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The emission of neutrons from heavy ion reactions is an important observable for studying the asymmetric nuclear equation of state and the reaction dynamics. A 20-unit neutron array has been developed and mounted on the compact spectrometer for heavy ion experiments (CSHINE) to measure the neutron spectra, neutron-neutron and neutron-proton correlation functions. Each unit consists of a $\rm 15\times 15\times 15~cm^3$ plastic scintillator coupled to a $ \phi=52 ~\rm mm$ photomultiplier. The Geant4 simulation with optical process is performed to investigate the time resolution and the neutron detection efficiency. The inherent time resolution of 212 ps is obtained by cosmic ray coincidence test. The n-$\gamma$ discrimination and time-of-flight performance are given by $\rm ^{252}Cf$ radioactive source test and beam test. The neutron energy spectra have been obtained in the angle range $30^\circ \le \theta_{\rm lab} \le 51^\circ$ in the beam experiment of $^{124}$Sn+$^{124}$Sn at 25 MeV/u with CSHINE., Comment: 8 pages, 11 figures

A trigger system has been designed and implemented for the HIRFL-CSR external target experiment (CEE), the spectrometer for studying nuclear matter properties with heavy ion collisions in the GeV energy region. The system adopts master-slave structure and serial data transmission mode using optical fiber to deal with different types of detectors and long-distance signal transmission. The trigger logic can be accessed based on command register and controlled by a remote computer. The overall field programmable gate array (FPGA) logic can be flexibly reconfigured online to match the physical requirements of the experiment. The trigger system has been tested in beam experiment. It is demonstrated that the trigger system functions correctly and meets the physical requirements of CEE.

Let $H$, $T$ and $C_n$ be a graph, a tree and a cycle of order $n$, respectively. Let $H^{(i)}$ be the complete join of $H$ and an empty graph on $i$ vertices. Then the Cartesian product $H\Box T$ of $H$ and $T$ can be obtained by applying zip product on $H^{(i)}$ and the graph produced by zip product repeatedly. Let $\textrm{cr}_{\Sigma}(H)$ denote the crossing number of $H$ in an arbitrary surface $\Sigma$. If $H$ satisfies certain connectivity condition, then $\textrm{cr}_{\Sigma}(H\Box T)$ is not less than the sum of the crossing numbers of its ``subgraphs". In this paper, we introduced a new concept of generalized periodic graphs, which contains $H\Box C_n$. For a generalized periodic graph $G$ and a function $f(t)$, where $t$ is the number of subgraphs in a decomposition of $G$, we gave a necessary and sufficient condition for $\textrm{cr}_{\Sigma}(G)\geq f(t)$. As an application, we confirmed a conjecture of Lin et al. on the crossing number of the generalized Petersen graph $P(4h+2,2h)$ in the plane. Based on the condition, algorithms are constructed to compute lower bounds on the crossing number of generalized periodic graphs in $\Sigma$. In special cases, it is possible to determine lower bounds on an infinite family of generalized periodic graphs, by determining a lower bound on the crossing number of a finite generalized periodic graph., Comment: 26 pages, 20 figures

Let $\textrm{cr}(G)$ denote the crossing number of a graph $G$. The well-known Zarankiewicz's conjecture (ZC) asserted $\textrm{cr}(K_{m,n})$ in 1954. In 1971, Harborth gave a conjecture (HC) on $\textrm{cr}(K_{x_1,...,x_n})$. HC on $K_{1,m,n}$ is verified if ZC is true by Ho et al. in 2021. In this paper, we showed the following results: If both $m$ and $n$ are even, then \[\textrm{cr}(K_{1,1,m,n})\geq \frac{1}{2}(\textrm{cr}(K_{m+1,n+3})+\textrm{cr}(K_{m+3,n+1})-mn-\frac{1}{4}(m^2+n^2));\] If both $m$ and $n$ are odd, then \[\textrm{cr}(K_{1,1,m,n})\geq \frac{1}{2}(\textrm{cr}(K_{1,m+1,n+1})+\textrm{cr}(K_{2,m,n})-\frac{1}{4}(m+1)(n+1)+1);\] If $m$ is even and $n$ is odd, then \begin{equation}\nonumber \begin{split} \textrm{cr}(K_{1,1,m,n})&\geq \frac{1}{4}(\textrm{cr}(K_{m+1,n+2})+\textrm{cr}(K_{m+3,n+2})+2\textrm{cr}(K_{2,m,n}) \\&-m(n+1)-\frac{1}{4}(n+1)^2). \end{split} \end{equation} The lower bounds in our result imply that if both $m$ and $n$ are even and ZC is true, then HC on $K_{1,1,m,n}$ holds; if at least one of $m$ and $n$ is odd and both ZC and HC on $K_{2,m,n}$ are true, then HC on $K_{1,1,m,n}$ holds., Comment: 22 pages, 11 figures

In a graph $G=(V,E)$, each vertex $v\in V$ is labelled with $0$, $1$ or $2$ such that each vertex labelled with $0$ is adjacent to at least one vertex labelled $2$ or two vertices labelled $1$. Such kind of labelling is called an Italian dominating function (IDF) of $G$. The weight of an IDF $f$ is $w(f)=\sum_{v\in V}f(v)$. The Italian domination number of $G$ is $\gamma_{I}(G)=\min_{f} w(f)$. Gao et al. (2019) have determined the value of $\gamma_I(P(n,3))$. In this article, we focus on the study of the Italian domination number of generalized Petersen graphs $P(n, k)$, $k\neq3$. We determine the values of $\gamma_I(P(n, 1))$, $\gamma_I(P(n, 2))$ and $\gamma_I(P(n, k))$ for $k\ge4$, $k\equiv2,3(\bmod5)$ and $n\equiv0(\bmod5)$. For other $P(n,k)$, we present a bound of $\gamma_I(P(n, k))$. With the obtained results, we partially solve the open problem presented by Bre\v{s}ar et al. (2007) by giving $P(n,1)$ is an example for which $\gamma_I=\gamma_{r2}$ and characterizing $P(n,2)$ for which $\gamma_I(P(n,2))=\gamma_{r2}(P(n,2))$. Moreover, our results imply $P(n,1)$ $(n\equiv0(\bmod\ 4))$ is Italian, $P(n,1)$ $(n\not\equiv0(\bmod\ 4))$ and $P(n,2)$ are not Italian.

The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup E(G_n)$ with $|F|\leq n-2$. We show that for arbitrary two different correct vertices $u$ and $v$, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^n-f_v-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_n-F$($n\geq5$).