### On the real cohomology of arithmetic groups and the rank conjecture for number fields

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This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established. This is the first time oscillation of these partial difference equations is discussed by employing frequency measures.

We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new non-monotone stepsize rule is used in the method. Without such an additional condition, the theorem of weak convergence is proved. Furthermore, with strongly pseudomonotone condition, the $R$-linear convergence rate of the method is established. The results obtained are applied to a variational inequality problem, and the convergence rate...

In this paper we develop analytical solutions for the Helmholtz and Laplace equations involving local fractional derivative operators. We implement the local fractional decomposition method (LFDM) for finding the exact solutions. The iteration procedure is based upon the local fractional derivative sense. The numerical results, whichwe present in this paper, show that the methodology used provides an efficient and simple tool for solving fractal phenomena arising in mathematical physics and engineering....

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