Anderson mixing (AM) is a powerful acceleration method for fixed-point iterations, but its computation requires storing many historical iterations. The extra memory footprint can be prohibitive when solving high-dimensional problems in a resource-limited machine. To reduce the memory overhead, we propose a novel class of short-term recurrence AM methods (ST-AM). The ST-AM methods only store two previous iterations with cheap corrections. We prove that the basic version of ST-AM is equivalent to the full-memory AM in strongly convex quadratic optimization, and with minor changes it has local linear convergence for solving general nonlinear fixed-point problems. We further analyze the convergence properties of the regularized ST-AM for nonconvex (stochastic) optimization. Finally, we apply ST-AM to several applications including solving root-finding problems and training neural networks. Experimental results show that ST-AM is competitive with the long-memory AM and outperforms many existing optimizers.

In this paper we show how to totally break the fully homomorphic encryption scheme of eprint 2017/458.

We consider a two-dimensional electron with an anomalous magnetic moment, <i>g</i>&gt;2, interacting with a nonzero magnetic field <i>B</i> perpendicular to the plane which gives rise to a flux <i>F</i>. Recent results about the discrete spectrum of the Pauli operator are extended to fields with the \mathcal{O}\left( {r^{ - 2 - \delta } } ight) decay at infinity: we show that if \mathcal{O}\left( {r^{ - 2 - \delta } } ight) exceeds an integer <i>N</i>, there is at least \mathcal{O}\left( {r^{ - 2 - \delta } } ight) bound states. Furthermore, we prove that weakly coupled bound states exist under mild regularity assumptions also in the zero flux case.

In this paper, we show the claims in the original Kipnis-Shamir attack on the HFE cryptosystems and the improved attack by Courtois that the complexity of the attacks is polynomial in terms of the number of variables are invalid. We present computer experiments and a theoretical argument using basic algebraic geometry to explain why it is so. Furthermore we show that even with the help of the powerful new Gröbner basis algorithm like F_4, the Kipnis-Shamir attack still should be exponential but not polynomial. This again is supported by our theoretical argument.

We show how the von Neumann theory of self-adjoint extensions can be used to investigatequantum systems the configuration space of which can be decomposed into parts of different dimensionalities. The method can be applied in many situations; we illustrate it on examples including point contact spectroscopy, nanotube systems, microwave resonators, or spin conductance oscillations.