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Anal. 173–197; Determinants of elliptic pseudodifferential operators. : On the non-commutative residue for pseudo-differential operators with log-polyhomogeneous symbols. Ann. Global Anal. Geom. , Manchon, D. : Stokes’ formulae on classical symbol valued forms and applications. : Renormalised Chen integrals for symbols on Rn and renormalised polyzeta functions. : From heat-operators to anomalies; a walk through various regularization techniques in mathematics and physics. de; Anomalies and regularisation techniques in mathematics and physics.

The symbol of √ −z is the Laplacian on S 1 reads for ξ ∈ R: , where |k|−z δk (ξ ). σz (x, ξ ) = k∈Z−{0} Proof. If Ht (x, y) = h t (x − y) denotes the heat-kernel of f ∈ C ∞ (S 1 , R) ∩ Ker ⊥ : √ −z ∞ 1 f = z 2 z t 2 −1 h t on S 1 we have for every f dt. 0 Taking Fourier transforms we get σz = ∞ 1 z 2 z t 2 −1 h t dt, 0 since h t f = h t · fˆ. We therefore need to compute the Fourier transform of h t and hence an explicit expression for the heat-kernel of the Laplace operator on S 1 . The heat kernel of the corresponding Laplace operator on R at time t is given by K t (x, y) = kt (x − y) with: x2 1 kt (x) := √ e− 4t , 4π t and when identifying S 1 with R/2π Z, the heat-kernel of the Laplacian on S 1 is given by Ht (x, y) = kt (x − y + 2π n).

G. [9]). The representation theory of G 2 gives rise to a second symmetric Ricci type tensor on G 2 -manifolds. Therefore one may consider two complementary Einstein equations. 7, with no compactness assumption, that if both Einstein conditions hold simultaneously on a G 2 -manifold with closed fundamental form then the fundamental form is parallel. Our main tool is the canonical connection of a G 2 -structure and its curvature. We will show that the Ricci tensor of the canonical connection is proportional to the Riemannian On the Geometry of Closed G2 -Structures 55 Ricci tensor.