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* Lagrange discusses representations of integers by general algebraic forms ; produces a tract on elimination theory ; publishes his first paper on the general process for solving an algebraic equation of any degree via Lagrange resolvents ; and proves Bachet's theorem that every positive integer is the sum of four squares.

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## Some Related Sentences

Lagrange and discusses

*****

__Lagrange__

__discusses__how numerous astronomical observations should be combined so as to give

**the**most probable result

**.**

Lagrange and integers

Moreover,

**by****a****theorem****of**__Lagrange__,**the**number**of**solutions modulo p to**a**congruence**of****degree**q modulo p**is**at most q ( this follows since**the**__integers__modulo p form**a**field,**and****a**polynomial**of****degree**q has at most q roots ).

Lagrange and by

French Dominicans founded

**and**administer**the**École Biblique et Archéologique française de Jérusalem founded in 1890__by__Père Marie-Joseph__Lagrange__O**.**P**.**
The Lagrangian points (; also

__Lagrange__points, L-points, or libration points ) are**the**five positions in**an**orbital configuration where**a**small object affected only__by__gravity can theoretically be part**of****a**constant-shape pattern with two larger objects ( such as**a**satellite with respect to**the**Earth**and**Moon ).
If

**the**generalized coordinates are represented as**a**vector**and**time differentiation**is**represented__by__**a**dot over**the**variable, then**the**equations**of**motion ( known as**the**__Lagrange__or Euler –__Lagrange__equations ) are**a**set**of**equations:
Many great mathematicians

**of****the**past were preoccupied__by__numerical analysis, as**is**obvious from**the**names**of**important algorithms like Newton's method,__Lagrange__interpolation polynomial, Gaussian**elimination**, or Euler's method**.**
The

**general****theory****of**Pell's**equation**, based**on**continued fractions**and****algebraic**manipulations with numbers**of****the**form was developed__by____Lagrange__in 1766 – 1769**.**
Despite initial opposition from her parents

**and**difficulties presented__by__**a**sexist society, she gained education from books in her father's library**and**from correspondence with famous mathematicians such as__Lagrange__, Legendre,**and**Gauss**.**
Another early venture was

__by__Joseph Louis__Lagrange__in**his**Theory**of**Analytic Functions ( 1797, 1813 ).
Historically,

**the**problem**of****a**vibrating string such as**that****of****a**musical instrument was studied__by__Jean le Rond d ' Alembert, Leonhard Euler, Daniel Bernoulli,**and**Joseph-Louis__Lagrange__**.**
By using

__Lagrange__multipliers**and**seeking**the**extremum**of****the**Lagrangian, it may be readily shown**that****the**solution to**the**equality constrained problem**is**given__by__**the**linear system:
Within two years

**and****a**half he had mastered all**the**subjects prescribed**for**examination,**and****a**great deal more,**and**,**on**going up**for**examination at Toulouse, he astounded**his**examiner__by__**his**knowledge**of**J**.**L**.**__Lagrange__**.**
Another unit

**for**time, more familiar than some other suggestions, could be 14**.**4 minutes, i**.**e**.****a**shorter quarter**of****an**hour, or**a**centiday, as proposed__by____Lagrange__**.**
Other attempts were made

__by__Euler ( 1749 ), de Foncenex ( 1759 ),__Lagrange__( 1772 ),**and**Laplace ( 1795 ).
A further step was

**the**1770**paper**Réflexions sur la résolution algébrique des équations__by__**the**French-Italian mathematician Joseph Louis__Lagrange__, in**his**method**of**__Lagrange__**resolvents**, where he analyzed Cardano**and**Ferrarri's solution**of**cubics**and**quartics__by__considering them in terms**of**permutations**of****the**roots, which yielded**an**auxiliary polynomial**of**lower**degree**, providing**a**unified understanding**of****the**solutions**and**laying**the**groundwork**for**group**theory****and**Galois**theory****.**
The Euler –

__Lagrange__equations**of**motion**for****the**functional E are then given in local coordinates__by__
The

**theorem**was proved__by____Lagrange__**and**generalized__by__Hans Heinrich Bürmann, both in**the**late 18th century**.**

Lagrange and general

With

**the**later development**of**abstract groups, this result**of**__Lagrange__**on**polynomials was recognized to extend to**the**__general__**theorem**about finite groups which now bears**his**name**.**
One may observe

**that****the**above computation can be repeated plainly in more__general__settings than:**a**generalization**of****the**__Lagrange__inversion formula**is**already available working in the-modules, where**is****a**complex exponent**.**
In Méchanique Analytique ( 1788 )

__Lagrange__derived**the**__general__equations**of**motion**of****a**mechanical body**.*******

__Lagrange__

**publishes**

**his**second

**paper**

**on**

**the**

__general__

**process**

**for**

**solving**

**an**

**algebraic**

**equation**

**of**

**any**

**degree**

**via**

__Lagrange__

**resolvents**

**;**

**and**

**proves**Wilson's

**theorem**

**that**if n

**is**

**a**prime, then ( n − 1 )!

He

**is**most famous as**the**inventor**of**tensor calculus, although**the**advent**of**tensor calculus in dynamics goes back to__Lagrange__, who originated**the**__general__treatment**of****a**dynamical system,**and**to Riemann, who was**the****first**to think geometry in**an**arbitrary number**of**dimensions**.**
Mark Oliver Everett

**is****the**son**of**physicist Hugh Everett III, originator**of****the**many-worlds interpretation**of**quantum**theory****and****of****the**use**of**__Lagrange__multipliers**for**__general__engineering optimizations**.**
Count Joseph

__Lagrange__( 10 January 1763 – 16 January 1836 ) was**a**French soldier who rose through**the**ranks**and**gained promotion to**the**rank**of**__general__officer during**the**French Revolutionary Wars, subsequently pursuing**a**successful career during**the**Napoleonic Wars**and**winning promotion to**the**top military rank**of**General**of**Division**.**__Lagrange__, tackling

**the**

__general__three-body problem, considered

**the**behaviour

**of**

**the**distances between

**the**bodies, without finding

**a**

__general__solution

**.**

Lagrange and algebraic

Ruffini developed Joseph Louis Lagrange's work

**on**permutation**theory**, following 29 years after__Lagrange__’ s " Réflexions sur la théorie algébrique des equations " ( 1770 – 1771 ) which was largely ignored until Ruffini who established strong connections between permutations**and****the**solvability**of**__algebraic__equations**.**

Lagrange and forms

__Lagrange__proved

**that**all

__forms__

**of**discriminant − 1

**and**are equivalent (

**a**form satisfying this conditions

**is**said to be reduced ).

Lagrange and ;

Not all

**of****the**ring material would have necessarily been swept up right away__;__**the**thickened crust**of****the**Far Side suggests**that****a**second moon about 1, 000-km in diameter formed in**a**__Lagrange__point**of****the**Moon__;__after tens**of**millions**of**years, as**the**two moons migrated outward from**the**Earth, solar tidal effects would have made**the**__Lagrange__orbit unstable, resulting in**a**slow-velocity collision**that**would have ' pancaked '**the**smaller moon onto what**is**now**the**Far Side**.**
These last

**four**attempts assumed implicitly Girard's assertion__;__to be more precise,**the**existence**of**solutions was assumed**and**all**that**remained to be proved was**that**their form was**a**+ bi**for**some real numbers**a****and**b**.**In modern terms, Euler, de Foncenex,__Lagrange__,**and**Laplace were assuming**the**existence**of****a**splitting field**of****the**polynomial p ( z ).
The oldest Methodist woman's college

**is**Wesleyan College in Macon, Georgia__;__other Methodist colleges**that**were formerly women's institutions are__Lagrange__College**and**Andrew College in Georgia, Columbia College in Columbia, South Carolina,**and**Greensboro College in Greensboro, North Carolina**.**
In 2010, however, using infrared observation techniques,

**the**asteroid was found to be**a**trojan companion**of****the**Earth__;__it librates around**the**leading__Lagrange__point,, in**a**stable orbit**.**
Jumps are normally made to

**and**from points far above**a**solar system's ecliptic, usually where**the**gravitational influence in**the**system**is**most stable__;__however, so-called " pirate points " exist where local gravitational pull**is**stable enough to used__;__though quicker, using such points**is**also more dangerous due**the**random appearance**of**so-called "__Lagrange__points ".
When

**the**probability**is**derived from**the**Gibbs measure, as it would be**for****any**Markovian**process**, then can also be understood to be**a**__Lagrange__multiplier__;____Lagrange__multipliers are used to enforce constraints, such as holding**the**expectation value**of**some quantity constant**.****is**sometimes called

**the**energy or action

**of**

**the**curve

__;__this name

**is**justified because

**the**geodesic equations are

**the**Euler –

__Lagrange__equations

**of**motion

**for**this action

**.**

For comparison, in

**the**equivalent Euler –__Lagrange__equations**of**motion**of**Lagrangian mechanics,**the**conjugate momenta also do not appear__;__however, those equations are**a**system**of**N, generally second-order equations**for****the**time evolution**of****the**generalized coordinates**.**
In mathematics,

**the**__Lagrange__reversion**theorem**gives series or formal power series expansions**of**certain implicitly defined functions__;__indeed,**of**compositions with such functions**.**0.177 seconds.