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  2. Upper and lower bounds - Wikipedia

    en.wikipedia.org/wiki/Upper_and_lower_bounds

    The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.

  3. Turán's brick factory problem - Wikipedia

    en.wikipedia.org/wiki/Turán's_brick_factory_problem

    More generally, it has been proven that every complete bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound. Closing the gap between this lower bound and the upper bound remains an open problem. [16]

  4. Crossing number inequality - Wikipedia

    en.wikipedia.org/wiki/Crossing_number_inequality

    For instance, the Szemerédi–Trotter theorem, an upper bound on the number of incidences that are possible between given numbers of points and lines in the plane, follows by constructing a graph whose vertices are the points and whose edges are the segments of lines between incident points. If there were more incidences than the Szemerédi ...

  5. Coin problem - Wikipedia

    en.wikipedia.org/wiki/Coin_problem

    Formulae [9] and fast algorithms [10] are known for three numbers though the calculations can be very tedious if done by hand. Simpler lower and upper bounds for Frobenius numbers for n = 3 have also been determined. The asymptotic lower bound due to Davison

  6. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...

  7. Directed set - Wikipedia

    en.wikipedia.org/wiki/Directed_set

    A subset of a poset is downward-directed if and only if its upper closure is a filter. Directed subsets are used in domain theory, which studies directed-complete partial orders. [5] These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of ...

  8. Least-upper-bound property - Wikipedia

    en.wikipedia.org/wiki/Least-upper-bound_property

    A real number x is called an upper bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real ...

  9. Nested intervals - Wikipedia

    en.wikipedia.org/wiki/Nested_intervals

    The construction follows a recursion by starting with any number , that is not an upper bound (e.g. =, where and an arbitrary upper bound of ). Given I n = [ a n , b n ] {\displaystyle I_{n}=[a_{n},b_{n}]} for some n ∈ N {\displaystyle n\in \mathbb {N} } one can compute the midpoint m n := a n + b n 2 {\displaystyle m_{n}:={\frac {a_{n}+b_{n ...

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