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This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design. [39] Minimal addition chains for sequences. [40] The complexity of minimal addition chains for individual numbers is unknown. [41] Modal logic S5-Satisfiability; Pancake sorting distance problem for strings [42]
In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances. It was posed by Paul Erdős in 1946 [ 1 ] [ 2 ] and almost proven by Larry Guth and Nets Katz in 2015.
The Erdős Distance Problem consists of twelve chapters and three appendices. [5]After an introductory chapter describing the formulation of the problem by Paul Erdős and Erdős's proof that the number of distances is always at least proportional to , the next six chapters cover the two-dimensional version of the problem.
Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the ...
Falconer (1985) proved that Borel sets with Hausdorff dimension greater than (+) / have distance sets with nonzero measure. [2] He motivated this result as a multidimensional generalization of the Steinhaus theorem, a previous result of Hugo Steinhaus proving that every set of real numbers with nonzero measure must have a difference set that contains an interval of the form (,) for some >. [3]
If the Erdős–Ulam problem has a positive solution, it would provide a counterexample to the Bombieri–Lang [4] [5] conjecture and to the abc conjecture. [6] It would also solve Harborth's conjecture, on the existence of drawings of planar graphs in which all distances are integers. If a dense rational-distance set exists, any straight-line ...
A systematic solution for the paths of geodesics was given by Legendre (1806) and Oriani (1806) (and subsequent papers in 1808 and 1810). The full solution for the direct problem (complete with computational tables and a worked out example) is given by Bessel (1825). During the 18th century geodesics were typically referred to as "shortest lines".
Given the coordinates of the two points (Φ 1, L 1) and (Φ 2, L 2), the inverse problem finds the azimuths α 1, α 2 and the ellipsoidal distance s. Calculate U 1, U 2 and L, and set initial value of λ = L. Then iteratively evaluate the following equations until λ converges: