Quadrant Model of Reality Book 38 Art and Philosophy

Art Chapter

QMRBattenburg markings or Battenberg markings are a pattern of high-visibility markings used primarily on the sides of emergency service vehicles in several European countries, Australia, New Zealand, and Hong Kong. The name comes from the similarity in appearance to the cross-section of a Battenberg cake. They are checkers/ quadrants

Painting Chapter

QMRThe Rome Statute of the International Criminal Court (often referred to as the International Criminal Court Statute or the Rome Statute) is the treaty that established the International Criminal Court (ICC). It was adopted at a diplomatic conference in Rome on 17 July 1998[5][6] and it entered into force on 1 July 2002.[2] As of 6 January 2015, 123 states are party to the statute.[2] Among other things, the statute establishes the court's functions, jurisdiction and structure. The Rome Statute established four core international crimes: genocide, crimes against humanity, war crimes, and the crime of aggression. Those crimes "shall not be subject to any statute of limitations".[7] Under the Rome Statute, the ICC can only investigate and prosecute the four core international crimes in situations where states are "unable" or "unwilling" to do so themselves. The court has jurisdiction over crimes only if they are committed in the territory of a state party or if they are committed by a national of a state party; an exception to this rule is that the ICC may also have jurisdiction over crimes if its jurisdiction is authorized by the United Nations Security Council.

The Rome Statute established four core international crimes: genocide, crimes against humanity, war crimes and the crime of aggression. Following years of negotiation, aimed at establishing a permanent international tribunal to prosecute individuals accused of genocide and other serious international crimes, such as crimes against humanity, war crimes and crimes of aggression, the United Nations General Assembly convened a five-week diplomatic conference in Rome in June 1998 "to finalize and adopt a convention on the establishment of an international criminal court".[8][9] On 17 July 1998, the Rome Statute was adopted by a vote of 120 to 7, with 21 countries abstaining.[5] Because the way each delegation voted was officially unrecorded, there is some dispute over the identity of the seven countries that voted against the treaty.[10] It is certain that the People's Republic of China, Israel, and the United States were three of the seven because they have publicly confirmed their negative votes; India, Indonesia, Iraq, Libya, Qatar, Russia, Saudi Arabia, Sudan, and Yemen have been identified by various observers and commentators as possible sources for the other four negative votes, with Iraq, Libya, Qatar, and Yemen being the four most commonly identified.[10]

QMRFountain of the four Rivers with Egyptian obelisk, in the middle of Piazza Navona

spiderman's costume is quadrants

QMRMillet created a series of four woodcuts entitled The Four Hours of the Day (1860). This idea is reportedly derived from the medieval "Book of Hours" that held that rural life was structured by the rhythms of God and nature.[35] In this work, the man and woman walk to the fields, silhouetted by the morning sun. The simple composition depicts the harshness of their lives. Van Gogh said of Millet’s work, "they seem to have been painted with the very earth they were going to till."[37] Van Goh made four copies of these four works by Millet.

QMRThe Four Philosophers is a 1611-12 painting by Peter Paul Rubens. It is now held in the Galleria Palatina of the Palazzo Pitti in Florence. It also features in the 1772 painting The Tribuna of the Uffizi by Zoffany. From left to right it shows Philippus Rubens (the painter's brother), Justus Lipsius, Jan Woverius and an unidentified pupil or friend of the other three. In the background is a bust of Seneca.

QMrAl-Farabi divides intellect into four categories: potential, actual, acquired and the Agent. The first three are the different states of the human intellect and the fourth is the Tenth Intellect (the moon) in his emanational cosmology. The potential intellect represents the capacity to think, which is shared by all human beings, and the actual intellect is an intellect engaged in the act of thinking. By thinking, al-Farabi means abstracting universal intelligibles from the sensory forms of objects which have been apprehended and retained in the individual's imagination.[60]

In his treatment of the human soul, al-Farabi draws on a basic Aristotelian outline, which is informed by the commentaries of later Greek thinkers. He says it is composed of four faculties: The appetitive (the desire for, or aversion to an object of sense), the sensitive (the perception by the senses of corporeal substances), the imaginative (the faculty which retains images of sensible objects after they have been perceived, and then separates and combines them for a number of ends), and the rational, which is the faculty of intellection.[66] It is the last of these which is unique to human beings and distinguishes them from plants and animals. It is also the only part of the soul to survive the death of the body. Noticeably absent from these scheme are internal senses, such as common sense, which would be discussed by later philosophers such as Avicenna and Averroes.[67][68]

Van Goh The work depicts happy life of a rural family: father, mother and child. Here the image seems bathed in yellow light like that of the Holy Family.[40] A lamp casts long shadows of many colors on the floor of the humble cottage. The painting includes soft shades of green and purple. The work was based on a print by Millet from his series, the four times of day.[41]

Night 1867 Fine Arts Museum, Boston

The End of the Day 1867-69 Memorial Art Gallery of the University of Rochester, Rochester Vincent van Gogh - Evening, The End of the Day (after Millet).jpg Man in a Field or Evening, the End of the Day November 1889 Menard Art Museum, Komaki, Japan (F649)

Noon: Rest 1866 Museum of Fine Arts, Boston Noon, rest from work - Van Gogh.jpeg The Siesta or Noon - Rest from Work 1890 Musée d'Orsay, Paris, France (F686)

Morning: Going to Work ca. 1858-60 Van Gogh Museum, Amsterdam, Netherlands Van Gogh - Auf dem Weg zum Feld (nach Millet).jpeg Morning: Peasant Couple Going to Work 1890 Hermitage Museum, St. Petersberg, Russia (F684)

QMRMillet created a series of woodcuts entitled The Four Hours of the Day (1860). This idea is reportedly derived from the medieval "Book of Hours" that held that rural life was structured by the rhythms of God and nature.[35] In this work, the man and woman walk to the fields, silhouetted by the morning sun. The simple composition depicts the harshness of their lives. Van Gogh said of Millet’s work, "they seem to have been painted with the very earth they were going to till."[37]

QMRIn 1882 Van Gogh had remarked that he found Honoré Daumier's The Four Ages of a Drinker both beautiful and soulful.[11] Van Gogh wrote to his brother Theo of Daumier's artistic perspective and humanity: "What impressed me so much at the time was something so stout and manly in Daumier's conception, something that made me think It must be good to think and to feel like that and to overlook or ignore a multitude of things and to concentrate on what makes us sit up and think and what touches us as human beings more directly and personally than meadows or clouds."[12] Daumier's artistic talents included painting, sculpting and creating lithographs. He was well known for his social and political commentary.[13] Van Gogh made Men Drinking after Daumier's work in Saint-Remy about February 1890.[12]

Music Chapter

QMR"Ten Little Indians" is an American children's rhyme. It has a Roud Folk Song Index number of 13512. The word Indian usually refers to Native Americans. The song is traditionally performed in the tune of the Irish folk song Michael Finnegan.[citation needed] It has four partsThe modern lyrics are: One little, two little, three little Indians Four little, five little, six little Indians Seven little, eight little, nine little Indians Ten little Indian boys. Ten little, nine little, eight little Indians Seven little, six little, five little Indians Four little, three little, two little Indians One little Indian boy.[1]

QMRFamily Four were a Swedish pop group who recorded during the 1960s and 1970s. They were made up of Berndt Öst, Marie Bergman, Agnetha Munther and Pierre Isacsson. They won Melodifestivalen twice, in 1971 with "Vita vidder" and in 1972 with "Härliga sommardag". They went on to represent Sweden in the Eurovision Song Contest on these two occasions. They finished sixth in 1971 and 13th in 1972.

QMRPage Four is a Danish boy band established in Copenhagen in November 2014 and made up of Lauritz Emil Christiansen, Jonas Eilskov, Stefan Hjort and Pelle Højer. The band gained fame uploading cover versions online and were eventually signed to Sony Music Entertainment Denmark. They are best known for their debut single "Sommer" co-written by Tim McEwan, Theis Andersen and Søren Ohrt Nissen.[1] The single charted on Tracklisten, the official Danish Singles Chart.

QMRThe European Free Trade Association (EFTA) is a common market consisting of four European countries that operates in parallel with – and is linked to – the European Union (EU).[1] The EFTA was established on 3 May 1960 as a trade bloc-alternative for European states who were either unable or unwilling to join the then-European Economic Community (EEC) which has now become the EU. The Stockholm Convention, establishing the EFTA, was signed on 4 January 1960 in the Swedish capital by seven countries (known as the "outer seven"). Today's EFTA members are Iceland, Liechtenstein, Norway, and Switzerland, of which the latter two were founding members. The initial Stockholm Convention was superseded by the Vaduz Convention, which enabled greater liberalisation of trade among the member states. EFTA states have jointly concluded free trade agreements with a number of other countries. All four current members of EFTA participate in the European Union's single market: Iceland, Liechtenstein and Norway through the Agreement on a European Economic Area (EEA) and Switzerland through a set of bilateral agreements. This development prompted the EFTA states to modernise their Convention to ensure that it will continue to provide a successful framework for the expansion and liberalization of trade among themselves and with the rest of the world.

QMRABBA (stylised ᗅᗺᗷᗅ) were a Swedish pop group who formed in Stockholm in 1972. With members Agnetha Fältskog, Björn Ulvaeus, Benny Andersson, and Anni-Frid Lyngstad, ABBA became one of the most commercially successful acts in the history of popular music, topping the charts worldwide from 1975 to 1982. They won the Eurovision Song Contest 1974 at the Dome in Brighton, UK, giving Sweden its first triumph in the contest, and were the most successful group ever to take part in the competition.

The constitution of 1809 divided the powers of government between the monarch and the Riksdag of the Estates, and after 1866 between the monarch and the new Riksdag. In 1866 all the Estates voted in favor of dissolution and at the same time to constitute a new assembly, Sveriges Riksdag (the Riksdag). The four former estates were abolished. The House of Nobility, Swedish: Riddarhuset, remains as a quasi-official representation of the Swedish nobility. The modern Centre Party which grew out of the Swedish farmers' movement, could be construed as a modern representation with a traditional bond to the Estate of the Peasants.

QMRWolfgang Amadeus Mozart completed his Symphony No. 41 in C major, K. 551, on 10 August 1788.[1] It was the last symphony that he composed, and also the longest. The work is nicknamed the Jupiter Symphony. This name stems not from Mozart but rather was likely coined by the impresario Johann Peter Salomon[2] in an early arrangement for piano. The four movements are arranged in the traditional symphonic form of the Classical era: Allegro vivace, 4/4 Andante cantabile, 3/4 in F major Menuetto: Allegretto - Trio, 3/4 Molto allegro, 2/2 The third movement, a Menuetto marked allegretto is similar to a Ländler, a popular Austrian folk dance form. Midway through the movement there is a chromatic progression in which sparse imitative textures are presented by the woodwinds (bars 43 to 51) before the full orchestra returns. In the trio section of the movement, the four-note figure that will form the main theme of the last movement appears prominently (bars 68 to 71), but on the seventh degree of the scale rather than the first, and in a minor key rather than a major, giving it a very different character. Finally, a remarkable characteristic of this symphony is the five-voice fugato (representing the five major themes) at the end of the fourth movement. But there are fugal sections throughout the movement either by developing one specific theme or by combining two or more themes together, as seen in the interplay between the woodwinds. The main theme consists of four notes: The four-note theme is a common plainchant motif which can be traced back at least as far as Josquin des Prez's Missa Pange lingua from the sixteenth century. It was very popular with Mozart. It makes a brief appearance as early as his first symphony in 1764. Later, he used it in the Credo of an early Missa Brevis in F major, the first movement of his 33rd symphony and trio of the minuet of this symphony.[6] Scholars are certain Mozart studied Michael Haydn's Symphony No. 28 in C major, which also has a fugato in its finale and whose coda he very closely paraphrases for his own coda. Charles Sherman speculates that Mozart also studied the younger Haydn's Symphony No. 23 in D major because he "often requested his father Leopold to send him the latest fugue that Haydn had written."[7] The Michael Haydn No. 39, written only a few weeks before Mozart's, also has a fugato in the finale, the theme of which begins with two whole notes. Sherman has pointed out other similarities between the two almost perfectly contemporaneous works. The four-note motif is also the main theme of the contrapuntal finale of Michael's elder brother Joseph's Symphony No. 13 in D major (1764).

QMRMotets[edit] Josquin's motet style varied from almost strictly homophonic settings with block chords and syllabic text declamation to highly ornate contrapuntal fantasias, to the psalm settings which combined these extremes with the addition of rhetorical figures and text-painting that foreshadowed the later development of the madrigal. He wrote many of his motets for four voices, an ensemble size which had become the compositional norm around 1500, and he also was a considerable innovator in writing motets for five and six voices.[42] No motets of more than six voices have been reliably attributed to Josquin. A passage from the psalm motet Domine ne in furore (Ps. 37). Three variants of a motive built on a major triad are introduced, each in paired imitation between two voices. About this sound Play (help·info) Almost all of Josquin's motets use some kind of compositional constraint on the process; they are not freely composed.[43] Some of them use a cantus firmus as a unifying device; some are canonic; some use a motto which repeats throughout; some use several of these methods. The motets that use canon can be roughly divided into two groups: those in which the canon is plainly designed to be heard and appreciated as such, and another group in which a canon is present, but almost impossible to hear, and seemingly written to be appreciated by the eye, and by connoisseurs.[44] Josquin frequently used imitation, especially paired imitation, in writing his motets, with sections akin to fugal expositions occurring on successive lines of the text he was setting. An example is his setting of Dominus regnavit (Psalm 93), for four voices; each of the lines of the psalm begins with a voice singing a new tune alone, quickly followed by entries of other three voices in imitation.[45] In writing polyphonic settings of psalms, Josquin was a pioneer, and psalm settings form a large proportion of the motets of his later years. Few composers prior to Josquin had written polyphonic psalm settings.[46] Some of Josquin's settings include the famous Miserere, written in Ferrara in 1503 or 1504 and most likely inspired by the recent execution of the reformist monk Girolamo Savonarola,[47] Memor esto verbi tui, based on Psalm 119, and two settings of De profundis (Psalm 130), both of which are often considered to be among his most significant accomplishments.[45][48]

Chansons and instrumental compositions[edit] "El grillo" redirects here. For the Guatemalan football coach, see Jorge Roldán. In the domain of secular music, Josquin left numerous French chansons, for from three to six voices, as well as a handful of Italian secular songs known as frottole, as well as some pieces which were probably intended for instrumental performance. Problems of attribution are even more acute with the chansons than they are with other portions of his output: while about 70 three and four-voice chansons were published under his name during his lifetime, only six of the more than thirty five- and six-voice chansons attributed to him were circulated under his name during the same time. Many of the attributions added after his death are considered to be unreliable, and much work has been done in the last decades of the 20th century to correct attributions on stylistic grounds.[49]

Masses[edit] Missa Ave maris stella (Rome, 1486–1495) (four voices) Missa D'ung aultre amer (four voices; authorship questioned by Jeremy Noble) Missa de Beata Virgine (around 1510) (four voices in parts I–II, five voices in parts III–V) Missa Di dadi (=N'aray je jamais) (four voices; authorship doubted by some scholars) Missa Faisant regretz (four voices) Missa Fortuna desperata (four voices) Missa Gaudeamus (four voices) Missa Hercules Dux Ferrariae (Ferrara, 1503/04) (four voices, six in Agnus III) Missa La sol fa re mi (four voices) Missa L'ami Baudichon (four voices) Missa L'homme armé sexti toni (four voices, six in Agnus III) Missa L'homme armé super voces musicales (four voices) Missa Malheur me bat (four voices, six in Agnus III) Missa Mater patris (four voices; authorship doubted by some scholars) Missa Pange lingua (Condé, around 1514) (four voices) Missa Sine nomine (four voices; canonic mass, originally titled "Missa Ad fugam") Doubtful works: Missa Ad fugam (four voices) Missa da pacem (four voices) Missa Una musque de Biscaya (Une mousse de Biscaye) (four voices)

QMrBuilding on Josquin's fugal treatment of the Pange Lingua hymn's third line in the Kyrie of the Missa Pange Lingua, the "Do-Re-Fa-Mi-Re-Do"-theme became one of the most famous in music history. Simon Lohet,[11] Michelangelo Rossi,[12] François Roberday,[13] Johann Caspar Ferdinand Fischer,[14] Johann Jakob Froberger,[15][16] Johann Caspar Kerll,[17] Johann Sebastian Bach,[18] and Johann Fux wrote fugues on it, and the latter's extensive elaborations in the Gradus ad Parnassum[19] made it known to every aspiring composer—among them Wolfgang Amadeus Mozart, who used its first four notes as the fugal subject for the last movement of his Symphony #41, the Jupiter Symphony.[20]

QMRJoseph Haydn's Symphony No. 13 in D major was written in 1763 for the orchestra of Haydn's patron, Prince Nikolaus Esterházy, in Eisenstadt. The work can be precisely dated thanks to a dated score in Haydn's own hand in the National Library of Budapest. Two other Haydn symphonies are known to have been written in the same year: the Symphony No. 12 and the Symphony No. 40. Movements[edit] A typical symphony at this time was written for a pairs of oboes and horns and strings, but the Eisenstadt orchestra had recently taken on two new horn players, and Haydn wrote this symphony for an expanded ensemble of one flute, two oboes, four horns, timpani and strings (violins divided into firsts and seconds), violas, cellos and double basses, with bassoon doubling the bass-line. The timpani part in the autograph score is not in Haydn's hand, but it is quite possibly authentic: he may have written it on a separate sheet, with somebody else adding it to the score at a later date. Allegro molto, 4/4 Adagio cantabile, 4/4 Menuet & Trio, 3/4 Finale. Allegro molto, 2/4 The D major third movement is a minuet and trio in ternary form, with the flute prominently featured in the trio. The last movement is based around a four-note figure which seems to foreshadow the last movement of Mozart's Jupiter Symphony:

QMRMercury4 is a Melbourne-based Australian boy band which, as of June 2004, has achieved three top 40 singles in Australia.

QMRAguilera is a soprano,[166] having a vocal range spanning four octaves (from Bb2 to C♯7)

QMRIn an interview for Billboard, Beyoncé stated that despite having another concept for the album, she was ultimately influenced by her fans to name the album 4. She described the number four as being "special" to her, as her and Jay-Z's birthday, several other family and friends' birthdays, and her wedding anniversary fall on the fourth day of the month. Hey my birthday is October 4

QMRFour the Record is the fourth studio album by American country recording artist Miranda Lambert, released on November 1, 2011, by Sony Music. It was well received by critics and sold over one million copies in the United States. The album's first single, "Baggage Claim," became Lambert's highest chart-debuting single at number 33 on Billboard's Hot Country Songs.[1] A deluxe edition of the album was also released, which included a bonus song and a DVD.[2]

Dance Chapter

qMRHyperdimension Neptunia (超次元ゲイム ネプテューヌ Chōjigen Geimu Neputyūnu?, lit. "Super Dimensional Game Neptune") is a video game series of role-playing games created and developed by Idea Factory. The series debuted in Japan on August 19, 2010 with the video game of the same name exclusively for the PlayStation 3, later re-released as an enhanced remake under the name Hyperdimension Neptunia Re;Birth 1 for the PlayStation Vita. Two sequels, Hyperdimension Neptunia Mk2 and Hyperdimension Neptunia Victory, in addition to the remake titles for both of them and three spin-offs on the PlayStation Vita, have also been released. Another sequel on the PlayStation 4, titled Megadimension Neptunia VII, was released in 2015. It has also branched off into a manga, light novel and anime media franchise. Games within the series takes place in the world of Gamindustri (ゲイムギョウ界 Geimugyō-kai?, a pun on ゲーム業界 Gēmu gyōkai, "game industry"), which is divided into four regions: Planeptune, Lastation, Lowee, and Leanbox. Each region is completely different from the others in appearance and atmosphere, with each representing a specific video game console. In the beginning of the story, the four goddesses are fighting each another for "shares" in a war known as the Console War.

QMRChariot racing (Greek: ἁρματοδρομία harmatodromia, Latin: ludi circenses) was one of the most popular ancient Greek, Roman, and Byzantine sports. Chariot racing often was dangerous to both driver and horse as they frequently suffered serious injury and even death, but generated strong spectator enthusiasm. Chariot races could be watched by women, while women were barred from watching many other sports. In the ancient Olympic Games, as well as the other Panhellenic Games, the sport was one of the main events. Each chariot was pulled by four horses.

QMRPluto velificans, with a Cupid attending his abduction of Proserpina in a four-horse chariot (Roman cinerary altar, Antonine Era, 2nd century)

QMRTwo consecutive strikes are referred to as a "double" (or a "Barney Rubble" to rhyme) aka the "rhino". Three strikes bowled consecutively is known as a "turkey". Any longer string of strikes is referred to by a number affixed to the word "bagger," as in "four-bagger" for four consecutive strikes. ESPN commentator Rob Stone created the name "hambone" to describe four consecutive strikes.[2]

QMrSection 8: Prejudice is a science fiction, first-person shooter video game developed by TimeGate Studios. It is the direct sequel to the 2009 game Section 8 and was demonstrated at Penny Arcade Expo East 2011.[2] Unlike its predecessor, Prejudice is a digital download-only title that contains more content than the previous game. It was released for Xbox Live Arcade on April 20, 2011, on PC May 4, 2011, and on PlayStation Network July 26, 2011.[3] The game features a single player campaign and several multiplayer modes. In multiplayer games teams compete to control control points and complete a range of mission objectives to gain victory. Contents [hide] 1 Plot 2 Gameplay 2.1 Multiplayer 3 Development 4 Reception 5 See also 6 References 7 External links Plot[edit] The single-player campaign explores the conflict between the 8th Armored Infantry and the Arm of Orion. The player again takes control of Alex Corde, the protagonist of the first game, who must complete eight objective-based missions taking place in four different environments.[4]

Literature Chapter

QMRKishōtenketsu (起承転結?) describes the structure and development of classic Chinese, Korean and Japanese narratives. It was originally used in Chinese poetry as a four-line composition, such as Qijue, and is also referred to as kishōtengō (起承転合?). The first Chinese character refers to the introduction or kiku (起句?), the next: development, shōku (承句?), the third: twist, tenku (転句?), and the last character indicates conclusion or kekku (結句?). 句 is the phrase (句 ku?), and gō (合?) means "meeting point of 起 and 転" for conclusion. The following is an example of how this might be applied to a fairytale. Introduction (ki): Topic toss or introduction, what characters appear, era, and other important information for understanding the setting of the story. Development (shō): Receives or follows on from the introduction and leads to the twist in the story. Major changes do not occur. Twist (ten): Turn or twist to another, new or unknown topic. This is the crux of the story, which is also referred to as the yama (ヤマ?) or climax. It has the biggest twist in the story. Conclusion (ketsu): Resultant, also referred to as the ochi (落ち?) or ending, it wraps up the story by bringing it to its conclusion. A specific example by the poet Sanyō Rai (頼山陽): Introduction (ki): Daughters of Itoya, in the Honmachi of Osaka. Development (shō): The elder daughter is sixteen and the younger one is fourteen. Twist (ten): Throughout history, generals (daimyo) killed the enemy with bows and arrows. Ketsu (結?): The daughters of Itoya kill with their eyes.[1] The same pattern is used to arrange arguments: Introduction (ki): In old times, copying information by hand was necessary. Some mistakes were made. Development (shō): Copying machines made it possible to make quick and accurate copies. Twist (ten): Traveling by car saves time, but you don't get much impression of the local beauty. Walking makes it a lot easier to appreciate nature close up. Conclusion (ketsu): Although photocopying is easier, copying by hand is sometimes better, because the information stays in your memory longer and can be used later. In the structure of narrative and yonkoma manga, and even for document and dissertation, the style in kishōtenketsu applies to sentence or sentences, and even clause to chapter as well as the phrase for understandable introduction to conclusion. The concept has also been used in game design, particularly in Nintendo's video games, most notably Super Mario games such as Super Mario Galaxy (2007) and Super Mario 3D World (2013); their designers Shigeru Miyamoto and Koichi Hayashida are known to utilize this concept for their game designs.[2]

QMRThe Foure Monarchies was the title of a long poem by Anne Bradstreet from 1650.[41] Title page of the 1678 edition of her poems .

QMRPines of Rome (Italian: Pini di Roma) is a symphonic poem written by the Italian composer Ottorino Respighi in 1924. It is the second orchestral work in his "Roman trilogy", preceded by Fountains of Rome (1917) and followed by Roman Festivals (1928). Each of the four movements depicts pine trees in different locations in Rome at different times of the day. The premiere took place at the Augusteo, Rome under the direction of Bernardino Molinari on 14 December 1924. Contents [hide] 1 Structure 1.1 Pines of the Villa Borghese (I pini di Villa Borghese: Allegretto vivace) 1.2 Pines Near a Catacomb (Pini presso una catacomba: Lento) 1.3 Pines of the Janiculum (I pini del Gianicolo: Lento) 1.4 Pines of the Appian Way (I pini della Via Appia: Tempo di marcia) 2 Instrumentation 3 Performances and recordings 4 Use in film and elsewhere 5 References 6 External links Structure[edit] This section includes a list of references, related reading or external links, but the sources of this section remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (August 2013) Pines of Rome 1. Pines of the Villa Borghese MENU0:00 2. Pines Near a Catacomb MENU0:00 3. Pines of the Janiculum MENU0:00 4. Pines of the Appian Way MENU0:00 Problems playing these files? See media help. Pines of Rome consists of four movements, each depicting pine trees located in different areas in the city of Rome at different times of the day: Pines of the Villa Borghese (I pini di Villa Borghese: Allegretto vivace)[edit] The first movement portrays children playing by the pine trees in the Villa Borghese gardens. The great Villa Borghese is a monument to the patronage of the Borghese family, who dominated the city in the early seventeenth century. It is a sunny morning and the children sing nursery rhymes and play soldiers. Pines Near a Catacomb (Pini presso una catacomba: Lento)[edit] The second movement is a majestic dirge, conjuring up the picture of a solitary chapel in the deserted Campagna; open land, with a few pine trees silhouetted against the sky. A hymn is heard (specifically, the Kyrie ad libitum 1, Clemens Rector; and the Sanctus from Mass IX, Cum jubilo), the sound rising and sinking again into some sort of catacomb, the subterranean cavern in which the dead are immured. Lower orchestral instruments, plus the organ pedal at 16' and 32' pitch, suggest the subterranean nature of the catacombs, while the trombones and horns represent priests chanting. Pines of the Janiculum (I pini del Gianicolo: Lento)[edit] The third is a nocturne set on the Janiculum hill. The full moon shines on the pines that grow on the hill of the temple of Janus, the double-faced god of doors and gates and of the new year. Respighi took the opportunity to have the sound of a nightingale recorded onto a phonograph and played at the movement's ending. This was something that had never been done before, and created discussion. The score also mentions a specific recording that references a Brunswick Panatrope record player. Pines of the Appian Way (I pini della Via Appia: Tempo di marcia)[edit] Respighi recalls the past glories of the Roman republic in a representation of dawn on the great military road leading into Rome. The final movement portrays pine trees along the Appian Way in the misty dawn as a triumphant legion advances along the Via Appia in the brilliance of the newly-rising sun. Respighi wanted the ground to tremble under the footsteps of his army and he instructs the organ to play bottom B flat on 8', 16' and 32' organ pedal. The score calls for buccine – ancient circular trumpets that are usually represented by modern flugelhorns. Trumpets peal and the consular army rises in triumph to the Capitoline Hill.

QMRThe Fourth Estate is a 1996 novel by Jeffrey Archer. It chronicles the lives of two media barons, Richard Armstrong and Keith Townsend, from their starkly contrasting childhoods to their ultimate battle to build the world's biggest media empire. The book is based on two real life media barons – Robert Maxwell and Rupert Murdoch,[1] who fought to control the newspaper market in Britain. (Murdoch had bought The Sun and News of the World and later Times Newspapers Ltd and Maxwell bought the Daily Mirror and the other newspapers in its group.). The concept of the fourth estate is in essence the press as a watchdog on other powerful institutions or "estates", the original three estates in England and later the United Kingdom being the Lords Spiritual (of the Church of England), the Lords Temporal, and the commons. The fourth estate is charged with keeping an honest watch on activities of the other states and itself. These duties would help democratic societies function properly, openly, and honestly. Debate still flourishes as to whether or not this ever operated (or operates) as it was intended. It also shows a battle between two strong characters from differing backgrounds, who are willing to take endless risks.

QMRThe Four Great Medieval Allegories Le Roman de la Rose. A major allegorical work, it had many lasting influences on western literature, creating entire new genres and development of vernacular languages. The Divine Comedy. Ranked amongst the greatest medieval works, both allegorically and as a work of literature; was (and remains) hugely popular. Piers Plowman. An encyclopedic array of allegorical devices. Dream-vision; pilgrimage; personification; satire; typological story structure (the dreamer's progress mirrors the progress of biblical history from the Fall of Adam to Apocalypse). Pearl. A plot based on an anagogical allegory; a dreamer is introduced to heavenly Jerusalem. Focus on the meaning of death. A religious response to Consolation of Philosophy.

QMRFour types of interpretation or allēgoria[edit] There were four categories of interpretation (or meaning) used in the Middle Ages, which had originated with the Bible commentators of the early Christian era.[2] The first is simply the literal interpretation of the events of the story for historical purposes with no underlying meaning. The second is called typological: it connects the events ofthe Old Testament with the New Testament; in particular drawing allegorical connections between the events of Christ's life with the stories of the Old Testament. The third is moral (or tropological), which is how one should act in the present, the "moral of the story". The fourth type of interpretation is anagogical, dealing with the future events of Christian history, heaven, hell, the last judgment; it deals with prophecies. Thus the four types of interpretation (or meaning) deal with past events (literal), the connection of past events with the present (typology), present events (moral), and the future (anagogical).[2] Dante describes interpreting through a "four-fold method" (or "allegory of the theologians") in his epistle to Can Grande della Scala. He says the "senses" of his work are not simple, but: “ Rather, it may be called "polysemous", that is, of many senses. A first sense derives from the letters themselves, and a second from the things signified by the letters. We call the first sense "literal" sense, the second the "allegorical", or "moral" or "anagogical". To clarify this method of treatment, consider this verse: When Israel went out of Egypt, the house of Jacob from a barbarous people: Judea was made his sanctuary, Israel his dominion (Psalm 113). Now if we examine the letters alone, the exodus of the children of Israel from Egypt in the time of Moses is signified; in the allegory, our redemption accomplished through Christ; in the moral sense, the conversion of the soul from the grief and misery of sin to the state of grace; in the anagogical sense, the exodus of the holy soul from slavery of this corruption to the freedom of eternal glory.. they can all be called allegorical.

QMRThe High and Late Middle Ages saw many allegorical works and techniques. There were four 'great' works from this period.[2] The Four Great Medieval Allegories Le Roman de la Rose. A major allegorical work, it had many lasting influences on western literature, creating entire new genres and development of vernacular languages. The Divine Comedy. Ranked amongst the greatest medieval works, both allegorically and as a work of literature; was (and remains) hugely popular. Piers Plowman. An encyclopedic array of allegorical devices. Dream-vision; pilgrimage; personification; satire; typological story structure (the dreamer's progress mirrors the progress of biblical history from the Fall of Adam to Apocalypse). Pearl. A plot based on an anagogical allegory; a dreamer is introduced to heavenly Jerusalem. Focus on the meaning of death. A religious response to Consolation of Philosophy.

Cinema Chapter

QMRRequiem for a Dream is a 2000 American psychological drama film directed by Darren Aronofsky and starring Ellen Burstyn, Jared Leto, Jennifer Connelly, and Marlon Wayans. The film is based on the novel of the same name by Hubert Selby, Jr., with whom Aronofsky wrote the screenplay. Burstyn was nominated for an Academy Award for Best Actress for her performance. The film was screened out of competition at the 2000 Cannes Film Festival.[3] The film depicts four different forms of drug addiction, which lead to the characters’ imprisonment in a world of delusion and reckless desperation that is subsequently overtaken by reality, thus leaving them as hollow shells of their former selves.[4]

Philosophy Chapter

QMRAmong the useful things which formed a portion of our equipage, was an India-rubber boat, 18 feet long, made somewhat in the form of a bark canoe of the northern lakes. The sides were formed by two airtight cylinders, eighteen inches in diameter, connected with others forming the bow and stern. To lessen the danger of accidents to the boat, these were divided into four different compartments, and the interior was sufficiently large to contain five or six persons, and a considerable weight of baggage." In 1848, General George Cullum, U.S. Army Corps of Engineers, introduced a rubber coated fabric inflatable bridge pontoon, which was used in the Mexican–American War and later on to a limited extent during the American Civil War.[3] An inflatable rubber boat, circa 1855. By 1855 there were numerous types of inflatable rubber boats in use, some made by Goodyear in the U.S. and other surprisingly modern looking boats by the Thomas Hancock Company in Britain. In 1866, four men crossed the Atlantic Ocean from New York to Britain on a three-tube raft called Nonpareil.[4]

Besides Adam Smith, prominent Scottish authors in the field of conjectural history included Adam Ferguson, David Hume, Lord Kames, John Millar, and Lord Monboddo, writing from the later 1750s to later 1770s.[31] Smith, Kames and Millar were content to adhere to the four stage theory.[32] Monboddo's stadial history was more complex, and very much more controversial. He included primates and feral children as material.[33] Robertson in his History of America moves between narrative and conjectural history.[34]

QMRMLC NAND flash is a flash memory technology using multiple levels per cell to allow more bits to be stored using the same number of transistors. In single-level cell (SLC) NAND flash technology, each cell can exist in one of two states, storing one bit of information per cell. Most MLC NAND flash memory has four possible states per cell, so it can store two bits of information per cell. This reduces the amount of margin separating the states and results in the possibility of more errors. Multi-level cells which are designed for low error rates are sometimes called enterprise MLC (eMLC).

QMRExtrinsic motivation comes from external sources. Deci and Ryan[15] developed Organismic Integration Theory (OIT), as a sub-theory of SDT, to explain the different ways extrinsically motivated behaviour is regulated. OIT details the different forms of extrinsic motivation and the contexts in which they come about. It is the context of such motivation that concerns the SDT theory as these contexts affect whether the motivations are internalised and so integrated into the sense of self. OIT describes four different types of extrinsic motivations that often vary in terms of their relative autonomy: Externally regulated behaviour: Is the least autonomous, it is performed because of external demand or possible reward. Such actions can be seen to have an externally perceived locus of causality.[10] Introjected regulation of behaviour: describes taking on regulations to behaviour but not fully accepting said regulations as your own. Deci and Ryan[19] claim such behaviour normally represents regulation by contingent self-esteem, citing ego involvement as a classic form of introjections.[20] This is the kind of behaviour where people feel motivated to demonstrate ability to maintain self-worth. While this is internally driven, introjected behavior has an external perceived locus of causality or not coming from one's self. Since the causality of the behavior is perceived as external, the behavior is considered non-self-determined. Regulation through identification: Is a more autonomously driven form of extrinsic motivation. It involves consciously valuing a goal or regulation so that said action is accepted as personally important. Integrated Regulation: Is the most autonomous kind of extrinsic motivation. Occurring when regulations are fully assimilated with self so they are included in a person's self evaluations and beliefs on personal needs. Because of this, integrated motivations share qualities with intrinsic motivation but are still classified as extrinsic because the goals that are trying to be achieved are for reasons extrinsic to the self, rather than the inherent enjoyment or interest in the task.

QMRFour units[edit] An understanding of the catena is established by distinguishing between the catena and other, similarly defined units. There are four units (including the catena) that are pertinent in this regard: string, catena, component, and constituent. The definition of the catena is repeated for easy comparison with the definitions of the other three units: String Any element (word or morph) or any combination of elements that is continuous in the horizontal dimension (x-axis) Catena Any element (word or morph) or any combination of elements that is continuous in the vertical dimension (y-axis) Component Any element (word or morph) or any combination of elements that is both a string and a catena Constituent Any component that is complete A component is complete if it includes all the elements that its root node dominates. The string and catena complement each other in an obvious way, and the definition of the constituent is essentially the same as one finds in most theories of syntax, where a constituent is understood to consist of any node plus all the nodes that that node dominates. These definitions will now be illustrated with the help of the following dependency tree. The capital letters serve to abbreviate the words:

QMRDuddo Five Stones (grid reference NT930437) is a stone circle north of Duddo in North Northumberland, approximately 4miles (6km) South of the Scottish Border. The stones were known as the Four Stones until 1903, when the fifth stone was re-erected to improve the skyline. There were originally seven stones, the empty sockets of two stones being found on the western side during excavation in the 1890s [1] The stones themselves are a soft sandstone and have become deeply fissured by natural weathering since erection in the Early Bronze Age approximately 4000 years ago The site of the Duddo Stones offers panoramic views of the Cheviot Hills to the South and the Lammermuir Hills to the north The circle is accessible via the B5364 road, through a gate and up a path. The stones are on private land with no formal right-of way, but the landowner has cleared a permissive path across the field to the stones.[2] The fifth is always questionable

QMrThe Georgia Guidestones is a granite monument erected in 1980 in Elbert County, Georgia, in the United States. A set of 10 guidelines is inscribed on the structure in eight modern languages, and a shorter message is inscribed at the top of the structure in four ancient language scripts: Babylonian, Classical Greek, Sanskrit, and Egyptian hieroglyphs. The monument stands at the highest point in Elbert County, about 90 miles (140 km) east of Atlanta, 45 miles (72 km) from Athens, and 9 miles (14 km) north of the center of Elberton. The stones are visible from Georgia Highway 77 (Hartwell Highway) and are reached by turning east on Guidestones Road. The structure is sometimes referred to as an "American Stonehenge".[1] The monument is 19 feet 3 inches (5.87 m) tall, made from six granite slabs weighing 237,746 pounds (107,840 kg) in all.[2] One slab stands in the center, with four arranged around it. A capstone lies on top of the five slabs, which are astronomically aligned. An additional stone tablet, which is set in the ground a short distance to the west of the structure, provides some notes on the history and purpose of the Guidestones. History[edit] The stones defaced with polyurethane paint and graffiti In June 1979, an unknown person or persons under the pseudonym R. C. Christian hired Elberton Granite Finishing Company to build the structure.[2] The land was apparently purchased by Elbert County on October 1, 1979,[3][4][non-primary source needed] although the Georgia Mountain Travel Association's history says the monument is located "on the farm of Mildred and Wayne Mullenix."[5] The monument was unveiled on March 22, 1980, before an audience variously described as 100[6] or 400 people.[2] In 2008, the stones were defaced with polyurethane paint and graffiti with slogans such as "Death to the new world order".[7] Wired magazine called the defacement "the first serious act of vandalism in the Guidestones' history".[2] In September 2014, an employee of the Elbert County maintenance department contacted the FBI when the stones were vandalised with graffiti including the phrase "I Am Isis, goddess of love".[8] Description[edit] Inscriptions[edit] A message consisting of a set of ten guidelines or principles is engraved on the Georgia Guidestones[9] in eight different languages, one language on each face of the four large upright stones. Moving clockwise around the structure from due north, these languages are: English, Spanish, Swahili, Hindi, Hebrew, Arabic, Chinese, and Russian. Maintain humanity under 500,000,000 in perpetual balance with nature. Guide reproduction wisely — improving fitness and diversity. Unite humanity with a living new language. Rule passion — faith — tradition — and all things with tempered reason. Protect people and nations with fair laws and just courts. Let all nations rule internally resolving external disputes in a world court. Avoid petty laws and useless officials. Balance personal rights with social duties. Prize truth — beauty — love — seeking harmony with the infinite. Be not a cancer on the earth — Leave room for nature — Leave room for nature. Explanatory tablet[edit] Georgia Guidestones 04.jpg Georgia Guidestones 19.jpg Georgia Guidestones 10.jpg A few feet to the west of the monument, an additional granite ledger has been set level with the ground. This tablet identifies the structure and the languages used on it, lists various facts about the size, weight, and astronomical features of the stones, the date it was installed, and the sponsors of the project. It also speaks of a time capsule buried under the tablet, but spaces on the stone reserved for filling in the dates on which the capsule was buried and is to be opened have not been inscribed, so it is uncertain if the time capsule was put in place. Each side of the tablet is perpendicular to one of the cardinal directions, and is inscribed so that the northern edge is the top of the inscription. At the center of each tablet edge is a small circle, each containing a letter representing the appropriate compass direction (N, S, E, W). The complete text of the explanatory tablet is detailed below. The tablet is somewhat inconsistent with respect to punctuation, misspells the word "pseudonym," and incorrectly uses the adjective "hieroglyphic" as a plural noun. The original spelling, punctuation, and line breaks in the text have been preserved in the transcription which follows (letter case is not). At the top center of the tablet is written: The Georgia Guidestones Center cluster erected March 22, 1980 Immediately below this is the outline of a square, inside which is written: Let these be guidestones to an Age of Reason Around the edges of the square are written the names of four ancient languages, one per edge. Starting from the top and proceeding clockwise, they are: Babylonian (in cuneiform script), Classical Greek, Sanskrit, and Ancient Egyptian (in hieroglyphs). On the left side of the tablet is the following column of text: Astronomic Features 1. channel through stone indicates celestial pole. 2. horizontal slot indicates annual travel of sun. 3. sunbeam through capstone marks noontime throughout the year Author: R.C. Christian (a pseudonyn) [sic] Sponsors: A small group of Americans who seek the Age of Reason Time Capsule Placed six feet below this spot On To Be Opened on The words appear as shown under the time capsule heading; no dates are engraved. Physical data[edit] On the right side of the tablet is the following column of text (metric conversions added): PHYSICAL DATA 1. OVERALL HEIGHT - 19 FEET 3 INCHES [5.87 m]. 2. TOTAL WEIGHT - 237,746 POUNDS [107,840 kg]. 3. FOUR MAJOR STONES ARE 16 FEET, FOUR INCHES [4.98 m] HIGH, EACH WEIGHING AN AVERAGE OF 42,437 POUNDS [19,249 kg]. 4. CENTER STONE IS 16 FEET, FOUR- INCHES [4.98 m] HIGH, WEIGHS 20,957 POUNDS [9,506 kg]. 5. CAPSTONE IS 9-FEET, 8-INCHES [2.95 m] LONG, 6-FEET, 6-INCHES [1.98 m] WIDE; 1-FOOT, 7-INCHES [0.48 m] THICK. WEIGHS 24,832 POUNDS [11,264 kg]. 6. SUPPORT STONES (BASES) 7-FEET, 4 INCHES [2.24 m] LONG 2-FEET [0.61 m] WIDE. 1 FOOT, 4-INCHES [0.41 m] THICK, EACH WEIGHING AN AVERAGE OF 4,875 POUNDS [2,211 kg]. 7. SUPPORT STONE (BASE) 4-FEET, 2½ INCHES [1.28 m] LONG, 2-FEET, 2-INCHES [0.66 m] WIDE, 1-FOOT, 7-INCHES [0.48 m] THICK. WEIGHT 2,707 POUNDS [1,228 kg]. 8. 951 CUBIC FEET [26.9 m³] GRANITE. 9. GRANITE QUARRIED FROM PYRAMID QUARRIES LOCATED 3 MILES [5 km] WEST OF ELBERTON, GEORGIA. Guidestone languages[edit] Below the two columns of text is written the caption "GUIDESTONE LANGUAGES", with a diagram of the granite slab layout beneath it. The names of eight modern languages are inscribed along the long edges of the projecting rectangles, one per edge. Starting from due north and moving clockwise around so that the upper edge of the northeast rectangle is listed first, they are English, Spanish, Swahili, Hindi, Hebrew, Arabic, Chinese, and Russian. At the bottom center of the tablet is the following text: Additional information available at Elberton Granite Museum & Exhibit College Avenue Elberton, Georgia Astronomical features[edit] The four outer stones are oriented to mark the limits of the 18.6 year lunar declination cycle.[5] The center column features a hole drilled at an angle from one side to the other, through which can be seen the North Star, a star whose position changes only very gradually over time. The same pillar has a slot carved through it which is aligned with the Sun's solstices and equinoxes. A 7/8" aperture in the capstone allows a ray of sun to pass through at noon each day, shining a beam on the center stone indicating the day of the year.[2] Documentary series[edit] The Georgia Guidestones was featured extensively in the July 2012 Travel Channel episode "Mysteries at the Museum: Monumental Mysteries Special" featuring Don Wildman.[10] Reception[edit] Yoko Ono and others have praised the inscribed messages as "a stirring call to rational thinking", while Wired.comstated that unspecified opponents have labeled them as the "Ten Commandments of the Antichrist".[2] The Guidestones have become a subject of interest for conspiracy theorists. One of them, an activist named Mark Dice, demanded that the Guidestones "be smashed into a million pieces, and then the rubble used for a construction project",[11] claiming that the Guidestones are of "a deep Satanic origin", and that R. C. Christian belongs to "a Luciferian secret society" related to the New World Order.[2] At the unveiling of the monument, a local minister proclaimed that he believed the monument was "for sun worshipers, for cult worship and for devil worship".[6] Others have suggested that the stones were commissioned by the Rosicrucians.[12] Computer analyst Van Smith said the monument's dimensions predicted the height of the Burj Khalifa, the tallest building in the world which opened in Dubai over thirty years after the Georgia Guidestones were designed. Smith said the builders of the Guidestones were likely aware of the Burj Khalifa project which he compared to the biblical Tower of Babel.[13] The most widely agreed-upon interpretation of the stones is that they describe the basic concepts required to rebuild a devastated civilization.[14] Brad Meltzer notes that the stones were built in 1979 at the height of the Cold War, and may have been intended as a message to the possible survivors of a World War III. The engraved suggestion to keep humanity's population below 500 million could have been made under the assumption that it had already been reduced below this number.[15]

QMRFour shire stone From Wikipedia, the free encyclopedia The Four shire stone at the meeting point of Warwickshire, Oxfordshire and Gloucestershire, and formerly also Worcestershire The Four shire stone is a boundary marker that marks the place where the four historic English counties of Worcestershire, Warwickshire, Oxfordshire and Gloucestershire once met. Since 1931, with a change to the boundaries of Worcestershire, only three counties have met at the stone. Boundary marker[edit] It is not a stone, but a nine-foot high monument, built from the local Cotswold stone. It is in the English midlands, a mile and a half east of the small town of Moreton-in-Marsh, at 51°59′15″N 1°39′57″WCoordinates: 51°59′15″N 1°39′57″W, grid reference SP231301. The existing structure was probably built in the 18th century, and is a grade II listed building.[1] There was an earlier "4 Shire Stone" on or near the site in 1675.[2] Detail of map from Philips' New handy general Atlas, 1921, showing four counties meeting at the Four shire stone. Worcestershire is shown in yellow, Warwickshire in green, Oxfordshire in violet[3] and Gloucestershire in pink. From the stone, you could go west into Gloucestershire, east into Warwickshire, south-east into Oxfordshire, or south into a small exclave of Worcestershire. Most of Worcestershire is to the north-west of the stone. Thus the order of the four counties around the stone was different from what one might expect from a map of England. The stone ceased to be the meeting-point of four shires in 1931, when the exclave of Worcestershire to its south was transferred to Gloucestershire, so since that date only three counties meet at the stone.[4] It is claimed that the Four shire stone inspired the "Three-Farthing Stone" in J. R. R. Tolkien's books The Hobbit and The Lord of the Rings. In those books, The Shire is divided into four farthings, three of which meet at the "Three-Farthing Stone".[5][6]

QMRThe Great Stone of Fourstones, or the "Big Stone" as it is known locally, is a glacial deposit on the moorlands of Tatham Fells, situated in North Yorkshire, England, near Bentham in the District of Craven, and 10 metres (11 yd) from the county border with Lancashire. The name suggests that there were once four stones, but now there is only one. The other three were possibly broken up for scythe sharpening stones,[1] or building stone, centuries ago. Large stones such as this were useful as boundary markers in the open countryside, and this one was used as a boundary marker for the Lancashire–Yorkshire boundary between Tatham and Bentham parishes.[2] A local myth tells of how the stone was dropped by the devil, on his way to build Devil's Bridge at nearby Kirkby Lonsdale.[3] The stone has 15 steps carved into the side of it to allow access to the top. It is not known when they were carved, but they are well worn from years of use.

QMRThe Analogy of the Divided Line (or Allegory of the Divided Line; Greek: γραμμὴ δίχα τετμημένη) is presented by the Greek philosopher Plato in the Republic (509d–511e). It is written as a dialogue between Glaucon and Socrates, in which the latter further elaborates upon the immediately preceding Analogy of the Sun at the former's request. Socrates asks Glaucon to not only envision this unequally bisected line but to imagine further bisecting each of the two segments. Socrates explains that the four resulting segments represent four separate 'affections' (παθήματα) of the psyche. The lower two sections are said to represent the visible while the higher two are said to represent the intelligible. These affections are described in succession as corresponding to increasing levels of reality and truth from conjecture (εἰκασία) to belief (πίστις) to thought (διάνοια) and finally to understanding (νόησις). Furthermore this Analogy not only elaborates a theory of the psyche but also presents metaphysical and epistemological views. This analogy is immediately followed by the Analogy of the Cave at 514a. Contents [hide] 1 Description 2 The visible world 3 The intelligible world 4 Tabular summary of the Divided Line 5 Metaphysical importance 6 Epistemological meaning 7 See also 8 Notes 9 External links Description[edit] The Divided Line – (AC) is generally taken as representing the visible world and (CE) as representing the intelligible world.[1] In The Republic (509d–510a), Plato describes the Divided Line this way: Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion,[2] and suppose the two main divisions to answer, one to the visible and the other to the intelligible, and then compare the subdivisions in respect of their clearness and want of clearness, and you will find that the first section in the sphere of the visible consists of images. And by images I mean, in the first place, shadows, and in the second place, reflections in water and in solid, smooth and polished bodies and the like: Do you understand? Yes, I understand. Imagine, now, the other section, of which this is only the resemblance, to include the animals which we see, and everything that grows or is made.[3] The visible world[edit] Thus AB represents shadows and reflections of physical things, and BC the physical things themselves. These correspond to two kinds of knowledge, the illusion (εἰκασία eikasia) of our ordinary, everyday experience, and belief (πίστις pistis) about discrete physical objects which cast their shadows.[4] In the Timaeus, the category of illusion includes all the "opinions of which the minds of ordinary people are full," while the natural sciences are included in the category of belief.[4] The intelligible world[edit] According to some translations,[2] the segment CE, representing the intelligible world, is divided into the same ratio as AC, giving the subdivisions CD and DE (it can be readily verified that CD must have the same length as BC:[5] There are two subdivisions, in the lower of which the soul uses the figures given by the former division as images; the enquiry can only be hypothetical, and instead of going upwards to a principle descends to the other end; in the higher of the two, the soul passes out of hypotheses, and goes up to a principle which is above hypotheses, making no use of images as in the former case, but proceeding only in and through the ideas themselves (510b).[3] Plato describes CD, the "lower" of these, as involving mathematical reasoning (διάνοια dianoia),[4] where abstract mathematical objects such as geometric lines are discussed. Such objects are outside the physical world (and are not to be confused with the drawings of those lines, which fall within the physical world BC). However, they are less important to Plato than the subjects of philosophical understanding (νόησις noesis), the "higher" of these two subdivisions (DE): And when I speak of the other division of the intelligible, you will understand me to speak of that other sort of knowledge which reason herself attains by the power of dialectic, using the hypotheses not as first principles, but only as hypotheses — that is to say, as steps and points of departure into a world which is above hypotheses, in order that she may soar beyond them to the first principle of the whole (511b).[3] Plato here is using the familiar relationship between ordinary objects and their shadows or reflections in order to illustrate the relationship between the physical world as a whole and the world of Ideas (Forms) as a whole. The former is made up of a series of passing reflections of the latter, which is eternal, more real and "true." Moreover, the knowledge that we have of the Ideas – when indeed we do have it – is of a higher order than knowledge of the mere physical world. In particular, knowledge of the forms leads to a knowledge of the Idea (Form) of the Good.[1] Tabular summary of the Divided Line[edit] Segment Type of knowledge or opinion Affection of the psyche Type of object Method of the psyche or eye Relative truth and reality DE Noesis (νόησις) Knowledge: understanding of only the Intelligible (νοητόν) Only Ideas, which are all given existence and truth by the Good itself (τὸ αὐτὸ ἀγαθόν) The Psyche examines all hypotheses by the Dialectic making no use of likenesses, always moving towards a First Principle Highest CD Dianoia (διάνοια) Knowledge: thought that recognizes but is not only of the Intelligible Some Ideas, specifically those of Geometry and Number The Psyche assumes hypotheses while making use of likenesses, always moving towards final conclusions High BC Pistis (πίστις) Opinion: belief concerning visible things visible things (ὁρατά) The eye makes probable predictions upon observing visible things low AB Eikasia (εἰκασία) Opinion: conjectures concerning likenesses likenesses of visible things (εἰκόνες) The eye makes guesses upon observing likenesses of visible things lowest Metaphysical importance[edit] The Allegory of the Divided Line is the cornerstone of Plato's metaphysical framework. This structure, well hidden in the middle of the Republic, a complex, multi-layered dialogue, illustrates the grand picture of Plato's metaphysics, epistemology, and ethics, all in one. It is not enough for the philosopher to understand the Ideas (Forms), he must also understand the relation of Ideas to all four levels of the structure to be able to know anything at all.[6][7][8] In the Republic, the philosopher must understand the Idea of Justice to live a just life or to organize and govern a just state.[9] The Divided Line also serves as our guide for most past and future metaphysics. The lowest level, which represents "the world of becoming and passing away" (Republic, 508d), is the metaphysical model for a Heraclitean philosophy of constant flux and for Protagorean philosophy of appearance and opinion. The second level, a world of fixed physical objects,[10][11] also became Aristotle's metaphysical model. The third level might be a Pythagorean level of mathematics. The fourth level is Plato's ideal Parmenidean reality, the world of highest level Ideas. Epistemological meaning[edit] Plato holds a very strict notion of knowledge. For example, he does not accept expertise about a subject, nor direct perception (see Theaetetus), nor true belief about the physical world (the Meno) as knowledge. It is not enough for the philosopher to understand the Ideas (Forms), he must also understand the relation of Ideas to all four levels of the structure to be able to know anything at all.[12] For this reason, in most of the "earlier Socratic" dialogues, Socrates denies knowledge both to himself and others. For the first level, "the world of becoming and passing away," Plato expressly denies the possibility of knowledge.[13] Constant change never stays the same, therefore, properties of objects must refer to different Ideas at different times. Note that for knowledge to be possible, which Plato believed, the other three levels must be unchanging. The third and fourth level, mathematics and Ideas, are already eternal and unchanging. However, to ensure that the second level objective, physical world is also unchanging, Plato, in the Republic, Book 4[14] introduces empirically derived[15][16][17] axiomatic restrictions that prohibit both motion and shifting perspectives.[10][18]

QMRHume accepts that ideas may be either the product of mere sensation, or of the imagination working in conjunction with sensation.[4] According to Hume, the creative faculty makes use of (at least) four mental operations which produce imaginings out of sense-impressions. These operations are compounding (or the addition of one idea onto another, such as a horn on a horse to create a unicorn); transposing (or the substitution of one part of a thing with the part from another, such as with the body of a man upon a horse to make a centaur); augmenting (as with the case of a giant, whose size has been augmented); and diminishing (as with Lilliputians, whose size has been diminished). (Hume 1974:317) In a later chapter, he also mentions the operations of mixing, separating, and dividing. (Hume 1974:340)

QMRThe four pillars policy is an Australian Government policy to maintain the separation of the four largest banks in Australia by rejecting any merger or acquisition between the four major banks.[1]

QMRA structuring into the first three of these dimensions was proposed by Binswanger on the basis of Heidegger's description of Umwelt and Mitwelt and his further notion of Eigenwelt. The fourth dimension was added by van Deurzen on the basis of Heidegger's description of a spiritual world (Überwelt) in Heidegger's later work.[5][6]

QMRFor piecewise linear manifolds, the Poincaré conjecture is true except possibly in dimension 4, where the answer is unknown, and equivalent to the smooth case. In other words, every compact PL manifold of dimension not equal to 4 that is homotopy equivalent to a sphere is PL isomorphic to a sphere.[1]

In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard n-sphere. The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected and closed. The Generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal recipients John Milnor, Steve Smale, Michael Freedman and Grigori Perelman. Contents [hide] 1 Status 2 History 3 Exotic spheres 4 PL 5 References Status[edit] Here is a summary of the status of the Generalized Poincaré conjecture in various settings. Top: true in all dimensions. PL: true in dimensions other than 4; unknown in dimension 4, where it is equivalent to Diff. Diff: false generally, true in some dimensions including 1,2,3,5, and 6. First known counterexample is in dimension 7. The case of dimension 4 is unsettled (as of 2011).

QMRFor piecewise linear manifolds, the Poincaré conjecture is true except possibly in dimension 4, where the answer is unknown, and equivalent to the smooth case. In other words, every compact PL manifold of dimension not equal to 4 that is homotopy equivalent to a sphere is PL isomorphic to a sphere.[1]

QMRIn mathematics, the Poincaré conjecture (/pwɛn.kɑːˈreɪ/ pwen-kar-ay; French: [pwɛ̃kaʁe])[1] is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Dimensions[edit] Main article: Generalized Poincaré conjecture The classification of closed surfaces gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the Generalized Poincaré conjecture: is a homotopy n-sphere homeomorphic to the n-sphere? A stronger assumption is necessary; in dimensions four and higher there are simply-connected, closed manifolds which are not homotopy equivilent to an n-sphere. Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem. In 1982 Michael Freedman proved the Poincaré conjecture in four dimension. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not diffeomorphic to the four-sphere. This so-called smooth Poincaré conjecture, in dimension four, remains open and is thought to be very difficult. Milnor's exotic spheres show that the smooth Poincaré conjecture is false in dimension seven, for example. These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the geometrization conjecture put it into a framework governing all 3-manifolds. John Morgan wrote:[13] It is my view that before Thurston's work on hyperbolic 3-manifolds and . . . the Geometrization conjecture there was no consensus among the experts as to whether the Poincaré conjecture was true or false. After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensus developed that the Poincaré conjecture (and the Geometrization conjecture) were true.

QMRIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

In mathematics[edit] In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates, but a single polar coordinate (the angle) would be sufficient, so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with the question "what makes En n-dimensional?" One answer is that to cover a fixed ball in En by small balls of radius ε, one needs on the order of ε−n such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En , they turn out to be different when one looks at more general spaces. A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4". Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schläfli's 1852 Theorie der vielfachen Kontinuität, Hamilton's 1843 discovery of the quaternions and the construction of the Cayley algebra marked the beginning of higher-dimensional geometry. The rest of this section examines some of the more important mathematical definitions of the dimensions. Complex dimension[edit] Main article: Complex dimension A complex number (x + iy) has a real part x and an imaginary part iy whose magnitude is y. A single complex coordinate system may be applied to an object having two real dimensions, for example an ordinary two-dimensional spherical surface, when given a complex metric, becomes a Riemann sphere of one complex dimension.[3] Complex dimensions appear in the study of complex manifolds and algebraic varieties. Dimension of a vector space[edit] Main article: Dimension (vector space) The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension. Manifolds[edit] A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold. For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point. In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to "work"; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied. Varieties[edit] Main article: Dimension of an algebraic variety The dimension of an algebraic variety may be defined in various equivalent ways. The most intuitive way is probably the dimension of the tangent space at any regular point. Another intuitive way is to define the dimension as the number of hyperplanes that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety. An algebraic set being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains V_0\subsetneq V_1\subsetneq \ldots \subsetneq V_d of sub-varieties of the given algebraic set (the length of such a chain is the number of "\subsetneq"). Each variety can be considered as an algebraic stack, and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Specifically, if V is a variety of dimension m and G is an algebraic group of dimension n acting on V, then the quotient stack [V/G] has dimension m−n.[4]

Krull dimension[edit] Main article: Krull dimension The Krull dimension of a commutative ring is the maximal length of chains of prime ideals in it, a chain of length n being a sequence \mathcal{P}_0\subsetneq \mathcal{P}_1\subsetneq \ldots \subsetneq\mathcal{P}_n of prime ideals related by inclusion. It is strongly related to the dimension of an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring of the polynomials on the variety. For an algebra over a field, the dimension as vector space is finite if and only if its Krull dimension is 0. Lebesgue covering dimension[edit] Main article: Lebesgue covering dimension For any normal topological space X, the Lebesgue covering dimension of X is defined to be n if n is the smallest integer for which the following holds: any open cover has an open refinement (a second open cover where each element is a subset of an element in the first cover) such that no point is included in more than n + 1 elements. In this case dim X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the dimension of X is said to be infinite, and one writes dim X = ∞. Moreover, X has dimension −1, i.e. dim X = −1 if and only if X is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open". Inductive dimension[edit] Main article: Inductive dimension An inductive definition of dimension can be created as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an (n + 1)-dimensional object by dragging an n-dimensional object in a new direction. The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that (n + 1)-dimensional balls have n-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets. Hausdorff dimension[edit] Main article: Hausdorff dimension For structurally complicated sets, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the dimensions considered above, can also attain non-integer real values.[5] The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have been found useful to describe many natural objects and phenomena.[6][page needed][7][page needed] Hilbert spaces[edit] Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the above dimensions coincide. In physics[edit] Spatial dimensions[edit] Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See Space and Cartesian coordinate system.) Number of dimensions Example co-ordinate systems 1 Number line Number line Angle Angle 2 Cartesian system (2d) Cartesian (two-dimensional) Polar system Polar Geographic system Latitude and longitude 3 Cartesian system (3d) Cartesian (three-dimensional) Cylindrical system Cylindrical Spherical system Spherical Time[edit] A temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move in one direction. The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy). The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space.

Networks and dimension[edit] Some complex networks are characterized by fractal dimensions.[11] The concept of dimension can be generalized to include networks embedded in space.[12] The dimension characterize their spatial constraints. In literature[edit] Main article: Fourth dimension in literature Science fiction texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones. One of the most heralded science fiction stories regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novella Flatland by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions." The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-Dimension Catapult (1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include Robert A. Heinlein's —And He Built a Crooked House (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract; and Alan E. Nourse's Tiger by the Tail and The Universe Between (both 1951). Another reference is Madeleine L'Engle's novel A Wrinkle In Time (1962), which uses the fifth dimension as a way for "tesseracting the universe" or "folding" space in order to move across it quickly. The fourth and fifth dimensions were also a key component of the book The Boy Who Reversed Himself by William Sleator. In philosophy[edit] Immanuel Kant, in 1783, wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain."[13] "Space has Four Dimensions" is a short story published in 1846 by German philosopher and experimental psychologist Gustav Fechner under the pseudonym "Dr. Mises". The protagonist in the tale is a shadow who is aware of and able to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time.[14] The story bears a strong similarity to the "Allegory of the Cave" presented in Plato's The Republic (c. 380 BC). Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled "The Philosophy of Hyperspace".[15] Linda Dalrymple Henderson coined the term "hyperspace philosophy", used to describe writing that uses higher dimensions to explore metaphysical themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art.[16] Examples of "hyperspace philosophers" include Charles Howard Hinton, the first writer, in 1888, to use the word "tesseract";[17] and the Russian esotericist P. D. Ouspensky.

QMR4+1 is a view model designed by Philippe Kruchten for "describing the architecture of software-intensive systems, based on the use of multiple, concurrent views".[1] The views are used to describe the system from the viewpoint of different stakeholders, such as end-users, developers and project managers. The four views of the model are logical, development, process and physical view. In addition selected use cases or scenarios are used to illustrate the architecture serving as the 'plus one' view. Hence the model contains 4+1 views:[1] Logical view : The logical view is concerned with the functionality that the system provides to end-users. UML Diagrams used to represent the logical view include Class diagram, Communication diagram, Sequence diagram.[2] Development view : The development view illustrates a system from a programmer's perspective and is concerned with software management. This view is also known as the implementation view. It uses the UML Component diagram to describe system components. UML Diagrams used to represent the development view include the Package diagram.[2] Process view : The process view deals with the dynamic aspects of the system, explains the system processes and how they communicate, and focuses on the runtime behavior of the system. The process view addresses concurrency, distribution, integrators, performance, and scalability, etc. UML Diagrams to represent process view include the Activity diagram.[2] Physical view : The physical view depicts the system from a system engineer's point of view. It is concerned with the topology of software components on the physical layer, as well as the physical connections between these components. This view is also known as the deployment view. UML Diagrams used to represent physical view include the Deployment diagram.[2] Scenarios : The description of an architecture is illustrated using a small set of use cases, or scenarios which become a fifth view. The scenarios describe sequences of interactions between objects, and between processes. They are used to identify architectural elements and to illustrate and validate the architecture design. They also serve as a starting point for tests of an architecture prototype. This view is also known as use case view.

QMrThe process focus of COBIT 4.1 is illustrated by a process model that subdivides IT into four domains (Plan and Organize, Acquire and Implement, Deliver and Support, and Monitor and Evaluate) and 34 processes in line with the responsibility areas of plan, build, run and monitor. It is positioned at a high level and has been aligned and harmonized with other, more detailed, IT standards and good practices such as COSO, ITIL, BiSL, ISO 27000, CMMI, TOGAF and PMBOK. COBIT acts as an integrator of these different guidance materials, summarizing key objectives under one umbrella framework that link the good practice models with governance and business requirements.

QMRSince Stephen Spewak's book called enterprise architecture planning (EAP) in 1993, and perhaps before then, it has been normal to recognise four types of architecture domain: Business architecture Data architecture Applications architecture Technology architecture Note that the applications architecture is about the application portfolio, not the internal architecture of a single application - which is often called the application architecture. Many EA frameworks combine data and application domains into a single layer, sitting below the business (usually a human activity system) and above the technology (the platform IT infrastructure). There are many variations on this theme. Typical architecture domains[edit] Typical architecture domains are listed below, and defined drawing on the definitions found in The British Computer Society's "Reference Model for Enterprise and Solution Architecture". Business architecture: The structure and behaviour of a business system (not necessarily related to computers). Covers business goals, business functions or capabilities, business processes and roles etc. Business functions and business processes are often mapped to the applications and data they need. Data architecture: The data structures used by a business and/or its applications. Descriptions of data in storage and data in motion. Descriptions of data stores, data groups and data items. Mappings of those data artifacts to data qualities, applications, locations etc. Applications architecture: The structure and behaviour of applications used in a business, focused on how they interact with each other and with users. Focused on the data consumed and produced by applications rather than their internal structure. In application portfolio management, the applications are usually mapped to business functions and to application platform technologies. Application (or Component) architecture: The internal structure, the modularisation of software, within an application. This is software architecture at the lowest level of granularity. It is usually below the level of modularisation that solution architects define. However, there is no rigid dividing line. Technical architecture or infrastructure architecture: The structure and behaviour of the technology infrastructure. Covers the client and server nodes of the hardware configuration, the infrastructure applications that run on them, the infrastructure services they offer to applications, the protocols and networks that connect applications and nodes.

QMREuler's sum of powers conjecture From Wikipedia, the free encyclopedia Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n kth powers of non-zero integers is itself a kth power, then n is greater than or equal to k. In symbols, the conjecture falsely states that if \sum_{i=1}^{n} a_i^k = b^k where n>1 and a_1, a_2, \dots, a_n, b are non-zero integers, then n\geq k. The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case n = 2: if a_1^k + a_2^k = b^k, then 2 \geq k. Although the conjecture holds for the case k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4 and k = 5. It is unknown whether the conjecture fails or holds for any value k ≥ 6. Contents [hide] 1 Background 2 Counterexamples 3 Generalizations 3.1 k = 4 3.2 k = 5 3.3 k = 7 3.4 k = 8 4 See also 5 References 6 External links Background[edit] Euler had an equality for four fourth powers 59^4 + 158^4 = 133^4 + 134^4; this however is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number 3^3+4^3+5^3=6^3 or the taxicab number 1729.[1][2] The general solution for: x_1^3+x_2^3=x_3^3+x_4^3 is x_1 = 1-(a-3b)(a^2+3b^2), x_2 = (a+3b)(a^2+3b^2)-1 x_3 = (a+3b)-(a^2+3b^2)^2, x_4 = (a^2+3b^2)^2-(a-3b) where a and b are any integers. Counterexamples[edit] Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for k = 5.[3] A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: 275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966), (−220)5 + 50275 + 62375 + 140685 = 141325 (Scher & Seidl, 1996), and 555 + 31835 + 289695 + 852825 = 853595 (Frye, 2004). In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the k = 4 case.[4] His smallest counterexample was 26824404 + 153656394 + 187967604 = 206156734. A particular case of Elkies' solutions can be reduced to the identity[5][6] (85v2 + 484v − 313)4 + (68v2 − 586v + 10)4 + (2u)4 = (357v2 − 204v + 363)4 where u2 = 22030 + 28849v − 56158v2 + 36941v3 − 31790v4. This is an elliptic curve with a rational point at v1 = −31/467. From this initial rational point, one can compute an infinite collection of others. Substituting v1 into the identity and removing common factors gives the numerical example cited above. In 1988, Roger Frye found the smallest possible counterexample 958004 + 2175194 + 4145604 = 4224814 for k = 4 by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000.[7] Generalizations[edit] Main article: Lander, Parkin, and Selfridge conjecture In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[8] that if k > 3 and \sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k, where ai ≠ bj are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m+n ≥ k. In the special case m = 1, the conjecture states that if \sum_{i=1}^{n} a_i^k = b^k (under the conditions given above) then n ≥ k − 1. The special case may be described as the problem of giving a partition of a perfect power into few like powers. For k = 4, 5, 7, 8 and n = k or k − 1, there are many known solutions. Some of these are listed below. There are no solutions for k = 6 where b ≤ 272580.[citation needed] k = 4[edit] 958004 + 2175194 + 4145604 = 4224814 (R. Frye, 1988)[4] 304 + 1204 + 2724 + 3154 = 3534 (R. Norrie, 1911)[8] k = 5[edit] 275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966) 195 + 435 + 465 + 475 + 675 = 725 (Lander, Parkin, Selfridge, smallest, 1967)[8] 75 + 435 + 575 + 805 + 1005 = 1075 (Sastry, 1934, third smallest)[8] k = 7[edit] 1277 + 2587 + 2667 + 4137 + 4307 + 4397 + 5257 = 5687 (M. Dodrill, 1999)[citation needed] k = 8[edit] 908 + 2238 + 4788 + 5248 + 7488 + 10888 + 11908 + 13248 = 14098 (S. Chase, 2000)[citation needed]

QMRA prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p, p+2, p+6, p+8}.[1] This represents the closest possible grouping of four primes larger than 3. Contents [hide] 1 Prime quadruplets 2 Prime quintuplets 3 Prime sextuplets 4 References Prime quadruplets[edit] The first eight prime quadruplets are: {5, 7, 11, 13}, {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, {1481, 1483, 1487, 1489}, {1871, 1873, 1877, 1879}, {2081, 2083, 2087, 2089} (sequence A007530 in OEIS) All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n. (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade. A prime quadruplet contains two pairs of twin primes or can be described as having two overlapping prime triplets. It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is 1, 3, 7, 26, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 (sequence A120120 in OEIS). As of December 2013 the largest known prime quadruplet has 3503 digits.[2] It starts with p = 2339662057597 × 103490 + 1. The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets: B_4 = \left(\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13}\right) + \left(\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19}\right) + \left(\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109}\right) + \cdots with value: B4 = 0.87058 83800 ± 0.00000 00005. This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4. The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed. Excluding the first prime quadruplet, the shortest possible distance between two quadruplets {p, p+2, p+6, p+8} and {q, q+2, q+6, q+8} is q - p = 30. The first occurrences of this are for p = 1006301, 2594951, 3919211, 9600551, 10531061, ... (OEIS A059925).

QMRIn linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if X is either a vector space of dimension n or more generally a matroid of rank n, with n disjoint bases Bi, then it is possible to arrange the elements of these bases into an n × n matrix in such a way that the rows of the matrix are exactly the given bases and the columns of the matrix are also bases. That is, it should be possible to find a second set of n disjoint bases Ci, each of which consists of one element from each of the bases Bi. Rota's basis conjecture has a simple formulation for points in the Euclidean plane: it states that, given three triangles with distinct vertices, with each triangle colored with one of three colors, it must be possible to regroup the nine triangle vertices into three "rainbow" triangles having one vertex of each color. The triangles are all required to be non-degenerate, meaning that they do not have all three vertices on a line. To see this as an instance of the basis conjecture, one may use either linear independence of the vectors (xi,yi,1) in a three-dimensional real vector space (where (xi,yi) are the Cartesian coordinates of the triangle vertices) or equivalently one may use a matroid of rank three in which a set S of points is independent if either |S| ≤ 2 or S forms the three vertices of a non-degenerate triangle. For this linear algebra and this matroid, the bases are exactly the non-degenerate triangles. Given the three input triangles and the three rainbow triangles, it is possible to arrange the nine vertices into a 3 × 3 matrix in which each row contains the vertices of one of the single-color triangles and each column contains the vertices of one of the rainbow triangles. Analogously, for points in three-dimensional Euclidean space, the conjecture states that the sixteen vertices of four non-degenerate tetrahedra of four different colors may be regrouped into four rainbow tetrahedra.

QMRInformally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly[1] and Ulam.[2][3] Contents [hide] 1 Formal statements 2 Verification 2.1 Reconstructible graph families 3 Recognizable properties 4 Reduction 5 Other structures 6 See also 7 Further reading 8 References Formal statements[edit] graph and the associated deck of single-vertex-deleted subgraphs. Note some of the cards show isomorphic graphs. Given a graph G = (V,E), a vertex-deleted subgraph of G is a subgraph formed by deleting exactly one vertex from G. Clearly, it is an induced subgraph of G. For a graph G, the deck of G, denoted D(G), is the multiset of all vertex-deleted subgraphs of G. Each graph in D(G) is called a card. Two graphs that have the same deck are said to be hypomorphic. With these definitions, the conjecture can be stated as: Reconstruction Conjecture: Any two hypomorphic graphs on at least three vertices are isomorphic. (The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.) Harary[4] suggested a stronger version of the conjecture: Set Reconstruction Conjecture: Any two graphs on at least four vertices with the same sets of vertex-deleted subgraphs are isomorphic. Given a graph G = (V,E), an edge-deleted subgraph of G is a subgraph formed by deleting exactly one edge from G. For a graph G, the edge-deck of G, denoted ED(G), is the multiset of all edge-deleted subgraphs of G. Each graph in ED(G) is called an edge-card. Edge Reconstruction Conjecture: (Harary, 1964)[4] Any two graphs with at least four edges and having the same edge-decks are isomorphic.

QMRThe Albertson conjecture is vacuously true for n ≤ 4: K4 has crossing number zero, and all graphs have crossing number greater than or equal to zero. The case n = 5 of Albertson's conjecture is equivalent to the four color theorem, that any planar graph can be colored with four or fewer colors, for the only graphs requiring fewer crossings than the one crossing of K5 are the planar graphs, and the conjecture implies that these should all be at most 4-chromatic. Through the efforts of several groups of authors the conjecture is now known to hold for all n ≤ 16.[4] For every integer c ≥ 6, Luiz and Richter presented a family of (c+1)-colour-critical graphs that do not contain a subdivision of the complete graph K(c+1) but have crossing number at least that of K(c+1).[5]

QMRIn differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.[1] Carathéodory did publish a paper on a related subject,[2] but never committed the Conjecture into writing. In,[3] John Edensor Littlewood mentions the Conjecture and Hamburger's contribution [4] as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in [5] the formal analogy of the Conjecture with the Four Vertex Theorem for plane curves. Modern references to the Conjecture are the problem list of Shing-Tung Yau,[6] the books of Marcel Berger,[7][8] as well as the books.[9][10][11][12]

QMRThe classical four-vertex theorem states that the curvature function of a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion. Contents [hide] 1 Examples 2 History 3 Converse 4 Application to mechanics 5 Discrete variations 6 Generalizations to space curve 7 See also 8 References 9 External links Examples[edit] An ellipse has exactly four vertices: two local maxima of curvature where it is crossed by the major axis of the ellipse, and two local minima of curvature where it is crossed by the minor axis. In a circle, every point is both a local maximum and a local minimum of curvature, so there are infinitely many vertices. History[edit] The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya.[1] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has 4th-order contact with the curve (in general the osculating circle has only 3rd-order contact with the curve). The four-vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument.[2] Converse[edit] The converse to the four-vertex theorem states that any continuous, real-valued function of the circle that has at least two local maxima and two local minima is the curvature function of a simple, closed plane curve. The converse was proved for strictly positive functions in 1971 by Herman Gluck as a special case of a general theorem on pre-assigning the curvature of n-spheres.[3] The full converse to the four-vertex theorem was proved by Björn Dahlberg shortly before his death in January 1998, and published posthumously.[4] Dahlberg's proof uses a winding number argument which is in some ways reminiscent of the standard topological proof of the Fundamental Theorem of Algebra.[5] Application to mechanics[edit] One corollary of the theorem is that a homogeneous, planar disk rolling on a horizontal surface under gravity has at least 4 balance points. A discrete version of this is that there cannot be a monostatic polygon. However in three dimensions there do exist monostatic polyhedra, and there also exists a convex, homogeneous object with exactly 2 balance points (one stable, and the other unstable), the Gömböc. illustration of the Four-vertex theorem at an ellipse Discrete variations[edit] There are several discrete versions of the four-vertex theorem, both for convex and non-convex polygons.[6] Here are some of them: (Bilinski) The sequence of angles of a convex equilateral polygon has at least four extrema. The sequence of side lengths of a convex equiangular polygon has at least four extrema. (Musin) A circle circumscribed around three consecutive vertices of the polygon is called extremal if it contains all remaining vertices of the polygon, or has none of them in its interior. A convex polygon is generic if it has no four vertices on the same circle. Then every generic convex polygon has at least four extremal circles. (Legendre–Cauchy) Two convex n-gons with equal corresponding side length have either zero or at least 4 sign changes in the cyclic sequence of the corresponding angle differences. (A.D. Alexandrov) Two convex n-gons with parallel corresponding sides and equal area have either zero or at least 4 sign changes in the cyclic sequence of the corresponding side lengths differences. Some of these variations are stronger than the other, and all of them imply the (usual) four-vertex theorem by a limit argument. Generalizations to space curve[edit] The stereographic projection from the sphere to the plane preserves critical points of geodesic curvature. Thus simple closed spherical curves have four vertices. Furthermore, on the sphere vertices of a curve correspond to points where its torsion vanishes. So for space curves a vertex is defined as a point of vanishing torsion. In 1994 V. D. Sedykh [7] showed that every simple closed space curve which lies on the boundary of a convex body has four vertices. In 2015 Mohammad Ghomi [8] generalized Sedykh's theorem to all curves which bound a locally convex disk.

In graph theory, the Hadwiger conjecture (or Hadwiger's conjecture) states that, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single vertex produces a complete graph Kk on k vertices as a minor of G. This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. Bollobás, Catlin & Erdős (1980) call it “one of the deepest unsolved problems in graph theory.”[1]

QMRA graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph (each subgraph has an edge connecting it to each other subgraph), illustrating the case k = 4 of Hadwiger's conjecture

QMRIn graph theory, Hedetniemi's conjecture, named after Stephen T. Hedetniemi, concerns the connection between graph coloring and the tensor product of graphs. This conjecture states that Clearly, any graph with a nonempty set of edges requires two colors; therefore, the conjecture is true whenever G or H is bipartite. It is also true when G or H is 3-colorable, for if both G and H contain an odd cycle then so does G × H. In the remaining cases, both factors of the tensor product require four or more colors. When both factors are 4-chromatic, El-Zahar & Sauer (1985) showed that their tensor product also requires four colors; therefore, Hedetniemi's conjecture is true for this case as well.

The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture.[citation needed] Using Vinogradov's method, Chudakov,[14] Van der Corput,[15] and Estermann[16] showed that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers which can be so written tends towards 1). In 1930, Lev Schnirelmann proved[17][18] that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant, see Schnirelmann density. Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800000. This result was subsequently enhanced by many authors; currently, the best known result is due to Olivier Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes. In fact, the proof of the weak Goldbach conjecture by Harald Helfgott [19] directly implies that every even number n ≥ 4 is the sum of at most four primes.[20]

QMRGoldbach number[edit] The number of ways an even number can be represented as the sum of two primes.[3] A Goldbach number is a positive integer that can be expressed as the sum of two odd primes.[4] Since four is the only even number greater than two that requires the even prime 2 to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.

QMRThe analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre. The rationality part of the conjectures was proved first by Bernard Dwork (1960), using p-adic methods. Grothendieck (1965) and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Artin for attacking the Weil conjectures, as outlined in Grothendieck (1960). Of the four conjectures the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of Serre (1960) of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles (Kleiman 1968). However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne (1974), using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.

QMR Four exponentials conjecture From Wikipedia, the free encyclopedia In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of a certain number of values of the exponential function. Contents [hide] 1 Statement 2 History 3 Corollaries 4 Sharp four exponentials conjecture 5 Strong four exponentials conjecture 6 Three exponentials conjecture 7 Bertrand's conjecture 8 Notes 9 References 10 External links Statement[edit] If x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental: e^{x_1y_1}, e^{x_1y_2}, e^{x_2y_1}, e^{x_2y_2}. An alternative way of stating the conjecture in terms of logarithms is the following. For 1 ≤ i,j ≤ 2 let λij be complex numbers such that exp(λij) are all algebraic. Suppose λ11 and λ12 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then \lambda_{11}\lambda_{22}\neq\lambda_{12}\lambda_{21}.\, An equivalent formulation in terms of linear algebra is the following. Let M be the 2×2 matrix M=\begin{pmatrix}\lambda_{11}&\lambda_{12} \\ \lambda_{21}&\lambda_{22}\end{pmatrix}, where exp(λij) is algebraic for 1 ≤ i,j ≤ 2. Suppose the two rows of M are linearly independent over the rational numbers, and the two columns of M are linearly independent over the rational numbers. Then the rank of M is 2. While a 2×2 matrix having linearly independent rows and columns usually means it has rank 2, in this case we require linear independence over a smaller field so the rank isn't forced to be 2. For example, the matrix \begin{pmatrix}1&\pi \\ \pi&\pi^2\end{pmatrix} has rows and columns that are linearly independent over the rational numbers, since π is irrational. But the rank of the matrix is 1. So in this case the conjecture would imply that at least one of e, eπ, and eπ ² is transcendental (which in this case is already known since e is transcendental). History[edit] The conjecture was considered as early as the early 1940s by Atle Selberg who never formally stated the conjecture.[1] A special case of the conjecture is mentioned in a 1944 paper of Leonidas Alaoglu and Paul Erdős who suggest that it had been considered by Carl Ludwig Siegel.[2] An equivalent statement was first mentioned in print by Theodor Schneider who set it as the first of eight important, open problems in transcendental number theory in 1957.[3] The related six exponentials theorem was first explicitly mentioned in the 1960s by Serge Lang[4] and Kanakanahalli Ramachandra,[5] and both also explicitly conjecture the above result.[6] Indeed, after proving the six exponentials theorem Lang mentions the difficulty in dropping the number of exponents from six to four — the proof used for six exponentials “just misses” when one tries to apply it to four. Corollaries[edit] Using Euler's identity this conjecture implies the transcendence of many numbers involving e and π. For example, taking x1 = 1, x2 = √2, y1 = iπ, and y2 = iπ√2, the conjecture — if true — implies that one of the following four numbers is transcendental: e^{i\pi}, e^{i\pi\sqrt{2}}, e^{i\pi\sqrt{2}}, e^{2i\pi}. The first of these is just −1, and the fourth is 1, so the conjecture implies that eiπ√2 is transcendental (which is already known, by consequence of the Gelfond–Schneider theorem). An open problem in number theory settled by the conjecture is the question of whether there exists a non-integral real number t such that both 2t and 3t are integers, or indeed such that at and bt are both integers for some pair of integers a and b that are multiplicatively independent over the integers. Values of t such that 2t is an integer are all of the form t = log2m for some integer m, while for 3t to be an integer, t must be of the form t = log3n for some integer n. By setting x1 = 1, x2 = t, y1 = log2, and y2 = log3, the four exponentials conjecture implies that if t is irrational then one of the following four numbers is transcendental: 2, 3, 2^t, 3^t.\, So if 2t and 3t are both integers then the conjecture implies that t must be a rational number. Since the only rational numbers t for which 2t is also rational are the integers, this implies that there are no non-integral real numbers t such that both 2t and 3t are integers. It is this consequence, for any two primes not just 2 and 3, that Alaoglu and Erdős desired in their paper as it would imply the conjecture that the quotient of two consecutive colossally abundant numbers is prime, extending Ramanujan's results on the quotients of consecutive superior highly composite number.[7] Sharp four exponentials conjecture[edit] The four exponentials conjecture reduces the pair and triplet of complex numbers in the hypotheses of the six exponentials theorem to two pairs. It is conjectured that this is also possible with the sharp six exponentials theorem, and this is the sharp four exponentials conjecture.[8] Specifically, this conjecture claims that if x1, x2, and y1, y2 are two pairs of complex numbers with each pair being linearly independent over the rational numbers, and if βij are four algebraic numbers for 1 ≤ i,j ≤ 2 such that the following four numbers are algebraic: e^{x_1 y_1-\beta_{11}}, e^{x_1 y_2-\beta_{12}}, e^{x_2 y_1-\beta_{21}}, e^{x_2 y_2-\beta_{22}}, then xi yj = βij for 1 ≤ i,j ≤ 2. So all four exponentials are in fact 1. This conjecture implies both the sharp six exponentials theorem, which requires a third x value, and the as yet unproven sharp five exponentials conjecture that requires a further exponential to be algebraic in its hypotheses. Strong four exponentials conjecture[edit] Logical implications between the various n-exponentials problems The logical implications between the various problems in this circle. Those in red are as yet unproven while those in blue are known results. The top most result refers to that discussed at Baker's theorem, while the lower two rows are detailed at the six exponentials theorem article. The strongest result that has been conjectured in this circle of problems is the strong four exponentials conjecture.[9] This result would imply both aforementioned conjectures concerning four exponentials as well as all the five and six exponentials conjectures and theorems, as illustrated to the right, and all the three exponentials conjectures detailed below. The statement of this conjecture deals with the vector space over the algebraic numbers generated by 1 and all logarithms of non-zero algebraic numbers, denoted here as L∗. So L∗ is the set of all complex numbers of the form \beta_0+\sum_{i=1}^n \beta_i\log\alpha_i, for some n ≥ 0, where all the βi and αi are algebraic and every branch of the logarithm is considered. The statement of the strong four exponentials conjecture is then as follows. Let x1, x2, and y1, y2 be two pairs of complex numbers with each pair being linearly independent over the algebraic numbers, then at least one of the four numbers xi yj for 1 ≤ i,j ≤ 2 is not in L∗. Three exponentials conjecture[edit] The four exponentials conjecture rules out a special case of non-trivial, homogeneous, quadratic relations between logarithms of algebraic numbers. But a conjectural extension of Baker's theorem implies that there should be no non-trivial algebraic relations between logarithms of algebraic numbers at all, homogeneous or not. One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture.[10] In its logarithmic form it is the following conjecture. Let λ1, λ2, and λ3 be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ1λ2 = γλ3. Then λ1λ2 = γλ3 = 0. The exponential form of this conjecture is the following. Let x1, x2, and y be non-zero complex numbers and let γ be a non-zero algebraic number. Then at least one of the following three numbers is transcendental: e^{x_1y}, e^{x_2y}, e^{\gamma x_1/x_2}. There is also a sharp three exponentials conjecture which claims that if x1, x2, and y are non-zero complex numbers and α, β1, β2, and γ are algebraic numbers such that the following three numbers are algebraic e^{x_1 y-\beta_1}, e^{x_2 y-\beta_2}, e^{(\gamma x_1/x_2)-\alpha}, then either x2y = β2 or γx1 = α x2. The strong three exponentials conjecture meanwhile states that if x1, x2, and y are non-zero complex numbers with x1y, x2y, and x1/x2 all transcendental, then at least one of the three numbers x1y, x2y, x1/x2 is not in L∗. As with the other results in this family, the strong three exponentials conjecture implies the sharp three exponentials conjecture which implies the three exponentials conjecture. However, the strong and sharp three exponentials conjectures are implied by their four exponentials counterparts, bucking the usual trend. And the three exponentials conjecture is neither implied by nor implies the four exponentials conjecture. The three exponentials conjecture, like the sharp five exponentials conjecture, would imply the transcendence of eπ² by letting (in the logarithmic version) λ1 = iπ, λ2 = −iπ, and γ = 1. Bertrand's conjecture[edit] Many of the theorems and results in transcendental number theory concerning the exponential function have analogues involving the modular function j. Writing q = e2πiτ for the nome and j(τ) = J(q), Daniel Bertrand conjectured that if q1 and q2 are non-zero algebraic numbers in the complex unit disc that are multiplicatively independent, then J(q1) and J(q2) are algebraically independent over the rational numbers.[11] Although not obviously related to the four exponentials conjecture, Bertrand's conjecture in fact implies a special case known as the weak four exponentials conjecture.[12] This conjecture states that if x1 and x2 are two positive real algebraic numbers, neither of them equal to 1, then π² and the product (logx1)(logx2) are linearly independent over the rational numbers. This corresponds to the special case of the four exponentials conjecture whereby y1 = iπ, y2 = −iπ, and x1 and x2 are real. Perhaps surprisingly, though, it is also a corollary of Bertrand's conjecture, suggesting there may be an approach to the full four exponentials conjecture via the modular function j.

QMRIn mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of a certain number of values of the exponential function. Statement[edit] If x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental: e^{x_1y_1}, e^{x_1y_2}, e^{x_2y_1}, e^{x_2y_2}. An alternative way of stating the conjecture in terms of logarithms is the following. For 1 ≤ i,j ≤ 2 let λij be complex numbers such that exp(λij) are all algebraic. Suppose λ11 and λ12 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then \lambda_{11}\lambda_{22}\neq\lambda_{12}\lambda_{21}.\, An equivalent formulation in terms of linear algebra is the following. Let M be the 2×2 matrix M=\begin{pmatrix}\lambda_{11}&\lambda_{12} \\ \lambda_{21}&\lambda_{22}\end{pmatrix}, where exp(λij) is algebraic for 1 ≤ i,j ≤ 2. Suppose the two rows of M are linearly independent over the rational numbers, and the two columns of M are linearly independent over the rational numbers. Then the rank of M is 2. While a 2×2 matrix having linearly independent rows and columns usually means it has rank 2, in this case we require linear independence over a smaller field so the rank isn't forced to be 2. For example, the matrix \begin{pmatrix}1&\pi \\ \pi&\pi^2\end{pmatrix} has rows and columns that are linearly independent over the rational numbers, since π is irrational. But the rank of the matrix is 1. So in this case the conjecture would imply that at least one of e, eπ, and eπ ² is transcendental (which in this case is already known since e is transcendental). History[edit] The conjecture was considered as early as the early 1940s by Atle Selberg who never formally stated the conjecture.[1] A special case of the conjecture is mentioned in a 1944 paper of Leonidas Alaoglu and Paul Erdős who suggest that it had been considered by Carl Ludwig Siegel.[2] An equivalent statement was first mentioned in print by Theodor Schneider who set it as the first of eight important, open problems in transcendental number theory in 1957.[3] The related six exponentials theorem was first explicitly mentioned in the 1960s by Serge Lang[4] and Kanakanahalli Ramachandra,[5] and both also explicitly conjecture the above result.[6] Indeed, after proving the six exponentials theorem Lang mentions the difficulty in dropping the number of exponents from six to four — the proof used for six exponentials “just misses” when one tries to apply it to four.

QMRThe Greek language distinguishes at least four different ways as to how the word love is used. Ancient Greek has four distinct words for love: agápe, éros, philía, and storgē. However, as with other languages, it has been historically difficult to separate the meanings of these words when used outside of their respective contexts. Nonetheless, the senses in which these words were generally used are as follows: Agápe (ἀγάπη agápē[1]) means "love: esp. brotherly love, charity; the love of God for man and of man for God."[2] Agape is used in ancient texts to denote feelings for one's children and the feelings for a spouse, and it was also used to refer to a love feast.[3] Agape is used by Christians to express the unconditional love of God for his children.[citation needed] [4] This type of love was further explained by Thomas Aquinas as "to will the good of another."[5] Éros (ἔρως érōs) means "love, mostly of the sexual passion."[6] The Modern Greek word "erotas" means "intimate love." Plato refined his own definition: Although eros is initially felt for a person, with contemplation it becomes an appreciation of the beauty within that person, or even becomes appreciation of beauty itself. Plato does not talk of physical attraction as a necessary part of love, hence the use of the word platonic to mean, "without physical attraction." In the Symposium, the most famous ancient work on the subject, Plato has Socrates argue that eros helps the soul recall knowledge of beauty, and contributes to an understanding of spiritual truth, the ideal "Form" of youthful beauty that leads us humans to feel erotic desire – thus suggesting that even that sensually based love aspires to the non-corporeal, spiritual plane of existence; that is, finding its truth, just like finding any truth, leads to transcendence.[7] Lovers and philosophers are all inspired to seek truth through the means of eros. Philia (φιλία philía) means "affectionate regard, friendship," usually "between equals."[8] It is a dispassionate virtuous love, a concept developed by Aristotle.[9] In his best-known work on ethics, Nicomachean Ethics, philia is expressed variously as loyalty to friends, family, and community, and requires virtue, equality, and familiarity. Furthermore, in the same text philos denotes a general type of love, used for love between family, between friends, a desire or enjoyment of an activity, as well as between lovers. Storge (στοργή storgē) means "love, affection" and "especially of parents and children"[10] It's the common or natural empathy, like that felt by parents for offspring.[11] Rarely used in ancient works, and then almost exclusively as a descriptor of relationships within the family. It is also known to express mere acceptance or putting up with situations, as in "loving" the tyrant.