 to 1993 prior to his presidency, won the 1993 presidential election. He served as president from 1993 to 1999. The Singapore Government officially regarded Ong's predecessor Wee Kim Wee as the first elected president on the basis that he held and exercised the powers of the elected president. This was a result of transitional provisions in the Constitution of Singapore in 2017, which were affirmed by the High Court following a legal challenge by then-presidential candidate Tan Cheng Bock. Tan appealed against this decision, but the appeal was dismissed by the Court of Appeal.

The sixth and the oldest to become president was S. R. Nathan, who served as the Ambassador of the United States from 1990 to 1996 prior to his presidency. He ran unopposed, and became president by virtue of being the sole candidate deemed qualified by the Presidential Elections Committee (PEC). He served his first term of office from 1999 to 2005 before being re-elected unopposed again for a second term, and he served until 2011.

After S. R. Nathan stepped down, Tony Tan, who served as Deputy Prime Minister from 1995 to 2005 prior to his presidency, won the 2011 presidential election by a narrow margin in a four-cornered fight. He was sworn in as the seventh president of Singapore on 1 September 2011. Ong Teng Cheong and Tony Tan are the only Deputy Prime Minister who became presidents.

In 2016, further amendments to the Constitution were passed providing for "reserved elections" for a particular ethnic community, if that community has not provided a president in the past five presidential terms.

The eighth president, Halimah Yacob, who served as Speaker of Parliament from 2013 to 2017, prior to her presidency, took office on 14 September 2017. She was the sole eligible candidate under the new reform terms which took effect earlier that year. Yacob was also the first Malay head of state in 47 years since the death of the first president of Singapore, Yusof Ishak, and the first female President of Singapore.

On 1 September 2023, Tharman Shanmugaratnam, who served as Senior Minister from 2019 to 2023, prior to his presidency, won the 2023 presidential election and was elected the ninth President of Singapore after receiving 70.41% of the votes cast.

Constitutional role
The president is the head of state of Singapore. In addition to being the head of state, it is also the function of the president to safeguard the national reserves and the integrity of the public service of Singapore. The executive authority of Singapore is vested in the president and exercisable by them or by the Cabinet or any minister authorised by the Cabinet. However, the Constitution vests "general direction and control of the Government" in the Cabinet. In most cases, the president is bound to exercise their powers in accordance with the advice of the Cabinet or of a minister acting under the general authority of the Cabinet.

However, the president is able to exercise some powers in their personal discretion such as preventing the government of the day from drawing on the reserves which were not accumulated during its current term of office, revoking or appointing any of the public offices under Article 22 of the Constitution such as Chief Justice, Attorney-General, Chief of Defence Force and Commissioner of Police, amongst others., exercising oversight over the Corrupt Practices Investigation Bureau (CPIB) and decisions of the Executive under the Internal Security Act and the Maintenance of Religious Harmony Act.

As a component of the legislature together with Parliament, the president is also jointly vested with legislative authority. The president's primary role in the exercise of legislative power to make laws is assenting to bills passed by Parliament.

As the president exercises this constitutional function in accordance with the Cabinet's advice and not in their personal discretion except in certain circumstances, they may not refuse to assent to bills that Parliament has validly passed. The words of enactment in Singapore Statutes are: "Be it enacted by the president with the advice and consent of the Parliament of Singapore, as follows:". The president usually opens each parliamentary session with an address drafted by the Cabinet setting out the government's agenda for the session, and may address Parliament and send messages to it.

The president has been called "Singapore's No. 1 diplomat". Ambassadors and high commissioners accredited to Singapore present their credentials to the president, and the president is called upon by visiting foreign dignitaries. In addition, the president contributes to the nation's external relations by undertaking overseas trips on Cabinet's advice. The president also serves as the ex officio chancellor of both the National University of Singapore and the Nanyang Technological University.

Presidents have also used the office to champion charitable causes. Wee Kim Wee promoted sports and volunteerism; Ong Teng Cheong promoted culture and the arts, particularly music; and S. R. Nathan established the President's Challenge with the Ministry of Community Development, Youth and Sports and its statutory board, the National Council of Social Service. As of 2011, the endeavour had raised more than S$100 million for charities supporting disabled and needy people.

Powers

The powers of the president are divided into those which the president may exercise in their own discretion, and those which must be exercised in accordance with the advice of the Cabinet or of a minister acting under the general authority of the Cabinet. In addition, the president is required to consult the Council of Presidential Advisers (CPA) when performing some of their functions. In other cases, they may consult the CPA if they wish to but is not bound to do so.

The Constitution confers on the president certain executive functions to block attempts by the government of the day to draw down reserves that it did not accumulate. Thus, a guarantee may only be given or a loan raised by the government if the president concurs, and their approval is also needed for budgets of specified statutory boards and state-owned companies that draw on their past reserves.

The president also possesses personal discretion to withhold assent to any bill in Parliament providing directly or indirectly for the direct or indirect variation, changing or increase in powers of the Central Provident Fund Board to invest moneys belonging to it; and the borrowing of money, the giving of any guarantee or the raising of any loan by the government if in the president's opinion the bill is likely to draw on reserves not accumulated by the Government during its current term of office. In addition, the president may withhold assent to any Supply Bill, Supplementary Supply Bill or Final Supply Bill for any financial year if in their opinion the estimates of revenue and expenditure, supplementary estimates or statement of excess are likely to lead to a drawing on past reserves.

The president is also empowered to approve changes to key political appointments, such as the chief justice, attorney-general, chairman and members of the Public Service Commission, chief of Defence Force and the commissioner of Police. They also appoint the prime minister, a Member of Parliament (MP) who, in their opinion, is likely to command the confidence of a majority of MPs. The president has certain powers of oversight over the Corrupt Practices Investigation Bureau and decisions of the Executive under the Internal Security Act and the Maintenance of Religious Harmony Act.

Under the Singapore Armed Forces Act, the president has the authority to raise and maintain the Singapore Armed Forces (SAF). The president also has the power to form, disband or amalgamate units within the SAF.

The term of office of the first elected president, Ong Teng Cheong, was marked by the differences between the government and him, concerning the extent of his discretionary fiscal powers. Discussions culminated in the government issuing a non-binding white paper entitled The Principles for Determining and Safeguarding the Accumulated Reserves of the Government and the Fifth Schedule Statutory Boards and Government Companies (1999).

In 2009, the government had to request for the approval from President S. R. Nathan to draw $4.9 billion from past financial reserves to meet current budget expenditure, the first time it had done so. The sum was used to fund the government's Resilience Package consisting of two schemes aimed at preserving jobs and businesses during the financial downturn.

The president also holds the prerogative to grant pardons.

Election

Qualifications
A person who wishes to run for the office of president has to fulfil stringent qualifications set out in the Constitution, which are as follows:
The president must be a citizen of Singapore.
The president must not be less than 45 years of age.
The president's name must appear in a current register of electors.
The president must be resident in Singapore at the date of their nomination for election, and must have been so resident for periods amounting in the aggregate to not less than ten years prior to that date.
The president must not be subject to any of the following disqualifications:
(a) being and having been found or declared to be of unsound mind;
(b) being an undischarged bankrupt;
(c) holding an office of profit;
(d) having been nominated for election to Parliament or the office of President or having acted as election agent to a person so nominated, failing to lodge any return of election expenses required by law within the time and in the manner so required;
(e) having been convicted of an offence by a court of law (whether in Singapore or not) and sentenced to imprisonment for a term of not less than one year or to a fine of not less than S$10,000 (or equivalent in foreign currency) and having not received a free pardon. The person if convicted by a foreign court of law shall not be disqualified unless the offence is also one which, had it been committed in Singapore, would have been punishable by a court of law in Singapore;
(f) having voluntarily acquired the citizenship of, or exercised rights of citizenship in, a foreign country, or having made a declaration of allegiance to a foreign country;
(g) being disqualified under any law relating to offences in connection with elections to Parliament or the office of President by reason of having been convicted of such an offence or having in proceedings relating to such an election been proved guilty of an act constituting such an offence.
The president must be a person of integrity, good character and reputation.
The president must not be a member of any political party on the date of their nomination for election.
The president must have for a period of not less than three years held office —
as Minister, Chief Justice, Speaker, Attorney-General, Chairman of the Public Service Commission, Auditor-General, Accountant-General or Permanent Secretary;
as chief executive officer (CEO) of a key statutory board or government company: the Central Provident Fund Board, the Housing and Development Board, the Jurong Town Corporation, the Monetary Authority of Singapore, Temasek Holdings, or GIC Private Limited (formerly known as the Government of Singapore Investment Corporation);
as the most senior executive of a company with an average of $500 million in shareholders' equity for the most recent three years in that office, and which is profitable after taxes; or
in any other similar or comparable position of seniority and responsibility in any other organisation or department of equivalent size or complexity in the public or private sector which has given him such experience and ability in administering and managing financial affairs as to enable him to carry out effectively the functions and duties of the office of president.

The strictness of these qualifications led to the 1999, 2005, and 2017 elections being walkovers as only one candidate had qualified on nomination day.

In November 2016, further amendments provide for "reserved elections" for a particular racial group (Chinese, Malay and Indian/other minority) — if that community has not been represented for five presidential terms. Other amendments were made to expand the list of key government companies eligible for the candidacy, and, for candidates using their private sector experience, the use of $500 million of shareholder equity instead of $100 million in paid-up capital. The changes went into effect in April 2017. Prime Minister Lee Hsien Loong later explained that while he expected the "reserved election" policy to be unpopular among the population, he believed it was "the right thing to do".

Election procedure

The president holds office for a term of six years from the date on which they assume office. The office falls vacant upon the expiry of the incumbent's term or if the president is for some reason unable to complete their term; for example, due to death, resignation, or removal from office for misconduct or mental or physical infirmity. If the office of president becomes vacant before the incumbent's term expires, a poll for an election must be held within six months. In other cases, an election can take place any time from three months before the expiry of the incumbent's term of office.

The procedure for elections is laid out in the Presidential Elections Act. The process begins when the prime minister issues a writ of election to the returning officer specifying the date and place of nomination day. Potential candidates must obtain certificates of eligibility from the Presidential Elections Committee (PEC), the function of which is to ensure that such persons have the necessary qualifications to be nominated as a candidate for the election. In particular, the PEC must be satisfied that the potential candidates are persons of integrity, good character and reputation; and if they have not previously held certain key government offices or acted as chairman of the board of directors or CEO of a company incorporated or registered under the Companies Act with shareholders' equity of at least $500 million, that they held a position of comparable seniority and responsibility in the public or private sector that has given them experience and ability in administering and managing financial affairs. The PEC consists of the chairman of the Public Service Commission, who is also the chairman of the PEC, the chairman of the Accounting and Corporate Regulatory Authority, and a member of the Presidential Council for Minority Rights. For the 2017 presidential election, the members of the PEC are Eddie Teo (chairman), Lim Soo Hoon, Chan Heng Chee, Po'ad Shaik Abu Bakar Mattar, Tay Yong Kwang, and Peter Seah.

In addition, candidates must obtain political donation certificates from the registrar of political donations stating that they have complied with the Political Donations Act, and file their nomination papers with the returning officer on nomination day. A deposit must also be paid. The candidate is declared to have been elected president if only one candidate is nominated. Otherwise, the returning officer issues a notice of contested election specifying when polling day will be.

During the election period, a candidate may not spend more than $600,000 or 30 cents for each person on the electoral register, whichever is greater. Permits must be obtained to hold election meetings and display posters and banners, and a number of acts are unlawful, including bribery, dissuading electors from voting, making false statements about candidates, treating and undue influence. Legal changes introduced in 2010 made the eve of polling day a "cooling-off day" – campaigning must not take place on that day and on polling day itself.

Polling day is a public holiday, and voting is compulsory. Voters must go to the polling stations assigned to them. After the poll closes, the presiding officer of each polling station seals the ballot boxes without opening them. Candidates or their polling agents may also affix their own seals to the ballot boxes. The ballot boxes are then taken to counting centres to be opened and the ballots counted. A candidate or his or her counting agent may ask the returning officer for a recount of votes if the difference between the number of votes for the candidate with the most votes and any other candidate's number of votes is 2% or less. After all counts, and recounts if any, have been completed, the returning officer ascertains whether the total number of electors registered to vote overseas is less than the difference between the number of votes for the two candidates with the highest number of votes. If so, the returning officer declares the candidate with the highest number of votes to be elected as president. If not, the overseas votes may be decisive. The returning officer then states the number of votes cast for each candidate and the date and location where the overseas votes will be counted.

Last contested election
The 2023 presidential election was the first election with a ballot since the 2011 election. The election was won by Tharman Shanmugaratnam with 1,746,427 (70.40%) of valid votes.

Assumption of office and disabilities
The person elected to the office of president assumes office on the day his predecessor ceases to hold office or, if the office is vacant, on the day following the election. Upon assumption of office, the president is required to take and subscribe in the presence of the chief justice or of another justice of the Supreme Court the Oath of Office, which states:

Once elected, the president shall:
 not hold any other office created or recognised by the Constitution;
 not actively engage in any commercial enterprise;
 not be a member of any political party; and
 not serve in Parliament.

Succession
In the case when the president is unable to perform their duties, their powers are temporarily transferred to the chairman of the Council of Presidential Advisers (CPA). If the chairman of the Council of Presidential Advisers is not available, the speaker of the Parliament performs the duties of the president. If both are unavailable the presidential functions are performed by an individual appointed by the Parliament.

Salary and entitlements

The Parliament of Singapore is required to provide a civil list for the maintenance of the president, and it does so by way of the Civil List and Gratuity Act. With effect from 17 February 2012, the sum under Class I of the list, which includes the president's personal pay ($1,568,900, known by the British term the "privy purse"), an entertainment allowance ($73,000) and an allowance for an acting president ($4,500), is $1,646,400. The privy purse was reduced from $4,267,500 after the president accepted the Ministerial Salaries Review Committee's recommendations on the matter.

The salaries for the president's personal staff (Class II) amount to $4,532,400. Speaking in Parliament on 10 March 2011, Minister for Finance Tharman Shanmugaratnam explained that this sum was to cater for the salaries of an additional staff officer to support the work of the Council of Presidential Advisers, and a butler manager; and to meet higher variable staff salary payments due to the nation's strong economic growth. The allowance for the Istana's household expenses (Class III) is $2,762,308, an increase from $694,000. This allowance is used to cover the maintenance of the Istana, vehicles, utilities and other supplies, as well as for ceremonies and celebrations. The increase was to cater for higher expenses for maintaining computer systems, buildings and land, and to account for inflation.

Class IV expenses for "special services" are $550,000. In previous years, this sum was used to cover various expenses such as the cost of replacing state cars and installing a new document repository. Overall, the current civil list of $9,491,100 represents a decrease of about 18% from the sum for the past fiscal year of $11,605,000.

List of presidents

See also
 Spouse of the President of Singapore
 Chief Justice of Singapore
 Prime Minister of Singapore

Notes

References

Citations

Sources

 .

Legislation
 .
  ("PEA").

Further reading

Articles
.
.

Books
.
.
.
.
.
.
.

News reports
.
.
.
.
.
.
.
.
.
.
.
.

External links

Lists of political office-holders in Singapore
Singapore
Presidents of Singapore
1965 establishments in Singapore

President of Singapore


In mathematics (particularly in abstract algebra), two mathematical structures are isomorphic when they are the same in some sense. More specifically, an isomorphism is a function between two structures that preserves the relationships between the parts. To indicate isomorphism between two structures   and , one often writes .

Using the language of category theory, this means that morphisms map to morphisms without breaking composition. An isomorphism is also a homomorphism that is one-to-one.

As an example, one can consider the operation of adding integers Z. The doubling function φ(x) = 2x maps elements of Z to elements of the even integers 2Z. Since φ(a+b) = 2(a+b) = 2a+2b = φ(a)+φ(b), adding in Z is structurally identical as adding in 2Z (which makes this an example of isomorphism).

Related pages

 Automorphism

References

Abstract algebra
Functions and mappings

Isomorphism


The Trenti is an imp-like creature that comes from the Cantabrian mythology, northern Spain. This annoying but not malicious goblin is very difficult to see because it lives deep in the forest and resembles mushrooms, leaves, and moss. Its most distinguishing features are a completely black face with green eyes. This creature tends to stay away from humans but it does like to jump out of the bushes and scare innocent walkers. Its favorite prank is to pull women's skirts.

References

Cantabrian legendary creatures
Imps
Goblins
Forest spirits

Trenti


Vacanze a Ischia (also known as Holiday Island) is a 1957 Italian comedy film directed by Mario Camerini.

Plot
The lives of several people intersect as they go on vacation to enjoy a pleasant holiday on the island. A lawyer spends it in agitation due to thinking he caused the death of a youngster catching coins in the water by the boat.  Then you have an engineer which is insulted by four youngsters making an atrocious prank, which makes him suspect that his wife has had a fling and she is with child that is not his. Then a group of young men chasing after the women for fun, but one of them falls in love with a nurse.  There is a Frenchman who comes for a restoration of his marriage but realizes that his wife is being wooed by a young islander.

Cast
 Vittorio De Sica: ingegnere Occhipinti
 Isabelle Corey: Caterina Lisotto
 Antonio Cifariello: Antonio
 Nadia Gray: Carla Occhipinti
 Myriam Bru: Denise Tissot
 Paolo Stoppa: avvocato Appicciato
 Susanne Cramer: Antonietta
 Raf Mattioli: Salvatore
 Peppino De Filippo: Battistella
 Maurizio Arena: Franco
 Bernard Dhéran: Pierre Tissot
 Nino Besozzi: Guido Lucarelli
 Giampiero Littera: Benito
 Hubert von Meyerinck: Colonel Manfredi
 Giuseppe Porelli: Judge
 Guglielmo Inglese: Cancelliere
 Ennio Girolami: Furio
 Marisa Merlini

References

External links

1957 films
Films directed by Mario Camerini
1957 comedy films
Italian comedy films
Films scored by Alessandro Cicognini
1950s Italian films

Vacanze a Ischia


In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

For example, the following laws, equations, and theories respect Lorentz symmetry:
 The kinematical laws of special relativity
 Maxwell's field equations in the theory of electromagnetism
 The Dirac equation in the theory of the electron
 The Standard Model of particle physics

The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity.

Basic properties
The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is the isotropy subgroup with respect to the origin of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations.
Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1,3), the matrix Lie group that preserves the quadratic form

on  (The vector space equipped with this quadratic form is sometimes written ). This quadratic form is, when put on matrix form (see classical orthogonal group), interpreted in physics as the metric tensor of Minkowski spacetime.

The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected. The identity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted . The restricted Lorentz group consists of those Lorentz transformations that preserve both the orientation of space and the direction of time. Its fundamental group has order 2, and its universal cover, the indefinite spin group , is isomorphic to both the special linear group  and to the symplectic group . These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably spinors. Thus, in relativistic quantum mechanics and in quantum field theory, it is very common to call  the Lorentz group, with the understanding that  is a specific representation (the vector representation) of it.

A recurrent representation of the action of the Lorentz group on Minkowski space uses biquaternions, which form a composition algebra. The isometry property of Lorentz transformations holds according to the composition property

Another property of the Lorentz group is conformality or preservation of angles. Lorentz boosts act by hyperbolic rotation of a spacetime plane, and such "rotations" preserve hyperbolic angle, the measure of rapidity used in relativity. Therefore the Lorentz group is a subgroup of the conformal group of spacetime.

Note that this article refers to  as the "Lorentz Group",  as the "Proper Lorentz Group", and  as the "Restricted Lorentz Group".  Many authors (especially in physics) use the name "Lorentz Group" for  (or sometimes even ) rather than .  When reading such authors it is important to keep clear exactly which they are referring to.

Connected components

Because it is a Lie group, the Lorentz group  is a group and also has a topological description as a smooth manifold. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.

The four connected components can be categorized by two transformation properties its elements have:
 Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector
 Some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein (tetrads)

Lorentz transformations that preserve the direction of time are called . The subgroup of orthochronous transformations is often denoted . Those that preserve orientation are called proper, and as linear transformations they have determinant .  (The improper Lorentz transformations have determinant .)  The subgroup of proper Lorentz transformations is denoted .

The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by . (Note that some authors refer to  or even  when they actually mean .)

The set of the four connected components can be given a group structure as the quotient group , which is isomorphic to the Klein four-group. Every element in  can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group
 {1, P, T, PT}

where P and T are the parity and time reversal operators:
 P = diag(1, −1, −1, −1)
 T = diag(−1, 1, 1, 1).

Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two
bits of information, which pick out one of the four connected components.  This pattern is typical of finite-dimensional Lie groups.

Restricted Lorentz group
The restricted Lorentz group  is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a continuous curve lying in the group. The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension, in this case with dimension six.

The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which are rotations in a hyperbolic space that includes a time-like direction). Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation (specified by 3 real parameters) and a boost (also specified by 3 real parameters), it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six-dimensional.  (See also the Lie algebra of the Lorentz group.)

The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO(3). The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.) A boost in some direction, or a rotation about some axis, generates a one-parameter subgroup.

Surfaces of transitivity

If a group  acts on a space , then a surface  is a surface of transitivity if  is invariant under  (i.e., ) and for any two points  there is a  such that . By definition of the Lorentz group, it preserves the quadratic form

The surfaces of transitivity of the orthochronous Lorentz group ,  acting on flat spacetime  are the following:
  is the upper branch of a hyperboloid of two sheets. Points on this sheet are separated from the origin by a future time-like vector.
  is the lower branch of this hyperboloid. Points on this sheet are the past time-like vectors.
  is the upper branch of the light cone, the future light cone.
  is the lower branch of the light cone, the past light cone.
  is a hyperboloid of one sheet. Points on this sheet are space-like separated from the origin.
The origin .

These surfaces are , so the images are not faithful, but they are faithful for the corresponding facts about . For the full Lorentz group, the surfaces of transitivity are only four since the transformation  takes an upper branch of a hyperboloid (cone) to a lower one and vice versa.

As symmetric spaces
An equivalent way to formulate the above surfaces of transitivity is as a symmetric space in the sense of Lie theory. For example, the upper sheet of the hyperboloid can be written as the quotient space , due to the orbit-stabilizer theorem. Furthermore, this upper sheet also provides a model for three-dimensional hyperbolic space.

Representations of the Lorentz group

These observations constitute a good starting point for finding all infinite-dimensional unitary representations of the Lorentz group, in fact, of the Poincaré group, using the method of induced representations. One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are called little groups by physicists. The problem is then essentially reduced to the easier problem of finding representations of the little groups. For example, a standard vector in one of the hyperbolas of two sheets could be suitably chosen as . For each , the vector pierces exactly one sheet. In this case the little group is , the rotation group, all of whose representations are known. The precise infinite-dimensional unitary representation under which a particle transforms is part of its classification. Not all representations can correspond to physical particles (as far as is known). Standard vectors on the one-sheeted hyperbolas would correspond to tachyons. Particles on the light cone are photons, and more hypothetically, gravitons. The "particle" corresponding to the origin is the vacuum.

Homomorphisms and isomorphisms

Several other groups are either homomorphic or isomorphic to the restricted Lorentz group SO+(1, 3). These homomorphisms play a key role in explaining various phenomena in physics.

 The special linear group SL(2, C) is a double covering of the restricted Lorentz group. This relationship is widely used to express the Lorentz invariance of the Dirac equation and the covariance of spinors. In other words, the (restricted) Lorentz group is isomorphic to SL(2, C)/
 The symplectic group Sp(2, C) is isomorphic to SL(2, C); it is used to construct Weyl spinors, as well as to explain how spinors can have a mass.
 The spin group Spin(1, 3) is isomorphic to SL(2, C); it is used to explain spin and spinors in terms of the Clifford algebra, thus making it clear how to generalize the Lorentz group to general settings in Riemannian geometry, including theories of supergravity and string theory.
 The restricted Lorentz group is isomorphic to the projective special linear group PSL(2, C) which is, in turn, isomorphic to the Möbius group, the symmetry group of conformal geometry on the Riemann sphere. This relationship is central to the classification of the subgroups of the Lorentz group according to an earlier classification scheme developed for the Möbius group.

The Weyl representation
The Weyl representation or spinor map is a pair of surjective homomorphisms from SL(2,C) to SO+(1, 3). They form a matched pair under parity transformations, corresponding to left and right chiral spinors.

One may define an action of SL(2,C) on Minkowski spacetime by writing a point of spacetime as a two-by-two Hermitian matrix in the form

in terms of Pauli matrices.

This presentation, the Weyl presentation, satisfies

Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as a real vector space) with Minkowski spacetime, in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An element  acts on the space of Hermitian matrices via

where  is the Hermitian transpose of . This action preserves the determinant and so SL(2,C) acts on Minkowski spacetime by (linear) isometries. The parity-inverted form of the above is

which transforms as

That this is the correct transformation follows by noting that

remains invariant under the above pair of transformations.

These maps are surjective, and kernel of either map is the two element subgroup ±I. By the first isomorphism theorem, the quotient group PSL(2, C) = SL(2, C) / {±I} is isomorphic to SO+(1, 3).

The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism of  These two distinct coverings corresponds to the two distinct chiral actions of the Lorentz group on spinors. The non-overlined form corresponds to right-handed spinors transforming as  while the overline form corresponds to left-handed spinors transforming as

It is important to observe that this pair of coverings does not survive quantization; when quantized, this leads to the peculiar phenomenon of the chiral anomaly. The classical (i.e., non-quantized) symmetries of the Lorentz group are broken by quantization; this is the content of the Atiyah–Singer index theorem.

Notational conventions
In physics, it is conventional to denote a Lorentz transformation  as  thus showing the matrix with spacetime indexes  A four-vector can be created from the Pauli matrices in two different ways: as  and as  The two forms are related by a parity transformation. Note that

Given a Lorentz transformation  the double-covering of the orthochronous Lorentz group by  given above can be written as

Dropping the  this takes the form

The parity conjugate form is

Proof
That the above is the correct form for indexed notation is not immediately obvious, partly because, when working in indexed notation, it is quite easy to accidentally confuse a Lorentz transform with its inverse, or its transpose. This confusion arises due to the identity  being difficult to recognize when written in indexed form. Lorentz transforms are not tensors under Lorentz transformations! Thus a direct proof of this identity is useful, for establishing its correctness. It can be demonstrated by starting with the identity

where  so that the above are just the usual Pauli matrices, and  is the matrix transpose, and  is complex conjugation. The matrix  is

Written as the four-vector, the relationship is

This transforms as

Taking one more transpose, one gets

The symplectic group
The symplectic group Sp(2, C) is isomorphic to SL(2, C). This isomorphism is constructed so as to preserve a symplectic bilinear form on  that is, to leave the form invariant under Lorentz transformations.  This may be articulated as follows. The symplectic group is defined as

where

Other common notations are  for this element; sometimes  is used, but this invites confusion with the idea of almost complex structures, which are not the same, as they transform differently.

Given a pair of Weyl spinors (two-component spinors)

the invariant bilinear form is conventionally written as

This form is invariant under the Lorentz group, so that for  one has

This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant mass term in Lagrangians. There are several notable properties to be called out that are important to physics. One is that  and so

The defining relation can be written as

which closely resembles the defining relation for the Lorentz group

where  is the metric tensor for Minkowski space and of course,  as before.

Covering groups
Since  is simply connected, it is the universal covering group of the restricted Lorentz group .  By restriction, there is a homomorphism .  Here, the special unitary group SU(2), which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group SO(3). Each of these covering maps are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient.  One often says that the restricted Lorentz group and the rotation group are doubly connected.  This means that the fundamental group of the each group is isomorphic to the two-element cyclic group Z2.

Twofold coverings are characteristic of spin groups.  Indeed, in addition to the double coverings
 Spin+(1, 3) = SL(2, C) → SO+(1, 3)
 Spin(3) = SU(2) → SO(3)

we have the double coverings
 Pin(1, 3) → O(1, 3)
 Spin(1, 3) → SO(1, 3)
 Spin+(1, 2) = SU(1, 1) → SO(1, 2)

These spinorial double coverings are constructed from Clifford algebras.

Topology
The left and right groups in the double covering
 SU(2) → SO(3)
are deformation retracts of the left and right groups, respectively, in the double covering
 SL(2, C) → SO+(1, 3).
But the homogeneous space SO+(1, 3)/SO(3) is homeomorphic to hyperbolic 3-space H3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H3.  Since the latter is homeomorphic to R3, while SO(3) is homeomorphic to three-dimensional real projective space RP3, we see that the restricted Lorentz group is locally homeomorphic to the product of RP3 with R3.  Since the base space is contractible, this can be extended to a global homeomorphism.

Conjugacy classes
Because the restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group PSL(2, C), its conjugacy classes also fall into five classes:
 Elliptic transformations
 Hyperbolic transformations
 Loxodromic transformations
 Parabolic transformations
 The trivial identity transformation

In the article on Möbius transformations, it is explained how this classification arises by considering the fixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here to null eigenspaces of restricted Lorentz transformations in their action on Minkowski spacetime.

An example of each type is given in the subsections below, along with the effect of the one-parameter subgroup it generates (e.g., on the appearance of the night sky).

The Möbius transformations are the conformal transformations of the Riemann sphere (or celestial sphere).  Then conjugating with an arbitrary element of SL(2,C) obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on the flow lines of the corresponding one-parameter subgroups is to transform the pattern seen in the examples by some conformal transformation. For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points still flow along circular arcs from one fixed point toward the other. The other cases are similar.

Elliptic
An elliptic element of SL(2, C) is

and has fixed points  = 0, ∞.  Writing the action as  and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation

This transformation then represents a rotation about the  axis, exp().  The one-parameter subgroup it generates is obtained by taking  to be a real variable, the rotation angle, instead of a constant.

The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles.  The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about the   axis as  increases. The angle doubling evident in the spinor map is a characteristic feature of spinorial double coverings.

Hyperbolic
A hyperbolic element of SL(2,C) is

and has fixed points  = 0, ∞.  Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin.

The spinor map converts this to the Lorentz transformation

This transformation represents a boost along the  axis with rapidity . The one-parameter subgroup it generates is obtained by taking  to be a real variable, instead of a constant.  The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points along longitudes away from the South pole and toward the North pole.

Loxodromic
A loxodromic element of SL(2, C) is

and has fixed points  = 0, ∞. The spinor map converts this to the Lorentz transformation

The one-parameter subgroup this generates is obtained by replacing η + iθ  with any real multiple of this complex constant. (If η, θ vary independently, then a two-dimensional abelian subgroup is obtained, consisting of simultaneous rotations about the  axis and boosts along the -axis; in contrast, the one-dimensional subgroup discussed here consists of those elements of this two-dimensional subgroup such that the rapidity of the boost and angle of the rotation have a fixed ratio.)

The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles).  They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves called loxodromes.  Each loxodrome spirals infinitely often around each pole.

Parabolic
A parabolic element of SL(2, C) is

and has the single fixed point  = ∞ on the Riemann sphere.  Under stereographic projection, it appears as an ordinary translation along the real axis.

The spinor map converts this to the matrix (representing a Lorentz transformation)

This generates a two-parameter abelian subgroup, which is obtained by considering  a complex variable rather than a constant.  The corresponding continuous transformations of the celestial sphere (except for the identity transformation) move points along a family of circles that are all tangent at the North pole to a certain great circle.  All points other than the North pole itself move along these circles.

Parabolic Lorentz transformations are often called null rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime.

The matrix given above yields the transformation

Now, without loss of generality, pick . Differentiating this transformation with respect to the now real group parameter  and evaluating at  produces the corresponding vector field (first order linear partial differential operator),

Apply this to a function , and demand that it stays invariant; i.e., it is annihilated by this transformation.  The solution of the resulting first order linear partial differential equation can be expressed in the form

where  is an arbitrary smooth function.  The arguments of  give three rational invariants describing how points (events) move under this parabolic transformation, as they themselves do not move,

Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime.  This curve is an orbit of the transformation.

The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinate , each orbit is the intersection of a null plane, , with a hyperboloid,  .  The case 3 = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes.

A particular null line lying on the light cone is left invariant; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above.  The other null lines through the origin are "swung around the cone" by the transformation.  Following the motion of one such null line as   increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.

A choice   instead, produces similar orbits, now with the roles of  and  interchanged.

Parabolic transformations lead to the gauge symmetry of massless particles (like photons) with helicity || ≥ 1. In the above explicit example, a massless particle moving in the  direction, so with 4-momentum , is not affected at all by the -boost and -rotation combination   defined below, in the "little group" of its motion. This is evident from the explicit transformation law discussed: like any light-like vector, P itself is now invariant; i.e., all traces or effects of  have disappeared.  1 = 2 = 3 = 0, in the special case discussed. (The other similar generator,  as well as it and z comprise altogether the little group of the light-like vector, isomorphic to (2).)

Appearance of the night sky
This isomorphism has the consequence that Möbius transformations of the Riemann sphere represent the way that Lorentz transformations change the appearance of the night sky, as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars".

Suppose the "fixed stars" live in Minkowski spacetime and are modeled by points on the celestial sphere. Then a given point on the celestial sphere can be associated with , a complex number that corresponds to the point on the Riemann sphere, and can be identified with a null vector (a light-like vector) in Minkowski space

or, in the Weyl representation (the spinor map), the Hermitian matrix

The set of real scalar multiples of this null vector, called a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the celestial sphere (equivalently, lines of sight) are identified with certain Hermitian matrices.

Projective geometry and different views of the 2-sphere

This picture emerges cleanly in the language of projective geometry. The (restricted) Lorentz group acts on the projective celestial sphere. This is the space of non-zero null vectors with  under the given quotient for projective spaces:  if  for . This is referred to as the celestial sphere as this allows us to rescale the time coordinate  to 1 after acting using a Lorentz transformation, ensuring the space-like part sits on the unit sphere.

From the Möbius side,  acts on complex projective space , which can be shown to be diffeomorphic to the 2-sphere - this is sometimes referred to as the Riemann sphere. The quotient on projective space leads to a quotient on the group .

Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If  is a  projective vector, it can be tensored with its Hermitian conjugate to produce a  Hermitian matrix. From elsewhere in this article we know this space of matrices can be viewed as 4-vectors. The space of matrices coming from turning each projective vector in the Riemann sphere into a matrix is known as the Bloch sphere.

Lie algebra

As with any Lie group, a useful way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group  is a matrix Lie group, its corresponding Lie algebra  is a matrix Lie algebra, which may be computed as
.

If  is the diagonal matrix with diagonal entries , then the Lie algebra  consists of  matrices  such that
.

Explicitly,  consists of  matrices of the form
,

where  are arbitrary real numbers. This Lie algebra is six dimensional. The subalgebra of  consisting of elements in which , , and  equal to zero is isomorphic to .

The full Lorentz group , the proper Lorentz group  and the proper orthochronous Lorentz group  (the component connected to the identity) all have the same Lie algebra, which is typically denoted .

Since the identity component of the Lorentz group is isomorphic to a finite quotient of  (see the section above on the connection of the Lorentz group to the Möbius group), the Lie algebra of the Lorentz group is isomorphic to the Lie algebra . As a complex Lie algebra  is three dimensional, but is six dimensional when viewed as a real Lie algebra.

Commutation relations of the Lorentz algebra
The standard basis matrices can be indexed as  where  take values in . These arise from taking only one of  to be one, and others zero, in turn. The components can be written as
.
The commutation relations are

There are different possible choices of convention in use. In physics, it is common to include a factor of  with the basis elements, which gives a factor of  in the commutation relations.

Then  generate boosts and  generate rotations.

The structure constants for the Lorentz algebra can be read off from the commutation relations. Any set of basis elements which satisfy these relations form a representation of the Lorentz algebra.

Generators of boosts and rotations
The Lorentz group can be thought of as a subgroup of the diffeomorphism group of R4 and therefore its Lie algebra can be identified with vector fields on R4. In particular, the vectors that generate isometries on a space are its Killing vectors, which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. We can write down a set of six generators:
 Vector fields on R4 generating three rotations i J,

 Vector fields on R4 generating three boosts i K,

The factor of  appears to ensure that the generators of rotations are Hermitian.

It may be helpful to briefly recall here how to obtain a one-parameter group from a vector field, written in the form of a first order linear partial differential operator such as

The corresponding initial value problem (consider  a function of a scalar  and solve  with some initial conditions) is

The solution can be written

or

where we easily recognize the one-parameter matrix group of rotations exp(i λ Jz) about the z axis.

Differentiating with respect to the group parameter  and setting it λ=0 in that result, we recover the standard matrix,

which corresponds to the vector field we started with.  This illustrates how to pass between matrix and vector field representations of elements of the Lie algebra. The exponential map plays this special role not only for the Lorentz group but for Lie groups in general.

Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectively η/2 (for the three boosts) or  iθ/2 (for the three rotations) times the three Pauli matrices

Generators of the Möbius  group
Another generating set arises via the isomorphism to the Möbius group.  The following table lists the six generators, in which
 The first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a real vector field on the Euclidean plane.
 The second column gives the corresponding one-parameter subgroup of Möbius transformations.
 The third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup).
 The fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.

Notice that the generators consist of
 Two parabolics (null rotations)
 One hyperbolic (boost in the  direction)
 Three elliptics (rotations about the x, y, z axes, respectively)

Worked example: rotation about the y-axis
Start with

Exponentiate:

This element of  represents the one-parameter subgroup of (elliptic) Möbius transformations:

Next,

The corresponding vector field on  (thought of as the image of  under stereographic projection) is

Writing , this becomes the vector field on

Returning to our element of , writing out the action  and collecting terms, we find that the image under the spinor map is the element of

Differentiating with respect to  at , yields the corresponding vector field on ,

This is evidently the generator of counterclockwise rotation about the -axis.

Subgroups of the Lorentz group
The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which the closed subgroups of the restricted Lorentz group can be listed, up to conjugacy.  (See the book by Hall cited below for the details.)  These can be readily expressed in terms of the generators  given in the table above.

The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
  generates a one-parameter subalgebra of parabolics SO(0, 1),
  generates a one-parameter subalgebra of boosts SO(1, 1),
  generates a one-parameter of rotations SO(2),
  (for any ) generates a one-parameter subalgebra of loxodromic transformations.
(Strictly speaking the last corresponds to infinitely many classes, since distinct  give different classes.)
The two-dimensional subalgebras are:
  generate an abelian subalgebra consisting entirely of parabolics,
  generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group Aff(1),
  generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.
The three-dimensional subalgebras use the Bianchi classification scheme:
  generate a Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of euclidean homotheties,
  generate a Bianchi VII subalgebra, isomorphic to the Lie algebra of E(2), the euclidean group,
 , where , generate a Bianchi VII subalgebra,
  generate a Bianchi VIII subalgebra, isomorphic to the Lie algebra of SL(2, R), the group of isometries of the hyperbolic plane,
  generate a Bianchi IX subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group.

The Bianchi types refer to the classification of three-dimensional Lie algebras by the Italian mathematician Luigi Bianchi.

The four-dimensional subalgebras are all conjugate to
  generate a subalgebra isomorphic to the Lie algebra of Sim(2), the group of Euclidean similitudes.

The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a closed subgroup of the restricted Lie group.  From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group.

As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or homogeneous spaces, have considerable mathematical interest. A few, brief descriptions:
 The group Sim(2) is the stabilizer of a null line; i.e., of a point on the Riemann sphere—so the homogeneous space SO+(1, 3)/Sim(2) is the Kleinian geometry that represents conformal geometry on the sphere S2.
 The (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous space SO+(1, 3)/SE(2) is the momentum space of a massless particle; geometrically, this Kleinian geometry represents the degenerate geometry of the light cone in Minkowski spacetime.
 The rotation group SO(3) is the stabilizer of a timelike vector, so the homogeneous space SO+(1, 3)/SO(3) is the momentum space of a massive particle; geometrically, this space is none other than three-dimensional hyperbolic space H3.

Generalization to higher dimensions

The concept of the Lorentz group has a natural generalization to spacetime of any number of dimensions. Mathematically, the Lorentz group of (n + 1)-dimensional Minkowski space is the indefinite orthogonal group O(n, 1) of linear transformations of Rn+1 that preserves the quadratic form

The group O(1, n) preserves the quadratic form

O(1, n) is isomorphic to O(n, 1), and both presentations of the Lorentz group are in use in the theoretical physics community.  The former is more common in literature related to gravity, while the latter is more common in particle physics literature.

A common notation for the vector space , equipped with this choice of quadratic form, is .

Many of the properties of the Lorentz group in four dimensions (where ) generalize straightforwardly to arbitrary n. For instance, the Lorentz group O(n, 1) has four connected components, and it acts by conformal transformations on the celestial (n−1)-sphere in (n+1)-dimensional Minkowski space. The identity component SO+(n, 1) is an SO(n)-bundle over hyperbolic n-space Hn.

The low-dimensional cases  and  are often useful as "toy models" for the physical case , while higher-dimensional Lorentz groups are used in physical theories such as string theory that posit the existence of hidden dimensions. The Lorentz group O(n, 1) is also the isometry group of n-dimensional de Sitter space dSn, which may be realized as the homogeneous space O(n, 1)/O(n − 1, 1). In particular O(4, 1) is the isometry group of the de Sitter universe dS4, a cosmological model.

See also

Notes

References

Reading List
 Emil Artin (1957) Geometric Algebra, chapter III: Symplectic and Orthogonal Geometry via Internet Archive, covers orthogonal groups O(p,q)
  A canonical reference; see chapters 1–6 for representations of the Lorentz group.
   An excellent resource for Lie theory, fiber bundles, spinorial coverings, and many other topics.
   See Lecture 11 for the irreducible representations of SL(2,C).

 .
  See Chapter 6 for the subalgebras of the Lie algebra of the Lorentz group.
   See also the  See Section 1.3 for a beautifully illustrated discussion of covering spaces.  See Section 3D for the topology of rotation groups.
 §41.3
  (Dover reprint edition.) An excellent reference on Minkowski spacetime and the Lorentz group.
   See Chapter 3 for a superbly illustrated discussion of Möbius transformations.

.

Lie groups
Special relativity
Group theory
Hendrik Lorentz

Lorentz group


William McCormick (1742–1815) was a Scottish-born merchant who lived in North Carolina, took the loyalist side in the American War of Independence, and played a leading role in efforts to win compensation from the British government for losses suffered as a result of the war.

Early life

William was born in Edinburgh, Scotland, on 8 April 1742, the fifth child (and third son) of Samuel McCormick, the General Examiner of Excise, and his wife Helen. He worked as a clerk in the office of the solicitor of excise in Edinburgh, including two years working for T. Wharton, one of the Commissioners of Excise. He was apparently headed for promotion when he decided to emigrate to America, leaving in early 1761.

Life in North Carolina

William's first job was working for Thomas MacKnight & Co in North Carolina.  MacKnight  was a Scotsman who had emigrated to North Carolina in 1755, and was almost certainly a relative of William's by marriage. William lived in the settlement of Windfield in the northeastern part of the province on the Pasquotank River. The Pasquotank had become an important artery of commerce, the main industries being transport, logging, and oyster harvesting. North Carolina was not heavily populated, but Windfield was on the coastal road to Virginia and well situated for trade. At some point in the 1760s William set up a company in his own name in which he had a 20% share, the balance being held by Thomas MacKnight (40%), James Parker (20%), and William Aitchison (20%). The company traded to Europe and the West Indies.

By the late 1760s, William was one of the leading members of North Carolina society, or certainly of Pasquotank County society. In November 1771, he was appointed Register (described in other places as Registrar) of Pasquotank County by Josiah Martin, governor of North Carolina (in office 1771-75). A month later, Governor Martin, 'reposing especial trust and confidence in your loyalty, courage, and good conduct', appointed William as a captain in the Pasquotank regiment of militia, commanded by Colonel John Lowry. William McCormick & Co owned a small port and cargo facility at Windfield on 25 acres of land, several ships, and two plantations covering a total of more than 1,000 acres, well stocked with cattle, pigs, and sheep. According to one of his employees, Caleb Church, William had a house with several well-furnished rooms, which he kept 'in a very genteel way, and at a great expense', and he travelled with 'two horses and a chair and servant' and had 'several clerks in his counting house'.

But the colonial era was drawing to a close.  Tensions began to build between Britain and the colonies in the early 1770s as Americans resisted the right of the British Parliament to govern without representation. Colonists were divided over which side to take during the buildup to the war, and during the war itself.  About 40-45 per cent supported the revolutionaries (or patriots), about 15-20 per cent—known as loyalists, Tories or 'King's Men'—supported the British Crown, and the balance stayed neutral.  William was clearly a loyalist, and was later described by one of his employees as 'one of the greatest Tories in the county.'  Loyalists may have agreed with some of the principles being pursued by the revolutionaries, but they did not want to go to war over the issue.

Life as a Loyalist

The first tangible effect of the unrest on William came in late 1775/early 1776 when the Belville and the Betsy, two ships (most likely brigantines, or two-masted merchant ships) owned by William McCormick & Co, were lost. The Belville (built in 1772) sailed from Windfield in December 1775 bound for Cadiz in Spain with a cargo of 500 pounds of beeswax, 61,900 pipe staves, 600 hoghead staves, and 3400 barrel staves. Because the Restraining Act (passed in 1775 in response to an American boycott of British goods, and prohibiting trade with all countries except Britain) did not extend to North Carolina (because the British government mistakenly thought that North Carolina was not part of the boycott), the ship was regularly cleared on her journeys, but this time she was intercepted by an American vessel soon after leaving.

Afraid that the ship would be captured by a British vessel that hove into sight, the captain of the American ship ordered that the cargo of the Belville be dumped overboard and the vessel be taken to New Bern. William gave bond and security to get the vessel back, she was reloaded, and she left again in early 1776. She was within a day's sail of Cadiz when she was taken by a British man of war off the coast of Spain, taken to Gibraltar, and condemned as the property of a colony in rebellion under the terms of the Prohibitory Act; this was passed in December 1775, coming into force on 1 January 1776, and aimed at destroying the American economy through a trade blockade. The ship was valued at £1500, the cargo at £600, and a slave on board at £80; everything was handed over to the captors as a prize.

The second lost ship, the Betsy, was captured by a British man of war in April/May 1776, ironically in the process of carrying provisions from North Carolina to British troops stationed in Norfolk, at the request of Lord Dunmore, governor of Virginia. As William and his partners later observed, they were 'plundered by the Americans for our loyalty on the one hand, and thus punished as partisans of rebellion on the other'.

As a loyalist who worked for the Crown, William was in a risky situation, particularly because the Second Continental Congress declared that the property of loyalists would be subject to seizure as a means of financing hostilities against England. The first North Carolina law providing for confiscation of property had been passed in November 1776, but commissioners had already been appointed to take an inventory of estates. An act of banishment against loyalists was passed in April 1777 which included the possible penalty of death for loyalists, and required everyone in the state either to take an oath of allegiance or to leave the state within 60 days. They could sell their property before leaving, otherwise it was forfeited.

William arranged for a brigantine named the Johnston  that he owned jointly with Thomas MacKnight to be repaired and loaded in case of the need for a hasty departure. The boat eventually set sail in September 1777 under the pretence that it was bound for Spain, but William told the other passengers that it was his plan to go to New York and seek protection. He arrived back in Britain in about November 1778, so in all it took 14 months to make a trip that should only have taken two months.

Return to Britain

Although the surrender at Yorktown in October 1781 did not mean the end of hostilities (Britain decided to suspend operations to recover the colonies in March 1782), within weeks every American state had passed a confiscation act, and loyalists had lost property worth millions of pounds. Article V of the 1783 Treaty of Paris ending the war provided that Congress would 'earnestly recommend' that states recognize the rightful owners of all confiscated lands and 'provide for the restitution of all estates, rights, and properties, which have been confiscated belonging to real British subjects'. Also, all other loyalists were to be allowed to return to their former homes for up to 12 months to recover their property or to buy back their land from the new owners. In the event, only South Carolina responded to the plea. Confiscation of property in North Carolina had begun in January 1779, and early in 1780 the property of William McCormick & Co of Pasquotank County was sold for £102,870.

By October 1782 more than 300 British-based American loyalists were receiving financial aid from Parliament. Two members of Parliament—John Wilmot and Daniel Parker Coke—were behind the passage of a 1783 Act of Parliament setting up a Royal Commission looking into the situation of the loyalists; this met sporadically between 1783 and 1788, and reported for the last time in May 1789. It gathered information and took statements with the object of providing advice to the government on who should be compensated. Their work was supported by the efforts of the loyalists themselves, who in March and July 1783 held meetings at the London Coffee House under the chairmanship of James Parker, and set up a committee (with Thomas MacKnight as one of the members) to decide who had legitimate claims; William was among the 40 or so chosen, with McCormick & Company listed separately.  A total of 3,225 claims were examined by the Commission, and the total final amount awarded was £3,033,091; a long way short of the more than £10.3 million claimed.

References

1742 births
1815 deaths
Businesspeople from Edinburgh
Loyalists in the American Revolution from North Carolina
Scottish emigrants to the United States

William McCormick (businessman)


Poulton-le-Sands is one of three small villages that combined to create Morecambe, Lancashire, England, the other two being Torrisholme and Bare. A local board of health was established in 1852, which, taking its name from Morecambe Bay, became the borough of Morecambe in 1902.

Poulton is known locally for its many murals which depict the origin of Poulton as a fishing village. Artist Patricia Haskey-Knowles completed several of these in the Morecambe Bay area.

Poulton was first mentioned in the Domesday Book as Poltune. It later became Poulton and remained this way for a number of centuries. The name 'Poulton' is likely a combination of Old English pull or pōl, meaning 'pool' and tūn meaning 'farmstead'.  It was towards the end of the eighteenth or the beginning of the nineteenth century that the Le-Sands was added.  This was to distinguish it from another Poulton located near Blackpool which is now Poulton-le-Fylde.

References

Villages in Lancashire
Geography of the City of Lancaster
Populated coastal places in Lancashire
Morecambe
Morecambe Bay

Poulton-le-Sands


Scottish Language Dictionaries (SLD), now Dictionaries of the Scots Language (DSL) is Scotland's lexicographical body for the Scots Language. DSL is responsible for the major Scots dictionaries, the Dictionary of the Older Scottish Tongue and the Scottish National Dictionary. Since 2004, all 22 volumes of these major texts have been available, free, via the internet, as the Dictionary of the Scots Language.

The organisation was formed in 2002 and continues the work of several generations of Scottish lexicographers. The current project team includes editorial staff from the Dictionary of the Older Scottish Tongue and from the Scottish National Dictionary Association.

In 2021, Scottish Language Dictionaries became an SCIO (Scottish Charitable Incorporated Organisation) and changed its name to Dictionaries of the Scots Language. It is a registered charity in Scotland with the OSCR number SC032910. DSL also undertakes a wide programme of educational work throughout Scotland, with people of all ages and abilities.

See also
 Dictionary of the Scots Language
 Dictionary of the Older Scottish Tongue
 Scottish National Dictionary

External links
 Dictionary of the Scots Language

Scots language
Charities based in Scotland
Companies based in Edinburgh
University of Edinburgh
Lexicography
Book publishing companies of Scotland
2002 establishments in Scotland
Companies established in 2002
Educational charities based in the United Kingdom

Scottish Language Dictionaries


Tsatsane is a community council located in the Quthing District of Lesotho. Its population in 2006 was 9,452.

Villages
The community of Tsatsane includes the villages of Ha 'MaKo (Tosing), Ha Choko, Ha Falatsa, Ha France, Ha Hlaela, Ha Hlanyane, Ha Lephahamela, Ha Liphapang, Ha Mafura (Ha Maphasa), Ha Mafura (Ha Moso), Ha Mafura (Khorong), Ha Mafura (Mokhatseng),  Ha Mafura (Moreneng), Ha Mafura (Sekhitsaneng), Ha Mafura (Tereseng), Ha Maleka, Ha Malibeng, Ha Mateisi, Ha Mathe, Ha Matiase, Ha Mofokeng, Ha Mongoli, Ha Montši, Ha Mosele, Ha Motsumi, Ha Mpapa, Ha Phoofo (Dalewe), Ha Raemile, Ha Ralebona (Dalewe), Ha Ramanasi, Ha Ranthoto, Ha Sekhobe, Ha Sekonyela, Ha Seliba, Ha Thaha, Ha Thibella, Koti-se-phola, Kueneng, Lahla-Nkobo, Lihlabeng (Ha Jobo), Lipeleng (Dalewe), Litšoeneng, Litšoeneng (Tosing), Mabele-a-tlala, Mahlanyeng, Mapeleng, Maqebeng, Masoothong, Matebeleng, Mateleng (Tosing), Matšela-habeli (Tiping), Mokema, Mokhoabong, Morataleng, Nonyana-e'a-mpitsa, Sekoaing, Selomong (Aupolasi), Seoling, Teraeng (Dalewe), Thaba-Ntšo, Tlokoeng and Tsekong.

References

External links
 Google map of community villages

Populated places in Quthing District

Tsatsane


Energy in Macau is related to all of the type of energy and its related infrastructure used in Macau, China. Energy-related affairs is administered under the Secretariat for Transport and Public Works of the Government of Macau.

Natural gas
Macau import its natural gas supply from Hengqin Island in Zhuhai, Guangdong. The pipeline has a capacity of 520 million m3 of natural gas per year. The sole gas importer in Macau is Sinosky Energy (Holdings) Co. Ltd. ().

Petroleum
Petroleum and other petrochemical products in Macau is handled by Nam Kwong Petroleum & Chemical Co., Ltd. (NKOIL; ) from its trading, storage, transportation, wholesale and retail, including the installation and maintenance of gas and fuel supply system.

Water supply
Water supply in Macau is handled by Macao Water.

See also
 Energy policy of China
 Electricity sector in Macau

References

Macau

Energy in Macau


this, self, and Me are keywords used in some computer programming languages to refer to the object, class, or other entity of which the currently running code is a part.  The entity referred to by these keywords thus depends on the execution context (such as which object is having its method called).  Different programming languages use these keywords in slightly different ways.  In languages where a keyword like "this" is mandatory, the keyword is the only way to access data and methods stored in the current object.  Where optional, they can disambiguate variables and functions with the same name.

Object-oriented programming
In many object-oriented programming languages, this (also called self or Me) is a variable that is used in instance methods to refer to the object on which they are working. The first OO language, SIMULA 67, used this to explicitly reference the local object.  C++ and languages which derive in style from it (such as Java, C#, D, and PHP) also generally use this. Smalltalk and others, such as Object Pascal, Perl, Python, Ruby, Rust, Objective-C, DataFlex and Swift, use self. Microsoft's Visual Basic uses Me.

The concept is similar in all languages: this is usually an immutable reference or pointer which refers to the current object; the current object often being the code that acts as 'parent' or 'invocant' to the property, method, sub-routine or function that contains the this keyword. After an object is properly constructed, or instantiated, this is always a valid reference. Some languages require it explicitly; others use lexical scoping to use it implicitly to make symbols within their class visible. Or alternatively, the current object referred to by this may be an independent code object that has called the function or method containing the keyword this. Such a thing happens, for example, when a JavaScript event handler attached to an HTML tag in a web page calls a function containing the keyword this stored in the global space outside the document object; in that context, this will refer to the page element within the document object, not the enclosing window object.

In some languages, for example C++, Java, and Raku this or self is a keyword, and the variable automatically exists in instance methods. In others, for example Python, Rust, and Perl 5, the first parameter of an instance method is such a reference. It needs to be specified explicitly. In Python and Perl, the parameter need not necessarily be named this or self; it can be named freely by the programmer like any other parameter. However, by informal convention, the first parameter of an instance method in Perl or Python is named self. Rust requires the self object to be called &self or self, depending on whether the invoked function borrows the invocant, or moves it in, respectively.

Static methods in C++ or Java are not associated with instances but classes, and so cannot use this, because there is no object.  In other languages, such as Ruby, Smalltalk, Objective-C, or Swift, the method is associated with a class object that is passed as this, and they are called class methods. For class methods, Python uses cls to access to the class object.

Subtleties and difficulties
When lexical scoping is used to infer this, the use of this in code, while not illegal, may raise warning bells to a maintenance programmer, although there are still legitimate uses of this in this case, such as referring to instance variables hidden by local variables of the same name, or if the method wants to return a reference to the current object, i.e. this, itself.

In some compilers (for example GCC), pointers to C++ instance methods can be directly cast to a pointer of another type, with an explicit this pointer parameter.

Open recursion
The dispatch semantics of this, namely that method calls on this are dynamically dispatched, is known as open recursion, and means that these methods can be overridden by derived classes or objects. By contrast, direct named recursion or anonymous recursion of a function uses closed recursion, with early binding. For example, in the following Perl code for the factorial, the token  is a reference to the current function:
use feature ":5.16";
sub {
    my $x = shift;
    $x == 0 ? 1 : $x * ->( $x - 1 );
}
By contrast, in C++ (using an explicit this for clarity, though not necessary) the this binds to the object itself, but if the class method was declared "virtual" i.e. polymorphic in the base, it's resolved via dynamic dispatch (late binding) so that derived classes can override it.
unsigned int factorial(unsigned int n)
{
  if (n == 0)
    return 1;
  else
    return n * this->factorial(n - 1);
}
This example is artificial, since this is direct recursion, so overriding the factorial method would override this function; more natural examples are when a method in a derived class calls the same method in a base class, or in cases of mutual recursion.

The fragile base class problem has been blamed on open recursion, with the suggestion that invoking methods on this default to closed recursion (static dispatch, early binding) rather than open recursion (dynamic dispatch, late binding), only using open recursion when it is specifically requested; external calls (not using this) would be dynamically dispatched as usual. The way this is solved in practice in the JDK is through a certain programmer discipline; this discipline has been formalized by C. Ruby and G. T. Leavens; it basically consists of the following rules:
 No code invokes public methods on this.
 Code that can be reused internally (by invocation from other methods of the same class) is encapsulated in a protected or private method; if it needs to be exposed directly to the users as well, then a wrapper public method calls the internal method.
 The previous recommendation can be relaxed for pure methods.

Implementations

C++

Early versions of C++ would let the this pointer be changed; by doing so a programmer could change which object a method was working on. This feature was eventually removed, and now this in C++ is an r-value.

Early versions of C++ did not include references and it has been suggested that had they been so in C++ from the beginning, this would have been a reference, not a pointer.

C++ lets objects destroy themselves with the source code statement: delete this.

C#

The  keyword this in C# works the same way as in Java, for reference types.  However, within C# value types, this has quite different semantics, being similar to an ordinary mutable variable reference, and can even occur on the left side of an assignment.

One use of this in C# is to allow reference to an outer field variable within a method that contains a local variable that has the same name. In such a situation, for example, the statement var n = localAndFieldname; within the method will assign the type and value of the local variable localAndFieldname to n, whereas the statement var n = this.localAndFieldname; will assign the type and value of the outer field variable to n.

D
In D this in a class, struct or union method refers to an immutable reference of the instance of the enclosing aggregate. Classes are reference types, structs and unions are value types. In the first version of D, the keyword  this is used as a pointer to the instance of the object the method is bound to, while in D2 it has the character of an implicit ref function argument.

Dylan
In the programming language Dylan, which is an object-oriented language that supports multimethods and doesn't have a concept of this, sending a message to an object is still kept in the syntax. The two forms below work in the same way; the differences are just syntactic sugar.
 object.method(param1, param2)

and

 method (object, param1, param2)

Eiffel
Within a class text, the current type is the type obtained from the current class. Within features (routines, commands and queries) of a class, one may use the keyword Current to reference the current class and its features. The use of the keyword Current is optional as the keyword Current is implied by simply referring to the name of the current class feature openly. For example: One might have a feature `foo' in a class MY_CLASS and refer to it by:

  class
     MY_CLASS

  feature -- Access

     foo: INTEGER

     my_function: INTEGER
        do
          Result := foo
       end

 end

Line #10 (above) has the implied reference to Current by the call to simple `foo'.

Line #10 (below) has the explicit reference to Current by the call to `Current.foo'.

  class
     MY_CLASS

  feature -- Access

     foo: INTEGER

     my_function: INTEGER
        do
           Result := Current.foo
       end

 end

Either approach is acceptable to the compiler, but the implied version (e.g. x := foo) is preferred as it is less verbose.

As with other languages, there are times when the use of the keyword Current is mandated, such as:

  class
     MY_CLASS

  feature -- Access

     my_command
           -- Create MY_OTHER_CLASS with `Current'
        local
           x: MY_OTHER_CLASS
       do
          create x.make_with_something (Current)
       end

 end

In the case of the code above, the call on line #11 to make_with_something is passing the current class by explicitly passing the keyword Current.

Java

The keyword this is a Java language keyword that represents the current instance of the class in which it appears. It is used to access class variables and methods.

Since all instance methods are virtual in Java, this can never be null.

JavaScript

In JavaScript, which is a programming or scripting language used extensively in web browsers, this is an important keyword, although what it evaluates to depends on where it is used.

 When used outside any function, in global space, this refers to the enclosing object, which in this case is the enclosing browser window, the window object.
 When used in a function defined in the global space, what the keyword this refers to depends on how the function is called. When such a function is called directly (e.g. f(x)), this will refer back to the global space in which the function is defined, and in which other global functions and variables may exist as well (or in strict mode, it is undefined). If a global function containing this is called as part of the event handler of an element in the document object, however, this will refer to the calling HTML element.
 When a method is called using the new keyword (e.g. var c = new Thing()) then within Thing  this refers to the Thing object itself.
 When a function is attached as a property of an object and called as a method of that object (e.g. obj.f(x)), this will refer to the object that the function is contained within. It is even possible to manually specify this when calling a function, by using the .call() or .apply() methods of the function object. For example, the method call obj.f(x) could also be written as obj.f.call(obj, x).

To work around the different meaning of this in nested functions such as DOM event handlers, it is a common idiom in JavaScript to save the this reference of the calling object in a variable (commonly called that or self), and then use the variable to refer to the calling object in nested functions.

For example:

// In this example $ is a reference to the jQuery library
$(".element").hover(function() {
    // Here, both this and that point to the element under the mouse cursor.
    var that = this;

    $(this).find('.elements').each(function() {
        // Here, this points to the DOM element being iterated.
        // However, that still points to the element under the mouse cursor.
        $(this).addClass("highlight");
    });
});

Notably, JavaScript makes use of both this and the related keyword self (in contrast to most other languages which tend to employ one or the other), with self being restricted specifically to web workers.

Finally, as a reliable way of specifically referencing the global (window or equivalent) object, JavaScript features the globalThis keyword.

Lua

In Lua, self is created as syntactic sugar when functions are defined using the : operator. When invoking a method using :, the object being indexed will be implicitly given as the first argument to the function being invoked.

For example, the following two functions are equivalent:

local obj = {}

function obj.foo(arg1, arg2)
  print(arg1, arg2) -- cannot use "self" here
end

function obj:bar(arg)
  print(self, arg) -- "self" is an implicit first argument before arg
end

-- All functions can be invoked both ways, with "." or with ":"

obj:foo("Foo") -- equivalent to obj.foo(obj, "Foo")
obj.bar(obj, "Bar") -- equivalent to obj:bar("Bar")

Lua itself is not object-oriented, but when combined with another feature called metatables, the use of self lets programmers define functions in a manner resembling object-oriented programming.

PowerShell
In PowerShell the special automatic variable $_ contains the current object in the pipeline object. You can use this variable in commands that perform an action on every object or on selected objects in a pipeline.

"one", "two", "three" | % { write $_ }Also starting with PowerShell 5.0, which adds a formal syntax to define classes and other user-defined types, $this variable describes the current instance of the object.

Python
In Python, there is no keyword for this. When a member function is called on an object, it invokes the member function with the same name on the object's class object, with the object automatically bound to the first argument of the function. Thus, the obligatory first parameter of instance methods serves as this; this parameter is conventionally named self, but can be named anything.

In class methods (created with the classmethod decorator), the first argument refers to the class object itself, and is conventionally called cls; these are primarily used for inheritable constructors, where the use of the class as a parameter allows subclassing the constructor. In static methods (created with the staticmethod decorator), no special first argument exists.

Rust
In Rust, types are declared separately from the functions associated with them. Functions designed to be analogous to instance methods in more traditionally object-oriented languages must explicitly take self as their first parameter. These functions can then be called using instance.method() syntax sugar. For example:

struct Foo {
    bar: i32,
}

impl Foo {
    fn new() -> Foo {
        Foo { bar: 0, }
    }
    fn refer(&self) {
        println!("{}", self.bar);
    }
    fn mutate(&mut self, baz: i32) {
        self.bar = baz;
    }
    fn consume(self) {
        self.refer();
    }
}

This defines a type, Foo, which has four associated functions. The first, Foo::new(), is not an instance function and must be specified with the type prefix. The remaining three all take a self parameter in a variety of ways, and can be called on a Foo instance using the dot-notation syntax sugar, which is equivalent to calling the type-qualified function name with an explicit self first parameter.

let mut foo = Foo::new(); // must called as a type-specified function
foo.refer(); // prints "0". Foo::refer() has read-only access to the foo instance
foo.mutate(5); // mutates foo in place, permitted by the &mut specification, need foo to be declared mut
foo.consume(); // prints "5" and destroys foo, as Foo::consume() takes full ownership of self

//  equivalent to foo.refer()
Foo::refer(foo); // compilation error: foo is out of scope

Self
The Self language is named after this use of "self".

Xbase++
Self is strictly used within methods of a class.
Another way to refer to Self is to use ::.

See also

References

Further reading
 Meyers, Scott, 1995. More Effective C++: 35 New Ways to Improve Your Programs and Designs.  Scott Meyers
 Stroustrup, Bjarne, 1994. The Design and Evolution of C++. Addison-Wesley Pub. Co.  Bjarne Stroustrup

Object-oriented programming

This (computer programming)


A Soldier's Daughter Never Cries is a semi-autobiographical novel by Kaylie Jones, who was the daughter of James Jones. It describes her childhood in Paris in the 1960s, her struggles adjusting to her parents' adoption of a French boy, Benoit, and the family's later cultural transition when they return to the United States.

This novel was first published in 1990 by Bantam Books. It was most recently reissued in 2003 by Akashic Books.

This novel was also adapted for the screen as A Soldier's Daughter Never Cries by James Ivory in 1998.

Footnotes

1990 American novels
American autobiographical novels
American novels adapted into films
Bantam Books books

A Soldier's Daughter Never Cries (novel)


Poring is a small tourist resort in Sabah, Malaysia. Located 40 km south-east of the Kinabalu National Park Headquarters, in the district of Ranau, Poring is situated in lowland rainforest, contrasting with the montane and submontane rainforest of Kinabalu National Park. The name Poring comes from a Kadazandusun word for a bamboo species found in the area.

Poring is known for its hot springs and the popular recreation and tourist complex is known as Poring Hot Springs. The springs are known for their therapeutic properties and the waters with their sulphuric minerals are reputed to ease aching muscles. The baths are favoured by returned climbers of Mount Kinabalu.

Features

 Butterfly farm: Borneo's first butterfly farm features a garden, nursery and hatchery for the purpose of research, education and the conservation of endangered species.
 Poring Orchid Conservation Centre: With 1,200 species of orchids found with Kinabalu Park, the centre has the largest collection of Sabah's endemic orchids.
 Kipungit Waterfall: Kipungit Waterfall is a 30-minute walk from Poring Hot Springs. The waterfall is smaller than the Laganan Waterfall, which is a 90-minute walk from Poring Hot Springs.
 Tropical Garden: Mousedeer and other deer are exhibited with colourful birds in the aviary.

 Poring Canopy Walkway: Visitors can stroll amidst the canopy of the Menggaris tree - the King of the Forest.  The Walkway is 157.8m long and 41m high.
 Accommodation: Poring Hot Spring features accommodation which used to be run by Sabah Parks under Kinabalu Nature Resorts in a very minimalist approach.  Since then, the management has been taken over by Sutera Sanctuary Lodges, which on the one hand has significantly modernised the accommodation facilities, but on the other, rocketed the prices of especially the refurbished accommodations.

References

External links
 Sabah Parks website
 ASEAN Heritage website

Mount Kinabalu
Protected areas of Sabah

Poring


Radioresistance is a property of some living things. It means they can resist the effects of gamma radiation, ultra-violet radiation, and similar types of radiation. A good example is the bacterium Deinococcus radiodurans. It has enzymes which repair the DNA strands which have been split by the radiation.

References

Ecology
Extremophiles

Radioresistance