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26077 results in Abstract analysis

Page 1245 of 1304

  • Shapes of worn stones

  • Part of
  • William J. Firey
  • Journal:
    Mathematika / Volume 21 / Issue 1 / June 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 1-11
    Print publication:
    June 1974

  • Often stones on beaches pounded by waves wear into quite smooth, regular shapes, sometimes apparently ellipsoidal and even spherical [8]. This paper begins with an idealization of this wearing process for materials isotropic with respect to wear, then develops an equation governing the idealized process, and goes on to show that a stone which is initially convex and centrally symmetric tends to assume a spherical shape as a consequence of the governing equation. This conclusion is predicated on the assumption that the mathematical conditions describing the wearing process are those of a well-posed problem.

  • On irregularities of integer sequences

  • S. L. G. Choi
  • Journal:
    Mathematika / Volume 21 / Issue 1 / June 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 101-106
    Print publication:
    June 1974

  • Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results is

    provided m ≥ log n. Related questions and results are also discussed.

  • The average size of gaps between primes

  • Carlos Julio Moreno
  • Journal:
    Mathematika / Volume 21 / Issue 1 / June 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 96-100
    Print publication:
    June 1974

  • Erdoső has asked whether there exists a positive constant c < 1 such that

    where pn is the n-th prime. In 1943 Selberg «3» proved, on the assumption of the Riemann Hypothesis, that

    uniformly for H > 0. By an elementary extension of Selberg's method we prove the following two results free of any unproved assumptions.

Page 1245 of 1304