By Patrangenaru V.

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**Additional resources for 5 dimensional strictly locally homogeneous Riemannian manifolds**

**Example text**

Since the structure does not possess dangling ends, the cause of the diﬀusion anomalies is connected with tortuosity of the typical paths available for diﬀusion, however, diﬀerently from the previous examples, these paths are numerous and non-equivalent. The numerical method allowing for obtaining exact results on the properties of diﬀusion is based on exact enumeration techniques, see e. g. [10]. The idea here is to calculate the displacement probability distribution based on the exact number of ways Wi;n the particle, starting at Anomalous Diffusion on Fractal Networks Anomalous Diffusion on Fractal Networks, Figure 2 A Sierpinski gasket: a its generator b a structure after 4th iteration.

2 t, does not change the form of this distribution. TheRmoments of the displacement scale according to hr k i D P(r; t)r k d d r k D const(k)t k/2 . Another important property of the the PDF is the fact that the return probability, i. e. the probability to be at time t at the origin of the motion, P(0; t), scales as P(0; t) / t d/2 ; (1) i. e. shows the behavior depending on the substrate’s dimension. The PDF P(r; t) is the solution of the Fick’s diﬀusion equation @ P(r; t) D kP(r; t) ; @t (2) where is a Laplace operator.

Proceeding as before, the CH Eq. (44) in an operator form is as follows: The components u n ; n x u0 (x; t) D ce 0 can be elegantly computed by ; 1 u1 (x; t) D L t (u0 (x; t)) x x t C L t 1 ( 3 A0 C 2B0 C C0 ) D c2 e x t; 1 L t u D u x x t 3uu x C2u x u x x Cuu x x x ; u(x; 0) D ce jxj ; (45) where the diﬀerential operator Lt is as deﬁned above. It is important to point out that the CH equation includes three nonlinear terms; therefore, we derive the following three sets of Adomian polynomials for the terms uu x ; u x u x x , and uu x x x : A 0 D u0 u0 x ; A 1 D u0 x u1 C u0 u1 x ; (46) and so on.

### 5 dimensional strictly locally homogeneous Riemannian manifolds by Patrangenaru V.

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