By Ince E.L.

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The fractional Laplacian, often known as the Riesz fractional spinoff, describes an strange diffusion approach linked to random tours. The Fractional Laplacian explores functions of the fractional Laplacian in technological know-how, engineering, and different components the place long-range interactions and conceptual or actual particle jumps leading to an abnormal diffusive or conductive flux are encountered.

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Here Y and Z are the transversal chosen above. Proof : We already know the existence of a normal form for each element. Suppose g0 p 1 g1 · · · p m gm and g0 pδ1 g1 · · · pδn gn are two normal forms representing the same element. Then 1 = g0 pδ1 g1 · · · pδn (gn g1−1 )p− m · · · g1−1 p− 1 g0−1 and so there must be a p-pinch. Since the normal forms are reduced this −1 implies δn = m . Moreover for δn = −1 we must have gn gm ∈ A and hence −1 gn = gm while for δn = 1 have gn gm ∈ B and hence gn = gm since the gi and gi belong to transversals.

If A is not finitely generated, then the corresponding HNN extension G ψ = S, t | D, t−1 at = ψ(a) (a ∈ A) is not finitely presented. We prove the second assertion. The proof of the first is similar using facts about amalgamated free products. We can assume that G = S | D is a finite presentation so that the given presentation of G is finite except ψ for the t−1 at = ψ(a) relations. Assume on the contrary that G ψ is finitely presented. Then, by the extraction theorem, some finite subset {D, t−1 a1 t = ψ(a1 ), .

We define a map θ : H → L by θ(h) = p−1 hp and a map ψ : K → L by ψ(k) = k. Then for a ∈ A we have θ(a) = p−1 ap = ϕ(a) ∈ B. But in H K A=B we have a = ϕ(a) ∈ K so that θ and ψ agree on the amalgamated subgroup. Hence they define a homomorphism from H K to L. Now using the A=B theory of HNN extensions it follows that both H and K are embedded in H K. A=B The reduction theorem for amalgamated free products follow from Britton’s Lemma. To see this, suppose w = h1 k1 · · · hm km is an alternating expression in H K.