text:
Aida

Aida (or Aïda, ) is a tragic opera in four acts by Giuseppe Verdi to an Italian libretto by Antonio Ghislanzoni. Set in the Old Kingdom of Egypt, it was commissioned by Cairo's Khedivial Opera House and had its première there on 24 December 1871, in a performance conducted by Giovanni Bottesini. Today the work holds a central place in the operatic canon, receiving performances every year around the world. At New York's Metropolitan Opera alone, Aida has been sung more than 1,100 times since 1886. Ghislanzoni's scheme follows a scenario often attributed to the French Egyptologist Auguste Mariette, but Verdi biographer Mary Jane Phillips-Matz argues that the source is actually Temistocle Solera.

Elements of the opera's genesis and sources
Isma'il Pasha, Khedive of Egypt, commissioned Verdi to write an opera to celebrate the opening of the Suez Canal, but Verdi declined. However, Auguste Mariette, a French Egyptologist, proposed to Khedive Pasha a plot for a celebratory opera set in ancient Egypt. Khedive Pasha referred Mariette to theatre manager Camille du Locle, who sent Mariette's story idea to Verdi. Eventually, Verdi agreed to compose an opera based on that story, for 150,000 francs.

Because the scenery and costumes were stuck in the French capital during the Siege of Paris (1870–71) of the ongoing Franco-Prussian War, the premiere was delayed and Verdi's Rigoletto was performed instead. The first opera performed at the Khedivial Opera House, Aida eventually premiered in Cairo on Christmas Eve of 1871.

Cairo premiere and initial success in Italy
Verdi originally chose to write a brief orchestral prelude instead of a full overture for the opera. He then composed an overture of the "potpourri" variety to replace the original prelude. However, in the end he decided not to have the overture performed because of its—his own words—"pretentious silliness". This overture, never used today, was given a rare broadcast performance by Arturo Toscanini and the NBC Symphony Orchestra on 30 March 1940, but was never commercially issued.

Aida met with great acclaim when it finally opened in Cairo on 24 December 1871. The costumes and accessories for the première were designed by Auguste Mariette, who also oversaw the design and construction of the sets, which were made in Paris by the Opéra's scene painters Auguste-Alfred Rubé and Philippe Chaperon (acts 1 and 4) and Édouard Desplechin and Jean-Baptiste Lavastre (acts 2 and 3), and shipped to Cairo. Although Verdi did not attend the premiere in Cairo, he was most dissatisfied with the fact that the audience consisted of invited dignitaries, politicians and critics, but no members of the general public. He therefore considered the Italian (and European) première, held at La Scala, Milan on 8 February 1872, and a performance in which he was heavily involved at every stage, to be its real première.

Verdi had also written the role of Aida for the voice of Teresa Stolz, who sang it for the first time at the Milan première. Verdi had asked her fiancé, Angelo Mariani, to conduct the Cairo première, but he declined, so Giovanni Bottesini filled the gap. The Milan Amneris, Maria Waldmann, was his favourite in the role and she repeated it a number of times at his request.

Aida was received with great enthusiasm at its Milan première. The opera was soon mounted at major opera houses throughout Italy, including the Teatro Regio di Parma (20 April 1872), the Teatro di San Carlo (30 March 1873), La Fenice (11 June 1873), the Teatro Regio di Torino (26 December 1874), the Teatro Comunale di Bologna (30 September 1877, with Giuseppina Pasqua as Amneris and Franco Novara as the King), and the Teatro Costanzi (8 October 1881, with Theresia Singer as Aida and Giulia Novelli as Amneris) among others.

Other 19th-century performances
Details of important national and other premières of Aida follow:
 * Argentina: 4 October 1873, at the original Teatro Colón, Buenos Aires, located at Rivadavia and Reconquista, then replaced by the headquarters of the Bank of the Argentine Nation.
 * United States: 26 November 1873, Academy of Music in New York City, with Ostava Torriani in the title role, Annie Louise Cary as Amneris, Italo Campanini as Radamès, Victor Maurel as Amonasro, and Evasio Scolara as the King
 * Germany: 20 April 1874, Berlin State Opera, with Mathilde Mallinger as Aida, Albert Niemann as Radamès, and Franz Betz as Amonasro
 * Austria: 29 April 1874, Vienna State Opera, with Amalie Materna as Amneris
 * Hungary: 10 April 1875, Hungarian State Opera House, Budapest
 * France: 22 April 1876, Théâtre-Lyrique Italien, Salle Ventadour, Paris, with almost the same cast as the Milan premiere, but with Édouard de Reszke making his debut as the King. This performance was conducted by Verdi.
 * United Kingdom: 22 June 1876, Royal Opera House, Covent Garden, with Adelina Patti as Aida, Ernesto Nicolini as Radamès, and Francesco Graziani as Amonasro
 * Australia: 6 September 1877, Royal Theatre, Melbourne
 * Munich: 1877, Bavarian State Opera, with Josephine Schefsky as Amneris
 * Stockholm: 16 February 1880, Royal Swedish Opera in Swedish, with Selma Ek in the title role
 * Palais Garnier, Paris: 22 March 1880, sung in French, with Gabrielle Krauss as Aida, Rosine Bloch as Amnéris, Henri Sellier as Radamès, Victor Maurel as Amonasro, Georges-François Menu as the King, and Auguste Boudouresque as Ramphis.
 * Metropolitan Opera, New York: 12 November 1886, conducted by Anton Seidl, with Therese Herbert-Förster (the wife of Victor Herbert) in the title role, Carl Zobel as Radamès, Marianne Brandt as Amneris, Adolf Robinson as Amonasro, Emil Fischer as Ramfis, and Georg Sieglitz as the King.
 * Rio de Janeiro: 30 June 1886, Theatro Lyrico Fluminense. During rehearsals, the performers of the Italian touring opera company had disagreements with the local conductor Leopoldo Miguez, described as "inept". After the failure of two replacement conductors, Arturo Toscanini, at the time a 19-year-old cellist who was assistant chorus master, was persuaded to conduct the performance. He conducted the entire opera from memory with great success—the start of a great career.

20th century and beyond

A complete concert version of the opera was given in New York City in 1949. Conducted by Toscanini with Herva Nelli as Aida and Richard Tucker as Radamès, it was televised on the NBC television network. Due to the length of the opera, it was divided into two telecasts, preserved on kinescopes, and later released on video by RCA and Testament. The audio portion of the broadcast, including some remakes in June 1954, was released on LP and CD by RCA Victor. Other notable performances from this period include a 1955 performance conducted by Tullio Serafin with Maria Callas as Aida and Richard Tucker as Radamès and a 1959 performance conducted by Herbert von Karajan with Renata Tebaldi as Aida and Carlo Bergonzi as Radamès.

La Scala mounted a lavish new production of Aida designed by Franco Zeffirelli for the opening night of its 2006/2007 season. The production starred Violeta Urmana in the title role and Roberto Alagna as Radamès. Alagna subsequently made the headlines when he was booed for his rendition of "Celeste Aida" during the second performance, walked off the stage, and was dismissed from the remainder of the run. The production continued to cause controversy in 2014 when Zeffirelli protested La Scala's rental of the production to the Astana Opera House in Kazakhstan without his permission. According to Zeffirelli, the move had doomed his production to an "infamous and brutal" fate. Aida continues to be a staple of the standard operatic repertoire. It is frequently performed in the Verona Arena, and is a staple of its renowned opera festival.

Instrumentation
3 flutes (3rd also piccolo), 2 oboes, English horn, 2 clarinets, bass clarinet, 2 bassoons, 4 horns, 2 trumpets, 3 trombones, cimbasso, timpani, triangle, bass drum, cymbals, tam-tam, harp, strings; on-stage banda: 6 Egyptian trumpets ("Aida trumpets"), military band, harp

Setting
The libretto does not specify a precise time period, so it is difficult to place the opera more specifically than the Old Kingdom. For the first production, Mariette went to great efforts to make the sets and costumes authentic. Considering the consistent artistic styles throughout the 3000-year history of ancient Egypt, a given production does not particularly need to choose a specific time period within the larger frame of ancient Egyptian history.

Synopsis
Backstory: The Egyptians have captured and enslaved Aida, an Ethiopian princess. An Egyptian military commander, Radamès, struggles to choose between his love for her and his loyalty to the King of Egypt. To complicate the story further, the King's daughter Amneris is in love with Radamès, although he does not return her feelings.

Act 1
Scene 1: A hall in the King's palace; through the rear gate the pyramids and temples of Memphis are visible

Ramfis, the high priest of Egypt, tells Radamès, the young warrior, that war with the Ethiopians seems inevitable, and Radamès hopes that he will be chosen as the Egyptian commander (Ramfis, Radamès: "Sì, corre voce l'Etiope ardisca" / Yes, it is rumored that Ethiopia dares once again to threaten our power).

Radamès dreams both of gaining victory on the battlefield and of Aida, an Ethiopian slave, with whom he is secretly in love (Radamès: "Se quel guerrier io fossi! ... Celeste Aida" / Heavenly Aida). Aida, who is also secretly in love with Radamès, is the captured daughter of the Ethiopian King Amonasro, but her Egyptian captors are unaware of her true identity. Her father has invaded Egypt to deliver her from servitude.

Amneris, the daughter of the Egyptian King, enters the hall. She too loves Radamès, but fears that his heart belongs to someone else (Radamès, Amneris: "Quale insolita gioia nel tuo sguardo" / In your looks I trace an unwonted joy).

Aida appears and, when Radamès sees her, Amneris notices that he looks disturbed. She suspects that Aida could be her rival, but is able to hide her jealousy and approach Aida (Amneris, Aida, Radamès: "Vieni, o diletta, appressati" / Come, O delight, come closer).

The King enters, along with the High Priest, Ramfis, and the whole palace court. A messenger announces that the Ethiopians, led by King Amonasro, are marching towards Thebes. The King declares war and proclaims that Radamès is the man chosen by the goddess Isis to be the leader of the army (The King, Messenger, Radamès, Aida, Amneris, Ramfis, chorus: "Alta cagion v'aduna .. Guerra, guerra, guerra!" / Oh fate o'er Egypt looming .. War, war, war!). Upon receiving this mandate from the King, Radamès proceeds to the temple of Vulcan to take up the sacred arms (The King, Radamès, Aida, Amneris, chorus: "Su! del Nilo al sacro lido" .. (reprise) "Guerra, guerra, guerra!" / On! Of Nilus' sacred river, guard the shores .. (reprise) War, war, war!).

Alone in the hall, Aida feels torn between her love for her father, her country, and Radamès (Aida: "Ritorna vincitor!" / Return a conqueror!).

Scene 2: Inside the Temple of Ptah

Solemn ceremonies and dances by the priestesses take place (High Priestess, chorus, Radamès: "Possente Ftha ... Tu che dal nulla" / O mighty Ptah). This is followed by the installation of Radamès to the office of commander-in-chief (High Priestess, chorus, Ramfis, Radamès: "Immenso Ftha .. Mortal, diletto ai Numi" / O mighty one, guard and protect!). All present in the temple pray fervently for the victory of Egypt and protection for their warriors ("Nume, custode e vindice"/ Hear us, O guardian deity).

Act 2
Scene 1: The chamber of Amneris

Dances and music to celebrate Radamès' victory take place (Chorus, Amneris: "Chi mai fra gli inni e i plausi" / Our songs his glory praising). However, Amneris is still in doubt about Radamès' love and wonders whether Aida is in love with him. She tries to forget her doubt, entertaining her worried heart with the dance of Moorish slaves (Chorus, Amneris: "Vieni: sul crin ti piovano" / Come bind your flowing tresses).

When Aida enters the chamber, Amneris asks everyone to leave. By falsely telling Aida that Radamès has died in the battle, she tricks her into professing her love for him. In grief, and shocked by the news, Aida confesses that her heart belongs to Radamès eternally (Amneris, Aida: "Fu la sorte dell'armi a' tuoi funesta" / The battle's outcome was cruel for your people).

This confession fires Amneris with rage, and she plans on taking revenge on Aida. Ignoring Aida's pleadings (Amneris, Aida, chorus: "Su! del Nilo al sacro lido" / Up! at the sacred shores of the Nile), Amneris leaves her alone in the chamber.

Scene 2: The grand gate of the city of Thebes

Radamès returns victorious and the troops march into the city (Chorus, Ramfis: "Gloria all'Egitto, ad Iside" / Glory to Egypt, [and] to Isis!).

The Egyptian king decrees that on this day the triumphant Radamès may have anything he wishes. The Ethiopian captives are led onstage in chains, Amonasro among them. Aida immediately rushes to her father, who whispers to her to conceal his true identity as King of Ethiopia from the Egyptians. Amonasro deceptively proclaims to the Egyptians that the Ethiopian king (referring to himself) has been slain in battle. Aida, Amonasro, and the captured Ethiopians plead with the Egyptian King for mercy, but Ramfis and the Egyptian priests call for their death (Aida, Amneris, Radamès, The King, Amonasro, chorus: "Che veggo! .. Egli? .. Mio padre! .. Anch'io pugnai .. Struggi, o Re, queste ciurme feroci" / What do I see?.. Is it he? My father? .. Destroy, O King, these ferocious creatures).

Claiming the reward promised by the King of Egypt, Radamès pleads with him to spare the lives of the prisoners and to set them free. The King grants Radamès' wish, and declares that he (Radamès) will be his (the King's) successor and will marry the King's daughter (Amneris). (Aida, Amneris, Radamès, Ramfis, The King, Amonasro, chorus: "O Re: pei sacri Numi! .. Gloria all'Egitto" / O King, by the sacred gods ... Glory to Egypt!). At Ramfis' suggestion to the King, Aida and Amonasro remain as hostages to ensure that the Ethiopians do not avenge their defeat.

Act 3
On the banks of the Nile, near the Temple of Isis

Prayers are said (Chorus, High Priestess, Ramfis, Amneris: "O tu che sei d'Osiride" / O thou who to Osiris art) on the eve of Amneris and Radamès' wedding in the Temple of Isis. Outside, Aida waits to meet with Radamès as they had planned (Aida: "Qui Radamès verra .. O patria mia" / Oh, my dear country!).

Amonasro appears and orders Aida to find out the location of the Egyptian army from Radamès. Aida, torn between her love for Radamès and her loyalty to her native land and to her father, reluctantly agrees. (Aida, Amonasro: "Ciel, mio padre! .. Rivedrai le foreste imbalsamate" / Once again shalt thou gaze). When Radamès arrives, Amonasro hides behind a rock and listens to their conversation.

Radamès affirms that he will marry Aida ("Pur ti riveggo, mia dolce Aida .. Nel fiero anelito"; "Fuggiam gli ardori inospiti .. Là, tra foreste vergini" / I see you again, my sweet Aida!), and Aida convinces him to flee to the desert with her.

To facilitate their escape, Radamès suggests using a secure route free from the risk of discovery and discloses the location his army plans to attack. Upon hearing this, Amonasro emerges from hiding and reveals his true identity. Radamès, horrified, realizes he has inadvertently divulged a critical military secret to the enemy. Meanwhile, Amneris and Ramfis leave the temple, spot Radamès in conference with the enemy, and summon the imperial guards. Amonasro, armed with a dagger, intends to kill Amneris and Ramfis before they can alert the guards, but Radamès disarms him. He swiftly instructs Amonasro to flee with Aida and surrenders himself to the guards, who arrest him as a traitor.

Act 4

''Scene 1: A hall in the Temple of Justice. To one side is the door leading to Radamès' prison cell''

Amneris desires to save Radamès ("L'aborrita rivale a me sfuggia" / My hated rival has escaped me). She calls for the guard to bring him to her.

She asks Radamès to deny the accusations, but Radamès, who does not wish to live without Aida, refuses. He is relieved to know Aida is still alive and hopes she has reached her own country (Amneris, Radamès: "Già i Sacerdoti adunansi" / Already the priests are assembling).

Offstage, Ramfis recites the charges against Radamès and calls on him to defend himself, but he stands mute, and is condemned to death as a traitor. Amneris, who remains onstage, protests that Radamès is innocent, and pleads with the priests to show mercy. The priests sentence him to be buried alive; Amneris weeps and curses the priests as he is taken away (Judgment scene, Amneris, Ramfis, and chorus: "Ahimè! .. morir mi sento .. Radamès, è deciso il tuo fato" / Alas .. I feel death .. Radamès, your fate is decided).

Scene 2: The lower portion of the stage shows the vault in the Temple of Ptah; the upper portion represents the temple itself

Radamès has been taken into the lower floor of the temple and sealed up in a dark vault, where he thinks that he is alone. As he hopes that Aida is in a safer place, he hears a sigh and then sees Aida. She has hidden herself in the vault in order to die with Radamès (Radamès: "La fatal pietra sovra me si chiuse" / The fatal stone now closes over me). They accept their terrible fate (Radamès: "Morir! Sì pura e bella" / To die! So pure and lovely!) and bid farewell to Earth and its sorrows (duet "Invan! Tutto è finito ... O terra addio"). Above the vault in the temple of Ptah, Amneris weeps and prays to the goddess Isis. In the vault below, Aida dies in Radamès' arms as the priests, offstage, pray to the god Ptah. (Chorus, Aida, Radamès, Amneris: "Immenso Ftha" / Almighty Ptah).

Adaptations
The 1952 Broadway musical My Darlin' Aida, set on a plantation in Tennessee in the first year of the American Civil War, is based on the opera and uses Verdi's music. The opera has been adapted for motion pictures on several occasions, most notably in a 1953 production which starred Lois Maxwell as Amneris and Sophia Loren as Aida, and a 1987 Swedish production. In both cases, the lead actors lip-synched to recordings by actual opera singers. In the case of the 1953 film, Ebe Stignani sang as Amneris, while Renata Tebaldi sang as Aida. The opera's story, but not its music, was used as the basis for a 1998 musical of the same name written by Elton John and Tim Rice. The opera has been portrayed in the 2001 Italian animated film Aida of the Trees (Aida degli alberi). The characters are seen as anthropomorphic creatures between the fictional kingdoms of Alborea and Petra as the star-crossed lovers must find a way to unify their worlds while facing off against the evil high priest Ramfis.

title: Aida


text:
Talk:Brain-wash

Move discussion in progress (2017)
There is a move discussion in progress on Talk:Mind control which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 23:00, 8 April 2017 (UTC)
 * As of this comment, discussion is at Talk:Brainwashing/Archive 4. Rotideypoc41352 (talk · contribs) 12:50, 15 June 2024 (UTC)

Move discussion in progress (2024)
There is a move discussion in progress on Talk:Mind control (disambiguation) which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 15:35, 4 June 2024 (UTC)

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

title: Talk:Brain-wash


text:
Micky Dolenz

George Michael Dolenz Jr. (born March 8, 1945) is an American musician and actor. He was the drummer and one of two primary vocalists for the pop rock band the Monkees (1966–1970, and reunions until 2021), and a co-star of the TV series The Monkees (1966–1968). Dolenz is the last surviving member of the band.

Life and entertainment career
Dolenz was born in Los Angeles, the son of actors George Dolenz and Janelle Johnson. He has three sisters, Gemma Marie ("Coco"), Deborah, and Kathleen ("Gina"). Gemma's nickname, Coco, is a shortened form of "Coco Sunshine", a nickname given to her as a child by Micky. Coco was a frequent guest on the set of The Monkees TV show and sometimes a guest performer on records by the Monkees, singing background vocals or duetting with Micky. She often performs as a member of Micky's backing band during his concerts.

Dolenz suffered from Perthes disease as a child, affecting his hip joint and right leg, leaving that leg weaker (and shorter) than the other. This resulted in Dolenz adapting an unorthodox drum setup – right-handed and left-footed – in his musical career.

Circus Boy
Dolenz began his show-business career in 1956 when he starred in a children's TV show called Circus Boy under the name Mickey Braddock. He played Corky, an orphaned water boy for the elephants in a one-ring circus at the start of the 20th century. The program ran for two seasons, after which Dolenz made sporadic appearances on network television shows and pursued his education. Dolenz went to Ulysses S. Grant High School in Valley Glen and graduated in 1962. In 1964, he was cast as Ed in the episode "Born of Kings and Angels" of the NBC education drama series Mr. Novak, starring James Franciscus as an idealistic Los Angeles teacher. Dolenz was attending college in Los Angeles when he was hired for the "drummer" role in NBC's The Monkees.

Early musical career
Dolenz originally had his own rock band called "Micky and the One-Nighters" in the early- to mid-1960s with himself as lead singer. He had already begun writing his own songs. According to Dolenz, his band's live stage act included rock songs, cover songs, and even some R&B. One of his favorite songs to sing was Chuck Berry's "Johnny B. Goode", which he sang at his Monkees audition, resulting in his being hired as one of the cast/band members. He recorded two 45s in 1965 that went unreleased until the Monkees' success in 1967. Issued on the Challenge label, the recordings were "Don't Do It" b/w "Plastic Symphony III" and "Huff Puff" b/w "Fate (Big Ben)". Neither B-side on the Challenge 45s is by Dolenz, but rather a band later credited as The Obvious.

The Monkees
In 1965, Dolenz was cast in the television sitcom The Monkees and became the drummer and a lead vocalist in the band created for the show. Tommy Boyce and Bobby Hart, writers of many of the Monkees' songs, observed quickly that when brought into the studio together, the four actors would try to make each other laugh. Because of this, the writers often brought in each singer individually. The antics escalated until Dolenz poured a cup of ice on Don Kirshner's head. At the time, Dolenz did not know Kirshner on sight.

During tension-filled times, Nesmith and Peter Tork voluntarily turned over lead vocal duties to Dolenz on their own compositions.

Dolenz wrote a few of the band's self-penned songs, most prominently "Randy Scouse Git" from the album Headquarters. He provided the lead vocals for such hits as "Last Train to Clarksville", "Pleasant Valley Sunday", and "I'm a Believer".

Dolenz purchased the third modular Moog synthesizer sold commercially. (The first two belonged to Wendy Carlos and Buck Owens.) His performance on the Monkees' song "Daily Nightly" (written by Nesmith), from the album Pisces, Aquarius, Capricorn & Jones Ltd., was one of the first uses of the synthesizer on a rock recording. He eventually sold his instrument to Bobby Sherman.

He is the last surviving member of the Monkees (after Davy Jones's death in 2012, Tork's in 2019, and Nesmith's in 2021). He is the only member of the Monkees who was part of every lineup from the band's inception, and the only member with contemporary recordings of his vocals on all studio albums.

Solo MGM recordings
The Moog synthesizer that Dolenz had bought proved vital when he composed a song entitled "Easy on You" in 1971; he began recording it in his home studio, playing acoustic guitar and drums, and using the Moog like a keyboard. With that song completed, he next invited former Monkee Peter Tork over to help with more recordings. Then, a fortuitous street encounter led to former Monkee stand-in David Price joining, as well as contributing a rock song he had written called "Oh Someone". With Dolenz on drums and vocals, Tork on bass, and Price on rhythm guitar, the song was completed in only two hours; subsequently, guitarist B.J. Jones came in two days later and added lead guitar. With these two songs recorded, Dolenz contacted his former high school friend Mike Curb, then the head of MGM Records; after playing the songs for Curb, Dolenz was immediately signed to MGM.

Dolenz recorded and released songs for MGM for about three years (with a few of the songs being credited to Starship, an ersatz group, not the later Jefferson Starship). After the first year, Dolenz's friend Harry Nilsson contributed his song "Daybreak", also arranging and producing the recording, which included Keith Allison on guitar, former Monkees producer Chip Douglas on bass, and steel-guitarist Orville "Red" Rhodes.

By early 1974, with no chart successes to date, Dolenz headed to England, and with Tony Scotti, he cut four songs for MGM: two rock classics, "Splish Splash" and "Purple People Eater", as well as "I Hate Rock and Roll" and a new song, "Wing Walker". Meanwhile, Mike Curb left MGM and joined Warner Bros. Records. Dolenz's association with MGM then ended (and those final four songs remained unreleased).

Dolenz, Jones, Boyce & Hart
In part because of reruns of The Monkees on Saturday mornings and in syndication, The Monkees Greatest Hits charted in 1976. The LP, issued by Arista (a subsidiary of Columbia Pictures), was actually a repackaging of a 1972 compilation LP called Re-Focus that had been issued by Arista's previous label imprint, Bell Records, also owned by Columbia Pictures.

Dolenz and Jones took advantage of this, joining ex-Monkees songwriters Tommy Boyce and Bobby Hart to tour the United States. They could not use the Monkees name for legal reasons, but from 1975 to 1977, as the "Golden Hits of The Monkees" show ("The Guys Who Wrote 'Em and the Guys Who Sang 'Em!"), they successfully performed in smaller venues such as state fairs and amusement parks, as well as making stops in Japan, Thailand, and Singapore. They also released an album of new material, appropriately called Dolenz, Jones, Boyce & Hart, and a live album, Concert in Japan, was released by Capitol in 1976.

A Christmas single (credited to Dolenz, Jones and Tork) was produced by Chip Douglas and released on his own label in 1976. The single featured Douglas's and Howard Kaylan's "Christmas Is My Time of Year" (originally recorded by a 1960s supergroup, The Christmas Spirit), with a B-side of Irving Berlin's "White Christmas" (Douglas released a remixed version of the single, with additional overdubbed instruments, in 1986). Tork also joined Dolenz, Jones, Boyce & Hart on stage at Disneyland on July 4, 1976, and also joined Dolenz and Jones on stage at the Starwood in Hollywood, California in 1977.

Notable stage work
In 1977, he performed with former bandmate Davy Jones in a stage production of the Harry Nilsson musical The Point! at London's Mermaid Theatre, playing and singing the part of the "Count's Kid" and the Leafman to Jones' starring role as Oblio (according to the CD booklet). An original cast recording was made and released. The comedic chemistry of Dolenz and Jones proved so strong that the show was revived in 1978 with Nilsson inserting additional comedy for the two, plus two more songs, with one of them ("Gotta Get Up") being sung by Dolenz and Jones together. The show was considered so good that it was planned to be revived again in 1979, but it proved cost-prohibitive. After the show's run, Dolenz remained in England and began directing for stage and television, as well as producing several of the shows he directed. From August to September 2006, Dolenz played Charlemagne at the Goodspeed Opera House for the revival of the musical Pippin in East Haddam, Connecticut. He also toured in that role. Also in the mid-2000s, Dolenz played the role of Zoser in the Broadway production of Elton John and Tim Rice's Aida.

After Monkees television and film career
After The Monkees television show ended, Dolenz continued performing providing voice-overs for a number of Saturday-morning cartoon series including The Funky Phantom, Partridge Family 2200 A.D., The Scooby-Doo Show, Butch Cassidy and the Sundance Kids, These Are the Days, Devlin, and Wonder Wheels (from The Skatebirds). Dolenz provided the voice of Arthur in the first season of the animated series The Tick. In 1972, Dolenz played Vance in the murder mystery film Night of the Strangler. He was featured in an episode of Adam-12, entitled "Dirt Duel" (season 5, episode 1), and an episode of Cannon, entitled "Bitter Legion" (season 2, episode 3). Dolenz provided the voice of Two-Face's twin henchmen Min and Max in the two-part episode "Two-Face" on Batman: The Animated Series. In a September 2006 radio interview, Dolenz reported that he was the current voice of Snuggle the Fabric Softener Bear at that time. In 2017, Dolenz returned to voice-over, providing the voice of Wendell the Love Grub, as well as singing the featured song, in the Cartoon Network series Mighty Magiswords half-hour episode, "The Saga of Robopiggeh!". Dolenz recorded the voice-acting remotely in New York weeks before his Good Times! tour.

Both Dolenz and Michael Nesmith auditioned for the role of Arthur "The Fonz" Fonzarelli on Happy Days, but neither was selected, as they were both taller than lead actor Ron Howard and co-stars Anson Williams and Don Most. In 1975, Dolenz acted in Linda Lovelace for President, starring Linda Lovelace.

In 1994–95, Dolenz played in two episodes of the sitcom Boy Meets World; in the first one (entitled "Band on the Run"), he played Norm, a bandmate of Alan Matthews. In 1995, he joined Davy Jones and Peter Tork in episode eight of the third season (entitled "Rave On"), although they did not play the Monkees, per se – Dolenz's character is "Gordy", while Davy Jones is "Reginald Fairfield" and Tork is "Jedidiah Lawrence". However, at the climax of the program, the three are put on stage together and perform the classic Buddy Holly song "Not Fade Away", and the Temptations' "My Girl". As an inside joke, actor Dave Madden, who had played the manager on The Partridge Family, cameoed as a manager; he suddenly appears, wanting to handle the "new" group, and tells them that they "could be bigger than The Beatles", which they all scoff at.

In 2007, he appeared in Rob Zombie's remake of Halloween as Derek Allan, the owner of a gun shop.

On January 29, 2011, Dolenz appeared in the Syfy Channel film Mega Python vs. Gatoroid, alongside Debbie Gibson and Tiffany. On February 21, 2015, he had a cameo as himself in the Adult Swim TV special Bagboy. In 2017, he appeared as himself on the sitcom Difficult People.

Directorial work
In 1980, Dolenz produced and directed the British television sitcom Metal Mickey, featuring a large metallic robot with the catch-phrase "boogie boogie". In 1981, he directed a short film based on the sketch "Balham, Gateway to the South", with Robbie Coltrane playing multiple roles. In the early 1980s, Dolenz directed a stage adaptation of Bugsy Malone. He was producer of the TV show Luna in 1983–84.

MTV reignites Monkee Mania
In 1986, a screening of the entire Monkees television series by MTV led to renewed interest in the band, followed by a single, "That Was Then, This Is Now", which reached No. 20 on the Billboard Hot 100 in the U.S. and No.41 in Canada, a 20th-anniversary tour, a greatest hits album, and a brand new LP, Pool It! in 1987. The band's original albums were reissued, and all hit the record charts at the same time.

Beginning in 1986, Dolenz joined the other ex-Monkees for several reunion tours and toured extensively as a solo artist.

WCBS-FM
On January 10, 2005, Dolenz replaced Dan Taylor as the morning disc jockey at oldies radio station WCBS-FM in New York. On June 3, 2005, Dolenz celebrated his 100th show with a special morning show at B.B. King's. That was also his last regular show at the station; at 5:00 pm, WCBS-FM announced that the station would replace its oldies format with a "Jack" format, eliminating the need for on-air disc jockeys.

However, WCBS-FM eventually returned to its oldies format on July 12, 2007, with Taylor re-assuming his role as the morning disc jockey the following day. Several months later, on February 3, 2008, Dolenz was invited back to the station to present his long-postponed 101st show and final in-studio appearance there by guest-hosting a three-hour broadcast during WCBS' Sunday evening "New York Radio Greats" program.

Solo work and further Monkees reunions
In 2009, Dolenz signed a deal to record an album of the classic songs of Carole King, titled King for a Day. The album (released on Gigitone Records) was produced by Jeffrey Foskett, who has worked extensively with Brian Wilson and played on Wilson's 2004 Grammy-winning Brian Wilson Presents Smile. King's songs "Pleasant Valley Sunday", "Sometime in the Morning", and "Porpoise Song" (Theme from Head) have emerged as signature songs from the Monkees. As of February 2010, he was appearing on stage in London in Hairspray with Michael Ball. The show also went on tour and had a successful run in Dublin, Ireland, during November 2010. In 2011, he rejoined Tork and Jones for An Evening with The Monkees: The 45th Anniversary Tour. After Jones' sudden death in February 2012, Dolenz and Tork reunited with Michael Nesmith for a 12-concert tour of the United States as a tribute. The three remaining Monkees toured again in 2013 and 2014 and Dolenz toured as a duo with Tork in 2015 and 2016.

Following Tork's death in 2019, Dolenz toured with Nesmith as "The Mike and Micky Show" in 2018 and 2019. On May 4, 2021, Dolenz and Nesmith announced "The Monkees Farewell Tour" which was the last for the group. The tour consisted of 40 US dates from September to November. The final show was held on November 14, 2021, at the Greek Theatre in Los Angeles.

On May 21, 2021, Dolenz released a solo album, Dolenz Sings Nesmith, featuring songs written by Nesmith and produced by Christian Nesmith.

On November 3, 2023 Dolenz released an EP of R.E.M. cover songs.

Other tours
In late 2019, Dolenz toured with Todd Rundgren, Jason Scheff, Christopher Cross, and Joey Molland of Badfinger, in celebration of the Beatles' White Album on the "It Was Fifty Years Ago Today – A Tribute to the Beatles' White Album" tour. Dolenz performed the Monkees' songs "I'm a Believer" and "Pleasant Valley Sunday".

Personal life
Dolenz has been married three times and is the father of four daughters. In 1967, while in the UK on tour with the Monkees, Dolenz met future wife Samantha Juste, a co-presenter on BBC TV's pop music show Top of the Pops. They married in 1968, and their daughter Ami Bluebell Dolenz was born on January 8, 1969; she became an actress who was particularly active in the 1980s and 1990s. Dolenz and Juste divorced in 1975, but remained close friends until her death following a stroke on February 5, 2014.

He married Trina Dow in 1977. They had three daughters: Charlotte Janelle (born August 8, 1981), Emily Claire (born July 25, 1983), and Georgia Rose (born September 3, 1984). They divorced in 1991. Trina Dow Dolenz has become a couples therapist, still using her married name. Dolenz married his third wife, Donna Quinter, in 2002.

Discography
Albums
 * Dolenz, Jones, Boyce & Hart (Capitol, 1976) – with Dolenz, Jones, Boyce & Hart
 * Concert in Japan (Capitol, 1976) – live, with Dolenz, Jones, Boyce & Hart
 * The Point! (MCA, 1977) – with the London cast of The Point! (MCA Records – VIM 6262; CD release 2016, Varèse Sarabande)
 * Micky Dolenz Puts You to Sleep (Kid Rhino, 1991)
 * Broadway Micky (Kid Rhino, 1994)
 * Demoiselle (self-released, 1998)
 * King for a Day (Gigatone, 2010)
 * Remember (Robo, 2012)
 * A Little Bit Broadway, a Little Bit Rock and Roll (Broadway, 2015) – live
 * An Evening With Peter Noone & Micky Dolenz (7A Records, 2016) – spoken word, live
 * The MGM Singles Collection – Expanded CD Edition (7A Records, 2016)
 * Out of Nowhere (7A Records, 2017)
 * Live in Japan (7A Records, 2020)
 * Dolenz Sings Nesmith (7A Records, 2021) – produced by Christian Nesmith
 * Demoiselle (7A Records, 2022) – expanded deluxe edition
 * Dolenz Sings R.E.M. (EP, 7a Records, 2023)

Singles
 * "Don't Do It"/"Plastic Symphony III" (Challenge, 1967) (recorded in 1965) US No. 75 CAN No. 74
 * "Huff Puff"/"Fate (Big Ben)" (Challenge, 1967) (recorded in 1965)
 * "Do It in the Name of Love"/"Lady Jane" (Bell, 1971) - with Davy Jones
 * "Easy on You"/"Oh Someone" (MGM, 1971)
 * "A Lover's Prayer"/"Unattended in the Dungeon" (MGM, 1972)
 * "Johnny B. Goode"/"It's Amazing to Me" (Lion, 1972) – with Starship
 * "Daybreak"/"Love War" (Romar, 1973)
 * "The Buddy Holly Tribute"/"Ooh, She's Young" (Romar, 1974)
 * "I Remember the Feeling"/"You and I" (Capitol, 1975) – with Dolenz, Jones, Boyce & Hart
 * "I Love You and I'm Glad That I Said It"/"Saving My Love for You" (Capitol, 1975) – with Dolenz, Jones, Boyce & Hart
 * "Christmas Is My Time of Year"/"White Christmas" (Harmony, 1976) – with Davy Jones & Peter Tork
 * "Lovelight"/"Alicia" (Chrysalis, 1979)
 * "To Be or Not to Be"/"Beverly Hills" (JAM, 1981)
 * "Tomorrow"/"Fat Sam's Grand Slam" (A&M, 1983) – with the Bugsy Malone Gang
 * "Chance of a Lifetime"/"Livin' on Lies" (7A Records, 2016)
 * "Porpoise Song"/"Good Morning Good Morning"/"Crying in the Rain"/"Randy Scouse Git" (7A Records, 2016) – with Christian Nesmith and Circe Link
 * "Sunny Girlfriend"/"Zor and Zam" (Live in Japan, 1982) (7A Records, 2016)

Stage

 * 1977–1978: The Point!, Mermaid Theatre, London, England (Role: Count's Kid / The Leafman)
 * 1983: Bugsy Malone, Her Majesty's Theatre, London, England (director)
 * 1994–1998: Grease, Eugene O'Neill Theatre, NYC (Role: Vince Fontaine – replacement)
 * 2004: Aida, Palace Theatre, NYC (Role: Zoser – replacement)
 * 2006: Pippin, Goodspeed Opera House, East Haddam, Connecticut (Role: Charlemagne)
 * 2010: Hairspray, Grand Canal Theatre, Dublin, Ireland (Role: Wilbur Turnblad – alternate)

title: Micky Dolenz


text:
Thrust

Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that system. The force applied on a surface in a direction perpendicular or normal to the surface is also called thrust. Force, and thus thrust, is measured using the International System of Units (SI) in newtons (symbol: N), and represents the amount needed to accelerate 1 kilogram of mass at the rate of 1 meter per second per second. In mechanical engineering, force orthogonal to the main load (such as in parallel helical gears) is referred to as static thrust.

Examples
A fixed-wing aircraft propulsion system generates forward thrust when air is pushed in the direction opposite to flight. This can be done by different means such as the spinning blades of a propeller, the propelling jet of a jet engine, or by ejecting hot gases from a rocket engine. Reverse thrust can be generated to aid braking after landing by reversing the pitch of variable-pitch propeller blades, or using a thrust reverser on a jet engine. Rotary wing aircraft use rotors and thrust vectoring V/STOL aircraft use propellers or engine thrust to support the weight of the aircraft and to provide forward propulsion.

A motorboat propeller generates thrust when it rotates and forces water backwards.

A rocket is propelled forward by a thrust equal in magnitude, but opposite in direction, to the time-rate of momentum change of the exhaust gas accelerated from the combustion chamber through the rocket engine nozzle. This is the exhaust velocity with respect to the rocket, times the time-rate at which the mass is expelled, or in mathematical terms:
 * $$\mathbf{T}=\mathbf{v}\frac{\mathrm{d}m}{\mathrm{d}t}$$

Where T is the thrust generated (force), $$\frac {\mathrm{d}m} {\mathrm{d}t}$$ is the rate of change of mass with respect to time (mass flow rate of exhaust), and v is the velocity of the exhaust gases measured relative to the rocket.

For vertical launch of a rocket the initial thrust at liftoff must be more than the weight.

Each of the three Space Shuttle Main Engines could produce a thrust of 1.8 meganewton, and each of the Space Shuttle's two Solid Rocket Boosters 14.7 MN, together 29.4 MN.

By contrast, the Simplified Aid for EVA Rescue (SAFER) has 24 thrusters of 3.56 N each.

In the air-breathing category, the AMT-USA AT-180 jet engine developed for radio-controlled aircraft produce 90 N (20 lbf) of thrust. The GE90-115B engine fitted on the Boeing 777-300ER, recognized by the Guinness Book of World Records as the "World's Most Powerful Commercial Jet Engine," has a thrust of 569 kN (127,900 lbf) until it was surpassed by the GE9X, fitted on the upcoming Boeing 777X, at 609 kN (134,300 lbf).

Thrust to power
The power needed to generate thrust and the force of the thrust can be related in a non-linear way. In general, $$\mathbf{P}^2 \propto \mathbf{T}^3$$. The proportionality constant varies, and can be solved for a uniform flow, where $$v_\infty$$ is the incoming air velocity, $$v_d$$ is the velocity at the actuator disc, and $$v_f$$ is the final exit velocity:

 * $$\frac{\mathrm{d}m}{\mathrm{d}t} = \rho A {v}$$

 * $$\mathbf{T} = \frac{\mathrm{d}m}{\mathrm{d}t} \left (v_f - v_\infty \right ), \frac{\mathrm{d}m}{\mathrm{d}t} = \rho A v_d$$

 * $$\mathbf{P} = \frac{1}{2} \frac{\mathrm{d}m}{\mathrm{d}t} (v_f^2 - v_\infty^2), \mathbf{P} = \mathbf{T}v_d$$

Solving for the velocity at the disc, $$v_d$$, we then have:
 * $$v_d = \frac{1}{2}(v_f - v_\infty)$$

When incoming air is accelerated from a standstill – for example when hovering – then $$v_\infty = 0$$, and we can find:

 * $$\mathbf{T} = \frac{1}{2} \rho A {v_f}^2, \mathbf{P} = \frac{1}{4} \rho A {v_f}^3$$

From here we can see the $$\mathbf{P}^2 \propto \mathbf{T}^3$$ relationship, finding:

 * $$\mathbf{P}^2 = \frac{\mathbf{T}^3}{2 \rho A}$$

The inverse of the proportionality constant, the "efficiency" of an otherwise-perfect thruster, is proportional to the area of the cross section of the propelled volume of fluid ($$A$$) and the density of the fluid ($$\rho$$). This helps to explain why moving through water is easier and why aircraft have much larger propellers than watercraft.

Thrust to propulsive power
A very common question is how to compare the thrust rating of a jet engine with the power rating of a piston engine. Such comparison is difficult, as these quantities are not equivalent. A piston engine does not move the aircraft by itself (the propeller does that), so piston engines are usually rated by how much power they deliver to the propeller. Except for changes in temperature and air pressure, this quantity depends basically on the throttle setting.

A jet engine has no propeller, so the propulsive power of a jet engine is determined from its thrust as follows. Power is the force (F) it takes to move something over some distance (d) divided by the time (t) it takes to move that distance:

 * $$\mathbf{P}=\mathbf{F}\frac{d}{t}$$

In case of a rocket or a jet aircraft, the force is exactly the thrust (T) produced by the engine. If the rocket or aircraft is moving at about a constant speed, then distance divided by time is just speed, so power is thrust times speed:

 * $$\mathbf{P}=\mathbf{T}{v}$$

This formula looks very surprising, but it is correct: the propulsive power (or power available ) of a jet engine increases with its speed. If the speed is zero, then the propulsive power is zero. If a jet aircraft is at full throttle but attached to a static test stand, then the jet engine produces no propulsive power, however thrust is still produced. The combination piston engine–propeller also has a propulsive power with exactly the same formula, and it will also be zero at zero speed – but that is for the engine–propeller set. The engine alone will continue to produce its rated power at a constant rate, whether the aircraft is moving or not.

Now, imagine the strong chain is broken, and the jet and the piston aircraft start to move. At low speeds:

The piston engine will have constant 100% power, and the propeller's thrust will vary with speed

The jet engine will have constant 100% thrust, and the engine's power will vary with speed

Excess thrust
If a powered aircraft is generating thrust T and experiencing drag D, the difference between the two, T − D, is termed the excess thrust. The instantaneous performance of the aircraft is mostly dependent on the excess thrust.

Excess thrust is a vector and is determined as the vector difference between the thrust vector and the drag vector.

Thrust axis
The thrust axis for an airplane is the line of action of the total thrust at any instant. It depends on the location, number, and characteristics of the jet engines or propellers. It usually differs from the drag axis. If so, the distance between the thrust axis and the drag axis will cause a moment that must be resisted by a change in the aerodynamic force on the horizontal stabiliser. Notably, the Boeing 737 MAX, with larger, lower-slung engines than previous 737 models, had a greater distance between the thrust axis and the drag axis, causing the nose to rise up in some flight regimes, necessitating a pitch-control system, MCAS. Early versions of MCAS malfunctioned in flight with catastrophic consequences, leading to the deaths of over 300 people in 2018 and 2019.

title: Thrust


text:
User:Joel Schlosberg

OK, I'm finally getting this page in shape after letting it sit for years.

Most of my current stuff can be found on my blog.

My contributions to Wikimedia Commons

For a long time, my page looked like this (and no, I'm no longer a junior at NYU, for starters):

I'm a junior at New York University, studying math.

I've created the articles about Ann Druyan, C. L. Moore, and the magazine  Unknown; and have added to the articles on the Game of Hex and Georges Méliès.

title: User:Joel Schlosberg


text:
Savoy knot

The Savoy knot, a type of decorative knot, is a heraldic knot used primarily in Italian heraldry. It is most notable for its appearance on the heraldic badge of the House of Savoy, where it is accompanied by the motto Stringe ma non costringe, "It tightens, but does not constrain". The Cavendish knot is an identical heraldic knot. In shape, the Savoy knot is comparable to a figure eight.

When used outside heraldry (as a real knot), it is known as a figure-eight knot.

The Savoy knot can also be seen on the Alfa Romeo automobile badge (founded and manufactured in Milan, Italy) up to 1943.

title: Savoy knot


text:
Collatz conjecture

The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to $295,000,000,000,000,000,000$, but no general proof has been found.

It is named after the mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics".

Statement of the problem
Consider the following operation on an arbitrary positive integer:

 * If the number is even, divide it by two.
 * If the number is odd, triple it and add one.

In modular arithmetic notation, define the function $x$ as follows: $$ f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2},\\[4px] 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$

Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.

In notation: $$ a_i = \begin{cases}n & \text{for } i = 0, \\ f(a_{i-1}) & \text{for } i > 0 \end{cases}$$ (that is: $3n + 1$ is the value of $y$ applied to $x$ recursively $y$ times; $3x + 1$).

The Collatz conjecture is: ''This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each'' $$n$$, there is some $$i$$ with $$a_i = 1$$.

If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.

The smallest $f$ such that $a_{i}$ is called the stopping time of $f$. Similarly, the smallest $n$ such that $a_{i} = f(n)$ is called the total stopping time of $i$. If one of the indexes $i$ or $n$ doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite.

The Collatz conjecture asserts that the total stopping time of every $k$ is finite. It is also equivalent to saying that every $a_{i} < a_{0}$ has a finite stopping time.

Since $a_{k} = 1$ is even whenever $n$ is odd, one may instead use the "shortcut" form of the Collatz function:

This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.

Empirical data
For instance, starting with $n ≥ 2$ and applying the function $3n + 1$ without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.

The number $n = 12$ takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The sequence for $f$, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1.

 * 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with:
 * 1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, ....

The starting values whose maximum trajectory point is greater than that of any smaller starting value are as follows:
 * 1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, ...

Number of steps for $i$ to reach 1 are
 * 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, ...

The starting value having the largest total stopping time while being
 * less than 10 is 9, which has 19 steps,
 * less than 100 is 97, which has 118 steps,
 * less than 1000 is 871, which has 178 steps,
 * less than 104 is 6171, which has 261 steps,
 * less than 105 is $k$, which has 350 steps,
 * less than 106 is $n$, which has 524 steps,
 * less than 107 is $n$, which has 685 steps,
 * less than 108 is $n$, which has 949 steps,
 * less than 109 is $77,031$, which has 986 steps,
 * less than 1010 is $837,799$, which has 1132 steps,
 * less than 1011 is $8,400,511$, which has 1228 steps,
 * less than 1012 is $63,728,127$, which has 1348 steps.

These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, $670,617,279$ has 1132 steps, as does $9,780,657,630$.

The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the powers of two since $n = 19$ is halved $75,128,138,247$ times to reach 1, and is never increased.

Supporting arguments
Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.

Experimental evidence
The conjecture has been checked by computer for all starting values up to 268 ≈ $989,345,275,647$. All values tested so far converge to 1.

This computer evidence is still not rigorous proof that the conjecture is true for all starting values, as counterexamples may be found when considering very large (or possibly immense) positive integers, as in the case of the disproven Pólya conjecture and Mertens conjecture.

However, such verifications may have other implications. Certain constraints on any non-trivial cycle, such as lower bounds on the length of the cycle, can be proven based on the value of the lowest term in the cycle. Therefore, computer searches to rule out cycles that have a small lowest term can strengthen these constraints.

A probabilistic heuristic
If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average $9,780,657,631$ of the previous one. (More precisely, the geometric mean of the ratios of outcomes is $9,780,657,630$.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the 2-adic extension of the Collatz process has two division steps for every multiplication step for almost all 2-adic starting values.)

Stopping times
As proven by Riho Terras, almost every positive integer has a finite stopping time. In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of parity vectors and uses the central limit theorem.

In 2019, Terence Tao improved this result by showing, using logarithmic density, that almost all (in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, Quanta Magazine wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".

Lower bounds
In a computer-aided proof, Krasikov and Lagarias showed that the number of integers in the interval $n = 27$ that eventually reach 1 is at least equal to $2^{n}$ for all sufficiently large $n$.

Cycles
In this part, consider the shortcut form of the Collatz function $$ f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \pmod{2},\\[4px] \frac{3n+1}{2} & \text{if } n \equiv 1 \pmod{2}. \end{cases}$$ A cycle is a sequence $x = 9663$ of distinct positive integers where $[1,x]$, $x^{0.84}$, ..., and $(a_{0}, a_{1}, ..., a_{q})$.

The only known cycle is $f(a_{0}) = a_{1}$ of period 2, called the trivial cycle.

Cycle length
The length of a non-trivial cycle is known to be at least $x$ (or $y$ without short-cut). If it can be shown that for all positive integers less than $$3 \times 2^{69}$$ the Collatz sequences reach 1, then this bound would raise to $y$ ($x$ without short-cut). In fact, Eliahou (1993) proved that the period $27,000,000$ of any non-trivial cycle is of the form $$p = 301994 a + 17087915 b + 85137581 c$$ where $y$, $295,000,000,000,000,000,000$ and $3⁄4$ are non-negative integers, $f(a_{1}) = a_{2}$ and $f(a_{q}) = a_{0}$. This result is based on the continued fraction expansion of $(1,2)$.

$3⁄4$-cycles
A $x$-cycle is a cycle that can be partitioned into $b ≥ 1$ contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers. For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1-cycle.

Steiner (1977) proved that there is no 1-cycle other than the trivial $ac = 0$. Simons (2005) used Steiner's method to prove that there is no 2-cycle. Simons and de Weger (2005) extended this proof up to 68-cycles; there is no $114,208,327,604$-cycle up to $ln 3⁄ln 2$. Hercher extended the method further and proved that there exists no k-cycle with $k$. As exhaustive computer searches continue, larger $(1; 2)$ values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.

In reverse
There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called Collatz graph. The Collatz graph is a graph defined by the inverse relation $$ R(n) = \begin{cases} \{2n\} & \text{if } n\equiv 0,1,2,3,5 \\[4px] \left\{2n,\frac{n-1}{3}\right\} & \text{if } n\equiv 4 \end{cases} \pmod 6. $$

So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer $186,265,759,595$, $k = 68$ if and only if $k≤91$. Equivalently, $k$ if and only if $n ≡ 1 (mod 2)$. Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function $217,976,794,617$ defined in the Statement of the problem section of this article).

When the relation $3n + 1 ≡ 4 (mod 6)$ of the function $355,504,839,929$ is replaced by the common substitute "shortcut" relation $n − 1⁄3 ≡ 1 (mod 2)$, the Collatz graph is defined by the inverse relation, $$ R(n) = \begin{cases} \{2n\} & \text{if } n\equiv 0,1 \\[4px] \left\{2n,\frac{2n-1}{3}\right\} & \text{if } n\equiv 2 \end{cases} \pmod 3. $$

For any integer $p$, $n ≡ 4 (mod 6)$ if and only if $3n + 1$. Equivalently, $3n + 1⁄2$ if and only if $n ≡ 1 (mod 2)$. Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above).

Alternatively, replace the $3n + 1⁄2 ≡ 2 (mod 3)$ with $2n − 1⁄3 ≡ 1 (mod 2)$ where $n ≡ 2 (mod 3)$ and $3n + 1$ is the highest power of 2 that divides $n⁄H(n)$ (with no remainder). The resulting function $a$ maps from odd numbers to odd numbers. Now suppose that for some odd number $b$, applying this operation $c$ times yields the number 1 (that is, $n = 3n + 1$). Then in binary, the number $k$ can be written as the concatenation of strings $H(n)$ where each $n$ is a finite and contiguous extract from the representation of $f(n) = 1$. The representation of $k$ therefore holds the repetends of $w_{k} w_{k−1} ... w_{1}$, where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs. Conjecturally, every binary string $k$ that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to $n$).

As an abstract machine that computes in base two
Repeated applications of the Collatz function can be represented as an abstract machine that handles strings of bits. The machine will perform the following three steps on any odd number until only one 1 remains:

 * 1) Append 1 to the (right) end of the number in binary (giving $w_{h}$);
 * 2) Add this to the original number by binary addition (giving $1⁄3^{h}$);
 * 3) Remove all trailing 0s (that is, repeatedly divide by 2 until the result is odd).

Example
The starting number 7 is written in base two as 111. The resulting Collatz sequence is:

111        1111         1011 0        10111       10001 0      100011      1101 00     11011    101 000   1011  1 0000

As a parity sequence
For this section, consider the shortcut form of the Collatz function $$ f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \\[4px] \frac{3n + 1}{2} & \text{if } n \equiv 1 \end{cases} \pmod{2}.$$

If $1⁄3^{h}$ is the parity of a number, that is $2n + 1$ and $2n + 1 + n = 3n + 1$, then we can define the Collatz parity sequence (or parity vector) for a number $f$ as $P(...)$, where $P(2n) = 0$, and $P(2n + 1) = 1$.

Which operation is performed, $p_{i} = P(a_{i})$ or $a_{0} = n$, depends on the parity. The parity sequence is the same as the sequence of operations.

Using this form for $a_{i+1} = f(a_{i})$, it can be shown that the parity sequences for two numbers $f$ and $n$ will agree in the first $f$ terms if and only if $n$ and $k$ are equivalent modulo $3n + 1⁄2$. This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different.

Applying the $n$ function $n$ times to the number $n⁄2$ will give the result $f(n)$, where $s$ is the result of applying the $s$ function $n$ times to $m$, and $n$ is how many increases were encountered during that sequence. For example, for $2^{k}$ there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is $n = 2^{k}a + b$; for $3^{c}a + d$ there is only 1 increase as 1 rises to 2 and falls to 1 so the result is $2^{5}a + 1$. When $k$ is $3^{3}a + 2$ then there will be $m$ rises and the result will be $2^{2}a + 1$. The power of 3 multiplying $n$ is independent of the value of $f$; it depends only on the behavior of $k$. This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations: for example, $3a + 1$ becomes $2^{k} − 1$ after two applications of $d$ and $3^{k}a + 3^{k} − 1$ becomes $4a + 1$ after four applications of $f$. Whether those smaller numbers continue to 1, however, depends on the value of $k$.

As a tag system
For the Collatz function in the shortcut form

$$ f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \\[4px] \frac{3n+1}{2} & \text{if } n \equiv 1. \end{cases} \pmod{2}$$

Hailstone sequences can be computed by the 2-tag system with production rules

In this system, the positive integer $b$ is represented by a string of $c$ copies of $b$, and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.)

The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of $k$ as the initial word, eventually halts (see Tag system for a worked example).

Iterating on all integers
An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of $a$. These cycles are listed here, starting with the well-known cycle for positive $a$:

Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.

The generalized Collatz conjecture is the assertion that every integer, under iteration by $b$, eventually falls into one of the four cycles above or the cycle 0 → 0. (The 0 → 0 cycle is only included for the sake of completeness.)

Iterating on rationals with odd denominators
The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of 2-adic integers, which contains the ring of rationals with odd denominators as a subring.

When using the "shortcut" definition of the Collatz map, it is known that any periodic parity sequence is generated by exactly one rational. Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture ).

If a parity cycle has length $f$ and includes odd numbers exactly $f$ times at indices $3a + 1$, then the unique rational which generates immediately and periodically this parity cycle is

For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction $$\frac{3^3 2^0 + 3^2 2^2 + 3^1 2^3 + 3^0 2^6}{2^7 - 3^4} = \frac{151}{47}$$ as the latter leads to the rational cycle $$\frac{151}{47} \rightarrow \frac{250}{47} \rightarrow \frac{125}{47} \rightarrow \frac{211}{47} \rightarrow \frac{340}{47} \rightarrow \frac{170}{47} \rightarrow \frac{85}{47} \rightarrow \frac{151}{47} .$$

Any cyclic permutation of (1 0 1 1 0 0 1) is associated to one of the above fractions. For instance, the cycle (0 1 1 0 0 1 1) is produced by the fraction $$\frac{3^3 2^1 + 3^2 2^2 + 3^1 2^5 + 3^0 2^6}{2^7 - 3^4} = \frac{250}{47}. $$

For a one-to-one correspondence, a parity cycle should be irreducible, that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction $a$ when reduced to lowest terms.

In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2, respectively).

If the odd denominator $n$ of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the "$16a + 3$ " generalization of the Collatz function $$ T_d(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2},\\[4px] \frac{3x+d}{2} & \text{if } x\equiv 1 \pmod{2}. \end{cases}$$

2-adic extension
The function $$ T(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2}\\[4px] \frac{3x+1}{2} & \text{if } x\equiv 1 \pmod{2} \end{cases}$$ is well-defined on the ring $$\mathbb{Z}_2$$ of 2-adic integers, where it is continuous and measure-preserving with respect to the 2-adic measure. Moreover, its dynamics is known to be ergodic.

Define the parity vector function $n$ acting on $$\mathbb{Z}_2$$ as $$ Q(x) = \sum_{k=0}^{\infty} \left( T^k (x) \mod 2 \right) 2^k .$$

The function $a$ is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost all trajectories are acyclic in $$\mathbb{Z}_2$$.

An equivalent formulation of the Collatz conjecture is that $$ Q\left(\mathbb{Z}^{+}\right) \subset \tfrac13 \mathbb{Z}.$$

Iterating on real or complex numbers

The Collatz map can be extended to the real line by choosing any function which evaluates to $$x/2$$ when $$x$$ is an even integer, and to either $$3x + 1$$ or $$(3x + 1)/2$$ (for the "shortcut" version) when $$x$$ is an odd integer. This is called an interpolating function. A simple way to do this is to pick two functions $$g_1$$ and $$g_2$$, where:
 * $$g_1(n) = \begin{cases}1, &n\text{ is even,}\\ 0, &n\text{ is odd,}\end{cases}$$
 * $$g_2(n) = \begin{cases}0, &n\text{ is even,}\\1, &n\text{ is odd,}\end{cases}$$

and use them as switches for our desired values:
 * $$f(x) \triangleq \frac{x}{2}\cdot g_1(x) \,+\, \frac{3x + 1}{2}\cdot g_2(x)$$.

One such choice is $$g_1(x) \triangleq \cos^2\left(\tfrac{\pi}{2} x\right)$$ and $$g_2(x) \triangleq \sin^2\left(\tfrac{\pi}{2} x\right)$$. The iterations of this map lead to a dynamical system, further investigated by Marc Chamberland. He showed that the conjecture does not hold for positive real numbers since there are infinitely many fixed points, as well as orbits escaping monotonically to infinity. The function $$f$$ has two attracting cycles of period $$2$$: $$(1;\,2)$$ and $$(1.1925...;\,2.1386...)$$. Moreover, the set of unbounded orbits is conjectured to be of measure $$0$$.

Letherman, Schleicher, and Wood extended the study to the complex plane. They used Chamberland's function for complex sine and cosine and added the extra term $$\tfrac{1}{\pi}\left(\tfrac12 - \cos(\pi z)\right)\sin(\pi z)\,+$$ $$h(z)\sin^2(\pi z)$$, where $$h(z)$$ is any entire function. Since this expression evaluates to zero for real integers, the extended function
 * $$\begin{align}f(z) \triangleq \;&\frac{z}{2}\cos^2\left(\frac{\pi}{2} z\right) + \frac{3z + 1}{2}\sin^2\left(\frac{\pi}{2} z\right) \, + \\

&\frac{1}{\pi}\left(\frac12 - \cos(\pi z)\right)\sin(\pi z) + h(z)\sin^2(\pi z)\end{align}$$

is an interpolation of the Collatz map to the complex plane. The reason for adding the extra term is to make all integers critical points of $$f$$. With this, they show that no integer is in a Baker domain, which implies that any integer is either eventually periodic or belongs to a wandering domain. They conjectured that the latter is not the case, which would make all integer orbits finite.

Most of the points have orbits that diverge to infinity. Coloring these points based on how fast they diverge produces the image on the left, for $$h(z) \triangleq 0$$. The inner black regions and the outer region are the Fatou components, and the boundary between them is the Julia set of $$f$$, which forms a fractal pattern, sometimes called a "Collatz fractal".

There are many other ways to define a complex interpolating function, such as using the complex exponential instead of sine and cosine:
 * $$f(z) \triangleq \frac{z}{2} + \frac14(2z + 1)\left(1 - e^{i\pi z}\right)$$,

which exhibit different dynamics. In this case, for instance, if $$\operatorname{Im}(z) \gg 1$$, then $$f(z) \approx z + \tfrac14$$. The corresponding Julia set, shown on the right, consists of uncountably many curves, called hairs, or rays.

Time–space tradeoff
The section As a parity sequence above gives a way to speed up simulation of the sequence. To jump ahead $a$ steps on each iteration (using the $f$ function from that section), break up the current number into two parts, $n$ (the $f$ least significant bits, interpreted as an integer), and $n$ (the rest of the bits as an integer). The result of jumping ahead $m$ is given by

The values of $$ (or better $9a + 2$) and $5⁄7$ can be precalculated for all possible $d$-bit numbers $Q$, where $a → bc$ is the result of applying the $Q$ function $k$ times to $f$, and $b → a$ is the number of odd numbers encountered on the way. For example, if $c → aaa$, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using

 * $b$(0...31, 5) = { 0, 3, 2, 2, 2, 2, 2, 4, 1, 4, 1, 3, 2, 2, 3, 4, 1, 2, 3, 3, 1, 1, 3, 3, 2, 3, 2, 4, 3, 3, 4, 5 },
 * $k$(0...31, 5) = { 0, 2, 1, 1, 2, 2, 2, 20, 1, 26, 1, 10, 4, 4, 13, 40, 2, 5, 17, 17, 2, 2, 20, 20, 8, 22, 8, 71, 26, 26, 80, 242 }.

This requires $k_{0} < ⋯ < k_{m−1}$ precomputation and storage to speed up the resulting calculation by a factor of $a$, a space–time tradeoff.

Modular restrictions
For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of $k$. If, for some given $c$ and $d$, the inequality

holds for all $k$, then the first counterexample, if it exists, cannot be $b$ modulo $3n + d$. For instance, the first counterexample must be odd because $f(2^{k}a + b) = 3^{c(b, k)}a + d(b, k)$, smaller than $3^{c}$; and it must be 3 mod 4 because $d(b, k)$, smaller than $c(b, k)$. For each starting value $f$ which is not a counterexample to the Collatz conjecture, there is a $k$ for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class. As $b$ increases, the search only needs to check those residues $c$ that are not eliminated by lower values of $d$. Only an exponentially small fraction of the residues survive. For example, the only surviving residues mod 32 are 7, 15, 27, and 31.

Syracuse function
If $k$ is an odd integer, then $k = 5$ is even, so $2^{k}$ with $f(2^{k}a + b) = 3^{c(b)}a + d(b) < 2^{k}a + b$ odd and $2^{k}$. The Syracuse function is the function $n$ from the set $b$ of positive odd integers into itself, for which $f(2n) = n$.

Some properties of the Syracuse function are:

 * For all $2n$, $f(4n + 1) = 3n + 1$. (Because $4n + 1$.)
 * In more generality: For all $3k + 1$ and odd $k$, $3k + 1 = 2^{a}k$. (Here $k$ is function iteration notation.)
 * For all odd $a$, $a ≥ 1$

The Collatz conjecture is equivalent to the statement that, for all $b$ in $a$, there exists an integer $f(k) = k$ such that $k ∈ I$.

Undecidable generalizations
In 1972, John Horton Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable.

Specifically, he considered functions of the form $$ {g(n) = a_i n + b_i} \text{ when } {n\equiv i \pmod P},$$ where $f(4k + 1) = f(k)$ are rational numbers which are so chosen that $3(4k + 1) + 1 = 12k + 4 = 4(3k + 1)$ is always an integer. The standard Collatz function is given by $p ≥ 1$, $f(2^{p}h − 1) = 2 × 3^{p − 1}h − 1$, $f$, $f(2h − 1) ≤ 3h − 1⁄2$, $n ≥ 1$. Conway proved that the problem

 * Given $k$ and $k$, does the sequence of iterates $f(k) = 1$ reach $a_{0}, b_{0}, ..., a_{P − 1}, b_{P − 1}$?

is undecidable, by representing the halting problem in this way.

Closer to the Collatz problem is the following universally quantified problem:

 * Given $b$, does the sequence of iterates $g(n)$ reach $P = 2$, for all $a_{0} = 1⁄2$?

Modifying the condition in this way can make a problem either harder or easier to solve (intuitively, it is harder to justify a positive answer but might be easier to justify a negative one). Kurtz and Simon proved that the universally quantified problem is, in fact, undecidable and even higher in the arithmetical hierarchy; specifically, it is $b_{0} = 0$-complete. This hardness result holds even if one restricts the class of functions $k$ by fixing the modulus $k$ to 6480.

Iterations of in a simplified version of this form, with all $$b_i$$ equal to zero, are formalized in an esoteric programming language called FRACTRAN.

In popular culture
In the movie Incendies, a graduate student in pure mathematics explains the Collatz conjecture to a group of undergraduates. She puts her studies on hold for a time to address some unresolved questions about her family's past. Late in the movie, the Collatz conjecture turns out to have foreshadowed a disturbing and difficult discovery that she makes about her family.

title: Collatz conjecture


text:
Cyprian

Cyprian (Thascius Caecilius Cyprianus; ca. 210 to 14 September 258 AD ) was a bishop of Carthage and an early Christian writer of Berber descent, many of whose Latin works are extant. He is recognized as a saint in the Western and Eastern churches.

He was born around the beginning of the 3rd century in North Africa, perhaps at Carthage, where he received a classical education. Soon after converting to Christianity, he became a bishop in 249. A controversial figure during his lifetime, his strong pastoral skills, firm conduct during the Novatianist heresy and outbreak of the Plague of Cyprian (named after him due to his description of it), and eventual martyrdom at Carthage established his reputation and proved his sanctity in the eyes of the Church.

His skillful Latin rhetoric led to his being considered the pre-eminent Latin writer of Western Christianity until Jerome and Augustine.

Early life
Cyprian was born into a rich pagan Roman African Carthaginian family sometime during the early third century. His original name was Thascius; he took the additional name Caecilius in memory of the priest to whom he owed his conversion. Before his conversion, he was a leading member of a legal fraternity in Carthage, an orator, a "pleader in the courts", and a teacher of rhetoric. After a "dissipated youth", Cyprian was baptized when he was thirty-five years old, c. 245 AD. After his baptism, he gave away a portion of his wealth to the poor of Carthage, as befitted a man of his status.

In the early days of his conversion, he wrote an Epistola ad Donatum de gratia Dei and the Testimoniorum Libri III that adhere closely to the models of Tertullian, who influenced his style and thinking. Cyprian described his own conversion and baptism in the following words:

Contested election as bishop of Carthage
Not long after his baptism he was ordained a deacon and soon afterwards a priest. Sometime between July 248 and April 249, he was elected bishop of Carthage, a popular choice among the poor who remembered his patronage as demonstrating good equestrian style. However, his rapid rise did not meet with the approval of senior members of the clergy in Carthage, an opposition that did not disappear during his episcopate.

Not long afterward, the entire community was put to an unwanted test. Christians in North Africa had not suffered persecution for many years; the Church was assured and lax. In early 250, the Decian persecution began. Emperor Decius issued an edict, the text of which is lost, ordering sacrifices to the gods to be made throughout the Empire. Jews were specifically exempted from that requirement. Cyprian chose to go into hiding, rather than face potential execution. While some clergy saw that decision as a sign of cowardice, Cyprian defended himself by saying that he had fled in order not to leave the faithful without a shepherd during the persecution and that his decision to continue to lead them, although from a distance, was in accordance with divine will. Moreover, he pointed to the actions of the Apostles and Jesus himself: "And therefore the Lord commanded us in the persecution to depart and to flee; and both taught that this should be done, and Himself did it. For as the crown is given by the condescension of God, and cannot be received unless the hour comes for accepting it, whoever abiding in Christ departs for a while does not deny his faith, but waits for the time...".

Controversy over the lapsed
The persecution was especially severe at Carthage, according to Church sources. Many Christians fell away and were thereafter referred to as "Lapsi" (fallen). The majority had obtained signed statements (libelli) certifying that they had sacrificed to the Roman gods to avoid persecution or confiscation of property. In some cases Christians had actually sacrificed, whether under torture or otherwise. Cyprian found those libellatici especially cowardly and demanded that they and the rest of the lapsi undergo public penance before being readmitted to the Church.

However, in Cyprian's absence, some priests disregarded his wishes by readmitting the lapsed to communion with little or no public penance. Some of the lapsi presented a second libellus purported to bear the signature of some martyr or confessor who, it was held, had the spiritual prestige to reaffirm individual Christians. That system was not limited to Carthage, but on a wider front by its charismatic nature, it clearly constituted a challenge to institutional authority in the Church, in particular to that of the bishop. Hundreds or even thousands of lapsi were readmitted that way against the express wishes of Cyprian and the majority of the Carthaginian clergy, who insisted upon earnest repentance.

A schism then broke out in Carthage, as the laxist party, led largely by the priests who had opposed Cyprian's election, attempted to block measures taken by him during his period of absence. After fourteen months, Cyprian returned to the diocese and in letters addressed to the other North African bishops defended having left his post. After issuing a tract, "De lapsis" (On the Fallen), he convoked a council of North African bishops at Carthage to consider the treatment of the lapsed, and the apparent schism of Felicissimus (251). Cyprian took a middle course between the followers of Novatus of Carthage, who were in favour of welcoming back all with little or no penance, and Novatian of Rome, who would not allow any of those who had lapsed to be reconciled. The council in the main sided with Cyprian and condemned Felicissimus though no acts of that council survive.

The schism continued as the laxists elected a certain Fortunatus as bishop in opposition to Cyprian. At the same time, the rigorist party in Rome, who refused reconciliation to any of the lapsed, elected Novatian as bishop of Rome in opposition to Pope Cornelius. The Novatianists also secured the election of a certain Maximus as a rival bishop of their own at Carthage. Cyprian now found himself wedged between laxists and rigorists, but the polarisation highlighted the firm but moderate position adopted by Cyprian and strengthened his influence by wearing down the numbers of his opponents. Moreover, his dedication during the time of a great plague and famine gained him still further popular support.

Cyprian comforted his brethren by writing his De mortalitate and in his De eleemosynis exhorted them to active charity towards the poor and set a personal example. He defended Christianity and the Christians in the apologia Ad Demetrianum, directed against a certain Demetrius, and countered pagan claims that Christians were the cause of the public calamities.

Persecution under Valerian
In late 256, a new persecution of the Christians broke out under Emperor Valerian, and Pope Sixtus II was executed in Rome.

In Africa, Cyprian prepared his people for the expected edict of persecution by his De exhortatione martyrii and set an example when he was brought before the Roman proconsul Aspasius Paternus (30 August 257). He refused to sacrifice to the pagan deities and firmly professed Christ.

The proconsul banished him to Curubis, now Korba, where, to the best of his ability, he comforted his flock and his banished clergy. In a vision, he believed he saw his approaching fate. When a year had passed, he was recalled and kept practically a prisoner in his own villa in expectation of severe measures after a new and more stringent imperial edict arrived, which Christian writers subsequently claimed demanded the execution of all Christian clerics.

On 13 September 258, Cyprian was imprisoned on the orders of the new proconsul, Galerius Maximus. The public examination of Cyprian by Galerius Maximus, on 14 September 258, has been preserved:

Galerius Maximus: "Are you Thascius Cyprianus?" Cyprian: "I am." Galerius: "The most sacred Emperors have commanded you to conform to the Roman rites." Cyprian: "I refuse." Galerius: "Take heed for yourself." Cyprian: "Do as you are bid; in so clear a case I may not take heed." Galerius, after briefly conferring with his judicial council, with much reluctance pronounced the following sentence: "You have long lived an irreligious life, and have drawn together a number of men bound by an unlawful association, and professed yourself an open enemy to the gods and the religion of Rome; and the pious, most sacred and august Emperors ... have endeavoured in vain to bring you back to conformity with their religious observances; whereas therefore you have been apprehended as principal and ringleader in these infamous crimes, you shall be made an example to those whom you have wickedly associated with you; the authority of law shall be ratified in your blood." He then read the sentence of the court from a written tablet: "It is the sentence of this court that Thascius Cyprianus be executed with the sword." Cyprian: "Thanks be to God."

The execution was carried out at once in an open place near the city. A vast multitude followed Cyprian on his last journey. He removed his garments without assistance, knelt down, and prayed. After he blindfolded himself, he was beheaded by the sword. The body was interred by Christians near the place of execution.

Cyprian's martyrdom was followed by the martyrdom of eight of his disciples in Carthage.

Writings
Cyprian's works were edited in volumes 3 and 4 of the Patrologia Latina. He was not a speculative theologian, his writings being always related to his pastoral ministry. The first major work was a monologue spoken to a friend called Ad Donatum, detailing his own conversion, the corruption of Roman government and the gladiatorial spectacles, and pointing to prayer as "the only refuge of the Christian". Another early written work was the Testimonia ad Quirinum. During his exile from Carthage Cyprian wrote his most famous treatise, De Ecclesiae Catholicae Unitate (On the Unity of the Catholic Church) and on returning to his see, he issued De Lapsis (On the Fallen). Another important work is his Treatise on the Lord's Prayer. Doubtless only part of his written output has survived, and this must apply especially to his correspondence, of which some sixty letters are extant, in addition to some of the letters he received.

Cyprian of Carthage is often confused with Cyprian of Antioch, reputedly a magician before his conversion. A number of grimoires, such as Libellus Magicus, are thus mistakenly attributed to Cyprian of Carthage.

Biography
Pontius the Deacon wrote a biography of Cyprian titled The Life and Passion of St. Cyprian, which details the saint's early life, his conversion, notable acts, and martyrdom under Valerian.

Sacraments
Cyprian believed in infant baptism and infant communion. Cyprian however spoke against the efficiency of baptism done by heretics and insisted on their rebaptism, and he believed that the Eucharist cannot be properly consecrated outside the church.

Cyprian was one of the earliest of the Church Fathers to enunciate clearly and unambiguously the doctrine of baptismal regeneration ("the idea that salvation happens at and by water baptism duly administered"): "While he attributed all the saving energy to the grace of God, he considered the 'laver of saving water' the instrument of God that makes a person 'born again,' receiving a new life and putting off what he had previously been. The 'water of new birth' animated him to new life by the Spirit of holiness working through it."

Church
Cyprian believed that the lapsed can be re-admitted to the Church after penance and he opposed the Novatians.

Cyprian believed the see of Peter (Rome) is the direct heir of Peter. While Cyprian believed that all the apostles were equal and that all the bishops followed the Apostles in succession, Cyprian emphasized the unity of the Church under a single cathedra (chair): "he [Jesus Christ] assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. Indeed, the others were that also which Peter was [i.e., apostles], but a primacy is given to Peter, whereby it is made clear that there is but one Church and one chair. So too, all [the apostles] are shepherds, and the flock is shown to be one, fed by all the apostles in single-minded accord."

Other
Cyprian was amillennial. Augustine argued that Cyprian taught the gift of perseverance. Cyprian argued that each day of the Genesis creation account consisted of 1000 years.

Veneration
Churches were erected over his tomb and over the place of his death. In later centuries, however, these churches were destroyed by the Vandals. The graves of such saints as Cyprian and Martin of Tours came to be regarded as "contact points between Heaven and Earth", and they became the centres of new, redefined, Christian urban communities. A surviving homily from Augustine on Cyprian's feast day indicates that his following was fairly widespread throughout Africa by the fourth century.

Charlemagne is said to have had the bones transferred to France; and Lyons, Arles, Venice, Compiègne, and Roenay in Flanders all have claimed to possess part of the martyr's relics.

The Catholic Church celebrates his feast day together with that of his good friend Pope Cornelius on 16 September, and in the Catholic Middle Ages the Sarum use observed it on the day of his death, 14 September. The Eastern Orthodox Church commemorates him on 31 August. Lutherans now commemorate him on 16 September, while Anglicans celebrate his feast usually either on 13 September (e.g. the Anglican Church of Australia) or 15 September (the present-day Church of England Calendar of saints) remembers him with a Lesser Festival.

English Translations of Works by St. Cyprian

 * St. Cyprian. The lapsed. The unity of the Catholic Church, translated and annotated by Maurice Bévenot, SJ), 1957 (Ancient Christian Writers, 25) ISBN 9780809102600
 * The Letters of St. Cyprian of Carthage, translated and annotated by Graeme Wilber Clarke), 4 vols., 1984-89 (Ancient Christian Writers, 43-44, 46-47) ISBN 9780809103416
 * Brent, Allen, editor and translator, "St Cyprian of Carthage: Selected Treatises," St. Vladimir's Seminary Press, 2007, ISBN 0-88141-312-7
 * Brent, Allen, editor and translator, "St Cyprian of Carthage: Selected Letters," St. Vladimir's Seminary Press, 2007, ISBN 0-88141-313-5
 * Campbell, Phillip, editor, "The Complete works of Saint Cyprian" Evolution Publishing, 2013, ISBN 1-935228-11-0
 * Christian Classics Ethereal Library: Cyprian texts
 * Christian Classics Ethereal Library: Cyprian texts

Others

 * Daniel, Robin, "This Holy Seed: Faith, Hope and Love in the Early Churches of North Africa", (Chester, Tamarisk Publications, 2010: from www.opaltrust.org) ISBN 0-9538565-3-4
 * J.M. Tebes, "Cyprian of Carthage: Christianity and Social World in the 3rd. century", Cuadernos de Teología 19, (2000)
 * J.M. Tebes, "Cyprian of Carthage: Christianity and Social World in the 3rd. century", Cuadernos de Teología 19, (2000)

title: Cyprian


text:
Amelia

Amelia may refer to:

Films

 * Amélia (film), a 2000 Brazilian film directed by Ana Carolina
 * Amelia (film), a 2009 film based on the life of Amelia Earhart

Literature

 * Amelia (magazine), a Swedish women's magazine
 * Amelia (novel), a 1751 sentimental novel by Henry Fielding
 * Amelia Bedelia, a series of US children's books
 * Amelia Jane, a series of books by Enid Blyton
 * Amelia Rules!, a series of American children's graphic novels

Music

 * Amelia (album), album by Mimi Webb
 * Amelia (opera), music by Daron Hagen; libretto by Gardner McFall; story by Stephen Wadsworth
 * "Amelia" (song), a song by Joni Mitchell on her 1976 album Hejira
 * "Amelia", a song by The Mission, from the album Carved in Sand
 * "Amelia", a song by the Cocteau Twins on their 1984 album Treasure
 * "Amelia", a song by Prism on their 1977 album Prism
 * "Amelia", a 1972 song by Wayne Cochran and The C.C. Riders

People

 * Amelia (given name), including people so named

 * Marco Amelia (born 1982), Italian football goalkeeper
 * Princess Amelia (disambiguation)
 * Amelie, Character in Les Miserable

United States

 * Amelia, Louisiana, a census-designated place in St. Mary Parish, Louisiana
 * Amelia, Nebraska, an unincorporated community in Holt County, Nebraska
 * Amelia, Ohio, a village in Clermont County, Ohio
 * Amelia, West Virginia
 * Amelia City, Florida, a town in Nassau County, Florida
 * Amelia County, Virginia
 * Amelia Courthouse, Virginia, a village in Amelia County near Richmond
 * Amelia Island, the southernmost of the Sea Islands, near Florida

Elsewhere

 * Amelia, Umbria, a town in Italy
 * Amelia Cove, a former settlement in Newfoundland and Labrador, Canada

Ships

 * Amelia (ship)
 * HMS Amelia, ships of the Royal Navy

Other uses

 * Amelia (birth defect)
 * Amelia (typeface)
 * Tropical Storm Amelia (disambiguation)
 * Amelia (company), an internet technology company

title: Amelia


text:
Valerian (herb)

Valerian (Valeriana officinalis, Caprifoliaceae) is a perennial flowering plant native to Europe and Asia. In the summer when the mature plant may have a height of 1.5 m, it bears sweetly scented pink or white flowers that attract many fly species, especially hoverflies of the genus Eristalis. It is consumed as food by the larvae of some Lepidoptera (butterfly and moth) species, including the grey pug.

Crude extract of valerian root may have sedative and anxiolytic effects, and is commonly sold in dietary supplement capsules to promote sleep, but clinical evidence that it is effective for this purpose is weak or inconclusive as yet. Its roots and leaves cause a catnip-like response in cats.

History
Valerian has been used as a herb in traditional medicine since at least the time of ancient Greece and Rome. Hippocrates described its properties, and Galen later prescribed it as a remedy for insomnia. In medieval Sweden, it was sometimes placed in the wedding clothes of a bridegroom to ward off the "envy" of the elves. In the 16th century, Pilgram Marpeck prescribed valerian tea for a sick woman.

John Gerard's Herball, or Generall Historie of Plantes, first published in 1597, states that his contemporaries found valerian "excellent for those burdened and for such as be troubled with croup and other like convulsions, and also for those that are bruised with falls". He says that the dried root was valued as a medicine by the poor in the north of England and the south of Scotland, such that "no brothes, pottages or phisicalle meates are woorth [worth] anything if Setwall [valerian] were not at one end".

The 17th century astrological botanist Nicholas Culpeper thought the plant was "under the influence of Mercury, and therefore hath a warming faculty". He recommended both herb and root, and said that "the root boiled with liquorice, raisons and aniseed is good for those troubled with cough. Also, it is of special value against the plague, the decoction thereof being drunk and the root smelled. The green herb being bruised and applied to the head taketh away pain and pricking thereof."

Etymology and common names
The name of the herb is derived from the personal name Valeria and the Latin verb valere (to be strong, healthy). Other names used for this plant include garden valerian (to distinguish it from other Valeriana species), garden heliotrope (although not related to Heliotropium), setwall and all-heal (which is also used for plants in the genus Stachys). Red valerian, often grown in gardens, is also sometimes referred to as "valerian", but is a different species (Centranthus ruber), from the same family but not very closely related. Valerian is also called cat's love for its catnip-like effects.

Phytochemicals
Known compounds detected in valerian include:
 * Alkaloids: actinidine, chatinine, shyanthine, valerianine, and valerine
 * Isovaleramide may be created in the extraction process.
 * Gamma-aminobutyric acid (GABA)
 * Valeric acid
 * Isovaleric acid
 * Iridoids, including valepotriates: isovaltrate and valtrate
 * Sesquiterpenes (contained in the volatile oil): valerenic acid, hydroxyvalerenic acid and acetoxyvalerenic acid
 * Flavanones: hesperidin, 6-methylapigenin, and linarin

Preparation
The chief constituent of valerian is a yellowish-green to brownish-yellow oil present in the dried root, varying in content from 0.5 to 2.0%. This variation in quantity may be determined by location; a dry, stony soil yields a root richer in oil than moist, fertile soil.

Traditional medicine
Although valerian is a common traditional medicine used for treating insomnia, there is no good evidence it is effective for this purpose. Valerian has not been shown to be helpful in treating restless leg syndrome or anxiety.

The European Medicines Agency (EMA) approved the health claim that valerian can be used as a traditional herb to relieve mild nervous tension and to aid sleep; the EMA stated that although there is insufficient evidence from clinical studies, its effectiveness as a dried extract is considered plausible.

The American Academy of Sleep Medicine's 2017 clinical practice guidelines recommended against the use of valerian in the treatment of insomnia due to poor effectiveness and low quality of evidence.

Oral forms
Oral forms are available in both standardized and unstandardized forms. Standardized products may be preferable considering the wide variation of the chemicals in the dried root, as noted above. When standardized, it is done so as a percentage of valerenic acid or valeric acid. For commonly used doses, valerian is generally recognized as safe (GRAS) in the United States.

Adverse effects
Because the compounds in valerian produce central nervous system depression, they should not be used with other depressants, such as ethanol (drinking alcohol), benzodiazepines, barbiturates, opiates, kava, or antihistamine drugs.

As an unregulated product, the concentration, contents, and potential contaminants in valerian preparations cannot be easily determined. Because of this uncertainty and the potential for toxicity in the fetus and hepatotoxicity in the mother, valerian use is discouraged during pregnancy. Headache and diarrhea have occurred among subjects using valerian in clinical studies.

Effect on cats
Valerian root is a cat attractant, containing attractant semiochemicals in a way similar to catnip, which can lead to a behaviour modification effect in cats. Its roots and leaves are one of three alternatives for the one-third of domesticated or medium-sized cats who do not feel the effects of catnip. Valerian root has also been reported to be attractive to rats and used to attract members of the family Canidae to traps.

Invasive species
Valerian is considered an invasive species in many locations outside its natural range, including the US state of Connecticut where it is officially banned, and in New Brunswick, Canada, where it is listed as a plant of concern.

title: Valerian (herb)