text:
John Hughes (filmmaker)

John Wilden Hughes Jr. (February 18, 1950 – August 6, 2009) was an American film director, producer and screenwriter. He began his career in 1970 as an author of humorous essays and stories for the National Lampoon magazine. He went on in Hollywood to write, produce and sometimes direct some of the most successful live-action comedy films of the 1980s and 1990s. He directed such films as Sixteen Candles, The Breakfast Club, Weird Science, Ferris Bueller's Day Off, Planes, Trains and Automobiles, She's Having a Baby, and Uncle Buck; and wrote the films National Lampoon's Vacation, Mr. Mom, Pretty in Pink, The Great Outdoors, National Lampoon's Christmas Vacation, Home Alone, Dutch, and Beethoven.

Most of Hughes' works were set in Chicago. He is best known for his coming-of-age teen comedy films with honest depictions of suburban teenage life. Many of his most enduring characters from these years were written for Molly Ringwald. While out on a walk one morning in New York City in the summer of 2009, Hughes suffered a fatal heart attack. His legacy after his death was honored by many, including at the 82nd Academy Awards by many actors he had worked with such as Ringwald, Matthew Broderick, Anthony Michael Hall, Chevy Chase, and Macaulay Culkin, among others. Actors whose careers Hughes helped launch include Michael Keaton, Anthony Michael Hall, Bill Paxton, Broderick, Culkin, and members of the Brat Pack group.

Early life and education
Hughes was born on February 18, 1950, in Lansing, Michigan, to Marion Crawford, who volunteered in charity work, and John Hughes Sr., who worked in sales. He was the only boy, and had three sisters. He spent the first twelve years of his life in Grosse Pointe, Michigan, where he was a fan of Detroit Red Wings right winger Gordie Howe. One of Howe's #9 jerseys, sent by Howe himself, was later prominently featured in Hughes's 1986 film Ferris Bueller's Day Off. Hughes described himself as "kind of quiet" as a kid.

"I grew up in a neighborhood that was mostly girls and old people. There weren't any boys my age, so I spent a lot of time by myself, imagining things. And every time we would get established somewhere, we would move. Life just started to get good in seventh grade, and then we moved to Chicago. I ended up in a really big high school, and I didn't know anybody. But then The Beatles came along (and) changed my whole life. And then Bob Dylan's Bringing It All Back Home came out and really changed me. Thursday I was one person, and Friday I was another. My heroes were Dylan, John Lennon and Picasso, because they each moved their particular medium forward, and when they got to the point where they were comfortable, they always moved on."

In 1963, Hughes's family moved to Northbrook, Illinois, a suburb of Chicago. There, his father found work selling roofing materials. Hughes attended Grove Middle School, later going on to Glenbrook North High School, which gave him inspiration for the films that eventually made his reputation. He met Nancy Ludwig, a cheerleader and his future wife, in high school. As a teenager, Hughes turned to movies as an escape. According to childhood friend Jackson Peterson, "His mom and dad criticized him a lot (...) She [Marion] would be critical of what John would want to do". Hughes was an avid fan of the Beatles, and according to several friends, he knew a lot about movies and the Rat Pack.

1970–1981: Rise to prominence
After dropping out of the University of Arizona, Hughes began selling jokes to well-established performers such as Rodney Dangerfield and Joan Rivers. Hughes used his jokes to get an entry-level job at Needham, Harper & Steers as an advertising copywriter in Chicago in 1970 and later in 1974 at Leo Burnett Worldwide. During this period, he created what became the famous Edge "Credit Card Shaving Test" ad campaign.

Hughes's work on the Virginia Slims account frequently took him to the Philip Morris headquarters in New York City, which allowed him to visit the offices of National Lampoon magazine. Soon thereafter, Hughes became a regular contributor; editor P. J. O'Rourke recalled that "John wrote so fast and so well that it was hard for a monthly magazine to keep up with him." One of Hughes's first stories, inspired by his family trips as a child, was "Vacation '58", later to become the basis for the film National Lampoon's Vacation. Among his other contributions to the Lampoon, the April Fools' Day stories "My Penis" and "My Vagina" gave an early indication of Hughes's ear for the particular rhythm of teenspeak, as well as for the various indignities of teenage life in general.

1982–1986: Breakthrough and teen films
His first credited screenplay, National Lampoon's Class Reunion, was written while he was still on staff at the magazine. The resulting film became the second disastrous attempt by the flagship to duplicate the runaway success of National Lampoon's Animal House. Hughes's next screenplay for the imprint, however, National Lampoon's Vacation, would become a major hit in 1983. This, along with the success of another Hughes script that same year, Mr. Mom, earned him a three-film deal with Universal Pictures.

Hughes's directorial debut, Sixteen Candles (1984), won almost unanimous praise when it was released in 1984, due in no small part to its more honest depiction of navigating adolescence and the social dynamics of high school life in stark contrast to the Porky's-inspired comedies made at the time. It was the first in a string of efforts about teenage life set in or around high school, including The Breakfast Club (1985), Weird Science (1985), and Ferris Bueller's Day Off (1986), all of which he wrote and directed, and Pretty in Pink (1986) and Some Kind of Wonderful (1987), which he wrote and produced.

1987–2008: Beyond teen movies
To avoid being pigeonholed as a maker of only teen movies, Hughes branched out in 1987 by writing, directing, and producing the hit comedy Planes, Trains and Automobiles starring Steve Martin and John Candy. His later output was not so well received critically, though films like Uncle Buck and National Lampoon's Christmas Vacation proved popular. His final film as a director was 1991's Curly Sue. By that time, in 1991, his John Hughes Entertainment production company had signed various deals with 20th Century Fox and Warner Bros.

Actor John Candy created many memorable roles in films written, directed or produced by Hughes, including National Lampoon's Vacation (1983), Planes, Trains and Automobiles (1987), The Great Outdoors (1988), Uncle Buck (1989), Home Alone (1990), Career Opportunities and Only the Lonely (both 1991). Over the years, Hughes and Candy developed a close friendship. Hughes was greatly shaken by Candy's sudden death from a heart attack in 1994. "He talked a lot about how much he loved Candy—if Candy had lived longer, I think John would have made more films as a director", says Vince Vaughn, a friend of Hughes.

Hughes's greatest commercial success came with Home Alone (1990), a film he wrote and produced about a child accidentally left behind when his family goes away for Christmas, forcing him to protect himself and his house from a pair of inept burglars. Hughes completed the first draft of Home Alone in just 9 days. Home Alone was the top-grossing film of 1990, and remains the most successful live-action family comedy of all time. He followed up with the sequels Home Alone 2: Lost in New York in 1992 and Home Alone 3 in 1997. Some of the subsequent films he wrote and produced during this time also contained elements of the Home Alone formula, including the successful Dennis the Menace (1993) and the box office flop Baby's Day Out (1994). He also wrote screenplays under the pseudonym Edmond Dantes (or Dantès), after the protagonist of Alexandre Dumas's novel The Count of Monte Cristo. Screenplays credited to the Dantes nom de plume include Maid in Manhattan, Drillbit Taylor and Beethoven.

Unproduced screenplays

 * National Lampoon's Jaws 3: People 0 – a parody sequel to the popular film series (1979)
 * Motorheads vs. Sportos, aka Just Like Romeo and Juliet, aka Suburban Westside Story
 * The History of Ohio from the Beginning of Time to the End of the Universe, also known as National Lampoon's Dacron, Ohio (1980; with P. J. O'Rourke)
 * The Joy of Sex: A Dirty Love Story (1982; some drafts with Dan Greenburg)
 * Debs – a satire on Texas debutantes (1983; Aaron Spelling Productions)
 * The New Kid (1986)
 * Oil and Vinegar – a soon-to-be-married man and a hitchhiking girl end up talking about their lives during the length of the car ride (1987)
 * Bartholomew Vs. Neff – a vehicle that would have starred Sylvester Stallone and John Candy as feuding neighbors (1991)
 * Black Cat Bone: The Return of Huckleberry Finn (1991)
 * The Nanny (1991)
 * The Bugster (1991)
 * Ball 'n' Chain (1991)
 * Live-action Peanuts film – Warner Bros. acquired the film rights to make a live-action Charlie Brown film, with Hughes set to both produce and write (1993)
 * The Pajama Game – Warner Bros planned a remake of the film.
 * Damn Yankees – Warner Bros planned a remake of the film.
 * The Bee – a feature-length Disney film that actor Daniel Stern was attached to direct (1994)
 * Tickets – Teens wait overnight for free tickets to a farewell concert (1996)
 * The Grigsbys Go Broke – a wealthy family lose their fortune, forcing them to move to the other side of the tracks. (2003)

Later life
In 1994, Hughes retired from the public eye and moved back to the Chicago area. The following year, Hughes and Ricardo Mestres, both of whom had production deals with Walt Disney Pictures, formed the short-lived joint venture production studio Great Oaks Entertainment. Hughes worked in Chicago, while Mestres was based in Los Angeles. The company produced the films Jack, 101 Dalmatians, and Flubber, but Hughes and Mestres ended their partnership in 1997. The 1998 film Reach the Rock, which was produced as part of the partnership between Hughes and Mestres, was subsequently credited as "a Gramercy Pictures release of a John Hughes and Ricardo Mestres production".

In the following years, Hughes rarely granted interviews to the media, save a select few in 1999 to promote the soundtrack album of Reach the Rock. The album was compiled by Hughes's son, John Hughes III, and released on his son's Chicago-based record label Hefty Records. He also recorded an audio commentary for the 1999 DVD release of Ferris Bueller's Day Off.

Personal life
In 1970, the then-20-year old Hughes married Nancy Ludwig, whom he had met in high school. Together they had two children: John Hughes III (born in 1976) and James Hughes (born in 1979). They were together until his death in 2009. Nancy Hughes died on September 15, 2019. Michael Weiss argued that Hughes's films expressed a Reagan Republican worldview. In response to this, P. J. O'Rourke wrote that:

"'I have no idea how, or if, John voted ... John and I never bothered to talk much about our politics. What we did talk about was the 20th century's dominant scrambled egghead bien pensant buttinski parlor pinko righty-tighty lefty-loosey nutfudge notion that middle-class American culture was junk, that middle-class Americans were passive dimbulbs, that America itself was a flop and that America's suburbs were a living hell almost beyond the power of John Cheever's words to describe ... We were becoming conservatives—in the most conservational sense. There were things that others before us had achieved and these were worth conserving ... Family was the most conservative thing about John. Walking across the family room in your stocking feet and stepping on a Lego (ouch!) was the fundamental building block of society.'"

Death
On August 5, 2009, Hughes and his wife traveled to New York City to visit their son James and their new grandson. James said his father appeared to be in good health that night and that the family had made plans for the next day. On the morning of August 6, Hughes was taking a walk close to his hotel on West 55th Street in Manhattan when he suffered a heart attack. He was rushed to Roosevelt Hospital, where he was pronounced dead at age 59. Hughes's funeral was held on August 11 in Chicago; he was buried at Lake Forest Cemetery. It was attended by his wife, his two children, and his grandchildren.

Legacy
The pilot episode of the NBC comedy Community, broadcast on September 17, 2009, was dedicated to Hughes. The episode included several references to The Breakfast Club and ended with a cover of "Don't You (Forget About Me)." The One Tree Hill episode titled "Don't You Forget About Me," broadcast on February 1, 2010, ended with a scene similar to the ending scene of Sixteen Candles. It also contained references to other Hughes movies such as Home Alone. The 2011 Bob's Burgers episode "Sheesh! Cab, Bob?" also paid homage to Sixteen Candles. The teen comedy Easy A (2010) starring Emma Stone paid tribute to Hughes and his films at the very end, where Stone's character states she wishes her life were a John Hughes movie, by showing various clips of Sixteen Candles, The Breakfast Club, and Ferris Bueller's Day Off.

After Hughes's death, many of those who knew him commented on the impact Hughes had on their lives and on the film industry. Molly Ringwald said, "I was stunned and incredibly sad to hear about the death of John Hughes. He was and will always be such an important part of my life. ... He will be missed – by me and by everyone that he has touched. My heart and all my thoughts are with his family now." Matthew Broderick also released his own statement, saying, "I am truly shocked and saddened by the news about my old friend John Hughes. He was a wonderful, very talented guy and my heart goes out to his family." The 82nd Academy Awards (2010) included a tribute to Hughes's work. A retrospective of clips from Hughes's films was followed by cast members from several of them, including Molly Ringwald, Matthew Broderick, Macaulay Culkin, Judd Nelson, Ally Sheedy, Anthony Michael Hall and Jon Cryer, gathering on stage to commemorate Hughes and his contributions to the film industry.

Hughes's work has also influenced a new generation of millennial filmmakers, including M. H. Murray of Teenagers fame, who has cited Hughes as one of his main influences. In interviews, Murray stated,  "I loved how John Hughes wrote teens ... They were flawed in this genuine sort of way." Kelly Fremon Craig, who wrote and directed The Edge of Seventeen, also cited Hughes as an influence.

Hughes is referenced in the song "Hello Chicago" by the collaborative project between Jesu and Sun Kil Moon, and appears on the album 30 Seconds To The Decline Of Planet Earth. Mark Kozelek recalls a phone conversation with Hughes in which Kozelek asked him for $15,000 in order to release his album Songs for a Blue Guitar (released by his band The Red House Painters). Hughes agreed, stating "You're young and on the rise, and I'm just an old man living in Chicago". British indie pop band The 1975 cites Hughes as an influence in the band's music. Maisie Peters released a song called "John Hughes Movie" in 2021. John Hughes's films served as inspiration for the style and tone of the Marvel Cinematic Universe film Spider-Man: Homecoming directed by Jon Watts, who took inspiration from films such as Ferris Bueller's Day Off.

Television
Writer

Books

 * National Lampoon Sunday Newspaper Parody (1978) (with P. J. O'Rourke)

title: John Hughes (filmmaker)


text:
Ramstein air show disaster

The Ramstein air show disaster occurred on Sunday, 28 August 1988 during the Flugtag '88 airshow at USAF Ramstein Air Base near Kaiserslautern, West Germany. Three aircraft of the Italian Air Force display team collided during their display, crashing to the ground in front of a crowd of about 300,000 people. There were 70 fatalities (67 spectators and 3 pilots), and 346 spectators sustained serious injuries in the resulting explosion and fire. Hundreds more had minor injuries. At the time, it was the deadliest air show accident in history until a 2002 crash at the Sknyliv air show that killed 77.

Background
Ten Aermacchi MB-339 PAN jets from the Italian Air Force display team, Frecce Tricolori, were performing their "pierced heart" (Italian: Cardioide, German: Durchstoßenes Herz) formation. In this formation, two groups of aircraft trace out a heart shape in front of the audience along the runway. As they reach the lower tip of the heart, the two groups pass each other parallel to the runway. The heart is then pierced by a lone aircraft, flying in the direction of spectators.

The crash

The mid-air collision took place as the two heart-forming groups passed each other and the heart-piercing aircraft hit them. One of the pilots finished the maneuver too early. The piercing aircraft crashed onto the runway and consequently both the fuselage and resulting fireball of aviation fuel tumbled into the spectator area, hitting the crowd and coming to rest against a refrigerated trailer being used to dispense ice cream to the various vendor booths in the area.

At the same time, one of the damaged aircraft from the heart-forming group crashed into the emergency medical evacuation UH-60 Black Hawk helicopter, injuring the helicopter's pilot, Captain Kim Strader. He died 20 days later, on Saturday, 17 September 1988, at Brooke Army Medical Center in Texas from burns he suffered in the accident.

The pilot of the aircraft that hit the helicopter ejected, but was killed as he hit the runway before his parachute opened. The third aircraft disintegrated in the collision and parts of it were strewn along the runway.

After the crash, the remaining aircraft regrouped and landed at Sembach Air Base.

Scope
Of the 31 people who died on impact, 28 had been hit by debris in the form of airplane parts, concertina wire, and items on the ground. Sixteen of the fatalities occurred in the days and weeks after the disaster due to severe burns; the last was the burned and injured helicopter pilot. About 500 people had to seek hospital treatment following the event, and over 600 people reported to the clinic that afternoon to donate blood.

Criticism
The disaster revealed serious shortcomings in the handling of large-scale medical emergencies by German civil and American military authorities. US military personnel did not immediately allow German ambulances onto the base, and the rescue work was generally hampered by a lack of efficiency and coordination. The rescue coordination center in Kaiserslautern was unaware of the disaster's scale as much as an hour after it occurred, even though several German medevac helicopters and ambulances had already arrived on site and left with patients. American helicopters and ambulances provided the quickest and largest means of evacuating burn victims, but lacked sufficient capacities for treating them, or had difficulty finding them. Further confusion was added by the American military's usage of different standards for intravenous catheters from German paramedics. A single standard was codified in 1995, updated with a newer version in 2013 and an amendment to the current standard in 2017.

Actions
A crisis counseling center was immediately established at the nearby Southside Base Chapel and remained open throughout the week. Base mental health professionals provided group and individual counseling in the following weeks, and they surveyed the response workers two months following the tragedy and again six months after the disaster to gauge recovery.

Investigation
Several different video recordings of the accident were made. They show that the "piercing" aircraft (Pony 10) came in too low and too fast at the crossing point with the other two groups (five aircraft on the left and four on the right) as they completed the heart-shaped figure. Lead pilot Lieutenant Colonel Ivo Nutarelli, flying Pony 10, was unable to correct his altitude or lower his speed, and collided with the leading airplane (Pony 1, piloted by Lt. Col. Mario Naldini) of the left formation "inside" the figure, destroying the plane's tail section with the front of his aircraft. Pony 1 then spiralled out of control, hitting the plane on its lower left (Pony 2, piloted by Captain Giorgio Alessio). Lt. Col. Naldini ejected but was killed as he hit the runway before his parachute opened. His plane crashed onto a taxiway near the runway, destroying a medevac helicopter and fatally injuring its pilot, Captain Kim Strader. Pony 2, the third plane involved in the disaster, was severely damaged from the impact with Pony 1, and crashed beside the runway, exploding in a fireball. Its pilot, Captain Alessio, died on impact.

Pony 10, the aircraft that started the crash, continued on a ballistic trajectory across the runway, completely out of control and in flames, its forward section destroyed by the impact with Pony 1. The plane hit the ground ahead of the spectator stands, exploding in a fireball and destroying a police vehicle parked inside the concertina-wire fence that defined the active runway area. The plane continued, cartwheeling for a distance before picking up the three-strand concertina-wire fence, crossing an emergency access road, slamming into the crowd, and hitting a parked ice cream van. The area of the crash, being centered on the flightline and as close to the airshow as civilian spectators could get, had been considered the "best seats in the house", and was densely packed. The entire incident, from the collision of the first two planes to the crash into the crowd, took less than seven seconds, leaving almost no time for spectators to escape. The low altitude of the maneuver (45 meters above the crowd) also contributed to the short time frame.

An examination of photos and footage from the disaster showed that Pony 10's landing gear came down at some point; it has been suggested that this could have been lowered intentionally as a last second effort by Lt. Col. Nutarelli to slow his plane down and avoid the impact, but there is no substantial evidence pointing to this; the undercarriage could have been lowered by a number of factors. In April 1991, Werner Reith, a German journalist from the newspaper Die Tageszeitung, suggested in an article that the Ramstein disaster could have been caused by some sudden technical problem—or even sabotage—in Nutarelli's plane. No supporting evidence could be collected. Reith pointed out that Lt. Col. Nutarelli and Lt. Col. Naldini were supposed to know details about another air disaster, the 1980 Ustica massacre, citing Italian press sources. Judge Rosario Priore, who was investigating the case at the time, found that they were performing training flights nearby minutes before the Ustica incident, but he definitely rejected their deaths as sabotage.

References in popular culture

 * In games
 * A similar disaster is portrayed in the German made strategy PC game Emergency: Fighters for Life as Mission 22

 * In literature
 * Both Ramstein Air Force Base and the Ramstein air disaster figure as plot points in Donna Leon's second Guido Brunetti novel, Death in a Strange Country (1993)

 * In music
 * The Neue Deutsche Härte band Rammstein was named after this catastrophe. The second "m" was initially added by mistake, but the band eventually embraced the misspelling as its literal translation is "ramming-stone." Rammstein's self-titled song (on the album Herzeleid (1995)) is also a reference to the event.
 * DJ and record producer Boris Brejcha was a 6-year old in the crowd, and was badly burned by the flames. He describes his style as high-tech minimal, as it reflects his subsequent isolation, using the Venetian Carnival mask as his logo.

 * In television
 * The disaster was featured on the 22 February 2008, episode of Shockwave, on The History Channel and the December 10, 2000 episode of World's Most Amazing Videos as well as an episode of Real TV.

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

title: Ramstein air show disaster


text:
Aude (river)

The Aude (Latin Atax ) is a river of southern France that is 224 km long. Its source is in the Pyrenees mountains then runs to Carcassonne and finally reaches the Mediterranean Sea near Narbonne. The river is navigable by raft or canoe for nearly all of its length. It is registered as essential to the Languedoc-Roussillon region.

The river gave its name to the Aude department.

Etymology
In antiquity, the Aude was called Atax by the Romans. Some authors in antiquity called the river Narbôn (e.g. Polybius).

In 1342 the Roussillon Cartulary of Alart called it the Auda or the Ribera d'Aude. In the Middle Ages the terms Adice, again Atax, Fluvium Atacis, Flumine Atace, Flumen Ataze, and Juxta Aditum fluvium were also used to designate the Aude. In all likelihood the current name comes from a gradual evolution of Atax given by Strabo (in his Geography, Book IV), a word borrowed from the Gallic term atacos meaning "spirited" or "very fast".

Course
The river has its source in the Massif of Carlit at the Lac d'Aude at an elevation of 2185 m in the commune of Les Angles (department of Pyrénées-Orientales) and flows parallel to the Tet. The Col de la Quillane, elevation 1713 m, marks the boundary of the watershed. The river then flows into the Mediterranean Sea a few kilometres from Narbonne at Grau de Vendres (near Cabanes-de-Fleury) on the border between the departments of Aude and Hérault.

At Carcassonne in the Pyrénées its course is oriented north-south. The Aude then presents the characteristics of a mountain stream as it traverses Capcir and feeds several reservoirs (Matemale, Puyvalador) before plunging into gorges (those of Saint-Georges are the most scenic) crossing ancient terrain. Starting from Axat, after being joined by the Aiguette on the right bank and the Rébenty on the left bank, the river flows through bands of pre-Pyrénéen limestone (in the Pierre-Lys Gorges) and passes through a series of small communes: Quillan, Espéraza, Couiza, Alet-les-Bains, then Limoux. Downstream of this great medieval city, the Aude flows eastwards. This elbow results from a Stream capture, the Aude having once flowed in the valley of the Hers, indicating a deepening by an ancient hydrology following the uplifting of the Pyrénées.

From Carcassonne, the river becomes calmer following the great tectonic furrow which separates the Pyrénées (Corbières) from the Massif Central (Montagne Noire), receiving from these heights a series of tributaries of which the main ones are the Orbieu on the right bank, and the Argent-Double and the Casse on the left bank. From this point, bordered by the Canal du Midi and sinuous amid vineyards, the Aude enters the broad alluvial plain of Narbonne with the main stream flowing to the Gulf of Lion and some waters diverted to a network of ancient ponds before flowing into the Mediterranean Sea.

Hydrography

The Aude has a catchment area of 6074 km2 that extends unevenly across six departments: Aude, Pyrénées-Orientales, Ariège, Haute-Garonne, Tarn, and Hérault (Haute-Garonne and Tarn are only affected by the course of tributaries and sub-tributaries of the river). Before Carcassonne, the Aude receives tributary streams of short length which often have the characteristics of mountain torrents.

Starting from the prefecture of Aude department in the lower valley the Aude river joins with the largest rivers, most of them from the Montagne Noire. From upstream to downstream, the main tributaries of the Aude, longer than 10 km, are:

 * the Lladura or Lladure (16 km), left bank at Réal;
 * the Galbe (14 km), left bank at Puyvalador;
 * the Bruyante (14 km), left bank at Usson-les-Bains;
 * the Aiguette (20 km), right bank at Sainte-Colombe-sur-Guette;
 * the Rébenty (34 km), left bank at Saint-Martin-Lys;
 * the Saint-Bertrand (19 km), right bank at Quillan;
 * the Ruisseau de Fa or Faby (14 km), left bank at Espéraza;
 * the Sals (20 km), right bank at Couiza;
 * the Ruisseau de Castillou, right bank at Alet-les-Bains;
 * the Véraza, right bank at Alet-les-Bains;
 * the Saint-Polycarpe (13 km), right bank at Limoux;
 * the Corneilla (22 km), left bank at Limoux;
 * the Cougaing (17 km), left bank at Limoux;
 * the Sou de Val de Daigne (30 km), left bank at Cépie;
 * the Lauquet (37 km), right bank at Couffoulens;
 * the Fresquel (63 km), left bank at Carcassonne;
 * the Trapel (19 km), left bank at Villedubert;
 * the Orbiel (41 km), left bank at Trèbes;
 * the Argent-Double (37 km), left bank at La Redorte;
 * the Ognon (23 km), left bank at Olonzac;
 * the Orbieu (84 km), right bank at Saint-Nazaire-d'Aude;
 * the Cesse (54 km), left bank at Saint-Marcel-sur-Aude.

Hydrology

In its upper reaches the Aude has a nivo-pluvial regime (with a maximum flow in spring linked to snowmelt). Then from Carcassonne where the average flow rate reaches 20.4 m3/s, the system is almost entirely rainfed (The flow at Grau de Vendres where it meets the Mediterranean is around 50 m3/s.)

The Aude is thus characterized in its lower course by a pluvio-nival regime of meridional type with high baseflow in summer, 9.8 m3/s in August at Moussan in the lower alluvial plain not far from the river mouth, against an average of 44.2 m3/s. Heavy autumnal rains enable a rapid rise in the flow rate which peaks in February, 78.6 m3/s and is supported by the spring snowmelt from the Pyrénées mountains.

Floods of 1999
Exceptional and sudden rainfall, so characteristic of the Mediterranean climate during the autumnal season, can be the cause of devastating floods such as those of 12 and 13 November 1999. The results of this event, which affected the lower valley of the Aude, were catastrophic: 35 victims, hundreds of people evacuated by helicopter or rescued from thousands of homes, businesses, and commercial premises which were more or less damaged, 5000 ha of vineyards more or less ruined, roads, drinking water networks, and sanitation were seriously affected. The scale of the disaster was due to the combination of two factors: Thunderstorms with rainfall on an unprecedented scale. Up to 620 mm fell in two days at Lézignan-Corbières, that is to say more than the average annual total. In addition, a strong storm in the Gulf of Lyon led to a rise of 80 cm in the sea level and, combined with the strong waves and wind, prevented the flow of water already swollen by the torrential rains.

Floods of 2018
Remnants of Hurricane Leslie fed a quasi-stationary cold front over southwestern France, generating heavy thunderstorms and leading to flash flooding in that area. The Aude rose to a height of 7 m, its highest level since 1891. Carcassonne received 160 - 180 mm of rainfall within five hours; water level in the city rose 8 m during that period. At least 14 people died because of the flash flood, mainly in the town of Villegailhenc, Aude.

Departments and towns crossed

 * Pyrénées-Orientales: in the region of Capcir to Matemale, and Formiguères
 * Hérault: Olonzac, Lespignan, Vendres
 * Aude: Axat, Belvianes-et-Cavirac, Quillan, Campagne-sur-Aude, Espéraza, Montazels, Couiza, Alet-les-Bains, Limoux, Carcassonne, Trèbes, Narbonne, Coursan, Cuxac-d'Aude, Saint-Nazaire-d'Aude
 * Ariège: Rouze

title: Aude (river)


text:
Talk:Lead and follow

History
I've cleared the discussion - see history. All concerns addressed long ago ! Gender-neutral terms used throughout, as far as I can see. Seems general enough to cover all dance forms well. --195.137.93.171 10:05, 15 September 2007 (UTC)

Jargon
I am concerned that jargon is used differently in different countries and dances.

Particularly : 'Compression' and Tension' - in scientific physical terms, these could mean forces or pressures (pressure = force divided by area). The article uses them to mean motion instead, which in scientific medical terms would be called 'Flexion' and 'Extension'.

In (some) Jive/Swing circles, 'Leverage' means 'tension': in scientific physical terms leverage means multiplying a force with a beam and a pivot/hinge ! In West Coast Swing, what this article calls 'leverage' or 'grounding' is known as 'anchoring' or 'posting' ! I'm not sure what the answer is, but edit wars are to be avoided ... --195.137.93.171 10:05, 15 September 2007 (UTC)

Re-structuring
The nested levels had become confused, and bits seemed duplicate and out-of-sequence.

I've also removed
 * In Caribbean dances the hip contact is more for the sensual touch, and quickly gets into the way of the more advanced figures.

because it had mostly been said already and I wasn't sure if it meant 'hip contact gets in the way' or 'helps you get into the way of doing' - apologies to the author, but written english is difficult to use clearly! --195.137.93.171 (talk) 10:41, 15 December 2007 (UTC)

Pulse
No mention that I could see of pulse or maintenance of the beat. Perhaps an oversight or not needed here, I'll have to read again. Essential part of leading and following in some Charleston and Balboa. Litui (talk) 23:02, 17 July 2008 (UTC)

Language and tone
I don't think this Lead and Follow page clearly explains how to actually lead or follow. While I agree with most everything in it. I'm not sure it really helps the novice and I feel that the general tone doesn't communicate how to actually dance. I don't mean to criticize and am not trying to offend anybody, my only concern is people being able to dance after reading about it so they can dance and have more fun:) And so I would like to dialog with people but haven't figured out wikipedia yet. And will be making changes over the next month and looking forward to the conversations to take placeJavacaliente (talk) 19:37, 13 March 2010 (UTC)

I organized some of the categories. While I know there are some messes that have been created i feel its better to at least start the clean up process rather than not do anything at all with the hopes of having it done "perfect". Javacaliente (talk) 09:41, 14 March 2010 (UTC)

Lead and Follow
Does the role of the lead and follow in terms of job description and how it is properly done have any relevance with which sex is the lead or follow? Because I don't see the relevance to it and would like to delete the conversation pertaining to local dance customs( meaning, which sex is lead and which sex is follow) and focus on the lead and follow and how it is done. Or conversely start a subsection and discuss that only there and leave the rest of the article to lead and follow terminology. Javacaliente (talk) 00:52, 15 March 2010 (UTC)

Circular reference
Hi,

anyone noticed that this article refers in the "References" section to a web page, which itself says that the text is "pulled" from this very same Wikipedia article?

Best regards 77.2.137.146 (talk) 01:51, 21 August 2013 (UTC)

History section please
How did "Lead and follow" develop? Have there been other kinds of dancing without "lead and follow" before?--89.12.139.120 (talk) 14:08, 15 April 2015 (UTC)

What's the name of the male partner? Camarero?
Thank you --SvenAERTS (talk) 07:07, 6 January 2019 (UTC)

title: Talk:Lead and follow


text:
Historical mystery

The historical mystery or historical whodunit is a subgenre of two literary genres, historical fiction and mystery fiction. These works are set in a time period considered historical from the author's perspective, and the central plot involves the solving of a mystery or crime (usually murder). Though works combining these genres have existed since at least the early 20th century, many credit Ellis Peters's Cadfael Chronicles (1977–1994) for popularizing what would become known as the historical mystery. The increasing popularity and prevalence of this type of fiction in subsequent decades has spawned a distinct subgenre recognized by the publishing industry and libraries. Publishers Weekly noted in 2010 of the genre, "The past decade has seen an explosion in both quantity and quality. Never before have so many historical mysteries been published, by so many gifted writers, and covering such a wide range of times and places." Editor Keith Kahla concurs, "From a small group of writers with a very specialized audience, the historical mystery has become a critically acclaimed, award-winning genre with a toehold on the New York Times bestseller list."

Since 1999, the British Crime Writers' Association has awarded the CWA Historical Dagger award to novels in the genre. The Left Coast Crime conference has presented its Bruce Alexander Memorial Historical Mystery award (for mysteries set prior to 1950) since 2004.

Origins
Though the term "whodunit" was coined sometime in the early 1930s, it has been argued that the detective story itself has its origins as early as the 429 BC Sophocles play Oedipus Rex and the 10th century tale "The Three Apples" from One Thousand and One Nights (Arabian Nights). During China's Ming dynasty (1368–1644), gong'an ("crime-case") folk novels were written in which government magistrates—primarily the historical Di Renjie of the Tang dynasty (618–907) and Bao Zheng of the Song dynasty (960–1279)—investigate cases and then as judges determine guilt and punishment. The stories were set in the past but contained many anachronisms. Robert van Gulik came across the 18th century anonymously written Chinese manuscript Di Gong An, in his view closer to the Western tradition of detective fiction than other gong'an tales and so more likely to appeal to non-Chinese readers, and in 1949 published it in English as Celebrated Cases of Judge Dee. He subsequently wrote his own Judge Dee stories (1951–1968) in the same style and time period.

Perhaps the first modern English work that can be classified as both historical fiction and a mystery however is the 1911 Melville Davisson Post story "The Angel of the Lord", which features amateur detective Uncle Abner in pre-American Civil War West Virginia. Barry Zeman of the Mystery Writers of America calls the Uncle Abner short stories "the starting point for true historical mysteries." In the 22 Uncle Abner tales Post wrote between 1911 and 1928, the character puzzles out local mysteries with his keen observation and knowledge of the Bible. It was not until 1943 that American mystery writer Lillian de la Torre did something similar in the story "The Great Seal of England", casting 18th century literary figures Samuel Johnson and James Boswell into Sherlock Holmes and Dr. Watson roles in what would become the first of her Dr. Sam: Johnson, Detector series of stories. In 1944, Agatha Christie published Death Comes as the End, a mystery novel set in ancient Egypt and the first full-length historical whodunit. In 1950, John Dickson Carr published the second full-length historical mystery novel called The Bride of Newgate, set at the close of the Napoleonic Wars.

Popularization
In 1970, Peter Lovesey began a series of novels featuring Sergeant Cribb, a Victorian-era police detective, and Elizabeth Peters's Amelia Peabody series (1975–2010) followed the adventures of the titular Victorian lady/archaeologist as she solved mysteries surrounding her excavations in early 20th century Egypt. But historical mystery stories remained an oddity until the late 1970s, with the success of Ellis Peters and her Cadfael Chronicles (1977–1994), featuring Benedictine monk Brother Cadfael and set in 12th century Shrewsbury. Umberto Eco's one-off The Name of the Rose (1980) also helped popularize the concept, and starting in 1979, author Anne Perry wrote two series of Victorian era mysteries featuring Thomas Pitt (1979–2013) and William Monk (1990–2013). However it was not until about 1990 that the genre's popularity expanded significantly with works such as Lindsey Davis's Falco and Flavia Albia novels (1989–2022), set in the Roman Empire of Vespasian; John Maddox Roberts's SPQR series (1990–2010) and Steven Saylor's Roma Sub Rosa novels (1991–2018), both set in the Roman Republic in the 1st century BC; and Paul Doherty's various series, including the Hugh Corbett medieval mysteries (1986–2010), the Sorrowful Mysteries of Brother Athelstan (1991–2012), and the Canterbury Tales of Mystery and Murder (1994–2012). For Mike Ashley'sThe Mammoth Book of Historical Detectives (1995), F. Gwynplaine MacIntyre wrote "Death in the Dawntime", a locked room mystery (or rather, sealed cave mystery) set in Australia around 35,000 BC, which Ashley suggests is the furthest in the past a historical mystery has been set to date. Diana Gabaldon began the Lord John series in 1998, casting a recurring secondary character from her Outlander series, Lord John Grey, as a nobleman-military officer-amateur detective in 18th century England. Using the pen name Ariana Franklin, Diana Norman wrote four Mistress of the Art of Death novels between 2007 and 2010, featuring 12th-century English medical examiner Adelia Aguilar.

Publishers Weekly noted in 2010 of the genre, "The past decade has seen an explosion in both quantity and quality. Never before have so many historical mysteries been published, by so many gifted writers, and covering such a wide range of times and places." Editor Keith Kahla concurs, "From a small group of writers with a very specialized audience, the historical mystery has become a critically acclaimed, award-winning genre with a toehold on the New York Times bestseller list."

Awards
In 1999, the British Crime Writers' Association awarded the first CWA Historical Dagger award to a novel in the genre. The award was called the Ellis Peters Historical Dagger through 2012. In 2014, Endeavour Press supported the award, which is called the Endeavour Historical Dagger for the 2014 and 2015 awards. The Left Coast Crime conference has presented its Bruce Alexander Memorial Historical Mystery award (for mysteries set prior to 1950) since 2004.

Variations
In an early twist of the genre, Josephine Tey's The Daughter of Time (1951) features a modern police detective who alleviates an extended hospital stay by investigating the 15th century case of Richard III of England and the Princes in the Tower. Georgette Heyer's The Talisman Ring (1936), set in 1793 England, is a Regency romance with elements of mystery that Jane Aiken Hodge called "very nearly a detective story in period costume". Many of Heyer's other historical romances have thriller elements but to a much lesser extent.

Other variations include mystery novels set in alternate history timelines or even fantasy worlds. These would include The Ultimate Solution (1973) by Eric Norden and Fatherland (1992) by Robert Harris, both being police procedurals set in alternate timelines where the Nazis won World War II; Randall Garrett's Lord Darcy series, taking place in a 20th-century in which magic is possible; and Phyllis Ann Karr's The Idylls of the Queen (1982), set in King Arthur's court as depicted in Arthurian myth and with no attempt at historical accuracy.

The genre would not include fiction which was contemporary at the time of writing, such as Arthur Conan Doyle's canonical Sherlock Holmes works set in Victorian England, or the Lord Peter Wimsey books by Dorothy L. Sayers set in the Interwar period. However, subsequent Holmes and Wimsey books written by other authors decades later could arguably be classified as historical mysteries.

List of fictional historical detectives
The following list consists of fictional historical detectives in chronological order of their time period setting:

title: Historical mystery


text:
Log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable $X$ is log-normally distributed, then $Y = ln(X)$ has a normal distribution. Equivalently, if $Y$ has a normal distribution, then the exponential function of $Y$, $X = exp(Y)$, has a log-normal distribution. A random variable that is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in the natural sciences, engineering, as well as medicine, economics and other fields. It can be applied to diverse quantities such as energies, concentrations, lengths, prices of financial instruments, and other metrics, while acknowledging the inherent uncertainty in all measurements.

The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.

A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate $X$—for which the mean and variance of $ln(X)$ are specified.

Generation and parameters
Let $$\ Z\ $$ be a standard normal variable, and let $$\mu$$ and $$\sigma$$ be two real numbers, with $$\sigma > 0$$. Then, the distribution of the random variable

 * $$ X = e^{\mu + \sigma Z} $$

is called the log-normal distribution with parameters $$\mu$$ and $$\sigma$$. These are the expected value (or mean) and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of $$\ X\ $$ itself.

This relationship is true regardless of the base of the logarithmic or exponential function: If $$\ \log_a(X)\ $$ is normally distributed, then so is $$\ \log_b(X)\ $$ for any two positive numbers $$\ a, b \neq 1 ~.$$ Likewise, if $$\ e^Y\ $$ is log-normally distributed, then so is $$\ a^Y\ ,$$ where $$0 < a \neq 1$$.

In order to produce a distribution with desired mean $$\mu_X$$ and variance $$\ \sigma_X^2\ ,$$ one uses $$\ \mu = \ln\left( \frac{ \mu_X^2 }{\ \sqrt{ \mu_X^2 + \sigma_X^2\ }\ }\right)\ $$ and $$\ \sigma^2 = \ln\left( 1 + \frac{\ \sigma_X^2\ }{ \mu_X^2 } \right) ~.$$

Alternatively, the "multiplicative" or "geometric" parameters $$\ \mu^* = e^\mu\ $$ and $$\ \sigma^* = e^\sigma\ $$ can be used. They have a more direct interpretation: $$\ \mu^*\ $$ is the median of the distribution, and $$\ \sigma^*\ $$ is useful for determining "scatter" intervals, see below.

Probability density function
A positive random variable $$\ X\ $$ is log-normally distributed (i.e., $$\ X \sim \operatorname{Lognormal}\left(\ \mu, \sigma^2\ \right)\ $$), if the natural logarithm of $$\ X\ $$ is normally distributed with mean $$ \mu$$ and variance $$\ \sigma^2\ :$$

 * $$ \ln(X) \sim \mathcal{N}(\mu,\sigma^2)$$

Let $$\ \Phi\ $$ and $$\ \varphi\ $$ be respectively the cumulative probability distribution function and the probability density function of the $$\ \mathcal{N}(\ 0, 1\ )\ $$ standard normal distribution, then we have that the probability density function of the log-normal distribution is given by:

\begin{align} f_X(x) & = \frac{ \rm{d} }{ {\rm d} x }\ \operatorname{\mathbb{P}_\mathit{X}}\,\!\bigl[\ X \le x\ \bigr] \\[6pt] & = \frac{ \rm{d} }{ {\rm d} x }\ \operatorname{\mathbb{P}_\mathit{X}}\,\!\bigl[\ \ln X \le \ln x\ \bigr] \\[6pt] & = \frac{ \rm{d} }{ {\rm d} x } \operatorname{\Phi}\!\!\left( \frac{\ \ln x -\mu\ }{ \sigma } \right) \\[6pt] & = \operatorname{\varphi}\!\left( \frac{\ln x - \mu} \sigma \right) \frac{ \rm{d} }{ {\rm d} x } \left( \frac{\ \ln x - \mu\ }{ \sigma }\right) \\[6pt] & = \operatorname{\varphi}\!\left( \frac{\ \ln x - \mu\ }{ \sigma } \right) \frac{ 1 }{\ \sigma\ x\ } \\[6pt] & = \frac{ 1 }{\ x\ \sigma\sqrt{2\ \pi\ }\ } \exp\left( -\frac{\ (\ln x-\mu)^2\ }{2\ \sigma^2} \right) ~. \end{align} $$

Cumulative distribution function
The cumulative distribution function is

 * $$ F_X(x) = \Phi\left( \frac{(\ln x) - \mu} \sigma \right) $$

where $$\ \Phi\ $$ is the cumulative distribution function of the standard normal distribution (i.e., $$\ \operatorname\mathcal{N}(\ 0,\ 1 )\ $$).

This may also be expressed as follows:

\frac12 \left[ 1 + \operatorname{erf} \left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) \right] = \frac12 \operatorname{erfc} \left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) $$

where $erfc$ is the complementary error function.

Multivariate log-normal
If $$\boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)$$ is a multivariate normal distribution, then $$Y_i=\exp(X_i)$$ has a multivariate log-normal distribution. The exponential is applied elementwise to the random vector $$\boldsymbol X$$. The mean of $$\boldsymbol Y$$ is

 * $$\operatorname{E}[\boldsymbol Y]_i=e^{\mu_i+\frac{1}{2}\Sigma_{ii}} ,$$

and its covariance matrix is

 * $$\operatorname{Var}[\boldsymbol Y]_{ij}=e^{\mu_i+\mu_j + \frac{1}{2}(\Sigma_{ii}+\Sigma_{jj}) }( e^{\Sigma_{ij}} - 1) . $$

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.

Characteristic function and moment generating function
All moments of the log-normal distribution exist and

 * $$\operatorname{E}[X^n]= e^{n\mu+n^2\sigma^2/2}$$

This can be derived by letting $$z=\tfrac{\ln(x) - (\mu+n\sigma^2)}{\sigma}$$ within the integral. However, the log-normal distribution is not determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero. Indeed, the expected value $$\operatorname{E}[e^{t X}]$$ is not defined for any positive value of the argument $$t$$, since the defining integral diverges.

The characteristic function $$\operatorname{E}[e^{i t X}]$$ is defined for real values of $t$, but is not defined for any complex value of $t$ that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor formal series diverges:

 * $$\sum_{n=0}^\infty \frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2}$$

However, a number of alternative divergent series representations have been obtained.

A closed-form formula for the characteristic function $$\varphi(t)$$ with $$t$$ in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by

 * $$\varphi(t)\approx\frac{\exp\left(-\frac{W^2(-it\sigma^2e^\mu) + 2W(-it\sigma^2e^\mu)}{2\sigma^2} \right)}{\sqrt{1+W(-it\sigma^2e^\mu)}}$$

where $$W$$ is the Lambert W function. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of $$\varphi$$.

Probability in different domains
The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method. (Matlab code)

Probabilities of functions of a log-normal variable
Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed. (Matlab code)

Geometric or multiplicative moments
The geometric or multiplicative mean of the log-normal distribution is $$\operatorname{GM}[X] = e^\mu = \mu^*$$. It equals the median. The geometric or multiplicative standard deviation is $$\operatorname{GSD}[X] = e^{\sigma} = \sigma^*$$.

By analogy with the arithmetic statistics, one can define a geometric variance, $$\operatorname{GVar}[X] = e^{\sigma^2}$$, and a geometric coefficient of variation, $$\operatorname{GCV}[X] = e^{\sigma} - 1$$, has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of $$\operatorname{CV}$$ itself (see also Coefficient of variation).

Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM–GM inequality and is a consequence of the logarithm being a concave function. In fact,

 * $$\operatorname{E}[X] = e^{\mu + \frac12 \sigma^2} = e^{\mu} \cdot \sqrt{e^{\sigma^2}} = \operatorname{GM}[X] \cdot \sqrt{\operatorname{GVar}[X]}.$$

In finance, the term $$e^{-\frac12\sigma^2}$$ is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

Arithmetic moments
For any real or complex number $n$, the $n$-th moment of a log-normally distributed variable $X$ is given by
 * $$\operatorname{E}[X^n] = e^{n\mu + \frac12n^2\sigma^2}.$$

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable $X$ are respectively given by:

 * $$\begin{align}

\operatorname{E}[X] & = e^{\mu + \tfrac{1}{2}\sigma^2}, \\[4pt] \operatorname{E}[X^2] & = e^{2\mu + 2\sigma^2}, \\[4pt] \operatorname{Var}[X] & = \operatorname{E}[X^2] - \operatorname{E}[X]^2 = (\operatorname{E}[X])^2(e^{\sigma^2} - 1) = e^{2\mu + \sigma^2} (e^{\sigma^2} - 1), \\[4pt] \operatorname{SD}[X] & = \sqrt{\operatorname{Var}[X]} = \operatorname{E}[X] \sqrt{e^{\sigma^2} - 1} = e^{\mu + \tfrac{1}{2}\sigma^2}\sqrt{e^{\sigma^2} - 1}, \end{align}$$

The arithmetic coefficient of variation $$\operatorname{CV}[X]$$ is the ratio $$\tfrac{\operatorname{SD}[X]}{\operatorname{E}[X]}$$. For a log-normal distribution it is equal to
 * $$\operatorname{CV}[X] = \sqrt{e^{\sigma^2} - 1}.$$

This estimate is sometimes referred to as the "geometric CV" (GCV), due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

The parameters $μ$ and $σ$ can be obtained, if the arithmetic mean and the arithmetic variance are known:

 * $$\begin{align}

\mu &= \ln \left(\frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{E}[X^2]}}\right) = \ln \left( \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{Var}[X] + \operatorname{E}[X]^2}} \right), \\[4pt] \sigma^2 &= \ln \left(\frac{\operatorname{E}[X^2]}{\operatorname{E}[X]^2}\right) =  \ln \left(1 + \frac{\operatorname{Var}[X]}{\operatorname{E}[X]^2}\right). \end{align}$$

A probability distribution is not uniquely determined by the moments $E[X^{n}] = e^{nμ + 1⁄2n^{2}σ^{2}}|undefined$ for $n ≥ 1$. That is, there exist other distributions with the same set of moments. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Mode, median, quantiles
The mode is the point of global maximum of the probability density function. In particular, by solving the equation $$(\ln f)'=0$$, we get that:

 * $$\operatorname{Mode}[X] = e^{\mu - \sigma^2}.$$

Since the log-transformed variable $$Y = \ln X$$ has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of $$X$$ are

 * $$q_X(\alpha) = e^{\mu+\sigma q_\Phi(\alpha)} =\mu^* (\sigma^*)^{q_\Phi(\alpha)},$$

where $$q_\Phi(\alpha)$$ is the quantile of the standard normal distribution.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,

 * $$\operatorname{Med}[X] = e^\mu = \mu^* ~.$$

Partial expectation
The partial expectation of a random variable $$X$$ with respect to a threshold $$k$$ is defined as

 * $$ g(k) = \int_k^\infty x f_X(x \mid X > k)\, dx  . $$

Alternatively, by using the definition of conditional expectation, it can be written as $$g(k)=\operatorname{E}[X\mid X>k] P(X>k)$$. For a log-normal random variable, the partial expectation is given by:

 * $$g(k) = \int_k^\infty x f_X(x \mid X > k)\, dx = e^{\mu+\tfrac{1}{2} \sigma^2}\, \Phi\!\left(\frac{\mu+\sigma^2-\ln k} \sigma \right) $$

where $$\Phi$$ is the normal cumulative distribution function. The derivation of the formula is provided in the Talk page. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.

Conditional expectation
The conditional expectation of a log-normal random variable $$X$$—with respect to a threshold $$k$$—is its partial expectation divided by the cumulative probability of being in that range:

 * $$\begin{align}

E[X\mid X<k] & =e^{\mu +\frac{\sigma^2}{2}}\cdot \frac{\Phi \left[\frac{\ln(k)-\mu -\sigma^2}{\sigma} \right]}{\Phi \left[\frac{\ln(k)-\mu}{\sigma} \right]} \\[8pt] E[X\mid X\geqslant k] &=e^{\mu +\frac{\sigma^2}{2}}\cdot \frac{\Phi \left[\frac{\mu +\sigma^2-\ln(k)} \sigma \right]}{1-\Phi \left[\frac{\ln(k)-\mu}{\sigma}\right]} \\ [8pt] E[X\mid X\in [k_1,k_2]] &=e^{\mu +\frac{\sigma^2}{2}}\cdot \frac{ \Phi \left[\frac{\ln(k_2)-\mu -\sigma^2} \sigma \right]-\Phi \left[\frac{\ln(k_1)-\mu -\sigma^2} \sigma \right] }{ \Phi \left[\frac{\ln(k_2)-\mu}{\sigma}\right]-\Phi \left[\frac{\ln(k_1)-\mu}{\sigma}\right] } \end{align}$$

Alternative parameterizations
In addition to the characterization by $$\mu, \sigma$$ or $$\mu^*, \sigma^*$$, here are multiple ways how the log-normal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions lists seven such forms:

 * LogNormal1(μ,σ) with mean, μ, and standard deviation, σ, both on the log-scale
 * $$P(x;\boldsymbol\mu,\boldsymbol\sigma)=\frac{1}{x \sigma \sqrt{2 \pi}} \exp\left[-\frac{(\ln x - \mu)^2}{2 \sigma^2}\right]$$
 * LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale
 * $$P(x;\boldsymbol\mu,\boldsymbol {v})=\frac{1}{x \sqrt{v} \sqrt{2 \pi}} \exp\left[-\frac{(\ln x - \mu)^2}{2 v}\right]$$
 * LogNormal3(m,σ) with median, m, on the natural scale and standard deviation, σ, on the log-scale
 * $$P(x;\boldsymbol m,\boldsymbol \sigma) =\frac{1}{x \sigma \sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2 \sigma^2}\right]$$
 * LogNormal4(m,cv) with median, m, and coefficient of variation, cv, both on the natural scale
 * $$P(x;\boldsymbol m,\boldsymbol {cv})= \frac{1}{x \sqrt{\ln(cv^2+1)} \sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2\ln(cv^2+1)}\right]$$
 * LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale
 * $$P(x;\boldsymbol\mu,\boldsymbol \tau)=\sqrt{\frac{\tau}{2 \pi}} \frac{1}{x} \exp\left[-\frac{\tau}{2}(\ln x-\mu)^2\right]$$
 * LogNormal6(m,σg) with median, m, and geometric standard deviation, σg, both on the natural scale
 * $$P(x;\boldsymbol m,\boldsymbol {\sigma_g})=\frac{1}{x \ln(\sigma_g)\sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2 \ln^2(\sigma_g)}\right]$$
 * LogNormal7(μN,σN) with mean, μN, and standard deviation, σN, both on the natural scale
 * $$P(x;\boldsymbol {\mu_N},\boldsymbol {\sigma_N})= \frac{1}{x \sqrt{2 \pi \ln\left(1+\sigma_N^2/\mu_N^2\right)}} \exp\left(-\frac{\Big[ \ln x - \ln\frac{\mu_N}{\sqrt{1+\sigma_N^2/\mu_N^2}}\Big]^2}{2\ln(1+\sigma_N^2/\mu_N^2)}\right)$$

Examples for re-parameterization
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM and PopED. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.

For the transition $$\operatorname{LN2}(\mu, v) \to \operatorname{LN7}(\mu_N, \sigma_N)$$ following formulas hold $\mu_N = \exp(\mu+v/2) $ and $\sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}$.

For the transition $$\operatorname{LN7}(\mu_N, \sigma_N) \to \operatorname{LN2}(\mu, v)$$ following formulas hold $\mu = \ln\left( \mu_N / \sqrt{1+\sigma_N^2/\mu_N^2} \right) $ and $ v = \ln(1+\sigma_N^2/\mu_N^2)$.

All remaining re-parameterisation formulas can be found in the specification document on the project website.

Multiple, reciprocal, power

 * Multiplication by a constant: If $$X \sim \operatorname{Lognormal}(\mu, \sigma^2)$$ then $$a X \sim \operatorname{Lognormal}( \mu + \ln a,\ \sigma^2)$$ for $$ a > 0. $$
 * Reciprocal: If $$X \sim \operatorname{Lognormal}(\mu, \sigma^2)$$ then $$\tfrac{1}{X} \sim \operatorname{Lognormal}(-\mu,\ \sigma^2).$$
 * Power: If $$X \sim \operatorname{Lognormal}(\mu, \sigma^2)$$ then $$X^a \sim \operatorname{Lognormal}(a\mu,\ a^2 \sigma^2)$$ for $$a \neq 0.$$

Multiplication and division of independent, log-normal random variables
If two independent, log-normal variables $$X_1$$ and $$X_2$$ are multiplied [divided], the product [ratio] is again log-normal, with parameters $$\mu=\mu_1+\mu_2$$ [$$\mu=\mu_1-\mu_2$$] and $$\sigma$$, where $$\sigma^2=\sigma_1^2+\sigma_2^2$$. This is easily generalized to the product of $$n$$ such variables.

More generally, if $$X_j \sim \operatorname{Lognormal} (\mu_j, \sigma_j^2)$$ are $$n$$ independent, log-normally distributed variables, then $$Y = \textstyle\prod_{j=1}^n X_j \sim \operatorname{Lognormal} \Big(\textstyle \sum_{j=1}^n\mu_j,\ \sum_{j=1}^n \sigma_j^2 \Big).$$

Multiplicative central limit theorem
The geometric or multiplicative mean of $$n$$ independent, identically distributed, positive random variables $$X_i$$ shows, for $$n \to\infty$$, approximately a log-normal distribution with parameters $$\mu = E[\ln(X_i)]$$ and $$\sigma^2 = \mbox{var}[\ln(X_i)]/n$$, assuming $$\sigma^2$$ is finite.

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of $$\ln(X_i)$$ to all have finite variance and satisfy the other conditions of any of the many variants of the central limit theorem.

This is commonly known as Gibrat's law.

Other
A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).

The harmonic $$H$$, geometric $$G$$ and arithmetic $$A$$ means of this distribution are related; such relation is given by

 * $$H = \frac{G^2} A.$$

Log-normal distributions are infinitely divisible, but they are not stable distributions, which can be easily drawn from.

Related distributions
\sigma^2_Z &= \ln\!\left[ \frac{\sum e^{2\mu_j+\sigma_j^2}(e^{\sigma_j^2}-1)}{(\sum e^{\mu_j+\sigma_j^2/2})^2} + 1\right], \\ \mu_Z &= \ln\!\left[ \sum e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}. \end{align}$$ In the case that all $$X_j$$ have the same variance parameter $$\sigma_j=\sigma$$, these formulas simplify to $$\begin{align} \sigma^2_Z &= \ln\!\left[ (e^{\sigma^2}-1)\frac{\sum e^{2\mu_j}}{(\sum e^{\mu_j})^2} + 1\right], \\ \mu_Z &= \ln\!\left[ \sum e^{\mu_j} \right] + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}. \end{align}$$ For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.
 * If $$X \sim \mathcal{N}(\mu, \sigma^2)$$ is a normal distribution, then $$\exp(X) \sim \operatorname{Lognormal}(\mu, \sigma^2).$$
 * If $$X \sim \operatorname{Lognormal}(\mu, \sigma^2)$$ is distributed log-normally, then $$\ln(X) \sim \mathcal{N}(\mu, \sigma^2)$$ is a normal random variable.
 * Let $$X_j \sim \operatorname{Lognormal}(\mu_j, \sigma_j^2)$$ be independent log-normally distributed variables with possibly varying $$\sigma$$ and $$\mu$$ parameters, and $Y = \sum_{j=1}^n X_j$ . The distribution of $$Y$$ has no closed-form expression, but can be reasonably approximated by another log-normal distribution $$Z$$ at the right tail. Its probability density function at the neighborhood of 0 has been characterized  and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematically justified by Marlow ) is obtained by matching the mean and variance of another log-normal distribution: $$\begin{align}

The sum of correlated log-normally distributed random variables can also be approximated by a log-normal distribution $$\begin{align} S_+ &= \operatorname{E}\left[\sum_i X_i \right] = \sum_i \operatorname{E}[X_i] = \sum_i e^{\mu_i + \sigma_i^2/2} \\	\sigma^2_{Z} &= 1/S_+^2 \, \sum_{i,j} \operatorname{cor}_{ij} \sigma_i \sigma_j \operatorname{E}[X_i] \operatorname{E}[X_j] = 1/S_+^2 \, \sum_{i,j} \operatorname{cor}_{ij} \sigma_i \sigma_j e^{\mu_i+\sigma_i^2/2} e^{\mu_j+\sigma_j^2/2} \\	\mu_Z &= \ln\left( S_+ \right) - \sigma_{Z}^2/2 \end{align}$$
 * If $$X \sim \operatorname{Lognormal}(\mu, \sigma^2)$$ then $$X+c$$ is said to have a Three-parameter log-normal distribution with support $$x\in (c, +\infty)$$. $$\operatorname{E}[X+c] = \operatorname{E}[X] + c$$, $$\operatorname{Var}[X+c] = \operatorname{Var}[X]$$.
 * The log-normal distribution is a special case of the semi-bounded Johnson's SU-distribution.
 * If $$X\mid Y \sim \operatorname{Rayleigh}(Y)$$ with $$ Y \sim \operatorname{Lognormal}(\mu, \sigma^2)$$, then $$ X \sim \operatorname{Suzuki}(\mu, \sigma)$$ (Suzuki distribution).
 * A substitute for the log-normal whose integral can be expressed in terms of more elementary functions can be obtained based on the logistic distribution to get an approximation for the CDF $$ F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} + 1\right]^{-1}.$$ This is a log-logistic distribution.

Estimation of parameters
For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. Note that $$L(\mu, \sigma) = \prod_{i=1}^n \frac 1 {x_i} \varphi_{\mu,\sigma} (\ln x_i),$$ where $$\varphi$$ is the density function of the normal distribution $$\mathcal N(\mu,\sigma^2)$$. Therefore, the log-likelihood function is $$ \ell (\mu,\sigma \mid x_1, x_2, \ldots, x_n) = - \sum _i \ln x_i + \ell_N (\mu, \sigma \mid \ln x_1, \ln x_2, \dots, \ln x_n).$$

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, $$\ell$$ and $$\ell_N$$, reach their maximum with the same $$\mu$$ and $$\sigma$$. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations $$\ln x_1, \ln x_2, \dots, \ln x_n)$$, $$\widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad \widehat \sigma^2 = \frac {\sum_i \left( \ln x_i - \widehat \mu \right)^2} {n}.$$

For finite n, the estimator for $$\mu$$ is unbiased, but the one for $$\sigma$$ is biased. As for the normal distribution, an unbiased estimator for $$\sigma$$ can be obtained by replacing the denominator n by n−1 in the equation for $$\widehat\sigma^2$$.

When the individual values $$x_1, x_2, \ldots, x_n$$ are not available, but the sample's mean $$\bar x$$ and standard deviation s is, then the Method of moments can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation $$\operatorname{E}[X]$$ and variance $$\operatorname{Var}[X]$$ for $$\mu$$ and $$\sigma$$: $$ \mu = \ln\left(\frac{ \bar x} {\sqrt{1+\widehat\sigma^2/\bar x^2} } \right), \qquad \sigma^2 = \ln\left(1 + {\widehat\sigma^2} / \bar x^2 \right).$$

Interval estimates
The most efficient way to obtain interval estimates when analyzing log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.

Prediction intervals
A basic example is given by prediction intervals: For the normal distribution, the interval $$[\mu-\sigma,\mu+\sigma]$$ contains approximately two thirds (68%) of the probability (or of a large sample), and $$[\mu-2\sigma,\mu+2\sigma]$$ contain 95%. Therefore, for a log-normal distribution, $$[\mu^*/\sigma^*,\mu^*\cdot\sigma^*]=[\mu^* {}^\times\!\!/ \sigma^*]$$ contains 2/3, and $$[\mu^*/(\sigma^*)^2,\mu^*\cdot(\sigma^*)^2] = [\mu^* {}^\times\!\!/ (\sigma^*)^2]$$ contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.

Confidence interval for μ*
Using the principle, note that a confidence interval for $$\mu$$ is $$[\widehat\mu \pm q \cdot \widehat\mathop{se}]$$, where $$\mathop{se} = \widehat\sigma / \sqrt{n}$$ is the standard error and q is the 97.5% quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for $$\mu^*$$ (the median), is: $$[\widehat\mu^* {}^\times\!\!/ (\operatorname{sem}^*)^q]$$ with $$\operatorname{sem}^*=(\widehat\sigma^*)^{1/\sqrt{n}}$$

Confidence interval for μ
The literature discusses several options for calculating the confidence interval for $$\mu$$ (the mean of the log-normal distribution). These include bootstrap as well as various other methods.

Extremal principle of entropy to fix the free parameter σ
In applications, $$\sigma$$ is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that $$\sigma = \frac 1 \sqrt{6} $$

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution. This relationship is determined by the base of natural logarithm, $$e = 2.718\ldots$$, and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For example, the log-normal function with such $$\sigma$$ fits well with the size of secondarily produced droplets during droplet impact and the spreading of an epidemic disease.

The value $\sigma = 1 \big/ \sqrt{6}$ is used to provide a probabilistic solution for the Drake equation.

Occurrence and applications
The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "Multiplicative Central Limit Theorem". This is also known as Gibrat's law, after Robert Gibrat (1904–1980)  who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal. Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.

A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.

Specific examples are given in the following subsections. contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas. is a review article on log-normal distributions in neuroscience, with annotated bibliography.

Human behavior

 * The length of comments posted in Internet discussion forums follows a log-normal distribution.
 * Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.
 * The length of chess games tends to follow a log-normal distribution.
 * Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.

Biology and medicine

 * Measures of size of living tissue (length, skin area, weight).
 * Incubation period of diseases.
 * Diameters of banana leaf spots, powdery mildew on barley.
 * For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production.
 * The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.
 * The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
 * The PacBio sequencing read length follows a log-normal distribution.
 * Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).
 * Several pharmacokinetic variables, such as Cmax, elimination half-life and the elimination rate constant.
 * In neuroscience, the distribution of firing rates across a population of neurons is often approximately log-normal. This has been first observed in the cortex and striatum and later in hippocampus and entorhinal cortex, and elsewhere in the brain. Also, intrinsic gain distributions and synaptic weight distributions appear to be log-normal as well.
 * Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.
 * In operating-rooms management, the distribution of surgery duration.
 * In the size of avalanches of fractures in the cytoskeleton of living cells, showing log-normal distributions, with significantly higher size in cancer cells than healthy ones.

Chemistry

 * Particle size distributions and molar mass distributions.
 * The concentration of rare elements in minerals.
 * Diameters of crystals in ice cream, oil drops in mayonnaise, pores in cocoa press cake.

Hydrology

 * In hydrology, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.

 * The image on the right, made with CumFreq, illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution.

 * The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.

Social sciences and demographics

 * In economics, there is evidence that the income of 97%–99% of the population is distributed log-normally.  (The distribution of higher-income individuals follows a Pareto distribution).
 * If an income distribution follows a log-normal distribution with standard deviation $$\sigma$$, then the Gini coefficient, commonly use to evaluate income inequality, can be computed as $$G = \operatorname{erf}\left(\frac{\sigma }{2 }\right)$$ where $$\operatorname{erf}$$ is the error function, since $$ G=2 \Phi \left(\frac{\sigma }{\sqrt{2}}\right)-1$$, where $$\Phi(x)$$ is the cumulative distribution function of a standard normal distribution.
 * In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoit Mandelbrot have argued that log-Lévy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed, stock price distributions typically exhibit a fat tail. The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the central limit theorem.
 * In scientometrics, the number of citations to journal articles and patents follows a discrete log-normal distribution.
 * City sizes (population) satisfy Gibrat's Law. The growth process of city sizes is proportionate and invariant with respect to size. From the central limit theorem therefore, the log of city size is normally distributed.
 * The number of sexual partners appears to be best described by a log-normal distribution.

Technology

 * In reliability analysis, the log-normal distribution is often used to model times to repair a maintainable system.
 * In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." Also, the random obstruction of radio signals due to large buildings and hills, called shadowing, is often modeled as a log-normal distribution.
 * Particle size distributions produced by comminution with random impacts, such as in ball milling.
 * The file size distribution of publicly available audio and video data files (MIME types) follows a log-normal distribution over five orders of magnitude.
 * File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.
 * Sizes of text-based emails (1990s) and multimedia-based emails (2000s).
 * In computer networks and Internet traffic analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.
 * in physical testing when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.

title: Log-normal distribution


text:
Gwenwynwyn

Gwenwynwyn ab Owain Cyfeiliog (died c. 1216) was the last major ruler of mid Wales before the completion of the Norman English invasion. He was one of few native rulers to represent a real threat to the rule of Llywelyn the Great.

Lineage
Gwenwynwyn ruled southern Powys (Powys Wenwynwyn) from 1195, and was given charge of the kingdom following the retirement of his father Owain Cyfeiliog. Prior to this, in 1187, he had made an attack on Carreghofa Castle with his brother Cadwallon, in the course of which they killed their father's cousin and former ally, Owain Fychan. He had assisted Maelgwn ap Rhys in taking Aberystwyth Castle and capturing Maelgwn's brother Gruffydd ap Rhys II, whom he handed over to the English. His military ambitions were temporarily thwarted by his failure at the siege of Painscastle, when Gruffydd returned to battle.

Owain Cyfeiliog had been a notable poet as well as a leader of the resistance against English invasion, and chose to enter the abbey of Strata Marcella in later life; he died in 1197, two years after becoming a monk. Gwenwynwyn continued his father's strategy and is thought to have been one of the Welsh rulers who made an agreement with King Richard I of England not to attack his kingdom while he was away on crusade. In 1199, he made peace with Richard's successor, John. He was rewarded with marriage to an English noblewoman, Margaret Corbet, the daughter of the Lord of the border territory of Caus, around 1200, as well as the grant of the manor of Ashford.

Conflict with Llywelyn the Great
Gwenwynwyn's possession of Powys Wenwynwyn brought him into conflict with Llywelyn the Great, ruler of Gwynedd, who was keen to extend his own jurisdiction over the whole of Wales. In 1202 there was an uneasy peace between the two Welsh princes as Gwenwynwyn began attacking Marcher lords, thus losing John's favour; this eventually resulted in his being imprisoned at Shrewsbury.

Conflict with King John
King John favoured Gwenwynwyn until a marriage alliance was made between Llywelyn and John's illegitimate daughter. The two native princes kept their distance until 1207. As a result of Gwenwynwyn's aggression, John took him prisoner in 1208 and confiscated his lands, while Llywelyn seized Ceredigion, Aberystwyth, and Powys.

Gwenwynwyn's lands were restored to him as a result of John's intervention after two years. Freed by the King, he recovered most of his possessions, and accompanied John on an expedition into Wales. However, his continued resentment towards the English led him into an alliance with Llywelyn, which lasted from 1212 until 1216, when John again restored some of Gwenwynwyn's property and the two Welsh princes fell out again. Gwenwynwyn was pursued by Llywelyn, who took his lands, and he took refuge in Chester.

Death and legacy
Llywelyn invaded Powys, backed by other leaders of the Welsh alliance against the English. Gwenwynwyn, having been driven out of his lands, is believed to have died or been killed later that same year. He was succeeded by Gruffydd ap Gwenwynwyn, his son by Margaret Corbet, who was eventually restored to his lands after Llywelyn's death, and continued to feud with the princes of Gwynedd.

title: Gwenwynwyn


text:
Narbonne

Narbonne (, ,   ; Narbona ; Narbo ; Late Latin:Narbona) is a commune in Southern France in the Occitanie region. It lies 849 km from Paris in the Aude department, of which it is a sub-prefecture. It is located about 15 km from the shores of the Mediterranean Sea and was historically a prosperous port.

From the 14th century it declined following a change in the course of the river Aude. It is marginally the largest commune in Aude, but the capital of the Aude department is the smaller commune of Carcassonne.

Geography
Narbonne is linked to the nearby Canal du Midi and the river Aude by the Canal de la Robine, which runs through the centre of town. It is very close to the A9 motorway, which connects Montpellier and Nîmes to Perpignan and, across the border, to Barcelona in Spain. There is also a recently renovated train station which serves the TGV to Spain, Paris and Calais, which in turn connects to the Eurostar. Narbonne is only 10 km from Narbonne Plage (beach), but it is only 2 km from the nearest open water, at La Nautique, although there is no sand, rather pebbles.

Etymology
The source of the town's original name of Narbo is lost in antiquity, and it may have referred to a hillfort from the Iron Age close to the location of the current settlement or its occupants. The earliest known record of the area comes from the Greek Hecataeus of Miletus in the fifth century BC, who identified it as a Celtic harbor and marketplace at that time, and called its inhabitants the Ναρβαῖοι. In ancient inscriptions the name is sometimes rendered in Latin and sometimes translated into Iberian as Nedhena.

Under the Romans
Narbonne was established in Gaul by the Roman Republic in 118 BC, as Colonia Narbo Martius, colloquially Narbo, and made into the capital of the newly established province of Gallia Transalpina. It was located on the Via Domitia, the first Roman road in Gaul, built at the time of the foundation of the colony, and connecting Italy to Spain. Geographically, Narbonne was therefore located at a very important crossroads because it was situated where the Via Domitia connected to the Via Aquitania, which led toward the Atlantic through Tolosa and Burdigala. In addition, it was crossed by the river Aude. Surviving members of Julius Caesar's Legio X Equestris were given lands in the area that today is called Narbonne.

Politically, Narbonne gained importance as a competitor to Massilia (Marseille). Julius Caesar settled veterans from his 10th Legion there and attempted to develop its port while Marseille was supporting Pompey. Among the products of Narbonne, its rosemary-flower honey was famous among Romans.

Later, the province of Gallia Transalpina was renamed Gallia Narbonensis after the city, which became its capital. Seat of a powerful administration, the city enjoyed economic and architectural expansion. At that point, the city is thought to have had 30,000–50,000 inhabitants, and may have had as many as 100,000.

Under the Visigoths
According to Hydatius, in 462 the city was handed over to the Visigoths by a local military leader in exchange for support; as a result Roman rule ended in the city. It was subsequently the capital of the Visigothic province of Septimania, the only territory from Gaul to fend off the Frankish thrust after the Battle of Vouille (507). In 531, Frankish king, Childebert I, invaded Septimania and defeated Visigothic king, Amalaric near Narbonne and occupied the city. However, after Childebert's continued invasion to Catalonia failed, Amalaric's successor Theudis was able to reclaim the rich province of Septimania, including Narbonne. Following the loss of Toledo and Barcelona in 711/712, the last two kings of the Visigoths, Agila II and Ardo retreated to Narbonne, where they were able to resist Muslim attacks until 716.

Under the Arabs
For 40 years, from 719 to 759, Narbonne was part of the Umayyad Caliphate. The Umayyad governor Al-Samh captured Narbonne from the Kingdom of Visigoths in 719.

Under the Carolingians
The Carolingian Pepin the Short conquered Narbonne from the Arabs in 759 after which it became part of the Carolingian Viscounty of Narbonne. He invited prominent Jews from the Caliphate of Baghdad to settle in Narbonne and establish a major Jewish learning center for Western Europe. In the 12th century, the court of Ermengarde of Narbonne (reigned 1134 to 1192) presided over one of the cultural centers where the spirit of courtly love was developed.

The historian Arthur J. Zuckerman wrote in 1973 the book A Jewish Princedom in Feudal France, presenting the thesis that from the 8th to 10 centuries AD there was a Jewish vassal princedom based in Narbonne, given to the Jews by the Carolingian king Pepin as a gift of gratitude for their cooperation in the Frankish conquest of Narbonne from Al-Andalus in the year 759. This is however controversial, the book having been criticized by other historians.

Under the Capets
In the 11th and 12th centuries, Narbonne was home to an important Jewish exegetical school, which played a pivotal role in the growth and development of the Zarphatic (Judæo-French) and Shuadit (Judæo-Provençal) languages. Jews had settled in Narbonne from about the 5th century, with a community that numbered about 2,000 people in the 12th century. At this time, Narbonne was frequently mentioned in Talmudic works in connection with its scholars. One source, Abraham ibn Daud of Toledo, gives them an importance similar to the exilarchs of Babylon. In the 12th and 13th centuries, the community went through a series of ups and downs before settling into extended decline.

Narbonne loses its river and port
Narbonne itself fell into a slow decline in the 14th century, for a variety of reasons. One was due to a change in the course of the river Aude, which caused increased silting of the navigational access. The river, known as the Atax in ancient times, had always had two main courses which split close to Salelles; one fork going south through Narbonne and then to the sea close to the Clappe Massif, the other heading east to the etang at Vendres close to the current mouth of the river well to the east of the city. The Romans had improved the navigability of the river by building a dam near Salelles and also by canalising the river as it passed through its marshy delta to the sea (then as now the canal was known as the Robine.) A major flood in 1320 swept the dam away. The Aude river had a long history of overflowing its banks. When it was a bustling port, the distance from the coast was approximately 5 to 10 km, but at that time the access to the sea was deep enough when the river was in full spate which made communication between port and city unreliable. However, goods could easily be transported by land and in shallow barges from the ports (there were several: a main port and forward ports for larger vessels; indeed the navigability from the sea into the étang and then into the river had been a perennial problem)

The changes to the long seashore which resulted from the silting up of the series of graus or openings which were interspersed between the islands which made up the shoreline (St. Martin; St. Lucie) had a more serious impact than the change in course of the river. Other causes of decline were the plague and the raid of Edward, the Black Prince, which caused much devastation. The growth of other ports was also a factor.

Narbonne Cathedral
Narbonne Cathedral, dedicated to Saints Justus and Pastor, provides stark evidence of Narbonne's sudden and dramatic change of fortunes when one sees at the rear of the structure the enormously ambitious building programme frozen in time, for the cathedral—still one of the tallest in France—was never finished. The reasons are many, but the most important is that the completed cathedral would have required demolishing the city wall. The 14th century also brought the plague and a host of reasons for retaining the 5th-century (pre-Visigothic) walls.

Yet the choir, side chapels, sacristy, and courtyard remain intact, and the cathedral, although no longer the seat of a bishop or archbishop, remains the primary place of worship for the Roman Catholic population of the city, and is a major tourist attraction.

Building of the Canal de la Robine

From the sixteenth century, eager to maintain a link to important trade, the people of Narbonne began costly work to the vestiges of the river Aude's access to the sea so that it would remain navigable to a limited draft vessel and also serve as a link with the Royal Canal. This major undertaking resulted in the construction of the Canal de la Robine, which was finally linked with the Canal du Midi (then known as the Royal Canal) via the Canal de Jonction in 1776.

In the 19th century, the canal system in the south of France had to compete with an expanding rail network, which could ship goods more quickly. The canals kept some importance as they were used to support the flourishing wine trade.

Despite its decline from Roman times, Narbonne held on to its vital but limited importance as a trading route. This has continued in more recent centuries.

Sights

 * The cathedral dating from 1272
 * The Palais des Archevêques, the Archbishop's Palace, and its donjon with views over Narbonne
 * Musée Archeologique, an archaeological museum in the town centre (currently closed - November 2019, most sections will be moved to new museum Narbo Via which is planned to open in September 2020)
 * Vestiges archéologiques du Clos de la Lombarde002.JPGClos de la Lombarde - an archaeological site presenting the vestiges of Roman townhouses, bath houses, workshops from the 1st century BC to the 3rd century AD and the first Christian basilica in Narbonne (3rd/4th century AD). Link to website: http://www.amiscloslombarde.fr/ (site in French and English)
 * The Roman Horreum, a former grain warehouse, built underground as a cryptoporticus
 * Remains of the Via Domitia in the city center
 * The canal, Canal de la Robine, running through the centre of the town
 * The Halles covered market operates every day. The busiest times are Sunday and Thursday mornings.
 * The nearby limestone massif known as La Clape and the beach at Narbonne plage

Sport
Narbonne is home to the rugby union team RC Narbonne founded in 1907. It is an historic team in France, Narbonne have twice won the French first division title and reached a European final in 2001. They play at the Parc des Sports Et de l'Amitié (capacity 12,000). They wear orange and black.

Transport
The Gare de Narbonne railway station offers direct connections to Paris, Barcelona, Toulouse, Marseille and many regional destinations. An extensive local system of buses and routes operated by Citibus.fr allow for easy public transport within Narbonne and surrounding communities. Travellers wishing to connect by plane arrive by airports in nearby Béziers, Carcassonne, Perpignan, Toulouse or Montpellier, as Narbonne does not have an airport.

Personalities

 * Carus, Roman emperor, from 282 to 283, known for his late victories against the Sassanid empire and the Germanic tribes
 * Varro Atacinus, Roman poet
 * Saint Sébastien, third-century Christian saint and martyr
 * Makhir of Narbonne, medieval Jewish scholar
 * Moshe ha-Darshan (11th century), chief of the yeshiva of Narbonne
 * Bonfilh, a Jewish troubadour from the city
 * Kalonymus ben Todros, (d. ca. 1194) was a Provençal rabbi who flourished at Narbonne in the second half of the twelfth century
 * Joseph Barsalou (physician) (1600-1669), apothecary and physician whose family was from Narbonne
 * Guillaume Barthez de Marmorières, (1707-1799), civil engineer
 * Jean-Joseph Cassanéa De Mondonville, (1711-1772) violinist and composer
 * Léon Blum was born in Paris but was elected as Deputy for Narbonne in 1929, re-elected in 1932 and 1936
 * Pierre Reverdy, surrealist poet
 * Anaïs Napoleón, French-Spanish photographer
 * Charles Trenet, singer-songwriter
 * Dimitri Szarzewski, rugby player
 * Camille Lacourt, World champion swimmer
 * Ateyaba, French hip-hop artist
 * Alexandre Baron, racing driver
 * Benjamin Lariche, racing driver
 * Rabbinic family of Benveniste

International relations
Narbonne is twinned with:
 * 🇮🇹 Aosta, Italy
 * 🇮🇹 Grosseto, Italy
 * Salford, England
 * 🇩🇪 Weilheim, Germany

title: Narbonne


text:
Via Domitia

The Via Domitia was the first Roman road built in Gaul, to link Italy and Hispania through Gallia Narbonensis, across what is now Southern France. The route that the Romans regularised and paved was ancient when they set out to survey it, and traces the mythic route travelled by Heracles.

The construction of the road was commissioned by Gnaeus Domitius Ahenobarbus, whose name it bore, following the defeat of the Allobroges and Averni by himself and Quintus Fabius Maximus Allobrogicus in 122 BCE. Domitius also established a fortified garrison at Narbo (modern Narbonne) on the coast, near Hispania, to guard construction of the road. It soon developed into a full Roman colony Colonia Narbo Martius. The lands on the western part of the route, beyond the River Rhône had been under the control of the Averni who, according to Strabo, had stretched their control to Narbo and the Pyrenees.

Crossing the Alps by the easiest passage, the Col de Montgenèvre (1850 m), the Via Domitia followed the valley of the Durance, crossed the Rhône at Beaucaire passed through Nîmes (Nemausus) then followed the coastal plain along the Gulf of Lion. At Narbonne, it met the Via Aquitania (which led toward the Atlantic Ocean through Toulouse and Bordeaux). Thus Narbonne was a crucial strategic crossroads of the Via Domitia and the Via Aquitania, and it was an accessible, but easily defensible port at that time. This "cusp point" in the Roman westwards expansion and ensuing supply, communication and fortification was a very important asset, and was treated as such (see Narbonne). In between the cities that it linked, the Via Domitia was provided with a series of mansiones at distances of a day's journey for a loaded cart, at which shelter, provender and fresh horses could be obtained for travellers on official business.

The route as it was in Late Antiquity is represented in schematic fashion on the Tabula Peutingeriana.

Route
This route can be traced on topographical maps overprinted with the ancient route, in G. Castellve, J.-B. Compsa, J. Kotarba and A. Pezin, eds. Voies romaines du Rhône à l'Èbre: Via Domitia et Via Augusta (DAF 61) Paris 1997. ''At Ruscino, the road separates in two: the Inland Route and the Coastal Route, which rejoin at La Junquera. ''
 * Briançon (Brigantio)
 * Chorges (Caturigomagus)
 * Gap (Vapincum)
 * Le Monetier Allemont (Alabons)
 * Embrun (Eburodunum)
 * Sisteron (Segustero)
 * Lurs (Alaunium)
 * Céreste (Catuiacia)
 * Apt (Apta Julia)
 * Notre Dame des Lumières (Ad Fines)
 * Cavaillon (Cabellio)
 * Saint-Rémy-de-Provence (Glanum)
 * Saint-Gabriel (Ernaginum)
 * Beaucaire (Ugernum)
 * Nîmes (Nemausus)
 * Ambrussum
 * Lunel-Vieil
 * Castelnau-le-Lez (Sextantio)
 * Montpellier route remains unknown
 * Montbazin (Forum Domitii)
 * Mèze
 * Pinet
 * Saint-Thibéry (Cessero) and its Roman bridge
 * Béziers (Baeterris)
 * Narbonne (Narbo Martius) At Narbonne, a section of the Via Domitia is exposed in the Place de l'Hôtel de Ville. The Via Domitia crossed the Atax (the Aude) by a seven-arched bridge at the site of the Pont des Marchands.
 * Fitou (Ad Viscensimum)
 * Salses (Ad Salsulae)
 * Perpignan
 * Ruscino

Coastal Route
 * Elne (Illiberis)
 * Saint-Cyprien
 * Argelès
 * Collioure
 * Port-Vendres (Portus Veneris)
 * Banyuls

Inland Route
 * Montescot
 * Le Boulou
 * Les Cluses (Clausurae)
 * Le Perthus (Col de Panissars), at the Trophy of Pompey

Rejoins at:
 * La Junquera (Deciana)

Here the Via Augusta begins.

Roman bridges
There are the remains of several Roman bridges along the road, including the Roman Bridge of Saint-Thibéry, the Pont Ambroix at Ambrussum, the Pont Julien and the Pont Serme.

title: Via Domitia


text:
Stormarn (district)

Stormarn is a district in Schleswig-Holstein, Germany. It is bounded by (from the north and clockwise) the districts of Segeberg and Ostholstein, the city of Lübeck, the district of Lauenburg, and the city-state of Hamburg.

History
In medieval times the name Stormarn was applied to a larger area, of which the present-day district is only the eastern half. It was the home of the Saxon tribe the Sturmarii.

Stormarn became a part of Holstein in the 12th century. When Schleswig-Holstein became a province of Prussia in 1867, the Prussian administration established the district of Stormarn, with Wandsbek as its capital. In 1937 the southwestern part of the district was incorporated into Hamburg, and the district lost half of its population. Since Wandsbek was now a borough of Hamburg, the capital was moved to Bad Oldesloe after the war. In 1970 Stormarn again lost a substantial portion of its territory, when the city of Norderstedt was founded in order to become a part of the Segeberg district.

Geography
The river Trave runs through the district before entering the city of Lübeck. In the southernmost portion of Stormarn there are the eastern suburbs of the Hamburg metropolitan area.

Coat of arms
The coat of arms displays a white swan, with a crown around its neck, on red ground. The swan has been the heraldic animal of Stormarn since the Middle Ages.

title: Stormarn (district)