Ferromagnetic Type-II Weyl Semimetal in Pyrite Chromium Dioxide

Magnetic topological materials have recently drawn significant importance and interest, due to their topologically nontrivial electronic structure within spontaneous magnetic moments and band inversion. Based on first-principles calculations, we propose that chromium dioxide, in its ferromagnetic pyrite structure, can realize one pair of type-II Weyl points between the $N$th and $(N+1)$th bands, where $N$ is the total number of valence electrons per unit cell. Other Weyl points between the $(N-1)$th and $N$th bands also appear close to the Fermi level due to the complex topological electronic band structure. The symmetry analysis elucidates that the Weyl points arise from a triply-degenerate point splitting due to the mirror reflection symmetry broken in the presence of spin-orbital coupling, which is equivalent to an applied magnetic field along the direction of magnetization. The Weyl points located on the magnetic axis are protected by the three-fold rotational symmetry. The corresponding Fermi arcs projected on both (001) and (110) surfaces are calculated as well and observed clearly. This finding opens a wide range of possible experimental realizations of type-II Weyl fermions in a system with time-reversal breaking.Comment: 8 pages, 5 figure

Narrative in design development

This paper describes the value of narrative used with ideation tools in aiding the rapid production of product concepts and designs for masters students of graphics, fine art, product and industrial design. The ideation tools used alongside narrative included elements of divergent and convergent thinking in combination with reverse engineering and functional analysis, and practical prototyping using a range of readily adapted artefacts. Narrative was introduced and used by the students in order to ensure the development of a context and purpose for the product, artefact or system developed or proposed and to stimulate original product concepts, ideas and thinking. The concept of narrative is familiar in design. Here however the concept was reinforced using structures associated with fictional narrative. Reverse engineering exploring the deconstruction and identification of function for each component in a product was used to aid students ensure practicality in their idea implementation. This paper describes positive experiences resulting from this activity, with a particular focus on the value of narrative in developing robust concepts. The use of physical prototyping provided tangible and instant feedback for divergent and convergent phases of idea development

The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kahn. A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq 5$ then $${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and recently proved for all $d$ by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most $3$ the search could be done by hand, but for the case of maximum degree $4$ or $5$, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics

Synthetic vision and emotion calculation in intelligent virtual human modeling

The virtual human technique already can provide vivid and believable human behaviour in more and more scenarios. Virtual humans are expected to replace real humans in hazardous situations to undertake tests and feed back valuable information. This paper will introduce a virtual human with a novel collision-based synthetic vision, short-term memory model and a capability to implement the emotion calculation and decision making. The virtual character based on this model can ‘see’ what is in his field of view (FOV) and remember those objects. After that, a group of affective computing equations have been introduced. These equations have been implemented into a proposed emotion calculation process to enlighten emotion for virtual intelligent huma