For any 2⩽r⩽∞,n⩾2, we prove the existence of an open set U of C^r‑self‑mappings of any n‑manifold so that a generic map f in U displays a fast growth of the number of periodic points: the number of its n‑periodic points grows as fast as asked. This complements the works of Martens–de Melo–van Strien, Kaloshin, Bonatti–Díaz–Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore, for any 1⩽r<∞ and any k⩾0, we prove the existence of an open set \hat U of k-parameter families in U such that for a generic (f_p)^p ∈ \hat U, for every ∥p∥⩽1, the map f_p displays a fast growth of the number of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.

Collecting paired training data is difficult in practice, but the unpaired samples broadly exist. Current approaches aim at generating synthesized training data from the unpaired samples by exploring the relationship between the corrupted and clean data. This work proposes LUD-VAE, a deep generative method to learn the joint probability density function from data sampled from marginal distributions. Our approach is based on a carefully designed probabilistic graphical model in which the clean and corrupted data domains are conditionally independent. Using variational inference, we maximize the evidence lower bound (ELBO) to estimate the joint probability density function. Furthermore, we show that the ELBO is computable without paired samples under the inference invariant assumption. This property provides the mathematical rationale of our approach in the unpaired setting. Finally, we apply our method to real-world image denoising and super-resolution tasks and train the models using the synthetic data generated by the LUD-VAE. Experimental results validate the advantages of our method over other learnable approaches.

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In this paper, we present a new algorithm, such that, for the small integer solution (SIS) problem, if the solution is bounded ( by an integer \beta in l_\infty norm, which we call a bounded SIS (BSIS) problem, {\it and if the difference between the row dimension n and the column dimension m of the corresponding matrix is relatively small with respect the row dimension m}, we can solve it easily with a complexity of polynomial in m.

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