Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics — including a large portion of modern physics — is usually explained with the help of bra-ket notation.

A modern classification for libraries, with simple notation, mnemonics, and alternatives.

The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation.

According to Margaret Bent, " Renaissance notation is under-prescriptive by our standards ; when translated into modern form it acquires a prescriptive weight that overspecifies and distorts its original openness.

Accidentals … may or may not have been notated, but what modern notation requires would then have been perfectly apparent without notation to a singer versed in counterpoint ".

The solution to one particular case of the cubic equation ( in modern notation ), was communicated to him by Niccolò Fontana Tartaglia ( who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long fight ), and the quartic was solved by Cardano's student Lodovico Ferrari.

* Otto Müller Verlag Salzburg 1969: Hildegard von Bingen: Lieder ( modern edition in adapted square notation )

This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.

He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.

He also introduced the modern notation for the trigonometric functions, the letter for the base of the natural logarithm ( now also known as Euler's number ), the Greek letter Σ for summations and the letter to denote the imaginary unit.

In modern notation the first is CO ( carbon monoxide ) and the second is CO < sub > 2 </ sub > ( carbon dioxide ).

* Proposition ✸ 54. 43 in a more modern notation ( Metamath )

The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which is from Peano's ε ) and implication (⊃, which is from Peano's reversed ' C '.

According to Margaret Bent ( 1998 ), " Renaissance notation is under-prescriptive by our standards ; when translated into modern form it acquires a prescriptive weight that overspecifies and distorts its original openness.

However, Renaissance musicians would have been highly trained in dyadic counterpoint and thus possessed this and other information necessary to read a score, " what modern notation requires would then have been perfectly apparent without notation to a singer versed in counterpoint.

Although Franciscus Vieta ( 1540 – 1603 ) gave the first notation of modern algebra, John Napier ( 1550 – 1617 ) invented logarithms, and Edmund Gunter ( 1581 – 1626 ) created the logarithmic scales ( lines, or rules ) upon which slide rules are based.

In modern notation, the free Boolean algebra on basic propositions p and q arranged in a Hasse diagram.

In 1659 Huygens derived the now well-known formula for the centripetal force, exerted by an object describing a circular motion, for instance on the string to which it is attached, in modern notation:

Huygens was the first to derive the formula for the period of an ideal mathematical pendulum ( with massless rod or cord and length much longer than it's swing ), in modern notation:

* All modern integrated development environments display variable types on demand, and automatically flag operations which use incompatible types, making the notation largely obsolete.

This is the only case in modern history of a people of Britannic origin submitting without continued struggle to what they view as foreign domination.

While sovereignty has roots in antiquity, in its present usage it is essentially modern.

Neither is primary experience understood according to the attitude of modern empiricism in which nothing is thought to be received other than signals of sensory qualities producing their responses in the appropriate sense organs.

Shakespeare's Shylock, too, is of dubious value in the modern world.

But a modern Oedipus who is doomed because he cannot oppose his own childhood is only pathetic, and for renouncing the mystery in favor of psychological truth he gives up the claim on our sympathies.

A characteristic expression of such concern and inquiry is found in Joseph P. Lyford's Introduction To The Agreeable Autocracies, a recent paperback study of the institutions of modern democratic society.

The men who speculate on these institutions have, for the most part, come to at least one common conclusion: that many of the great enterprises and associations around which our democracy is formed are in themselves autocratic in nature, and possessed of power which can be used to frustrate the citizen who is trying to assert his individuality in the modern world ''.

As Sandburg said at the time: `` It is as ancient as the medieval European ballads brought to the Appalachian Mountains, it is as modern as skyscrapers, the Volstead Act, and the latest oil well gusher ''.

Here we may observe that at least one modern philosophy of history is built on the assumption that ideas are the primary objectives of the historian's research.

The Chicago contingent of modern critics follow Aristotle so far in this direction that it is hard to see how they can compare one poem with another for the purpose of evaluation.

Diplomatic is another area for which the dawn of the twentieth century marks the beginning of modern standards of scholarship.

This is certainly an irrational dogmatism, in which the modern mind attempts to understand the spirit of the sixteenth century on twentieth-century terms.

Rexroth is a longtime jazz buff, a name-dropper of jazz heroes, and a student of traditional as well as modern jazz.

In addition to his experiments in reading poetry to jazz, Patchen is beginning to use the figure of the modern jazz musician as a myth hero in the same way he used the figure of the private detective a decade ago.

This angry and exasperated stance which Patchen has maintained in his poetry for almost fifteen years has been successfully modulated into a kind of woe that is as effective as anger and still expresses his disapproval of the modern world.

But the firm has recognized the tight dollar and the tourist's desire to visit the `` smaller, less-traveled and relatively inexpensive countries '', and is now prepared to teach modern Greek and Portuguese through recordings.

Perhaps there is more truth than we are wont to admit in the conviction of that ornament of Tarheelia, Robert Ruark's grandfather, who was persuaded that the great curse of the modern world is `` all this gallivantin' ''.

The possibility of recall into the Army is part of the price that a modern American has to pay for the enviable heritage of liberty which he enjoys.

Under modern conditions, this is especially true of the ready reserve.

However, its modern one-story layout is designed to increase our production capacity, permit more efficient manufacturing, and substantially reduce current repair and maintenance costs.

He described the piece as a " rhapsodic ballet " because it was written freely and is more modern than his previous works.

The abacus was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and elsewhere.

One modern scholar has written " It is almost certain not only that at no time in his life did he ever see, let alone command, a Roman army, but that, throughout the twenty-three years of his reign, he never went within five hundred miles of a legion ".

So-called " autobiographies " of modern professional athletes and media celebrities — and to a lesser extent about politicians, generally written by a ghostwriter, are routinely published.

Examples include Edward Elgar's Great is the Lord ( 1912 ) and Give unto the Lord ( 1914 ) ( both with orchestral accompaniment ), Benjamin Britten's Rejoice in the Lamb ( 1943 ) ( a modern example of a multi-movement anthem and today heard mainly as a concert piece ), and, on a much smaller scale, Ralph Vaughan Williams ' O taste and see ( 1952 ) ( written for the coronation of Queen Elizabeth II ).

He outlined his conception of the play as a " modern tragedy " in a note written in Rome on 19 October 1878.

Most modern scholars believe it was written around 95 AD, with some believing it dates from around 70 AD.

Most modern day biblical scholars assert that the Book of Lamentations was written by one or more authors in Judah, shortly after the Babylonian conquest of Jerusalem in 586 BC ; and was penned as a response to Babylonian Exile, the intense suffering of the people of Judah, and the complete and utter destruction of Jerusalem.

However, most modern scholars believe the book of Zechariah was written by at least two different people.

He is also traditionally associated with the Epistle of Barnabas, although modern scholars think it more likely that that epistle was written in Alexandria in the 130s.

BeOS was built for digital media work and was written to take advantage of modern hardware facilities such as symmetric multiprocessing by utilizing modular I / O bandwidth, pervasive multithreading, preemptive multitasking and a 64-bit journaling file system known as BFS.

BeOS was optimized for digital media work and was written to take advantage of modern hardware facilities such as symmetric multiprocessing by utilizing modular I / O bandwidth, pervasive multithreading, preemptive multitasking and a 64-bit journaling file system known as BFS.

Generally, every modern written constitution confers specific powers to an organization or institutional entity, established upon the primary condition that it abides by the said constitution's limitations.

In the modern world, written calendars are no longer an essential part of such systems, as the advent of accurate clocks has made it possible to record time independently of astronomical events.

However, the possibility of actually constructing a conscious machine was probably first discussed by Ada Lovelace, in a set of notes written in 1842 about the Analytical Engine invented by Charles Babbage, a precursor ( never built ) to modern electronic computers.

The Biblia pauperum (" Paupers ' Bible "), a tradition of picture Bibles beginning in the later Middle Ages, sometimes depicted Biblical events with words spoken by the figures in the miniatures written on scrolls coming out of their mouths — which makes them to some extent ancestors of the modern cartoon strips.

The Commentaries of Gaius ( written between 130 and 180 AD ) on the Twelve Tables treated furtum ( in modern parlance: " theft ") as a tort.

According to Emile Benveniste ( 1954 ), the earliest written occurrence in English of civilisation in its modern sense may be found in Adam Ferguson's An Essay on the History of Civil Society ( Edinburgh, 1767 – p. 2 ): " Not only the individual advances from infancy to manhood, but the species itself from rudeness to civilisation.

The most well known version is that written by Gerald Gardner, and includes material paraphrased works by Aleister Crowley, primarily from Liber ALThe Book of the Law ( particularly from Ch 1, spoken by Nuit, the Star Goddess ), and from his Liber XV: the Gnostic Mass as well as Liber LXV ( Liber Cordis Cincti Serpente, or the Book of the Heart Girt with the Serpent ), thus linking modern Wicca irrevocably to the cosmology and revelations of Thelema.

T. S. Eliot called Collins's novel The Moonstone ( 1868 ) " the first, the longest, and the best of modern English detective novels ... in a genre invented by Collins and not by Poe ", and Dorothy L. Sayers called it " probably the very finest detective story ever written ".

Part Two of Don Quixote is often regarded as the birth of modern literature, as it explores the concept of a character understanding that he is being written about.

In Old Castilian the letter x represented the sound written with sh in modern English, so the name was originally pronounced " ".

BIND version 9 was written from scratch and now has a security record comparable to other modern DNS software.

The Tao Te Ching is written in Classical Chinese, which can be difficult to understand completely, even for well-educated native speakers of modern Chinese.

In jazz, pizzicato is the norm, except for some solos and also occasional written parts in modern jazz that call for bowing.