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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Furthermore, for the case that is an arbitrary convex polygon, the global relation can be solved numerically in a straightforward way, for example using MATLAB. Also, for the case that is a convex polygon, the Fokas method constructs an integral representation in the Fourier complex plane. By using this representation together with the global ...
The seven selected problems span a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science. Unlike Hilbert's problems, the problems selected by the Clay Institute were already renowned among ...
[5] The subdivision of the polygon into triangles forms a planar graph, and Euler's formula + = gives an equation that applies to the number of vertices, edges, and faces of any planar graph. The vertices are just the grid points of the polygon; there are = + of them. The faces are the triangles of the subdivision, and the single region of the ...
Van der Waerden [11] cites the polynomial f(x) = x 5 − x − 1. By the rational root theorem, this has no rational zeroes. Neither does it have linear factors modulo 2 or 3. The Galois group of f(x) modulo 2 is cyclic of order 6, because f(x) modulo 2 factors into polynomials of orders 2 and 3, (x 2 + x + 1)(x 3 + x 2 + 1).
Some problems may be treated as belonging to either of the categories, depending on the context. For example, consider the following problem. Point in polygon: Decide whether a point is inside or outside a given polygon. In many applications this problem is treated as a single-shot one, i.e., belonging to the first class.
The Delian problem, for instance, was to construct a length x so that the cube of side x contained the same volume as the rectangular box a 2 b for given sides a and b. Menaechmus ( c. 350 BC ) considered the problem geometrically by intersecting the pair of plane conics ay = x 2 and xy = ab . [ 2 ]
It can be proven by forming a 3-connected planar graph with the given set of polygon faces, and then applying Steinitz's theorem to find a polyhedral realization of that graph. [ 3 ] László Lovász has shown a correspondence between polyhedral representations of graphs and matrices realizing the Colin de Verdière graph invariants of the same ...