2-( v,k ,1) Designs and PSL (3, q ) where q is Odd by Shifeng D.

By Shifeng D.

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For b = a, U is a Legendre process. 2. Yt = sin2 equation dX = dt B √ t + arcsin 2 Xt , where Xt is the solution of the √ √ X(1 − X)(a − b + (a + b + 1) cos(B 2 + 2 arcsin X)) √ √ , sin(B 2 + 2 arcsin X) with X0 = x. 3. For a = b = − we have 2 √ Bt Yt = sin2 √ + arcsin x , 0 ≤ t < ζ. 4, and [21], Chapter 5. We conclude with two examples. I. , 1 W u(x) = x(1 − x)u′′ (x) + ( − x)u′ (x), 2 with D(W ) = DV (W ) described in Section 2. 78)). √ Now let fn (x) = (arcsin x)n , 0 ≤ x ≤ 1, n ≥ 1. 3, √ n Bt un (t, x) := Efn (Yt ) = E √ + arcsin x 2 [n/2] = j=0 n (2j)!

Let a, b ≤ −1 or −1 < a, b < 0. Then ϕ satisfies both (8) and (9). If a function f ∈ C[0, 1] also satisfies both (8) and (9), then for x ∈ [0, 1], t ≥ 0, |T (t)f (x) − f (0)(1 − ϕ(x)) − f (1)ϕ(x)| ≤ (Cf Kϕ + Cϕ Kf )e(a+b)t x−a (1 − x)−b . (10) 4. Let a = b = −1; then ϕ(x) = x. Let (T (t)) be the corresponding semigroup. 5). The generator of (S(t)) is W , which 2 means that S(t) = T (t/2). From (10) we get |S(t)f (x) − f (0)(1 − x) − f (1)x| ≤ (Cf + Kf )e−t x(1 − x). (11) Another proof of (11) can be obtained using the approximation of S(t) by iterates of classical Bernstein operators.

4, the following statements hold true: (i) Each operator T (t) (t ≥ 0) maps increasing continuous functions into increasing continuous functions as well as convex continuous functions into convex continuous functions. (ii) For every 0 < α ≤ 1, M ≥ 0 and t ≥ 0 T (t)(Lip(α, M )) ⊂ Lip(α, M exp(−(a + b + 2)αt)). Proof. We need only to prove (ii). Let f ∈ Lip(α, M ) and t ≥ 0. Consider a sequence (k(n))n≥1 of positive integers such that k(n) n → 2t. Replacing, if necessary, f by f /M , we can always assume that M = 1.

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