By Roger C. Lyndon

From the stories: "This booklet [...] defines the limits of the topic now known as combinatorial crew concept. [...] it's a substantial fulfillment to have focused a survey of the topic into 339 pages. [...] a helpful and great addition to the literature, containing many effects no longer formerly on hand in a ebook. it is going to absolutely develop into a typical reference." Mathematical reports

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**Example text**

10), we have λi ∼ i λ /nλ+1 , for all i = 1, . . 9) i=1 c > 0, constant independent of n. , Szabados–Vértesi [100]) n k f − Fn (f ) I ≤ c ω1 f ; k λ−2 , if λ > 0. 3 we obtain n i α λ−2 1 + i , if λ > 0, |dn | ≥ cλγ / n2 nα n i=0 where the constant c (with cλ > 0) is independent of n but depends on f . If for example 0 < λ + α < 1, by ni=0 i α+λ−2 ≤ c, we obtain |dn | ≥ cλγ , n2−α c, λ > 0. (3) Two open questions appear in a natural way: (i) If n is odd then ﬁnd other points of weak preservation of partial strict-convexity for Fn (f ); (ii) What happens if n is even?

Proof. By the standard technique we obtain ω1 (Sn,λ,Lm (f ); h) ≤ Cm,f max{h||Sn,λ,Lm (f )(x)||, ||Sn,λ,Lm (f )(x) − f (x)|| + ω1 (f ; h)}, for all n ∈ N, h ∈ (0, 1). 4, the standard method gives the optimal choice for n, as solution of the equation hF1 (n, h) = F2 (n, h). Remark. 4 contains many global smoothness preservation properties, depending on the relations satisﬁed by λ, m and α. ” For example, let us suppose that λ > m + 2, λ is even and m > 1−α α . We get nEλm (n) = n(m+1)(1−α) < nm and therefore we get the equation hnm = −1 nm−(m+1)α , which gives n = h (m+1)α and ω1 (Sn,λ,Lm (f ); h) −1 ≤ Ch × (h (m+1)α )m = Ch[(m+1)α−m]/[(m+1)α] , ∀n ∈ N, 0 < h < 1.

X − xk )4 we obtain (4) hk (0) = hk (0) = hk (0) = hk (0) = 0 and hk (0) 54 2 Partial Shape Preservation, Univariate Case = 2 24 1 + xk2 (n2 − 1) > 0. 3 xk4 (4) (4) This implies hn+1−k (0) = hk (0) and therefore (n−1)/2 n (4) Kn(4) (f )(0) = (4) hk (0)f (xk ) = [f (xj ) + f (xn+1−j )]hj (0) j =1 i=1 (n−1)/2 (4) 2 n 2hj (0) + f (0)h n+1 (0) = f (0) +f (0)h n+1 (0) > f (0) 2 j =1 hi (0) = 0. i=1 (i) Because Kn (f )(0) = 0, i =, 1, 2, 3, it follows that 0 is minimum point for Kn (f )(x) and Kn (f ) is strictly convex in a neighborhood of 0, V (0).