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The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G.
[9] [10] Finding the capacity of an information-stable finite state machine channel. [11] In network coding, determining whether a network is solvable. [12] [13] Determining whether a player has a winning strategy in a game of Magic: The Gathering. [14] Planning in a partially observable Markov decision process.
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.
An arrangement of nine points (related to the Pappus configuration) forming ten 3-point lines.. In discrete geometry, the original orchard-planting problem (or the tree-planting problem) asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane.
In addition to S(2,3,9), Kramer and Mesner examined other systems that could be derived from S(5,6,12) and found that there could be up to 2 disjoint S(5,6,12) systems, up to 2 disjoint S(4,5,11) systems, and up to 5 disjoint S(3,4,10) systems. All such sets of 2 or 5 are respectively isomorphic to each other.
The dynamic optimality conjecture: Do splay trees have a bounded competitive ratio?; Can a depth-first search tree be constructed in NC?; Can the fast Fourier transform be computed in o(n log n) time?
Tile(1,1) from Smith, Myers, Kaplan & Goodmann-Strauss on the left. A spectre is obtained by modifying the edges of this polygon as in the middle and right example. In May 2023 the same team (Smith, Myers, Kaplan, and Goodman-Strauss) posted a new preprint about a family of shapes, called "spectres" and related to the "hat", each of which can ...
The disk covering problem asks for the smallest real number such that disks of radius () can be arranged in such a way as to cover the unit disk.Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.