Errata

• Bouligand dimension and almost Lipschitz embeddings, Pacific J. Math. 202 (2002), no. 2, 459--474.

  Theorem 3.2 on the Bouligand (or Assouad) dimension of the product of sets is wrong as stated. In fact there exists compact sets A and B such that

  dim_b(A x B) < dim_b(A) + dim_b(B),

  see Larman [Proc. London Math. Soc. (3) 17 (1967), 178--192].

  The proof in the paper is correct when A = B, in which case

  dim_b(A x A) = 2dim_b(A).

  Only this result is used in the remainder of the paper.

• Continuous Data Assimilation Using General Interpolant Observables, Journal of Nonlinear Science, 2013, pp. 1--27.

  Lemma 1 misquoted the uniform Gronwall inequality from Jones and Titi [Lemma 4.1, Physica D (1992), 165--174]. The correct statement should be

  Lemma 1. Let α be a locally integrable real valued function on (0,∞) satisfying for some 0 < T < ∞ the following conditions:

  liminf_t→∞ ∫_t^t+T α(s)ds = γ > 0

  and

  limsup_t→∞ ∫_t^t+T max(−α(s),0)ds = Γ < ∞.

  Suppose Y is an absolutely continuous non-negative function on (0,∞) such that

  dY/dt + αY ≤ 0

  almost everywhere on (0,∞). Then Y → 0 as t → ∞.

  Subsequent applications of the uniform Gronwall inequality in the paper are correct, except the words limsup should read liminf. In particular, this error does not effect any results or proofs appearing in the paper.

  In the appendix the definition of ρ is missing a minus sign and for |ξ| < 1 should read

  ρ(ξ) = K₀ exp(−1/(1−|ξ|²))