To characterize the complete structure of the Fucik spectrum of the p-Laplacian on higher dimensional domains is a long-standing problem. In this paper, we study the p-Laplacian with integrable potentials on an interval under the Dirichlet or the Neumann boundary conditions. Based on the strong continuity and continuous differentiability of solutions in potentials, we will give a comprehensive characterization of the corresponding Fucik spectra: each of them is composed of two trivial lines and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral curves are precisely described by using eigenvalues of the p-Laplacian with potentials; and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent on potentials in a certain sense.

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We show that in the generic case the smooth noncommutative tori associated with two$n$×$n$real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO($n$,$n$∣$Z$) action.

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Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional N = 2 SCFT. The SeibergWitten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding SeibergWitten differential is given by the GelfandLeray form of K. Saitos primitive form. Our result also extends the SeibergWitten solution to include irrelevant deformations.