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.

In modern notation, this is written by placing the expressions on either side of an equals sign (=), for example

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.

Suppose, he says, that the tables were turned, and we were in the Soviets' position: `` There would be more than 2,000 modern Soviet fighters, all better than ours, stationed at 250 bases in Mexico and the Caribbean.

The way croquet is depicted in paintings and books says much about popular perceptions of the game, though little about the reality of modern play.

Reviewing the film for Scientific American, John Rennie says " The term is a curious throwback, because in modern biology almost no one relies solely on Darwin's original ideas ...

Garcia-Ballester says the following of Galen ’ s use of prognosis: " In modern medicine, we are used to distinguishing between the diagnostic judgment ( the scientific knowledge of what a patient has ) and the prognostic judgment ( the conjecture about what will happen to him.

" Austrian School economist Hans-Hermann Hoppe says that " the 1849 article " The Production of Security " is probably the single most important contribution to the modern theory of anarcho-capitalism.

Michael Palin says that the various separatist movements were modelled on " modern resistance groups, all with obscure acronyms which they can never remember and their conflicting agendas ".

He says that "... an indubitable residuum of things Persian in the Mysteries and a better knowledge of what constituted actual Mazdaism have allowed modern scholars to postulate for Roman Mithraism a continuing Iranian theology.

He cites his modern influences as Harry Enfield ( who he says without meeting he would not have been doing what he does now ), and the approach of Reeves and Mortimer who he thinks are " far and away the best comedians that we have had in this country for a long while.

One modern academic theory of religion, social constructionism, says that religion is a modern concept that suggests all spiritual practice and worship follows a model similar to the Abrahamic religions as an orientation system that helps to interpret reality and define human beings, and thus religion, as a concept, has been applied inappropriately to non-Western cultures that are not based upon such systems, or in which these systems are a substantially simpler construct.

When ( modern ) Klingon Worf and his crewmates see an original-series Klingon, his crewmates ask about the stunning difference in appearance ; Worf says tersely that the matter is something Klingons " do not discuss with outsiders ".

Perhaps its most important and prominent usage in scientific literature was Nobel laureate Theodor W. Hänsch's reference in a 1979 Scientific American article on spectroscopy where he says that " the spectrum of the hydrogen atoms has proved to be the Rosetta stone of modern physics: once this pattern of lines had been deciphered much else could also be understood ".

( The truism that says, diversify your investments, may have worked for the Elizabethan play market as for the modern stock market.

As the Dictionary of the Middle Ages says, " William's achievements in assembling and evaluating sources, and in writing in excellent and original Latin a critical and judicious ( if chronologically faulty ) narrative, make him an outstanding historian, superior by medieval, and not inferior by modern, standards of scholarship.

Paul Davis, another modern historian who addresses both sides in the debate over whether or not this Battle truly determined the direction of history, as Watson claims, or merely was a relatively minor raid, as Cardini writes, says " whether Charles Martel saved Europe for Christianity is a matter of some debate.

The second, " Disciplinary Punishment ," is what Foucault says is practiced in the modern era.

The earliest precursor to modern complex systems theory can be found in the classical political economy of the Scottish Enlightenment, later developed by the Austrian school of economics, which says that order in market systems is spontaneous ( or emergent ) in that it is the result of human action, but not the execution of any human design.

The author of the book Planet Simpson, Chris Turner, said that " if the institution of the News has a single iconic face on The Simpsons, it's Brockman's " and that " in Brockman's journalism, we see some of the modern news media's ugliest biases ", of which he says are glibness, amplification, and sensationalism.

On the other hand, a contemporary source says that one general Leo of the Armeniakon theme was punished for his humiliating defeat by the Arabs during which he also lost the salaries of his thematic units ( a modern scholar suggests that this Leo is not the same with the emperor ).

One of his revisionist modern biographers, however, Miriam Griffin says in her biography of Seneca that " the evidence for Seneca's life before his exile in 41 is so slight, and the potential interest of these years, for social history as well as for biography, is so great that few writers on Seneca have resisted the temptation to eke out knowledge with imagination.

A legend says that many bones were lost along the roadway that encompasses much of modern U. S. Route 322, and that every January 1 ( Wayne's birthday ), his ghost wanders the highway searching for his lost bones.

In modern notation it says that given quantities p, q, r and s, then p: q > r: s if there are positive integers m and n so that np > mq and nr ≤ ms.

" film " says Pilger, " is about the struggle of people to free themselves from a modern form of slavery ".

Yet the original Latin text, published by Georg Weitz in the Monumenta Germaniae Historica in 1883, contains a very significant difference from modern translations: it says the ruler of the Rhos was named not chaganus, but chacanus.