TY - JOUR

T1 - Arbitrarily slow approach to limiting behavior

AU - Golden, K.

AU - Goldstein, S.

PY - 1991/5

Y1 - 1991/5

N2 - Let f(k, t): RN × [0, ∞)→R be jointly continuous in k and t, limt → ∞ f(k, t) = F(k)with discontinuous for a dense set Γ of k's. It is proven that there exists a dense set T of k's such that, for k ∈ Γ, f(k, t) − F(k) approaches 0 arbitrarily slowly, i.e., roughly speaking, more slowly than any expressible function g(t) → 0. This result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media, and, in fact, implies that there is no law whatsoever governing the asymptotics.

AB - Let f(k, t): RN × [0, ∞)→R be jointly continuous in k and t, limt → ∞ f(k, t) = F(k)with discontinuous for a dense set Γ of k's. It is proven that there exists a dense set T of k's such that, for k ∈ Γ, f(k, t) − F(k) approaches 0 arbitrarily slowly, i.e., roughly speaking, more slowly than any expressible function g(t) → 0. This result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media, and, in fact, implies that there is no law whatsoever governing the asymptotics.

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U2 - https://doi.org/10.1090/S0002-9939-1991-1050020-6

DO - https://doi.org/10.1090/S0002-9939-1991-1050020-6

M3 - Article

VL - 112

SP - 109

EP - 119

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -