Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients.

Babakhani, A.; Enteghami, E.

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In the first part, we investigate the singular BVP ${\frac{d}{d t}}^{c} D^{α} u + {(a / t)}^{c} D^{α} u = ℋ u$ , u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $ℋ$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems ${\frac{d}{d t}}^{c} D^{α_{n}} u + {(a / t)}^{c} D^{α_{n}} u = f (t, u,^{c} D^{β_{n}} u)$ , u(0) = A, u(1) = B, ${^{c} D^{α_{n}} u (t)|}_{t = 0} = 0$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t,...

Distortion mismatch in the quantization of probability measures

Siegfried Graf, Harald Luschgy, Gilles Pagès (2008)

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We elucidate the asymptotics of the -quantization error induced by a sequence of -optimal -quantizers of a probability distribution on $ℝ^{d}$ when . In particular we show that under natural assumptions, the optimal rate is preserved as long as (and for every in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration on $ℝ^{d}$ and on the Wiener space.