By Hernandez D. B., Spigler R.

Numerical balance of either particular and implicit Runge-Kutta equipment for fixing traditional differential equations with an additive noise time period is studied. the idea that of numerical balance of deterministic schemes is prolonged to the stochastic case, and a stochastic analogue of Dahlquist's A-stability is proposed. it really is proven that the discretization of the waft time period by myself controls the A-stability of the full scheme. The quantitative impression of implicitness upon A-stability is additionally investigated, and balance areas are given for a kin of implicit Runge-Kutta equipment with optimum order of convergence.

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On the prescribed scalar curvature problem on 4-manifolds, Duke Mathematical Journal, 84 (1996), 633-677. , Group representations arising from Lorentz conformal geometry, Journal of Functional Analysis, 74 (1987), 199-291. , Differential operators canonically associated to a conformal structure, Mathematica Scandinavica, 57-2 (1985), 293-345. , Estimates and extremal problems for the log-determinant on 4-manifolds, Communications in Mathematical Physics, 149 (1992), 241-262. , On a fourth order PDE in conformal geometry, Preprint, 1997.

5 Suppose n ≥ 5, and suppose v ∈ C 4 (B1 \ {0}) satisfies the conditions  2  in B2 \ {0}, ∆ v = 0 v≥0 in B2 \ {0},   ∆v ≥ 0 in B2 \ {0}. Then there exist a1 , a2 ≥ 0 a function b ∈ C ∞ (B1 ) with ∆2 b = 0 such that v(x) = a1 |x|4−n + a2 |x|2−n + b(x) x ∈ B 21 \ {0}. Proof Set w = ∆v. 4 with a0 = 0, so there exist a3 ≥ 0 and a function d ∈ C ∞ (B1 ) with ∆d = 0 such that w(x) = a3 |x|2−n + d(x) x ∈ B1 \ {0}. Define v˜ : B1 \ {0} to be v˜(x) = a3 |x|4−n + ∆−1 d(x) 2 (n − 4) x ∈ B1 \ {0}. where ∆−1 d denotes a classical solution of ∆u = d in B1 .

9] Bahri A. An invariant for Yamabe type flows with applications to scalar curvature problems in higher dimensions, Duke Mathematical Journal, 81 (1996), 323-466. , The Scalar-Curvature problem on the standard three-dimensional sphere, Journal of Functional Analysis, 95 (1991), 106-172. , On the prescribed scalar curvature problem on 4-manifolds, Duke Mathematical Journal, 84 (1996), 633-677. , Group representations arising from Lorentz conformal geometry, Journal of Functional Analysis, 74 (1987), 199-291.

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A -stability of Runge-Kutta methods for systems with additive noise by Hernandez D. B., Spigler R.


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