We discuss the idea of using continuous dynamical systems to model general high-dimensional nonlinear functions used in machine learning. We also discuss the connection with deep learning.

In this paper, we develop a lattice Boltzmann model for a class of one-dimensional nonlinear wave equations, including the second-order hyperbolic telegraph equation, the nonlinear Klein–Gordon equation, the damped and undamped sine-Gordon equation and double sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman–Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The results of numerical examples have been compared with the analytical solutions to confirm the good accuracy and the applicability of our scheme.

Let $(\Sigma ,g)$ be a compact Riemannian surface, $p_j\in \Sigma $, $\beta _{j}>-1$, for $j=1,\cdots ,m$. Denote $\beta =\min \{0,\beta _1,\cdots ,\beta _{m}\}$. Let $H\in C^0(\Sigma )$ be a positive function and $h(x)=H(x)\left( d_g(x,p_j)\right) ^{2\beta _j}$, where $d_g(x,p_j)$ denotes the geodesic distance between x and $p_j$ for each $j=1,\cdots ,m$. In this paper, using a method of blow-up analysis, we prove that the functional

While the convergence of alternating direction method (ADM) for two separable variables has been established for years, the validity of its direct generalization to more than two blocks has been studying now. In this paper, we propose an additional requirement on the constraints, i.e., the pair-wise linear constraints and establish the convergence of ADM for more than two blocks. Then we apply our approach to two kinds of optimization problems. We also show several numerical experiments to verify the rationality of proposed algorithm.

Let $\sigma =\{\sigma _{i}\ |\ i\in I\}$ be some partition of the set $\mathbb {P}$ of all primes and G a finite group. A set ${{{\mathcal {H}}}}$ of subgroups of G is said to be a complete Hall $\sigma $ -set of G if every member $\ne 1$ of ${{{\mathcal {H}}}}$ is a Hall $\sigma _{i}$-subgroup of G for some $i\in I$ and ${{\mathcal {H}}}$ contains exactly one Hall $\sigma _{i}$-subgroup of G for every i such that $\sigma _{i}\cap \pi (G)\ne \emptyset $. A subgroup A of G is said to be ${{\mathcal {H}}}$-permutable if A permutes with all members of the complete Hall $\sigma $-set ${{{\mathcal {H}}}}$ of G. In this paper, we study the structure of G under the assuming that some subgroups of G are ${{\mathcal {H}}}$-permutable.

The Jewett–Krieger theorem states that each ergodic system has a strictly ergodic topological model. In this article, we show that for an ergodic system one may require more properties on its strictly ergodic model. For example, the orbit closure of points in diagonal under face transforms may be also strictly ergodic. As an application, we show the pointwise convergence of ergodic averages along cubes, which was firstly proved by Assani (J Anal Math 110:241–269, 2010).