A Course Of Pure Mathematics by G. Hardy

By G. Hardy

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16. GI (H) Fig. 17. GtJ (H) At last, we note that our argument goes through to combinatorial and probabilistic transitional (p, q)-Laplacians of locally finite K¨ahler graphs which act on the set of square summable functions on the set of vertices. Also, if we replace probabilistic weights by cardinalities of auxiliary paths in the definitions of adjacency and probabilistic transition operators, we can get similar results. In the forthcoming paper, we study first eigenvalue of (p, q)-Laplacians corresponding to the results in [6 ,7 ,9 ] and in their references.

Magnetic mean operators on a K¨ahler manifold, in Topics in 2. Almost Hermitian geometry and related fields, Y. G. Rio, H. Hashimoto, T. Koda & T. , 30–40, (World Scientific, Singapore, 2005). 3. , A discrete model for K¨ahler magnetic fields on a complex hyperbolic space, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, K. S. Gerdjikov & S. Dimiev eds, 1–9 (World Scientific, Singapore, 2009). , A theorem of Hadamard-Cartan type for K¨ahler magnetic 4. fields, J.

N) is called a path, ❇❍ ◗◗ ❇✑ ✂ ✂✑ ❇ or more precisely an n-step path. When there is r ◗❇r ✂ ❇ ✂✑ j0 with vj0 −1 = vj0 +1 , we say a path γ contains a ❅ ❅✂r ❇r backtracking. It is considered that paths without Fig. 1. Petersen graph backtracking on a graph correspond to geodesics on a Riemannian manifold. For a path γ we denote its origin v0 by o(γ) and its terminus vn by t(γ). Though a 1-step path, which is an edge, does not have direction, we use these notations. Given two paths σ = (v0 , .

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