On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}}\chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with $a\neq 0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^*_{q^s}}\chi^{(s)}(ac^u+bc^{-1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided also.Comment: 18 page

Self-Organized Cooperative Criticality in Coupled Complex Systems

We show that the coupled complex systems can evolve into a new kind of self-organized critical state where each subsystem is not critical, however, they cooperate to be critical. This criticality is different from the classical BTW criticality where the single system itself evolves into a critical state. We also find that the outflows can be accumulated in the coupled systems. This will lead to the emergency of spatiotemporal intermittency in the critical state

DOC: Deep Open Classification of Text Documents

Traditional supervised learning makes the closed-world assumption that the classes appeared in the test data must have appeared in training. This also applies to text learning or text classification. As learning is used increasingly in dynamic open environments where some new/test documents may not belong to any of the training classes, identifying these novel documents during classification presents an important problem. This problem is called open-world classification or open classification. This paper proposes a novel deep learning based approach. It outperforms existing state-of-the-art techniques dramatically.Comment: accepted at EMNLP 201

A Family of Binary Sequences with Optimal Correlation Property and Large Linear Span

A family of binary sequences is presented and proved to have optimal correlation property and large linear span. It includes the small set of Kasami sequences, No sequence set and TN sequence set as special cases. An explicit lower bound expression on the linear span of sequences in the family is given. With suitable choices of parameters, it is proved that the family has exponentially larger linear spans than both No sequences and TN sequences. A class of ideal autocorrelation sequences is also constructed and proved to have large linear span.Comment: 21 page