Zeros of polynomials over finite Witt rings

Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order $q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two positive integers $m,n$, define a box $\mathcal B_m$ to be a subset of $\mathbb{Z}_q^n$ with $q^{nm}$ elements such that $\mathcal B_m$ modulo $p^m$ is equal to $(\mathbb{Z}_q/p^m \mathbb{Z}_q)^n$. For a collection of nonconstant polynomials $f_1,\dots,f_s\in \mathbb{Z}_q[x_1,\ldots,x_n]$ and positive integers $m_1,\dots,m_s$, define the set of common zeros inside the box $\mathcal B_m$ to be V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\mbox{ for all } 1\leq i\leq s\}. It is an interesting problem to give the sharp estimates for the $p$-divisibility of $|V|$. This problem has been partially solved for the three cases: (i) $m=m_1=\cdots=m_s=1$, which is just the Ax-Katz theorem, (ii) $m=m_1=\cdots=m_s>1$, which was solved by Katz, Marshal and Ramage, and (iii) $m=1$, and $m_1,\dots,m_s\geq 1$, which was recently solved by Cao, Wan and Grynkiewicz. Based on the multi-fold addition and multiplication of the finite Witt rings over $\mathbb{F}_q$, we investigate the remaining unconsidered case of $m>1$ and $m\neq m_j$ for some $1\leq j\leq s$, and finally provide a complete answer to this problem

Avatar Knowledge Distillation: Self-ensemble Teacher Paradigm with Uncertainty

Knowledge distillation is an effective paradigm for boosting the performance of pocket-size model, especially when multiple teacher models are available, the student would break the upper limit again. However, it is not economical to train diverse teacher models for the disposable distillation. In this paper, we introduce a new concept dubbed Avatars for distillation, which are the inference ensemble models derived from the teacher. Concretely, (1) For each iteration of distillation training, various Avatars are generated by a perturbation transformation. We validate that Avatars own higher upper limit of working capacity and teaching ability, aiding the student model in learning diverse and receptive knowledge perspectives from the teacher model. (2) During the distillation, we propose an uncertainty-aware factor from the variance of statistical differences between the vanilla teacher and Avatars, to adjust Avatars' contribution on knowledge transfer adaptively. Avatar Knowledge Distillation AKD is fundamentally different from existing methods and refines with the innovative view of unequal training. Comprehensive experiments demonstrate the effectiveness of our Avatars mechanism, which polishes up the state-of-the-art distillation methods for dense prediction without more extra computational cost. The AKD brings at most 0.7 AP gains on COCO 2017 for Object Detection and 1.83 mIoU gains on Cityscapes for Semantic Segmentation, respectively.Comment: Accepted by ACM MM 202

Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. First, the well-posedness, which implies global existence of classical solutions, is proved. Then, we establish critical conditions for the destabilization of the coexistence equilibrium via Turing/Turing-Turing bifurcations by describing the first Turing bifurcation curve; we also theoretically predict possible bistable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that the coexistence equilibrium can also be destabilized via Hopf, Hopf-Hopf and Turing-Hopf bifurcations; also possible stable/bistable spatially inhomogeneous staggered periodic patterns and bistable spatially inhomogeneous synchronous periodic patterns are theoretically predicted. Finally, numerical experiments also support theoretical predictions and partially extend them. In a word, theoretical analyses indicate that, on the one hand, strong predator-taxis can eliminate spatial patterns caused by self-diffusion; on the other hand, the joint effects of predator-taxis and conversion time delay can induce complex survival patterns, e.g., bistable spatially heterogeneous staggered/synchronous periodic patterns, thus diversifying populations' survival patterns