The dual code of any (δ+αu2)-constacyclic code over F2m[u]∕〈u4〉 of oddly even length

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $R={\mathbb{F}}_{2^m}\left[u\right]∕〈u^4〉={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+u^2{\mathbb{F}}_{2^m}+u^3{\mathbb{F}}_{2^m}$ $(u^4=0)$ which is a finite chain ring, and $n$ is an odd positive integer. For any $\delta ,\alpha \in {\mathbb{F}}_{2^m}^{\times }$, an explicit representation for the dual code of any $(\delta +\alpha u^2)$-constacyclic code over $R$ of length $2n$ is given. And some dual codes of $(1+u^2)$-constacyclic codes over $R$ of length 14 are constructed. For the case of $\delta =1$, all distinct self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of length $2n$ are determined.     

浅析嵇康“越名教而任自然”的道德意蕴

嵇康是魏晋时期著名的思想家之一。在司马氏利用名教控制天下的情况下，嵇康提出“越名教而任自然”的主张来反对虚伪名教的束缚，提倡个性解放，主张万物要遵循自然规律与法则，顺应其自然本性发展。实际上嵇康并不是反对一切道德和名教，而是通过“任自然”来回归到原初时代。顺应自然和人之性情的名教，是要超越现在而回归过去无欲、无贪的状态。     

Two-Dimensional Germanium Sulfide Nanosheets as an Ultra-Stable and High Capacity Anode for Lithium Ion Batteries

Lithium ion batteries (LIBs) at present still suffer from low rate capability and poor cycle life during fast ion insertion/extraction processes. Searching for high-capacity and stable anode materials is still an ongoing challenge. Herein, a facile strategy for the synthesis of ultrathin GeS₂ nanosheets with the thickness of 1.1 nm is reported. When used as anodes for LIBs, the two-dimensional (2D) structure can effectively increase the electrode/electrolyte interface area, facilitate the ion transport, and buffer the volume expansion. Benefiting from these merits, the as-synthesized GeS₂ nanosheets deliver high specific capacity (1335 mAh g⁻¹ at 0.15 A g⁻¹), extraordinary rate performance (337 mAh g⁻¹ at 15 A g⁻¹) and stable cycling performance (974 mAh g⁻¹ after 200 cycles at 0.5 A g⁻¹). Importantly, our fabricated Li-ion full cells manifest an impressive specific capacity of 577 mAh g⁻¹ after 50 cycles at 0.1 A g⁻¹ and a high energy density of 361 Wh kg⁻¹ at a power density of 346 W kg⁻¹. Furthermore, the electrochemical reaction mechanism is investigated by the means of ex-situ high-resolution transmission electron microscopy. These results suggest that GeS₂ can use to be an alternative anode material and encourage more efforts to develop other high-performance LIBs anodes.     

Linear codes associated to skew-symmetric determinantal varieties

In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum distances of these codes are determined and a recursive formula for the weight of a general codeword in these codes is given.     

Molybdenum Disulfide Based Nanomaterials for Rechargeable Batteries

The rapid development of electrochemical energy storage systems requires new electrode materials with high performance. As a two-dimensional material, molybdenum disulfide (MoS₂) has attracted increasing interest in energy storage applications due to its layered structure, tunable physical and chemical properties, and high capacity. In this review, the atomic structures and properties of different phases of MoS₂ are first introduced. Then, typical synthetic methods for MoS₂ and MoS₂-based composites are presented. Furthermore, the recent progress in the design of diverse MoS₂-based micro/nanostructures for rechargeable batteries, including lithium-ion, lithium-sulfur, sodium-ion, potassium-ion, and multivalent-ion batteries, is overviewed. Additionally, the roles of advanced in situ/operando techniques and theoretical calculations in elucidating fundamental insights into the structural and electrochemical processes taking place in these materials during battery operation are illustrated. Finally, a perspective is given on how the properties of MoS₂-based electrode materials are further improved and how they can find widespread application in the next-generation electrochemical energy-storage systems.     

New lower bounds on the size of (n,r)‐arcs in PG(2,q)

An $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ is a set of $n$ points such that each line contains at most $r$ of the selected points. It is well known that $\left(n,r\right)$‐arcs in $\text{PG}\left(2,q\right)$ correspond to projective linear codes. Let $m_r\left(2,q\right)$ denote the maximal number $n$ of points for which an $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ exists. In this paper we obtain improved lower bounds on $m_r\left(2,q\right)$ by explicitly constructing $\left(n,r\right)$‐arcs. Some of the constructed $\left(n,r\right)$‐arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.