In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

For either Euler or Tait-Bryan angles, it is very simple to convert from an intrinsic ( rotating axes ) to an extrinsic ( static axes ) convention, and vice-versa: just swap the order of the operations.

simple: Ulf von Euler

The Verlet integrator offers greater stability, as well as other properties that are important in physical systems such as time-reversibility and preservation of the symplectic form on phase space, at no significant additional cost over the simple Euler method.

The number of edges of the arrangement is at most n < sup > 2 </ sup >, as may be seen either by using the Euler characteristic to calculate it from the numbers of vertices and cells, or by observing that each line is partitioned into at most n edges by the other n − 1 lines ; again, this worst-case bound is achieved for simple arrangements.

In particular, it can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζ < sub > K </ sub >( s ) = 0 and 0 < Re ( s ) < 1, then Re ( s ) = 1 / 2.

The description of the Hasse – Weil zeta function up to finitely many factors of its Euler product is relatively simple.

It is remarkable that this rule replaces the fairly complicated function of all three Euler angles, time derivatives of Euler angles, and inertia moments ( characterizing the rigid rotor ) by a simple differential operator that does not depend on time or inertia moments and differentiates to one Euler angle only.

Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution.

Actually this simple use of " quaternions " was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares.

Thābit's formula was rediscovered by Fermat ( 1601 – 1665 ) and Descartes ( 1596 – 1650 ), to whom it is sometimes ascribed, and extended by Euler ( 1707 – 1783 ).

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

It was Euler ( presumably around 1740 ) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him.

The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 ( and later generalized as Darboux's formula ).

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

This view was further elaborated by Belidor ( representation of rough surfaces with spherical asperities, 1737 ) and Leonhard Euler ( 1750 ) who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction ..

Euler was born on April 15, 1707, in Basel to Paul Euler, a pastor of the Reformed Church.

Paul Euler was a friend of the Bernoulli family — Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard.

Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician.

As a result, it was made especially attractive to foreign scholars like Euler.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731.

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece.

This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy.

For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.

This direct relationship between curved streamlines and pressure differences was derived from Newton's second law by Leonard Euler in 1754:

It is named for the Dutch-Swiss mathematician and scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others.

Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

Clenshaw – Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler – Maclaurin approach is very accurate ( in that particular case the Euler – Maclaurin formula takes the form of a discrete cosine transform ).

Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler ( 1708 – 1783 ) in the 18th century.

This step of the argument is very similar to the usual proof that the Riemann zeta function has no zeros with real part greater than 1, by writing it as an Euler product.

Leonhard Euler gave a formulation of the action principle in 1744, in very recognizable terms, in the Additamentum 2 to his Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes.

This similarity between linear coordinates and angular coordinates makes Euler angles very intuitive, but unfortunately they suffer from the gimbal lock problem.

This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.

* = 0. 57735, very close to the Euler – Mascheroni constant, 0. 57721 ...

His short stay as a postdoctoral student in Dale's laboratory was very fruitful: in 1931 he discovered with John H. Gaddum an important autopharmacological principle, substance P. After returning to Stockholm, von Euler pursued further this line of research, and successively discovered four other important endogenous active substances, prostaglandin, vesiglandin ( 1935 ), piperidine ( 1942 ) and noradrenaline ( 1946 ).

Often it is very difficult to determine the exact buckling load in complex structures using the Euler formula, due to the difficulty in deciding the constant K. Therefore, maximum buckling load often is approximated using energy conservation.

The Euler force is typically ignored because its magnitude is very small.

Finally, identify this sum of indices as the Euler characteristic of M. To do that, construct a very specific vector field on M using a triangulation of M for which it is clear that the sum of indices is equal to the Euler characteristic.

Imre Lakatos studied very sophisticated kinds of translations of mathematical ( e. g., the Euler formula for polyhedra ) and scientific theories.

The elastica theory is a theory of mechanics of solid materials developed by Leonhard Euler that allows for very large scale elastic deflections of structures.