Based on Perelman’s entropy monotonicity, uniform logarithmic Sobolev inequalities along the Ricci flow are derived. Then uniform Sobolev inequalities along the Ricci flow are derived via harmonic analysis of the integral transform of the relevant heat operator. These inequalities are fundamental analytic properties of the Ricci flow. They are also extended to the volume-normalized Ricci flow and the Kähler–Ricci flow.

In recent years, there has been a growing usage of sparse representations in signal processing. This paper revisits the K-SVD, an algorithm for designing overcomplete dictionaries for sparse and redundant representations. We present a new approach to solve dictionary learning models by combining the alternating direction method of multipliers and the orthogonal matching pursuit. The experimental results show that our approach can reliably obtain better learned dictionary elements and outperform other algorithms.

It is known that a minimal prime system is either a subshift or with a connected phase space (Keynes and Newton Trans Am Math Soc 217:237–255, 1976). We show that a double minimal system is a subshift; this implies immediately that no non-periodic map has 4-fold topological minimal self-joinings. We also prove that a POD system is either uniformly rigid or is a subshift.

In this short note, we prove that an almost calibrated Lagrangian translating soliton must be a plane if it has weighted integrable mean curvature vector or weighted quadratic area growth. Similar results are also true for symplectic translating solitons.

Suppose that $G$ is a finite $p$-group. If all subgroups of index $p^t$ of $G$ are abelian and at least one subgroup of index $p^{t-1}$ of $G$ is not abelian, then $G$ is called an ${\mathcal {A}}_t$-group. We use ${\mathcal {A}}_0$-group to denote an abelian group. From the definition, we know every finite non-abelian $p$-group can be regarded as an ${\mathcal {A}}_t$-group for some positive integer $t$. ${\mathcal {A}}_1$-groups and ${\mathcal {A}}_2$-groups have been classified. Classifying ${\mathcal {A}}_3$-groups is an old problem. In this paper, some general properties about ${\mathcal {A}}_t$-groups are given. ${\mathcal {A}}_3$-groups are completely classified up to isomorphism. Moreover, we determine the Frattini subgroup, the derived subgroup and the center of every ${\mathcal {A}}_3$-group, and give the number of ${\mathcal {A}}_1$-subgroups and the triple $(\mu _0,\mu _1,\mu _2)$ of every ${\mathcal {A}}_3$-group, where $\mu _i$ denotes the number of ${\mathcal {A}}_i$-subgroups of index $p$ of ${\mathcal {A}}_3$-groups.