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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 mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between and at some point within the interval.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
In topology, 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. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it.". More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.
In mathematics, the Seifert–Van Kampen theorem of algebraic topology, sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space in terms of the fundamental groups of two open, path-connected subspaces that cover . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring of a triangulation of an -dimensional simplex contains a cell whose vertices all have different colors.
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0).
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each one of them in half (with respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. This is even possible if the objects overlap.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, the Poincaré–Hopf theorem is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In graph theory, a knight's graph, or a knight's tour graph, is a graph that represents all legal moves of the knight chess piece on a chessboard. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other. More specifically, an knight's graph is a knight's graph of an chessboard. Its vertices can be represented as the points of the Euclidean plane whose Cartesian coordinates are integers with and , and with two vertices connected by an edge when their Euclidean distance is .
In graph theory, the hypercube graphQn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Qn has 2n vertices, 2n – 1n edges, and is a regular graph with n edges touching each vertex.
Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques of an undirected graph.
Masonic chess is a chess variant invented by George R. Dekle Sr. in 1983. The game is played on a modified chessboard whereby even-numbered ranks are indented to the right—resembling masonry brickwork. The moves of the pieces are adapted to the new geometry; in other respects the game is the same as chess.
Cross chess is a chess variant invented by George R. Dekle Sr. in 1982. The game is played on a board comprising 61 cross-shaped cells, with players each having an extra rook, knight, and pawn in addition to the standard number of chess pieces. Pieces move in the context of a gameboard with hexagonal cells, but Cross chess has its own definition of ranks and diagonals.
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intuitively, it states that a set of points that is large in its Hausdorff dimension must determine a set of distances that is large in measure. More precisely, if is a compact set of points in -dimensional Euclidean space whose Hausdorff dimension is strictly greater than , then the conjecture states that the set of distances between pairs of points in must have nonzero Lebesgue measure.
In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. It says as follows:
References
- 1 2 Kulpa, Władysław; Socha, Lesƚaw; Turzański, Marian (2000). "Steinhaus chessboard theorem". Acta Universitatis Carolinae. Mathematica et Physica. 041 (2): 47–50. ISSN 0001-7140.
- ↑ Gale, David (December 1979). "The Game of Hex and the Brouwer Fixed-Point Theorem". The American Mathematical Monthly. 86 (10): 818–827. doi:10.1080/00029890.1979.11994922. ISSN 0002-9890.
- ↑ Tkacz, Przemysław; Turzański, Marian (2008-01-01). "An n-dimensional version of Steinhaus' chessboard theorem". Topology and its Applications. Proceedings of the Tenth Prague Symposium on General Topology and its Relations to Modern Analysis and Algebra. 155 (4): 354–361. doi:10.1016/j.topol.2007.07.005. ISSN 0166-8641.
- ↑ Ahlbach, Connor (2013-05-12). "A Discrete Approach to the Poincare-Miranda Theorem". HMC Senior Theses.
- ↑ Morawiec, Adam (2019-03-01). "On Rooks, Marriages, and Matchings or Steinhaus via Hall". Graphs and Combinatorics. 35 (2): 503–511. doi:10.1007/s00373-019-02015-4. ISSN 1435-5914.