Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every sufficiently large odd number is the sum of three primes.

In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly ( see Vinogradov's theorem ) that all sufficiently large odd numbers can be expressed as the sum of three primes.

With the help of this method, Vinogradov tackled questions such as the ternary Goldbach problem in 1937 ( using Vinogradov's theorem ), and the zero-free region for the Riemann zeta function.

* Vinogradov's theorem, in math

These techniques are known as the circle method, and give explicit upper bounds for the function G ( k ), the smallest number of k < sup > th </ sup > powers needed, such as Vinogradov's bound

Another hypothetical was the staging of the Haitian struggle for independence as depicted in Anatolii Vinogradov's The Black Consul, influenced as well by John Vandercook's Black Majesty.

In analytic number theory, Vinogradov's method refers to his main problem-solving technique, applied to central questions involving the estimation of exponential sums.

Vinogradov applied finite sums to Waring's problem in 1926, and the general trigometric sum method became known as " the circle method of Hardy, Littlewood and Ramanujan, in the form of Vinogradov's trigonometric sums.

There was much debate as to whether Vinogradov's staging of Gusev's version would be retained or whether Sergeyev's version would be reinstated.

In spite of being a great success, Vinogradov's production was taken out of the repertory after his departure as director of the Kirov / Mariinsky Ballet in 1995, and to date the company have no production of the full-length work in their repertory.

Euclid's original proof adds a third: the two lengths are not prime to one another.

Some folk etymologies argue that the original forms of these symbols indicated their value through the number of angles they contained, however there is no proof of any such origin.

Suggested locations for backups are a fireproof, waterproof, and heat proof safe, or in a separate, offsite location than that in which the original files are contained.

Investors in the VOC were issued paper certificates as proof of share ownership, and were able to trade their shares on the original Amsterdam stock exchange.

The original versions of his papers contained " many technical errors of varying degree "; when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception — a claimed proof of the Continuum hypothesis.

The original proof is based on the Taylor series expansions of the exponential function e < sup > z </ sup > ( where z is a complex number ) and of sin x and cos x for real numbers x ( see below ).

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.

Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.

In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof.

While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof was discovered in 1980 by American mathematician Donald J. Newman.

They initially implied the proof was their own achievement based on the " Hamilton-Perelman theory ", but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest " exposition of Hamilton – Perelman's proof ".

The original completeness proof applies to all classical models, not some special proper subclass of intended ones.

We will now follow the strategy of David Hilbert ( 1862 – 1943 ) who gave a simplification of the original proof of Charles Hermite.

Where a title has been enacted into positive law, however, a court may neither permit nor require proof of the underlying original statutes.

The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

His original proof relied partly on Hamilton's work on the Ricci flow.

In his work on the divine proportion, H. E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art — the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor.

Forty other figures similar to 5 and 6 and the original data used in the construction of all figures and tables in this monograph have been included in the Appendix.

But because teeth sometimes may drift back to their original position, a retaining appliance is used to lock them in place.

The term " droid ", coined by George Lucas for the original Star Wars film and now used widely within science fiction, originated as an abridgment of " android ", but has been used by Lucas and others to mean any robot, including distinctly non-human form machines like R2-D2.

Beginning with Reginald Fessenden's audio demonstrations in 1906, it was also the original method used for audio radio transmissions, and remains in use today by many forms of communication —" AM " is often used to refer to the mediumwave broadcast band ( see AM radio ).

On the 2007 deluxe edition release, it is revealed that parts of this tape were used for the 1976 Griffith Park Planetarium launch of the original album, the 1987 remix, and various radio spots, all of which were included as bonus material.

Unlike his translation of the Pastoral Care, Alfred here deals very freely with his original and though the late Dr. G. Schepss showed that many of the additions to the text are to be traced not to Alfred himself, but to the glosses and commentaries which he used, still there is much in the work which is solely Alfred's and highly characteristic of his style.

The project included collection and identification of all stone fragments, even small ones, from the Acropolis and its slopes and the attempt was made to restore as much as possible using reassembled original material-with new marble from Mount Penteli used sparingly.

AppleTalk was released for the original Macintosh in 1985, and was the primary protocol used by Apple devices through the 1980s and 90s.

Of great surprise to many fans was a brand new color scheme ; apparently the original colors used by the franchise since Major League Baseball awarded it to Jerry Colangelo's ownership group in 1995 were to be discontinued.

Because the original designer had left the company another employee completely redesigned most of the system, ( adding a display snow remover circuit, true 80 / 64 column text mode support, ( with different size letters for TRS-80 and CP / M mode, so that in TRS-80 mode the full screen was also used, not just a 64x16 portion of the 80x25 screen ) with an improved font set ( adding " gray scale " version of the TRS-80 mozaik graphics and many special PETSCII like characters ), and a more flexible and reliable floppy disk controller and keyboard interface plus many other small improvements ), also an enclosure was developed for the main computer system, ( in the form of a 19-inch rack for the Eurocards ) and for two floppy disk drives and the power supply.

The Artistic License refers most commonly to the original Artistic License ( version 1. 0 ), a software license used for certain free and open source software packages, most notably the standard Perl implementation and most CPAN modules, which are dual-licensed under the Artistic License and the GNU General Public License ( GPL ).

** The Artistic License ( the original Artistic License 1. 0, the one which is still used by Perl and CPAN )

* The original order (), used for lettering, derives from the order of the Phoenician alphabet, and is therefore similar to the order of other Phoenician-derived alphabets, such as the Hebrew alphabet.

The concept of playing solo steel-string guitar in a concert setting was introduced in the early 1960s by such performers as Davey Graham and John Fahey, who used country blues fingerpicking techniques to compose original compositions with structures somewhat like European classical music.

A cappella has been used as the sole orchestration for original works of Musical Theater that have had commercial runs Off-Broadway ( theaters in New York City with 99 to 500 seats ) only three times.

This style, called " tabbed " or " ridged ", used the same basic design as the original cartridges with the addition of two small tabs on the cartridge's underside to aid in removal.

There were three main kinds of harpoon that the Aleut ’ s used: a simple harpoon, with a head that kept its original position in the animal after striking, a compound ( toggle-head ) harpoon in which the head took a horizontal position in the animal after penetration, and the throwing-lance used to kill large animals.

This was used after Sam Raimi was unable to use his original title " The Medieval Dead.

Until 1945, Bulgarian orthography did not reveal this alternation and used the original Old Slavic Cyrillic letter yat (), which was commonly called двойно е ( dvoyno e ) at the time, to express the historical yat vowel or at least root vowels displaying the ya – e alternation.

He is also credited with the first port of the GNU C compiler to the INTEL 386 architecture, used in the original implementation of the Linux kernel.

Backplanes have grown in complexity from the simple Industry Standard Architecture ( ISA ) ( used in the original IBM PC ) or S-100 style where all the connectors were connected to a common bus.

This discovery has shed much light on the differences between the two versions ; while it was previously maintained that the Greek Septuagint ( the version used by the earliest Christians ) was only a poor translation, professor Emanuel Tov, senior editor of the Dead Sea Scrolls ' publication, wrote that the Masoretic edition either represents a substantial rewriting of the original Hebrew, or there had previously been two different versions of the text.