We correct errors in the proof of [7, Theorem 19] and in the description of [7, Table 1]. We also fix some typos and notational issues. We thank Thomas Kahle for pointing out the mistake in the table and Robert Krone for finding the gap in our original proof. We begin with a comment to avoid confusion in the proof of [7, Proposition 13]. Let dlex be the partial ordering of [P]k (as defined in [...

where σ T (x) = κ(x,x) − k1:T (x)K−1k1:T (x) and this bound is tight. Moreover, σ T (x) is the posterior predictive variance of a Gaussian process with the same kernel. Lemma 3 (Adapted from Proposition 1 of de Freitas et al. (2012)). Let κ : R × R → R be a kernel that is twice differentiable along the diagonal {(x,x) |x ∈ RD}, with L defined as in Lemma 1.1, and f be an element of the RKHS wit...

In this appendix, we prove Theorem 4 and Lemmas 7 – 12 in order. A.1. Proof of Theorem 4. We first need a lemma for perturbation bound of square root matrices. Lemma 16. Let A, B be positive semi-definite matrices, and then for any unitarily invariant norm ï¿¿·ï¿¿, ï¿¿A 1/2 − B 1/2 ï¿¿ ≤ 1 σ min (A 1/2) + σ min (B 1/2) ï¿¿A − Bï¿¿. Proof. The proof essentially follows the idea of [27]. Let D = ...

We introduce the categories of algebraic σ-varieties and σ-groups over a difference field (K,σ). Under a “linearly σ-closed” assumption on (K,σ) we prove an isotriviality theorem for σ-groups. This theorem yields immediately the key lemma in a proof of the Manin-Mumford conjecture. The present paper uses crucially ideas from [10] but in a model theory free manner. The applications to Manin-Mumf...

In [1] we introduced a class of positive recurrent Markov chains, named tame chains. A perfect simulation algorithm, based on the method of dominated CFTP, was then shown to exist in principle for such chains. The construction of a suitable dominating process was flawed, in that it relied on an incorrectly stated lemma ([1], Lemma 6). This claimed that a geometrically ergodic chain, subsampled ...

and Applied Analysis 3 Lemma 1.4 see 2 . Let X ⊃ φ be a BK-space and Y any of the spaces c0, c, or ∞. If A ∈ X,Y , then ‖LA‖ ‖A‖ X, ∞ sup n ‖An‖X < ∞, 1.4 where ‖A‖ X, ∞ denotes the operator norm for the matrix A ∈ X, ∞ . Sargent 3 defined the following sequence spaces. Let C denote the space whose elements are finite sets of distinct positive integers. Given any element σ of C, we denote by c ...

and Applied Analysis 3 Combining Lemma 2 in [45] and Lemma 1 in [28], one can obtain the remaining proof easily. Remark 4. WhenT → 0, system (2) degenerates to a general continuous-time positive switched system as follows: ?̇? (t) = A σ(t) x (t) + A dσ(t) x (t − d (t)) + D σ(t) w (t) , x (t) = φ (t) , t ∈ [−d, 0] z (t) = C σ(t) x (t) + E σ(t) w (t) , (5) where d(t) denotes the time-varying delay...

Lemma 2. For any r∗ ∈ (r, r̃) and ε > 0, there exist η̄ > 0 and ρ̄ < r/r∗ such that for any (η, ρ) < (η̄, ρ̄), conditions (3)-(2) below admit a solution (x0, x̂, θ0, θ00) that satisfies θ0 ≤ θ00, |x0 − x∗| < ε, ̄̄θ0 − θ∗ ̄̄ < ε, ̄̄θ00 − θ∗∗ ̄̄ < ε, and x̂ < −1/ε. 1−Ψ(x−θ σ ) = r − (r − ρr∗)[Ψ(x 0−θ0 σ )−Ψ(x 0−θ00 σ )] (1) 1−Ψ( x̂−θ0 σ ) = r + [r∗ρ+ r∗(1− ρ) exp( r ∗−r η )− r][Ψ( x̂−θ 0 σ )−Ψ( x̂−θ 00 σ )] (2) θ0...

and Applied Analysis 3 Lemma 2.1 see 8, 9 . 1 If x ∈ L1 0, 1 , ρ > σ > 0, and n ∈ N, then IIx t I x t , DtIx t Iρ−σx t , 2.4 DtIx t x t , d dtn Dtx t Dt x t . 2.5 2 If ν > 0, σ > 0, then Dttσ−1 Γ σ Γ σ − ν t σ−ν−1. 2.6 Lemma 2.2 see 8 . Assume that x ∈ L1 0, 1 and μ > 0. Then IDtx t x t c1tμ−1 c2tμ−2 · · · cntμ−n, 2.7 where ci ∈ R i 1, 2, . . . , n , n is the smallest integer greater than or eq...

ion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. Abstraction of Γ, x : N `M : Nion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. nat× σ′ (γ × nat)× σ′ γ × σ nat× (γ × σ′) Abstraction of Γ, x : N `M : Nion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. nat× τ γ × σ ∀σ.∃φ.∀τ . φ× σ (φ× nat)× τ nat× (φ× τ) λx:X.M Abstraction of Γ, x : N `M : Nion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. nat× τ γ ...