In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a nonempty compact convex set to itself, there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset of Euclidean space to itself.
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary of non-zero length. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain.
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
In mathematics, Hall's marriage theorem, proved by Philip Hall, is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and sufficient condition for an object to exist:
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume ,
In mathematics, and more specifically number theory, the superfactorial of a positive integer is the product of the first factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.
In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other. This technique can be useful as a way of finding a formula for the number of elements of certain sets, by corresponding them with other sets that are easier to count. Additionally, the nature of the bijection itself often provides powerful insights into each or both of the sets.
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons.
In mathematics, the "happy ending problem" is the following statement:
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world.
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.
In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.
András Hajnal was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.
In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002.
Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area , it can be translated so that it includes at least points of the integer lattice. Equivalently, every bounded set of area contains a set of points whose coordinates all differ by integers.
In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after Eugen Netto.
