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Let $p,q$ be two integers with $p\geq q$. Given a finite graph $F$ with no isolated vertices, the generalized Ramsey achievement game of $F$ on the complete graph $K_n$, denoted by $(p,q;K_n,F,+)$, is played by two players called Alice and Bob. In each round, Alice firstly chooses $p$ uncolored edges $e_1,e_2,...,e_p$ and colors it blue, then Bob chooses $q$ uncolored edge $f_1,f_2,...,f_q$ and colors it red; the player who can first complete the formation of $F$ in his (or her) color is the winner. The generalized achievement number of $F$, denoted by ${a}(p,q;F)$ is defined to be the smallest $n$ for which Alice has a winning strategy. If $p=q=1$, then it is denoted by ${a}(F)$, which is the classical achievement number of $F$ introduced by Harary in 1982. If Alice aims to form a blue $F$, and the goal of Bob is to try to stop him, this kind of game is called the first player game by Bollob\'{a}s. Let ${a}^*(F)$ be the smallest positive integer $n$ for which Alice has a winning strategy in the first player game. A conjecture due to Harary states that the minimum value of ${a}(T)$ is realized when $T$ is a path and the maximum value of ${a}(T)$ is realized when $T$ is a star among all trees $T$ of order $n$. He also asked which graphs $F$ satisfy $a^*(F)=a(F)$? In this paper, we proved that $n\leq {a}(p,q;T)\leq n+q\left\lfloor (n-2)/p \right\rfloor$ for all trees $T$ of order $n$, and obtained a lower bound of ${a}(p,q;K_{1,n-1})$, where $K_{1,n-1}$ is a star. We proved that the minimum value of ${a}(T)$ is realized when $T$ is a path which gives a positive solution to the first part of Harary's conjecture, and ${a}(T)\leq 2n-2$ for all trees of order $n$. We also proved that for $n\geq 3$, we have $2n-2-\sqrt{(4n-8)\ln (4n-4)}\leq a(K_{1,n-1})\leq 2n-2$ with the help of a theorem of Alon, Krivelevich, Spencer and Szab\'o. We proved that $a^*(P_n)=a(P_n)$ for a path $P_n$., Comment: 13 pages

Combination of recent measurements for $h \to Z \gamma$ from ALTLAS and CMS shows an excess that the ratio of the observed and standard model (SM) predicted branching ratios $\mu = (\sigma\cdot{\cal B})_{\mathrm{obs}}/(\sigma\cdot{\cal B})_{\mathrm{SM}}$ is $2.2\pm 0.7$. If confirmed, it is a signal of new physics (NP) beyond SM. We study NP explanation for this excess. In general, for a given model it also affects the process $h \to \gamma \gamma$. Since measured branching ratio for this process agrees with SM prediction well, the model is constrained severely. We find that a minimally fermion singlets and doublet extended NP model can explain simultaneously the current data for $h \to Z \gamma$ and $h\to \gamma\gamma$. There are two solutions. One is the SM amplitude $c_Z^{\text{SM}}$ is enhanced by $\delta c_Z$ for $h \to Z \gamma$ to the observed value, but the $h \to \gamma \gamma$ amplitude $c_\gamma^{\text{SM}} +\delta c_\gamma$ is decreased to $-c_\gamma^{\text{SM}}$ to give the observed branching ratio. This seems to be a contrived solution that although cannot be ruled out simply using branching ratio measurements. We, however, find another solution which naturally enhances the $h \to Z \gamma$ to the measured value, but keeps the $h \to \gamma\gamma$ close to its SM prediction. We also comment on some phenomenology of these new fermions., Comment: 8 pages, 7 figures

We identified a new ultra-distant comet C/2019 E3 (ATLAS) exhibiting preperihelion cometary activity at heliocentric distances $\gtrsim\!20$ au, making it the fourth member of this population after C/2010 U3 (Boattini), C/2014 UN$_{271}$ (Bernardinelli-Bernstein), and C/2017 K2 (PANSTARRS). From serendipitous archival data, we conducted analyses of the comet, finding that the activity was consistent with steady-state behaviour, suggestive of sublimation of supervolatiles, that the cross-section of dust increased gradually on the inbound leg of the orbit, varying with heliocentric distances as $r_{\rm H}^{-1.5 \pm 0.4}$, and that the dust was produced at a rate of $\gtrsim\!10^2$ kg s$^{-1}$ within the observed timespan. Our modelling of the largely symmetric morphology of the comet suggests that the dust environment was likely dominated by mm-scale dust grains ejected at speeds $\lesssim\!0.4$ m s$^{-1}$ from the sunlit hemisphere of the nucleus. Assuming a typical geometric albedo of 0.05 and adopting several simplistic thermophysical models, we estimated the nucleus to be at least $\sim\!3$ km across. We also measured the colour of the comet to be consistent with other long-period comets, except being slightly bluer in $g-r$. With our astrometric measurements, we determined an improved orbit of the comet, based upon which we derived that the comet is dynamically new and that its perihelion distance will further shrink due to the Galactic tide. We conclude the paper by comparing the known characteristics of the known ultra-distant comets., Comment: 21 pages, 8 figures, 4 tables, 2 appendices. Accepted for publication in AJ

Due to finite masses and mixing, for neutrinos propagation in space-time, there is a chiral oscillation between left- and right- chiral neutrinos, besides the usual oscillation between different generations. The probability of chiral oscillation is suppressed by a factor of $m^2/E^2$ making the effect small for relativistic neutrinos. However, for non-relativistic neutrinos, this effect can be significant. In matter, the equation of motion is modified. When neutrinos produced in weak interaction pass through the matter, the eigen-energies are split into two different ones depending on the helicity of the neutrino. This results in different oscillation behavior for neutrinos with different helicity, in particular there is a new resonant effect related to the helicity state of neutrino different than the usual MSW effect. For Majorana neutrinos, chiral oscillation also depends on Majorana phases., Comment: 7 pages, 3 figures, published version

In 1982, Harary introduced the concept of Ramsey achievement game on graphs. Given a graph $F$ with no isolated vertices. Consider the following game played on the complete graph $K_n$ by two players Alice and Bob. First, Alice colors one of the edges of $K_n$ blue, then Bob colors a different edge red, and so on. The first player who can complete the formation of $F$ in his color is the winner. The minimum $n$ for which Alice has a winning strategy is the achievement number of $F$, denoted by $a(F)$. If we replace $K_n$ in the game by the completed bipartite graph $K_{n,n}$, we get the bipartite achievement number, denoted by $\operatorname{ba}(F)$. In his seminal paper, Harary proposed an open problem of determining bipartite achievement numbers for trees. In this paper, we correct $\operatorname{ba}(mK_2)=m+1$ to $m$ and disprove $\operatorname{ba}(K_{1,m})=2m-2$ from Erickson and Harary, and extend their results on bipartite achievement numbers. We also find the exact values of achievement numbers for matchings, and the exact values or upper and lower bounds of bipartite achievement numbers on matchings, stars, and double stars. Our upper bounds are obtained by deriving efficient winning strategies for Alice., Comment: 21 pages

Fujita and Kawarabayashi [J. Combin. Theory, Ser. B 98 (2008), 805--811] conjectured that for all positive integers $k$, $m$, there is a (least) non-negative integer $f_{k}(m)$ such that every $k$-connected graph $G$ with $\delta(G)\geq \lfloor\frac{3k}{2}\rfloor+ f_{k}(m)-1$ contains a connected subgraph $W$ of order $m$ such that $G-V(W)$ is still $k$-connected. Mader confirmed Fujita-Kawarabayashi's conjecture by proving $f_{k}(m)=m$ and $W$ is a path. In this paper, the authors will confirm Fujita-Kawarabayashi's conjecture again by proving $f_{k}(m)=m$ and $W$ is a spider by a new method, where a spider is a tree with at most one vertex of degree at least three. Meanwhile, this result will verify a conjecture proposed by Mader [J. Graph Theory 65 (2010), 61--69] for the case of the spider., Comment: 7 pages

In Type-II seesaw model, an electroweak triplet scalar field $\Delta$ with a non-zero vacuum expectation value (vev) $v_\Delta$ is introduced to facilitate the generation of small neutrino masses. A non-zero $v_\Delta$ also affects the W mass through the electroweak $\rho$ parameter, making it to be less than 1 as predicted by standard model (SM). The component fields in $\Delta$ come along introduce additional contributions to reduce the SM rare neutrino trident scattering cross section. These fields also induce new processes not existed in SM, such as $l_i \to \overline{ l_j} l_k l_l$ and $l_i \to l_j \gamma$. There are severe constraints on these processes which limit the effects on neutrino trident scattering and the $\rho$ parameter and therefore the W mass. The newly measured W mass by CDF makes the central value of $\rho$ parameter to be larger than 1, even larger than previously expected. Combining neutrinoless double beta decay, direct neutrino mass and oscillation data, we find a lower limit for $v_\Delta$ as a function of the triplet scalar mass $m_\Delta$, $v_\Delta > (6.3 \sim 8.4) \mathrm{eV} (100 \mathrm{GeV}/m_\Delta)$. To have significant effect on $\rho$ in this model, $v_\Delta$ needs to be in the range of a GeV or so. However this implies a very small $m_\Delta$ which is ruled out by data. We conclude that the effect of triplet vev $v_\Delta$ on the W mass can be neglected. We also find that at 3$\sigma$ level, the deviation of the ratio for Type-II Seesaw to SM neutrino trident scattering cross section predictions is reduced to be below 1, but is restricted to be larger than 0.98., Comment: 9 pages, 5 figures, match the published version in PLB

The recently proposed Graph Convolutional Networks (GCNs) have achieved significantly superior performance on various graph-related tasks, such as node classification and recommendation. However, currently researches on GCN models usually recursively aggregate the information from all the neighbors or randomly sampled neighbor subsets, without explicitly identifying whether the aggregated neighbors provide useful information during the graph convolution. In this paper, we theoretically analyze the affection of the neighbor quality over GCN models' performance and propose the Neighbor Enhanced Graph Convolutional Network (NEGCN) framework to boost the performance of existing GCN models. Our contribution is three-fold. First, we at the first time propose the concept of neighbor quality for both node classification and recommendation tasks in a general theoretical framework. Specifically, for node classification, we propose three propositions to theoretically analyze how the neighbor quality affects the node classification performance of GCN models. Second, based on the three proposed propositions, we introduce the graph refinement process including specially designed neighbor evaluation methods to increase the neighbor quality so as to boost both the node classification and recommendation tasks. Third, we conduct extensive node classification and recommendation experiments on several benchmark datasets. The experimental results verify that our proposed NEGCN framework can significantly enhance the performance for various typical GCN models on both node classification and recommendation tasks., Comment: 29 pages, 3 figures, 7 tables. Accepted to Knowledge-Based Systems

To address the issues of superconducting and charge properties in high-T$_c$ cuprates, we perform a quantum Monte Carlo study of an extended three-band Emery model, which explicitly includes attractive interaction $V_{OO}$ between oxygen orbitals. In the physically relevant parameter range, we find that $V_{OO}$ acts to strongly enhance the long-range part of d-wave pairing correlation, with a clear tendency to form long-range superconducting order in the thermodynamic limit. Simultaneously, increasing $|V_{OO}|$ renders a rapid increase of the nematic charge structure factor at most of wavevectors, especially near $\textbf{q}=(0,0)$, indicating a dramatic enhancement of nematicity and charge density waves. Our findings suggest that the attraction between oxygen orbitals in high-T$_c$ cuprates is a common thread linking their superconducting and charge properties., Comment: 5pages, 4 figures