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List of state partition proposals in the United States

Numerous state partition proposals have been put forward since the 1776 establishment of the United States that would partition an existing U.S. state or states so that a particular region might either join another state or create a new state. Article IV, Section 3, Clause 1 of the United States Constitution, often called the New States Clause, grants to the United States Congress the authority to admit new states into the United States beyond the thirteen that existed when the Constitution went into effect (June 21, 1788, after ratification by nine of the thirteen states). It also includes a stipulation originally designed to give Eastern states that still had Western land claims, which included Virginia, North Carolina, and Georgia, a veto over whether their western counties could become states.

"New States may be admitted by the Congress into this Union; but no new State shall be formed or erected within the Jurisdiction of any other State; nor any State be formed by the Junction of two or more States, or Parts of States, without the Consent of the Legislatures of the States concerned as well as of the Congress."

The clause has served the same function since then whenever a proposal to partition an existing state or states has come before Congress. New breakaway states are permitted to join the Union only with the proper consents. Of the 37 states admitted to the Union by Congress, three were set off from an already existing state: Another state that may fit into this category is Vermont, which existed as a de facto but unrecognized sovereign state from 1777 to 1791. The region had been a subject of a territorial dispute between New York and New Hampshire during the colonial period, which royal authorities had resolved in favor of New York. As the State of New York continued to claim Vermont's territory under that ruling after independence, the Continental Congress never recognized Vermont as an independent state. In 1790, after negotiating the common boundary between the two states and Vermont agreeing to pay New York $30,000, New York relinquished its land grant claim and consented to Vermont becoming part of the Union. Vasan Kesavan and Michael Stokes Paulsen assert that "although Vermont was admitted into the Union with New York's consent, it is not at all clear that New York's consent was constitutionally necessary. While Vermont was within the territory claimed by New York, the preponderance of evidence suggests that Vermont was not within the jurisdiction of New York."
 * Kentucky – 1792, was a part of Virginia
 * Maine – 1820, was a part of Massachusetts
 * West Virginia – 1863, was a part of Virginia

The following is a list of substantive proposals, both successful and unsuccessful, put forward since the nation's founding to partition or set off a portion of an existing U.S. state or states so that the region might either join another state or create a new state. Proposals to secede from the Union and proposals to create states from either organized incorporated or unorganized U.S. territories are not included. Land cessions made by several individual states to the federal government in the 18th and the 19th centuries also are not listed.

Arizona

 * In February 2011, Tucson politicians and activists formed the group "Start our State," to advocate secession for Pima County and other southern counties to create a state called "Baja Arizona". The group wanted the Pima County Board of Supervisors to put the issue on the 2012 ballot, but it was rejected by the board due to lack of authority, so the group circulated petitions. Interest in secession grew when Republican governor Jan Brewer and her allies enacted Arizona SB 1070, regarding illegal immigration. Furthermore, the state had passed laws affecting Tucson elections and how the city bids for public works projects.

Colorado

 * In the mid-1930s, the Walsenburg World-Independent proposed that Huerfano County secede from the state. This was a pet project of Sam T. Taylor, a sports editor, who went on to become a long-serving state senator, continuing to pursue the idea unsuccessfully.
 * In 1973, nearby Costilla County had expressed interest in seceding from Colorado and joining New Mexico.
 * On June 6, 2013, commissioners in Weld County announced a proposal to secede with seven other counties to form the state of North Colorado, citing concerns with state policy and recently enacted legislation relating to the region's main economic drivers, including agriculture and energy. The bid was motivated by a belief that the urban population centers of the State weren't concerned with the economic interests of more rural areas.  The commissioners stated that they would hold public meetings to gather input before crafting a ballot initiative by August 1, and that the proposal had aroused preliminary interest from fellow commissioners in Morgan, Logan, Sedgwick, Phillips, Washington, Yuma and Kit Carson counties.  Eleven counties were involved in the partition movement by election day, though Moffat County was looking into joining Wyoming rather than North Colorado. Voters rejected partition in six of the eleven counties, including Weld and Moffat, and supported it (without binding authority) in five.

Florida

 * Politicians in the South Florida metropolitan area have made proposals to split Florida into two states, North Florida and South Florida. One such proposal was made in 2008 by the North Lauderdale commissioners. The proposal was revived in 2014 when South Miami City Commission passed a South Florida state resolution on October 7 and sent it to counties in the proposed state's area.

Georgia

 * The Pierce County Republican Party placed a question on the county's May 2018 primary ballot asking Republican voters if they wanted all Georgia counties south of Macon to join together to “form the 51st state of South Georgia”. The question failed with only 27% of voters agreeing with it.

Idaho

 * In the mid and late 1860s, there was a proposal centered on Lewiston in northern Idaho for the Territory of Columbia to be formed in the Inland Northwest from parts of what is now eastern Washington, northern Idaho and western Montana.
 * Longstanding political rivalry along with geographic isolation led to proposals in the early 20th century that portions of northern Idaho and eastern Washington be united into a new state named Lincoln.
 * In 2020, "Move Oregon's Border for a Greater Idaho" proposed breaking off most of Oregon's area and some of Northern California and join it with Idaho. Even if passed by voters, it would still need approval from all three state legislatures.    In 2021, five counties in eastern Oregon voted to "require county officials to take steps to promote" adding the counties to Idaho. In May 2022, voters in   Douglas and Josephine counties rejected an advisory vote, causing the proponents to scale back the scope of the proposal and issue a “less ambitious” map.  The reduced scope includes only eastern Oregon, does not include any California territory, and only includes a little more than a third of the original map’s inhabitants. Eleven Oregon counties have approved some version of the proposal. In February, 2023, the House State Affairs committee of the Idaho House of Representatives approved a resolution to authorize the legislature to discuss moving the state border with Oregon lawmakers.

Illinois

 * Between 1840 and 1842, several northern counties in Illinois, including Jo Daviess County, Stephenson County, Winnebago County and Boone County, voted to reattach to Wisconsin, from which the counties were ceded to Illinois by Congress in 1818. The split was precipitated by the mutual antagonism between northerners and southerners due to social and political differences. The split was never realized due to lack of support from Chicago and Cook County, as the benefits of the Illinois and Michigan Canal linking northern to central and southern Illinois outweighed secession.
 * In 1861, the southern region of Illinois, known as Little Egypt, proposed secession due to cultural and political differences from Chicago and much of Central and Northern Illinois.
 * In 1925, Cook County considered secession to create the state of Chicago.
 * In the early 1970s, residents in western Illinois were upset over the allocation of state funds for transportation, prompting a student at Western Illinois University to declare 16 counties the Republic of Forgottonia. Although the declaration was meant to be a joke, the secession idea was picked up by the Western Illinois Regional Council, until State Representative Doug Kane showed that the counties had received funding that was more than what they paid in state taxes.
 * In November 2011, State Representatives Bill Mitchell and Adam Brown introduced a proposal to make Cook County a state of its own. They felt that all of Illinois outside of Cook County should become a separate state, due to Chicago's "dictating its views" to the rest of the state.
 * On February 7, 2019, State Representative Brad Halbrook, with co-sponsors Representatives Chris Miller and Darren Bailey, filed a resolution that urges the United States Congress to declare the City of Chicago the 51st state of the United States of America and separate it from the rest of Illinois.

Kansas

 * In 1992, a group in southwestern Kansas advocated the secession of a number of counties in that region from the state. Nominally headed by Don O. Concannon, former Chairman of the Kansas Republican Party and gubernatorial candidate from Hugoton, the group gave the new state the name "West Kansas", a state bird (pheasant), and a state flower (yucca). The proposal was in reaction to laws raising real estate taxes, and shifting state education funding away from rural school districts and into more urban areas. Though organizers arranged for a series of straw polls that demonstrated widespread support for secession in nine counties, the movement died out by the mid-1990s.

Maine

 * Maine was initially part of the Commonwealth of Massachusetts before being admitted to the Union as a state in 1820. However, its boundary with British North America (now Canada) had been in dispute for several decades. In 1827, John Baker unilaterally declared the disputed territory (now part of Aroostook County) to be the "Republic of Madawaska". The declaration was rejected by Maine in 1831. Following the undeclared Aroostook War in 1838–1839, the United States and United Kingdom signed the Webster–Ashburton Treaty on August 9, 1842, to settle the border issue.
 * In 1998 and again in 2005, state representative Henry Joy proposed legislation to partition Maine into northern and southern states. He cited concern for the rural northern part, encompassing Maine's 2nd congressional district, being affected by "anti-business policies" and "overzealous environmental safeguards". Reflecting his political opinion of the trends there, Joy suggested the southern half be named "Northern Massachusetts" and the northern half remain "Maine", though others have suggested "Acadia" for the northern half.

Maryland

 * Westsylvania, proposed during the American Revolution, would have been created from parts of Pennsylvania and Virginia and a small part of Maryland.
 * In 1998, state legislator Richard F. Colburn proposed to the Maryland General Assembly that a referendum be held to allow nine counties representing the Eastern Shore to secede from the state. They would invite counties from Delaware and Virginia to form the state of Delmarva.
 * In September 2009, Frederick County Commissioner John L. Thompson Jr. proposed that the county secede from Maryland because the county pays more to state government in taxes than it receives in services and benefits. The proposal was rejected by the other commissioners in the county.
 * In February 2014, it was reported that residents from Western Maryland started petitions to form a new state, citing taxes and gun control as issues. Possible names for such a proposed state included Liberty, Antietam, Appalachia, and Augusta.
 * In October 2021, six Republican lawmakers from Garrett, Allegany, and Washington Counties in Western Maryland sent a letter to the legislative leaders of West Virginia to ask if said counties seceded from Maryland, could they be annexed into West Virginia.

Massachusetts

 * The state's exclave District of Maine had proposed secession multiple times in the early 19th century. Long-standing disagreements over land speculation and settlements led Maine residents and their allies in Massachusetts proper to force an 1807 vote in the Massachusetts General Court on permitting Maine to separate; the vote failed. Separatist sentiment in Maine was stoked during the War of 1812 when pro-British Massachusetts merchants opposed the war, and refused to defend Maine from British invaders. Finally, on June 19, 1819, the Massachusetts General Court passed enabling legislation separating the "District of Maine" from the rest of the state (an action approved by the voters in Maine on July 19, 1819, by 17,001 to 7,132).  Then, on February 25, 1820, the court passed a follow-up measure officially accepting the fact of Maine's imminent statehood. Maine became the 23rd state on March 15, 1820, as part of the Missouri Compromise, which also geographically limited the spread of slavery and enabled the admission to statehood of Missouri the following year.
 * During the abolitionist era some supporters of William Lloyd Garrison sought the secession of Essex County from the state.
 * Boston Corner, in the southwestern corner of the state, was ceded to the state of New York in 1857, due to Massachusetts being unable to administer the hamlet.
 * A 1919 tax-protest proposal filed in the state legislature would have created an independent State of Boston.
 * In 1977, the islands of Martha's Vineyard, Nantucket, and the Elizabeth Islands proposed to separate from Massachusetts because of a redistricting bill that would have deprived Dukes County, consisting of Martha's Vineyard and the Elizabeth Islands, and Nantucket County of separate representation in the General Court. At local town meetings, culminating in the All-Island Selectmen's Association Conference, residents and community leaders voted in favor of secession with an "overwhelming majority". When the Nantucket state representative filed a bill with the Massachusetts Legislature, Connecticut governor Ella T. Grasso suggested that the islands join her state. Additionally, the legislatures of New Hampshire, Rhode Island and Vermont each supported the islands' annexation to their state. Although the redistricting bill passed, the state representatives pledged to assign aides for the two counties that would report to their state representative, and the area received much positive publicity.

Michigan

 * Several times between 1858 and 1957, Michigan's Upper Peninsula and parts of Wisconsin have engaged in talks about forming a fifty-first state called "Superior."
 * In 1979 a group called Citizens for Secession attempted to prompt leaders to move Cass County to Indiana and change the name to "Michiana County".

Minnesota

 * In the mid 19th century there had been intermittent advocacy for the Arrowhead of Minnesota, the four northeast counties of the state adjacent to Lake Superior, to join with northwestern Wisconsin and the Upper Peninsula of Michigan to form a new state to be named "North Country" or Superior, with Duluth as its capital.
 * In March 2021, HF 2423 was introduced to the state legislature with the goal of establishing a process for counties to vote to leave the state of Minnesota and instead join one of Minnesota's bordering states. A petition was also started alongside the introduction of the proposal with the primary goal of counties outside the Twin Cities area joining South Dakota, but also encouraging joining Iowa or Wisconsin. The bill was introduced in response to Governor Tim Walz's response to the COVID-19 pandemic, in which Minnesota was shut down and restricted to a far stricter extent and for a longer duration of time than all its neighboring states. South Dakota, which remains the only state in the country to never issue any business restrictions at any time was viewed in a major contrast to Minnesota's restrictions. Shortly after the bill's introduction, South Dakota Governor Kristi Noem tweeted in favor of the bill.

Mississippi
During the American Civil War of 1861–65, in Jones County, Mississippi, Newton Knight, a deserter from the Confederate army, organized a militia of fellow deserters and escaped slaves and declared Jones County to be the Free State of Jones. They successfully prevented Confederate authorities from enforcing conscription, taxation, and slavery within the county, and hoped for admission to the United States as a new state.

Missouri

 * McDonald County declared itself the McDonald Territory for a brief period of time in July 1961. The movement was half in jest, after a city within the county was accidentally omitted from an official Missouri map.

Montana

 * In 1939, a secessionist movement proposed the State of Absaroka, to be formed from portions of Montana, adjacent areas of Wyoming, and parts of South Dakota. Motivated by opposition to New Deal politics and a desire to bring tourism to the area, the craze was reflected in state automobile license plates bearing the name; a "Miss Absaroka" contest held in that year; and a minor league baseball team called the Absaroka Eagles.

Nebraska

 * In the 1890s, residents of the Nebraska Panhandle threatened secession when the state refused to enact water laws that would encourage irrigation.

Nevada

 * In 2020, residents of rural Nevada proposed breaking away from the state due to the strong liberal influence of Clark County on the politics of the state. The movement proposes the state of "New Nevada". Proponents of "New Nevada", alongside those of "New California", filed an amicus brief in the Supreme Court case Texas v. Pennsylvania.

New Hampshire

 * In 2001, the communities of Newington and Rye considered secession in response to the enactment of a uniform statewide property tax.

New Jersey

 * From 1674–1702 the colonial Province of New Jersey was divided into East Jersey, largely consisting of today's North Jersey, and West Jersey, largely consisting of South Jersey. The two regions maintain cultural differences with North Jersey largely being dominated by New York City, and South Jersey by Philadelphia. In 1980 a local journalist in Mount Holly, Albert Freeman, wrote an editorial calling for secession, initially a joke, this movement gained momentum leading to the Egg Harbor council voting to secede and form the state of South Jersey. The issue eventually caught the attention of local lawmakers, upset with the construction of the Meadowlands Sports Complex in North Jersey, while the reconstruction of the Garden State Park Racetrack in South Jersey was blocked. A non-binding referendum was held in six southern counties, with the exclusion of Camden and Gloucester counties, which passed with 51% of voters in favor of secession. The only county to vote against secession was Ocean County.

New York

 * In the New York City mayoral election of 1969, writer Norman Mailer ran in the Democratic Party primary on a ticket with columnist Jimmy Breslin, who ran for City Council President. Part of their joint platform was a proposal that New York City should secede from New York State and become the 51st state. At around the same time, a public-affairs series on the local educational TV station, WNET-TV, channel 13, was called The Fifty-First State.
 * In the 1990s, Randy Kuhl, from rural upstate Hammondsport, had advocated secession by regularly proposing bills to that effect while he was a state senator. His 1999 bill would have New York City, Long Island, Westchester and Rockland Counties become a separate state of New York, while the rest of the counties would be grouped as West New York.
 * From 2007 to 2009, Long Island residents discussed secession on the grounds that their tax money is not used to fund programs in their counties. Proposals were made for the entire island (Kings, Queens, Nassau, and Suffolk counties) and for just the two suburban counties (Nassau and Suffolk).
 * State senators Joseph Robach, Dale Volker, and Michael Ranzenhofer, all Republicans from western New York, proposed a nonbinding referendum to gauge support for dividing the state in November 2009. The referendum was again proposed by Stephen Hawley in 2013 and 2015, with members of the Long Island delegation to the state legislature also backing the 2015 bill.
 * 2010 gubernatorial candidate Carl Paladino's supporters, Rus Thompson and James Ostrowski in the Buffalo region, have supported secession of western New York from New York City and its nearby counties. Fred Smerlas, in discussing a potential platform for a 2010 Congressional run from western New York, stated that he would make the separation of New York City and upstate a top priority: "My first act if I ever got elected would be to take a big saw and cut New York City off."
 * Fifteen towns in the Southern Tier of New York proposed potential secession in 2015 to join Pennsylvania.

North Carolina

 * In 1784, the western counties of Greene, Washington, Sullivan, and parts of Hawkins County, formed the provisional State of Franklin, with Revolutionary War hero John Sevier as governor. However, since the state was not recognized by the Congress of the Confederation, it disbanded and joined North Carolina. In 1790, North Carolina relinquished the region to the federal government, creating the Southwest Territory. In 1796, the territory would be admitted to the Union as the State of Tennessee, with Sevier as its first governor.

Oregon

 * In 1941, counties in southwestern Oregon joined counties of Northern California to secede as the State of Jefferson. The movement was centered in rural communities who felt ignored by political leaders in more urban areas.
 * In 2020, "Move Oregon's Border For a Greater Idaho" proposed breaking off most of Oregon's area and some of Northern California and join it with Idaho. The areas proposed to break off of Oregon and California vote Republican but in a state whose legislatures are dominated by Democrats. Douglas and Josephine counties in Oregon approved language for petitions to put a measure on the ballot. Even if passed by voters, it would still need approval from all three state legislatures.
 * In 2021, five counties in eastern Oregon voted to "require county officials to take steps to promote" adding the counties to Idaho. In 2022, two more counties voted in favor of being added to Idaho. As of 2024, 13 counties have approved ballot measures in favor of Greater Idaho.

Pennsylvania

 * Westsylvania was proposed as the 14th state during the American Revolutionary War. It would have been located primarily in what is now West Virginia, southwestern Pennsylvania, and small parts of Kentucky, Maryland, and Virginia.
 * More recent commentators have occasionally proposed separating the Philadelphia region from the rest of Pennsylvania, either as a state unto itself or as part of New Jersey.

Rhode Island

 * In 1984, the town of New Shoreham, coterminous with Block Island, threatened to secede because the state had denied them the ability to ban or to control the use of mopeds on the island. Both Massachusetts and Connecticut were reported as having interest in annexing the island. After the town voted to put the issue on the state ballot for June, the Rhode Island government eventually compromised by allowing the island to control the number of mopeds on the island.

South Dakota

 * Parts of South Dakota were proposed as part of Absaroka, a territory inspired by opposition to New Deal politics and a desire for increased tourism.

Tennessee

 * In 1861, after Tennessee joined the Confederacy, Scott County passed a proclamation to secede from Tennessee and form the "Free and Independent State of Scott" in order to support the Union. When it was discovered in 1986 that this county law was still on the books, the proclamation was finally repealed. The county then petitioned the state of Tennessee for readmission, even though the original secession had not been recognized by either the state or federal government.

Texas

 * Under the joint resolution of Congress, the Republic of Texas joined the Union with the right to partition itself into as many as five states. As a result, Texas "divisionists" would occasionally propose partitioning in its early decades.

Utah

 * State of Deseret, partly settled by Mormons before it was purchased from Mexico by the United States, had proposed boundaries far larger than the eventual Utah Territory.
 * In 2002, the United States House of Representatives voted to allow Wendover to leave the state and join Nevada, merging with the city of West Wendover. However, Nevada Senator Harry Reid blocked the bill's consideration in the Senate, citing that it would affect the investments of the casinos in the border town.
 * In 2008, state representative Neal Hendrickson proposed Joint Resolution 6 (HJR006): "the creation of a separate state, consisting of the southern portion of the present state of Utah with a northern boundary stretching east and west across the present state of Utah at the southern border of Utah County". Hendrickson's bill cited differences between the urban and heavily populated northern part of the state, with the less populous and mostly rural south. The bill did not pass.

Vermont

 * In 2004 and 2005, the town of Killington voted to secede from Vermont to join New Hampshire, despite being situated in the center of the state; the symbolic votes, taken at the yearly town meeting, were a protest against a rise in property taxes in Vermont. A similar motion was attempted in Winhall, but was voted down.

Virginia

 * Westsylvania, proposed as the 14th state during the American Revolutionary War, included areas that were then part of Virginia but mostly later became West Virginia and Kentucky. It would have been located primarily in what is now West Virginia, southwestern Pennsylvania, and small parts of Kentucky, Maryland, and Virginia.
 * The District of Kentucky: Fayette, Jefferson, and Lincoln (all formerly part of Kentucky County) sought on numerous occasions to split from Virginia, beginning in the 1780s. Ten constitutional conventions were held in the Constitution Square Courthouse in Danville between 1784 and 1792. In 1790, Kentucky's delegates accepted Virginia's terms of separation, and a state constitution was drafted at the final convention in April 1792. The Virginia General Assembly adopted legislation on December 18, 1789, separating its "District of Kentucky" from the rest of the State and approving its statehood. Kentucky became the 15th state in the Union on June 1, 1792.
 * West Virginia: Tensions within Virginia had been building; the western counties felt ignored and uncared for by the Richmond government. This broke into open rebellion after Virginia voted to secede from the Union. Several Trans–Allegheny region counties voted to secede from the state after Virginia joined the Confederate States of America at the beginning of the Civil War on April 17, 1861. Unionist leaders in Wheeling set up a new State government for Virginia under the Wheeling Convention that was recognized by the U.S. Government in Washington. On May 13, 1862, the General Assembly of the Restored Government of Virginia passed an act granting permission for creation of West Virginia, and the secessionist area wrote a constitution. It was admitted to the Union as West Virginia on June 20, 1863, the 35th state. Support for the Confederacy and the Union was about evenly divided in the new State and a guerrilla war lasted until 1865. Later, by its ruling in Virginia v. West Virginia (1871), the Supreme Court implicitly affirmed that the breakaway Virginia counties did have the proper consents necessary to become a separate state.
 * Northern Virginia: Given the difference between Northern Virginia (NoVa) and the rest of Virginia (RoVa), some have proposed separating the two parts of the Commonwealth.

Washington

 * Present-day Washington is geographically divided into Eastern and Western regions by the Cascade Range. In the original proposal to establish Washington Territory, it was bounded on the east by the Columbia River. Since 1861, some eastern residents have proposed forming a new state, sometimes in combination with the Idaho Panhandle and relatively small parts of Montana and Oregon (consisting of the greater market area of Spokane). Names proposed include East Washington, Lincoln, and Cascadia (a name sometimes applied to the Pacific Northwest region as a whole). Reasons given are the distinct needs of urban and rural populations and political differences over water rights, labor law, and taxation.
 * In 2015, a bill was introduced in the Washington House of Representatives to create a task force for studying the impacts of adjusting the boundary lines of Washington to create two new states with one state east and one state west of the Cascade mountain range. In the same legislative session, a similar bill was introduced that would have included parts of Oregon in the newly created states. In 2017, legislators introduced a petition to the federal government regarding the creation of the state of Liberty, which would have a western boundary following the crest of the Cascade mountains and the western borders of Okanogan, Chelan, Kittitas, Yakima, and Klickitat counties, and whose eastern, northern, and southern boundaries would be the existing state borders. A bill and a federal petition to establish the state of Liberty were again introduced in 2019.  None of these bills or petitions have been passed by the Washington House.

Wisconsin

 * On July 21, 1967, some residents of the village of Winneconne, Wisconsin said they were seceding, in protest over the village's omission from the state's official highway map, to become what they called the "Sovereign State of Winneconne". After the village was restored to the highway map, the secessionists in the village of Winneconne reconciled with the state. The village has since celebrated the event with an annual Sovereign State Days celebration.
 * Portions of the northern counties were included in proposals for the State of Superior.

Wyoming

 * Parts of northern Wyoming were considered in a proposal for the State of Absaroka.


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Eg, Afghanistan

Eg is a town of almost 2,500 residents in Bamyan Province, Afghanistan. It is located at 34.0667 N latitude, 67.1167 longitude at an altitude of approximately 2800 meters (9200 feet), near the towns of Pshin Mazar, Zergak and Zerko.

Climate
The town has a subarctic climate (Köppen: Dsc) with mild, dry summers and cold, long winters.

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

title: Eg, Afghanistan


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Raymond Moriyama

Raymond Junichi Moriyama (October 11, 1929 – September 1, 2023) was a Canadian architect.

The private practice in Toronto he co-founded with Ted Teshima, Moriyama & Teshima Architects, was renowned for designing many major buildings across the world, including the Canadian War Museum and the Canadian Embassy in Tokyo. Moriyama's focus was on humane architecture with the pursuit of true ideals, democracy, and unanimity of all people.

Early life and education
Born in Vancouver, British Columbia, Raymond Moriyama suffered burns as a four-year-old and was sometimes teased about his scars. During the eight months he spent bedridden after the accident, he saw an architect coming and going from a nearby construction site "with a blueprint under his arm and a pipe in his mouth." Moriyama decided then and there that he would become an architect.

Moriyama's father was an outspoken pacifist who was arrested and imprisoned for his activism. Moriyama was then twelve; his mother, who was pregnant at the time, was left with him and his two sisters to run the family hardware store. Shortly after, Moriyama and his family were forced out of Vancouver and confined to an internment camp in the Slocan Valley of British Columbia during World War II. Japanese Canadians on the West Coast were classified as security threats, in a policy similar to that of the United States. Moriyama said these years were influential in his later career.

Moriyama described his experiences in internment camps as miserable. During this time, his mother experienced a miscarriage, which Moriyama grieved as the loss of a potential younger brother. He looked for a place for escape and solitude, and decided to build a treehouse outside of camp as a lookout point. He made friends with Canadian farmers who supplied him with lumber and tools to build. Moriyama describes his experience of finding escape as such: "In despair, I decided to bathe in the Slocan River on the other side of a little mountain away from the camp. The water was glacial, but it was better than hot tears. To see who might be coming, I built an observation platform. Soon I found myself wanting to build my first architectural project, a tree house, without being found out by the RCMP. I used just an axe as a hammer, an old borrowed saw, six spikes, some nails, a rope, and mostly branches and scraps from the lumberyard. It was hard work building it by myself, and it was a lesson in economy of material and means""That tree house, when finished, was beautiful. It was my university, my place of solace, a place to think and learn." After the war, his family reunited with his father and they resettled in Hamilton, Ontario, where he attended Westdale Secondary School and worked in a pottery factory. Ambidextrous, Moriyama was able to finish his piecework quickly, and his bosses allowed him to use his extra time to study for school.

During his years in university, Moriyama ran into his childhood friend Sachi from Vancouver, and the two began dating. They married in 1954, and together had five children, including two sons who also become architects, Ajon and Jason Moriyama.

Moriyama received a Bachelor of Architecture degree from the University of Toronto in 1954, and a Master of Architecture degree in civic and town planning from the School of Architecture at McGill University in 1957.

After he retired in 2003, Moriyama's sons, Ajon and Jason, became principals at Moriyama & Teshima Architects. In 2013, Ajon Moriyama founded Ajon Moriyama Architect, which takes on independent work in Toronto.

Career
Moriyama's first large project as an independent architect was the Ontario Science Centre in Toronto, with design starting in 1964, the building being finished in 1969.

After years of working independently as an architect, Moriyama established his Toronto-based firm in 1958 and in 1970 was joined by Ted Teshima and is now Moriyama & Teshima Architects. Some of their notable early projects include the Scarborough Civic Centre from 1973, and the Metropolitan Toronto Reference Library from 1977. Both of these projects won Governor General's Medals. Moriyama has been heavily involved in bringing a Japanese cultural influence to Western society. He is a part of the Japanese Order of the Rising Sun, as well as the Order of Ontario. Many of Moriyama's architectural awards recognize his excellency in materiality, landscape, and urban design. Some of his core values as an architect involve the principles of designing for human scale and human functionality. Ted Teshima retired in 2006, and died in 2016.

In 1999, Moriyama was the key architect for designing of the National Museum of Saudi Arabia in Riyadh.

From 2001 to 2007, Moriyama served as the Chancellor of Brock University in St Catharines, Ontario. He has designed several buildings at Brock University from the 1970s onwards.

In 2012 he created a $200,000 endowment with the Royal Architectural Institute of Canada called the Moriyama RAIC International Prize.

Moriyama retired in 2003. He died in Toronto on September 1, 2023, at the age of 93.

Honours
In 1985, Moriyama was made an Officer of the Order of Canada (OC) and promoted to Companion of the same order (CC) in 2008. He was also inducted into the Order of Ontario in 1992.

In 1997, he received the RAIC Gold Medal, Canadian Architecture's highest honour.

In 2003, Moriyama was made a member of the Order of the Rising Sun, 4th class, a Japanese award given in recognition for his services to Japanese culture in Canada.

In 2007, he was honoured with a postage stamp by Canada Post featuring his design for the Ontario Science Centre.

In 2009, Moriyama was one of nine laureates to receive a Governor General's Award in Visual and Media Arts.

In 2010, he won the Sakura Award, for his impact and dedication in the promotion of Japanese culture worldwide.

In 2012, Moriyama received a Queen Elizabeth II Diamond Jubilee Medal. He also created a $200,000 endowment with the Royal Architectural Institute of Canada called the Moriyama RAIC International Prize.

In June 2013, Moriyama received an honorary doctorate from the University of Calgary.

Ontario Science Centre, Toronto
The Ontario Science Centre was Moriyama's first large-scale project as an independent architect. Until its closure in June 2024 it functioned as a place for visitors to experiment hands-on with installations. He had it built in 1969, incorporating Confuciuan philosophies into the design, building on the ideas of tactile and experiential learning.

The glass facade of the front entrance has a large, exposed steel cable structure. This immediately involves visitors experiencing the engineering and science behind architecture, an example of experiential learning in itself. The interior lends itself to flexibility, with large open spaces fit for various installations, exhibits, and large crowds.

Canadian embassy, Tokyo
Moriyama designed the Canadian embassy in Tokyo during his time with Moriyama & Teshima. Built in 1991, the prominent concept of this building is the "tree house", inspired by Moriyama childhood years spent in Japanese internment camps. His intention was to enable a place of social connection and welcoming for all those visiting the Embassy: Moriyama replicated the feeling of youthfulness and magic of the treehouse he first built while in camp. Its form resembles that of a temple. On the lower half, a large solid mass of concrete includes deep set windows, and above it, an asymmetrical glass peaked roof creates a light, sculptural contrast. Adjacent to the building is the Akasaka Imperial Grounds and the Takahashi Memorial Park. The building was limited in the amount of shadows it was allowed to cast over the park, a restriction that led to its final shape.

Bata Shoe Museum, Toronto
Moriyama & Teshima designed the Bata Shoe Museum. Built in 1995, it is recognizable by its angled trapezoidal limestone walls, and pyramid-like glazing on the front face. Its interior reflects the angularity of the outside, with large open spaces for exhibitions with tilted walls; this interior is mostly wood, with "floating" display boxes with hidden supports, similar to the grand wooden staircase in the room's centre.

This project won the City of Toronto Urban Design Award of Excellence in 1997.

Other projects

 * Limberlost Place, George Brown College (2022)
 * Canadian Port of Entry at the Gordie Howe International Border Crossing, Gordie Howe International Bridge (under construction as of 2022)
 * Toronto French School additions (2015)
 * Canadian War Museum (2005)
 * National Museum of Saudi Arabia in Riyadh (1999)
 * Seneca College's Seneca@York campus Stephen E. Quinlan Building in Toronto (1999)
 * John Labatt Visual Arts Centre, The University of Western Ontario, London, Ontario (1993)
 * Ottawa City Hall (1990)
 * North York Central Library (1987)
 * Canada Centre Building (1985)
 * Science North, Sudbury, Ontario (1984)
 * Museum London in London, Ontario (1980)
 * Peterborough Public Library, Peterborough, Ontario (1980)
 * Toronto Reference Library (1977)
 * Lester B. Pearson Collegiate Institute (1977)
 * Albert Campbell Collegiate Institute (1976)
 * Burnhamthorpe Library and Burnhamthorphe Library Theatre (now Maja Prentice Theatre), Mississauga (1976)
 * Scarborough Civic Centre (1973)
 * L'Amoreaux Collegiate Institute (1971)
 * John McCrae Senior Public School, Scarborough, Ontario (1969)
 * Health Services Building, University of Waterloo (1968)
 * Harvest Bible Chapel, Markham (1967)
 * Civic Garden Centre (now Toronto Botanical Garden), Toronto (1965)
 * Japanese Canadian Cultural Center (1963)

title: Raymond Moriyama


text:
Talk:Trevor Lucas

Discussion
the previous edit by IP 147.153.57.187 was myself, the computer I was using apparently did not log me on for the post. Anyhow the track listing added was from the source already listed. It is not complete, but is complete up to the last date listed. Also several singles that are listed have a question mark in their title. This is because I currently have no information on the title of the B side of those albums.

Also a question of preference, I have the information of what record company first produced each album and wondered if that should be listed or not, I currently have not listed any of them. --Jake 19:53, Apr 11, 2005 (UTC)

External links modified
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title: Talk:Trevor Lucas


text:
Springfield Armory, Inc.

Springfield Armory, Inc., is an American commercial firearms manufacturer and importer based in Geneseo, Illinois. Founded in 1974 by Bob Reese and family, the company produces rifles such as the M1A and imports handguns such as the XD series and Hellcat. The company is unrelated to the like-named former manufacturer of US military firearms in Massachusetts, Springfield Armory.

Formation
Elmer C. Ballance began using the name "Springfield Armory" through his company (LH Manufacturing) dedicated to the first civilian production of the M14 rifle, calling it the M1A. Ballance began in San Antonio, Texas, and soon after moved to Devine, Texas, where the company gained momentum and popularity. In 1974, Ballance sold the company to the Reese family, who had a well-established production shop in Illinois and experience base. The company then expanded its market into other firearms, including the M1911. After further success, the company began to branch into even more types of firearms. The company, now run by Dennis Reese (formerly along with his brother Tom Reese), produces numerous firearms in many styles and models, ranging from the AR-pattern SAINT rifles and pistols, to the M1A family, to a wide range of 1911s, to the compact 911 .380 and 9mm pistols, and also a broad selection of polymer-framed pistols. The latter group includes the micro 9mm Hellcat, released in September 2019, which has proven to be popular in the CCW market.

The company has previously (as late as 2017) used a motto, "The First Name in American Firearms", in reference to the original and operationally unrelated Springfield Armory of Springfield, Massachusetts, which was founded in 1777 and closed in 1968. Those two dates are included in a timeline on the company's website, which notes that in 1974 "a passionate family by the name of Reese rescued" the Springfield Armory name.

Rifles
The company's rifle offerings include the M1A, the Hellion (imported VHS-2), and the SAINT line of AR-15 style rifles and short-barreled rifles.

The M1A line includes offerings such as standard, loaded, SOCOM, national match, and tanker models. In 2016, the first SAINT rifle was introduced, with a pistol variant following a year later. Variants at different price-points, marketed as SAINT Victor and SAINT Edge, have also been added.

In 2020, they released the all-new Model 2020 Waypoint hunting rifle.

In 2022, Springfield announced they would be importing a civilian version of the HS Produkt VHS-2 under the name of the Springfield Hellion.

Previously, the company offered its own versions of the M1 Garand, Beretta BM59, FN FAL, HK 91 and M6 Scout.

Handguns
The company's handgun offerings include M1911 pistols, the 911 pistol, and the polymer-frame XD (X-treme Duty) pistol series. The M1911 line includes the Range Officer series, the EMP, Mil-Spec, Garrison, TRP, Ronin and Emissary versions.

The XD-M variant was added in 2007, and the compact concealed carry XD-S line was added in 2012. Second generation XD offerings, branded as XD Mod.2, were introduced in 2014. The XD-E, a variant with an exposed hammer and a double action / single action (DA/SA) trigger system, followed in 2017. In 2018, the XD-S Mod.2, a modernized and upgraded version of the original XD-S, was introduced. In 2019, the company introduced the XD-M Elite line of pistols, which takes the XD-M and adds features such as the META (Match Enhanced Trigger Assembly), extended magazine wells and capacities up to 22 rounds in 9mm, ambidextrous slide locks, and enhanced slide grasping grooves.

In 2018, the company introduced the 911 pistol. The Hellcat micro-compact pistol was introduced in 2019, featuring 11- and 13-round magazines.

In 2021, the company introduced the SA-35 pistol, which is their version of the Browning Hi-Power.

In 2022, the company introduced the 1911 Prodigy DS.

In 2023, the company introduced the Echelon 9mm pistol.

Other activities
Springfield Custom, the company's customization shop, offered a broad line of custom-grade firearms including various M1911 offerings, and upgrades to the company's standard M1911 and XD offerings.

Through 2017, the company had received at least eight Golden Bullseye awards from American Rifleman magazine, the first having been for the XD sub-compact pistol in 2003.

Rob Leatham, a professional shooter who has won multiple national and world titles, has been the captain of "Team Springfield", the practical shooting team, since its inception in 1985.

The co-founder and CEO Dennis Reese accepted NRA Publications’ esteemed Golden Bullseye Pioneer Award at the 2022 NRA Annual Meetings & Exhibits held in Houston, Texas.

Political involvement
In 2006, the company played a major role in vocally opposing then Governor of Illinois Rod Blagojevich's proposed assault weapons ban that threatened to ban firearm manufacturing in Illinois. Co-owner Dennis Reese responded to the proposed legislation by saying "if this passes, we're out of town."

On July 13, 2007, at the 100th anniversary of the National Matches held at Camp Perry in northern Ohio, Reese announced that the company would provide $1 million towards the NRA's "Competitive Shooting & National Championships Endowment" and challenged other firearms manufacturers to donate to the endowment.

In 2017, the company faced controversy during consideration in the Illinois state legislature of a bill (SB-1657) that would restrict firearms transfers within the state. An organization that Springfield Armory and Rock River Arms—both based in Illinois—provided funding to, the Illinois Firearms Manufacturers Association (IFMA), initially opposed the bill. Following efforts by IFMA's lobbyist, the bill was amended to exempt Illinois-based manufacturers, apparently in exchange for IFMA dropping its opposition to the bill. This action created "tremendous customer backlash" as the deal was made "at the expense of everyday gun owners". This was followed by the IFMA being dissolved and the companies subsequently opposing SB-1657, which was eventually vetoed.

Springfield Armory quickly reacted once the actions of the IFMA were revealed to the company, with the company releasing the following statement from Reese: "I can tell you now, we at Springfield Armory are unequivocally 100 percent against this bill and will continue to work with the NRA and others to ensure that it is defeated. Springfield Armory ... was not aware of the actions taken by trade association, IFMA, until after the fact."

title: Springfield Armory, Inc.


text:
Talk:Équinoxe

Use in documentary
Equinoxe Part IV was used in a science documentary some years ago, and I'm wracking my brain trying to remember which one. If you know, please add it to this article. dkliman (talk) 15:34, 4 May 2008 (UTC)

 * Equinoxe parts 1, 3 and 4 were apparently used in "Cosmos"

 * The website http://cosmic_voyager.tripod.com/cosmosindex.htm contains a list of the music featured in the original US TV broadcasts.
 * The music used in other versions can and does differ for various reasons that are explained there. —Preceding unsigned comment added by 86.25.121.201 (talk) 05:58, 3 January 2009 (UTC)

Instruments list
Re: "Fairlight CMI Digital Sampling Synthesiser" that someone added – did this exist in 1978? — Preceding unsigned comment added by Ricadus (talk • contribs) 18:29, 4 May 2008 (UTC)

 * No, the frst well-known album on which a Fairlight CMI was used was Peter Gabriel in 1980. — Preceding unsigned comment added by 95.149.173.73 (talk) 20:10, 5 February 2019 (UTC)
 * Kind of irrelevant now, as the Fairlight has long since been removed from the list. Richard3120 (talk) 01:20, 6 February 2019 (UTC)

Name of the album
It should be Equinoxe, without the accent. Some releases of the album:

Germany CD

MFSL CD

France LP

Canada CD —Preceding unsigned comment added by 95.74.246.99 (talk) 19:36, 29 December 2009 (UTC)

The title is, without accent, in discogs you can see the list of releases, including referring *scans*.(E-Kartoffel (talk) 13:09, 11 April 2010 (UTC))

It is without accent, because:
 * Equinoxe is a title, or a name, with a first capital letter. In common french language, accents on letters vanish when capitalized. Within a french sentence about an equinox, it would be spelled équinoxe (only if it is not the first word in the sentence). When you speak about the work titled Equinoxe, it is spelled without accent.
 * Sadly the discogs.com community does not understand that OXYGENE (all capitalized, as used on many releases) stands for Oxygène, this is truly their fault. the wikipedia community is better informed ... (E-Kartoffel (talk) 13:11, 12 April 2010 (UTC))
 * seems to go against that. I would like to revert; does anybody disagree? -- Red rose64 (talk) 17:00, 15 May 2010 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: Page moved. Ucucha 02:37, 24 February 2010 (UTC)

Équinoxe → Equinoxe — Equinoxe should be listed without accent, see here a well researched list of releases: http://www.discogs.com/Jean-Michel-Jarre-Equinoxe/master/49172 User:E-Kartoffel 10:53, 16 February 2010 (UTC)

Even on the French-language pages on the artist's website, it's without the accent, though the accents for other releases such as Oxygène are present. I wouldn't put much stock in the website you cited -- it indiscriminately strips all diacritics (at least on the Jarre pages I looked at) and therefore tells us nothing about how to write the name of this particular album in English. The artist's website is the most authoritative thing we'll get on something like this (as opposed to, say, the name of a city), so we can move to Equinoxe safely without having to consider more questionable sources. --Atemperman (talk) 20:28, 16 February 2010 (UTC)

 * This may be relevant: "“É”, the upper-case version of “é”, has lost some use in French due to typographical reasons. “E” can be sometimes used in its place, when the accented “é” is unavailable “E” is sometimes used to represent it. For example “ÉCOLE” becomes "ECOLE"." —   AjaxSmack   02:57, 17 February 2010 (UTC)
 * Does that mean that in French works of reference, words with accents on capital letters are treated as if the accent weren't there? Or in some typographical styles, are hte accents there but "invisible"?  FWIW, fr.wikipedia.org/wiki/Ecole redirects to fr.wikipedia.org/École.  --Atemperman (talk) 16:00, 19 February 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

cosmos TV series Carl Sagan
Parts of Equinox also used in the above. Perhaps it should be included in the article. Jakescows (talk) 13:03, 1 September 2017 (UTC)
 * Yes it was used in the TV series, but we would need a reliable source to use as a reference in order to add it. Richard3120 (talk) 13:08, 1 September 2017 (UTC)

title: Talk:Équinoxe


text:
MV TSgt John A. Chapman

MV TSgt John A. Chapman (T-AK-323) was a Buffalo Soldier-class container ship. She was one of Military Sealift Command's Prepositioning Program.

Built in 1978 by Chantier naval de La Ciotat in la Ciotat, France, she was originally named Merlin. On 8 April 2005, she was renamed for Pope Air Force Base combat controller Technical Sergeant John A. Chapman, a posthumous Air Force Cross recipient. He was awarded the Air Force Cross for his actions during the Battle of Takur Ghar. In 2018, Chapman's Air Force Cross was upgraded to the Medal of Honor.

TSgt John A. Chapman carried Air Force munitions. She featured climate-controlled cocoons on her weather decks which protect additional cargo from the marine environment. She was owned and operated by Sealift Incorporated under charter to MSC.

She was scrapped on 4 September 2014 at Alang.

title: MV TSgt John A. Chapman


text:
Wikipedia:Articles for deletion/Died to death

 This page is an archive of the discussion about the proposed deletion of the article below. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was delete. Mindspillage (spill yours?) 21:49, 24 Apr 2005 (UTC)

Died to death
Delete Vanity page. --Randolph 19:56, 11 Apr 2005 (UTC)

 * Delete. I'm a bit bemused that this is VfDed — it was marked as a speedy, being a defamatory load of nonsense. If you insist on this lengthy process, though, I've at least deleted the defamatory stuff, just leaving the semi-literate nonsense. Mel Etitis  ( Μελ Ετητης ) 21:02, 11 Apr 2005 (UTC)
 * Delete. Sometimes I try to be nice, but this is utter crap. -- 8^D gab 21:08, 2005 Apr 11 (UTC)
 * Delete, patent nonsense. Firebug 21:44, 11 Apr 2005 (UTC)
 * Speedy delete as personal attack. - Lucky 6.9 22:55, 11 Apr 2005 (UTC)
 * Speedy delete as a personal attack and/or nonsense, whichever takes your fancy, but delete it. Thryduulf 23:19, 11 Apr 2005 (UTC)
 * Speedy delete since it does seem to be a personal attack. Zscout370 00:42, 12 Apr 2005 (UTC)
 * Delete, this reads more like a personal attack than a vanity page. Megan1967 10:42, 12 Apr 2005 (UTC)
 * Speedily Delete what a load of rubbish. Brequinda 10:49, 12 Apr 2005 (UTC)
 * Speedily delete, preferebly faster than the current land speed record. I know what it's like to have an article deleted, but this is total crap. Forgotmytea
 * This page is now preserved as an archive of the debate and, like some other VfD subpages, is no longer 'live'. Subsequent comments on the issue, the deletion, or the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.


text:
Talk:Naked singularity

Undo (then redo) of large addition from a few days ago.
I just undid this edit: https://en.wikipedia.org/w/index.php?title=Naked_singularity&diff=prev&oldid=1185401257 and the one before it, for a a number of reasons. First of all, approximately none of the text had anything to do with naked singularities; it appears to describe the process of stellar collapse. This is the wrong article for such descriptions. Another reason is that one of the two edits garbled the lede (failed to follow rudimentary WP style guidelines.) Finally, the edit made some dubious claims, suggesting that naked singularities happen all the time... which they don't. Perhaps there's some good content in there that could be moved to an article on stellar collapse, but I dunno. 67.198.37.16 (talk) 09:27, 18 November 2023 (UTC)

 * Arghh. Maybe I did the wrong thing. After the third reading, its not so bad. It was the garbed lede that really messed things up. So I'm going to ... undo my undo. Sorry for the mess. Ooof. 67.198.37.16 (talk) 09:39, 18 November 2023 (UTC)
 * You have ruined this article, please do not edit it if you don't have any idea of the field. I am a researcher working in this field and we have discussed with many professors before editing and mentioning this definition of naked singularities in the Wikipedia. This is the only precise definition of the naked singularity which can cover all types of naked singularities. If you don't have the idea of the types of naked singularities and then please read the first citation mentioned. Recently, Professor Penrose himself discussed the issue in an interview with the New Scientists where he clearly mentioned that collapse can end in black holes as well as naked singularities. If you need more popular or technical articles then please do ask us before editing the Wikipedia.
 * https://www.scientificamerican.com/article/naked-singularities-extreme-physics-special/
 * https://www.scientificamerican.com/article/naked-singularities-extreme-physics-special/
 * https://medium.com/@jayverma4455/what-is-at-the-centre-of-our-galaxy-tussle-of-the-black-hole-with-its-sibling-9d58778c640a
 * Please do read this pop-articles which can clear you understanding of the topic. Jayverma2210 (talk) 10:37, 21 November 2023 (UTC)

 * My apologies; I made a mistake. I left a note on your talk page explaining how I got to making that mistake (it was late at night). More or less all of you changes are just fine, but for one issue: Wikipedia has some strict standards for article style. I have tried to revise the lede so that it is both technically correct, and also conforms to the manual of style. If the lede still contains errors, I apologize; please do correct them. However, please also adhere to the proper style. 67.198.37.16 (talk) 05:46, 25 November 2023 (UTC)
 * Please tell me exactly which part you find is in against of Wikipedia standards... Jayverma2210 (talk) 09:10, 29 November 2023 (UTC)

The Manual of Style states that the opening few sentences should state what the article is about. For this particular article, the opening paragraph should read something like this:

 * In general relativity, a naked singularity is a hypothesized singularity in the solution to the Einstein field equations that is not hidden behind an event horizon. Naked singularities were first proposed and discussed in 19xx by so-and-so X Y and Z. They are hotly debated, as future time-like geodesics leaving the singularity create issues with causality.

Or something similar to that. Perhaps you will find that what I wrote is not technically accurate -- please do fix it as needed.

This is a formulaic introduction: a 1-2 sentence description of what it is, 1-2 sentences about history, 1-2 sentences about the epistemic status. All wikipedia articles start with this kind of recipe. The remainder of the intro paragraphs are fine. 67.198.37.16 (talk) 07:47, 30 January 2024 (UTC)

title: Talk:Naked singularity


text:
Red letter day

A red letter day (sometimes hyphenated as red-letter day) is any day of special significance or opportunity. Its roots are in classical antiquity; for instance, important days are indicated in red in a calendar dating from the Roman Republic (509–27 BC).

In medieval manuscripts, initial capitals and highlighted words (known as rubrics) were written in red ink. The practice was continued after the invention of the printing press, including in Catholic liturgical books. Many calendars still indicate special dates, festivals and holidays in red instead of black.

In the universities of the UK, scarlet days are when doctors may wear their scarlet 'festal' or full dress gowns instead of their undress ('black') gown.

In Norway, Sweden, Hong Kong, South Korea, Indonesia and some Latin American countries, a public holiday is sometimes referred to as "red day" (rød dag, röd dag, 빨간 날, 紅日, tanggal merah), as it is printed in red in calendars.

Legal
On red letter days, judges of the English High Court (King's Bench Division) wear, at sittings of the Court of Law, their scarlet robes (see court dress). Red letter days for these purposes are a fixed selection of saints' days (sometimes coinciding with the traditional start or end dates of the legal terms during which sittings of the High Court take place) and of national celebrations, mostly associated with senior members of the British royal family (and, therefore, changing from generation to generation).

Current red letter days in the United Kingdom
The list of red letter days currently observed in the United Kingdom (and on which, if a weekday, judges of the English High Court (King's Bench Division) traditionally wear, at sittings of the Court of Law, their scarlet robes) is as follows:

October November December January February March April May June July
 * 18th (St Luke)
 * 28th (St Simon and St Jude)
 * 1st (All Saints)
 * 14th (King Charles III, born, 1948)
 * 2nd Saturday in November (Lord Mayor's Day)
 * 30th (St Andrew)
 * 21st (St Thomas)
 * 25th (Conversion of St Paul)
 * 2nd (Candlemas)
 * 6th (Queen's Accession, 1952)
 * Moveable (Ash Wednesday)
 * 24th (St Mathias)
 * 1st (St David)
 * 25th (Lady Day)
 * 21st (Queen Elizabeth II born, 1926)
 * 25th (St Mark)
 * 1st (St Philip and St James)
 * Moveable (Ascension Day)
 * 2nd (Coronation of Queen Elizabeth II, 1953)
 * 2nd Saturday in June (Queen's Official Birthday)
 * 10th (Duke of Edinburgh born, 1921)
 * 11th (St Barnabas)
 * 24th (St John the Baptist)
 * 29th (St Peter)
 * 25th (St James)

Days which will never fall within the legal term are not red letter days (examples being Christmas Day and Easter Day).

Former red letter days in the United Kingdom
A comparison can be drawn with the Red Letter Days listed 100 years ago in Dress and Insignia Worn at His Majesty's Court (1921), which are on the same principle (a fixed selection of saints’ days, plus days honouring senior members of the Royal Family), except that the modern list adds the national saints of Wales and Scotland (St David and St Andrew, not listed in 1921) although not the national saint of England (St George). The 1921 listing, in full, is:

Red Letter Days on which the Judges of the High Court (King's Bench Division) wear, at Sittings of the Court of Law, their Scarlet Robes:
 * January 25. (Conversion of Saint Paul.)
 * February 2. (Purification of the Blessed Virgin Mary.)
 * February 24. (Saint Matthias.)
 * Ash Wednesday. (Movable.)
 * March 25. (Annunciation of the Blessed Virgin Mary.)
 * April 25. (Saint Mark.)
 * May 1. (Saint Philip and Saint James.)
 * Ascension Day. (Holy Thursday.) (Movable.)
 * May 6. (The King's Accession.)
 * May 26. (The Queen's Birthday.)
 * June 3. (The King's Birthday.)
 * June 11. (Saint Barnabas.)
 * June 22. (The King's Coronation.)
 * June 23. (Prince of Wales's Birthday.)
 * June 24. (Saint John the Baptist.)
 * June 29. Saint Peter.
 * Celebration of His Majesty's Birthday. (Movable.)
 * June 25. (Saint James.)
 * October 18. (Saint Luke.)
 * October 28. (Saint Simon and Saint Jude.)
 * November 1. (All Saints.)
 * November 9. (Lord Mayor's Day.)
 * November 30. (Saint Andrew.)
 * December 1. (Birthday of Queen Alexandra.)
 * December 21. (Saint Thomas.)

title: Red letter day


text:
Mathieu function

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation

\frac{d^2y}{dx^2} + (a - 2q\cos(2x))y = 0, $$ where $a, q$ are real-valued parameters. Since we may add $π/2$ to $x$ to change the sign of $q$, it is a usual convention to set $q ≥ 0$.

They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads. They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation (PDE) boundary value problems possessing elliptic symmetry.

Mathieu functions
In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of $$a$$ and $$q$$. When no confusion can arise, other authors use the term to refer specifically to $$\pi$$- or $$2\pi$$-periodic solutions, which exist only for special values of $$a$$ and $$q$$. More precisely, for given (real) $$q$$ such periodic solutions exist for an infinite number of values of $$a$$, called characteristic numbers, conventionally indexed as two separate sequences $$a_n(q)$$ and $$b_n(q)$$, for $$n = 1, 2, 3, \ldots$$. The corresponding functions are denoted $$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$, respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind.

As a result of assuming that $$q$$ is real, both the characteristic numbers and associated functions are real-valued.

$$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$ can be further classified by parity and periodicity (both with respect to $$x$$), as follows:

 * {| class="wikitable"

! Function !! Parity !! Period
 * $$\text{ce}_n, \ n \text{ even}$$
 * even
 * $$ \pi$$
 * $$\text{ce}_n, \ n \text{ odd}$$
 * even
 * $$2 \pi$$
 * $$\text{se}_n, \ n \text{ even}$$
 * odd
 * $$\pi$$
 * $$\text{se}_n, \ n \text{ odd}$$
 * odd
 * $$2 \pi$$
 * }
 * $$\text{se}_n, \ n \text{ odd}$$
 * odd
 * $$2 \pi$$
 * }

The indexing with the integer $$n$$, besides serving to arrange the characteristic numbers in ascending order, is convenient in that $$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$ become proportional to $$\cos nx$$ and $$\sin nx$$ as $$q \rightarrow 0$$. With $$n$$ being an integer, this gives rise to the classification of $$\text{ce}_n$$ and $$\text{se}_n$$ as Mathieu functions (of the first kind) of integral order. For general $$a$$ and $$q$$, solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions.

Modified Mathieu functions
Closely related are the modified Mathieu functions, also known as radial Mathieu functions, which are solutions of Mathieu's modified differential equation
 * $$ \frac{d^2y}{dx^2} - (a - 2q\cosh 2x)y = 0,$$

which can be related to the original Mathieu equation by taking $$x \to \pm {\rm i} x$$. Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by $$\text{Ce}_n(x, q)$$ and $$\text{Se}_n(x, q)$$, are defined from

\begin{align} \text{Ce}_n(x, q) &= \text{ce}_n({\rm i} x, q). \\ \text{Se}_n(x, q) &= -{\rm i}\,\text{se}_n({\rm i} x, q). \end{align} $$ These functions are real-valued when $$x$$ is real.

Normalization
A common normalization, which will be adopted throughout this article, is to demand

\int_{0}^{2\pi} \text{ce}_n(x, q)^2 dx = \int_{0}^{2\pi} \text{se}_n(x, q)^2 dx = \pi $$

as well as require $$\text{ce}_n(x, q) \rightarrow +\cos nx$$ and $$\text{se}_n(x, q) \rightarrow +\sin nx$$ as $$q \rightarrow 0$$.

Stability
The Mathieu equation has two parameters. For almost all choices of parameter, by Floquet theory (see next section), any solution either converges to zero or diverges to infinity.

Parametrize Mathieu equation as $$\ddot x + k(1-m \cos(t))x = 0$$, where $$k \in \R, m \geq 0$$. The regions of stability and instability are separated by curves

$$m(k) = \begin{cases} 2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, & k<0 ; \\[4pt] \frac{1}{4} \left[\sqrt{(9-4 k)(13-20 k)}-(9-4 k) \right], & k<\frac{1}{4} ; \\[10pt] \frac{1}{4} \left[9-4 k \mp \sqrt{(9-4 k)(13-20 k)} \right], & \frac{1}{4}1. \end{cases}$$

Floquet theory
Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called Floquet theory. The central result is Floquet's theorem:

$$

It is natural to associate the characteristic numbers $$a(q)$$ with those values of $$a$$ which result in $$\sigma = \pm 1$$. Note, however, that the theorem only guarantees the existence of at least one solution satisfying $$y(x + \pi) = \sigma y(x)$$, when Mathieu's equation in fact has two independent solutions for any given $$a$$, $$q$$. Indeed, it turns out that with $$a$$ equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period $$\pi$$ or $$2 \pi$$), and this solution is one of the $$\text{ce}_n(x, q)$$, $$\text{se}_n(x, q)$$. The other solution is nonperiodic, denoted $$\text{fe}_n(x, q)$$ and $$\text{ge}_n(x, q)$$, respectively, and referred to as a Mathieu function of the second kind. This result can be formally stated as Ince's theorem:

$$

An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form
 * $$F(a, q, x) = \exp(i \mu \,x) \, P(a, q, x),$$

where $$\mu$$ is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and $$P$$ is a complex valued function periodic in $$x$$ with period $$\pi$$. An example $$P(a, q, x)$$ is plotted to the right.

Second kind
Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if $$a$$ is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic. The periodic solution is one of the $$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$, called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either $$\text{fe}_n(x, q)$$ and $$\text{ge}_n(x, q)$$, respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions are unstable, that is, they diverge as $$z \rightarrow \pm \infty$$.

The second solutions corresponding to the modified Mathieu functions $$\text{Ce}_n(x, q)$$ and $$\text{Se}_n(x, q)$$ are naturally defined as $$\text{Fe}_n(x, q) = -i \text{fe}_n(xi, q)$$ and $$\text{Ge}_n(x, q) = \text{ge}_n(xi, q)$$.

Fractional order
Mathieu functions of fractional order can be defined as those solutions $$\text{ce}_{p}(x, q)$$ and $$\text{se}_{p}(x, q)$$, $$p$$ a non-integer, which turn into $$\cos p x$$ and $$\sin p x$$ as $$q \rightarrow 0$$. If $$p$$ is irrational, they are non-periodic; however, they remain bounded as $$x \rightarrow \infty$$.

An important property of the solutions $$\text{ce}_{p}(x, q)$$ and $$\text{se}_{p}(x, q)$$, for $$p$$ non-integer, is that they exist for the same value of $$a$$. In contrast, when $$p$$ is an integer, $$\text{ce}_{p}(x, q)$$ and $$\text{se}_{p}(x, q)$$ never occur for the same value of $$a$$. (See Ince's Theorem above.)

These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly.

 * {| class="wikitable"

! Order !! First kind !! Second kind ($$p$$ non-integral)
 * +Classification of Mathieu functions
 * Integral
 * $$\text{ce}_n(x, q)$$
 * $$\text{fe}_n(x, q)$$
 * Integral
 * $$\text{se}_n(x, q)$$
 * $$\text{ge}_n(x, q)$$
 * Fractional
 * $$\text{ge}_n(x, q)$$
 * Fractional
 * Fractional
 * $$\text{ce}_{p}(x, q)$$
 * $$\text{se}_{p}(x, q)$$
 * }

First kind
Mathieu functions of the first kind can be represented as Fourier series:

\begin{align} \text{ce}_{2n}(x, q) &= \sum_{r=0}^{\infty} A^{(2n)}_{2r}(q) \cos (2 r x) \\ \text{ce}_{2n+1}(x, q) &= \sum_{r=0}^{\infty} A^{(2n+1)}_{2r+1}(q) \cos \left[ (2r+1) x \right] \\ \text{se}_{2n+1}(x, q) &= \sum_{r=0}^{\infty} B^{(2n+1)}_{2r+1}(q) \sin \left[(2r+1) x\right] \\ \text{se}_{2n+2}(x, q) &= \sum_{r=0}^{\infty} B^{(2n+2)}_{2r+2}(q) \sin \left[(2r+2) x\right] \\ \end{align} $$ The expansion coefficients $$A^{(i)}_{j}(q)$$ and $$B^{(i)}_{j}(q)$$ are functions of $$q$$ but independent of $$x$$. By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. For instance, for each $$\text{ce}_{2n}$$ one finds

\begin{align} a A_0 - q A_2 &= 0 \\ (a - 4) A_2 - q (A_4 + 2 A_0) &= 0 \\ (a - 4 r^2) A_{2r} - q (A_{2r+2} + A_{2r-2}) &= 0, \quad r \geq 2 \end{align} $$ Being a second-order recurrence in the index $$2r$$, one can always find two independent solutions $$X_{2r}$$ and $$Y_{2r}$$ such that the general solution can be expressed as a linear combination of the two: $$A_{2r} = c_1 X_{2r} + c_2 Y_{2r}$$. Moreover, in this particular case, an asymptotic analysis shows that one possible choice of fundamental solutions has the property

\begin{align} X_{2r} &= r^{-2r-1} \left(-\frac{e^2 q}{4} \right)^r \left[1 + \mathcal{O}(r^{-1}) \right] \\ Y_{2r} &= r^{2r-1} \left(-\frac{4}{e^2 q} \right)^r \left[1 + \mathcal{O}(r^{-1}) \right] \end{align} $$ In particular, $$X_{2r}$$ is finite whereas $$Y_{2r}$$ diverges. Writing $$A_{2r} = c_1 X_{2r} + c_2 Y_{2r}$$, we therefore see that in order for the Fourier series representation of $$\text{ce}_{2n}$$ to converge, $$a$$ must be chosen such that $$c_2 = 0.$$ These choices of $$a$$ correspond to the characteristic numbers.

In general, however, the solution of a three-term recurrence with variable coefficients cannot be represented in a simple manner, and hence there is no simple way to determine $$a$$ from the condition $$c_2 = 0$$. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients $$A_{2r}$$ by numerically iterating the recurrence towards increasing $$r$$. The reason is that as long as $$a$$ only approximates a characteristic number, $$c_2$$ is not identically $$0$$ and the divergent solution $$Y_{2r}$$ eventually dominates for large enough $$r$$.

To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a continued fraction expansion, casting the recurrence as a matrix eigenvalue problem, or implementing a backwards recurrence algorithm. The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.

In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as Mathematica, Maple, MATLAB, and SciPy. For small values of $$q$$ and low order $$n$$, they can also be expressed perturbatively as power series of $$q$$, which can be useful in physical applications.

Second kind
There are several ways to represent Mathieu functions of the second kind. One representation is in terms of Bessel functions:

\begin{align} \text{fe}_{2n}(x, q) &= -\frac{\pi \gamma_n}{2} \sum_{r=0}^{\infty} (-1)^{r+n} A^{(2n)}_{2r}(-q) \ \text{Im}[J_r(\sqrt{q}e^{ix}) Y_r(\sqrt{q}e^{-ix})], \quad \text{where } \gamma_n = \left\{ \begin{array}{cc} \sqrt{2}, & \text{ if } n = 0 \\ 2n, & \text{ if } n \geq 1 \end{array} \right. \\ \text{fe}_{2n+1}(x, q) &= \frac{\pi \sqrt{q}}{2} \sum_{r=0}^{\infty} (-1)^{r+n} A^{(2n+1)}_{2r+1}(-q) \ \text{Im}[J_r(\sqrt{q}e^{ix}) Y_{r+1}(\sqrt{q}e^{-ix}) + J_{r+1}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \\ \text{ge}_{2n+1}(x, q) &= -\frac{\pi \sqrt{q}}{2} \sum_{r=0}^{\infty} (-1)^{r+n} B^{(2n+1)}_{2r+1}(-q) \ \text{Re}[J_r(\sqrt{q}e^{ix}) Y_{r+1}(\sqrt{q}e^{-ix}) - J_{r+1}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \\ \text{ge}_{2n+2}(x, q) &= -\frac{\pi q}{4(n+1)} \sum_{r=0}^{\infty} (-1)^{r+n} B^{(2n+2)}_{2r+2}(-q) \ \text{Re}[J_r(\sqrt{q}e^{ix}) Y_{r+2}(\sqrt{q}e^{-ix}) - J_{r+2}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \end{align} $$ where $$n, q > 0$$, and $$J_r(x)$$ and $$Y_r(x)$$ are Bessel functions of the first and second kind.

Modified functions
A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series. For large $$n$$ and $$q$$, the form of the series must be chosen carefully to avoid subtraction errors.

Properties
There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other special functions, the solutions of Mathieu's equation cannot in general be expressed in terms of hypergeometric functions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable $$t=\cos(x)$$:

 * $$ (1-t^2)\frac{d^2y}{dt^2} - t\, \frac{d y}{dt} + (a + 2q (1- 2t^2)) \, y=0.$$

Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.

Qualitative behavior

For small $$q$$, $$\text{ce}_{n}$$ and $$\text{se}_{n}$$ behave similarly to $$\cos nx$$ and $$\sin nx$$. For arbitrary $$q$$, they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real $$q$$, $$\text{ce}_{m}(x, q)$$ and $$\text{se}_{m+1}(x, q)$$ have exactly $$m$$ simple zeros in $$0 < x < \pi$$, and as $$q \rightarrow \infty$$ the zeros cluster about $$x = \pi / 2 $$.

For $$q > 0$$ and as $$x \rightarrow \infty$$ the modified Mathieu functions tend to behave as damped periodic functions.

In the following, the $$A$$ and $$B$$ factors from the Fourier expansions for $$\text{ce}_{n}$$ and $$\text{se}_{n}$$ may be referenced (see Explicit representation and computation). They depend on $$q$$ and $$n$$ but are independent of $$x$$.

Reflections and translations
Due to their parity and periodicity, $$\text{ce}_n$$ and $$\text{se}_n$$ have simple properties under reflections and translations by multiples of $$\pi$$:

\begin{align} &\text{ce}_n(x + \pi) = (-1)^n \text{ce}_n(x) \\ &\text{se}_n(x + \pi) = (-1)^n \text{se}_n(x) \\ &\text{ce}_n(x + \pi / 2) = (-1)^n \text{ce}_n(-x + \pi / 2) \\ &\text{se}_{n+1}(x + \pi / 2) = (-1)^n \text{se}_{n+1}(-x + \pi / 2) \end{align} $$

One can also write functions with negative $$q$$ in terms of those with positive $$q$$:

\begin{align} &\text{ce}_{2n+1}(x, -q) = (-1)^n \text{se}_{2n+1}(-x + \pi/2, q) \\ &\text{ce}_{2n+2}(x, -q) = (-1)^n \text{ce}_{2n+2}(-x + \pi/2, q) \\ &\text{se}_{2n+1}(x, -q) = (-1)^n \text{ce}_{2n+1}(-x + \pi/2, q) \\ &\text{se}_{2n+2}(x, -q) = (-1)^n \text{se}_{2n+2}(-x + \pi/2, q) \end{align} $$

Moreover,

\begin{align} &a_{2n+1}(q) = b_{2n+1}(-q)\\ &b_{2n+2}(q) = b_{2n+2}(-q) \end{align} $$

Orthogonality and completeness
Like their trigonometric counterparts $$\cos nx$$ and $$\sin nx$$, the periodic Mathieu functions $$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$ satisfy orthogonality relations

\begin{align} &\int_0^{2\pi} \text{ce}_n \text{ce}_m \,dx = \int_0^{2\pi} \text{se}_n \text{se}_m \, dx = \delta_{nm} \pi \\ &\int_0^{2\pi} \text{ce}_n \text{se}_m \,dx = 0 \end{align} $$

Moreover, with $$q$$ fixed and $$a$$ treated as the eigenvalue, the Mathieu equation is of Sturm–Liouville form. This implies that the eigenfunctions $$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$ form a complete set, i.e. any $$\pi$$- or $$2 \pi$$-periodic function of $$x$$ can be expanded as a series in $$\text{ce}_n(x, q)$$ and $$\text{se}_n(x, q)$$.

Integral identities
Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels $$\chi(x, x')$$ that are solutions of

\frac{\partial^2 \chi}{\partial x^2} - \frac{\partial^2 \chi}{\partial x'^2} = 2 q \left(\cos 2x - \cos 2x' \right) \chi $$ More precisely, if $$\phi(x)$$ solves Mathieu's equation with given $$a$$ and $$q$$, then the integral

\psi(x) \equiv \int_C \chi(x, x') \phi(x') dx' $$ where $$C$$ is a path in the complex plane, also solves Mathieu's equation with the same $$a$$ and $$q$$, provided the following conditions are met:

 * $$\chi(x, x')$$ solves $$\frac{\partial^2 \chi}{\partial x^2} - \frac{\partial^2 \chi}{\partial x'^2} = 2 q \left(\cos 2x - \cos 2x' \right) \chi$$
 * In the regions under consideration, $$\psi(x)$$ exists and $$\chi(x, x')$$ is analytic
 * $$\left( \phi \frac{\partial \chi}{\partial x'} - \frac{\partial \phi}{\partial x'} \chi \right)$$ has the same value at the endpoints of $$C$$

Using an appropriate change of variables, the equation for $$\chi$$ can be transformed into the wave equation and solved. For instance, one solution is $$\chi(x, x') = \sinh (2 q^{1/2} \sin x \sin x' )$$. Examples of identities obtained in this way are

\begin{align} \text{se}_{2n+1}(x, q) &= \frac{\text{se}'_{2n+1}(0, q)}{\pi q^{1/2} B_1^{(2n+1)}} \int_0^{\pi} \sinh (2 q^{1/2} \sin x \sin x' ) \text{se}_{2n+1}(x', q) dx' \qquad (q > 0) \\ \text{Ce}_{2n}(x, q) &= \frac{\text{ce}_{2n}(\pi / 2, q)}{\pi A_0^{(2n)}} \int_0^{\pi} \cos (2 q^{1/2} \cosh x \cos x' ) \text{ce}_{2n}(x', q) dx' \qquad \ \ \ (q > 0) \end{align} $$

Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.

There also exist integral relations between functions of the first and second kind, for instance:

\text{fe}_{2n}(x, q) = 2n \int_0^x \text{ce}_{2n}(\tau, -q) \ J_0 \left( \sqrt{2q(\cos 2x - \cos 2\tau)} \right) d \tau, \qquad n \geq 1 $$ valid for any complex $$x$$ and real $$q$$.

Asymptotic expansions
The following asymptotic expansions hold for $$q > 0$$, $$\text{Im}(x) = 0$$, $$\text{Re}(x) \rightarrow \infty$$, and $$2 q^{1/2} \cosh x \simeq q^{1/2} e^x$$:

\begin{align} \text{Ce}_{2n}(x,q) &\sim \left(\frac{2}{\pi q^{1/2}} \right)^{1/2} \frac{\text{ce}_{2n}(0, q)\text{ce}_{2n}(\pi/2, q)}{A_0^{(2n)}} \cdot e^{-x/2} \sin \left( q^{1/2} e^x + \frac{\pi}{4} \right) \\ \text{Ce}_{2n+1}(x,q) &\sim \left(\frac{2}{\pi q^{3/2}} \right)^{1/2} \frac{\text{ce}_{2n+1}(0, q)\text{ce}'_{2n+1}(\pi/2, q)}{A_1^{(2n+1)}} \cdot e^{-x/2} \cos \left( q^{1/2} e^x + \frac{\pi}{4} \right) \\ \text{Se}_{2n+1}(x,q) &\sim -\left(\frac{2}{\pi q^{3/2}} \right)^{1/2} \frac{\text{se}'_{2n+1}(0, q)\text{se}_{2n+1}(\pi/2, q)}{B_1^{(2n+1)}} \cdot e^{-x/2} \cos \left( q^{1/2} e^x + \frac{\pi}{4} \right) \\ \text{Se}_{2n+2}(x,q) &\sim \left(\frac{2}{\pi q^{5/2}} \right)^{1/2} \frac{\text{se}'_{2n+2}(0, q)\text{se}'_{2n+2}(\pi/2, q)}{B_2^{(2n+2)}} \cdot e^{-x/2} \sin \left( q^{1/2} e^x + \frac{\pi}{4} \right) \end{align} $$

Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for $$\text{Fe}_{n}$$ and $$\text{Ge}_{n}$$; these also decay exponentially for large real argument.

For the even and odd periodic Mathieu functions $$ ce, se$$ and the associated characteristic numbers $$a$$ one can also derive asymptotic expansions for large $$q$$. For the characteristic numbers in particular, one has with $$N$$ approximately an odd integer, i.e. $$N\approx N_0= 2n+1, n =1,2,3,...,$$

\begin{align} a(N) ={}& -2q + 2q^{1/2}N -\frac{1}{2^3}(N^2+1) -\frac{1}{2^7q^{1/2}}N(N^2 +3) -\frac{1}{2^{12}q}(5N^4 + 34N^2 +9) \\ & -\frac{1}{2^{17}q^{3/2}}N(33N^4 +410N^2 +405) -\frac{1}{2^{20}q^2}(63N^6 + 1260N^4 + 2943N^2 +41807) + \mathcal{O}(q^{-5/2}) \end{align} $$ Observe the symmetry here in replacing $$ q^{1/2}$$ and $$N$$ by $$-q^{1/2}$$ and $$- N$$, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order $$|q|^{-7/2}$$. Here $$N$$ is only approximately an odd integer because in the limit of $$q\to \infty$$ all minimum segments of the periodic potential $$ \cos 2x$$ become effectively independent harmonic oscillators (hence $$N_0$$ an odd integer). By decreasing $$q$$, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers $$ a \to a_{\mp}$$ (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions (in quantum mechanics this provides the splitting of the eigenvalues into energy bands). The boundary conditions are:
 * $$ \left(\frac{dce_{N_0-1}}{dx}\right)_{\pi/2}=0,\;\; ce_{N_0}(\pi/2)=0, \;\; \left(\frac{dse_{N_0}}{dx}\right)_{\pi/2}=0, \;\; se_{N_0+1}(\pi/2)=0.$$

Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for $$a$$ one obtains
 * $$ N-N_0 = \mp 2\left(\frac{2}{\pi}\right)^{1/2} \frac{(16q^{1/2})^{N_0/2}e^{-4q^{1/2}}}{[\frac{1}{2}(N_0-1)]!}\left[1-\frac{3(N_0^2+1)}{2^6q^{1/2}} + \frac{1}{2^{13}q}(9N_0^4 -40N_0^3 +18N_0^2 -136N_0 + 9) + \dots  \right].$$

The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.
 * $$ a(N) = a(N_0) + (N-N_0) \left(\frac{\partial a}{\partial N}\right)_{N_0} + \cdots . $$

Insertion of the appropriate expressions above yields the result
 * $$\begin{align}

a(N)\to a_{\mp}(N_0) = {} & -2q + 2q^{1/2}N_0 -\frac{1}{2^3}(N^2_0+1) - \frac{1}{2^7q^{1/2}}N_0(N_0^2+3) - \frac{1}{2^{12}q}(5N_0^4+34N_0^2 +9) - \cdots \\ & \mp \frac{(16q^{1/2})^{N_0/2+1}e^{-4q^{1/2}}}{(8\pi)^{1/2}[\frac{1}{2}(N_0-1)]!}\bigg[1-\frac{N_0}{2^6q^{1/2}}(3N_0^2+8N_0+3) + \cdots\bigg]. \end{align} $$ For $$N_0=1,3,5,\dots$$ these are the eigenvalues associated with the even Mathieu eigenfunctions $$ ce_{N_0} $$ or $$ce_{N_0-1}$$ (i.e. with upper, minus sign) and odd Mathieu eigenfunctions $$se_{N_0+1}$$ or $$se_{N_0}$$ (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in or.

Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for Lamé functions and prolate and oblate spheroidal wave functions.

Applications
Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below.

Partial differential equations
Mathieu functions arise when separation of variables in elliptic coordinates is applied to 1) the Laplace equation in 3 dimensions, and 2) the Helmholtz equation in either 2 or 3 dimensions. Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in computational electromagnetics they can be used to analyze the scattering of electromagnetic waves off elliptic cylinders, and wave propagation in elliptic waveguides. In general relativity, an exact plane wave solution to the Einstein field equation can be given in terms of Mathieu functions.

More recently, Mathieu functions have been used to solve a special case of the Smoluchowski equation, describing the steady-state statistics of self-propelled particles.

The remainder of this section details the analysis for the two-dimensional Helmholtz equation. In rectangular coordinates, the Helmholtz equation is

\left(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} \right) \psi + k^2 \psi = 0, $$ Elliptic coordinates are defined by

\begin{align} x &= c \cosh \mu \cos \nu \\ y &= c \sinh \mu \sin \nu \end{align} $$ where $$0 \leq \mu < \infty$$, $$0 \leq \nu < 2\pi$$, and $$c$$ is a positive constant. The Helmholtz equation in these coordinates is

\frac{1}{c^2 (\sinh^2 \mu + \sin^2 \nu)} \left(\frac{\partial^2}{\partial \mu^2} +\frac{\partial^2}{\partial \nu^2} \right) \psi + k^2 \psi = 0 $$ The constant $$\mu$$ curves are confocal ellipses with focal length $$c$$; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via $$\psi(\mu, \nu) = F(\mu) G(\nu)$$ yields the Mathieu equations

\begin{align} &\frac{d^2F}{d \mu^2} - \left(a - \frac{c^2 k^2}{2} \cosh 2 \mu \right) F = 0 \\ &\frac{d^2G}{d \nu^2} + \left(a - \frac{c^2 k^2}{2} \cos 2\nu \right) G = 0 \\ \end{align} $$ where $$a$$ is a separation constant.

As a specific physical example, the Helmholtz equation can be interpreted as describing normal modes of an elastic membrane under uniform tension. In this case, the following physical conditions are imposed:
 * Periodicity with respect to $$\nu$$, i.e. $$\psi(\mu, \nu) = \psi(\mu, \nu + 2 \pi)$$
 * Continuity of displacement across the interfocal line: $$\psi(0, \nu) = \psi(0, -\nu)$$
 * Continuity of derivative across the interfocal line: $$\psi_{\mu}(0, \nu) = -\psi_{\mu}(0, -\nu)$$

For given $$k$$, this restricts the solutions to those of the form $$\text{Ce}_{n}(\mu, q)\text{ce}_n(\nu, q)$$ and $$\text{Se}_{n}(\mu, q)\text{se}_n(\nu, q)$$, where $$q = c^2 k^2 / 4$$. This is the same as restricting allowable values of $$a$$, for given $$k$$. Restrictions on $$k$$ then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by $$\mu = \mu_0 > 0$$. For instance, clamping the membrane at $$\mu = \mu_0$$ imposes $$\psi(\mu_0, \nu) = 0$$, which in turn requires

\begin{align} \text{Ce}_{n}(\mu_0, q) = 0 \\ \text{Se}_{n}(\mu_0, q) = 0 \end{align} $$ These conditions define the normal modes of the system.

Dynamical problems
In dynamical problems with periodically varying forces, the equation of motion sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics. A classic example along these lines is the inverted pendulum. Other examples are
 * vibrations of a string with periodically varying tension
 * stability of railroad rails as trains drive over them
 * seasonally forced population dynamics
 * the phenomenon of parametric resonance in forced oscillators
 * motion of ions in a quadrupole ion trap
 * the Stark effect for a rotating electric dipole
 * the Floquet theory of the stability of limit cycles

Quantum mechanics
Mathieu functions play a role in certain quantum mechanical systems, particularly those with spatially periodic potentials such as the quantum pendulum and crystalline lattices.

The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential $$V(r) = g^2/r^4$$ the radial Schrödinger equation
 * $$ \frac{d^2y}{dr^2} + \left[ k^2 - \frac{\ell(\ell+1)}{r^2} -\frac{g^2}{r^4} \right] y = 0 $$

can be converted into the equation
 * $$ \frac{d^2\varphi}{dz^2} + \left[ 2h^2 \cosh 2z - \left(\ell+\frac 1 2 \right)^2\right] \varphi = 0. $$

The transformation is achieved with the following substitutions
 * $$ y = r^{1/2} \varphi, r=\gamma e^z, \gamma = \frac{ig}{h}, h^2 = ikg, h = e^{I\pi/4}(kg)^{1/2}. $$

By solving the Schrödinger equation (for this particular potential) in terms of solutions of the modified Mathieu equation, scattering properties such as the S-matrix and the absorptivity can be obtained.

title: Mathieu function


text:
HMS K3

HMS K3 was the lead ship of the British K-class submarines. She was laid down on 21 May 1915 by Vickers, Barrow-in-Furness. She was commissioned on 4 August 1916.

In December 1916, K3, with the future King George VI aboard, uncontrollably dived. The ship plunged to 150 feet with the stern and propellers raised above the waves. It took twenty minutes to free the ship from the sea bed mud and surface successfully.

On 9 January 1917, K3’s boiler room was flooded in the North Sea. K3 was involved in an accident with the 4th Light Cruiser Squadron, that led to the sinking of HMS K1, in November 1917. K3 was also involved in the "Battle of May Island" in 1918. On 2 May 1918, K3 yet again uncontrollably dived to 266 feet which crushed part of the hull.

K3 was sold on 26 October 1921 to the Barking Ship Breaking Company for scrapping in London.

title: HMS K3


text:
Innishannon

Innishannon or Inishannon is a large village on the main Cork–Bandon road (N71) in County Cork, Ireland. Situated on the River Bandon, the village has grown due to its proximity to Cork city (20 km to the north-east), and is now a dormitory town for city workers. As of 2022, it had a population of 1,043.

History
Inishannon village is located at and developed around an important crossing-point on the River Bandon. Formerly controlled by the de Barry family, the area was used as a ferry point on the river from at least the early medieval period. Inishannon received a market and fair grant in 1256, and was given a royal charter in 1412. Writing in the mid-18th century, the antiquarian Charles Smith described Inishannon as "formerly walled and a place of some note". Innishannon Tower, the remains of a mid-18th century church, are built on the site the much earlier medieval parish church of Inishannon.

In 1837, Inishannon village had a population of approximately 650 people. By the 2016 census, Innishannon had a population of 1,043, a near threefold increase in the 25 years since the 1991 census, when the village had 319 inhabitants. As of 2022, it had a population of over 1000 people.

Events
Innishannon Steam and Vintage Rally is held in Innishannon annually in June. This event continues on from the old Upton Steam Rally that was held on the old St. Patricks School grounds. The Innishannon Steam and Vintage Rally was formed in 1998, and attracts upwards of 1,000 exhibits and approximately 60,000 visitors every year. Since 1998, the rally's organisers have raised over one million euro for the Irish Cancer Society.

Transport
The area was previously served by the Cork and Bandon Railway. Upton and Innishannon railway station opened in August 1849 and closed in April 1961.

The village lies on the N71 secondary road between Cork and Bandon. It is on several bus routes.

Innishannon parish

The parish of Innishannon stretches from the nearby Dromkeen to close to Aherla and over to Kilmacsimon in the east. The parish includes the village of Crossbarry. It also includes John Coleman's house in Togher Upper. The parish has four schools; Scoil Eoin in the village of Innishannon itself, Knockavilla to the north of the parish opposite St. Patrick's Church - the second church of the parish, Gurrane National School (sometimes called Gurranes) near Crossbarry, and Castleack National School near the parish's boundary with Bandon.

Amenities
The village has two food stores, a doctor's surgery, a dentist, a pharmacy, a butcher, a hairdresser, a café, a credit union, a fast food restaurant, a Chinese restaurant, a car sales garage and three public houses.

Innishannon's Gaelic Athletic Association pitch, home to Valley Rovers GAA club, is sometimes flooded because of its proximity to the river. The local soccer club is Innishvilla AFC.

People
Innishannon is home of the author Alice Taylor who wrote the bestselling To School Through the Fields, and Quench the Lamp, as well as many other novels and collections of poetry.

Valley Rovers GAA club has provided the national Gaelic Athletic Association organisation with two presidents, Seán McCarthy and Con Murphy.

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AirLink Express

AirLink Express is an air charter airline based in Jamaica. The airline is located at Sir Donald Sangster International Airport (MBJ), Montego Bay. The airline offers charter and scheduled service to every airport in Jamaica. As of June 2018 the Jamaica Gleaner reports that the airline uses a Cessna 208B Grand Caravan

Destinations
AirLink Express has domestic services from its base at Sangster International in Montego Bay to Negril Aerodrome

The company offers chartered flights from Montego Bay to Ian Fleming International Airport in Ocho Rios and Ken Jones Aerodrome in Port Antonio.

It previously provided scheduled services between Sangster International and Tinson Pen Aerodrome in Kingston.

Fleet
AirLink Express fleet consisted of

 * Cessna 206
 * Cessna 207
 * Cessna 208
 * Beech 1900D
 * Beech 1900C

title: AirLink Express


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User:Gold Stur

Neva4get
On January 4, 2006, this pristine and perfect article was maliciously vandalised through the bizarre psychic wizardry of the Ku Klux Klan and Harry Potter. As with 9/11, Pearl Harbor, the Alamo, and the sinking of the USS Maine... let us NEVA4GET!!

About me
I'm an African-Swiss dinosaur born in California though I currently reside in Ohio. I like to play video games and play with myself. Some of my favorite music artists and bands include GG Allin, Black Flag, Pink Floyd, Bad Brains, Dead Kennedys, The Screamers, the Beastie Boys, Circle Jerks, etc. I am not punk. Some of my heroes include Charles Whitman, Lee Harvey Oswald whom was the only shooter, Woo Bum-Kon, Leon Czolgosz, John Wilkes Booth, etc. Uh... I also skateboard. I am in awe of George Orwell. And I'm an atheist and anarcho-syndicalist.

title: User:Gold Stur


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Spion Kop, Nottinghamshire

Spion Kop is a small residential and former industrial area in Nottinghamshire, England, stretching for a few hundred metres on both sides of the main A60 road surrounded by open farmland. It is in the civil parish of Warsop.

It is located about a mile to the south of Warsop on the A60, Mansfield Road. It is a settlement built and named after the Battle of Spion Kop which took place during the Second Boer War in Natal, South Africa, in January 1900. A major military figure in the conflict was John Talbot Coke, grandson of D'Ewes Coke, born at Mansfield Woodhouse, a well-known Nottinghamshire industrialist and clergyman. At Mansfield Woodhouse a Coke Street was renamed Newhaven Avenue.

The one residential side-street adjoining the main A60 road formerly known as George Street has been renamed Mosscar Close.

A modern, large-scale mixed-residential development has been built on the extensive site of the old Wood Brothers timber business on Mansfield Road following a successful planning application to Mansfield District Council in 2011.

title: Spion Kop, Nottinghamshire


text: User talk:68.60.167.90

title: User talk:68.60.167.90