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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. Plastic limit theorems - Wikipedia

    en.wikipedia.org/wiki/Plastic_limit_theorems

    The two plastic limit theorems apply to any elastic-perfectly plastic body or assemblage of bodies. Lower limit theorem: If an equilibrium distribution of stress can be found which balances the applied load and nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure.

  4. 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 ...

  5. Limit inferior and limit superior - Wikipedia

    en.wikipedia.org/wiki/Limit_inferior_and_limit...

    Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.

  6. Divergence of the sum of the reciprocals of the primes

    en.wikipedia.org/wiki/Divergence_of_the_sum_of...

    We will now derive an upper and a lower estimate for | M x |, the number of elements in M x. For large x, these bounds will turn out to be contradictory. Upper estimate Every n in M x can be written as n = m 2 r with positive integers m and r, where r is square-free.

  7. Limits of integration - Wikipedia

    en.wikipedia.org/wiki/Limits_of_integration

    In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral () of a Riemann integrable function f {\displaystyle f} defined on a closed and bounded interval are the real numbers a {\displaystyle a} and b {\displaystyle b} , in which a {\displaystyle a} is called the lower limit and b {\displaystyle ...

  8. Bounded set - Wikipedia

    en.wikipedia.org/wiki/Bounded_set

    The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.) A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval.

  9. Chebyshev's inequality - Wikipedia

    en.wikipedia.org/wiki/Chebyshev's_inequality

    Navarro [17] proved that these bounds are sharp, that is, they are the best possible bounds for that regions when we just know the mean and the covariance matrix of X. Stellato et al. [ 18 ] showed that this multivariate version of the Chebyshev inequality can be easily derived analytically as a special case of Vandenberghe et al. [ 19 ] where ...

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