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We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann map$f$is topologically conjugate to a holomorphic map$g$, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then$g$is Collet-Eckmann, too.

This paper is devoted to the theoretical and numerical investigation of the local minimum problem in an inverse boundary value problem. We provide a mathematical analysis for the objective function of the optimal transportation type in the context of seismic full waveform inversion. In particular, we prove that the gradient obtained using the adjoint-state method does not depend on the specific choice of the Kantorovich potentials. Moreover, our frequency analysis results show a decreasing sensitivity of the reconstruction as the data misfit is concentrated in the high-frequency part. This confirms previous observations in many numerical experiments. We also propose a new method using the softplus encoding, which maps the seismic data into probability measures and, therefore, the Wasserstein metric can be applied. The softplus encoding retains the convexity of the data misfit with respect to translations and provides a parameter to tune the landscape of the objective function. The effectiveness of the proposed method is demonstrated numerically on an inversion task with the benchmark Marmousi model.

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Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence, and a system related to so-called A_1 A_1-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Backlund transformation of the Somos-4 recurrence. Related topics about Somos polynomials are also studied.