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.

Political economy was the earlier name for the subject, but economists in the latter 19th century suggested ' economics ' as a shorter term for ' economic science ' that also avoided a narrow political-interest connotation and as similar in form to ' mathematics ', ' ethics ', and so forth.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).

There is some dispute over priority of various ideas: Newton's Principia is certainly the seminal work and has been tremendously influential, and the systematic mathematics therein did not and could not have been stated earlier because calculus had not been developed.

Somewhat earlier, exploration of mathematical practice and quasi-empiricism in mathematics from the 1950s to 1980s had sought alternatives to metamathematics in social behaviours around mathematics itself: for instance, Paul Erdős's simultaneous belief in Platonism and a single " big book " in which all proofs existed, combined with his personal obsessive need or decision to collaborate with the widest possible number of other mathematicians.

It was not till after the publication of this work that Jevons became acquainted with the applications of mathematics to political economy made by earlier writers, notably Antoine Augustin Cournot and HH Gossen.

At the end of Part One, Durham reveals the full extent of his plan to Maria: after taking his earlier self-experiments to their logical conclusion, he became convinced of something he came to call the Dust Theory, which holds that there is no difference, even in principle, between physics and mathematics, and that all mathematically possible structures exist, among them our physics and therefore our spacetime.

A similar geoheliocentric model was also earlier proposed by Nilakantha Somayaji of the Kerala school of astronomy and mathematics.

These attempts were basically an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections.

In the earlier years of the Italian school under Castelnuovo, the standards of rigor were as high as most areas of mathematics.

Rather, it is correct to say that mathematicians have gradually developed more encompassing abstractions which do not invalidate earlier mathematics but do often reinterpret them in new and expanded contexts.

In his earlier years he showed an aptitude for mathematics, but eventually he devoted himself to the study of natural history and of medicine, and in 1824 he was appointed assistant naturalist to his father.

In 1997, the eastern end of the mathematics floor was dedicated to Dr. Richard Rothenberg, the math-department chairman who had died from a sudden heart attack earlier that year.

Pupils sit Standard Grade exams at the age of fifteen / sixteen, sometimes earlier, most often for up to eight subjects including compulsory exams in English, mathematics, a foreign language, a science subject and a social subject ; it is now required by the Scottish Parliament to have two hours of physical education a week.

His reputation and the influence of Sir William Boswell, the English resident, with the States-General procured his election in 1644 to the chair of mathematics in Amsterdam, after an earlier attempt immediately after Martin van den Hove left for Leiden had failed.

The earlier stages of mathematical biology were dominated by mathematical biophysics, described as the application of mathematics in biophysics, often involving specific physical / mathematical models of biosystems and their components or compartments.

The attribution, as often in mathematics, can be debated: this rule had been found 100 years earlier by Johannes Kepler, and in German is the so-called Keplersche Fassregel.

* In modern mathematics texts, scholia are marginal notes which may amplify a line of reasoning or compare it with proofs given earlier.

, one of Gauss's last students and a historian of mathematics, who was summarizing a remark made by Gauss about Eisenstein in a conversation many years earlier.

Hellenistic scholars frequently employed the principles developed in earlier Greek thought: the application of mathematics and deliberate empirical research, in their scientific investigations.

In this remake, Saiga Junki, a high school student with a gift in mathematics, learns that his archaeologist father, who disappeared years earlier, has died.