This began in his doctoral work leading to the Mordell – Weil theorem ( 1928, and shortly applied in Siegel's theorem on integral points ).

Siegel's theorem on integral points occurs in Chapter 28.

He notes that the content of the book is largely versions of the Mordell-Weil theorem, Thue-Siegel-Roth theorem, Siegel's theorem, with a treatment of Hilbert's irreducibility theorem and applications ( in the style of Siegel ).

Leaving aside issues of generality, and a completely different style, the major mathematical difference between the two books is that Lang used abelian varieties and offered a proof of Siegel's theorem, while Mordell noted that the proof " is of a very advanced character " ( p. 263 ).

Siegel's work spans analytic number theory ; and his theorem on the finiteness of the integer points of curves, for genus > 1, is historically important as a major general result on diophantine equations, when the field was essentially undeveloped.

Siegel's location work and his use of actual prisoners as extras in the film made a lasting impression on Peckinpah.

Lorre made nine movies altogether with Sydney Greenstreet counting The Maltese Falcon and Casablanca, most of them variations on the latter film, including Background to Danger ( 1943, with George Raft ); Passage to Marseille ( 1944, reteaming them with Casablanca stars Humphrey Bogart and Claude Rains ); The Mask of Dimitrios ( 1944, with character actor Greenstreet receiving top billing ); The Conspirators ( 1944, with Hedy Lamarr and Paul Henreid ); Hollywood Canteen ( 1944 ); Three Strangers ( 1946 ), a suspense film about three people who are joint partners on a winning lottery ticket starring top-billed Greenstreet, Geraldine Fitzgerald, and third-billed Lorre cast against type by director Jean Negulesco as the romantic lead ; and Greenstreet and Lorre's final film together, suspense thriller The Verdict ( 1946 ), director Don Siegel's first movie, with Greenstreet and Lorre finally billed first and second, respectively.

* Hail, American Development, containing 178 poems, including 32 translations —" all with the same incomparable sensibility at work saying things nobody else could say ", wrote Kenneth Rexroth in the New York Times Book Review ,; adding, " Siegel's translations of Baudelaire and his commentaries on them rank him with the most understanding of the Baudelaire critics in any language ".

Kiley turned the role down and Wanger cast two relative newcomers in the lead roles: Kevin McCarthy, who had just starred in Siegel's An Annapolis Story, and Dana Wynter, who had done several major dramatic roles on television.

Considered the second television movie, Don Siegel's The Hanged Man was broadcast by the National Broadcasting Co. on November 18, 1964.

Fabian was contracted to 20th Century-Fox beginning with Don Siegel's Hound-Dog Man, based on Fred Gipson's novel.

To illustrate, Sullivan knocks the first few beats of " Shave-and-a-Haircut " on the door of Siegel's tomb, to receive a moment later the response " Two-Bits " knocked from the inside of the tomb.

She has made few films since, the best-known being Don Siegel's The Black Windmill ( 1974 ), Nijinsky ( 1980 ), Priest of Love ( 1981 ), Peter Greenaway's The Draughtsman's Contract ( 1982 ), Federico Fellini's E la Nave Va ( And the Ship Sails On 1983 ), A Dry White Season ( 1989 ) with Marlon Brando and Nuns on the Run ( 1990 ; a rare comedic role ).

The enraged mobsters demanded Siegel's head on a platter.

Siegel's work on number theory, diophantine equations, and celestial mechanics in particular won him numerous honours.

In the early 1970s Weil gave a series of seminars on the history of number theory prior to the 20th century and he remarked that Siegel once told him that when the first person discovered the simplest case of Faulhaber's formula then, in Siegel's words, " Es gefiel dem lieben Gott.

Two years later, a CAT scan revealed a lesion on Siegel's left lung.

* Kevin Smith's Response Contains mp3 of Smith's confrontation with Siegel on the Opie and Anthony show on July 18, 2006 following Siegel's harshly negative review of Clerks II.

The Aesthetic Realism Theatre Company, composed of actors, singers and musicians, has appeared throughout the country in both musical performances and dramatic productions based on Eli Siegel's lectures on Shakespeare, Moliere, Richard Brinsley Sheridan, Ibsen, Strindberg, Eugene O ’ Neill, George Kelly, Susan Glaspell, and others.

Suffering from a damaged reputation due to alcohol and drug addictions noted most recently on the set of his 1978 film Convoy, Peckinpah had been given the opportunity to do second unit work on Don Siegel's Jinxed!

Siegel's music has featured on nationally syndicated shows such as NPR's All Things Considered and Mountain Stage.

Some horror films during this period, such as The Thing from Another World ( 1951 ) and Don Siegel's Invasion of the Body Snatchers ( 1956 ), managed to channel the paranoia of the Cold War into atmospheric creepiness.

Under Siegel's stint, the show went from last to first in the ratings for a brief period, and from 1977 until his departure in 1978 it was known as The Stanley Siegel Show.

Don Siegel's The Black Windmill ( 1974 ), derived from the Clive Egleton novel Seven Days to a Killing, with Michael Caine as MI-5 operative John Tarrant, an espionage agent whose son has been kidnapped.

" As such, Wallace drew inspiration from another pod film: Don Siegel's Invasion of the Body Snatchers ( 1956 ).

The Siegel estate claims that the original " Superboy " character published by DC Comics is an independent creation that used ideas from Jerry Siegel's original rejected pitch and was created without his consent.

Issues regarding the amount of monies owed Siegel's estate and whether the claim the estate has extends to derivative works such as movie versions will be settled at trial, although any compensation would only be owed from works published since 1999.

The Siegel estate claims that the original " Superboy " character published by DC Comics is an independent creation that used ideas from Jerry Siegel's original rejected pitch and was created without his consent.

Siegel's second wife, Jane Kessler, died from a brain tumor in 1982.

After a compilation album in the spring of 1981 and the single release of " You ( You Are My Soul )" from Fight Fight Fight, the band signed to Ralph Siegel's Jupiter Records.

IPO research firm Renaissance Capital responded to Siegel's and other academics ' arguments in a white paper of theirs from 2010, saying that IPOs outperform the broader market when weighted relative to their market cap.

* Argument principle, a theorem in complex analysis about meromorphic functions inside and on a closed contour

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The distance formula on the plane follows from the Pythagorean theorem.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as the Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.

* The extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.

( Gelfand – Naimark theorem ) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.

The first proof relies on a theorem about products of limits to show that the derivative exists.

The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.

Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface.