In On the Infinite, David Hilbert hypothesized that the Ackermann function was not primitive recursive, but it was Ackermann, Hilbert ’ s personal secretary and former student, who actually proved the hypothesis in his paper On Hilbert ’ s Construction of the Real Numbers.

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

J. B. Curtis in his 1979 paper " On Job's Response to Yahweh ", argues that Job's final responses to Yahweh are a total rejection of Yahweh rather than an expression of repentance, and translates Job 42: 6 as " Therefore I feel loathing contempt and revulsion ( toward you, O God ); and I am sorry for frail man.

It began in the 1948 paper, " On the Problem of the Molecular Theory of Superconductivity " where Fritz London proposed that the phenomenological London equations may be consequences of the coherence of a quantum state.

Rebuffed by William Thiselton-Dyer, the Director at Kew, because of her gender and her amateur status, Beatrix wrote up her conclusions and submitted a paper On the Germination of the Spores of the Agaricineae to the Linnean Society in 1897.

On June 30, 1945, before ENIAC was made, mathematician John von Neumann distributed the paper entitled First Draft of a Report on the EDVAC.

Building on the work of Clausius, between the years 1873-76 the American mathematical physicist Willard Gibbs published a series of three papers, the most famous one being the paper On the Equilibrium of Heterogeneous Substances.

On paper, using classical notation, this is usually done all in one step.

On the other hand, Emil Post's 1936 paper had appeared and was certified independent of Turing's work.

Turing adds another definition, Rosser equates all three: Within just a short time, Turing's 1936 – 37 paper " On Computable Numbers, with an Application to the Entscheidungsproblem " appeared.

In his 1868 paper " On Governors ", J. C. Maxwell ( who discovered the Maxwell electromagnetic field equations ) was able to explain instabilities exhibited by the flyball governor using differential equations to describe the control system.

His first paper, " On a recent formation of freshwater limestone in Forfarshire ", was presented in 1822.

On one hand, there are transmitted versions and commentaries that date back two millennia ; on the other, there are ancient bamboo, silk, and paper manuscripts that archeologists discovered in the last century.

Such was the time taken in peer-reviewing the paper for Nature that this was preceded by a 1972 essay by Maynard Smith in a book of essays titled On Evolution.

On the local level, a butcher may commonly break down larger animal meat into smaller manageable cuts, and pre-wrap them for commercial sale or wrap them to order in butcher paper.

On September 22, 2011, a paper from the OPERA Collaboration indicated detection of 17 and 28 GeV muon neutrinos, sent 730 kilometers ( 454 miles ) from CERN near Geneva, Switzerland to the Gran Sasso National Laboratory in Italy, traveling faster than light by a factor of 2. 48 × 10 < sup >− 5 </ sup > ( approximately 1 in 40, 000 ), a statistic with 6. 0-sigma significance.

The first paper from his years of study in Paris appeared in Frankfurt am Main in 1816, under the title of Über das Conjugationssystem der Sanskritsprache in Vergleichung mit jenem der griechischen, lateinischen, persischen und germanischen Sprache ( On the Conjugation System of Sanskrit in comparison with that of Greek, Latin, Persian and Germanic ) ( Windischmann contributed a preface ).

Three years later came the paper that would make his name famous, Sur les équations du mouvement relatif des systèmes de corps ( On the equations of relative motion of a system of bodies ).

On awakening he appeared to himself to have a distinct recollection of the whole, and taking his pen, ink, and paper, instantly and eagerly wrote down the lines that are here preserved.

After reading from Purchas's book, " The Author continued for about three hours in a profound sleep, at least of the external senses, during which time he had the most vivid confidence, that he could not have composed less than from two or three hundred lines ... On Awaking he appeared to himself to have a distinct recollection of the whole, and taking his pen, ink, and paper, instantly and eagerly wrote down the lines that are here preserved.

In 1917, Albert Einstein established the theoretical foundations for the laser and the maser in the paper Zur Quantentheorie der Strahlung ( On the Quantum Theory of Radiation ); via a re-derivation of Max Planck ’ s law of radiation, conceptually based upon probability coefficients ( Einstein coefficients ) for the absorption, spontaneous emission, and stimulated emission of electromagnetic radiation ; in 1928, Rudolf W. Ladenburg confirmed the existences of the phenomena of stimulated emission and negative absorption ; in 1939, Valentin A. Fabrikant predicted the use of stimulated emission to amplify “ short ” waves ; in 1947, Willis E. Lamb and R. C. Retherford found apparent stimulated emission in hydrogen spectra and effected the first demonstration of stimulated emission ; in 1950, Alfred Kastler ( Nobel Prize for Physics 1966 ) proposed the method of optical pumping, experimentally confirmed, two years later, by Brossel, Kastler, and Winter.

In his 1861 paper On Physical Lines of Force he modelled these magnetic lines of force using a sea of molecular vortices that he considered to be partly made of aether and partly made of ordinary matter.

Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, " On Physical Lines of Force ", which he published between 1861 and 1862.

On the truck bed there was nothing smaller than a piece of rusty machinery ; ;

On the question of admission to Veterans Administration hospitals of service-connected and non-service-connected disabled veterans, it must be recognized that there are many men who are greatly affected by war service.

On this point there was fairly general agreement that assessors would like to do more than they are doing now.

On the contrary, there are fewer colds and smaller doctor bills.

On the other hand, there are a few antithyroid drugs of this same general type, such as resorcinol, possessing no reducing activity and possibly acting through formation of a complex with molecular iodine.

On the surfaces of both lungs there were emphysematous blebs measuring up to 3 cm. in diameter.

On the surface of the right kidney there were also 2 yellow, firm, friable raised areas measuring up to 2 cm. in diameter.

On C there is a Af correspondence in which the Af points cut from C by a general line, l, of the pencil correspond to the point of intersection of the image of L and the plane of the pencil.

On the other hand, there is a counterbalancing purpose in education which is to pass on the advantages of the parents to their children.

On the theory before us, such relief would be groundless, for in that suffering itself there was nothing bad at all, and hence in its nonoccurrence there would be nothing to be relieved about.

On the one hand, there is a sense of not having moved beyond the ambiance of their high school.

On the south it ended sharply as though the lava had been cut off there suddenly.

On the one hand, there are ecumenists who are so stirred by the crises of the church in its encounter with the world at large that they have no eyes for what the church is doing in their own town.

On the wall of this square there are delightful bas-reliefs of musical instruments.

On the upper Mississippi the Illinois post was to be established near Kaskaskia, and dependent posts were to be built on the Missouri, `` where there are mines in abundance ''.

On his side of the road there were two farm hands, well back in a field, leaning against a plow.

On the third occasion -- another Big Four summit session at Paris a year ago -- there was no problem of an illusory `` spirit ''.

On the contrary, even in the heart of `` the Bible belt '' itself, as can be attested by any one who is called to work there, the industrial and technological revolutions have long been under way, together with the corresponding changes in man's picture of himself and his world.

On the beach, there were pale girls and not-so-pale girls.

On the occasion of a pestilence in the 430s BCE, Apollo's first temple at Rome was established in the Flaminian fields, replacing an older cult site there known as the " Apollinare ".

On April 28 he moved to Los Angeles where he had family and remained there until May when, suspected by local Union authorities, he evaded arrest and joined the Los Angeles Mounted Rifles as a private, leaving Warner's Ranch May 27.

** On every infinite-dimensional topological vector space there is a discontinuous linear map.

On 31 August, in the great Charles Alcock-edited magazine Cricket: A Weekly Record of The Game, there appeared a mock obituary:

On January 14, 1863, the Alcotts received a telegram that Louisa was sick ; Bronson immediately went to bring her home, briefly meeting Abraham Lincoln while there.