title: Horace

text: Horace left Rome, possibly after his father's death, and continued his formal education in Athens, a great centre of learning in the ancient world, where he arrived at nineteen years of age, enrolling in The Academy. Founded by Plato, The Academy was now dominated by Epicureans and Stoics, whose theories and practises made a deep impression on the young man from Venusia. Meanwhile, he mixed and lounged about with the elite of Roman youth, such as Marcus, the idle son of Cicero, and the Pompeius to whom he later addressed a poem. It was in Athens too that he probably acquired deep familiarity with the ancient tradition of Greek lyric poetry, at that time largely the preserve of grammarians and academic specialists (access to such material was easier in Athens than in Rome, where the public libraries had yet to be built by Asinius Pollio and Augustus).


title: Horace

text: Rome's troubles following the assassination of Julius Caesar were soon to catch up with him. Marcus Junius Brutus came to Athens seeking support for the republican cause. Brutus was fêted around town in grand receptions and he made a point of attending academic lectures, all the while recruiting supporters among the young men studying there, including Horace. An educated young Roman could begin military service high in the ranks and Horace was made tribunus militum (one of six senior officers of a typical legion), a post usually reserved for men of senatorial or equestrian rank and which seems to have inspired jealousy among his well-born confederates. He learned the basics of military life while on the march, particularly in the wilds of northern Greece, whose rugged scenery became a backdrop to some of his later poems. It was there in 42 BC that Octavian (later Augustus) and his associate Mark Antony crushed the republican forces at the Battle of Philippi. Horace later recorded it as a day of embarrassment for himself, when he fled without his shield, but allowance should be made for his self-deprecating humour. Moreover, the incident allowed him to identify himself with some famous poets who had long ago abandoned their shields in battle, notably his heroes Alcaeus and Archilochus. The comparison with the latter poet is uncanny: Archilochus lost his shield in a part of Thrace near Philippi, and he was deeply involved in the Greek colonization of Thasos, where Horace's die-hard comrades finally surrendered.


title: Horace

text: Octavian offered an early amnesty to his opponents and Horace quickly accepted it. On returning to Italy, he was confronted with yet another loss: his father's estate in Venusia was one of many throughout Italy to be confiscated for the settlement of veterans (Virgil lost his estate in the north about the same time). Horace later claimed that he was reduced to poverty and this led him to try his hand at poetry. In reality, there was no money to be had from versifying. At best, it offered future prospects through contacts with other poets and their patrons among the rich. Meanwhile, he obtained the sinecure of "scriba quaestorius", a civil service position at the "aerarium" or Treasury, profitable enough to be purchased even by members of the "ordo equester" and not very demanding in its work-load, since tasks could be delegated to "scribae" or permanent clerks. It was about this time that he began writing his "Satires" and "Epodes".


title: Horace

text: He describes in glowing terms the country villa which his patron, Maecenas, had given him in a letter to his friend Quintius:


title: Horace

text: The remains of Horace's Villa are situated on a wooded hillside above the river at Licenza, which joins the Aniene as it flows on to Tivoli.


title: Horace

text: The "Epodes" belong to iambic poetry. Iambic poetry features insulting and obscene language; sometimes, it is referred to as "blame poetry". "Blame poetry", or "shame poetry", is poetry written to blame and shame fellow citizens into a sense of their social obligations. Each poem normally has a archetype person Horace decides to shame, or teach a lesson to. Horace modelled these poems on the poetry of Archilochus. Social bonds in Rome had been decaying since the destruction of Carthage a little more than a hundred years earlier, due to the vast wealth that could be gained by plunder and corruption. These social ills were magnified by rivalry between Julius Caesar, Mark Antony and confederates like Sextus Pompey, all jockeying for a bigger share of the spoils. One modern scholar has counted a dozen civil wars in the hundred years leading up to 31 BC, including the Spartacus rebellion, eight years before Horace's birth. As the heirs to Hellenistic culture, Horace and his fellow Romans were not well prepared to deal with these problems:


title: Horace

text: Horace's Hellenistic background is clear in his Satires, even though the genre was unique to Latin literature. He brought to it a style and outlook suited to the social and ethical issues confronting Rome but he changed its role from public, social engagement to private meditation. Meanwhile, he was beginning to interest Octavian's supporters, a gradual process described by him in one of his satires. The way was opened for him by his friend, the poet Virgil, who had gained admission into the privileged circle around Maecenas, Octavian's lieutenant, following the success of his "Eclogues". An introduction soon followed and, after a discreet interval, Horace too was accepted. He depicted the process as an honourable one, based on merit and mutual respect, eventually leading to true friendship, and there is reason to believe that his relationship was genuinely friendly, not just with Maecenas but afterwards with Augustus as well. On the other hand, the poet has been unsympathetically described by one scholar as "a sharp and rising young man, with an eye to the main chance." There were advantages on both sides: Horace gained encouragement and material support, the politicians gained a hold on a potential dissident. His republican sympathies, and his role at Philippi, may have caused him some pangs of remorse over his new status. However most Romans considered the civil wars to be the result of "contentio dignitatis", or rivalry between the foremost families of the city, and he too seems to have accepted the principate as Rome's last hope for much needed peace.


title: Horace

text: In 37 BC, Horace accompanied Maecenas on a journey to Brundisium, described in one of his poems as a series of amusing incidents and charming encounters with other friends along the way, such as Virgil. In fact the journey was political in its motivation, with Maecenas en route to negotiatie the Treaty of Tarentum with Antony, a fact Horace artfully keeps from the reader (political issues are largely avoided in the first book of satires). Horace was probably also with Maecenas on one of Octavian's naval expeditions against the piratical Sextus Pompeius, which ended in a disastrous storm off Palinurus in 36 BC, briefly alluded to by Horace in terms of near-drowning. There are also some indications in his verses that he was with Maecenas at the Battle of Actium in 31 BC, where Octavian defeated his great rival, Antony. By then Horace had already received from Maecenas the famous gift of his Sabine farm, probably not long after the publication of the first book of "Satires". The gift, which included income from five tenants, may have ended his career at the Treasury, or at least allowed him to give it less time and energy. It signalled his identification with the Octavian regime yet, in the second book of "Satires" that soon followed, he continued the apolitical stance of the first book. By this time, he had attained the status of "eques Romanus" (Roman 'cavalryman', 'knight'), perhaps as a result of his work at the Treasury.


title: Horace

text: "Odes" 1–3 were the next focus for his artistic creativity. He adapted their forms and themes from Greek lyric poetry of the seventh and sixth centuries BC. The fragmented nature of the Greek world had enabled his literary heroes to express themselves freely and his semi-retirement from the Treasury in Rome to his own estate in the Sabine hills perhaps empowered him to some extent also yet even when his lyrics touched on public affairs they reinforced the importance of private life. Nevertheless, his work in the period 30–27 BC began to show his closeness to the regime and his sensitivity to its developing ideology. In "Odes" 1.2, for example, he eulogized Octavian in hyperboles that echo Hellenistic court poetry. The name "Augustus", which Octavian assumed in 27 January BC, is first attested in "Odes" 3.3 and 3.5. In the period 27–24 BC, political allusions in the "Odes" concentrated on foreign wars in Britain (1.35), Arabia (1.29) Spain (3.8) and Parthia (2.2). He greeted Augustus on his return to Rome in 24 BC as a beloved ruler upon whose good health he depended for his own happiness (3.14).


title: Horace

text: The public reception of "Odes" 1–3 disappointed him, however. He attributed the lack of success to jealousy among imperial courtiers and to his isolation from literary cliques. Perhaps it was disappointment that led him to put aside the genre in favour of verse letters. He addressed his first book of "Epistles" to a variety of friends and acquaintances in an urbane style reflecting his new social status as a knight. In the opening poem, he professed a deeper interest in moral philosophy than poetry but, though the collection demonstrates a leaning towards stoic theory, it reveals no sustained thinking about ethics. Maecenas was still the dominant confidante but Horace had now begun to assert his own independence, suavely declining constant invitations to attend his patron. In the final poem of the first book of "Epistles", he revealed himself to be forty-four years old in the consulship of Lollius and Lepidus i.e. 21 BC, and "of small stature, fond of the sun, prematurely grey, quick-tempered but easily placated".


title: Horace

text: According to Suetonius, the second book of "Epistles" was prompted by Augustus, who desired a verse epistle to be addressed to himself. Augustus was in fact a prolific letter-writer and he once asked Horace to be his personal secretary. Horace refused the secretarial role but complied with the emperor's request for a verse letter. The letter to Augustus may have been slow in coming, being published possibly as late as 11 BC. It celebrated, among other things, the 15 BC military victories of his stepsons, Drusus and Tiberius, yet it and the following letter were largely devoted to literary theory and criticism. The literary theme was explored still further in "Ars Poetica", published separately but written in the form of an epistle and sometimes referred to as "Epistles" 2.3 (possibly the last poem he ever wrote). He was also commissioned to write odes commemorating the victories of Drusus and Tiberius and one to be sung in a temple of Apollo for the Secular Games, a long-abandoned festival that Augustus revived in accordance with his policy of recreating ancient customs ("Carmen Saeculare").


title: Horace

text: Suetonius recorded some gossip about Horace's sexual activities late in life, claiming that the walls of his bedchamber were covered with obscene pictures and mirrors, so that he saw erotica wherever he looked. The poet died at 56 years of age, not long after his friend Maecenas, near whose tomb he was laid to rest. Both men bequeathed their property to Augustus, an honour that the emperor expected of his friends.


title: Horace

text: The dating of Horace's works isn't known precisely and scholars often debate the exact order in which they were first 'published'. There are persuasive arguments for the following chronology:


title: Horace

text: Horace composed in traditional metres borrowed from Archaic Greece, employing hexameters in his "Satires" and "Epistles", and iambs in his "Epodes", all of which were relatively easy to adapt into Latin forms. His "Odes" featured more complex measures, including alcaics and sapphics, which were sometimes a difficult fit for Latin structure and syntax. Despite these traditional metres, he presented himself as a partisan in the development of a new and sophisticated style. He was influenced in particular by Hellenistic aesthetics of brevity, elegance and polish, as modelled in the work of Callimachus.


title: Horace

text: In modern literary theory, a distinction is often made between immediate personal experience ("Urerlebnis") and experience mediated by cultural vectors such as literature, philosophy and the visual arts ("Bildungserlebnis"). The distinction has little relevance for Horace however since his personal and literary experiences are implicated in each other. "Satires" 1.5, for example, recounts in detail a real trip Horace made with Virgil and some of his other literary friends, and which parallels a Satire by Lucilius, his predecessor. Unlike much Hellenistic-inspired literature, however, his poetry was not composed for a small coterie of admirers and fellow poets, nor does it rely on abstruse allusions for many of its effects. Though elitist in its literary standards, it was written for a wide audience, as a public form of art. Ambivalence also characterizes his literary persona, since his presentation of himself as part of a small community of philosophically aware people, seeking true peace of mind while shunning vices like greed, was well adapted to Augustus's plans to reform public morality, corrupted by greed—his personal plea for moderation was part of the emperor's grand message to the nation.


title: Horace

text: Horace generally followed the examples of poets established as classics in different genres, such as Archilochus in the "Epodes", Lucilius in the "Satires" and Alcaeus in the "Odes", later broadening his scope for the sake of variation and because his models weren't actually suited to the realities confronting him. Archilochus and Alcaeus were aristocratic Greeks whose poetry had a social and religious function that was immediately intelligible to their audiences but which became a mere artifice or literary motif when transposed to Rome. However, the artifice of the "Odes" is also integral to their success, since they could now accommodate a wide range of emotional effects, and the blend of Greek and Roman elements adds a sense of detachment and universality. Horace proudly claimed to introduce into Latin the spirit and iambic poetry of Archilochus but (unlike Archilochus) without persecuting anyone ("Epistles" 1.19.23–25). It was no idle boast. His "Epodes" were modelled on the verses of the Greek poet, as 'blame poetry', yet he avoided targeting real scapegoats. Whereas Archilochus presented himself as a serious and vigorous opponent of wrong-doers, Horace aimed for comic effects and adopted the persona of a weak and ineffectual critic of his times (as symbolized for example in his surrender to the witch Canidia in the final epode). He also claimed to be the first to introduce into Latin the lyrical methods of Alcaeus ("Epistles" 1.19.32–33) and he actually was the first Latin poet to make consistent use of Alcaic meters and themes: love, politics and the symposium. He imitated other Greek lyric poets as well, employing a 'motto' technique, beginning each ode with some reference to a Greek original and then diverging from it.


title: Horace

text: The satirical poet Lucilius was a senator's son who could castigate his peers with impunity. Horace was a mere freedman's son who had to tread carefully. Lucilius was a rugged patriot and a significant voice in Roman self-awareness, endearing himself to his countrymen by his blunt frankness and explicit politics. His work expressed genuine freedom or libertas. His style included 'metrical vandalism' and looseness of structure. Horace instead adopted an oblique and ironic style of satire, ridiculing stock characters and anonymous targets. His libertas was the private freedom of a philosophical outlook, not a political or social privilege. His "Satires" are relatively easy-going in their use of meter (relative to the tight lyric meters of the "Odes") but formal and highly controlled relative to the poems of Lucilius, whom Horace mocked for his sloppy standards ("Satires" 1.10.56–61)


title: Horace

text: The "Epistles" may be considered among Horace's most innovative works. There was nothing like it in Greek or Roman literature. Occasionally poems had had some resemblance to letters, including an elegiac poem from Solon to Mimnermus and some lyrical poems from Pindar to Hieron of Syracuse. Lucilius had composed a satire in the form of a letter, and some epistolary poems were composed by Catullus and Propertius. But nobody before Horace had ever composed an entire collection of verse letters, let alone letters with a focus on philosophical problems. The sophisticated and flexible style that he had developed in his "Satires" was adapted to the more serious needs of this new genre. Such refinement of style was not unusual for Horace. His craftsmanship as a wordsmith is apparent even in his earliest attempts at this or that kind of poetry, but his handling of each genre tended to improve over time as he adapted it to his own needs. Thus for example it is generally agreed that his second book of "Satires", where human folly is revealed through dialogue between characters, is superior to the first, where he propounds his ethics in monologues. Nevertheless, the first book includes some of his most popular poems.


title: Horace

text: Horace developed a number of inter-related themes throughout his poetic career, including politics, love, philosophy and ethics, his own social role, as well as poetry itself. His "Epodes" and "Satires" are forms of 'blame poetry' and both have a natural affinity with the moralising and diatribes of Cynicism. This often takes the form of allusions to the work and philosophy of Bion of Borysthenes but it is as much a literary game as a philosophical alignment.


title: Horace

text: By the time he composed his "Epistles", he was a critic of Cynicism along with all impractical and "high-falutin" philosophy in general.


title: Horace

text: The "Satires" also include a strong element of Epicureanism, with frequent allusions to the Epicurean poet Lucretius. So for example the Epicurean sentiment "carpe diem" is the inspiration behind Horace's repeated punning on his own name ("Horatius ~ hora") in "Satires" 2.6. The "Satires" also feature some Stoic, Peripatetic and Platonic ("Dialogues") elements. In short, the "Satires" present a medley of philosophical programmes, dished up in no particular order—a style of argument typical of the genre.


title: Horace

text: The "Odes" display a wide range of topics. Over time, he becomes more confident about his political voice. Although he is often thought of as an overly intellectual lover, he is ingenious in representing passion. The "Odes" weave various philosophical strands together, with allusions and statements of doctrine present in about a third of the "Odes" Books 1–3, ranging from the flippant (1.22, 3.28) to the solemn (2.10, 3.2, 3.3). Epicureanism is the dominant influence, characterising about twice as many of these odes as Stoicism.


title: Horace

text: A group of odes combines these two influences in tense relationships, such as "Odes" 1.7, praising Stoic virility and devotion to public duty while also advocating private pleasures among friends. While generally favouring the Epicurean lifestyle, the lyric poet is as eclectic as the satiric poet, and in "Odes" 2.10 even proposes Aristotle's golden mean as a remedy for Rome's political troubles.


title: Horace

text: Many of Horace's poems also contain much reflection on genre, the lyric tradition, and the function of poetry. "Odes" 4, thought to be composed at the emperor's request, takes the themes of the first three books of "Odes" to a new level. This book shows greater poetic confidence after the public performance of his "Carmen saeculare" or "Century hymn" at a public festival orchestrated by Augustus. In it, Horace addresses the emperor Augustus directly with more confidence and proclaims his power to grant poetic immortality to those he praises. It is the least philosophical collection of his verses, excepting the twelfth ode, addressed to the dead Virgil as if he were living. In that ode, the epic poet and the lyric poet are aligned with Stoicism and Epicureanism respectively, in a mood of bitter-sweet pathos.


title: Horace

text: The first poem of the "Epistles" sets the philosophical tone for the rest of the collection: "So now I put aside both verses and all those other games: What is true and what befits is my care, this my question, this my whole concern." His poetic renunciation of poetry in favour of philosophy is intended to be ambiguous. Ambiguity is the hallmark of the "Epistles". It is uncertain if those being addressed by the self-mocking poet-philosopher are being honoured or criticised. Though he emerges as an Epicurean, it is on the understanding that philosophical preferences, like political and social choices, are a matter of personal taste. Thus he depicts the ups and downs of the philosophical life more realistically than do most philosophers.


title: Horace

text: The reception of Horace's work has varied from one epoch to another and varied markedly even in his own lifetime. "Odes" 1–3 were not well received when first 'published' in Rome, yet Augustus later commissioned a ceremonial ode for the Centennial Games in 17 BC and also encouraged the publication of "Odes" 4, after which Horace's reputation as Rome's premier lyricist was assured. His Odes were to become the best received of all his poems in ancient times, acquiring a classic status that discouraged imitation: no other poet produced a comparable body of lyrics in the four centuries that followed (though that might also be attributed to social causes, particularly the parasitism that Italy was sinking into). In the seventeenth and eighteenth centuries, ode-writing became highly fashionable in England and a large number of aspiring poets imitated Horace both in English and in Latin.


title: Horace

text: In a verse epistle to Augustus (Epistle 2.1), in 12 BC, Horace argued for classic status to be awarded to contemporary poets, including Virgil and apparently himself. In the final poem of his third book of Odes he claimed to have created for himself a monument more durable than bronze ("Exegi monumentum aere perennius", "Carmina" 3.30.1). For one modern scholar, however, Horace's personal qualities are more notable than the monumental quality of his achievement:


title: Horace

text: Yet for men like Wilfred Owen, scarred by experiences of World War I, his poetry stood for discredited values:


title: Horace

text: The same motto, "Dulce et decorum est pro patria mori", had been adapted to the ethos of martyrdom in the lyrics of early Christian poets like Prudentius.


title: Horace

text: These preliminary comments touch on a small sample of developments in the reception of Horace's work. More developments are covered epoch by epoch in the following sections.


title: Horace

text: Horace's influence can be observed in the work of his near contemporaries, Ovid and Propertius. Ovid followed his example in creating a completely natural style of expression in hexameter verse, and Propertius cheekily mimicked him in his third book of elegies. His "Epistles" provided them both with a model for their own verse letters and it also shaped Ovid's exile poetry.


title: Horace

text: His influence had a perverse aspect. As mentioned before, the brilliance of his "Odes" may have discouraged imitation. Conversely, they may have created a vogue for the lyrics of the archaic Greek poet Pindar, due to the fact that Horace had neglected that style of lyric (see Influence and Legacy of Pindar). The iambic genre seems almost to have disappeared after publication of Horace's "Epodes". Ovid's "Ibis" was a rare attempt at the form but it was inspired mainly by Callimachus, and there are some iambic elements in Martial but the main influence there was Catullus. A revival of popular interest in the satires of Lucilius may have been inspired by Horace's criticism of his unpolished style. Both Horace and Lucilius were considered good role-models by Persius, who critiqued his own satires as lacking both the acerbity of Lucillius and the gentler touch of Horace. Juvenal's caustic satire was influenced mainly by Lucilius but Horace by then was a school classic and Juvenal could refer to him respectfully and in a round-about way as ""the Venusine lamp"".


title: Horace

text: Statius paid homage to Horace by composing one poem in Sapphic and one in Alcaic meter (the verse forms most often associated with "Odes"), which he included in his collection of occasional poems, "Silvae". Ancient scholars wrote commentaries on the lyric meters of the "Odes", including the scholarly poet Caesius Bassus. By a process called "derivatio", he varied established meters through the addition or omission of syllables, a technique borrowed by Seneca the Younger when adapting Horatian meters to the stage.


title: Horace

text: Horace's poems continued to be school texts into late antiquity. Works attributed to Helenius Acro and Pomponius Porphyrio are the remnants of a much larger body of Horatian scholarship. Porphyrio arranged the poems in non-chronological order, beginning with the "Odes", because of their general popularity and their appeal to scholars (the "Odes" were to retain this privileged position in the medieval manuscript tradition and thus in modern editions also). Horace was often evoked by poets of the fourth century, such as Ausonius and Claudian. Prudentius presented himself as a Christian Horace, adapting Horatian meters to his own poetry and giving Horatian motifs a Christian tone. On the other hand, St Jerome, modelled an uncompromising response to the pagan Horace, observing: ""What harmony can there be between Christ and the Devil? What has Horace to do with the Psalter?"" By the early sixth century, Horace and Prudentius were both part of a classical heritage that was struggling to survive the disorder of the times. Boethius, the last major author of classical Latin literature, could still take inspiration from Horace, sometimes mediated by Senecan tragedy. It can be argued that Horace's influence extended beyond poetry to dignify core themes and values of the early Christian era, such as self-sufficiency, inner contentment and courage.


title: Horace

text: Classical texts almost ceased being copied in the period between the mid sixth century and the Carolingian revival. Horace's work probably survived in just two or three books imported into northern Europe from Italy. These became the ancestors of six extant manuscripts dated to the ninth century. Two of those six manuscripts are French in origin, one was produced in Alsace, and the other three show Irish influence but were probably written in continental monasteries (Lombardy for example). By the last half of the ninth century, it was not uncommon for literate people to have direct experience of Horace's poetry. His influence on the Carolingian Renaissance can be found in the poems of Heiric of Auxerre and in some manuscripts marked with neumes, mysterious notations that may have been an aid to the memorization and discussion of his lyric meters. "Ode" is neumed with the melody of a hymn to John the Baptist, "Ut queant laxis", composed in Sapphic stanzas. This hymn later became the basis of the solfege system ("Do, re, mi...")an association with western music quite appropriate for a lyric poet like Horace, though the language of the hymn is mainly Prudentian. Lyons argues that the melody in question was linked with Horace's Ode well before Guido d'Arezzo fitted Ut queant laxis to it. However, the melody is unlikely to be a survivor from classical times, although Ovid testifies to Horace's use of the lyre while performing his Odes.


title: Horace

text: The German scholar, Ludwig Traube, once dubbed the tenth and eleventh centuries "The age of Horace" ("aetas Horatiana"), and placed it between the "aetas Vergiliana" of the eighth and ninth centuries, and the "aetas Ovidiana" of the twelfth and thirteenth centuries, a distinction supposed to reflect the dominant classical Latin influences of those times. Such a distinction is over-schematized since Horace was a substantial influence in the ninth century as well. Traube had focused too much on Horace's "Satires". Almost all of Horace's work found favour in the Medieval period. In fact medieval scholars were also guilty of over-schematism, associating Horace's different genres with the different ages of man. A twelfth-century scholar encapsulated the theory: "...Horace wrote four different kinds of poems on account of the four ages, the "Odes" for boys, the "Ars Poetica" for young men, the "Satires" for mature men, the "Epistles" for old and complete men." It was even thought that Horace had composed his works in the order in which they had been placed by ancient scholars. Despite its naivety, the schematism involved an appreciation of Horace's works as a collection, the "Ars Poetica", "Satires" and "Epistles" appearing to find favour as well as the "Odes". The later Middle Ages however gave special significance to "Satires" and "Epistles", being considered Horace's mature works. Dante referred to Horace as "Orazio satiro", and he awarded him a privileged position in the first circle of Hell, with Homer, Ovid and Lucan.


title: Horace

text: Horace's popularity is revealed in the large number of quotes from all his works found in almost every genre of medieval literature, and also in the number of poets imitating him in quantitative Latin meter. The most prolific imitator of his "Odes" was the Bavarian monk, Metellus of Tegernsee, who dedicated his work to the patron saint of Tegernsee Abbey, St Quirinus, around the year 1170. He imitated all Horace's lyrical meters then followed these up with imitations of other meters used by Prudentius and Boethius, indicating that variety, as first modelled by Horace, was considered a fundamental aspect of the lyric genre. The content of his poems however was restricted to simple piety. Among the most successful imitators of "Satires" and "Epistles" was another Germanic author, calling himself Sextus Amarcius, around 1100, who composed four books, the first two exemplifying vices, the second pair mainly virtues.


title: Horace

text: Petrarch is a key figure in the imitation of Horace in accentual meters. His verse letters in Latin were modelled on the "Epistles" and he wrote a letter to Horace in the form of an ode. However he also borrowed from Horace when composing his Italian sonnets. One modern scholar has speculated that authors who imitated Horace in accentual rhythms (including stressed Latin and vernacular languages) may have considered their work a natural sequel to Horace's metrical variety. In France, Horace and Pindar were the poetic models for a group of vernacular authors called the Pléiade, including for example Pierre de Ronsard and Joachim du Bellay. Montaigne made constant and inventive use of Horatian quotes. The vernacular languages were dominant in Spain and Portugal in the sixteenth century, where Horace's influence is notable in the works of such authors as Garcilaso de la Vega, Juan Boscán, Sá de Miranda, Antonio Ferreira and Fray Luis de León, the last writing odes on the Horatian theme "beatus ille" ("happy the man"). The sixteenth century in western Europe was also an age of translations (except in Germany, where Horace wasn't translated into the vernacular until well into the seventeenth century). The first English translator was Thomas Drant, who placed translations of Jeremiah and Horace side by side in "Medicinable Morall", 1566. That was also the year that the Scot George Buchanan paraphrased the Psalms in a Horatian setting. Ben Jonson put Horace on the stage in 1601 in "Poetaster", along with other classical Latin authors, giving them all their own verses to speak in translation. Horace's part evinces the independent spirit, moral earnestness and critical insight that many readers look for in his poems.


title: Horace

text: During the seventeenth and eighteenth centuries, or the Age of Enlightenment, neoclassical culture was pervasive. English literature in the middle of that period has been dubbed Augustan. It is not always easy to distinguish Horace's influence during those centuries (the mixing of influences is shown for example in one poet's pseudonym, "Horace Juvenal"). However a measure of his influence can be found in the diversity of the people interested in his works, both among readers and authors.


title: Horace

text: New editions of his works were published almost yearly. There were three new editions in 1612 (two in Leiden, one in Frankfurt) and again in 1699 (Utrecht, Barcelona, Cambridge). Cheap editions were plentiful and fine editions were also produced, including one whose entire text was engraved by John Pine in copperplate. The poet James Thomson owned five editions of Horace's work and the physician James Douglas had five hundred books with Horace-related titles. Horace was often commended in periodicals such as The Spectator, as a hallmark of good judgement, moderation and manliness, a focus for moralising. His verses offered a fund of mottoes, such as "simplex munditiis" (elegance in simplicity), "splendide mendax" (nobly untruthful), "sapere aude" (dare to know), "nunc est bibendum" (now is the time to drink), "carpe diem" (seize the day, perhaps the only one still in common use today). These were quoted even in works as prosaic as Edmund Quincy's "A treatise of hemp-husbandry" (1765). The fictional hero Tom Jones recited his verses with feeling. His works were also used to justify commonplace themes, such as patriotic obedience, as in James Parry's English lines from an Oxford University collection in 1736:


title: Horace

text: Horatian-style lyrics were increasingly typical of Oxford and Cambridge verse collections for this period, most of them in Latin but some like the previous ode in English. John Milton's Lycidas first appeared in such a collection. It has few Horatian echoes yet Milton's associations with Horace were lifelong. He composed a controversial version of "Odes" 1.5, and Paradise Lost includes references to Horace's 'Roman' "Odes" 3.1–6 (Book 7 for example begins with echoes of "Odes" 3.4). Yet Horace's lyrics could offer inspiration to libertines as well as moralists, and neo-Latin sometimes served as a kind of discrete veil for the risqué. Thus for example Benjamin Loveling authored a catalogue of Drury Lane and Covent Garden prostitutes, in Sapphic stanzas, and an encomium for a dying lady "of salacious memory". Some Latin imitations of Horace were politically subversive, such as a marriage ode by Anthony Alsop that included a rallying cry for the Jacobite cause. On the other hand, Andrew Marvell took inspiration from Horace's "Odes" 1.37 to compose his English masterpiece Horatian Ode upon Cromwell's Return from Ireland, in which subtly nuanced reflections on the execution of Charles I echo Horace's ambiguous response to the death of Cleopatra (Marvell's ode was suppressed in spite of its subtlety and only began to be widely published in 1776). Samuel Johnson took particular pleasure in reading "The Odes". Alexander Pope wrote direct "Imitations" of Horace (published with the original Latin alongside) and also echoed him in "Essays" and The Rape of the Lock. He even emerged as "a quite Horatian Homer" in his translation of the "Iliad". Horace appealed also to female poets, such as Anna Seward ("Original sonnets on various subjects, and odes paraphrased from Horace", 1799) and Elizabeth Tollet, who composed a Latin ode in Sapphic meter to celebrate her brother's return from overseas, with tea and coffee substituted for the wine of Horace's sympotic settings:


title: Horace

text: Horace's "Ars Poetica" is second only to Aristotle's "Poetics" in its influence on literary theory and criticism. Milton recommended both works in his treatise "of Education". Horace's "Satires" and "Epistles" however also had a huge impact, influencing theorists and critics such as John Dryden. There was considerable debate over the value of different lyrical forms for contemporary poets, as represented on one hand by the kind of four-line stanzas made familiar by Horace's Sapphic and Alcaic "Odes" and, on the other, the loosely structured Pindarics associated with the odes of Pindar. Translations occasionally involved scholars in the dilemmas of censorship. Thus Christopher Smart entirely omitted "Odes" and re-numbered the remaining odes. He also removed the ending of "Odes" . Thomas Creech printed "Epodes" and in the original Latin but left out their English translations. Philip Francis left out both the English and Latin for those same two epodes, a gap in the numbering the only indication that something was amiss. French editions of Horace were influential in England and these too were regularly bowdlerized.


title: Horace

text: Most European nations had their own 'Horaces': thus for example Friedrich von Hagedorn was called "The German Horace" and Maciej Kazimierz Sarbiewski "The Polish Horace" (the latter was much imitated by English poets such as Henry Vaughan and Abraham Cowley). Pope Urban VIII wrote voluminously in Horatian meters, including an ode on gout.


title: Horace

text: Horace maintained a central role in the education of English-speaking elites right up until the 1960s. A pedantic emphasis on the formal aspects of language-learning at the expense of literary appreciation may have made him unpopular in some quarters yet it also confirmed his influencea tension in his reception that underlies Byron's famous lines from "Childe Harold" (Canto iv, 77):


title: Horace

text: William Wordsworth's mature poetry, including the preface to "Lyrical Ballads", reveals Horace's influence in its rejection of false ornament and he once expressed "a wish / to meet the shade of Horace...". John Keats echoed the opening of Horace's "Epodes" 14 in the opening lines of "Ode to a Nightingale".


title: Horace

text: The Roman poet was presented in the nineteenth century as an honorary English gentleman. William Thackeray produced a version of "Odes" in which Horace's 'boy' became 'Lucy', and Gerard Manley Hopkins translated the boy innocently as 'child'. Horace was translated by Sir Theodore Martin (biographer of Prince Albert) but minus some ungentlemanly verses, such as the erotic "Odes" and "Epodes" 8 and 12. Edward Bulwer-Lytton produced a popular translation and William Gladstone also wrote translations during his last days as Prime Minister.


title: Horace

text: Edward FitzGerald's "Rubaiyat of Omar Khayyam", though formally derived from the Persian "ruba'i", nevertheless shows a strong Horatian influence, since, as one modern scholar has observed, ""...the quatrains inevitably recall the stanzas of the 'Odes', as does the narrating first person of the world-weary, ageing Epicurean Omar himself, mixing sympotic exhortation and 'carpe diem' with splendid moralising and 'memento mori' nihilism."" Matthew Arnold advised a friend in verse not to worry about politics, an echo of "Odes" , yet later became a critic of Horace's inadequacies relative to Greek poets, as role models of Victorian virtues, observing: ""If human life were complete without faith, without enthusiasm, without energy, Horace...would be the perfect interpreter of human life."" Christina Rossetti composed a sonnet depicting a woman willing her own death steadily, drawing on Horace's depiction of 'Glycera' in "Odes" and Cleopatra in "Odes" . A. E. Housman considered "Odes" , in Archilochian couplets, the most beautiful poem of antiquity and yet he generally shared Horace's penchant for quatrains, being readily adapted to his own elegiac and melancholy strain. The most famous poem of Ernest Dowson took its title and its heroine's name from a line of "Odes" , "Non sum qualis eram bonae sub regno Cynarae", as well as its motif of nostalgia for a former flame. Kipling wrote a famous parody of the "Odes", satirising their stylistic idiosyncrasies and especially the extraordinary syntax, but he also used Horace's Roman patriotism as a focus for British imperialism, as in the story "Regulus" in the school collection "Stalky & Co.", which he based on "Odes" . Wilfred Owen's famous poem, quoted above, incorporated Horatian text to question patriotism while ignoring the rules of Latin scansion. However, there were few other echoes of Horace in the war period, possibly because war is not actually a major theme of Horace's work.


title: Horace

text: Both W.H.Auden and Louis MacNeice began their careers as teachers of classics and both responded as poets to Horace's influence. Auden for example evoked the fragile world of the 1930s in terms echoing "Odes" , where Horace advises a friend not to let worries about frontier wars interfere with current pleasures.


title: Horace

text: The American poet Robert Frost echoed Horace's "Satires" in the conversational and sententious idiom of some of his longer poems, such as "The Lesson for Today" (1941), and also in his gentle advocacy of life on the farm, as in "Hyla Brook" (1916), evoking Horace's "fons Bandusiae" in "Ode" . Now at the start of the third millennium, poets are still absorbing and re-configuring the Horatian influence, sometimes in translation (such as a 2002 English/American edition of the "Odes" by thirty-six poets) and sometimes as inspiration for their own work (such as a 2003 collection of odes by a New Zealand poet).


title: Horace

text: Horace's "Epodes" have largely been ignored in the modern era, excepting those with political associations of historical significance. The obscene qualities of some of the poems have repulsed even scholars yet more recently a better understanding of the nature of Iambic poetry has led to a re-evaluation of the "whole" collection. A re-appraisal of the "Epodes" also appears in creative adaptations by recent poets (such as a 2004 collection of poems that relocates the ancient context to a 1950s industrial town).


title: Horace

text: The Oxford Latin Course textbooks use the life of Horace to illustrate an average Roman's life in the late Republic to Early Empire.


title: Nyquist–Shannon sampling theorem

text: The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of "samples" to capture all the information from a continuous-time signal of finite bandwidth.


title: Nyquist–Shannon sampling theorem

text: Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are band-limited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.


title: Nyquist–Shannon sampling theorem

text: Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known (see below and compressed sensing). In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions. The fidelity of these reconstructions can be verified and quantified utilizing Bochner's theorem.


title: Nyquist–Shannon sampling theorem

text: The name "Nyquist–Shannon sampling theorem" honours Harry Nyquist and Claude Shannon, but the theorem was also previously discovered by E. T. Whittaker (published in 1915) and Shannon cited Whittaker's paper in his work. The theorem is thus also known by the names "Whittaker–Shannon sampling theorem", "Whittaker–Shannon", and "Whittaker–Nyquist–Shannon", and may also be referred to as the "cardinal theorem of interpolation".


title: Nyquist–Shannon sampling theorem

text: Sampling is a process of converting a signal (for example, a function of continuous time or space) into a sequence of values (a function of discrete time or space). Shannon's version of the theorem states:


title: Nyquist–Shannon sampling theorem

text: A sufficient sample-rate is therefore anything larger than formula_1 samples per second. Equivalently, for a given sample rate formula_2, perfect reconstruction is guaranteed possible for a bandlimit formula_3.


title: Nyquist–Shannon sampling theorem

text: When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing. Modern statements of the theorem are sometimes careful to explicitly state that formula_4 must contain no sinusoidal component at exactly frequency formula_5 or that formula_6 must be strictly less than ½ the sample rate. The threshold formula_1 is called the Nyquist rate and is an attribute of the continuous-time input formula_4 to be sampled. The sample rate must exceed the Nyquist rate for the samples to suffice to represent formula_9  The threshold formula_10 is called the Nyquist frequency and is an attribute of the sampling equipment. All meaningful frequency components of the properly sampled formula_4 exist below the Nyquist frequency. The condition described by these inequalities is called the "Nyquist criterion", or sometimes the "Raabe condition". The theorem is also applicable to functions of other domains, such as space, in the case of a digitized image. The only change, in the case of other domains, is the units of measure attributed to formula_12 formula_13 and formula_14


title: Nyquist–Shannon sampling theorem

text: The symbol formula_15 is customarily used to represent the interval between samples and is called the "sample period" or "sampling interval". The samples of function formula_4 are commonly denoted by formula_17 (alternatively formula_18 in older signal processing literature), for all integer values of formula_19  Another convenient definition is formula_20 which preserves the energy of the signal as formula_21 varies.


title: Nyquist–Shannon sampling theorem

text: A mathematically ideal way to interpolate the sequence involves the use of sinc functions. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original location of the sample formula_22 with the amplitude of the sinc function scaled to the sample value, formula_23 Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method is to convolve one sinc function with a series of Dirac delta pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as "interpolation error".


title: Nyquist–Shannon sampling theorem

text: Practical digital-to-analog converters produce neither scaled and delayed sinc functions, nor ideal Dirac pulses. Instead they produce a piecewise-constant sequence of scaled and delayed rectangular pulses (the zero-order hold), usually followed by a lowpass filter (called an "anti-imaging filter") to remove spurious high-frequency replicas (images) of the original baseband signal.


title: Nyquist–Shannon sampling theorem

text: the Poisson summation formula indicates that the samples, formula_27, of formula_4 are sufficient to create a periodic summation of formula_25. The result is:


title: Nyquist–Shannon sampling theorem

text: which is a periodic function and its equivalent representation as a Fourier series, whose coefficients are formula_31 This function is also known as the discrete-time Fourier transform (DTFT) of the sample sequence.


title: Nyquist–Shannon sampling theorem

text: As depicted, copies of formula_25 are shifted by multiples of the sampling rate formula_2 and combined by addition. For a band-limited function  formula_34  and sufficiently large formula_13 it is possible for the copies to remain distinct from each other. But if the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous formula_36 Any frequency component above formula_10 is indistinguishable from a lower-frequency component, called an "alias", associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), formula_4 is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a lowpass filter, and in this application it is called an anti-aliasing filter.


title: Nyquist–Shannon sampling theorem

text: When there is no overlap of the copies (also known as "images") of formula_25, the formula_40 term of can be recovered by the product:


title: Nyquist–Shannon sampling theorem

text: All that remains is to derive the formula for reconstruction. formula_45 need not be precisely defined in the region formula_46 because formula_47 is zero in that region. However, the worst case is when formula_48 the Nyquist frequency. A function that is sufficient for that and all less severe cases is:


title: Nyquist–Shannon sampling theorem

text: Poisson shows that the Fourier series in produces the periodic summation of formula_25, regardless of formula_2 and formula_6. Shannon, however, only derives the series coefficients for the case formula_60. Virtually quoting Shannon's original paper:


title: Nyquist–Shannon sampling theorem

text: Shannon's proof of the theorem is complete at that point, but he goes on to discuss reconstruction via sinc functions, what we now call the Whittaker–Shannon interpolation formula as discussed above. He does not derive or prove the properties of the sinc function, but these would have been familiar to engineers reading his works at the time, since the Fourier pair relationship between rect (the rectangular function) and sinc was well known.


title: Nyquist–Shannon sampling theorem

text: As in the other proof, the existence of the Fourier transform of the original signal is assumed, so the proof does not say whether the sampling theorem extends to bandlimited stationary random processes.


title: Nyquist–Shannon sampling theorem

text: The sampling theorem is usually formulated for functions of a single variable. Consequently, the theorem is directly applicable to time-dependent signals and is normally formulated in that context. However, the sampling theorem can be extended in a straightforward way to functions of arbitrarily many variables. Grayscale images, for example, are often represented as two-dimensional arrays (or matrices) of real numbers representing the relative intensities of pixels (picture elements) located at the intersections of row and column sample locations. As a result, images require two independent variables, or indices, to specify each pixel uniquely—one for the row, and one for the column.


title: Nyquist–Shannon sampling theorem

text: Color images typically consist of a composite of three separate grayscale images, one to represent each of the three primary colors—red, green, and blue, or "RGB" for short. Other colorspaces using 3-vectors for colors include HSV, CIELAB, XYZ, etc. Some colorspaces such as cyan, magenta, yellow, and black (CMYK) may represent color by four dimensions. All of these are treated as vector-valued functions over a two-dimensional sampled domain.


title: Nyquist–Shannon sampling theorem

text: Similar to one-dimensional discrete-time signals, images can also suffer from aliasing if the sampling resolution, or pixel density, is inadequate. For example, a digital photograph of a striped shirt with high frequencies (in other words, the distance between the stripes is small), can cause aliasing of the shirt when it is sampled by the camera's image sensor. The aliasing appears as a moiré pattern. The "solution" to higher sampling in the spatial domain for this case would be to move closer to the shirt, use a higher resolution sensor, or to optically blur the image before acquiring it with the sensor using an optical low-pass filter.


title: Nyquist–Shannon sampling theorem

text: Another example is shown to the right in the brick patterns. The top image shows the effects when the sampling theorem's condition is not satisfied. When software rescales an image (the same process that creates the thumbnail shown in the lower image) it, in effect, runs the image through a low-pass filter first and then downsamples the image to result in a smaller image that does not exhibit the moiré pattern. The top image is what happens when the image is downsampled without low-pass filtering: aliasing results.


title: Nyquist–Shannon sampling theorem

text: The sampling theorem applies to camera systems, where the scene and lens constitute an analog spatial signal source, and the image sensor is a spatial sampling device. Each of these components is characterized by a modulation transfer function (MTF), representing the precise resolution (spatial bandwidth) available in that component. Effects of aliasing or blurring can occur when the lens MTF and sensor MTF are mismatched. When the optical image which is sampled by the sensor device contains higher spatial frequencies than the sensor, the under sampling acts as a low-pass filter to reduce or eliminate aliasing. When the area of the sampling spot (the size of the pixel sensor) is not large enough to provide sufficient spatial anti-aliasing, a separate anti-aliasing filter (optical low-pass filter) may be included in a camera system to reduce the MTF of the optical image. Instead of requiring an optical filter, the graphics processing unit of smartphone cameras performs digital signal processing to remove aliasing with a digital filter. Digital filters also apply sharpening to amplify the contrast from the lens at high spatial frequencies, which otherwise falls off rapidly at diffraction limits.


title: Nyquist–Shannon sampling theorem

text: The sampling theorem also applies to post-processing digital images, such as to up or down sampling. Effects of aliasing, blurring, and sharpening may be adjusted with digital filtering implemented in software, which necessarily follows the theoretical principles.


title: Nyquist–Shannon sampling theorem

text: To illustrate the necessity of formula_83, consider the family of sinusoids generated by different values of formula_84 in this formula:


title: Nyquist–Shannon sampling theorem

text: . That sort of ambiguity is the reason for the "strict" inequality of the sampling theorem's condition.


title: Nyquist–Shannon sampling theorem

text: That is, a sufficient no-loss condition for sampling signals that do not have baseband components exists that involves the "width" of the non-zero frequency interval as opposed to its highest frequency component. See "Sampling (signal processing)" for more details and examples.


title: Nyquist–Shannon sampling theorem

text: For example, in order to sample the FM radio signals in the frequency range of 100–102 MHz, it is not necessary to sample at 204 MHz (twice the upper frequency), but rather it is sufficient to sample at 4 MHz (twice the width of the frequency interval).


title: Nyquist–Shannon sampling theorem

text: A bandpass condition is that "X"("f") = 0, for all nonnegative "f" outside the open band of frequencies:


title: Nyquist–Shannon sampling theorem

text: for some nonnegative integer "N". This formulation includes the normal baseband condition as the case "N"=0.


title: Nyquist–Shannon sampling theorem

text: The corresponding interpolation function is the impulse response of an ideal brick-wall bandpass filter (as opposed to the ideal brick-wall lowpass filter used above) with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses:


title: Nyquist–Shannon sampling theorem

text: Other generalizations, for example to signals occupying multiple non-contiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse. That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.


title: Nyquist–Shannon sampling theorem

text: The sampling theory of Shannon can be generalized for the case of nonuniform sampling, that is, samples not taken equally spaced in time. The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction.


title: Nyquist–Shannon sampling theorem

text: The general theory for non-baseband and nonuniform samples was developed in 1967 by Henry Landau. He proved that the average sampling rate (uniform or otherwise) must be twice the "occupied" bandwidth of the signal, assuming it is "a priori" known what portion of the spectrum was occupied.


title: Nyquist–Shannon sampling theorem

text: In the late 1990s, this work was partially extended to cover signals whose amount of occupied bandwidth was known, but the actual occupied portion of the spectrum was unknown. In the 2000s, a complete theory was developed


title: Nyquist–Shannon sampling theorem

text: (see the section Sampling below the Nyquist rate under additional restrictions below) using compressed sensing. In particular, the theory, using signal processing language, is described in this 2009 paper. They show, among other things, that if the frequency locations are unknown, then it is necessary to sample at least at twice the Nyquist criteria; in other words, you must pay at least a factor of 2 for not knowing the location of the spectrum. Note that minimum sampling requirements do not necessarily guarantee stability.


title: Nyquist–Shannon sampling theorem

text: The Nyquist–Shannon sampling theorem provides a sufficient condition for the sampling and reconstruction of a band-limited signal. When reconstruction is done via the Whittaker–Shannon interpolation formula, the Nyquist criterion is also a necessary condition to avoid aliasing, in the sense that if samples are taken at a slower rate than twice the band limit, then there are some signals that will not be correctly reconstructed. However, if further restrictions are imposed on the signal, then the Nyquist criterion may no longer be a necessary condition.


title: Nyquist–Shannon sampling theorem

text: A non-trivial example of exploiting extra assumptions about the signal is given by the recent field of compressed sensing, which allows for full reconstruction with a sub-Nyquist sampling rate. Specifically, this applies to signals that are sparse (or compressible) in some domain. As an example, compressed sensing deals with signals that may have a low over-all bandwidth (say, the "effective" bandwidth "EB"), but the frequency locations are unknown, rather than all together in a single band, so that the passband technique does not apply. In other words, the frequency spectrum is sparse. Traditionally, the necessary sampling rate is thus 2"B". Using compressed sensing techniques, the signal could be perfectly reconstructed if it is sampled at a rate slightly lower than 2"EB". With this approach, reconstruction is no longer given by a formula, but instead by the solution to a linear optimization program.


title: Nyquist–Shannon sampling theorem

text: Another example where sub-Nyquist sampling is optimal arises under the additional constraint that the samples are quantized in an optimal manner, as in a combined system of sampling and optimal lossy compression. This setting is relevant in cases where the joint effect of sampling and quantization is to be considered, and can provide a lower bound for the minimal reconstruction error that can be attained in sampling and quantizing a random signal. For stationary Gaussian random signals, this lower bound is usually attained at a sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimal quantization.


title: Nyquist–Shannon sampling theorem

text: The sampling theorem was implied by the work of Harry Nyquist in 1928, in which he showed that up to 2"B" independent pulse samples could be sent through a system of bandwidth "B"; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step-response sine integral; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a "Küpfmüller filter" (but seldom so in English).


title: Nyquist–Shannon sampling theorem

text: The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon. V. A. Kotelnikov published similar results in 1933, as did the mathematician


title: Nyquist–Shannon sampling theorem

text: E. T. Whittaker in 1915, J. M. Whittaker in 1935, and Gabor in 1946 ("Theory of communication"). In 1999, the Eduard Rhein Foundation awarded Kotelnikov their Basic Research Award "for the first theoretically exact formulation of the sampling theorem".


title: Nyquist–Shannon sampling theorem

text: In 1948 and 1949, Claude E. Shannon published – 16 years after Vladimir Kotelnikov – the two revolutionary articles in which he founded the information theory. In Shannon 1948 the sampling theorem is formulated as “Theorem 13”: Let "f"("t") contain no frequencies over W. Then


title: Nyquist–Shannon sampling theorem

text: It was not until these articles were published that the theorem known as “Shannon’s sampling theorem” became common property among communication engineers, although Shannon himself writes that this is a fact which is common knowledge in the communication art. A few lines further on, however, he adds: "but in spite of its evident importance, [it] seems not to have appeared explicitly in the literature of communication theory".


title: Nyquist–Shannon sampling theorem

text: Others who have independently discovered or played roles in the development of the sampling theorem have been discussed in several historical articles, for example, by Jerri and by Lüke. For example, Lüke points out that H. Raabe, an assistant to Küpfmüller, proved the theorem in his 1939 Ph.D. dissertation; the term "Raabe condition" came to be associated with the criterion for unambiguous representation (sampling rate greater than twice the bandwidth). Meijering mentions several other discoverers and names in a paragraph and pair of footnotes:


title: Nyquist–Shannon sampling theorem

text: As pointed out by Higgins [135], the sampling theorem should really be considered in two parts, as done above: the first stating the fact that a bandlimited function is completely determined by its samples, the second describing how to reconstruct the function using its samples. Both parts of the sampling theorem were given in a somewhat different form by J. M. Whittaker [350, 351, 353] and before him also by Ogura [241, 242]. They were probably not aware of the fact that the first part of the theorem had been stated as early as 1897 by Borel [25]. As we have seen, Borel also used around that time what became known as the cardinal series. However, he appears not to have made the link [135]. In later years it became known that the sampling theorem had been presented before Shannon to the Russian communication community by Kotel'nikov [173]. In more implicit, verbal form, it had also been described in the German literature by Raabe [257]. Several authors [33, 205] have mentioned that Someya [296] introduced the theorem in the Japanese literature parallel to Shannon. In the English literature, Weston [347] introduced it independently of Shannon around the same time.


title: Nyquist–Shannon sampling theorem

text: Exactly how, when, or why Harry Nyquist had his name attached to the sampling theorem remains obscure. The term "Nyquist Sampling Theorem" (capitalized thus) appeared as early as 1959 in a book from his former employer, Bell Labs, and appeared again in 1963, and not capitalized in 1965. It had been called the "Shannon Sampling Theorem" as early as 1954, but also just "the sampling theorem" by several other books in the early 1950s.


title: Nyquist–Shannon sampling theorem

text: In 1958, Blackman and Tukey cited Nyquist's 1928 article as a reference for "the sampling theorem of information theory", even though that article does not treat sampling and reconstruction of continuous signals as others did. Their glossary of terms includes these entries:


title: Nyquist–Shannon sampling theorem

text: When Shannon stated and proved the sampling theorem in his 1949 article, according to Meijering, "he referred to the critical sampling interval formula_94 as the "Nyquist interval" corresponding to the band "W", in recognition of Nyquist’s discovery of the fundamental importance of this interval in connection with telegraphy". This explains Nyquist's name on the critical interval, but not on the theorem.


title: Nyquist–Shannon sampling theorem

text: According to the OED, this may be the origin of the term "Nyquist rate". In Black's usage, it is not a sampling rate, but a signaling rate.


title: Kedarnath

text: Kedarnath is a town and Nagar Panchayat in Rudraprayag district of Uttarakhand, India, known primarily for the Kedarnath Temple. It is approximately 86 kilometres from Rudraprayag, the district headquarter. Kedarnath is the most remote of the four Chota Char Dham pilgrimage sites. It is located in the Himalayas, about above sea level near the Chorabari Glacier, which is the source of the Mandakini river. The town is flanked by snow-capped peaks, most prominently the Kedarnath Mountain. The nearest road head is at Gaurikund about 16 km away. The town suffered extensive destruction during June 2013 from the Flash Floods caused by torrential rains in Uttarakhand state.


title: Kedarnath

text: The name "Kedarnath" means "the Lord of the Field". It is derived from the Sanskrit words "kedara" ("field") and "natha" ("lord"). The text "Kashi Kedara Mahatmya" states that it is so-called because "the crop of liberation" grows here.


title: Kedarnath

text: Kedarnath has been a pilgrimage centre since ancient times.The temple's construction is credited to the Pandava brothers mentioned in the "Mahabharata". However, the "Mahabharata" does not mention any place called Kedarnath. One of the earliest references to Kedarnath occurs in the "Skanda Purana" (c. 7th-8th century), which names Kedara (Kedarnath) as the place where Lord Shiva released the holy waters of Ganga from his matted hair, resulting in the formation of the Ganges River.


title: Kedarnath

text: According to the hagiographies based on Madhava's "Sankshepa-Shankara-Vijaya", the 8th century philosopher Adi Shankaracharya died near the Kedarnath mountains; although other hagiographies, based on Anandagiri's "Prachina-Shankara-Vijaya", state that he died at Kanchipuram. The ruins of a monument marking the purported resting place of Adi Shankaracharya are located at Kedarnath. Kedarnath was definitely a prominent pilgrimage centre by the 12th century when it is mentioned in "Kritya-kalpataru" written by the Gahadavala minister Bhatta Lakshmidhara.


title: Kedarnath

text: Kedarnath is located at a distance of 223 km from Rishikesh in Uttarakhand and close to the source of the Mandakini River at the height of above sea level. The township is built on a barren stretch of land on the shores of Mandakini river. The surrounding scenery of the Himalayas and green pastures makes it a very attractive place for pilgrimage and trekking. Behind the town and the Kedarnath Temple, stands the majestic Kedarnath peak at , the Kedar Dome at and other peaks of the range.


title: Kedarnath

text: India census, Kedarnath has a population of 612 as for 2011. Males constitute 99% of the population and females 1%. Kedarnath has an average literacy rate of 63%: male literacy is 63%, and female literacy is 36%. Kedarnath, none of the population is under six years of age. The floating population from May to October every year is more than 5000 per day with the pilgrim influx rising upto 10 lacs (1 million) in 2022.


title: Kedarnath

text: The Kedarnath Temple is closed during the winter months due to heavy snowfall. For six months, from November to April, the palanquin with the Utsava Murti (Idol) of Kedarnath and of the Madhyamaheshwar Temple is brought to the Omkareshwar Temple in Ukhimath, near Guptakashi. Priests and other summer-time residents also move to nearby villages to cope with the winter. Around 360 families of the Tirtha Purohit of 55 villages and other nearby villages are dependent on the town for livelihood.


title: Kedarnath

text: According to the Köppen-Geiger climate classification system, Kedarnath's climate is monsoon-influenced subarctic climate ("Dwc"), bordering a uniform rainfall subarctic climate ("Dfc") with mild, rainy summers and cold, snowy winters.


title: Kedarnath

text: On 16 June 2013, at approximately 7:30 P.M., a landslide occurred near the Kedarnath Temple. Water from the Chorabari Tal or Gandhi Tal down Mandakini River also washed away everything in its path at about 8:30 p.m. On 17 June 2013 at approximately 6:40 a.m., waters rushed down from Chorabari Tal or Gandhi Sarovar, bringing along with its flow, a huge amount of silt, rocks, and boulders. A huge boulder got stuck behind the Kedarnath Temple, protecting it from the ravages of the flood. The flood-water gushed on both sides of the temple, destroying everything in its path. Thus in the middle of the pilgrimage season, torrential rains, cloud bursts, and resulting flash floods nearly destroyed the town of Kedarnath. The town was the worst affected area by the floods. Thousands of people were killed, and thousands of others (mostly pilgrims) were reported missing or stranded due to landslides around Kedarnath. Although the surrounding area and compound of the Kedarnath Temple were destroyed, the temple itself survived.


title: Kedarnath

text: The rescue operation resulted in more than 100,000 people being airlifted with the help of mainly the Private Helicopter Operators, who began the rescue mission voluntarily without any clear directives from the State Government or the Ministry of Defence. The Indian Army and Indian Air Force helicopters arrived much after the Private Helicopter Operators had already begun the massive air-rescue mission. Dare-devil helicopter pilots, mostly ex-Indian Air Force and Ex-Army Aviation officers, flew relentlessly. Capt. Unni Krishnan from Prabhatam Aviation & Capt. Bhatnagar from Premair were few such outstanding pilots who landed at the Sh. Kedarnath Ji right-ridge at dusk hours - 1910hrs (almost 35 minutes after 'sunset') to pick-up the 'last' batch of survivors for the day. The NDRF represented by a commandant, and another junior officer arrived at the 'right-ridge' of the town bordering the Mandakini River, unprepared with a malfunctioning satellite phone. However, the next day, they brought in more men and supplies. The first Indian Army officer to arrive at the Sh. Kedarnath Ji 'right-ridge' was a Captain from the Assam Rifles Regiment. He single-handedly displayed exemplary courage in rescuing many survivors by climbing up steep slopes and fractures along the Rambara ridge-line. The Indian Army later launched a massive rescue effort with thousands of its brave-heart men and vital equipment. The Indo-Tibetan Border Police and the reserve battalions of the Uttarakhand Police displayed outstanding courage in the rescue mission. A Eurocopter AS350 B3 helicopter, each, of the private helicopter operators - Prabhatam Aviation & Simm Samm Aviation, were lost during the rescue mission without any reported casualties. An Indian Air Force helicopter (Mi 17) also crashed during this exercise, killing all 20 people on board (all of them were soldiers involved in relief and rescue work). The Air Force dropped logs to build pyres for mass cremations of the victims. It was reported that previously uncollected bodies were still being found one year after the tragedy.


title: Kedarnath

text: After the floods, the Government of India decided to provide a flood-free infrastructure plan for the town. This involves:


title: Kedarnath

text: Other than the Kedarnath Temple, on the eastern side of the town is Bhairavnath Temple, and the deity of this temple, Bhairavnath, is believed to protect the town during the winter months.


title: Kedarnath

text: About 6 km upstream from the town, lies Chorabari Tal, a lake cum glacier also called Gandhi Sarovar. Near Kedarnath, there is a cliff called Bhairav Jhamp. Other places of interest include the Kedarnath Wildlife Sanctuary, Adi Shankaracharya Samadhi, and Rudra meditation cave.


title: Swedish Army

text: Svea Life Guards dates back to the year 1521, when the men of Dalarna chose 16 young able men as body guards for the insurgent nobleman Gustav Vasa in the Swedish War of Liberation against the Danish-dominated Union of Kalmar, thus making the present-day Life Guards one of the world's oldest regiments still on active duty.


title: Swedish Army

text: In 1901, Sweden introduced conscription. The conscription system was abolished in 2010 but reinstated in 2017.


title: Swedish Army

text: The peace-time organisation of the Swedish Army is divided into a number of regiments for the different branches. The number of active regiments has been reduced since the end of the Cold War. However the Swedish Army has begun to expand once again. The regiment forms training organizations that train the various battalions of the army and home guard.


title: Swedish Army

text: The Swedish Armed Forces recently underwent a transformation from conscription-based recruitment to a professional defence organisation. This is part of a larger goal to abandon the mass army from the Cold War and develop an army better suited to modern maneuver warfare and at the same time retain a higher readiness. Since 2014, the Swedish army has had around 50,000 soldiers in either full-time or part-time duty, with eight mechanized infantry battalions instantly available at any time and the full force of 71 battalions ready to be deployed within one week. The regular army consists of 8 mechanised maneuver battalions, 19 support battalions of different kinds including artillery battalions, anti-aircraft battalions, combat engineer battalions, rangers, logistics battalions and 4 reserve heavy armoured battalions and 40 territorial defence battalions. The battalion is the core unit but all units are completely modular and can be arranged in combat teams from company to brigade level with different units depending on the task. There are a total of 6 permanent staffs under the central command capable of handling large battlegroups, 4 regional staffs and 2 brigade staffs.


title: Swedish Army

text: Until 1937 the King was the formal Head of the Army, and until 1974 of the Armed Forces at large. In 1937, the Chief of the Army position and headquarters staff (, "CA") was created to lead the Army in peacetime. Following a larger reorganisation of the Swedish Armed Forces in 1994, CA ceased to exist as an independent agency, and all of the Armed Forces was centralized under the Supreme Commander ("ÖB"). Instead, the post Chief of Army Staff () was created at the then newly instituted Swedish Armed Forces Headquarters ("HKV").


title: Swedish Army

text: In 1998, the Swedish Armed Forces was yet again reorganized. Most of the duties of the Chief of Army Staff were transferred to the newly instituted post of "Inspector General of the Army" (). The post is similar to that of the "Inspector General of the Swedish Navy" () and the "Inspector General of the Swedish Air Force" (), later renamed to "Inspector of the Army" (). In 2014, the Chief of Army (, "AC") position was reinstated.


title: Swedish Army

text: Swedish Army regiments are tasked with training conscripts for the operational battalions of the army's rapid reaction organisation. The Gotland Regiment is the only regiment, which also trains Home Guard troops. The currently active regiments and their main peacetime subordinate units are:


title: Swedish Army

text: The Swedish Army distinguishes an administrative from an operational structure. The administrative structure includes the peacetime depot units ("depåförband"), which are responsible for the training, equipment and combat readiness of the forces. Depot units are the army regiments, the air force flotillas, the naval flotillas or the armed forces centers. The depot units generate wartime units ("krigsförband"), which are transferred to their respective wartime formations, so normally the Swedish army regiments are composed of administrative and training companies during peacetime and their battalions are only formed for exercises and during wartime. The operational units in their entirety are known as the operational organisation of the armed forces ("Försvarsmaktens Insatsorganisation" ("FM IO" or just "IO", followed by the year it has been introduced in, as "IO 2018" or just "IO 18" for example) in Swedish). The operational organisation of the army plans for the training regiments to form two combat brigades (2nd and 3rd) and a number of independent combat battalions within a few days. This plan, however, was considered by the Supreme Commander to be impossible due to the economic situation at the time.


title: Swedish Army

text: In 2013, the Armed Forces issued a statement saying that the reorganisation would only suffice for a reasonable defence of Swedish territory for one week. The force was to include the following units:


title: Swedish Army

text: The following Armed Forces' establishments provide additional units for the Rapid Reaction Organisation:


title: Swedish Army

text: The Home Guard (Hemvärnet) consists of 40 battalions with a total of 22,000 men. Many of the soldiers have served abroad with regular army units.


title: Swedish Army

text: The Northern Military Region is responsible for supporting Home Guard units in, Jämtland County, Norrbotten County, Västerbotten County, and Västernorrland County with training and administrative resources.


title: Swedish Army

text: The Central Military Region is responsible for supporting Home Guard units in Dalarna County, Gävleborg County, Södermanland County, Stockholm County, Uppsala County, and Västmanland County with training and administrative resources.


title: Swedish Army

text: The Southern Military Region is responsible for supporting Home Guard units in Blekinge County, Jönköping County, Kalmar County, Kronoberg County, Skåne County, and Östergötland County with training and administrative resources.


title: Swedish Army

text: The Western Military Region is responsible for supporting Home Guard units in Halland County, Värmland County, Västra Götaland County, and Örebro County with training and administrative resources.


title: Swedish Army

text: The Home Guard battalion on the island of Gotland falls under the Gotland Regiment, which also commands the active 181st Armored Battalion.


title: Swedish Army

text: The army on active service during most of the 20th century only consisted of conscript receiving their basic training (at the most one year for privates), and conscripts called up for refreshers (at the most one month for privates) with intervals of at least 4–6 years. During WWII smaller or larger parts of the wartime establishment were also called up for periodical stand-by duty (up to 6 months). The figures for total strength below refer to the total force which could be mobilised.


title: Swedish Army

text: Between the introduction of universal conscription in 1902 until the start of World War II, the army was usually maintained at a consistent strength of 100,000 men, with two-thirds of the force being conscripts for two years. From 1942 onwards, the Swedish government embarked upon a massive and ambitious militarisation program in which conscription was strictly enforced and compulsory service was extended. The basic training for privates was set at 12 months, for future conscript sergeants intended to become platoon leaders (mainly sixth form graduates) 18 months, and for future conscript officers 24 months. These periods were gradually reduced, and the ranks were raised, until it was the same period for all conscripts shortly before conscription was suspended in 2009. This combined with propaganda about conscription being a part of social duty and defending the Swedish principle of "folkhemmet", led to an army a size of about 700,000 active duty soldiers that could be mobilised in late 1945. Since the late winter of 1945 the size of the army was slowly decreased as entire reserve battalions and brigades were gradually demobilised, and by late 1947 the size of the army was around 170,000 soldiers who could be mobilised and was planned to stabilise at such a quantity of personnel.


title: Swedish Army

text: However, due to the rise in tensions between the East and West over the political landscape of Europe, the threat from the Soviet Union in 1949 and 1950, coinciding with the start of the Cold War, led to a return to the militaristic policy by the Swedish government. From 1950 until around 1976 the size of the army was at an average of 250,000 soldiers with a peak of 400,000 mobilisable soldiers during the late 1950s and early 1960s. The rules were badly enforced, but dodging the draft was punishable with imprisonment.


title: Swedish Army

text: During the 1980s the size of the army was around 180,000 soldiers and was slowly increased as time progressed until around 1988. The end of the Cold War led to a massive restructuring of the Swedish Army. Every year after 1988, the Army discharged around 40,000 conscripts and recruited only 20,000, so that by 1995 the size was down to 80,000 soldiers. Around this time the compulsory service obligation was further reduced to 10 months, reserve service became more flexible, and changes made in enforcement so that forceful enforcement became withdrawn as policy. By 2004 the size of the Swedish Army was down to 60,000 soldiers, and in 2013, three years after the end of conscription, the size was at an all-time low of just 16,000 soldiers, though the army plans to reach a level of 50,000 professional soldiers by 2020, mostly through a large media campaigns. A number of previously disbanded regiments will also be re-raised (Dalarna Regiment, Västernorrland Regiment, Norrland Dragoon Regiment, and Bergslagen Artillery Regiment) with the Jämtland Ranger Regiment re-raised as a battalion of Västernorrland Regiment.


title: Swedish Army

text: From the 17th century until 1901, Swedish Army recruitment was based upon the allotment system. In 1812, conscription was introduced for all males between age 20 and 25 to serve in the armed forces twelve days a year, increased in 1858 to four weeks per two years. The allotment system was abolished in 1901 and replaced with universal conscription for all males. All personnel were drafted as conscripts for a year of conscription, after which the unit the soldier trained with was put in reserve. Upon completion of conscript service with sufficient service marks, conscripts are eligible to apply for commissioned officer training, NCO/Warrant Officer or from 2007 stay in the Army as a professional private, mainly to be employed in the Nordic Battle Group. The army has employed soldiers for UN service on short time contracts since the 1950s for service abroad.


title: Swedish Army

text: From July 2010 until 2017, the Swedish Army was an all-professional fighting force. The government announced on 2 March 2017 that conscription was going to be reinstated. Of a pool of around 13,000 men and women born 1999, 4,000 were going to be selected for conscription starting January 2018. The government stated that the number of conscripts may increase in response to foreign events.


title: Japan men's national volleyball team

text: The Japan men's national volleyball team () represents Japan in international volleyball competitions and friendly matches, governed by Japan Volleyball Association. Their nickname is ""RYUJIN NIPPON (龍神 NIPPON)"", meaning "Japanese Dragon God" or "Dragon God of Japan" in Japanese. Japan won a bronze medal at the Tokyo 1964 Olympic Games, followed by silver at Mexico 1968 and a gold at Munich 1972. The years after this have been less successful. Japan missed three successive Olympic Games before returning at Beijing 2008, where they failed to win a match. Japan then failed to qualify for the London 2012 Olympic Games and also missed Rio 2016 after finishing seventh at the World Olympic Qualification Tournament in Tokyo.


title: Japan men's national volleyball team

text: They also failed to qualify for the 2014 FIVB World Championship the first edition they had missed in 54 years. They had previously won FIVB World Championship bronze medals in 1970 and 1974. Japan silver medalists at the 1969 and 1977 World Cup. Japan has been less successful in the World League era. The team has won the Asian Championships nine times and the Asian Games eight times.


title: Japan men's national volleyball team

text: In August 2021, at 2020 Summer Olympic Games in Tokyo, Japan team advanced to the Quarterfinals round and eventually finished at seventh place, the highest rank since 29 years.


title: Japan men's national volleyball team

text: Primary sponsors include: main sponsors like Shiseido other sponsors: Japanet, Molten, All Nippon Airways, Suntory, JTEKT, Nisshin Steel, Nippon Life, Marudai, Mizuno, Meiji Seika, Descente, Mikasa and Hisamitsu Pharmaceutical.


title: Japan men's national volleyball team

text: Tokyo Metropolitan Gymnasium and Yoyogi National Gymnasium Japan national team training and hosting venues.


title: Democratic republic

text: A democratic republic is a form of government operating on principles adopted from a republic and a democracy. As a cross between two exceedingly similar systems, democratic republics may function on principles shared by both republics and democracies.


title: Democratic republic

text: While not all democracies are republics (constitutional monarchies) and not all republics are democracies, common definitions of the terms "democracy" and "republic" often feature overlapping concerns, suggesting that many democracies function as republics, and many republics operate on democratic principles, as shown by these definitions from the "Oxford English Dictionary":


title: Democratic republic

text: Eugene Volokh of the UCLA School of Law notes that the United States exemplifies the varied nature of a constitutional republica country where some decisions (often local) are made by direct democratic processes, while others (often federal) are made by democratically elected representatives. As with many large systems, US governance is incompletely described by any single term. It also employs the concept, for instance, of a constitutional republic in which a court system is involved in matters of jurisprudence. However, these republics were, indeed, democratic republics that used Soviet democracy, a complicated form of indirect democracy.


title: Democratic republic

text: As with other democracies, not all persons in a democratic republic are necessarily citizens, and not all citizens are necessarily entitled to vote. Suffrage is commonly restricted by criteria such as voting age and sometimes by felony or imprisonment status.


title: Democratic republic

text: Prior to the American Revolution in what is now the United States—and before the coming of age of the "crowned republics" of constitutional monarchies in the U.K. and other European countries—"democracy" and "republic" were "used more or less interchangeably", and the concepts associated with representative democracy (and hence with a democratic republic) are suggested by John Adams (writing in 1784):


title: Democratic republic

text: The Republic of China (Taiwan) claims to be the oldest of Asia's democratic republics, though its recent history of democratic process is largely linked only to Taiwan.


title: Democratic republic

text: Likewise, Africa's oldest democratic republic, Liberia (formed in 1822), has had its political stability rocked by periodic violence and coups.


title: Democratic republic

text: Starting in the 20th century after World War II, many countries used the term "democratic republic" in their official names—most of which were Marxist-Leninist, or socialist, one-party states—that did not allow political opposition, free press or other democratic norms and institutions.


title: Democratic republic

text: These include states no longer in existence or who have changed their governmental systems and official names, (almost all Marxist-Leninist):<br>


title: Democratic republic

text: the German Democratic Republic (aka East Germany), the Somali Democratic Republic, the Democratic Republic of Vietnam (aka North Vietnam), the People's Democratic Republic of Yemen (aka South Yemen), the Democratic Republic of Afghanistan and the People's Democratic Republic of Ethiopia.


title: Democratic republic

text: States which (as of 2022) use the term "Democratic Republic" in their official names also include many that do not hold free elections and have been rated as "undemocratic" or "unfree" by organizations that gave such ratings. Algeria, Democratic Republic of the Congo, Ethiopia, North Korea, Laos, and Nepal, do not hold free elections and are rated as undemocratic "hybrid regimes" or "authoritarian regimes" by the Economist Intelligence Unit's Democracy Index, and "not free" by (the U.S.-based, U.S.-government-funded non-governmental organization) Freedom House.


title: Democratic republic

text: In addition there are a few countries which use the term "Democratic Republic" in the name and have a good track of holding free or relatively free general elections and were rated "flawed democracy" or "full democracy" in the Democracy Index, such as the Democratic Republic of Timor-Leste, the Democratic Republic of São Tomé and Príncipe and the Democratic Socialist Republic of Sri Lanka.


title: Percentage

text: In mathematics, a percentage (from , "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number (pure number); it has no unit of measurement.


title: Percentage

text: For example, 45% (read as "forty-five per cent") is equal to the fraction , the ratio 45:55 (or 45:100 when comparing to the total rather than the other portion), or 0.45.


title: Percentage

text: If 50% of the total number of students in the class are male, that means that 50 out of every 100 students are male. If there are 500 students, then 250 of them are male.


title: Percentage

text: An increase of $0.15 on a price of $2.50 is an increase by a fraction of = 0.06. Expressed as a percentage, this is a 6% increase.


title: Percentage

text: While many percentage values are between 0 and 100, there is no mathematical restriction and percentages may take on other values. For example, it is common to refer to 111% or −35%, especially for percent changes and comparisons.


title: Percentage

text: In Ancient Rome, long before the existence of the decimal system, computations were often made in fractions in the multiples of . For example, Augustus levied a tax of on goods sold at auction known as "centesima rerum venalium". Computation with these fractions was equivalent to computing percentages.


title: Percentage

text: As denominations of money grew in the Middle Ages, computations with a denominator of 100 became increasingly standard, such that from the late 15th century to the early 16th century, it became common for arithmetic texts to include such computations. Many of these texts applied these methods to profit and loss, interest rates, and the Rule of Three. By the 17th century, it was standard to quote interest rates in hundredths.


title: Percentage

text: The sign for "percent" evolved by gradual contraction of the Italian term "per cento", meaning "for a hundred". The "per" was often abbreviated as "p."—eventually disappeared entirely. The "cento" was contracted to two circles separated by a horizontal line, from which the modern "%" symbol is derived.


title: Percentage

text: The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, one first computes the ratio = 0.04, and then multiplies by 100 to obtain 4%. The percent value can also be found by multiplying first instead of later, so in this example, the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%.


title: Percentage

text: To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:


title: Percentage

text: It is not correct to divide by 100 and use the percent sign at the same time; it would literally imply division by 10,000. For example, , not , which actually is . A term such as would also be incorrect, since it would be read as 1 percent, even if the intent was to say 100%.


title: Percentage

text: Whenever communicating about a percentage, it is important to specify what it is relative to (i.e., what is the total that corresponds to 100%). The following problem illustrates this point.


title: Percentage

text: We are asked to compute the ratio of female computer science majors to all computer science majors. We know that 60% of all students are female, and among these 5% are computer science majors, so we conclude that × = or 3% of all students are female computer science majors. Dividing this by the 10% of all students that are computer science majors, we arrive at the answer: = or 30% of all computer science majors are female.


title: Percentage

text: The calculation of percentages is carried out and taught in different ways depending on the prerequisites and requirements. In this way, the usual formulas can be obtained with proportions, which saves them from having to remember them. In so-called mental arithmetic, the intermediary question is usually asked what 100% or 1% is (corresponds to).


title: Percentage

text: 42 kg is 7%. How much is (corresponds to) 100%?Given are "W" (percentage) and "p" % (percentage).We are looking for "G" (basic value).


title: Percentage

text: Due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the "initial value" of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).


title: Percentage

text: In general, a change of percent in a quantity results in a final amount that is 100 +  percent of the original amount (equivalently, (1 + 0.01) times the original amount).


title: Percentage

text: Percent changes applied sequentially "do not add up" in the usual way. For example, if the 10% increase in price considered earlier (on the $200 item, raising its price to $220) is followed by a 10% decrease in the price (a decrease of $22), then the final price will be $198—"not" the original price of $200. The reason for this apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to "different" quantities ($200 and $220, respectively), and thus do not "cancel out".


title: Percentage

text: In general, if an increase of percent is followed by a decrease of percent, and the initial amount was , the final amount is ; hence the net change is an overall decrease by percent "of" percent (the square of the original percent change when expressed as a decimal number). Thus, in the above example, after an increase and decrease of , the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200. The net change is the same for a decrease of percent, followed by an increase of percent; the final amount is .


title: Percentage

text: This can be expanded for a case where one does not have the same percent change. If the initial amount leads to a percent change , and the second percent change is , then the final amount is . To change the above example, after an increase of and decrease of , the final amount, $209, is 4.5% more than the initial amount of $200.


title: Percentage

text: In the case of interest rates, a very common but ambiguous way to say that an interest rate rose from 10% per annum to 15% per annum, for example, is to say that the interest rate increased by 5%, which could "theoretically" mean that it increased from 10% per annum to 10.05% per annum. It is clearer to say that the interest rate increased by 5 percentage points (pp). The same confusion between the different concepts of percent(age) and percentage points can potentially cause a major misunderstanding when journalists report about election results, for example, expressing both new results and differences with earlier results as percentages. For example, if a party obtains 41% of the vote and this is said to be a 2.5% increase, does that mean the earlier result was 40% (since 41 = ) or 38.5% (since 41 = )?


title: Percentage

text: In financial markets, it is common to refer to an increase of one percentage point (e.g. from 3% per annum to 4% per annum) as an increase of "100 basis points".