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Key West International Airport

Key West International Airport is an international airport located in the City of Key West in Monroe County, Florida, United States, 2 mi east of the main commercial center of Key West.

The relatively short 5076 ft runway limits the maximum size of aircraft that can serve the airport, along with restricting the weight of flights departing.

History
Key West's aviation history began in 1913 with a flight to Cuba by Augustin Parla. In 1928, Pan American Airways began scheduled flights from Key West. The main runway at Meacham Field was pressed into U.S. Army use after the Pearl Harbor attack, as well as into U.S. Navy use later in World War II as an alternative to the Trumbo Point seaplane base and the main Naval Air Station for fixed-wing and lighter-than-air (blimp) aircraft on Boca Chica Key. After the war, the city took over what became Key West Municipal Airport. In January 1953, the city gave Monroe County the title to Meacham Field, allowing the county to apply for Federal Aviation Administration grants. Around the same time, the airport became Key West International Airport.

National Airlines began flights to Miami in 1944 with Lockheed Lodestar twin prop aircraft, although the airport did not have a paved runway until around 1956. National served Key West for nearly 25 years and later operated Convair 340 and Convair 440 prop aircraft, as well as Lockheed L-188 Electra turboprops, into the airport. In 1968, National began the first jet flights into Key West with Boeing 727-100s, providing nonstop service to Miami. By 1969, National was operating daily 727 jet service direct to Washington National Airport, Philadelphia International Airport, and John F. Kennedy International Airport via intermediate stops in Miami, West Palm Beach, and Orlando.

Several other airlines also began operating jet service into Key West. In 1979, Air Florida was operating five nonstop flights a day to Miami with Boeing 737 jetliners. In 1987, Eastern Airlines was operating daily mainline Boeing 727-100 jet service nonstop to Miami. By 1989, Piedmont Airlines was operating six nonstop flights a day to Miami with Fokker F28 Fellowship twin jets. This F28 jet service was then continued by USAir following its acquisition of and merger with Piedmont.

A number of commuter and regional airlines also served Key West with turboprop and prop aircraft during the 1970s and 1980s primarily with nonstop flights to Miami but also with nonstop service to Fort Lauderdale, Naples, and Tampa. According to the Official Airline Guide (OAG), these air carriers included Air Florida Commuter, Dolphin Airlines, Gull Air, Jetstream International Airlines, Pro Air Services, Provincetown-Boston Airlines (and its successor Bar Harbor Airlines), Southeast Airlines, and Southern Express. Turboprop aircraft operated into the airport included the British Aerospace BAe Jetstream 31, Beechcraft 1900C, Beechcraft 1900D, Beechcraft C99, CASA 212 Aviocar, Embraer EMB-110 Bandeirante, Embraer EMB-120 Brasilia, Nord 262 and Saab 340. Piston engine twin prop aircraft flown by commuter air carriers serving Key West included the Douglas DC-3, Martin 4-0-4, and Piper Navajo.

Delta Connection carrier Comair began service to Delta Air Lines' hub in Orlando in 1988 and also briefly had service to Tampa and Fort Lauderdale as well. American Eagle Airlines began service from Key West to American Airlines' hub in Miami in 1989. American Eagle operated Saab 340 and ATR-72 propjets into the airport before introducing regional jet service in the 2010s.

By the early 1990s, both USAir and Eastern Air Lines ended jet service to Key West, and all commercial service to Key West would be operated by propeller aircraft for the rest of the decade. In 1993, Cape Air began service to Key West with flights to Naples using Cessna 402s. Service expanded in 1996 with flights to Fort Lauderdale and Fort Myers (the latter of which continued operating until 2013). Other airlines operating at Key West in the early 1990s included Airways International (with flights to Fort Lauderdale and Miami), Pan Am Express (with flights to Miami), Paradise Island Airlines (operating code sharing service for Carnival Air Lines to Fort Lauderdale), and USAir Express (later known as US Airways Express with flights to Miami and Tampa).

Gulfstream International Airlines also began service to Key West in the early 1990s with flights to Miami and Tampa. Gulfstream would later operate as Continental Connection, the regional brand of the now-defunct Continental Airlines. Gulfstream has since been rebranded as Silver Airways, who still serves Key West today.

In 2002, Delta Connection carrier Atlantic Southeast Airlines began regional jet service to Delta's hub in Atlanta. This brought jet service back to Key West as well as Key West's first non-stop flight to a destination outside of Florida since 1961. Delta Connection's service to Orlando was also upgraded to regional jets at the same time, though Orlando service was discontinued by 2008.

Main line jet service returned to Key West in 2009 when AirTran Airways, Key West's first low-cost carrier, started flights to Orlando International Airport on Boeing 737-700s. Around the same time, Delta Air Lines upgraded some of their Atlanta flights to main line service on Boeing 737s. The airport's runway, which was 4801 ft at the time, was the shortest runway to be regularly used by 737s in North America.

AirTran later added flights to Tampa in 2011. By the end of 2012, Southwest Airlines took over AirTran's flights to Orlando and Tampa as part of the merger of the two airlines. Southwest also added service to New Orleans in March 2013. However, in 2014, Southwest discontinued all service to the airport.

On July 15, 2017, Key West International Airport was awarded a grant of $6.5 million by the FAA to assist in a $10 million runway project. The project added 227 ft to the runway for takeoffs and landings as well as added 10 ft-wide shoulders paved on each side of the runway. Construction work began in January 2018 and all construction was done at night.

Facilities
Key West International Airport covers 334 acres (135 ha) at an elevation of 3 feet (1 m). Its one runway, 09/27, is 5,076 feet long and 100 feet wide (1,547 x 30 m) and has an asphalt surface.

The length of the runway limits the size of aircraft that can serve the airport. Currently the largest aircraft that serve the airport are the Airbus A319 and the similarly sized Boeing 737-700, both shortened variants of the popular narrow-body mainline aircraft. Additionally, flights departing from EYW often have weight restrictions.

The airport has two terminals designed by Mark Mosko and Dwane Stark of URS; Mosko also worked on Baltimore–Washington International Airport. The older ground-level terminal building opened in 1957 and now serves arriving passengers. The terminal was expanded with the addition of a second building elevated over the parking lot in February 2009. With an area of about 30000 sqft, it more than doubled the airport's terminal space. The newer building includes an elevated roadway and houses ticketing, check-in, and the airport's security checkpoint. The older building was then renovated with the former ticketing area becoming an expanded departure gate lounge, and the baggage claim area was then expanded into the former departure lounge. The two buildings are connected by an enclosed walkway.

Parking for 300 vehicles is at ground level beneath the newer terminal—150 spaces for rental cars and 150 for the public.

In November 2022, the airport broke ground on the construction of a new $98.8 million passenger concourse. This concourse will be nearly 50000 ft2 in size and feature a new departures area, an upgraded security checkpoint, additional baggage areas, and new gates featuring seven jet bridges. The project is expected to be completed in early 2025.

In the year ending December 31, 2021, the airport had 66,550 aircraft operations, averaging 182 per day: 55% general aviation, 34% airline, 10% air taxi, and <1% military. At the time, 59 aircraft were based at the airport: 39 single-engine, 13 multi-engine, 3 jet, 3 helicopter and 1 ultralight.

Airlines and destinations
Allegiant, American Airlines, and Delta Air Lines operate mainline jets into Key West, primarily the Airbus A319. JetBlue operates the Airbus A220. United Airlines operates the Boeing 737-700. American Eagle, Delta Connection, and United Express operate regional jets, including the CRJ700 and Embraer E-Jets. Silver Airways operates ATR 42 and ATR 72 turboprop aircraft.

Annual traffic
Key West's traffic was generally fairly stagnant to start the new millennium but gradually began increasing at the end of the 2000s with the addition of the new terminal and the introduction of low-cost jet service operated by AirTran, as well as mainline jet service by Delta.

When Southwest acquired AirTran in 2011, it continued to operate flights from the airport, first under the AirTran brand and then under the Southwest brand with Boeing 737-700 jetliners. Southwest Airlines ended service to Key West Airport in 2014.

Accidents and incidents

 * On April 25, 1959, a Vickers Viscount of Cubana de Aviación was hijacked on a flight from Varadero to Havana. The aircraft landed at Key West.
 * On March 19, 2003, Aerotaxi Flight 882, operated by Douglas DC-3C CU-T1192, was hijacked on a flight from Rafael Cabrera Airport in Nueva Gerona, Cuba to José Martí International Airport in Havana. The six hijackers were detained upon the plane's landing at Key West.
 * On April 2, 2003, a Cubana de Aviación flight scheduled from Siguanea Airport to José Martí International Airport, operated by a Antonov An-24, was hijacked and landed in Key West. As of 2021, the plane had remained in Key West and was being used as a training facility for fire and police.
 * On October 31, 2011, a Gulfstream G150 carrying NASCAR team owner Rick Hendrick ran off the end of Key West's runway after experiencing a loss in braking action upon landing. The jet, owned by NASCAR driver Jimmie Johnson, suffered nose gear damage.
 * On November 2, 2011, a Cessna Citation crash landed in Key West. The flight, which originated in Fort Lauderdale, had a brake failure upon landing. Two pilots were on board along with two passengers. Only minor injuries were reported. The aircraft was stopped by the airport's newly installed EMAS system.

title: Key West International Airport


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Maritime Museum of the Atlantic

The Maritime Museum of the Atlantic is a maritime museum located in downtown Halifax, Nova Scotia, Canada.

The museum is a member institution of the Nova Scotia Museum and is the oldest and largest maritime museum in Canada with a collection of over 30,000 artifacts including 70 small craft and a steamship: the CSS Acadia, a 180-foot steam-powered hydrographic survey ship launched in 1913.

History
The museum was founded in 1948. It was first known as the Maritime Museum of Canada and located at HMC Dockyard, the naval base on Halifax Harbour. Several naval officers served as volunteer chairs of the museum until 1959 when Niels Jannasch was hired as the museum's founding director, serving until 1985. The museum moved through several locations over the next three decades before its current building was constructed in 1981 as part of a waterfront redevelopment program. The museum received the CSS Acadia in 1982. Today the museum is part of the Nova Scotia Museum system.

Location
The museum was one of the first attractions to open on the redeveloped Halifax Waterfront. Its location provides the museum with several piers and boatsheds, as well as a strategic view of the Halifax Harbour, which looks seaward towards the Harbourmaster office and Georges Island and across to Dartmouth. Among its facilities is the restored 1880s Robertson Store ship chandler building, as well as modern exhibit galleries in the Devonian Wing (the modern museum building). HMCS Sackville, a World War II Flower-class corvette is docked adjacent to the museum in the summer months but is not owned or administered by the museum.

Collection
In addition to the over 30,000 artifacts, the museum also has a collection of 30,000 photographs as well as a large collection of charts and rare books. A reference library, open to the public, is named after the museum's founding director, The Niels Jannasch Library. The museum has Canada's largest collection of ship portraits including the oldest ship portrait in Canada as well as a large collection of ship models including the original production models of the television show Theodore Tugboat. Ongoing restoration of Whim, a 1937 C-class sloop can be found in one of the boatsheds on the wharf behind the museum. In addition to this current restoration project, the boatsheds house some of the museum's small craft collection. During the summer months three boats in the working small craft collection can be found moored next to the CSS Acadia. In July 2017 the museum also completed restoration of the small schooner Hebridee II.

Exhibits
Public galleries include the Days of Sail, the Age of Steam, Small Craft, the Canadian Navy, the Halifax Explosion, and Shipwrecks. A special permanent exhibit explores the sinking of the RMS Titanic with an emphasis on Nova Scotia's connection to recovering the bodies of Titanic victims. The museum has the world's foremost collection of wooden artifacts from Titanic, including one of the few surviving deck chairs. The Titanic exhibit also includes a child's pair of shoes, which helped identify Titanic's "unknown child" as Sidney Leslie Goodwin.

The adjacent exhibit "Shipwreck Treasures of Nova Scotia" explores the many other shipwrecks off the coast of Nova Scotia including archaeological discoveries on naval shipwrecks in Louisbourg Harbour and an unknown 1750s schooner at Lower Prospect, Nova Scotia. The results of treasure hunting are also featured in a section on Cape Breton treasure wrecks which has displays of weapons, instruments, gold and silver from wrecks such as the 1711 HMS Feversham, the 1725 wreck of Chameau and the 1761 wreck of the ship Auguste. The Age of Steam gallery includes a special display on Samuel Cunard, the Nova Scotian who created the Cunard Line. The restored 1880s Robertson building includes the fully restored Robertson ship chandlerly which features hands on foghorns, ropes and ship fittings.

The Navy gallery includes the "Convoy Exhibit" about the Battle of the Atlantic which includes the Canadian Merchant Navy Book of Remembrance. Monuments to the Canadian and Norwegian Merchant Navy are located just outside the museum along with a unique children's playground in the shape of a submarine.

The museum also has a changing exhibits gallery. A 2009 exhibit Ship of Fate: The Tragic Voyage of the St. Louis was the first Canadian exhibit to explore the 1939 voyage of the Jewish refugee ship MS St. Louis. The museum became the first museum in North America to present an exhibit about the lives of gay seafarers in 2011 when it presented Hello Sailor: Gay Life on the Ocean Waves, adapted from an exhibit developed at the Merseyside Maritime Museum in Liverpool, England. The 2012 exhibit explores the experiences of the cable ships based in Halifax who recovered most of the victims of the RMS Titanic sinking.

Vessels

 * CSS Acadia, served as research from 1913 to 1968 and as a WWI and WWII patrol ship
 * 70 small craft displayed in a Small Craft Gallery and boat sheds
 * Working collection vessels which sail from the museum wharves include the ketch Elson Perry, the sloop Windekilda; and S-Class sloop Valkyrie
 * Schooner Hebridee II

The corvette HMCS Sackville (K181) is not part of the Maritime Museum of the Atlantic but is located adjacent to the museum in the summer and works with the museum to interpret the Royal Canadian Navy.

Event location

The museum's location on the Halifax waterfront has made the museum the site of several significant public events. In addition to being a stop on most Canadian federal election campaigns, the museum hosted meetings from the 1995 G7 Summit, as well a September 11 commemorative event in 2006 attended by Canada's Minister of Foreign Affairs, Peter MacKay and United States Secretary of State, Condoleezza Rice. The museum hosts an annual commemoration of the Battle of the Atlantic on the first Sunday of every May and Canadian Merchant Navy day every September 3.

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

title: Maritime Museum of the Atlantic


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Antonin Nompar de Caumont, 1st Duke of Lauzun

Antonin Nompar de Caumont, 1st Duke of Lauzun (, 1632 – 19 November 1723) was a French courtier and soldier. He was the only love interest of the "greatest heiress in Europe", Anne Marie Louise d'Orléans, Duchess of Montpensier, cousin of Louis XIV.

He is often noted for his command of a French expeditionary Brigade which served alongside the Jacobite Irish Army during the Williamite War. After defeat at the Battle of the Boyne in July 1690 his brigade retreated to Galway where it was evacuated to France. Nonetheless, he remained an influential figure at the Jacobite court in exile, particularly with Mary of Modena.

Biography
He was the son of Gabriel de Caumont, Count of Lauzun, and his wife Charlotte, daughter of Henri Nompar de Caumont, Duke of La Force. He was brought up with the children of his relative, the maréchal-duc Antoine III de Gramont. One daughter, Catherine Charlotte, afterwards princess of Monaco by marriage to Prince Louis I of Monaco, was the object of the one passion of Lauzun's life.

He entered the army, and served under Turenne, also his kinsman, and in 1655 succeeded his father as commander of the cent gentilshommes de la maison de roi. Then known as the comte de Puyguilhem (or Péguilin, as contemporaries simplified his name), he rose rapidly in Louis XIV's favour, became colonel of the royal regiment of dragoons, and was gazetted maréchal de camp. He and Madame de Monaco belonged to the côterie of the young Duchess of Orléans. His rough wit and skill in practical jokes pleased Louis XIV, but his jealousy and violence were the causes of his undoing. He prevented a meeting between Louis XIV and Madame de Monaco, and jealousy in that matter, rather than hostility to Louise de La Vallière, led him to promote Madame de Montespan's intrigues with the king. He asked that lady to secure for him the post of grand-master of the artillery, and on Louis's refusal to give him the appointment, he turned his back on the king, broke his sword and swore that never again would he serve a monarch who had broken his word. The result was a short sojourn in the Bastille, but he soon returned to his function as court buffoon.

Meanwhile, Anne, Duchess of Montpensier (La Grande Mademoiselle) had fallen in love with the little man, whose ugliness seems to have exercised a certain fascination over many women. He naturally encouraged one of the greatest heiresses in Europe, and the wedding was fixed for December 21, 1670. Then, on the 18th, Louis sent for his cousin and forbade the marriage. Madame de Montespan had never forgiven Lauzun's fury when she failed to procure the grand-mastership of the artillery, and now, with Louvois, secured his arrest. He was removed in November 1671 from the Bastille to Pignerol, where excessive precautions were taken to ensure that he was firmly held. He was eventually allowed to meet another prisoner, Fouquet, but before that time he managed to find a way through the chimney into Fouquet's room, and on another occasion succeeded in reaching the courtyard in safety. Another fellow-prisoner, from communication with whom he was supposed to be rigorously excluded, was Eustache Dauger (see Man in the Iron Mask), who occasionally served as valet to Fouquet.

In his reports to his superiors in Paris, the prison governor, Bénigne Dauvergne de Saint-Mars, tells how Lauzun displayed evidence of deranged behaviour at this time: his cell was in constant disarray and he grew his beard to the point that it gave him a wild appearance. An attempted escape was foiled when, on emerging from his tunnel, he came across a maid who raised the alarm.

It was now intimated to Mademoiselle that Lauzun's restoration to liberty depended on her immediate settlement of the principality of Dombes, the county of Eu and the duchy of Aumale – three properties assigned by her to Lauzun – on Louis Auguste de Bourbon, Duke of Maine, eldest legitimised son of Louis XIV and Madame de Montespan. She gave way, but Lauzun, even after ten years of imprisonment, refused to sign the documents, when he was brought to Bourbon for the purpose. A short term of imprisonment at Chalon-sur-Saône made him change his mind, but when he was set free Louis XIV was still set against the marriage, which is supposed to have taken place secretly.

Married or not, Lauzun was openly courting Nicolas Fouquet's daughter, whom he had seen at Pignerol. He was to be restored to his place at court, and to marry Mademoiselle Fouquet, who, however, became Duchess d'Uzès in 1683.

In 1685 Lauzun went to England to seek his fortune under James II, whom he had served as Duke of York in Flanders. He rapidly gained great influence at the English court. In 1688 he was again in England, and arranged the journey into exile of Mary of Modena and the infant prince, whom he accompanied to Calais, where he received strict instructions from Louis to bring them "on any pretext" to Vincennes.

In the late autumn of 1689 he was put in command of the expedition fitted out at Brest for service in Ireland, and he sailed in the following year. Lauzun was honest, a quality not too common in James II's officials in Ireland, but had no experience of the field, and he blindly followed Richard Talbot, Earl of Tyrconnel. After the battle of the Boyne they fled to Limerick, and thence to the west, leaving Patrick Sarsfield to show a brave front. In September they sailed for France, and on their arrival at Versailles Lauzun found that his failure had destroyed any prospect of a return of Louis XIV's favour.

Mademoiselle died in 1693, and two years later Lauzun married Genevieve de Durfort, a child of fourteen, daughter of the maréchal-duc de Lorges. Queen Mary, through whose interest Lauzun secured his dukedom, retained her faith in him, and it was he who in 1715, more than a quarter of a century after the flight from Whitehall, brought her the news of the Battle of Sheriffmuir. After his death, the dukedom of Lauzun fell to his niece's husband, Charles Armand de Gontaut.

See the letters of Madame de Sévigné and the memoirs of Saint-Simon, who was Lauzun's wife's brother-in-law.


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Plasmodium

Plasmodium is a genus of unicellular eukaryotes that are obligate parasites of vertebrates and insects. The life cycles of Plasmodium species involve development in a blood-feeding insect host which then injects parasites into a vertebrate host during a blood meal. Parasites grow within a vertebrate body tissue (often the liver) before entering the bloodstream to infect red blood cells. The ensuing destruction of host red blood cells can result in malaria. During this infection, some parasites are picked up by a blood-feeding insect (mosquitoes in majority cases), continuing the life cycle.

Plasmodium is a member of the phylum Apicomplexa, a large group of parasitic eukaryotes. Within Apicomplexa, Plasmodium is in the order Haemosporida and family Plasmodiidae. Over 200 species of Plasmodium have been described, many of which have been subdivided into 14 subgenera based on parasite morphology and host range. Evolutionary relationships among different Plasmodium species do not always follow taxonomic boundaries; some species that are morphologically similar or infect the same host turn out to be distantly related.

Species of Plasmodium are distributed globally wherever suitable hosts are found. Insect hosts are most frequently mosquitoes of the genera Culex and Anopheles. Vertebrate hosts include reptiles, birds, and mammals. Plasmodium parasites were first identified in the late 19th century by Charles Laveran. Over the course of the 20th century, many other species were discovered in various hosts and classified, including five species that regularly infect humans: P. vivax, P. falciparum, P. malariae, P. ovale, and P. knowlesi. P. falciparum is by far the most lethal in humans, resulting in hundreds of thousands of deaths per year. A number of drugs have been developed to treat Plasmodium infection; however, the parasites have evolved resistance to each drug developed.

Although the parasite can also infect people via blood transfusion, this is very rare, and Plasmodium cannot be spread from person to person. Some of subspecies of Plasmodium are obligate intracellular parasites.

Description

The genus Plasmodium consists of all eukaryotes in the phylum Apicomplexa that both undergo the asexual replication process of merogony inside host red blood cells and produce the crystalline pigment hemozoin as a byproduct of digesting host hemoglobin. Plasmodium species contain many features that are common to other eukaryotes, and some that are unique to their phylum or genus. The Plasmodium genome is separated into 14 chromosomes contained in the nucleus. Plasmodium parasites maintain a single copy of their genome through much of the life cycle, doubling the genome only for a brief sexual exchange within the midgut of the insect host. Attached to the nucleus is the endoplasmic reticulum (ER), which functions similarly to the ER in other eukaryotes. Proteins are trafficked from the ER to the Golgi apparatus which generally consists of a single membrane-bound compartment in Apicomplexans. From here, proteins are trafficked to various cellular compartments or to the cell surface.

Like other apicomplexans, Plasmodium species have several cellular structures at the apical end of the parasite that serve as specialized organelles for secreting effectors into the host. The most prominent are the bulbous rhoptries which contain parasite proteins involved in invading the host cell and modifying the host once inside. Adjacent to the rhoptries are smaller structures termed micronemes that contain parasite proteins required for motility as well as recognizing and attaching to host cells. Spread throughout the parasite are secretory vesicles called dense granules that contain parasite proteins involved in modifying the membrane that separates the parasite from the host, termed the parasitophorous vacuole.

Species of Plasmodium also contain two large membrane-bound organelles of endosymbiotic origin, the mitochondrion and the apicoplast, both of which play key roles in the parasite's metabolism. Unlike mammalian cells which contain many mitochondria, Plasmodium cells contain a single large mitochondrion that coordinates its division with that of the Plasmodium cell. Like in other eukaryotes, the Plasmodium mitochondrion is capable of generating energy in the form of ATP via the citric acid cycle; however, this function is only required for parasite survival in the insect host, and is not needed for growth in red blood cells. A second organelle, the apicoplast, is derived from a secondary endosymbiosis event, in this case the acquisition of a red alga by the Plasmodium ancestor. The apicoplast is involved in the synthesis of various metabolic precursors, including fatty acids, isoprenoids, iron-sulphur clusters, and components of the heme biosynthesis pathway.

Life cycle

The life cycle of Plasmodium involves several distinct stages in the insect and vertebrate hosts. Parasites are generally introduced into a vertebrate host by the bite of an insect host (generally a mosquito, with the exception of some Plasmodium species of reptiles). Parasites first infect the liver or other tissue, where they undergo a single large round of replication before exiting the host cell to infect erythrocytes. At this point, some species of Plasmodium of primates can form a long-lived dormant stage called a hypnozoite. It can remain in the liver for more than a year. However, for most Plasmodium species, the parasites in infected liver cells are only what are called merozoites. After emerging from the liver, they enter red blood cells, as explained above. They then go through continuous cycles of erythrocyte infection, while a small percentage of parasites differentiate into a sexual stage called a gametocyte which is picked up by an insect host taking a blood meal. In some hosts, invasion of erythrocytes by Plasmodium species can result in disease, called malaria. This can sometimes be severe, rapidly followed by death of the host (e.g. P. falciparum in humans). In other hosts, Plasmodium infection can apparently be asymptomatic.

Even when humans have such subclinical plasmodial infections, there can nevertheless be very large numbers of multiplying parasites concealed in, particularly, the spleen and bone marrow. Certainly, this applies in the case of P. vivax. These hidden parasites (in addition to hypnozoites) are thought to be the origin of instances of recurrent P. vivax malaria.

Within the red blood cells, the merozoites grow first to a ring-shaped form and then to a larger form called a trophozoite. Trophozoites then mature to schizonts which divide several times to produce new merozoites. The infected red blood cell eventually bursts, allowing the new merozoites to travel within the bloodstream to infect new red blood cells. Most merozoites continue this replicative cycle, however some merozoites upon infecting red blood cells differentiate into male or female sexual forms called gametocytes. These gametocytes circulate in the blood until they are taken up when a mosquito feeds on the infected vertebrate host, taking up blood which includes the gametocytes.

In the mosquito, the gametocytes move along with the blood meal to the mosquito's midgut. Here the gametocytes develop into male and female gametes which fertilize each other, forming a zygote. Zygotes then develop into a motile form called an ookinete, which penetrates the wall of the midgut. Upon traversing the midgut wall, the ookinete embeds into the gut's exterior membrane and develops into an oocyst. Oocysts divide many times to produce large numbers of small elongated sporozoites. These sporozoites migrate to the salivary glands of the mosquito where they can be injected into the blood of the next host the mosquito bites, repeating the cycle.

Taxonomy
Plasmodium belongs to the phylum Apicomplexa, a taxonomic group of single-celled parasites with characteristic secretory organelles at one end of the cell. Within Apicomplexa, Plasmodium is within the order Haemosporida, a group that includes all apicomplexans that live within blood cells. Based on the presence of the pigment hemozoin and the method of asexual reproduction, the order is further split into four families, of which Plasmodium is in the family Plasmodiidae.

The genus Plasmodium consists of over 200 species, generally described on the basis of their appearance in blood smears of infected vertebrates. These species have been categorized on the basis of their morphology and host range into 14 subgenera:

 * Subgenus Asiamoeba (Telford, 1988) – reptiles
 * Subgenus Bennettinia (Valkiunas, 1997) – birds
 * Subgenus Carinamoeba (Garnham, 1966) – reptiles
 * Subgenus Giovannolaia (Corradetti, et al. 1963) – birds
 * Subgenus Haemamoeba (Corradetti, et al. 1963) – birds
 * Subgenus Huffia (Corradetti, et al. 1963) – birds
 * Subgenus Lacertamoeba (Telford, 1988) – reptiles
 * Subgenus Laverania (Bray, 1958) – great apes, humans
 * Subgenus Novyella (Corradetti, et al. 1963) – birds
 * Subgenus Ophidiella (Telford, 1988) – reptiles
 * Subgenus Paraplasmodium (Telford, 1988) – reptiles
 * Subgenus Plasmodium (Bray, 1955) – monkeys and apes
 * Subgenus Sauramoeba (Garnham, 1966) – reptiles
 * Subgenus Vinckeia (Garnham, 1964) – mammals inc. primates

Species infecting monkeys and apes with the exceptions of P. falciparum and P. reichenowi (which together make up the subgenus Laverania) are classified in the subgenus Plasmodium. Parasites infecting other mammals including some primates (lemurs and others) are classified in the subgenus Vinckeia. The five subgenera Bennettinia, Giovannolaia, Haemamoeba, Huffia, and Novyella contain the known avian malarial species. The remaining subgenera: Asiamoeba, Carinamoeba, Lacertamoeba, Ophidiella, Paraplasmodium, and Sauramoeba contain the diverse groups of parasites found to infect reptiles.

Phylogeny
More recent studies of Plasmodium species using molecular methods have implied that the group's evolution has not perfectly followed taxonomy. Many Plasmodium species that are morphologically similar or infect the same hosts turn out to be only distantly related. In the 1990s, several studies sought to evaluate evolutionary relationships of Plasmodium species by comparing ribosomal RNA and a surface protein gene from various species, finding the human parasite P. falciparum to be more closely related to avian parasites than to other parasites of primates. However, later studies sampling more Plasmodium species found the parasites of mammals to form a clade along with the genus Hepatocystis, while the parasites of birds or lizards appear to form a separate clade with evolutionary relationships not following the subgenera:

Estimates for when different Plasmodium lineages diverged have differed broadly. Estimates for the diversification of the order Haemosporida range from around 16.2 million to 100 million years ago. There has been particular interest in dating the divergence of the human parasite P. falciparum from other Plasmodium lineages due to its medical importance. For this, estimated dates range from 110,000 to 2.5 million years ago.

Distribution
Plasmodium species are distributed globally. All Plasmodium species are parasitic and must pass between a vertebrate host and an insect host to complete their life cycles. Different species of Plasmodium display different host ranges, with some species restricted to a single vertebrate and insect host, while other species can infect several species of vertebrates and/or insects.

Vertebrates

Plasmodium parasites have been described in a broad array of vertebrate hosts including reptiles, birds, and mammals. While many species can infect more than one vertebrate host, they are generally specific to one of these classes (such as birds).

Humans are primarily infected by five species of Plasmodium, with the overwhelming majority of severe disease and death caused by Plasmodium falciparum. Some species that infect humans can also infect other primates, and zoonoses of certain species (e.g. P. knowlesi) from other primates to humans are common. Non-human primates also contain a variety of Plasmodium species that do not generally infect humans. Some of these can cause severe disease in primates, while others can remain in the host for prolonged periods without causing disease. Many other mammals also carry Plasmodium species, such as a variety of rodents, ungulates, and bats. Again, some species of Plasmodium can cause severe disease in some of these hosts, while many appear not to.

Over 150 species of Plasmodium infect a broad variety of birds. In general each species of Plasmodium infects one to a few species of birds. Plasmodium parasites that infect birds tend to persist in a given host for years or for the life time of the host, although in some cases Plasmodium infections can result in severe illness and rapid death. Unlike with Plasmodium species infecting mammals, those infecting birds are distributed across the globe.

Species from several subgenera of Plasmodium infect diverse reptiles. Plasmodium parasites have been described in most lizard families and, like avian parasites, are spread worldwide. Again, parasites can result either in severe disease or be apparently asymptomatic depending on the parasite and the host.

A number of drugs have been developed over the years to control Plasmodium infection in vertebrate hosts, particularly in humans. Quinine was used as a frontline antimalarial from the 17th century until widespread resistance emerged in the early 20th century. Resistance to quinine spurred the development of a broad array of antimalarial medications through the 20th century including chloroquine, proguanil, atovaquone, sulfadoxine/pyrimethamine, mefloquine, and artemisinin. In all cases, parasites resistant to a given drug have emerged within a few decades of the drugs deployment. To combat this, antimalarial drugs are frequently used in combination, with artemisinin combination therapies currently the gold standard for treatment. In general, antimalarial drugs target the life stages of Plasmodium parasites that reside within vertebrate red blood cells, as these are the stages that tend to cause disease. However, drugs targeting other stages of the parasite life cycle are under development in order to prevent infection in travelers and to prevent transmission of sexual stages to insect hosts.

Insects

In addition to a vertebrate host, all Plasmodium species also infect a bloodsucking insect host, generally a mosquito (although some reptile-infecting parasites are transmitted by sandflies). Mosquitoes of the genera Culex, Anopheles, Culiseta, Mansonia and Aedes act as insect hosts for various Plasmodium species. The best studied of these are the Anopheles mosquitoes which host the Plasmodium parasites of human malaria, as well as Culex mosquitoes which host the Plasmodium species that cause malaria in birds. Only female mosquitoes are infected with Plasmodium, since only they feed on the blood of vertebrate hosts. Different species affect their insect hosts differently. Sometimes, insects infected with Plasmodium have reduced lifespan and reduced ability to produce offspring. Further, some species of Plasmodium appear to cause insects to prefer to bite infected vertebrate hosts over non-infected hosts.

History
Charles Louis Alphonse Laveran first described parasites in the blood of malaria patients in 1880. He named the parasite Oscillaria malariae. In 1885, zoologists Ettore Marchiafava and Angelo Celli reexamined the parasite and termed it a member of a new genus, Plasmodium, named for the resemblance to the multinucleate cells of slime molds of the same name. The fact that several species may be involved in causing different forms of malaria was first recognized by Camillo Golgi in 1886. Soon thereafter, Giovanni Batista Grassi and Raimondo Filetti named the parasites causing two different types of human malaria Plasmodium vivax and Plasmodium malariae. In 1897, William Welch identified and named Plasmodium falciparum. This was followed by the recognition of the other two species of Plasmodium which infect humans: Plasmodium ovale (1922) and Plasmodium knowlesi (identified in long-tailed macaques in 1931; in humans in 1965). The contribution of insect hosts to the Plasmodium life cycle was described in 1897 by Ronald Ross and in 1899 by Giovanni Batista Grassi, Amico Bignami and Giuseppe Bastianelli.

In 1966, Cyril Garnham proposed separating Plasmodium into nine subgenera based on host specificity and parasite morphology. This included four subgenera that had previously been proposed for bird-infecting Plasmodium species by A. Corradetti in 1963. This scheme was expanded upon by Sam R. Telford in 1988 when he reclassified Plasmodium parasites that infect reptiles, adding five subgenera. In 1997, G. Valkiunas reclassified the bird-infecting Plasmodium species adding a fifth subgenus: Bennettinia.

title: Plasmodium


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Andrew McCulloch (drummer)

Andrew McCulloch (born 19 November 1945, Bournemouth) is an English drummer who worked with Fields, Greenslade, Manfred Mann Chapter Three, Anthony Phillips, Peter Banks, the Crazy World of Arthur Brown and King Crimson in the 1970s before becoming a yachtmaster.

Biography
He spent his childhood in Hiroshima and elsewhere outside the UK. He began playing the drums at the age of 18 after returning to England. Joined Bournemouth's band Shy Limbs. He recorded for the first time in autumn 1968, and 9 May 1969 was his record debut.

After drumming for Manfred Mann in early 1970, he became Michael Giles' replacement in King Crimson. He appeared on just one King Crimson album, the jazz-influenced Lizard, released in 1970.

He joined Greenslade at its inception in 1972, remaining with them until their dissolution in 1976. He then guested on several releases, including "Opus One" with the London Philharmonic Orchestra, before leaving the music business. Despite at the time being referred to in The Guardian as being one of the most skillful and inventive drummers working anywhere in the jazz or rock spectrum, he left to become more involved with his other love, sailing.

As an RYA Yachtmaster instructor he skippered and taught on several yachts around Greece, Turkey, Cyprus, Italy, France, Spain, Lebanon and Antigua.

He now charters his own yachts around the Greek Islands in the summer season, as well as teaching, skippering and examining.

title: Andrew McCulloch (drummer)


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Françoise-Athénaïs de Rochechouart, Marquise de Montespan

Françoise-Athénaïs de Rochechouart de Mortemart, Marquise of Montespan (5 October 1640 – 27 May 1707), commonly known as Madame de Montespan, was a French noblewoman and the most celebrated royal mistress of King Louis XIV. During their romantic relationship, which lasted from the late 1660s to the late 1670s, she was sometimes referred to as the "true Queen of France" due to the pervasiveness of her influence at court.

Born into the House of Rochechouart, one of the oldest noble families of France, Françoise-Athénaïs married the Marquis of Montespan in 1663. She then became a maid of honour to Princess Henrietta, Duchess of Orléans, and later a lady-in-waiting to Queen Maria Theresa. Noted for her great beauty and wit, she carefully cultivated a relationship with Louis XIV and eventually supplanted Louise de La Vallière as his favourite. She had seven children by the king, six of which survived infancy and were later legitimised.

Madame de Montespan's alleged involvement in the Affair of the Poisons, which began in 1677, severely damaged her reputation and resulted in her fall from royal favour. In 1691, she withdrew to the convent of Filles de Saint-Joseph in Paris, but left thirteen years later when she relocated to the Château d'Oiron. She devoted the rest of her life to charity, patronage and penance until her death in 1707 at the age of 66. She is an ancestress of several royal houses in Europe, including those of Spain, Italy, Bulgaria, Portugal, Belgium and Luxembourg.

Early life
Françoise-Athénaïs was born in October 1640 and was baptised on 5 October 1640 at the Château of Lussac-les-Châteaux, today’s Vienne department, in the Nouvelle-Aquitaine region in France. Françoise (as a précieuse, she later adopted the name "Athénaïs"), or more formally, Mademoiselle de Tonnay-Charente, possessed the blood of two of the oldest noble families of France through her parents, Gabriel de Rochechouart, the Duke of Mortemart and the Prince of Tonnay-Charente, and Diane de Grandseigne, a lady-in-waiting to Anne of Austria, Queen consort of France.

From her father, she inherited the famous Mortemart esprit ("wit"). As a young girl, she often travelled with her mother between the family estates and the court at the Louvre in Paris. At the age of 12, she began her formal education at the Convent of St Mary at Saintes, where her sister Gabrielle had started hers almost a decade earlier. She was very religious and took Communion once a week, a practice that she would continue as a young woman.

At the age of 20, Françoise-Athénaïs became a maid-of-honour to the king's sister-in-law, Princess Henrietta, Duchess of Orléans, who was known at court by the traditional honorific of Madame. Later, because of the relationship between her mother and the queen dowager, Anne of Austria, Françoise-Athénaïs was appointed to be a lady-in-waiting to the king's wife, Maria Theresa of Spain.

Marriage
On 28 January 1663, Françoise-Athénaïs married Louis Henri de Pardaillan de Gondrin, Marquis of Montespan, who was one year her junior. Madame de La Fayette says in her Histoire de madame Henriette d'Angleterre that Françoise-Athénaïs was in love with another young man, Louis de La Trémoille, who was the elder son and heir to the Duc de Noirmoutier (one of the leaders of the Fronde). However, La Trémoille had to flee to Spain after a disastrous duel, and Françoise-Athénaïs was betrothed to Montespan. The wedding ceremony took place in a chapel at the Église Saint-Eustache, Paris. Françoise later recounted that as she had neglected to bring along the proper kneeling cushions for the ceremony, the couple had to kneel on dog cushions. She soon became pregnant with her first child, Christine. Two weeks after her daughter's birth she danced in a Court Ballet, and less than a year later her second child was born.

The couple lived in a small house close to the Louvre, which allowed Madame de Montespan to attend court and carry out her duties there as a lady-in-waiting to the Duchess of Orléans. She quickly established herself as the "reigning beauty of the court". Beauty, however, was only one of Madame de Montespan's many charms. She was a cultured and amusing conversationalist, who won the admiration of such literary figures as letter-writer Madame de Sévigné and diarist Saint-Simon. In addition, she kept abreast of political events. This had the effect of making her even more appealing to men of intellect and power. She was courted by a number of suitors including Louis de Buade de Frontenac and Charles Auguste de la Fare.

Rise as Maîtresse-en-titre
Madame de Montespan astounded the court by openly resenting the position of Queen Maria Theresa of Spain. The daughter of King Philip IV of Spain and Elisabeth de France, the Queen's Spanish title, before her marriage, was Infanta María Teresa de Austria. In France, she was known as Marie-Thérèse d'Autriche. A scandal arose when the Duchess of Montausier, Governess of the Children of France and lady-in-waiting to the Queen, was accused of acting as a go-between in order to secure the governorship of the Dauphin for her husband, the Duke of Montausier.

By 1666, Madame de Montespan was trying to take the place of Louis XIV’s current mistress, Louise de La Vallière. Using her wit and charm, she sought to ingratiate herself with the king. She also became close to the Dauphin, whose affection for her never wavered. Even though Louise de La Vallière knew that Montespan was trying to conquer the King's heart and reportedly laughed at her miserable efforts, she definitely underestimated her new rival. Montespan cleverly cultivated friendships with both Louise and Queen Maria Theresa and when both ladies were pregnant, Madame de Montespan was asked to help them entertain the King during private dinners. Soon they regretted their decision, for Montespan now cultivated an intimate relationship with the King. Madame de Montespan was also said to have seduced the King by dropping her towel obligingly when she spotted Louis spying on her while she showered. Shortly after, Louise's position was diminished to second place. To conceal his new relationship, the King placed the ladies in connected rooms so he could have access to both. Louise left court and joined a convent perhaps through regret and religiosity or because she had no other option. The spotlight belonged now to the twenty-five-year-old Athénaïs de Montespan.

She also became friends at court with another lady-in-waiting to the queen, Louise Boyer, the wife of Anne Jules de Noailles. Montespan's youngest son, the Count of Toulouse, would later marry one of Boyer's granddaughters.

Illegitimate children
The first of Madame de Montespan's seven children with the king was born in 1669. The newborn child, a girl, is thought to have been named Louise-Françoise. The upbringing of this first child (and subsequent children) was entrusted to one of Madame de Montespan's friends, Madame Scarron (the future marquise de Maintenon). A son, Louis-Auguste, was born in 1670. When the third child, Louis-César, was born in 1672, a house was purchased for Scarron and the children on the Rue Vaugirard.

In 1673, the couple's three living illegitimate children were legitimised by Louis XIV and given the royal surname of de Bourbon. The eldest, a son, Louis-Auguste de Bourbon, became the duc du Maine; the second child, a son, Louis-César de Bourbon, became the comte de Vexin; and the third, a daughter, Louise-Françoise de Bourbon, became Mademoiselle de Nantes and, in 1685, married the son of the head of the House of Condé, a cadet branch of the reigning House of Bourbon. As Madame de Montespan spent most of her time immersed in the social whirl of the court, the three had little contact with their busy mother and spent most of their childhood with their governess, Madame Scarron.

In 1674, an official separation with her husband was declared by the Procureur général Achille de Harlay, assisted by six judges at the Châtelet.

Due to her role in royal adultery, the Roman Catholic Church soon became her adversary. In 1675, the priest Lécuyer refused to give her absolution, which was necessary for her to take Easter communion, a requisite for all Catholics. Father Lécuyer raged,

"Is this the Madame that scandalises all France? Go abandon your shocking life and then come throw yourself at the feet of the ministers of Jesus Christ."

The King appealed to the priest's superiors, but the Church refused to yield to the king's demands. After a short separation, the King and Madame de Montespan resumed their relationship, resulting in the birth of two more children, Françoise Marie de Bourbon, Mademoiselle de Blois, in 1677, and Louis-Alexandre de Bourbon, comte de Toulouse, in 1678. Both were to be legitimised in 1681.

Affaire des Poisons
The Affaire des Poisons, which erupted in September 1677, was to be the beginning of the end of the reign of La Montespan. Suspicion that Madame de Montespan might be capable of murder or worse began when the King's eye strayed to another beauty, the Duchess of Fontanges. Madame de Montespan's relegation to the position of superintendent of the Queen's household as a result brought matters to a head. Before any further developments in her romance with the King could occur, Mlle de Fontanges died in 1681. Many at the time suspected that she had been poisoned by her rival, although none could prove it. It is now believed that Mlle de Fontanges died from natural causes.

Long assumed to have been involved in the infamous Affaire des Poisons, Madame de Montespan has never been conclusively implicated. Gabriel Nicolas de La Reynie, Paris' first Lieutenant General of Police and the chief judge of the court, before whom the famous poisoning cases were brought, heard testimony that placed Madame de Montespan's first visits to the so-called witch Catherine Monvoisin, known as La Voisin, in 1665. In 1666, Madame de Montespan supposedly went so far as to allow a priest, Étienne Guibourg, to perform a black mass over her nude body in a blood-soaked ceremony, which also, allegedly, included infant sacrifice. Whatever the truth in these allegations, in July 1667, Madame de Montespan became the king's new mistress even though Louise de La Vallière was carrying his child, Louis de Bourbon, comte de Vermandois.

In addition to seeking Louis' love, some charged Madame de Montespan with also conspiring to kill him, but inconsistencies in this testimony suggest that the royal mistress was innocent of these charges. However, suspicion was thrown onto Madame de Montespan because the name of her maid, Mlle Desœillets, was frequently mentioned in connection with La Voisin in the evidence brought before the Chambre Ardente.

Indeed, if anyone was attempting to kill the king, it was more likely Claude de Vin des Œillets, who had an illegitimate child fathered but not publicly acknowledged by Louis. Presumably, the maid resented the loss of Louis' attention. Olympia Mancini, Countess of Soissons, herself a former mistress of the king and well-known intrigante, was also implicated in the conspiracy.

From the end of 1680 onwards, Louvois, Jean-Baptiste Colbert, and Madame de Maintenon all helped to hush up the affair in order to prevent further scandal about the mother of the king's legitimised children. Concerning the king's need to avoid shocking scandal, Police Chief La Reynie said: "the enormity of their crimes proved their safeguard."

Exile
In 1691, no longer in royal favour, Madame de Montespan retired to the Filles de Saint-Joseph convent, in the rue Saint-Dominique in Paris, with a pension of half a million francs. In gratitude for her departure, the king made her brother the duc de Vivonne, a marshal of France. Louis had previously made one of her sisters, Gabrielle, whose vows were only four years old, the abbess of the wealthy Fontevraud Abbey. Gabrielle was appointed abbess in 1670 and it is said that her beauty, wit, and knowledge surpassed Athénaïs’ qualities. The King wanted her to stay for longer periods at court but Gabrielle always declined and stayed only for very short times.

In 1700, Madame de Montespan acquired the Château d'Oiron, thereby abandoning the hospice of Fontevraud from 1703 and moved there in 1704 after the death of her sister Gabrielle. In her long retirement, Madame de Montespan donated vast sums to hospitals and charities. She was also a generous patron of the arts and letters and befriended Corneille, Racine, and La Fontaine.

Death
The last years of Madame de Montespan's life were given up to a very severe penance. Real sorrow over her death was felt by her three youngest children. She died on 27 May 1707 at the age of almost sixty-seven while taking the waters at Bourbon-l'Archambault in order to try to heal an illness. The king forbade her children to wear mourning for her.

Appearance and personality
Athénaïs was considered "astonishingly beautiful" by the standards of her time. She had large blue eyes, long, thick, corn-coloured hair that fell in curls about her shoulders, and a curvaceous, voluptuous body. She was droll, amusing and used her considerable wit to mock others.

She also had an extravagant and demanding nature and possessed enough charm to get what she wanted. She was expensive and glorious, like the Palace of Versailles itself. Her apartments were filled with pet animals and thousands of flowers and she had a private gallery. Costly jewels were showered upon her and she was highly discriminating regarding the quality of the gems, returning them if they did not meet her exacting standards. She was given the nickname Quanto ("How much", in Italian). Her love for food and her numerous pregnancies caused her to gain weight in her late thirties.

Château de Clagny
The Château de Clagny in Versailles was built between 1674 and 1680 from the drawings of Jules Hardouin-Mansart, Premier architecte du Roi, (First architect of the King), on land bought by Louis XIV in 1665. Madame de Sévigné wrote that its construction employed 1,200 workers and the cost was no less than 2 million "livres". The royal gardener André Le Nôtre created the gardens, which looked west toward the much larger palace of Versailles, of which Clagny was a smaller version. The Château de Clagny was also famed for its gallery. In 1685, Louis XIV gave the magnificent palace to Madame de Montespan. At her death, her oldest son, the duc du Maine, inherited Clagny and, in turn, passed it on to his son, Louis-Auguste de Bourbon, prince de Dombes. The château reverted to the French crown in 1766 and was demolished in 1769.

Trianon de porcelaine
Louis XIV also had a pleasure pavilion, called the Trianon de Porcelaine near Versailles, purchased and built for Madame de Montespan, surrounded by gardens, on the site of the former hamlet of Trianon. It was meant as a hideaway for the couple. Because of the fragility of the earthenware tiles used in its construction, the Trianon de porcelaine was demolished in 1687 and replaced by the Grand Trianon of pink marble (marbre rose des Pyrénées).

In popular culture
In the 1979 comedy movie Love at First Bite, Dracula's servant Renfield mentions one of his master's past loves, the Countess de Montespan.


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La Voisin

Catherine Monvoisin, or Montvoisin, née Deshayes, known as "La Voisin" (c. 1640 – 22 February 1680), was a French fortune teller, commissioned poisoner, and professional provider of alleged sorcery. She was the head of a network of fortune tellers in Paris providing poison, aphrodisiacs, abortion, purported magical services and the arranging of black masses, with clients among the aristocracy and became the central figure in the famous affaire des poisons. Her purported organization of commissioned black magic and poison murder was suspected to have killed 1,000 people, but it is believed that upwards of 2,500 people might have been murdered.

Life
Little is known of Catherine Deshayes' early life. She learned fortune telling as a child and later married Antoine Monvoisin, who was active as a jeweller and silk merchant with a shop at Pont-Marie in Paris.

When her husband's trade business led to bankruptcy, La Voisin supported the family by practising chiromancy and face-reading. In addition to being a fortune teller, she was also active as a midwife, which developed into providing abortions. Her business as a fortune teller gradually developed into manufacturing and selling purported magical objects and potions, arranging black masses and selling aphrodisiacs and poison to profit from her clients' wishes upon their future.

From the late 1660s, La Voisin had become a wealthy and famous fortune teller with clients among the highest aristocracy of France. Among her clients were Olympia Mancini, comtesse de Soissons; Marie Anne Mancini, duchess de Bouillon; Elizabeth, comtesse de Gramont, and François-Henri de Montmorency, duc de Luxembourg.

She resided at Villeneuve-sur-Gravois, where she received her clients all day, and entertained the Parisian upper-class society at parties with violin music in her garden at night. La Voisin regularly attended service at the church of the Jansenist abbé de Sant-Amour, principal of the Paris University, and the godmother of her daughter was the noblewoman Mme de la Roche-Guyon.

She supported a family of six, including her husband, her mother and her children. She was known to have at least six lovers: the executioner Andre Guillaume, Monsieur Latour, vicomte de Cousserans, the count de Labatie, the alchemist Blessis, the architect Fauchet and the magician Adam Lesage. At one point, Adam Lesage tried to induce her to kill her husband, but while he was initially successful, she changed her mind and aborted the process.

La Voisin was interested in science and alchemy and financed several private projects and enterprises, some of them concocted by con artists who tried to swindle her. She was known to suffer from alcoholism, was apparently abused by Latour, and engaged in several conflicts with her rival, the poisoner Marie Bosse.

Abortion provider
La Voisin apparently started to include abortions, illegal at the time, for-profit within her services as a midwife, and her clients eventually included wealthy members of the aristocracy. She had a network of abortion providers working for her, notably Catherine Lepère, who stated that she received her clients from La Voisin, who referred clients to her and took the majority of the profit as a fee.

Marie Bosse claimed that fetuses, who had been aborted late in the pregnancy, were burned in a furnace at the house of La Voisin and buried in the garden of her house. However, as Louis XIV gave the order that the part of La Voisin's enterprise which had to do with abortions should not be pursued further, this part of her business is the least investigated and unknown one, and the claims of Marie Bosse therefore remain unconfirmed.

Fortune telling
La Voisin later said that as a fortune teller, she had merely used and developed what God had given her. She stated that she was taught the art of fortune telling at the age of nine, and that after her husband became ruined, she decided to profit by it.

She developed her art by studying the modern methods of physiology, and the art of reading the client's future by studying their face and hands.

She spent a great deal of money in order to provide an atmosphere which would make her clients more inclined to believe her prophecies: for example, she had a special robe of crimson red velvet embroidered with eagles in gold made for a price of 1,500 livres to perform in.

In 1665 or 1666, her divination was questioned by the Congregation of the Mission at the Saint Vincent de Paul's order and she was called for questioning, but La Voisin defended herself successfully before the professors at Sorbonne University and was allowed to continue her business as a fortune teller.

Professional sorcery
Her business as a fortune teller gradually developed into a business of professional alleged black magic. During her activity as a fortune teller, she noticed similarities among her clients' wishes about their future: almost all wanted to have someone fall in love with them, that someone would die so that they might inherit, or that their spouses would die so that they might marry someone else.

La Voisin decided to profit financially from her clients' wishes by offering services of purported magic to make their wishes come true. Initially, she told her clients that their wish would come true if it was also the will of God. Secondly, she started to recommend to her clients some actions that would make their dreams come true. These actions were initially to visit the church of some particular saint; then she started to sell amulets, and gradually, she recommended more and more alleged magical objects or rituals of various kinds.

For those clients who wished for someone to fall in love with them, she manufactured love powders: the bones of toads, the teeth of moles, Spanish fly, iron filings, human blood and mummy, and the dust of human remains were among the alleged ingredients of the love powders concocted by La Voisin.

Finally, her most radical and expensive recommended practices were the black mass, which she arranged for clients for profit, during which the client could pray to Satan for their wish to come true. During some of these masses, a woman performed as an altar, upon which a bowl was placed: a baby was held above the bowl, and the blood from it was poured into the bowl. Whether the baby in question was actually killed on this occasion, or whether the baby was already dead by natural causes, such as being stillborn, could vary.

She had several associates working for her arranging and participating in her professional magic services, notably Adam Lesage, who performed alleged magical tasks; and the priest Étienne Guibourg and abbé Mariotte, who officiated at the black masses.

Poison commission
La Voisin finally took the step from purported magical potions, amulets and rituals, to selling aphrodisiacs to those who wished for someone to fall in love with them, and fatal poison was given to those who wished for someone to die.

The art of poisoning had become a regular science at the time, having been perfected, in part, by Giulia Tofana, a professional female poisoner in Italy, only a few decades before La Voisin. La Voisin provided a large variety of poisons for her clients and had a network of poison providers working for her, notably the apothecary Catherine Trianon.

Connection to Madame de Montespan

The most important client of La Voisin was Madame de Montespan, the official royal mistress to King Louis XIV of France. Their contact was often performed through the companion of Montespan, Claude de Vin des Œillets. Montespan was alleged to have hired La Voisin in 1667 to arrange a black mass. This mass was celebrated in a house in Rue de la Tannerie. Adam Lesage and abbé Mariotte officiated, while Montespan prayed to win the love of the king. The same year, Montespan became the official mistress of the King, and after this, she employed La Voisin whenever a problem occurred in her relationship with the King.

In 1673, when the King's interest in Montespan seemed to wane, Montespan again employed La Voisin, who provided a series of black masses officiated by Étienne Guibourg. On at least one occasion, Montespan herself acted as the human altar during the mass. La Voisin also provided Montespan with an aphrodisiac, with which Montespan drugged the King. During the King's affair with Madame de Soubise, Montespan used an aphrodisiac provided by Voisin's colleague Françoise Filastre and made by Louis Galet in Normandy.

In 1677, Montespan made it clear that if the King should abandon her, she would have him killed. When the King entered into a relationship with Angélique de Fontanges in 1679, Montespan called for La Voisin and asked her to have both the King and Fontanges killed. La Voisin hesitated but was eventually convinced to agree. At the house of her colleague, Catherine Trianon, La Voisin constructed a plan to kill the King together with the poisoners Trianon, Bertrand and Romani, the last being the fiancé of her daughter. Trianon was unwilling to participate and tried to make her change her mind by constructing an ill-fated fortune for her, but Voisin refused to change her mind. The group decided to murder the King by poisoning a petition, to be delivered to his own hands.

On 5 March 1679, La Voisin visited the royal court in Saint-Germain to deliver the petition. That day, however, there were too many petitioners and the King did not take their petitions, which foiled her plan. Upon her return to her home in Paris, she was castigated by a group of monks. She handed the petition to her daughter Marguerite Montvoisin and asked her to burn it, which she also did. The next day, she made plans to visit Catherine Trianon after mass, to plan the next murder attempt upon Louis XIV.

Arrest
The death of the King's sister-in-law, the duchesse d'Orléans, had been falsely attributed to poison, and the crimes of Madame de Brinvilliers (executed in 1676) and her accomplices were still fresh in the public mind. In parallel, a riot took place where people accused witches of abducting children for the black masses, and priests reported that a growing number of people were confessing to poisoning in their confessions.

In 1677, the fortune teller Magdelaine de La Grange was arrested for poisoning, and claimed that she had information about crimes of high importance. The arrest of the successful fortune teller and poisoner Marie Bosse and Marie Vigoreaux in January 1679 made the police aware that there existed a network of fortune tellers in Paris who dealt in the distribution of poison.

On 12 March 1679, La Voisin was arrested outside Notre-Dame de Bonne-Nouvelle after having heard mass, just before her meeting with Catherine Trianon. In April 1679, a commission appointed to inquire into the subject and to prosecute the offenders met for the first time. Its proceedings, including some suppressed in the official records, are preserved in the notes of one of the official court reporters, Gabriel Nicolas de la Reynie.

Investigation and trial
At the arrest of La Voisin, her maid Margot stated that the arrest would mean the end of a number of people at all levels of society. La Voisin was imprisoned at Vincennes, where she was subjected to questioning. On 27 December 1679, Louis XIV issued an order that the whole network should be exterminated by all methods regardless of rank, sex, or age. The arrest of La Voisin was followed by the arrest of her daughter Marguerite Monvoisin, Guibourg, Lesage, Bertrand, Romani and the rest of her network of associates.

La Voisin was never subjected to torture: although a formal order was issued permitting the use of torture, it was made clear that the order was not to be put in effect, and consequently, it was never made use of. The reason is suggested to have been the fear that she might reveal the names of influential people if she was questioned under torture. Aware of her alcoholism, however, her interrogators reportedly did not limit her access to alcohol but instead kept her in a state of drunkenness during her interrogations. Her confrontations with other accused, particularly those with Adam Lesage and Marie Bosse, was especially effective. Initially, she claimed to have referred all those clients wishing to buy poison to Marie Bosse. In March, however, she named Marguerite Leféron and Francoise de Dreux as clients, and on 10 October, she admitted having sold poison and magical services to several members of the royal court; she also described the development of her career.

La Voisin denied ever meeting or serving Montespan. She did admit that "Paris is full of this kind of thing and there is an infinite number of people engaged in this evil trade" but refused to mention further clients nor did she mention having arranged or participated in black masses. She once mentioned to the guards that the question she feared most was that they would ask her about her visits to the royal court. It is likely that she was referring to Montespan as her client and her attempt at murdering the King, and that she feared that such a confession should result in her execution for regicide. Her list of clients, the arranging of the black masses, her connection to Montespan and the murder attempt on the King were not revealed until after her death, when it was stated by her daughter and confirmed by the testimonies of her former associates.

She also denied claims of using her oven to burn the bones of aborted infants that were too large for the grave area in her garden. She also counterclaimed any usage of poisons and that she only possessed purgatives for personal use or for her family. Another accusation she denied includes injecting syringes with fatal liquids into the bodies of pregnant women to "empty" them, after which the aborted foetuses were subsequently buried in the same manner at the garden grave.

Trial and execution
On 17 February 1680, La Voisin was put on trial, and was convicted of witchcraft and sentenced to execution by burning two days later. In the days following her verdict until her execution, she was formally interrogated under torture, and the official document stated that she was tortured enough to beg for mercy. However, while an official permit for torture was issued, the police were given to understand that the permit was not to be used, and Gabriel Nicolas de la Reynie stated that La Voisin was in reality never subjected to torture. Madame de Sevigne, who observed La Voisin shortly before her execution, described her as being able to move so freely and so seemingly healthy that it did not seem possible for her to have been subjected to torture. These final interrogations did not reveal anything new.

La Voisin was executed in public on the Place de Grève in Paris on 22 February 1680. On her way to her execution, she reportedly pushed away the priest, and when fastened on the stake, she desperately pushed away the hay which was piled up around her.

In July, her daughter Marguerite Monvoisin revealed her connection to Montespan, which was confirmed by the statements of the other accused. This caused the monarch to eventually close the investigation, seal the testimonies and place the remaining accused outside of the public justice system by imprisoning them under a lettre de cachet.

Film

 * Viviane Romance portrayed La Voisin in the 1955 film The Affair of Poisons.
 * Anémone portrayed La Voisin in the 1997 film Marquise.
 * La Voisin was the basis for the character portrayed by Suzanne Clément in the 2017 TV series Versailles.

Literature
La Voisin and her organisation have been described in several literary works:
 * W. Branch Johnson: "'The Age Of Arsenic'. Being an account of the life, trial and execution of Catherine Montvoison, known as La Voison, and of her vile associates and credulous Clients of both high and low Degree: together with the Relation of their various Transactions in Poison, Abortion, and Black or Satanic Masses, with other Details concerning sundry Manners and Habits of the Times and with but little Moralising thereon: the Whole comprising a curious and momentous Episode in the Reign of King Louis XIV of France" (1932).
 * She is also mentioned in Anne Rice's novel "The Vampire Lestat."
 * Brad Stieger: 'The Hypnotist' (1979)
 * Judith Merkle Riley: The Oracle Glass (1994)
 * Anne en Serge Golon: Angélique en de koning
 * Shelby Mahurin: Serpent & Dove
 * Danielle L. Jensen: Hidden Huntress
 * Addie Thorley: An Affair of Poisons
 * Lana Popovic:Poison Priestess

Manga
ボアザン (Boazan), by Takatoo Rui, is a manga partially based on La Voisin.

Music
Belgian black metal band Enthroned use La Voisin as the basis for the song "Graced by Evil Blood" on their 2002 album Carnage in Worlds Beyond.

title: La Voisin


text:
Talk:Rise of the New Imperialism

Why don't you try naming this page as the 'History of New Imperialism' rather than 'Rise...' so that it justifies sections before, after, and during especially? KRS

Move to Rise of the New Imperialism? --Jiang

Causes of Imperialism Causes of New Imperialism

Unreadable paragraph
I have removed the last paragraph in Securing foreign trade is unreadable. It is one long sentence with nonsensical grammar:
 * In these non-industrial regions (such as the Russian and Ottoman empires), which were the principal sources of surplus French capital, and other overseas territories that lacked both the knowledge and the power to direct the capital flow, served to colonies rather than develop them, destroying native industries and creating dangerous political and economic pressures which would, in time, produce the so-called 'north/south divide'.

I'm not qualified to fix it myself. AdamRetchless 21:57, 4 Apr 2005 (UTC)

Links
i added some links..... - Bagel7

title: Talk:Rise of the New Imperialism


text:
Talk:TS

I have changed this page to redirect to the article Tourette syndrome instead of the User:TimShell as TS appears to be a fairly common abbreviation of the disorder and is arguably what more people are looking for when they type in "TS"

What does TS mean?
What does TS mean, I see it in role plays all the time. What does it mean. I’m bored of wondering what it means 209.52.88.22 (talk) 07:41, 27 December 2021 (UTC)

Taylor Swift?
Why not?? 2600:6C50:787F:FF3B:6C09:3AEB:899:93EC (talk) 11:35, 14 November 2022 (UTC)

I second this 71.83.192.5 (talk) 04:27, 9 April 2023 (UTC)

 * Because MOS:DABINITIALS (see also: WP:DABABBREV). Just because someone's initials are TS doesn't mean they're widely known by those initials. Hoof Hearted (talk) 19:07, 23 February 2024 (UTC)

title: Talk:TS


text:
Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.

Statements
For functions of a single variable, the theorem states that if $$f$$ is a continuously differentiable function with nonzero derivative at the point $$a$$; then $$f$$ is injective (or bijective onto the image) in a neighborhood of $$a$$, the inverse is continuously differentiable near $$b=f(a)$$, and the derivative of the inverse function at $$b$$ is the reciprocal of the derivative of $$f$$ at $$a$$: $$\bigl(f^{-1}\bigr)'(b) = \frac{1}{f'(a)} = \frac{1}{f'(f^{-1}(b))}.$$

It can happen that a function $$f$$ may be injective near a point $$a$$ while $$f'(a) = 0$$. An example is $$f(x) = (x - a)^3$$. In fact, for such a function, the inverse cannot be differentiable at $$b = f(a)$$, since if $$f^{-1}$$ were differentiable at $$b$$, then, by the chain rule, $$1 = (f^{-1} \circ f)'(a) = (f^{-1})'(b)f'(a)$$, which implies $$f'(a) \ne 0$$. (The situation is different for holomorphic functions; see below.)

For functions of more than one variable, the theorem states that if $$f$$ is a continuously differentiable function from an open subset $$A$$ of $$\mathbb{R}^n$$ into $$\R^n$$, and the derivative $$f'(a)$$ is invertible at a point $a$ (that is, the determinant of the Jacobian matrix of $a$ at $a$ is non-zero), then there exist neighborhoods $$U$$ of $$a$$ in $$A$$ and $$V$$ of $$b = f(a)$$ such that $$f(U) \subset V$$ and $$f : U \to V$$ is bijective. Writing $$f=(f_1,\ldots,f_n)$$, this means that the system of $a$ equations $$y_i = f_i(x_1, \dots, x_n)$$ has a unique solution for $$x_1, \dots, x_n$$ in terms of $$y_1, \dots, y_n$$ when $$x \in U, y \in V$$. Note that the theorem does not say $$f$$ is bijective onto the image where $$f'$$ is invertible but that it is locally bijective where $$f'$$ is invertible.

Moreover, the theorem says that the inverse function $$f^{-1} : V \to U$$ is continuously differentiable, and its derivative at $$b=f(a)$$ is the inverse map of $$f'(a)$$; i.e.,
 * $$(f^{-1})'(b) = f'(a)^{-1}.$$

In other words, if $$Jf^{-1}(b), Jf(a)$$ are the Jacobian matrices representing $$(f^{-1})'(b), f'(a)$$, this means:
 * $$Jf^{-1}(b) = Jf(a)^{-1}.$$

The hard part of the theorem is the existence and differentiability of $$f^{-1}$$. Assuming this, the inverse derivative formula follows from the chain rule applied to $$f^{-1}\circ f = I$$. (Indeed, $$1=I'(a) = (f^{-1} \circ f)'(a) = (f^{-1})'(b) \circ f'(a).$$) Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if $$f$$ is continuously $$k$$ times differentiable, with invertible derivative at the point $f$, then the inverse is also continuously $$k$$ times differentiable. Here $$k$$ is a positive integer or $$\infty$$.

There are two variants of the inverse function theorem. Given a continuously differentiable map $$f : U \to \mathbb{R}^m$$, the first is and the second is
 * The derivative $$f'(a)$$ is surjective (i.e., the Jacobian matrix representing it has rank $$m$$) if and only if there exists a continuously differentiable function $$g$$ on a neighborhood $$V$$ of $$b = f(a)$$ such $$f \circ g = I$$ near $$b$$,
 * The derivative $$f'(a)$$ is injective if and only if there exists a continuously differentiable function $$g$$ on a neighborhood $$V$$ of $$b = f(a)$$ such $$g \circ f = I$$ near $$a$$.

In the first case (when $$f'(a)$$ is surjective), the point $$b = f(a)$$ is called a regular value. Since $$m = \dim \ker(f'(a)) + \dim \operatorname{im}(f'(a))$$, the first case is equivalent to saying $$b = f(a)$$ is not in the image of critical points $$a$$ (a critical point is a point $$a$$ such that the kernel of $$f'(a)$$ is nonzero). The statement in the first case is a special case of the submersion theorem.

These variants are restatements of the inverse functions theorem. Indeed, in the first case when $$f'(a)$$ is surjective, we can find an (injective) linear map $$T$$ such that $$f'(a) \circ T = I$$. Define $$h(x) = a + Tx$$ so that we have:
 * $$(f \circ h)'(0) = f'(a) \circ T = I.$$

Thus, by the inverse function theorem, $$f \circ h$$ has inverse near $$0$$; i.e., $$f \circ h \circ (f \circ h)^{-1} = I$$ near $$b$$. The second case ($$f'(a)$$ is injective) is seen in the similar way.

Example
Consider the vector-valued function $$F:\mathbb{R}^2\to\mathbb{R}^2\!$$ defined by:

F(x,y)= \begin{bmatrix} {e^x \cos y}\\ {e^x \sin y}\\ \end{bmatrix}. $$ The Jacobian matrix is:

J_F(x,y)= \begin{bmatrix} {e^x \cos y} & {-e^x \sin y}\\ {e^x \sin y} & {e^x \cos y}\\ \end{bmatrix} $$ with Jacobian determinant:

\det J_F(x,y)= e^{2x} \cos^2 y + e^{2x} \sin^2 y= e^{2x}. \,\!$$

The determinant $$e^{2x}\!$$ is nonzero everywhere. Thus the theorem guarantees that, for every point $a$ in $$\mathbb{R}^2\!$$, there exists a neighborhood about $n$ over which $a$ is invertible. This does not mean $p$ is invertible over its entire domain: in this case $p$ is not even injective since it is periodic: $$F(x,y)=F(x,y+2\pi)\!$$.

Counter-example
If one drops the assumption that the derivative is continuous, the function no longer need be invertible. For example $$f(x) = x + 2x^2\sin(\tfrac1x)$$ and $$f(0)= 0$$ has discontinuous derivative $$f'\!(x) = 1 -2\cos(\tfrac1x) + 4x\sin(\tfrac1x)$$ and $$f'\!(0) = 1$$, which vanishes arbitrarily close to $$x=0$$. These critical points are local max/min points of $$f$$, so $$f$$ is not one-to-one (and not invertible) on any interval containing $$x=0$$. Intuitively, the slope $$f'\!(0)=1$$ does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.

Methods of proof
As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed-point theorem (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations).

Since the fixed point theorem applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem (see Generalizations below).

An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set. This approach has an advantage that the proof generalizes to a situation where there is no Cauchy completeness (see ).

Yet another proof uses Newton's method, which has the advantage of providing an effective version of the theorem: bounds on the derivative of the function imply an estimate of the size of the neighborhood on which the function is invertible.

A proof using successive approximation
To prove existence, it can be assumed after an affine transformation that $$f(0)=0$$ and $$f^\prime(0)=I$$, so that $$ a=b=0$$.

By the mean value theorem for vector-valued functions, for a differentiable function $$u:[0,1]\to\mathbb R^m$$, $\|u(1)-u(0)\|\le \sup_{0\le t\le 1} \|u^\prime(t)\|$. Setting $$u(t)=f(x+t(x^\prime -x)) - x-t(x^\prime-x)$$, it follows that

 * $$\|f(x) - f(x^\prime) - x + x^\prime\| \le \|x -x^\prime\|\,\sup_{0\le t \le 1} \|f^\prime(x+t(x^\prime -x))-I\|.$$

Now choose $$\delta>0$$ so that $\|f'(x) - I\| < {1\over 2}$ for $$\|x\|< \delta$$. Suppose that $$\|y\|<\delta/2$$ and define $$x_n$$ inductively by $$x_0=0$$ and $$ x_{n+1}=x_n + y - f(x_n)$$. The assumptions show that if $$ \|x\|, \,\, \|x^\prime\| < \delta$$ then

 * $$\|f(x)-f(x^\prime) - x + x^\prime\| \le \|x-x^\prime\|/2$$.

In particular $$f(x)=f(x^\prime)$$ implies $$x=x^\prime$$. In the inductive scheme $$\|x_n\| <\delta$$ and $$\|x_{n+1} - x_n\| < \delta/2^n$$. Thus $$(x_n)$$ is a Cauchy sequence tending to $$x$$. By construction $$f(x)=y$$ as required.

To check that $$g=f^{-1}$$ is C1, write $$g(y+k) = x+h$$ so that $$f(x+h)=f(x)+k$$. By the inequalities above, $$\|h-k\| <\|h\|/2$$ so that $$\|h\|/2<\|k\| < 2\|h\|$$. On the other hand if $$A=f^\prime(x)$$, then $$\|A-I\|<1/2$$. Using the geometric series for $$B=I-A$$, it follows that $$\|A^{-1}\| < 2$$. But then

 * $$ {\|g(y+k) -g(y) - f^\prime(g(y))^{-1}k \| \over \|k\|}

= {\|h -f^\prime(x)^{-1}[f(x+h)-f(x)]\| \over \|k\|} \le 4 {\|f(x+h) - f(x) -f^\prime(x)h\|\over \|h\|} $$

tends to 0 as $$k$$ and $$h$$ tend to 0, proving that $$g$$ is C1 with $$g^\prime(y)=f^\prime(g(y))^{-1}$$.

The proof above is presented for a finite-dimensional space, but applies equally well for Banach spaces. If an invertible function $$f$$ is Ck with $$k>1$$, then so too is its inverse. This follows by induction using the fact that the map $$F(A)=A^{-1}$$ on operators is Ck for any $$k$$ (in the finite-dimensional case this is an elementary fact because the inverse of a matrix is given as the adjugate matrix divided by its determinant). The method of proof here can be found in the books of Henri Cartan, Jean Dieudonné, Serge Lang, Roger Godement and Lars Hörmander.

A proof using the contraction mapping principle
Here is a proof based on the contraction mapping theorem. Specifically, following T. Tao, it uses the following consequence of the contraction mapping theorem.

Basically, the lemma says that a small perturbation of the identity map by a contraction map is injective and preserves a ball in some sense. Assuming the lemma for a moment, we prove the theorem first. As in the above proof, it is enough to prove the special case when $$a = 0, b = f(a) = 0$$ and $$f'(0) = I$$. Let $$g = f - I$$. The mean value inequality applied to $$t \mapsto g(x + t(y - x))$$ says:
 * $$|g(y) - g(x)| \le |y-x|\sup_{0 < t < 1} |g'(x + t(y - x))|.$$

Since $$g'(0) = I - I = 0$$ and $$g'$$ is continuous, we can find an $$r > 0$$ such that
 * $$|g(y) - g(x)| \le 2^{-1}|y-x|$$

for all $$x, y$$ in $$B(0, r)$$. Then the early lemma says that $$f = g + I$$ is injective on $$B(0, r)$$ and $$B(0, r/2) \subset f(B(0, r))$$. Then
 * $$f : U = B(0, r) \cap f^{-1}(B(0, r/2)) \to V = B(0, r/2)$$

is bijective and thus has an inverse. Next, we show the inverse $$f^{-1}$$ is continuously differentiable (this part of the argument is the same as that in the previous proof). This time, let $$g = f^{-1}$$ denote the inverse of $$f$$ and $$A = f'(x)$$. For $$x = g(y)$$, we write $$g(y + k) = x + h$$ or $$y + k = f(x+h)$$. Now, by the early estimate, we have
 * $$|h - k| = |f(x+h) - f(x) - h| \le |h|/2$$

and so $$|h|/2 \le |k|$$. Writing $$\| \cdot \|$$ for the operator norm,
 * $$|g(y+k) - g(y) - A^{-1} k| = |h - A^{-1}(f(x + h) - f(x))| \le \|A^{-1}\||Ah - f(x+h) + f(x)|.$$

As $$k \to 0$$, we have $$h \to 0$$ and $$|h|/|k|$$ is bounded. Hence, $$g$$ is differentiable at $$y$$ with the derivative $$g'(y) = f'(g(y))^{-1}$$. Also, $$g'$$ is the same as the composition $$\iota \circ f' \circ g$$ where $$\iota : T \mapsto T^{-1}$$; so $$g'$$ is continuous.

It remains to show the lemma. First, we have:
 * $$|x - y| - |f(x) - f(y)| \le |g(x) - g(y)| \le c|x - y|,$$

which is to say
 * $$(1 - c)|x - y| \le |f(x) - f(y)|.$$

This proves the first part. Next, we show $$f(B(0, r)) \supset B(0, (1-c)r)$$. The idea is to note that this is equivalent to, given a point $$y$$ in $$B(0, (1-c) r)$$, find a fixed point of the map
 * $$F : \overline{B}(0, r') \to \overline{B}(0, r'), \, x \mapsto y - g(x)$$

where $$0 < r' < r$$ such that $$|y| \le (1-c)r'$$ and the bar means a closed ball. To find a fixed point, we use the contraction mapping theorem and checking that $$F$$ is a well-defined strict-contraction mapping is straightforward. Finally, we have: $$f(B(0, r)) \subset B(0, (1+c)r)$$ since
 * $$|f(x)| = |x + g(x) - g(0)| \le (1+c)|x|. \square$$

As might be clear, this proof is not substantially different from the previous one, as the proof of the contraction mapping theorem is by successive approximation.

Implicit function theorem
The inverse function theorem can be used to solve a system of equations
 * $$\begin{align}

&f_1(x) = y_1 \\ &\quad \vdots\\ &f_n(x) = y_n,\end{align}$$ i.e., expressing $$y_1, \dots, y_n$$ as functions of $$x = (x_1, \dots, x_n)$$, provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations:
 * $$\begin{align}

&f_1(x, y) = 0 \\ &\quad \vdots\\ &f_n(x, y) = 0\end{align}$$ for $$y$$ in terms of $$x$$. Though more general, the theorem is actually a consequence of the inverse function theorem. First, the precise statement of the implicit function theorem is as follows: To see this, consider the map $$F(x, y) = (x, f(x, y))$$. By the inverse function theorem, $$F : U \times V \to W$$ has the inverse $$G$$ for some neighborhoods $$U, V, W$$. We then have:
 * given a map $$f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m$$, if $$f(a, b) = 0$$, $$f$$ is continuously differentiable in a neighborhood of $$(a, b)$$ and the derivative of $$y \mapsto f(a, y)$$ at $$b$$ is invertible, then there exists a differentiable map $$g : U \to V$$ for some neighborhoods $$U, V$$ of $$a, b$$ such that $$f(x, g(x)) = 0$$. Moreover, if $$f(x, y) = 0, x \in U, y \in V$$, then $$y = g(x)$$; i.e., $$g(x)$$ is a unique solution.
 * $$(x, y) = F(G_1(x, y), G_2(x, y)) = (G_1(x, y), f(G_1(x, y), G_2(x, y)),$$

implying $$x = G_1(x, y)$$ and $$y = f(x, G_2(x, y)).$$ Thus $$g(x) = G_2(x, 0)$$ has the required property. $$\square$$

Giving a manifold structure
In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold. Indeed, let $$f : U \to \mathbb{R}^r$$ be such a smooth map from an open subset of $$\mathbb{R}^n$$ (since the result is local, there is no loss of generality with considering such a map). Fix a point $$a$$ in $$f^{-1}(b)$$ and then, by permuting the coordinates on $$\mathbb{R}^n$$, assume the matrix $$\left [ \frac{\partial f_i}{\partial x_j}(a) \right]_{1 \le i, j \le r}$$ has rank $$r$$. Then the map $$F : U \to \mathbb{R}^r \times \mathbb{R}^{n-r} = \mathbb{R}^n, \, x \mapsto (f(x), x_{r+1}, \dots, x_n)$$ is such that $$F'(a)$$ has rank $$n$$. Hence, by the inverse function theorem, we find the smooth inverse $$G$$ of $$F$$ defined in a neighborhood $$V \times W$$ of $$(b, a_{r+1}, \dots, a_n)$$. We then have
 * $$x = (F \circ G)(x) = (f(G(x)), G_{r+1}(x), \dots, G_n(x)),$$

which implies
 * $$(f \circ G)(x_1, \dots, x_n) = (x_1, \dots, x_r).$$

That is, after the change of coordinates by $$G$$, $$f$$ is a coordinate projection (this fact is known as the submersion theorem). Moreover, since $$G : V \times W \to U' = G(V \times W)$$ is bijective, the map
 * $$g = G(b, \cdot) : W \to f^{-1}(b) \cap U', \, (x_{r+1}, \dots, x_n) \mapsto G(b, x_{r+1}, \dots, x_n)$$

is bijective with the smooth inverse. That is to say, $$g$$ gives a local parametrization of $$f^{-1}(b)$$ around $$a$$. Hence, $$f^{-1}(b)$$ is a manifold. $$\square$$ (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.)

More generally, the theorem shows that if a smooth map $$f : P \to E$$ is transversal to a submanifold $$M \subset E$$, then the pre-image $$f^{-1}(M) \hookrightarrow P$$ is a submanifold.

Global version
The inverse function theorem is a local result; it applies to each point. A priori, the theorem thus only shows the function $$f$$ is locally bijective (or locally diffeomorphic of some class). The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.

Proof: First assume $$X$$ is compact. If the conclusion of the theorem is false, we can find two sequences $$x_i \ne y_i$$ such that $$f(x_i) = f(y_i)$$ and $$x_i, y_i$$ each converge to some points $$x, y$$ in $$A$$. Since $$f$$ is injective on $$A$$, $$x = y$$. Now, if $$i$$ is large enough, $$x_i, y_i$$ are in a neighborhood of $$x = y$$ where $$f$$ is injective; thus, $$x_i = y_i$$, a contradiction.

In general, consider the set $$E = \{ (x, y) \in X^2 \mid x \ne y, f(x) = f(y) \}$$. It is disjoint from $$S \times S$$ for any subset $$S \subset X$$ where $$f$$ is injective. Let $$X_1 \subset X_2 \subset \cdots $$ be an increasing sequence of compact subsets with union $$X$$ and with $$X_i$$ contained in the interior of $$X_{i+1}$$. Then, by the first part of the proof, for each $$i$$, we can find a neighborhood $$U_i$$ of $$A \cap X_i$$ such that $$U_i^2 \subset X^2 - E$$. Then $$U = \bigcup_i U_i$$ has the required property. $$\square$$ (See also for an alternative approach.)

The lemma implies the following (a sort of) global version of the inverse function theorem:

Note that if $$A$$ is a point, then the above is the usual inverse function theorem.

Holomorphic inverse function theorem
There is a version of the inverse function theorem for holomorphic maps.

The theorem follows from the usual inverse function theorem. Indeed, let $$J_{\mathbb{R}}(f)$$ denote the Jacobian matrix of $$f$$ in variables $$x_i, y_i$$ and $$J(f)$$ for that in $$z_j, \overline{z}_j$$. Then we have $$\det J_{\mathbb{R}}(f) = |\det J(f)|^2$$, which is nonzero by assumption. Hence, by the usual inverse function theorem, $$f$$ is injective near $$0$$ with continuously differentiable inverse. By chain rule, with $$w = f(z)$$,
 * $$\frac{\partial}{\partial \overline{z}_j} (f_j^{-1} \circ f)(z) = \sum_k \frac{\partial f_j^{-1}}{\partial w_k}(w) \frac{\partial f_k}{\partial \overline{z}_j}(z) + \sum_k \frac{\partial f_j^{-1}}{\partial \overline{w}_k}(w) \frac{\partial \overline{f}_k}{\partial \overline{z}_j}(z)$$

where the left-hand side and the first term on the right vanish since $$f_j^{-1} \circ f$$ and $$f_k$$ are holomorphic. Thus, $$\frac{\partial f_j^{-1}}{\partial \overline{w}_k}(w) = 0$$ for each $$k$$. $$\square$$

Similarly, there is the implicit function theorem for holomorphic functions.

As already noted earlier, it can happen that an injective smooth function has the inverse that is not smooth (e.g., $$f(x) = x^3$$ in a real variable). This is not the case for holomorphic functions because of:

Formulations for manifolds
The inverse function theorem can be rephrased in terms of differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map $$F: M \to N$$ (of class $$C^1$$), if the differential of $$F$$,
 * $$dF_p: T_p M \to T_{F(p)} N$$

is a linear isomorphism at a point $$p$$ in $$M$$ then there exists an open neighborhood $$U$$ of $$p$$ such that
 * $$F|_U: U \to F(U)$$

is a diffeomorphism. Note that this implies that the connected components of $F$ and $F$ containing p and F(p) have the same dimension, as is already directly implied from the assumption that dFp is an isomorphism. If the derivative of $F$ is an isomorphism at all points $M$ in $N$ then the map $F$ is a local diffeomorphism.

Banach spaces
The inverse function theorem can also be generalized to differentiable maps between Banach spaces $p$ and $M$. Let $F$ be an open neighbourhood of the origin in $X$ and $$F: U \to Y\!$$ a continuously differentiable function, and assume that the Fréchet derivative $$dF_0: X \to Y\!$$ of $Y$ at 0 is a bounded linear isomorphism of $U$ onto $X$. Then there exists an open neighbourhood $F$ of $$F(0)\!$$ in $X$ and a continuously differentiable map $$G: V \to X\!$$ such that $$F(G(y)) = y$$ for all $Y$ in $V$. Moreover, $$G(y)\!$$ is the only sufficiently small solution $Y$ of the equation $$F(x) = y\!$$.

There is also the inverse function theorem for Banach manifolds.

Constant rank theorem
The inverse function theorem (and the implicit function theorem) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal form near that point. Specifically, if $$F:M\to N$$ has constant rank near a point $$p\in M\!$$, then there are open neighborhoods $y$ of $V$ and $x$ of $$F(p)\!$$ and there are diffeomorphisms $$u:T_pM\to U\!$$ and $$v:T_{F(p)}N\to V\!$$ such that $$F(U)\subseteq V\!$$ and such that the derivative $$dF_p:T_pM\to T_{F(p)}N\!$$ is equal to $$v^{-1}\circ F\circ u\!$$. That is, $U$ "looks like" its derivative near $p$. The set of points $$p\in M$$ such that the rank is constant in a neighborhood of $$p$$ is an open dense subset of $V$; this is a consequence of semicontinuity of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of $F$ is injective (resp. surjective) at a point $p$, it is also injective (resp. surjective) in a neighborhood of $M$, and hence the rank of $F$ is constant on that neighborhood, and the constant rank theorem applies.

Polynomial functions
If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true or false, even in the case of two variables. This is a major open problem in the theory of polynomials.

Selections
When $$f: \mathbb{R}^n \to \mathbb{R}^m$$ with $$m\leq n$$, $$f$$ is $$k$$ times continuously differentiable, and the Jacobian $$A=\nabla f(\overline{x})$$ at a point $$\overline{x}$$ is of rank $$m$$, the inverse of $$f$$ may not be unique. However, there exists a local selection function $$s$$ such that $$f(s(y)) = y$$ for all $$y$$ in a neighborhood of $$\overline{y} = f(\overline{x})$$, $$s(\overline{y}) = \overline{x}$$, $$s$$ is $$k$$ times continuously differentiable in this neighborhood, and $$\nabla s(\overline{y}) = A^T(A A^T)^{-1}$$ ($$\nabla s(\overline{y})$$ is the Moore–Penrose pseudoinverse of $$A$$).

Over a real closed field
The inverse function theorem also holds over a real closed field k (or an O-minimal structure). Precisely, the theorem holds for a semialgebraic (or definable) map between open subsets of $$k^n$$ that is continuously differentiable.

The usual proof of the IFT uses Banach's fixed point theorem, which relies on the Cauchy completeness. That part of the argument is replaced by the use of the extreme value theorem, which does not need completeness. Explicitly, in, the Cauchy completeness is used only to establish the inclusion $$B(0, r/2) \subset f(B(0, r))$$. Here, we shall directly show $$B(0, r/4) \subset f(B(0, r))$$ instead (which is enough). Given a point $$y$$ in $$B(0, r/4)$$, consider the function $$P(x) = |f(x) - y|^2$$ defined on a neighborhood of $$\overline{B}(0, r)$$. If $$P'(x) = 0$$, then $$0 = P'(x) = 2[f_1(x) - y_1 \cdots f_n(x) - y_n]f'(x)$$ and so $$f(x) = y$$, since $$f'(x)$$ is invertible. Now, by the extreme value theorem, $$P$$ admits a minimal at some point $$x_0$$ on the closed ball $$\overline{B}(0, r)$$, which can be shown to lie in $$B(0, r)$$ using $$2^{-1}|x| \le |f(x)|$$. Since $$P'(x_0) = 0$$, $$f(x_0) = y$$, which proves the claimed inclusion. $$\square$$

title: Inverse function theorem


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Talk:Fred Walker

Is this the same Fred Walker that created Vegemite? Perhaps he changed his citizenship prior to serving in WWII? --Misfit 17:53, 3 Aug 2003 (UTC)

 * No different person, the Vegemite guy died in 1935 . BTW he didn't actually invent Vegemite, it was the scientist Cyril P. Callsiter who was responsible for working out how to turn brewery waste into something palatable. Walker just marketed it. -- Tim Starling 00:52, Aug 21, 2003 (UTC)

title: Talk:Fred Walker