In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

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.

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.

* Crystallographic restriction theorem, in mathematics

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development.

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

In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

In mathematics, the Hahn – Banach theorem is a central tool in functional analysis.

Of course, our understanding of what the theorem really means gains in profundity as the mathematics around the theorem grows.

In mathematics, the Poincaré conjecture ( ; ) is a theorem about the characterization of the three-dimensional sphere ( 3-sphere ), which is the hypersphere that bounds the unit ball in four-dimensional space.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics.

Some, on the other hand, may be called " deep ": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.

Jürgen Schmidhuber, however, says the " set of mathematical structures " is not even well-defined, and admits only universe representations describable by constructive mathematics, that is, computer programs.

In mathematics, the principle that says that if the number of players is one more than the number of chairs, then one player is left standing, is the pigeonhole principle.

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

Key to the study is the realization that the mathematics of such jams, which the researchers call " jamitons ," are strikingly similar to the equations that describe detonation waves produced by explosions, says Aslan Kasimov, lecturer in MIT's Department of Mathematics.

Laurent says: What are mathematics helpful for?

He says certain advocates of the new topics " ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones " ( p. 17 ).

In mathematics ( specifically linear algebra ), the Woodbury matrix identity, named after Max A. Woodbury says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix.

In mathematics, a Galois extension is an algebraic field extension E / F satisfying certain conditions ( described below ); one also says that the extension is Galois.

In mathematics, the multinomial theorem says how to expand a power of a sum in terms of powers of the terms in that sum.

The theorems of mathematics, he says, apply absolutely to all things, from things divine to original matter.

In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves ( up to linear equivalence ) has a one-dimensional subspace on which it is positive definite ( not uniquely determined ), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.

In algebraic topology, a branch of mathematics, the excision theorem is a useful theorem about relative homology — given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out ( excise ) U from both spaces such that the relative homologies of the pairs ( X, A ) and ( X

In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, " fails badly to be invertible ".

In an example of an elementary mathematics in the Suàn shù shū, the square root is approximated by using an " excess and deficiency " method which says to " combine the excess and deficiency as the divisor ; ( taking ) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend.

He supports this with an analysis of the mathematics of ancient maps and of their accuracy, which he says surpassed instrumentation available at the time of the map's drafting.

Charlie's research often interferes with his relationships: as with Amita on their first date, for all they could talk about is mathematics ; Fleinhardt says that it is a common interest and they should not struggle to avoid the subject.

This leads to Charlie being confrontational, but she calms him when she says she just wants him to be " the Sean Connery of the mathematics department.

In mathematics, the Riemann series theorem ( also called the Riemann rearrangement theorem ), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges.

He states that his 6 or so ( famous ) examples of paradoxes ( antinomies ) are all examples of impredicative definition, and says that Poincaré ( 1905 – 6, 1908 ) and Russel ( 1906, 1910 ) " enunciated the cause of the paradoxes to lie in these impredicative definitions " ( p. 42 ), however, " parts of mathematics we want to retain, particularly analysis, also contain impredicative definitions.

Nemeth says that this skill allowed him to succeed in mathematics, during an era without much technology, during which even Braille was difficult to use in mathematics.

The madman says, while playing Beethoven's 9th Symphony, that mathematics are wrong, that 1 + 1 does not equal 2 but a bigger 1, and illustrates it with two drops of olive oil.

In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number t and any vectors x and y in R < sup > n </ sup >, the following inequality holds: