text:
Prince Igor

Prince Igor (Князь Игорь, ) is an opera in four acts with a prologue, written and composed by Alexander Borodin. The composer adapted the libretto from the early Russian epic The Lay of Igor's Host, which recounts the campaign of the 12th-century prince Igor Svyatoslavich against the invading Cuman ("Polovtsian") tribes in 1185. He also incorporated material drawn from two medieval Kievan chronicles. The opera was left unfinished upon the composer's death in 1887 and was edited and completed by Nikolai Rimsky-Korsakov and Alexander Glazunov. It was first performed in St. Petersburg, Russia, in 1890.

Original composition: 1869–1887
After briefly considering Lev Mei's The Tsar's Bride as a subject (later taken up in 1898 by Nikolai Rimsky-Korsakov, his 9th opera), Borodin began looking for a new project for his first opera. Vladimir Stasov, critic and advisor to The Mighty Handful, suggested The Lay of Igor's Host, a 12th century epic prose poem, and sent Borodin a scenario for a three-act opera on 30 April 1869. Initially, Borodin found the proposition intriguing, but daunting:

"Your outline is so complete that everything seems clear to me and suits me perfectly. But will I manage to carry out my own task to the end? Bah! As they say here, 'He who is afraid of the wolf doesn't go into the woods!' So I shall give it a try ..."

After collecting material from literary sources, Borodin began composition in September 1869 with initial versions of Yaroslavna's arioso and Konchakovna's cavatina, and sketched the Polovtsian Dances and March of the Polovtsy. He soon began to have doubts and ceased composing. He expressed his misgivings in a letter to his wife: "There is too little drama here, and no movement ... To me, opera without drama, in the strict sense, is unnatural." This began a period of about four years in which he proceeded no further on Prince Igor, but began diverting materials for the opera into his other works, the Symphony No. 2 in B minor (1869–1876) and the collaborative opera-ballet Mlada (1872).

The Mlada project was soon aborted, and Borodin, like the other members of The Mighty Handful who were involved – César Cui, Modest Mussorgsky, and Rimsky-Korsakov – thought about ways to recycle the music he contributed. Of the eight numbers he had composed for Act 4 of Mlada, those that eventually found their way into (or back into) Prince Igor included No. 1 (Prologue: The opening C major chorus), No. 2 (material for Yaroslavna's arioso and Igor's aria), No. 3 (Prologue: The eclipse), No. 4 (Act 3: The trio), and No. 8 (Act 4: The closing chorus).

Borodin returned to Prince Igor in 1874, inspired by the success of his colleagues Rimsky-Korsakov and Mussorgsky in the staging of their historical operas, The Maid of Pskov (1873) and Boris Godunov (1874). This period also marks the creation of two new characters, the deserters Skula and Yeroshka, who have much in common with the rogue monks Varlaam and Misail in Boris Godunov.

In his memoirs, Rimsky-Korsakov mentions an 1876 concert at which Borodin's "closing chorus" was performed, the first public performance of any music from Prince Igor identified by him:

"... Borodin's closing chorus ['Glory to the beautiful Sun'] ..., which, in the epilogue of the opera (subsequently removed) extolled Igor's exploits, was shifted by the author himself to the prologue of the opera, of which it now forms a part. At present this chorus extolls Igor as he starts on his expedition against the Polovtsy. The episodes of the solar eclipse, of the parting from Yaroslavna, etc., divide it into halves which fringe the entire prologue. In those days this whole middle part was non-existent, and the chorus formed one unbroken number of rather considerable dimensions."

The idea of a choral epilogue in the original scenario was no doubt inspired by the example of A Life for the Tsar by Mikhail Glinka, to whose memory Prince Igor is dedicated.

Borodin's primary occupation was chemistry, including research and teaching. However, he also spent much time in support of women's causes, much to the consternation of his fellow composers, who felt he should devote his time and talent to music. In 1876, a frustrated Stasov gave up hope that Borodin would ever finish Prince Igor, and offered his scenario to Rimsky-Korsakov. Rimsky-Korsakov instead assisted Borodin in orchestrating important numbers in preparation for concert performance; for example, the Polovtsian Dances in 1879:

"There was no end of waiting for the orchestration of the Polovtsian Dances, and yet they had been announced and rehearsed by me with the chorus. It was high time to copy out the parts. In despair I heaped reproaches on Borodin. He, too, was none too happy. At last, giving up all hope, I offered to help him with the orchestration. Thereupon he came to my house in the evening, bringing with him the hardly touched score of the Polovtsian Dances; and the three of us—he, Anatoly Lyadov, and I – took it apart and began to score it in hot haste. To gain time, we wrote in pencil and not in ink. Thus we sat at work until late at night. The finished sheets of the score Borodin covered with liquid gelatine, to keep our pencil marks intact; and in order to have the sheets dry the sooner, he hung them out like washing on lines in my study. Thus the number was ready and passed on to the copyist. The orchestration of the closing chorus I did almost single-handed ...'"

Borodin worked on Prince Igor, off and on, for almost 18 years.

Posthumous completion and orchestration: 1887–1888
Borodin died suddenly in 1887, leaving Prince Igor incomplete. Rimsky-Korsakov and Stasov went to Borodin's home, collected his scores, and brought them to Rimsky-Korsakov's house.

"Glazunov and I together sorted all the manuscripts ... In the first place there was the unfinished Prince Igor. Certain numbers of the opera, such as the first chorus, the dance of the Polovtsy, Yaroslavna's Lament, the recitative and song of Vladimir Galitsky, Konchak's aria, the arias of Konchakovna and Prince Vladimir Igorevich, as well as the closing chorus, had been finished and orchestrated by the composer. Much else existed in the form of finished piano sketches; all the rest was in fragmentary rough draft only, while a good deal simply did not exist. For Acts II and III (in the camp of the Polovtsy) there was no adequate libretto – no scenario, even – there were only scattered verses and musical sketches, or finished numbers that showed no connection between them. The synopsis of these acts I knew full well from talks and discussions with Borodin, although in his projects he had been changing a great deal, striking things out and putting them back again. The smallest bulk of composed music proved to be in Act III. Glazunov and I settled the matter as follows between us: He was to fill in all the gaps in Act III and write down from memory the Overture played so often by the composer, while I was to orchestrate, finish composing, and systematize all the rest that had been left unfinished and unorchestrated by Borodin."

The often-repeated account that Glazunov reconstructed and orchestrated the overture from memory after hearing the composer play it at the piano is true only in part. The following statement by Glazunov himself clarifies the matter:

"The overture was composed by me roughly according to Borodin's plan. I took the themes from the corresponding numbers of the opera and was fortunate enough to find the canonic ending of the second subject among the composer's sketches. I slightly altered the fanfares for the overture ... The bass progression in the middle I found noted down on a scrap of paper, and the combination of the two themes (Igor's aria and a phrase from the trio) was also discovered among the composer's papers. A few bars at the very end were composed by me."

Musical analysis
Central to the opera is the way the Russians are distinguished from the Polovtsians through melodic characterization. While Borodin uses features of Russian folk music to represent his compatriots, he uses chromaticism, melismas and appoggiaturas—among other techniques—represent their 'heathen' opponents. These methods had already been used by Glinka and others to portray Orientalism in Russian music.

Performance history
"'During the season of 1888–9 the Directorate of Imperial Theatres began to lead us a fine dance with the production of Prince Igor, which had been finished, published, and forwarded to the proper authorities. We were led by the nose the following season as well, with constant postponements of production for some reason or other.' 'On October 23, 1890, Prince Igor was produced at last, rehearsed fairly well by K. A. Kuchera, as Nápravník had declined the honor of conducting Borodin's opera. Both Glazunov and I were pleased with our orchestration and additions. The cuts later introduced by the Directorate in Act 3 of the opera did it considerable harm. The unscrupulousness of the Mariinsky Theatre subsequently went to the length of omitting Act 3 altogether. Taken all in all, the opera was a success and attracted ardent admirers, particularly among the younger generation.'"

The world premiere was given in St. Petersburg on 4 November (23 October O.S.), 1890 at the Mariinsky Theatre. Set designers were Yanov, Andreyev, and Bocharov, while Lev Ivanov was balletmaster. Moscow premieres followed in 1892 by the Russian Opera Society, conducted by Iosif Pribik. The Bolshoi Theatre premiere was given in 1898 and was conducted by Ulrikh Avranek

Other notable premieres were given in Prague in 1899 and in Paris in 1909, with a Sergei Diaghilev production featuring Feodor Chaliapin as Galitsky and Maria Nikolaevna Kuznetsova as Yaroslavna. Ivanov's choreography was revived by Mikhail Fokin (and in that form can be seen in the 1969 film. London saw the same production in 1914 conducted by Thomas Beecham, again with Chaliapin as Galitsky. In 1915 the United States premiere took place at the Metropolitan Opera, staged in Italian and conducted by Giorgio Polacco. The first performance in English was at Covent Garden on 26 July 1919, with Miriam Licette as Yaroslavna.

In January and February 2009 there was a production at the Aalto Theatre by the Essen Opera. While some aspects of the production may have been unusual, one critic noted that "placing the (Polovtsian) Dances as a Finale is an elegant idea, ... the director Andrejs Zagars and the conductor Noam Zur have thus presented a musically and dramaturgically coherent Prince Igor. Heartfelt applause for a worthwhile evening at the opera."

In 2011 there was a concert performance in Moscow by Helikon Opera, based on Pavel Lamm's reconstruction. A new edition based on 92 surviving manuscripts by Borodin was completed by musicologist Anna Bulycheva and published in 2012.

In 2014, the Metropolitan Opera in New York City staged a reconceived version, sung in Russian for the first time there. Director Dmitri Tcherniakov and conductor Gianandrea Noseda removed most of the melodies contributed by Rimsky-Korsakov and Glazunov, although they retained those composers' orchestrations. They added many fragments by Borodin that Rimsky-Korsakov and Glazunov had omitted, basing their work on many decades of musicological research. They rearranged the order in which some of the material appeared, in some cases taking account of notes left by Borodin. The overall conception made the opera more of a psychological drama about Prince Igor and his state of mind, given the deep depression he went into following his soldiers' loss to the Polovtsians. The entire opera was reordered: after the prologue, in which the solar eclipse was taken as a bad omen, Act 1 presented a dream sequence dealing with the relation of Igor and his son with the Polovtsian general and his daughter in the Polovtsian camp. The second act largely dealt with the antics of Prince Galitsky in Putivyl and ended with the destruction of the city. The third act ended with Prince Igor coming out of his depression to begin the rebuilding of the destroyed city. This production starred Russian bass Ildar Abdrazakov in the title role with Ukrainian soprano Oksana Dyka as Yaroslavna. The performances in New York included a worldwide HD broadcast. The production was jointly produced with De Nederlandse Opera of Amsterdam.

At the beginning of the Opening Ceremony of the Winter Olympics in Sochi, Russia, in February 2014, some of Borodin's music from this opera was played while an eclipsed sun, crescent-shaped, drifted across the upper levels of the center of the stadium, showing the basis of Russian history in the Prince Igor story.

Publication history

 * 1885, three arias, piano-vocal score, edition by Borodin, W. W. Bessel, St. Petersburg
 * 1888, piano-vocal score, edition by Rimsky-Korsakov & Glazunov, M. P. Belyayev, Leipzig
 * 1888, full score, edition by Rimsky-Korsakov & Glazunov, M. P. Belyayev, Leipzig
 * 1953, piano-vocal score, edition by Rimsky-Korsakov & Glazunov, Muzgiz, Moscow
 * 1954, full score, edition by Rimsky-Korsakov & Glazunov, Muzgiz, Moscow
 * 2012, piano-vocal score, the original version, edited by Bulycheva, Classica-XXI, Moscow

Roles
Note:
 * The actual given name of the historical Yaroslavna is Yefrosinya (Ефросинья, Euphrosina). Yaroslavna is a patronymic, meaning "daughter of Yaroslav". Konchakovna's name is similarly derived.
 * Yaroslavna's brother, Vladimir Yaroslavich, is often called "Prince Galitsky" (Князь Галицкий), leading to the misconception that he was a prince by the name of Galitsky. In fact, he was a son of Prince of Galich Yaroslav Osmomysl. Prince Galitsky is a title meaning "Prince of Galich".

Synopsis
Time: The year 1185

Place: The city of Putivl (prologue, Acts 1 and 4); a Polovtsian camp (Acts 2 and 3)

Note: ''As discussed in this article, Borodin's final decision on the order of the first two acts is unclear. The traditional grouping presented here is that of the Rimsky-Korsakov-Glazunov edition. In many productions, Act 3 is omitted.''

Prologue
The cathedral square in Putivl

Prince Igor is about to set out on a campaign against the Cumans/Polovtsy and their Khans who have previously attacked the Russian lands. The people sing his praise and that of his son, the other leaders and the army (Chorus: "Glory to the beautiful Sun"). A solar eclipse takes place to general consternation. Two soldiers Skula and Yeroshka desert feeling sure that Vladimir Yaroslavich, Prince Galitsky, will offer them work more to their liking. Although Yaroslavna, Igor's wife, takes the eclipse for a bad omen, Igor insists that honour demands that he go to war. He leaves her to the care of her brother, Prince Galitsky, who tells of his gratitude to Igor for sheltering him after he was banished from his own home by his father and brothers. The people sing a great chorus of praise (Chorus: "Glory to the multitude of stars") as the host sets out on their campaign against the Polovtsy.

Act 1
Scene 1: Vladimir Galitsky's court in Putivl

Galitsky's followers sing his praise. Skula and Yeroshka are now working as gudok-players. They entertain the followers and all sing of how Galitsky and his men abducted a young woman and how she pleaded to be allowed to return to her father without being dishonoured. The prince arrives and sings of how, if he were Prince of Putivl, he would drink and feast all day while dispensing judgment and have the prettiest maidens with him all night (Galitsky's Song). The treasury would be spent on himself and his men while his sister would be praying in a monastery. A group of young women beg the prince to restore their abducted friend. He threatens them and drives them away, saying how she now lives in luxury in his quarters and does not have to work. The prince returns to his rooms having sent for wine for his followers. The gudok players and the prince's followers mock the women. They wonder what might happen if Yaroslavna hears of what happens, but then realise she would be helpless with all her men gone to war. They sing of how they are all drunkards and are supported by Galitsky. The men decide to go to the town square to declare Galitsky the Prince of Putivl, leaving just the two drunk musicians behind.

Scene 2: A room in Yaroslavna's palace

Yaroslavna is alone worrying about why she has not heard from Igor and his companions (Yaroslavna's Arioso). She sings of her tearful nights and nightmares and reminisces about when she was happy with Igor by her side. The nurse brings in the young women who tell Yaroslavna of their abducted friend. They are reluctant at first to reveal the culprit but eventually name Galitsky and talk of how he and his drunken followers cause trouble around Putivl. Galitsky enters and the women run away. Yaroslavna questions him as to the truth of their story and he mocks her saying she should treat him as a guest in her house. She threatens him with what Igor will do on his return, but Galitsky replies that he can seize the throne whenever he wants. Yaroslavna accuses him of repeating the betrayal that he carried out against their father, but he replies that he was only joking and asks if she has a lover now her husband is away. She threatens him with sending him back to their father. He replies that he will return the girl but will take another later and leaves. The council of boyars arrive to inform Yaroslavna that the Polovtsy under Khan Gzak are about to attack Putivl. Igor's army has been utterly destroyed and he has been wounded and captured with his son and brother. After a moment of faintness, Yaroslavna orders messengers sent to the city's allies, but the Boyars report that the roads are cut, some towns are in revolt and their princes will be captured. The Boyars say that they will organise the defence but Galitsky returns with his followers to demand that a new Prince be chosen. His retinue say it should be him as he is Yaroslavna's brother and Igor's brother-in-law. The boyars refuse. The argument is interrupted by the sight of flames and the sound of crying women. Some of the boyars flee; some join the battle, others guard the Princess. They call the attack God's judgment.

Act 2
Evening in the Polovtsian Camp

Polovtsian maidens sing comparing love to a flower that droops in the heat of the day and is revived by night. They dance together (Dance of the Polovtsian Maidens). Konchakovna joins in the singing hoping that her own lover will join her soon (Konchakovna's Cavatina). The Russian prisoners arrive from their day's work and express their gratitude when fed by Konchakovna and the maidens. Their guards retire for the night leaving just Ovlur, a Christian, in charge. Vladimir, son of Igor, sings of his hope that his love will soon join him now that the day is fading (Vladimir's Cavatina). His love is Konchakovna. She comes and the two sing of their love and their desire to marry (Love Duet). While her father will consent to the marriage, they know that his will not. They part when they hear Igor coming. He sings of his disgrace and torment at being captured with his followers dead (Prince Igor's Aria). Only his wife, he feels, will be loyal. He hopes for the chance to regain his honour. Ovlur urges Igor to escape and the prince agrees to think about it. Khan Konchak asks him if all is well (Konchak's Aria) and he replies that the falcon cannot live in captivity. Konchak says that as Igor did not ask for mercy he is not a prisoner but an honoured guest equal to a Khan. Igor reminds him that he too knows what it is to be a captive. Konchak offers Igor freedom if he will promise not to wage war on him again, but he refuses saying he cannot lie. Konchak regrets that they were not born to be allies. They would then have captured all of Russia. He summons the Polovtsian slaves to entertain Igor and himself and offers Igor his choice of them. As the slaves dance the Polovtsy sing of Konchak's glory (Polovtsian Dances).

Act 3
The Polovtsian camp

The Polovtsian army returns in triumph singing the praise of Khan Gzak (Polovtsian March). Konchak sings of the sack of Putivl and other victories and confidently predicts that they will soon capture all of Russia. Igor and his son Vladimir have their worst fears confirmed by the new captives. Vladimir and the other prisoners urge Igor to escape, but he is at first reluctant, singing of his shame and saying that it is the duty of the other Russian princes to save the homeland (Igor's Monologue, Mariinsky edition only). Ovlur now arrives to say that he has prepared horses for Igor and Vladimir and Igor now agrees to escape. The distressed Konchakovna comes, challenging Vladimir to show his love by either taking her with him or by staying. Igor urges his son to come, but Vladimir feels unable to leave Konchakovna who threatens to wake the camp. Eventually Igor flees alone and Konchakovna sounds the alarm. She and her father refuse to let the Polovtsy kill Vladimir. Instead Konchak orders the death of the guards and marries Vladimir to his daughter. As for Igor, Konchak thinks more of him for his escape.

Act 4
Dawn in Putivl

Yaroslavna weeps at her separation from Igor and the defeat of his army, blaming the very elements themselves for helping the enemy (Yaroslavna's Lament). Peasant women blame not the wind but Khan Gzak for the devastation. As Yaroslavna looks around to acknowledge the destruction, she sees two riders in the distance who turn out to be Igor and Ovlur. The two lovers sing of their joy of being reunited and of the expectation that Igor will lead the Russians to victory against the Khan. Unaware of Igor's return, Skula and Yeroshka, the drunken gudok players, sing a song that mocks him. Then they notice him in the distance. After a moment of panic about what will happen to them, Skula says that they should rely on their cunning and decides on a plan that will save them. They ring the church bells to summon a crowd. Although people at first treat them with suspicion, the gudok players manage to convince the crowd that Igor has returned and the boyars that they are loyal followers of the true prince and not Galitsky. All joyously celebrate Igor's return.

Principal arias and numbers

 * Overture

Prologue
 * Chorus: "Glory to the beautiful Sun", «Солнцу красному слава!» (People of Putivl)
 * Chorus: "Glory to the multitude of stars", «Частым звёздочкам слава!» (People of Putivl)

Act 1
 * Song: "If only I had the honor", «Только б мне дождаться чести» (Galitsky)
 * Arioso: "A long time has passed", «Немало времени прошло с тех пор» (Yaroslavna)

Act 2
 * Dance: "Dance of the Polovtsian Maidens", «Пляска половецких девушек» (Orchestra)
 * Cavatina: "The light of day fades", «Меркнет свет дневной» (Konchakovna)
 * Cavatina: "Slowly the day died away", «Медленно день угасал» (Vladimir)
 * Duet: "Is that you, my Vladimir?", «Ты ли, Владимир мой?» (Konchakovna, Vladimir)
 * Aria: "No sleep, no rest for my tormented soul", «Ни сна, ни отдыха измученной душе» (Igor)
 * Aria: "Are you well, Prince?", «Здоров ли, князь?» (Konchak)
 * Polovtsian Dances: "Fly away on the wings of the wind", «Улетай на крыльях ветра» (Slaves, Konchak)

Act 3
 * March: "Polovtsian March", «Половецкий марш» (Orchestra)
 * Trio: "Vladimir! Is all this really true?", «Владимир! Ужель все это правда?» (Konchakovna, Vladimir, Igor)

Act 4
 * Aria: "Oh, I weep", «Ах, плачу я» (Yaroslavna)
 * Chorus: "God heard our prayers", «Знать, господь мольбы услышал» (People of Putivl)

Both the Overture to Prince Igor and the "Polovtsian Dances" (from Act II) are well-known concert standards. Together with the "Polovtsian March", they form the so-called "suite" from the opera.

Critical analysis
Prince Igor is a staple of Russian opera, but has not travelled well abroad. One obvious reason is the Russian language, although translation into Italian was once a solution.

Another explanation for the failure to gain acceptance is its lack of unity resulting from its unfinished state. Despite the skill and efforts of editors Rimsky-Korsakov and Glazunov, the opera is still episodic and dramatically static, a problem of which the composer himself was aware when he embarked on composition (see quote above in "Composition History"). This is partly a consequence of Borodin's failure to complete a libretto before beginning composition of the music—the same problem that plagued his colleague Mussorgsky in the composition of Khovanshchina. Both composers wrote their librettos piece by piece while composing the music, both lost sight of the overall narrative thread of their operas, and both wound up with pages and pages of music that needed to be sacrificed to assemble a cohesive whole. Also, both died before finishing their operas, leaving the task of completion, editing, and orchestration in both cases to Rimsky-Korsakov.

Performance practice
One of the main considerations when performing Prince Igor is the question of whether to include Act 3, much of which was composed by Glazunov. The practice of omitting it was mentioned as early as 1909 in Rimsky-Korsakov's memoirs. Many productions leave Act 3 out because it "fails to carry conviction both musically and dramatically." On the other hand, maintaining the act has certain benefits. It contains some fine pages (e.g., the "Polovtsian March"), provides an important link in the narrative (Igor's escape, Vladimir's fate), and is the origin of some of the memorable themes first heard in the overture (the trio, brass fanfares). Fortunately, the option of omitting the fine overture, also known to have been composed by Glazunov, is seldom considered.

Recently, the question of the best sequence of scenes in which to perform the opera has gained some prominence. Borodin did not complete a libretto before composing the music to Prince Igor. The opera has traditionally been performed in the edition made by Rimsky-Korsakov and Glazunov. It will be obvious that the positions to which they assigned the Prologue, Act 3, and Act 4 cannot be changed if the story is to make sense. However, because the events of Act 1 and Act 2 overlap and are independent of one another, Act 2 may just as well precede Act 1 without any loss of coherence. Soviet musicologists Pavel Lamm and Arnold Sokhor reported the existence of a written plan (now in Glinka's Musical Culture Museum, Moscow), in Borodin's hand, that specified this sequence of scenes:

 * 1) The omen from heaven (Prologue)
 * 2) Imprisonment (Act 2)
 * 3) Galitsky's court (Act 1, Scene 1)
 * 4) Yaroslavna's palace (Act 1, Scene 2)
 * 5) Escape (Act 3)
 * 6) Return (Act 4)

Sokhor assessed the plan as not written later than 1883. The 1993 recording of Prince Igor by Valery Gergiev with the Kirov Opera features a new edition of the score with additions commissioned from composer Yuri Faliek for a production at the Mariinsky Theatre, adopting this hypothetical original sequence. The authors of the notes to the recording assert that this order better balances the musical structure of the score by alternating the acts in the Russian and Polovtsian settings with their distinctive musical atmospheres.

Despite this justification, there is reason to maintain the traditional sequence. Act II contains most of the numbers for which the work is known today, with Igor's brooding and impassioned aria ("Oh give me freedom") at the center, flanked by Vladimir's cavatina and Konchak's aria, not to mention the rousing conclusion provided by the Polovtsian Dances. Moving its wealth of arias and dances from the center of the work to near the beginning may weaken the opera's structure.

The "Mariinsky edition" makes other important changes and additions to the score. Although much of the material composed or orchestrated by Glazunov and Rimsky-Korsakov is retained, there are additions culled from the unpublished vocal score by Pavel Lamm, orchestrated and linked by Faliek. The changes include:
 * About 200 bars added to the scene in Yaroslavna's palace which make explicit Galitsky's rebellion
 * Various additions and removals from Act 3, including the restoration of a monologue for Igor composed by Borodin in 1875. A review in Gramophone highlights how the newly added monologue "helps to give a weighty focus to Act 3, otherwise a phenomenal feat of reconstruction on Glazunov's part, but somehow insubstantial".
 * A different final chorus for Act 4, "Glory to the multitude of stars", a repeat of material from the Prologue. This idea is historically justified, as Borodin had originally placed this chorus at the end of the opera in the form of an epilogue [see the quote by Rimsky-Korsakov above under Composition history]. This regrettably necessitates the elimination of Borodin's subsequent chorus, "God heard our prayers".

Outside Russia, the opera has often been given in languages other than Russian. For example, the 1960 recording under Lovro von Matačić is sung in German, the 1964 recording under Armando La Rosa Parodi is in Italian and the 1982 radio broadcast of Opera North conducted by David Lloyd-Jones is in English. On the other hand, the 1962 Oscar Danon and the 1990 Bernard Haitink recordings are Western performances sung in Russian.

Structure

 * This is a sortable table. Click on the button next to the criterion you would like to use to sort the information.
 * The numbers are given according to the traditional Rimsky-Korsakov-Glazunov edition.
 * The dates refer to composition, not orchestration. Where a pair of dates differ, a large gap (more than one year) may indicate an interruption of composition or a revision of the musical number.
 * In No.1 (the Prologue), the Eclipse scene (301 bars) was orchestrated by Rimsky-Korsakov and the remainder by Borodin.

Recordings
This is a list of studio recordings. A comprehensive list of all recordings of Prince Igor may be found at operadis-opera-discography.org.uk

Audio
 * 1936-38, Lev Steinberg, Aleksander Orlov and Alexander Melik-Pashayev (conductor), Bolshoi Theatre Orchestra and Chorus, Alexander Baturin (Igor), Xenia Derzhinskaya (Yaroslavna), Ivan Kozlovsky (Vladimir), Aleksandr Pirogov (Galitsky), Maxim Mikhailov (Konchak), Elizaveta Antonova (Konchakovna) (abridged over 32 sides)
 * 1941, Alexander Melik-Pashayev (conductor), Bolshoi Theatre Orchestra and Chorus, Alexander Baturin (Igor), Sofia Panova (Yaroslavna), Ivan Kozlovsky (Vladimir), Alexander Pirogov (Galitsky), Maxim Mikhailov (Konchak), Nadezhda Obukhova (Konchakovna) - Melodiya
 * 1952, Alexander Melik-Pashayev (conductor), Bolshoy Theatre Orchestra and Chorus, Andrey Ivanov (Igor), Yelena Smolenskaya (Yaroslavna), Sergey Lemeshev (Vladimir), Alexander Pirogov (Galitsky), Mark Reyzen (Konchak), Vera Borisenko (Konchakovna) - Melodiya
 * 1955, Oskar Danon (conductor), Belgrade National Opera Orchestra and Chorus; Dušan Popović (Igor), Valerija Heybalova (Yaroslavna), Noni Zunec (Vladimir), Žarko Cvejić (Galitsky, Konchak), Melanija Bugarinović (Konchakovna) - Decca / London
 * 1966, Jerzy Semkow (conductor), National Opera Theatre of Sofia; Constantin Chekerliiski (Igor), Julia Wiener (Yaroslavna), Todor Todorov (Vladimir), Boris Christoff (Galitsky, Konchak), Reni Penkova (Konchakovna) - EMI /Angel / Pathé
 * 1969, Mark Ermler (conductor), Bolshoy Theatre Orchestra and Chorus; Ivan Petrov (Igor), Tatyana Tugarinova (Yaroslavna), Vladimir Atlantov (Vladimir), Artur Eisen (Galitsky), Aleksandr Vedernikov (Konchak), Elena Obraztsova (Konchakovna) - Melodiya
 * 1990, Emil Tchakarov (conductor), Sofia Festival Orchestra and National Opera Chorus, Boris Martinovich (Igor), Stefka Evstatieva (Yaroslavna), Kaludi Kaludov (Vladimir), Nicola Ghiuselev (Galitsky), Nicolai Ghiaurov (Konchak), Alexandrina Milcheva-Nonova (Konchakovna) - Sony 44878
 * 1993, Valery Gergiev (conductor), Kirov Opera Orchestra and Chorus; Mikhail Kit (Igor), Galina Gorchakova (Yaroslavna), Gegham Grigoryan (Vladimir), Vladimir Ognovienko (Galitsky), Bulat Minjelkiev (Konchak), Olga Borodina (Konchakovna) - Philips 442–537–2.

Video
 * 1981 Evgeny Nesterenko (Prince Igor), Elena Kurovskaya (Jaroslavna), Vladimir Sherbakov (Vladimir Igorevich), Alexander Vedernikov (Prince Galitsky), Boris Morozov (Konchak), Tamara Sinyavskaya (Konchakovna), Vladimir Petrov (Ovlur), Valery Yaroslavtsev (Skula), Konstantin Baskov (Yeroshka), Nina Grigorieva (Nurse), Margarita Miglau (Polovtsian Maiden) Orchestra & Chorus of the Bolshoi Theatre, Mark Ermler.
 * 1993 Nikolai Putilin, Galina Gorchakova, Evgeny Akimov, Sergey Aleksashkin, Vladimir Vaneev, Olga Borodina, Kirov Opera & Ballet, Valery Gergiev

Popular culture
In the American musical Kismet (1953), most of the score was adapted from works by Borodin. Themes from the Polovtsian Dances from Prince Igor were used extensively and the "Gliding Dance of the Maidens" provided the melody for the popular hit song "Stranger in Paradise".

In The Simpsons episode "Simpson Tide", the Boyars' Chorus (Act 1, Scene 2) plays while tanks emerge from parade floats during a peace parade on Red Square in front of Saint Basil's, soldiers walk out of a building, the Berlin Wall re-erects itself out of the ground, and Lenin rises from his grave, saying "Rrr! Must Crush Capitalism, Rrr!". The "Gliding dance of the Maidens" is also played during the Valentine's Day Itchy and Scratchy cartoon featured in The Simpsons episode "I Love Lisa".

Musical quotations from the Polovtsian Dances can be heard:
 * in Warren G's 1997 hit "Prince Igor", and in the track "Echoes" on Pink Floyd's 1971 album, "Meddle"
 * Masterplan – Lonely Winds Of War (2010)
 * Arash feat. Helena – Angels Lullaby (2021)
 * Olivia Addams x Dylan Fuentes — Telepathy (2022)

title: Prince Igor


text:
Talk:Accelerating expansion of the universe

More theories for the accelerating expansion
In the section "alternative theories" the sentence "Räsänen does not consider the model likely" is not backed up by the reference added to that sentence. In the New Scientist article referred to Räsänen in stead is quoted as thinking that his theory is a plausible explanation for the accelerated expansion. His theory is just in its initial stages and the toy models he considers have some flaws vis à vis the modelling of the real universe.

More theories for the accelerating expansion
I'm not in favor of any one theory as there is no reason, or evidence, yet to believe any one of them is better in a lot of cases. The article in its current form is not very encompassing of the topic and could use significantly more work to add to it. It could even be there is no mysterious acceleration happening since that is based on an axiom that there is no special place or time in the universe (which is believed with no evidence whatsoever that it is actually true). This could already be argued against for a few reasons such as the time we live in is special since we are able to observe enough space to see other galaxies. In the future most intelligent civilizations may only be able to see a few local galaxies and therefore come to a very different conclusion on what happened in the timeline of the universe, perhaps not even being able to form a timeline of the universe since they won't be able to see the expansion occurring. Anyway there are a lot more theories and information should be added based on the Copernicus test of no special place existing and the idea that there is no reason we could not exist in a special place. I admit that it could be a very small chance but perhaps the universe is infinite and a special place is necessary for advanced life to form so it wouldn't matter how unlikely it is because it would happen an infinite amount of times. Or, for reasons unknown, the highly improbable happened regardless of how poor the probability is. Science becomes limited when we hold something to be true only because we want it tone true. — Preceding unsigned comment added by 2601:B:A3C0:7:A819:A15:7CD:529 (talk) 17:37, 3 September 2013 (UTC)

So I would at the very least suggest a disclaimer be placed stating that the article for this subject needs a lot more work to be anywhere near completion. I would also just like to say I don't support theories that are not provable or falsifiable. They do have to be logically put together and have predictions that canbe tested in some way. So the theory that the universe is only a year old, presumably by quantum fluctuations since including religious dogma would just start to muddle the science behind physics, for example but everything just appeared into existence in a way to make us believe its older. — Preceding unsigned comment added by 2601:B:A3C0:7:A819:A15:7CD:529 (talk) 17:49, 3 September 2013 (UTC)

An alternative theory: I recently submitted to the university of utah astrophysics group my theory entitled "An Engineered Universe: The Origin and Structure of the Universe". What I attempted to do is imagine the state of affairs before the big bang and postulate what was there and how structure was derived from it with the goal of reproducing the properties of our universe. My background is physics, engineering and I worked for 35.5 years in structural analysis with my interest focused on constitutive modeling. Constitutive modeling related the stress-strain response back to the material's internal constitutive structures. What seemed apparent to me is that our space is a composite structure. So I postulated the properties of the elementary constituent and proceeded developing structure. The approach continually surprized me as I slowly realized how different features/properties of our universe emerged. The randomness of quantum behavior, quantum foam, quantum entanglement, elements of the uncertainty principle, distance/separation, the exponential growth of space, a quantum theory of gravity, the big bang, separation of matter and anti-matter, charge balance, some dark matter, a potential route to gravity force neutralization, a possible approach to travel without inertia, and I ended on the accelerating universe with the slowing of time as contributing to the apparent acceleration. In this theory, an embryonic universe builds for ~2 trillion year (1.88T which was calculated from the physical constants and the generally accepted properties of our universe) and then is forced to split (big bang) into two 3D branes in the bulk - using string theory nomenclature. (It has features somewhat similar to the Ekpyrotic Universe model.) The structure of these two spaces have inherent differences leading to slight asymmetric decay nature of some particles. The randomness of the CMB occurs naturally. The theory does not permit worm holes and a singularity never occurs during the bang. There results two time metrics, one for growth of space and one for our electromagnetic existence. The interaction of the two leads to increased acceleration of space. spacemaker@comcast.net — Preceding unsigned comment added by 71.195.249.185 (talk) 18:05, 8 February 2014 (UTC)

Protection
I've temporarily semi-protected this page to cut off an edit war that has already exceeded the three revert rule. I agree that some of the material that 89.110.* has been adding is problematic in that it espouses a minority viewpoint on interpreting the evolution of the universe that doesn't deserve the prominence he is giving it (WP:WEIGHT). However, some of his edits, including grammatical changes and additional background info (e.g. mentioning the Nobel prize in the intro) are probably reasonable. I would encourage the parties to review the edits, restore any of 89.110.* content that is reasonable, and discuss the points of disagreement. Dragons flight (talk) 21:08, 24 February 2015 (UTC)
 * Thanks. I have restored much of the IP warrior's lead and edited it for neutrality etc., but I am on a bleepy thing so cannot easily do this for the body of the article. — Cheers, Steelpillow (Talk) 21:32, 24 February 2015 (UTC)

longer time?
Evidence for acceleration - Supernova observation -

"In an accelerating universe, the universe was expanding more slowly in the past than it is today, which means it took a longer time to expand from 2/3 to 1.0 times its present size compared to a non-accelerating universe." Should this not be "it took a shorter time", since expansion is accelerating?

Also, the use of the word "accelerating" to describe an increased rate of expansion is confusing, since in physics a decreasing rate of expansion would be just as accurately described as an acceleration. It seems imprecise to express it this way, with the implicit assumption that acceleration would necessarily mean an increased rate. — Preceding unsigned comment added by 70.59.95.151 (talk) 16:52, 15 July 2018 (UTC)

Error in Acceleration time?
FWIW - following discussion seems particularly relevant to the "Accelerating expansion of the universe" article (see copy below):

Copied from Talk:Timeline of the formation of the Universe

"Error in Acceleration time?"

You have an error in the acceleration time. That acceleration started 6.2 billion years ago, that is, 7.6 billion years after the Big Bang. [Note: Preceding unsigned comment added by 190.215.80.188 (talk) 12:32, 5 October 2015 (UTC)]
 * FWIW - the above comment about the time the universe expansion began accelerating seems supported by astrophysicist Ethan Siegel "... about 7.8 billion years old, or about 6 billion years ago, some 1.5 billion years before our Solar System formed." - however, a note by astronomer Joshua Frieman at CalTech notes => "the Universe began accelerating at redshift z ~ 0.4 and age t ~ 10 Gyr." [Note: "9.4" Gyr? determined using a cosmological calculator. ] - estimated time that expanding universe began accelerating may need clarification - comments from others welcome of course - in any case - Enjoy! :) Drbogdan (talk) 16:17, 1 April 2016 (UTC)
 * BRIEF Followup: The "7.8 billion years old" age of the accelerating expanding universe seems to be supported by comments made on the following talk page => "User talk:I9Q79oL78KiL0QTFHgyc" - see copy below:

Copied from "User talk:I9Q79oL78KiL0QTFHgyc":

"HELP: Clarify Time Expanding Universe Accelerated?"

Hello Josh:

IF POSSIBLE => may need help to clarify the best known Time the expanding universe began accelerating - this refers to several Wikipedia pages including "Accelerating expansion of the universe", "Timeline of the formation of the Universe" and "Talk:Timeline of the formation of the Universe" - Times presented on these pages (and related ones) seem to vary and include "7.8 billion years old"/astrophysicist ESiegel ; "10 Gyr" (z ~ 0.4)/astronomer JAFrieman ; "9.4 Gyr" (determined from z = 0.4 per Fermilab calculator) ; perhaps other values as well - any help to clarify the Time the expanding universe began accelerating would be appreciated - in any case - Enjoy! :) Drbogdan (talk) 12:44, 2 April 2016 (UTC)
 * 10 Gyrs is approximately right. It is going to occur at a redshift that solves the equation $$\Omega_\Lambda = \Omega_m (1 + z)^3$$ Roughly, $$\Omega_\Lambda = 0.7$$ and $$\Omega_m = 0.3$$, so it's actually closer to $$z=0.3$$ which I think puts it closer to the 7.8 billion year estimate. jps (talk) 13:15, 2 April 2016 (UTC)
 * @jps - Thank you *very much* for your comments - and clarifications - they're *greatly* appreciated - Thanks again - and - Enjoy! :) Drbogdan (talk) 13:21, 2 April 2016 (UTC)

 * In any case - Enjoy! :) Drbogdan (talk) 13:50, 2 April 2016 (UTC)

Hope the discussion copied above helps in some way - in any case - Enjoy! :) Drbogdan (talk) 16:16, 2 April 2016 (UTC)

Mechanism
I have removed a section added by 91.122.8.180 called "Mechanism" that looks to be original research. The first part talking about gravitational collapse is referenced, but the subsequent parts do not appear to have support from any reference. The addition can be seen in https://en.wikipedia.org/w/index.php?title=Accelerating_expansion_of_the_universe&oldid=715685352, so do people here agree that this section should not be in the article? Graeme Bartlett (talk) 11:18, 17 April 2016 (UTC)
 * The second part is a restatement of the first. 91.122.8.180 (talk) 11:24, 17 April 2016 (UTC)

Only marginal evidence ~3 sigma
"Marginal evidence for cosmic acceleration from Type Ia supernovae", J. T. Nielsen, A. Guffanti & S. Sarkar - estimated evidence for accelerating expansion with only ~3 sigma, which is insufficient. Article in under CC-BY 4.0: http://www.nature.com/articles/srep35596 OverQuantum (talk) 10:36, 24 October 2016 (UTC)
 * Agree saomeone should add this. that theory alwSergey Woropaew (talk) 10:29, 25 February 2018 (UTC)ays looke waek
 * It's already in the article (ref #42). Banedon (talk) 20:48, 25 February 2018 (UTC)

Criticism of supernova observational data
The essential query I would like to make is "Are the supernova observations data beyond the critical threshold that would lend credence to the idea of the universe's accelerating expansion?". The below site seems to suggest that such a 'reasonable doubt' criteria is not met, at least in regards to the Supernova data. I think the article should state this.

According to : http://www.ox.ac.uk/news/science-blog/universe-expanding-accelerating-rate-%E2%80%93-or-it "We analysed the latest catalogue of 740 Type Ia supernovae – over ten times bigger than the original samples on which the discovery claim was based – and found that the evidence for accelerated expansion is, at most, what physicists call "3 sigma". This is far short of the 5 sigma standard required to claim a discovery of fundamental significance."

Of course, it is possible that the Universe is expanding at an accelerating rate (especially given the other observations), but, according to this website, the issue is not resolved beyond a reasonable doubt in relation to the Supernova data. This may influence the class of viable theories for the accelerating rate of expansion of the universe.

ASavantDude (talk) 20:28, 23 March 2018 (UTC)
 * Already in article - "Yet other possibilities are that the accelerated expansion of the universe is an illusion caused by the relative motion of us to the rest of the universe, or that the supernovae sample size used wasn't large enough (emphasis mine)." I'll add that Prof. Sarkar's work isn't widely accepted. Banedon (talk) 03:00, 24 March 2018 (UTC)

Supernova observation and Accelerated Expansion
Article says:

"So a supernova with a measured redshift z = 0.5 implies the universe was 1/(1 + 0.5) = 0.67 of its present size when the supernova exploded. In an accelerating universe, the universe was expanding more slowly in the past than it is today, which means it took a longer time to expand from two thirds its present size to its present size, compared to a non-accelerating universe with the same present-day value of the Hubble constant."

This seems problematic to me. This would be true of a decelerating universe. This age of universe is: 0.67 *13.8= 9.2 billion years at constant expansion rate and it will take it 4.6 billion years to reach current size. A decelerating universe will take more time to reach current size, while accelerating one will be bigger; which is other way to say that it took more than 9.2 billion years to reach 0.67 and less than 4.6 to to reach 1.0.

This question was posted before by somebody else but was not answered. The editor who added this section is no more active. If no explanation comes in next few days, I will change it to: which means it took a shorter time to expand from two thirds its present size to its present size, compared to a non-accelerating universe. Shorter time, bigger size, fainter nova--  AhmadLX  ( talk ) 03:43, 20 May 2018 (UTC)

 * , could you guys please shed some light on this? Thank you--  AhmadLX  ( talk ) 18:51, 21 May 2018 (UTC)
 * The current text is correct. The key is "with the same present-day value of the Hubble constant".  That is equivalent to saying "with the same rate of expansion today".  A decelerating universe was expanding faster in the past than it is now.  An accelerating universe is the opposite.  So a decelerating universe covers the "distance" from z=.5 to z=0 faster than an accelerating one.   Waleswatcher  ( talk ) 19:31, 21 May 2018 (UTC)
 * Hi, thanks for your response. But you are assuming acceleration started very recently. And that way, the two versions of the universe (decelerating and accelerating) won't converge at z=0, which they must (they would differ in size). Also, acceleration started around 4 billion years ago.  AhmadLX  ( talk ) 20:05, 21 May 2018 (UTC)
 * "you are assuming acceleration started very recently" - no, I'm not assuming that. Maybe an analogy will help.  Suppose I tell you a car is going 60mph (one mile per minute) right now.  If the car neither accelerated nor decelerated over the last minute, then it took it one minute to travel the last mile before now.  If it decelerated over the last minute, then it took less than a minute to travel that last mile.  If it accelerated (over a sufficiently long time), then it took it more than a minute to travel that last mile.  If it accelerated for a while and then decelerated, then it depends.  Right?
 * "And that way, the two versions of the universe (decelerating and accelerating) won't converge at z=0, which they must (they would differ in size)." Not so.  Set a = a_0 today.  Set \dot a/a = H_0 today.  Now with those same a and H today, consider two different pasts, one that is accelerating and one decelerating.  (By the way before I wrote my first response I did this exercise and worked out the math, just to be sure.)
 * "Also, acceleration started around 4 billion years ago." That is true and makes it more complicated, but it would only matter if the universe was decelerating faster prior to 4 billion years ago then in the decelerating always scenario (think about the car).  Waleswatcher  ( talk ) 20:57, 21 May 2018 (UTC)
 * "If it decelerated over the last minute, then it took less than a minute to travel that last mile.". If it decelerated it will take more time to cover the last mile, I guess :) But I figured out the mistake that I was making. I was constraining decelerating universe to have same age as  accelerating one . Thanks anyway. Maybe you can add a note on this in the article?  AhmadLX  ( talk ) 22:41, 21 May 2018 (UTC)
 * No, it will take less! Remember, we are comparing two "trajectories" with the same final speed and same final location, and asking how long it took to cover the last mile before reaching that final location. If you decelerated, it means you were going faster before, so less time.
 * You mean a note explaining this in more detail? I'm a little leery of that, we should probably have a reliable source to summarize rather than adding our own explanations.  Waleswatcher  ( talk ) 00:13, 22 May 2018 (UTC)
 * Yeah but it is confusing as it stands. A note doesn't necessarily require ref I think. Current description is unsourced anyway; risk of getting cn tagged will be reduced if there is note explaining apparent confusion. thanks.  AhmadLX  ( talk ) 02:16, 25 May 2018 (UTC)

"Accelerating expansion" is incorrect.
This article repeats one of the most common misconceptions about the accelerated expansion of the universe. It's unfortunate because it's a misconception which is repeated in a great many sources which talk about the expansion.

To elucidate what is wrong about this description, a little bit of math helps. In simplified form, the first Friedmann equation can be written as:
 * $$ \frac{\dot{a}^2}{a^2} = H^2 = \frac{8 \pi G \rho + \Lambda c^2}{3} $$

Here, H is the rate of expansion, a is the scale factor, Λ is the cosmological constant, ρ is the matter/energy density of the contents of the universe, G is the gravitational constant, and c is the speed of light (note that I'm ignoring curvature here, as it's not relevant to the discussion).

As the universe expands, the density ρ decreases (except in the hypothetical Big Rip cosmology which is typically considered highly unlikely). Since Λ is a constant, and ρ only decreases, it is impossible for the rate of expansion H to increase.

What is referred to as the accelerated expansion, then, refers not at all to the rate of expansion H, but instead to the fact that distances between objects are increasing at an accelerated rate. This can be represented mathematically as $$\ddot{a} > 0$$.

I propose the following changes to this article:
 * 1) Rename it to "Accelerated Expansion of the Universe", redirect the old page to the new one.
 * 2) Clean up the wording to eliminate all reference to the rate of expansion increasing. Some of the math could also be made more clear.
 * 3) Add a section which explains and refers to this misconception (or, at a minimum, a sentence stating the misconception with a citation that goes into detail). I don't currently have a good citation which refers to this misconception, but I could hunt for one.

Kbdick24 (talk) 23:31, 10 November 2018 (UTC)

 * It's just a matter of how one defines "rate of expansion". I would say that if $$\ddot{a} > 0$$, the rate of expansion is accelerating. Eric Kvaalen (talk) 07:48, 20 April 2021 (UTC)
 * Using "accelerated" will be factually incorrect - as long as the second derivative of the scale factor is positive "accelerating" is true. The explanation that the objects accelerate but the first derivative at a fixed distance is (most probably) dropping should suffice. Kerguelen Rogue Avon (talk) 19:00, 29 September 2022 (UTC)

The proper noun used as the name of our universe is Universe
For other universes we write universe. (How does that sound different?) Our Universe is pronounced in a conceited way. — Preceding unsigned comment added by 2A02:587:410E:FF00:B1B0:8DDB:B80B:7431 (talk) 07:46, 8 March 2019 (UTC)

Lead sentence
Current lead sentence is dreadful. Better just to say that H will never drop below a certain value. cheers, Michael C. Price talk 18:09, 27 March 2019 (UTC)

"laymans section"
I think this page, and pages like it, need a section,or links to pages that simplify some of these things. For instance, a 10 year old kid coming here to read this for whatever reason, probably won't understand half of it. An example is this sentence 'an accelerated expansion can be accounted for by a positive value of the cosmological constant Λ,' see that funny symbol, im guessing it's Greek, i aint got a clue, but even if it is, knowing that still doesn't add meaning to it to me in the context of this sentence, and a kid trading this is just gonna see a weird symbol. A lot of wikipedia pages about things like this or quantum mechanics or, scientific equations for ex,seen to be written under the using that the reader knows what they are looking at. But they won't. I love reading new stuff, and live for learning new things about absolutely anything on here, but i constantly find myself, especially in regards to scientific and mathematical explanations, not understanding half of the symbols used, or how to actually put something in reading into words. Take that symbol from earlier. How do you word that symbol in your head when reading it when you dont know what it means? And i think it would be greatly beneficial, for thickos like myself lol, and especially for younger readers of wikipedia pages, if there was a so called laymans section, or simplified section /explanation of the subject. Would also love to know what that upside down v is and how you pronounce it.lol D0S81 (talk) 02:01, 4 March 2020 (UTC)

 * The trouble, is seems to me, is that scientists, or science enthusiasts, don't take science seriously. If they did, they would understand that there is no "ego" as it is ordinarily conceived. Darwin was right, humans are animals. Or to put it differently, they are both flesh and blood. If you take the imaginary "soul" out of the picture, what's left? The self doesn't have any continuous existence in time. As humans, we certainly don't have the "virtue" and "honesty" that we ascribe to ourselves. Arguably, the human mind is a self-serving lying machine, capable of adding two-plus-two or engineering a rocket with a mission to Pluto and beyond, but untrustworthy when it comes to emotionally-charged issues.
 * Wikipedia should be an information source for everyone, including, as you say, 10-year-olds trying to understand the universe. But it is monopolized by pedantic types who are showing off to each other. It's as if the questioner (in this case the person searching the internet for answers to a question) is ignored while the "experts" huddle together in a circle competing against each other.
 * Some articles, such as this one, should be written in two versions: one for lay people, and another for advanced readers. The layperson's verson should appear first, and at the top of the page there could be a link to the "advanced" version.
 * James 2600:8801:BE31:D300:A8E7:F7D4:F07D:1027 (talk) 17:04, 17 April 2022 (UTC)

Doubt about the acceleration
I have added the following paragraph:
 * In 2019 a paper was published claiming that the conclusion in favor of an accelerating expansion is unfounded. The authors anayzed the Joint Light-curve Analysis catalog of Type Ia supernovas, containing ten times as many supernovas as were used in the 1998 analyses, and found that there was a large dipole in the relationship between luminosity distance and redshift, falling off exponentially with distance (decreasing by a factor of e over 100 megaparsecs). When including this dipole into the fit, there was little evidence for a "monopole", that is, for an isotropic acceleration in all directions. The dipole is roughly aligned with our motion with respect to the cosmic microwave background, but not exactly, with the supernovas being dimmer at a given redshift in the direction of Hydra.

Last night my edit was quickly reverted with the comment "This sort of claim needs additional citations." I have added four more references, but I object to the idea that we should exclude a paragraph like this just because the conclusion has not been widely accepted. The analysis is good. The authors are not lying. The paper was peer reviewed. We should not exclude such information just because it goes against some Nobel prizes. We are Wikipedia, not the Nobel committee. Note that I did not dare to write "the acceleration is wrong", I just reported that a new analysis has been done, and the results call into question the whole idea of the acceleration. I don't believe that Wikipedia has a policy of excluding the mention of any research that goes against the received wisdom. It is certainly important if the conclusions are true. Eric Kvaalen (talk) 07:48, 20 April 2021 (UTC)

 * I object to including this. The relevant policy is WP:FRINGE. To quote, "a Wikipedia article should not make a fringe theory appear more notable or more widely accepted than it is". It's certainly important if the conclusions are true, but since few cosmologists are taking the conclusions seriously, they don't deserve to be included. I am going to revert. Banedon (talk) 07:59, 20 April 2021 (UTC)

 * I object to your reversion. I asked that my edit not be reverted until consensus be reached, and you ignored that. I did not make the theory appear more widely accepted than it is, and it is certainly notable. We should not be taking sides with the "establishment" in ignoring facts that go against their long-standing idées fixes. Our readers should be told that the evidence for a global acceleration is weak when one analyzes a larger set of data and when one allows for a dipole. Why should Wikipedia deprive its "poor readers" of such facts? Eric Kvaalen (talk) 06:04, 21 April 2021 (UTC)
 * Relevant policy here is WP:BRD - I do not need to wait for consensus that the material should be removed before reverting. As for why it should be removed, we deprive our "poor readers" of such facts for the same reason we do not include claims such as how the Michelson-Morley experiment detected the aether back in the early 20th century. There are lots of fringe theories; if we include one then there's an argument to include all of them and the article will become very bloated. The article should cover only the mainstream. If you agree that the theory is not widely accepted, then please give an explanation why WP:FRINGE does not apply. Banedon (talk) 03:00, 26 April 2021 (UTC)

81.156.186.152 (talk) 23:04, 2 January 2022 (UTC) I think, for a more complete article, it would be appropriate to add whatever well considered doubts there may be over the cosmic acceleration, as well as the strong reason that favours a cosmic acceleration; that, prior to the apparent discovery of the acceleration, the calculable age of the visible universe was too young to fit with its apparent state of evolution. An acceleration made the visible universe older and seemed to answer that mystery. (MarkJamesBridger)

81.156.186.139 (talk) 10:00, 3 January 2022 (UTC)The article should mention the original simple reason why a cosmic acceleration was not expected prior to its discovery (and the reason why there may still be doubts about it). It was naturally assumed that the expansion would be slowing down due to the effect of gravity. An acceleration against gravity, for whatever reason, agent or dark energy is used to ’explain’ that, means something that is doing work continuously, and therefore adding energy to the whole system, breaking the conservation of energy principle. The only way to escape this breach of the laws of physics is consider the cause of the acceleration not to be something that works against gravity but an effect of gravity itself. This was Einstein’s idea when he first used the Cosmological constant to hold what he believed was a steady state universe. In framing the cosmological constant within general relativity, one is saying it is an effect of gravity, rather than some force working against gravity. General relativity may describe the curvature of space-time but not the direction in which gravity may go, i.e. whether it is a push or a pull. That leaves the option open for the cosmological constant to be adjusted to fit whatever is believed or observed. This however is a somewhat arbitrary explanation and a deeper explanation of gravity may require that it has one direction, that it is always a pull or an attraction. In that case a model of the universe is required where the accelerating expansion is not gravity repelling things on the grand scale but gravity falling towards things on a grand scale. This is possible in a multiverse where there is no expansion between the neighbouring cosmoses/universes (thus a constant overall density), so that when they individually expand they will be falling towards each other and eventually overlapping. At some stage in the evolution of our own ‘universe’ we may see increasing evidence for this, given improving telescopes, to overturn the assumption that our universe is heading for a cold dark death of ever diminishing density. (Mark James Bridger) 81.156.186.139 (talk) 10:00, 3 January 2022 (UTC)

The article should also note, as a historical fact, that the theory that the cosmic acceleration is an effect of gravity (in this constant density model of a multiverse) is the only theory that was predictive, i.e. it was proposed in advance of the 1998 observations. To be specific it was recorded as a text in an oil on wood panel painting titled The Eternal and Infinite Universe, which was publicly exhibited in a gallery named The Jelly Leg'd Chicken gallery in Reading, Bershire, UK in January 1995 in a group exhibition. I am the artist of the painting and have witnesses to affirm this. The fact/ possibility that the cosmic acceleration was only predicted by an artist seems to agree with yet another of Einstein's expressed thoughts; that 'the greatest scientists are artists as well'!. (Mark James Bridger) 81.156.186.139 (talk) 11:37, 3 January 2022 (UTC)

Science
Lemaître law,[1] is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth. The velocity of the galaxies has been determined by their redshift, a shift of the light they emit toward the red end of the visible spectrum. 120.29.104.58 (talk) 13:41, 6 September 2022 (UTC)

Black holes could be the source of dark energy causing the accelerating expansion of the universe.
This should be added to the article - the whole article needs a facelift. It is 9 years old and a lot has changed or has become part of the consensus of our understanding. In the last few days (February 15, 2023) scientists have published an article sumarized and explained here:

https://earthsky.org/space/black-holes-as-the-source-of-dark-energy/?utm_source=EarthSky+News&utm_campaign=aca33d4354-EMAIL_CAMPAIGN_2018_02_02_COPY_01&utm_medium=email&utm_term=0_c643945d79-aca33d4354-395634733&mc_cid=aca33d4354&mc_eid=429423be84 2600:4040:2C32:B400:E065:8E67:E2FC:22D1 (talk) 16:01, 18 February 2023 (UTC)
 * As I wrote in the Dark Energy article, I think this is too soon. The research is new, and there are a lot of people skeptical about it, not least because they don't understand it. It's not clear for example how black holes can drive accelerating expansion, or if some/any of the evidence for accelerating expansion is being challenged. For example, does this new paper actually argue that the universe's expansion isn't accelerating? It does seem like the new paper only purports to explain accelerating expansion, in which case this article is still up-to-date. Banedon (talk) 07:39, 20 February 2023 (UTC)

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


text:
The Abduction of Figaro

The Abduction of Figaro is a comic opera in three acts, described as "A Simply Grand Opera by P. D. Q. Bach", by Peter Schickele. It is a parody of opera in general, and the title is a play on two operas by Wolfgang Amadeus Mozart: The Abduction from the Seraglio, K. 384, and The Marriage of Figaro, K. 492. Those two operas, as well as Così fan tutte and Don Giovanni, and Gilbert and Sullivan's The Pirates of Penzance are among the core inspirations for the piece. The Abduction of Figaro is numbered S. 384, 492 in Schickele's catalogue of works.

Schickele was commissioned to "discover" this opera by the Minnesota Opera, where the piece premiered on April 27 and 28, 1984. In addition to parodying Mozart, the music incorporates diverse influences and musical quotes, from traditional camp songs like "Found a Peanut" to popular songs like "Macho Man" by the Village People. The opera has been released on VHS and DVD.

Synopsis of scenes
1. Introductory remarks by Prof. Peter Schickele 2. Opening credits 3. Overture Act 1: A town on the seacoast of Spain or Italy or somewhere Scene 1: Figaro's bedroom in the palace of Count Almamater 4. Introduction: "Found a peanut!" 5. Recitative: "Ah, dear husband" Aria: "Stay with me" 6. Recitative: "Suzanna" 7. Recitative: "Dog!" Aria: "Perfidy, thy name is Donald" 8. Recitative: "I am distraught" 9. Quartet: "Love is gone" Scene 2: A courtyard of the palace 10. Recitative: "Well, here we are" 11. Aria: "Behold, fair maiden" 12. Recitative: "Just a moment" Duet: "Thy lofty tree" Scene 3: Figaro's bedroom 13. Recitative: "And here is my husband" 14. Recitative: "Hold it!" Aria: "My name is Captain Kadd" 15. Recitative: "Now that you've heard" 16. Sextet: "What a downer!" Scene 4: The courtyard 17. Recitative: "Schlepporello" Aria: "No man" Scene 5: At the dock 18. Recitative: "What a strange turn of events" 19. Quintet and chorus: "Ah, though we must part" 20. Act 1 finale

21. Introduction to act 2 Act 2: Somewhere in the Turkish Empire Scene 1: At the seashore 22. Duet: "God be praised" Scene 2: In front of the Pasha's palace 23. Aria: "Fish gotta swim" 24. Chorus: "Hey, make way" 25. Dance of the Seven Pails 26. Recitative: "Your immenseness" 27. Duet, chorus, and dialogue: "Who is the highest" 28. Quartet: "May I introduce" Scene 3: A courtyard of the palace 29. Dialogue and recitative: "Why?" Aria and dialogue: "Macho, macho" 30. Cavatina and dialogue: "You can beat me" 31. Act 2 finale

Act 3: A tropical forest 32. Ballet 33. Trio and dialogue: "A magic forest" 34. Duet and dialogue: "I am a swineherd" 35. Finale (part 1) and dialogue 36. Aria and dialogue: "Why, oh why" 37. Finale (part 2) 38. Curtain calls and closing credits 39. Closing remarks by Prof. Schickele

DVD bonus tracks
 * Excerpts from the Gross Concerto for Divers Flutes and Orchestra, S. −2
 * Professor Peter Schickele in conversation with Gordon Hunt

title: The Abduction of Figaro


text:
User talk:Skeeter451

Ah, another sporting wikipedian has joined our ranks. Welcome to the 'pedia! So you are 205.138.230.xxx - I like what you have done to the place. --User:maveric149

Anthro wikiproject?
I thought you might be interested in helping me set up a wikiproject for anthropology. If so, send me a message. Ungovernable ForceThe Wiki Kitchen! 11:38, 15 August 2006 (UTC)
 * I just made a proposal on the "list of proposed wikiprojects" page. If you're interested, you can sign up at our entry and on the temporary project page. Thanks. Ungovernable ForceThe Wiki Kitchen! 21:00, 17 August 2006 (UTC)

Featured article candidate review: Buffy article
Hi

Just letting WikiProject Buffyverse members know that the article 'Buffy the Vampire Slayer' has recently been nominated as a candidate to become a featured article. Should it become a featured article, it will be possible for the article to appear on the Wikipedia main page on March 10th 2007, the 10th anniversary of Buffy (the premiere, "Welcome to the Hellmouth" aired March 10th 1997).

Any feedback you can offer to improve the article and/or to either object or support the nomination would be wonderful:

Featured article candidates/Buffy the Vampire Slayer

-- Paxomen 17:31, 21 September 2006 (UTC)

Image copyright problem with Image:Cheese_salers_fromage_zoom.jpg
Thanks for uploading Image:Cheese_salers_fromage_zoom.jpg. The image has been identified as not specifying the copyright status of the image, which is required by Wikipedia's policy on images. If you don't indicate the copyright status of the image on the image's description page, using an appropriate copyright tag, it may be deleted some time in the next seven days. If you have uploaded other images, please verify that you have provided copyright information for them as well.

For more information on using images, see the following pages:
 * Image use policy
 * Image copyright tags

This is an automated notice by OrphanBot. For assistance on the image use policy, see Media copyright questions. 06:08, 15 June 2007 (UTC)

Orphaned non-free image (Image:The Journeyer MM Cover.jpg)
Thanks for uploading Image:The Journeyer MM Cover.jpg. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

If you have uploaded other unlicensed media, please check whether they're used in any articles or not. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any non-free images not used in any articles will be deleted after seven days, as described on criteria for speedy deletion. Thank you. BetacommandBot 00:49, 8 August 2007 (UTC)

Survey
Hi Skeeter451!

I have put together a survey for female editors of Wikipedia (and related projects) in order to explore, in greater detail, women's experiences and roles within the Wikimedia movement. It'd be wonderful if you could participate!

It's an independent survey, done by me, as a fellow volunteer Wikimedian. It is not being done on behalf of the Wikimedia Foundation. I hope you'll participate!

Just click this link to participate in this survey, via Google!

Any questions or concerns, feel free to email me or stop by my user talk page. Also, feel free to share this any other female Wikimedians you may know. It is in English, but any language Wikimedia participants are encouraged to participate. I appreciate your contributions - to the survey and to Wikipedia! Thank you! SarahStierch (talk) 01:23, 30 September 2011 (UTC)

Orphaned non-free image File:The Journeyer MM Cover.jpg
 Thanks for uploading File:The Journeyer MM Cover.jpg. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

Note that any non-free images not used in any articles will be deleted after seven days, as described in the criteria for speedy deletion. Thank you. Stefan2 (talk) 21:27, 28 May 2014 (UTC)

Women in Red World Contest
Hi. We're into the last five days of the Women in Red World Contest. There's a new bonus prize of $200 worth of books of your choice to win for creating the most new women biographies between 0:00 on the 26th and 23:59 on 30th November. If you've been contributing to the contest, thank you for your support, we've produced over 2000 articles. If you haven't contributed yet, we would appreciate you taking the time to add entries to our articles achievements list by the end of the month. Thank you, and if participating, good luck with the finale!

Invitation to join the Fifteen Year Society
Dear ,

I'd like to extend a cordial invitation to you to join the Fifteen Year Society, an informal group for editors who've been participating in the Wikipedia project for fifteen years or more.

Best regards, Urhixidur (talk) 02:40, 9 June 2019 (UTC)

title: User talk:Skeeter451


text:
Finite difference

A finite difference is a mathematical expression of the form $f (x + b) − f (x + a)$. If a finite difference is divided by $b − a$, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.

The difference operator, commonly denoted $$\Delta$$ is the operator that maps a function $f$ to the function $$\Delta[f]$$ defined by $$\Delta[f](x) = f(x+1)-f(x).$$ A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.

In numerical analysis, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives". Finite difference approximations are finite difference quotients in the terminology employed above.

Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (c. 1592) and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals.

Basic types
Three basic types are commonly considered: forward, backward, and central finite differences.

A forward difference, denoted $$\Delta_h[f],$$ of a function $f$ is a function defined as $$ \Delta_h[f](x) = f(x + h) - f(x). $$

Depending on the application, the spacing $h$ may be variable or constant. When omitted, $h$ is taken to be 1; that is, $$ \Delta[f](x) = \Delta_1[f](x) =f(x+1)-f(x) .$$

A backward difference uses the function values at $x$ and $x − h$, instead of the values at $x + h$ and $x$: $$ \nabla_h[f](x) = f(x) - f(x-h)=\Delta_h[f](x-h). $$

Finally, the central difference is given by $$ \delta_h[f](x) = f(x+\tfrac{h}2)-f(x-\tfrac{h}2)=\Delta_{h/2}[f](x)+\nabla_{h/2}[f](x).$$

Relation with derivatives
Finite difference is often used as an approximation of the derivative, typically in numerical differentiation.

The derivative of a function $f$ at a point $x$ is defined by the limit $$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. $$

If $h$ has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written $$ \frac{f(x + h) - f(x)}{h} = \frac{\Delta_h[f](x)}{h}. $$

Hence, the forward difference divided by $h$ approximates the derivative when $h$ is small. The error in this approximation can be derived from Taylor's theorem. Assuming that $f$ is twice differentiable, we have $$ \frac{\Delta_h[f](x)}{h} - f'(x) = o(h)\to 0 \quad \text{as }h \to 0. $$

The same formula holds for the backward difference: $$ \frac{\nabla_h[f](x)}{h} - f'(x) = o(h)\to 0 \quad \text{as }h \to 0. $$

However, the central (also called centered) difference yields a more accurate approximation. If $f$ is three times differentiable, $$ \frac{\delta_h[f](x)}{h} - f'(x) = o\left(h^2\right). $$

The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If $f (nh) = 1$ for $n$ odd, and $f (nh) = 2$  for $n$ even, then $f ′(nh) = 0$ if it is calculated with the central difference scheme. This is particularly troublesome if the domain of $f$ is discrete. See also Symmetric derivative.

Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).

Higher-order differences
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for $f ′(x + h⁄2)$  and $f ′(x − h⁄2)$   and applying a central difference formula for the derivative of $f ′$ at $x$, we obtain the central difference approximation of the second derivative of $f$:

 * Second-order central :$$ f''(x) \approx \frac{\delta_h^2[f](x)}{h^2} = \frac{ \frac{f(x+h) - f(x)}{h} - \frac{f(x) - f(x-h)}{h} }{h} = \frac{f(x+h) - 2 f(x) + f(x-h)}{h^{2}} . $$

Similarly we can apply other differencing formulas in a recursive manner.

 * Second order forward :$$ f''(x) \approx \frac{\Delta_h^2[f](x)}{h^2} = \frac{ \frac{f(x+2h) - f(x+h)}{h} - \frac{f(x+h) - f(x)}{h} }{h} = \frac{f(x+2h) - 2 f(x+h) + f(x)}{h^{2}} . $$
 * Second order backward :$$ f''(x) \approx \frac{\nabla_h^2[f](x)}{h^2} = \frac{ \frac{f(x) - f(x-h)}{h} - \frac{f(x-h) - f(x-2h)}{h} }{h} = \frac{f(x) - 2 f(x-h) + f(x - 2h)}{h^{2}} . $$

More generally, the $n$-th order forward, backward, and central differences are given by, respectively,

 * Forward :$$\Delta^n_h[f](x) = \sum_{i = 0}^{n} (-1)^{n-i} \binom{n}{i} f\bigl(x + i h\bigr),$$
 * Backward :$$\nabla^n_h[f](x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),$$
 * Central :$$\delta^n_h[f](x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f\left(x + \left(\frac{n}{2} - i\right) h\right).$$

These equations use binomial coefficients after the summation sign shown as $( n i )$. Each row of Pascal's triangle provides the coefficient for each value of $i$.

Note that the central difference will, for odd $n$, have $h$ multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of $δ^{n}[ f ](x − h⁄2)$ and $δ^{n}[ f ](x + h⁄2)$.

Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties. Forward differences may be evaluated using the Nörlund–Rice integral. The integral representation for these types of series is interesting, because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large $n$.

The relationship of these higher-order differences with the respective derivatives is straightforward, $$\frac{d^n f}{d x^n}(x) = \frac{\Delta_h^n[f](x)}{h^n}+o(h) = \frac{\nabla_h^n[f](x)}{h^n}+o(h) = \frac{\delta_h^n[f](x)}{h^n} + o\left(h^2\right).$$

Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order $h$. However, the combination $$ \frac{\Delta_h[f](x) - \frac12 \Delta_h^2[f](x)}{h} = - \frac{f(x+2h)-4f(x+h)+3f(x)}{2h} $$ approximates $f ′(x)$ up to a term of order $h^{2}$. This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below.

If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.

Polynomials
For a given polynomial of degree $n ≥ 1$, expressed in the function $P(x)$, with real numbers $a ≠ 0$ and $b$ and lower order terms (if any) marked as $l.o.t.$: $$P(x) = ax^n + bx^{n-1} + l.o.t.$$

After $n$ pairwise differences, the following result can be achieved, where $h ≠ 0$ is a real number marking the arithmetic difference: $$\Delta_h^n [P](x) = ah^nn!$$

Only the coefficient of the highest-order term remains. As this result is constant with respect to $x$, any further pairwise differences will have the value $0$.

Base case
Let $Q(x)$ be a polynomial of degree $1$: $$\Delta_h [Q](x) = Q(x + h) - Q(x) = [a(x + h) + b] - [ax + b] = ah = ah^11!$$

This proves it for the base case.

Inductive step
Let $R(x)$ be a polynomial of degree $m − 1$ where $m ≥ 2$ and the coefficient of the highest-order term be $a ≠ 0$. Assuming the following holds true for all polynomials of degree $m − 1$: $$\Delta_h^{m-1} [R](x) = ah^{m-1}(m-1)!$$

Let $S(x)$ be a polynomial of degree $m$. With one pairwise difference: $$\Delta_h [S](x) = [a(x+h)^{m} + b(x+h)^{m-1} + \text{l.o.t.}] - [ax^m + bx^{m-1} + \text{l.o.t.}] = ahmx^{m-1} + \text{l.o.t.} = T(x)$$

As $ahm ≠ 0$, this results in a polynomial $T(x)$ of degree $m − 1$, with $ahm$ as the coefficient of the highest-order term. Given the assumption above and $m − 1$ pairwise differences (resulting in a total of $m$ pairwise differences for $S(x)$), it can be found that: $$\Delta_h^{m-1} [T](x) = ahm \cdot h^{m-1}(m-1)! = a h^m m!$$

This completes the proof.

Application
This identity can be used to find the lowest-degree polynomial that intercepts a number of points $(x, y)$ where the difference on the x-axis from one point to the next is a constant $h ≠ 0$. For example, given the following points:

We can use a differences table, where for all cells to the right of the first $y$, the following relation to the cells in the column immediately to the left exists for a cell $(a + 1, b + 1)$, with the top-leftmost cell being at coordinate $(0, 0)$: $$(a+1, b+1) = (a, b+1) - (a, b)$$

To find the first term, the following table can be used:

This arrives at a constant $x$. The arithmetic difference is $y$, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree $Δy$. Thus, using the identity above: $$648 = a \cdot 3^3 \cdot 3! = a \cdot 27 \cdot 6 = a \cdot 162$$

Solving for $Δ^{2}y$, it can be found to have the value $Δ^{3}y$. Thus, the first term of the polynomial is $648$.

Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again:

Here, the constant is achieved after only two pairwise differences, thus the following result: $$-306 = a \cdot 3^2 \cdot 2! = a \cdot 18$$

Solving for $h = 3$, which is $3$, the polynomial's second term is $a$.

Moving on to the next term, by subtracting out the second term:

Thus the constant is achieved after only one pairwise difference: $$108 = a \cdot 3^1 \cdot 1! = a \cdot 3$$

It can be found that $4$ and thus the third term of the polynomial is $4x^{3}$. Subtracting out the third term:

Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant $Δy$. Thus, the lowest-degree polynomial intercepting all the points in the first table is found: $$4x^3 - 17x^2 + 36x - 19$$

Arbitrarily sized kernels
Using linear algebra one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid. This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side. Finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order may be constructed.

Properties

 * For all positive $x$ and $y$ $$\Delta^n_{kh} (f, x) = \sum\limits_{i_1=0}^{k-1} \sum\limits_{i_2=0}^{k-1} \cdots \sum\limits_{i_n=0}^{k-1} \Delta^n_h \left(f, x+i_1h+i_2h+\cdots+i_nh\right).$$
 * Leibniz rule: $$\Delta^n_h (fg, x) = \sum\limits_{k=0}^n \binom{n}{k} \Delta^k_h (f, x) \Delta^{n-k}_h(g, x+kh).$$

In differential equations
An important application of finite differences is in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary and partial differential equations. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference methods.

Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc.

Newton's series
The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton; in essence, it is  the Gregory–Newton interpolation formula (named after Isaac Newton and James Gregory), first published in his Principia Mathematica in 1687,  namely the discrete analog of the continuous Taylor expansion,

which holds for any polynomial function $k$ and for many (but not all) analytic functions. (It does not hold when $n$ is exponential type $$\pi$$. This is easily seen, as the sine function vanishes at integer multiples of $$\pi$$; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Yet clearly, the sine function is not zero.) Here, the expression $$\binom{x}{k} = \frac{(x)_k}{k!}$$ is the binomial coefficient, and $$(x)_k=x(x-1)(x-2)\cdots(x-k+1)$$ is the "falling factorial" or "lower factorial", while the empty product $Δ^{2}y$ is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of $4 − 4(1)^{3} = 4 − 4 = 0$ of the generalization below.

Note the formal correspondence of this result to Taylor's theorem. Historically, this, as well as the Chu–Vandermonde identity, $$(x+y)_n=\sum_{k=0}^n \binom{n}{k} (x)_{n-k} \,(y)_k ,$$ (following from it, and corresponding to the binomial theorem), are included in the observations that matured to the system of umbral calculus.

Newton series expansions can be superior to Taylor series expansions when applied to discrete quantities like quantum spins (see Holstein–Primakoff transformation), bosonic operator functions or discrete counting statistics.

To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the Fibonacci sequence $109 − 4(4)^{3} = 109 − 256 = −147$ One can find a polynomial that reproduces these values, by first computing a difference table, and then substituting the differences that correspond to $772 − 4(7)^{3} = 772 − 1372 = −600$ (underlined) into the formula as follows, $$ \begin{matrix}

\begin{array}{|c||c|c|c|} \hline x & f=\Delta^0 & \Delta^1 & \Delta^2 \\ \hline 1&\underline{2}& & \\ & &\underline{0}& \\ 2&2& &\underline{2} \\ & &2& \\ 3&4& & \\ \hline \end{array}

&

\quad \begin{align} f(x) & =\Delta^0 \cdot 1 +\Delta^1 \cdot \dfrac{(x-x_0)_1}{1!} + \Delta^2 \cdot \dfrac{(x-x_0)_2}{2!} \quad (x_0=1)\\ \\ & =2 \cdot 1 + 0 \cdot \dfrac{x-1}{1} + 2 \cdot \dfrac{(x-1)(x-2)}{2} \\ \\ & =2 + (x-1)(x-2) \\ \end{align} \end{matrix} $$

For the case of nonuniform steps in the values of $f$, Newton computes the divided differences, $$\Delta _{j,0}=y_j,\qquad \Delta _{j,k}=\frac{\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_j}\quad \ni \quad \left\{ k>0,\; j\le \max \left( j \right)-k \right\},\qquad \Delta 0_k = \Delta _{0,k}$$ the series of products, $${P_0}=1,\quad \quad P_{k+1}=P_k\cdot \left( \xi -x_k \right) ,$$ and the resulting polynomial is the scalar product, $$f(\xi ) = \Delta 0 \cdot P\left( \xi \right).$$

In analysis with $f$-adic numbers, Mahler's theorem states that the assumption that $x$ is a polynomial function can be weakened all the way to the assumption that $p$ is merely continuous.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist.

The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of suitably scaled forward differences.

In a compressed and slightly more general form and equidistant nodes the formula reads $$f(x) = \sum_{k=0}\binom{\frac{x-a}h}{k} \sum_{j=0}^k (-1)^{k-j}\binom{k}{j}f(a+j h).$$

Calculus of finite differences
The forward difference can be considered as an operator, called the difference operator, which maps the function $f$ to $2641 − 4(10)^{3} = 2641 − 4000 = −1359$. This operator amounts to $$\Delta_h = T_h-I, $$ where $6364 − 4(13)^{3} = 6364 − 8788 = −2424$ is the shift operator with step h, defined by $a$, and $f$ is the identity operator.

The finite difference of higher orders can be defined in recursive manner as $−17$. Another equivalent definition is $−17x^{2}$.

The difference operator $x$ is a linear operator, as such it satisfies $y$.

It also satisfies a special Leibniz rule indicated above, $Δy$. Similar statements hold for the backward and central differences.

Formally applying the Taylor series with respect to $f$, yields the formula $$ \Delta_h = hD + \frac{1}{2!} h^2D^2 + \frac{1}{3!} h^3D^3 + \cdots = e^{hD} - I, $$ where $I$ denotes the continuum derivative operator, mapping $h$ to its derivative $0 − (−17(1)^{2}) = 0 + 17 = 17$. The expansion is valid when both sides act on analytic functions, for sufficiently small $D$. Thus, $−147 − (−17(4)^{2}) = −147 + 272 = 125$, and formally inverting the exponential yields $$ hD = \ln(1+\Delta_h) = \Delta_h - \tfrac{1}{2} \, \Delta_h^2 + \tfrac{1}{3} \, \Delta_h^3 - \cdots. $$ This formula holds in the sense that both operators give the same result when applied to a polynomial.

Even for analytic functions, the series on the right is not guaranteed to converge; it may be an asymptotic series. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to $−600 − (−17(7)^{2}) = −600 + 833 = 233$ mentioned at the end of the section .

The analogous formulas for the backward and central difference operators are $$ hD = -\ln(1-\nabla_h) \quad\text{and}\quad hD = 2 \operatorname{arsinh}\left(\tfrac12 \, \delta_h\right). $$

The calculus of finite differences is related to the umbral calculus of combinatorics. This remarkably systematic correspondence is due to the identity of the commutators of the umbral quantities to their continuum analogs ($−1359 − (−17(10)^{2}) = −1359 + 1700 = 341$ limits),

A large number of formal differential relations of standard calculus involving functions $−2424 − (−17(13)^{2}) = −2424 + 2873 = 449$ thus map systematically to umbral finite-difference analogs involving $a = 36$.

For instance, the umbral analog of a monomial $f$ is a generalization of the above falling factorial (Pochhammer k-symbol), $$(x)_n\equiv \left(xT_h^{-1}\right)^n=x (x-h) (x-2h) \cdots \bigl(x-(n-1)h\bigr),$$ so that $$\frac{\Delta_h}{h} (x)_n=n (x)_{n-1} ,$$ hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function $36x$ in such symbols), and so on.

For example, the umbral sine is $$\sin \left(x\,T_h^{-1}\right) = x -\frac{(x)_3}{3!} + \frac{(x)_5}{5!} - \frac{(x)_7}{7!} + \cdots$$

As in the continuum limit, the eigenfunction of $x$ also happens to be an exponential,

 * $$\frac{\Delta_h}{h}(1+\lambda h)^\frac{x}{h} =\frac{\Delta_h}{h} e^{\ln (1+\lambda h) \frac{x}{h}}= \lambda e^{\ln (1+\lambda h) \frac{x}{h}} ,$$

and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials. This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols.

Thus, for instance, the Dirac delta function maps to its umbral correspondent, the cardinal sine function, $$\delta (x) \mapsto \frac{\sin \left[ \frac{\pi}{2}\left(1+\frac{x}{h}\right) \right]}{ \pi (x+h) },$$ and so forth. Difference equations can often be solved with techniques very similar to those for solving differential equations.

The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator.

Rules for calculus of finite difference operators
Analogous to rules for finding the derivative, we have:
 * Constant rule: If $h$ is a constant, then $$\Delta c = 0$$
 * Linearity: if $x^{n}$ and $c$ are constants, $$\Delta (a\ f + b\ g) = a \ \Delta f + b \ \Delta g$$

All of the above rules apply equally well to any difference operator as to $y$, including $17 − 36(1) = 17 − 36 = −19$ and $125 − 36(4) = 125 − 144 = −19$.

\Delta (f g) &= f \,\Delta g + g \,\Delta f + \Delta f \,\Delta g \\[4pt] \nabla (f g) &= f \,\nabla g + g \,\nabla f - \nabla f \,\nabla g \end{align}$$ \sum_{n=a}^b \Delta f(n) &= f(b+1)-f(a) \\ \sum_{n=a}^{b} \nabla f(n) &= f(b)-f(a-1) \end{align}$$
 * Product rule: $$ \begin{align}
 * Quotient rule: $$\nabla \left( \frac{f}{g} \right) = \left. \left( \det \begin{bmatrix} \nabla f & \nabla g \\ f & g \end{bmatrix} \right) \right/ \left( g \cdot \det {\begin{bmatrix} g & \nabla g \\ 1 & 1 \end{bmatrix}}\right) $$ or $$\nabla\left( \frac{f}{g} \right)= \frac {g \,\nabla f - f \,\nabla g}{g \cdot (g - \nabla g)}$$
 * Summation rules: $$\begin{align}

See references.

Generalizations

 * A generalized finite difference is usually defined as $$\Delta_h^\mu[f](x) = \sum_{k=0}^N \mu_k f(x+kh),$$ where $233 − 36(7) = 233 − 252 = −19$ is its coefficient vector. An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series. Another way of generalization is making coefficients $341 − 36(10) = 341 − 360 = −19$ depend on point $a$: $449 − 36(13) = 449 − 468 = −19$, thus considering weighted finite difference. Also one may make the step $b$ depend on point $x$: $−19$. Such generalizations are useful for constructing different modulus of continuity.
 * The generalized difference can be seen as the polynomial rings $(x)_{0}$. It leads to difference algebras.
 * Difference operator generalizes to Möbius inversion over a partially ordered set.
 * As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function $h$; for the difference operator, $x$ is the sequence (1, −1, 0, 0, 0, …).

Multivariate finite differences
Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables.

Some partial derivative approximations are: $$\begin{align} f_{x}(x,y) &\approx  \frac{f(x+h ,y) - f(x-h,y)}{2h} \\ f_{y}(x,y) &\approx  \frac{f(x,y+k ) - f(x,y-k)}{2k} \\ f_{xx}(x,y) &\approx \frac{f(x+h ,y) - 2 f(x,y) + f(x-h,y)}{h^2} \\ f_{yy}(x,y) &\approx \frac{f(x,y+k) - 2 f(x,y) + f(x,y-k)}{k^2} \\ f_{xy}(x,y) &\approx \frac{f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k)}{4hk}. \end{align}$$

Alternatively, for applications in which the computation of $μ$ is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is $$ f_{xy}(x,y) \approx \frac{f(x+h, y+k) - f(x+h, y) - f(x, y+k) + 2 f(x,y) - f(x-h, y) - f(x, y-k) + f(x-h, y-k)}{2hk},$$ since the only values to compute that are not already needed for the previous four equations are $x, h = 1$  and $f = 2, 2, 4, ...$.

title: Finite difference