* Monge's theorem

( Monge's 1781 memoir is also the earliest known anticipation of Linear Programming type of problems, in particular of the transportation problem.

If the radical center lies outside of all three circles, then it is the center of the unique circle ( the radical circle ) that intersects the three given circles orthogonally ; the construction of this orthogonal circle corresponds to Monge's problem.

Monge's name is one of the 72 names inscribed on the base of the Eiffel Tower.

* Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in R < sup > 3 </ sup >

The Mongean shuffle, or Monge's shuffle, is performed as follows ( by a right-handed person ): Start with the unshuffled deck in the left hand and transfer the top card to the right.

At first his solution was not accepted, since it had not taken the time judged to be necessary, but upon examination the value of the work was recognized, and Monge's exceptional abilities were recognized.

Unsurprisingly the French Revolution completely changed the course of Monge's career.

Monge's paper gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner ; the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795.

Monge's results had been anticipated by Henry Cavendish.

Leçons données aux écoles normales ( Descriptive Geometry ): a transcription of Monge's lectures.

The album produced the hit single, a cover of Yolandita Monge's " Quitame Ese Hombre ", which hit No. 1 and earned Montenegro various award nominations.

After another album, she starred an impressive run of 13 shows at the Luis A. Ferré Performing Arts Center of San Juan, following Yolandita Monge's record of 12 consecutive shows.

During Monge's term, Costa Rica proclaims an alignment with all " western democracies " and begins to work closely to the governments of Honduras, El Salvador and Guatemala, while its relationship with Nicaragua continued to deteriorate.

Monge's protocols allow an imaginary object to be drawn in such a way that it may be 3-D modeled.

See also invariance theorem.

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

Automated theorem proving ( also known as ATP or automated deduction ) is the proving of mathematical theorems by a computer program.

The binomial theorem also holds for two commuting elements of a Banach algebra.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

It is also an example of a more generalized version of the central limit theorem that is characteristic of all stable distributions, of which the Cauchy distribution is a special case.

Special cases of the Chinese remainder theorem were also known to Brahmagupta ( 7th century ), and appear in Fibonacci's Liber Abaci ( 1202 ).

Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R < sup > n </ sup > ( for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth ).

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument ( changing only that the minimal counterexample requires 6 colors ) and use Kempe chains in the degree 5 situation to prove the five color theorem.

The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs ( possibly with an uncountable number of vertices ) for which every finite subgraph is planar.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim – Skolem theorem and the compactness theorem.

" Financial economics ", at least formally, also considers investment under " certainty " ( Fisher separation theorem, " theory of investment value ", Modigliani-Miller theorem ) and hence also contributes to corporate finance theory.

Then Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.

The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes.

The Banach-Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of C ().

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

The sampling theorem asserts that, given such a bandlimited signal, the uniformly spaced discrete samples are a complete representation of the signal as long as the sampling rate is larger than twice the bandwidth B.

The prime number theorem asserts that an integer m selected at random has roughly a chance of being prime.

Note that the Bernstein-von Mises theorem asserts here the asymptotic convergence to the " true " distribution because the probability space corresponding to the discrete set of events is finite ( see above section on asymptotic behaviour of the posterior ).

In particular, the theorem asserts that integers called the quotient q and remainder r always exist and that they are uniquely determined by the dividend a and divisor d, with d ≠ 0.

This fact follows from the famous Hilbert's basis theorem named after mathematician David Hilbert ; the theorem asserts that if R is any Noetherian ring ( such as, for instance, ), R is also a Noetherian ring.

Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor − 1 also asserts uniqueness of the representation for p < sub > i </ sub > prime and e < sub > i </ sub > positive, a primary decomposition of ( n ) is essentially unique.

Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each integer can be written in essentially only one way as 1 or − 1 multiplied by a product of positive prime numbers.

According to Allen Weiss, in Mirrors of Infinity, this optical effect is a result of the use of the tenth theorem of Euclid ’ s Optics which asserts that “ the most distant parts of planes situated below the eye appear to be the most elevated .” In Fouquet ’ s time, interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden and shows the distortion of the grottos previously seen as sculptural.

A remarkable theorem of L. Claborn ( Claborn 1966 ) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C. R.

For example the fundamental theorem of Galois theory asserts that there is a one-to-one correspondence between extensions of a field and subgroups of the field's Galois group.

Fermat's theorem asserts that if p is prime, and coprime to a, then a < sup > p − 1 </ sup >

A fundamental theorem due to Jacobi asserts that q can be brought to a diagonal form

The Jordan curve theorem asserts that every Jordan curve divides the plane into an " interior " region bounded by the curve and an " exterior " region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere.

The Hopf – Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if

The first part of the Peter – Weyl theorem asserts (; ):

The theorem now asserts that the set of functions

Nevertheless, a group of results known under the general name Jordan-Hölder theorem asserts that whenever composition series exist, the isomorphism classes of simple pieces ( although, perhaps, not their location in the composition series in question ) and their multiplicities are uniquely determined.

If S is closed, bounded, and n-dimensional, and if p is a point in S, then p is k-extreme for some k < n. The theorem asserts that p is a convex combination of extreme points.

More precisely, find necessary and sufficient conditions on the tuple ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) and ( y < sub > 1 </ sub >, ..., y < sub > n </ sub >) separately, so that there is an element of R with the property that x < sub > i </ sub >· r = y < sub > i </ sub > for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.