This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it.

He corrected some mistakes of Lorentz and proved the Lorentz covariance of the electromagnetic equations.

Like the tenth spatial dimension, the approximate equations in the original five superstring models proved too weak to reveal membranes.

His writings present a fresh, often geometric approach to traditional mathematical topics like ordinary differential equations, and his many textbooks have proved influential in the development of new areas of mathematics.

Niels Henrik Abel ( 1802 – 1829 ), a Norwegian, and Évariste Galois, ( 1811 – 1832 ) a Frenchman, investigated into the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four ( Abel – Ruffini theorem ).

In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker – Weber theorem, however without offering a definitive proof ( the theorem was proved completely much later by David Hilbert ).

Finally Niels Henrik Abel proved, in 1824, that the equation of degree 5 and equations of higher degree are not always solvable using radicals.

Real Analysis: He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditions.

proved that a given number k + 2 is prime if and only if the following system of 14 Diophantine equations has a solution in the natural numbers:

These equations can be proved through straightforward matrix multiplication and application of trigonometric identities.

In mathematical physics, he proved the fundamental uniqueness theorems for the heat equation and studied Volterra integral equations.

The equivalence between the category of connected real Lie groups and finite dimensional real Lie algebras is called usually ( in the literature of the second half of 20th century ) Cartan's or Cartan-Lie theorem as it is proved by Élie Cartan whereas S. Lie has proved earlier just the infinitesimal version ( local solvability of Maurer-Cartan equations ( see Maurer-Cartan form ) or the equivalence between the finite dimensional Lie algebras and the category of local Lie groups ).

In the case of a chaotic dynamical system, which the equations under turbulent conditions are thought to be, this means that the system can have only one strange attractor, a result that has yet to be proved for the Navier-Stokes equations.

The theorem is proved with the exact time reversible dynamical equations of motion and the Axiom of Causality.

In this work, he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above.

Later on, Guido Stampacchia proved his generalization to the Lax – Milgram theorem in in order to study the regularity problem for partial differential equations and coined the name " variational inequality " for all the problems involving inequalities of this kind.

Refinements of the method have allowed results to be proved about the solutions of homogeneous Diophantine equations, as long as the number of variables k is large relative to the degree d ( see Birch's theorem for example ).

The reaction has been proved to obey these equations in solutions with pH ’ s of between 0 and 13. 5.

) It was a controversial open problem for many years to show that these six equations really were irreducible for generic values of the parameters ( they are sometimes reducible for special parameter values ; see below ), but this was finally proved by and.

In 1945, Bogolyubov proved a fundamental theorem on the existence and basic properties of a one-parameter integral manifold for a system of non-linear differential equations.

He used the blanket for late morning naps when hosts of the night had gone off to jobs and proved reluctant to leave him in their small rooms with their few possessions.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

The Peter family proved wonderful and helpful friends in the following days, Mrs. Peter, little Esther, and Raoul, who generously lent me his sleeping bag for my `` Watch on the Rhine ''.

for if this can be proved we shall surely be the gainers -- I mean, if there is a use in poetry as well as a delight ''.

Because of these involvements in the matter at stake, Boniface lacked the impartiality that is supposed to be an essential qualification for the position of arbiter, and in retrospect that would seem to be sufficient reason why the English embassies to the Curia proved so fruitless.

In fact, the world heard nothing but sabre-rattling, the same exercises which proved futile for the predecessors of the current President.

introduction of the `` dialogue '' principle proved strikingly effective at the thirty-fourth annual meeting of the Catholic Association for International Peace in Washington the last weekend in October.

Without exception Hino's brothers turned to either one or both of their father's occupations, but Hino showed a talent for neither and instead spent most of his time on the beach where he repaired nets and proved immensely popular as a storyteller.

Devoted to the improvement in business climate and increase in industrial construction in Rhode Island, it has proved a valuable mailing piece for this Division.

Here, this happy, roving son of good fortune proved that he could accept the disciplines of a new social-economic order fighting for its very existence and ideals in a truculent world.

Besides being most convenient, the line `` soon proved a good investment for the owners ''.

And she felt amply rewarded for her suffering when the evidence of Lee's quack shenanigans, gathered by the tape recorder under her friend's clothing, proved adequate in court for convicting Franklin D. Lee.

Alas, so it almost proved for us, too.

In this tangle of conflicting claims, the patent-sharing scheme adopted by the A.L.A.M. at its founding proved to be the best device for avoiding or mitigating the burdens of incessant litigation.

More than 300 teenagers last Sunday proved there is and as many more are expected to prove it again for Jim Kern and his wife Lynn from 4 to 8 p.m. Sunday at First Presbyterian Church.

We are left helpless to cope with it because we do not dare speak of it as anything real for fear that to do so would imply a commitment to that which has already been discredited and proved false.

It is the belief of Old Babylonian scholars such as Carruccio that Old Babylonians " may have used the abacus for the operations of addition and subtraction ; however, this primitive device proved difficult to use for more complex calculations ".

The stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of the Earth ’ s axis.

The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and " vigorous debate " ensued until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.

Nevertheless, the increasing strength of the Arians proved a formidable task for Ambrose.

The investment proved a great success and a source of profit for Woodruff and Carnegie.

The English longbow proved its worth for the first time in Continental warfare at the Battle of Crécy.

The Weil conjecture on Tamagawa numbers proved resistant for many years.