Given an elliptic curve , flat Ek-bundles over  are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing  as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an Ad-singularity containing  as an anti-canonical curve and Kac–Moody  Ek-bundles over  with k = 8−d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding  Ek-bundles over  can be reduced to Ek-bundles.

The construction of the Q-operator for twisted affine superalgebra Uq(C(2)(2)) is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek <i>et al.</i> in Taiwan. J.Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo <i>et al.</i> (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J.Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J.Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive

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We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface.