In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of pa
ths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.
Inspired by works of Casteras (Pacific J. Math., 2015), Li-Zhu (Calc. Var., 2019) and Sun-Zhu (Calc. Var., 2020), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely $$ left{begin{array}{lll} partial_tphi(
u)=Delta u-Q+rho frac{e^u}{int_Ve^udmu}[1.5ex] u(cdot,0)=u_0, end{array}right. $$ where $Delta$ is the standard graph Laplacian, $rho$ is a real number, $Q:Vrightarrowmathbb{R}$ is a function satisfying $int_VQdmu=rho$, and $phi:mathbb{R}rightarrowmathbb{R}$ is one of certain smooth functions including $phi(s)=e^s$. We prove that for any initial data $u_0$ and any $rhoinmathbb{R}$, there exists a unique solution $u:Vtimes[0,+infty)rightarrowmathbb{R}$ of the above heat flow; moreover, $u(x,t)$ converges to some function $u_infty:Vrightarrowmathbb{R}$ uniformly in $xin V$ as $trightarrow+infty$, and $u_infty$ is a solution of the mean field equation $$Delta u_infty-Q+rhofrac{e^{u_infty}}{int_Ve^{u_infty}dmu}=0.$$ Though $G$ is a finite graph, this result is still unexpected, even in the special case $Qequiv 0$. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz-Simon type inequality and use it to conclude the convergence of the heat flow.
Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schrodinger equation, the mean f
ield equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.
In this paper, we consider the problem of synthesis of maximally permissive covert damage-reachable attackers in the setup where the model of the supervisor is unknown to the adversary but the adversary has recorded a (prefix-closed) finite set of ob
servations of the runs of the closed-loop system. The synthesized attacker needs to ensure both the damage-reachability and the covertness against all the supervisors which are consistent with the given set of observations. There is a gap between the de facto maximal permissiveness, assuming the model of the supervisor is known, and the maximal permissiveness that can be attained with a limited knowledge of the model of the supervisor, from the adversarys point of view. We consider the setup where the attacker can exercise sensor replacement/deletion attacks and actuator enablement/disablement attacks. The solution methodology proposed in this work is to reduce the synthesis of maximally permissive covert damage-reachable attackers, given the model of the plant and the finite set of observations, to the synthesis of maximally permissive safe supervisors for certain transformed plant, which shows the decidability of the observation-assisted covert attacker synthesis problem. The effectiveness of our approach is illustrated on a water tank example adapted from the literature.
Recent studies have demonstrated a perceivable improvement on the performance of neural machine translation by applying cross-lingual language model pretraining (Lample and Conneau, 2019), especially the Translation Language Modeling (TLM). To allevi
ate the need for expensive parallel corpora by TLM, in this work, we incorporate the translation information from dictionaries into the pretraining process and propose a novel Bilingual Dictionary-based Language Model (BDLM). We evaluate our BDLM in Chinese, English, and Romanian. For Chinese-English, we obtained a 55.0 BLEU on WMT-News19 (Tiedemann, 2012) and a 24.3 BLEU on WMT20 news-commentary, outperforming the Vanilla Transformer (Vaswani et al., 2017) by more than 8.4 BLEU and 2.3 BLEU, respectively. According to our results, the BDLM also has advantages on convergence speed and predicting rare words. The increase in BLEU for WMT16 Romanian-English also shows its effectiveness in low-resources language translation.
The photoproduction of bottomonium-like states $Z_{b}(10610)$ and $Z_{b}(10650)$ via $gamma p$ scattering is studied within an effectiv Lagrangian approach and the vector-meson-dominance model. The Regge model is employed to calculate the photoproduc
tion of $Z_{b}$ states via $t$-channel with $pi$ exchange.The numerical results show that the values of the total cross-sections of $Z_{b}(10610)$ and $Z_{b}(10650)$ can reach 0.09 nb and 0.02 nb, respectively, near the center of mass energy of 22 GeV. The experimental measurements and studies on the photoproduction of $Z_{b}$ states near energy region around $Wsimeq 22$ GeV is suggested. Moreover, with the help of eSTARlight and STARlight programs, one obtains the cross-sections and event numbers of $Z_{b}(10610)$ production in electron-ion collision (EIC) and Ultraperipheral collisions (UPCs). The results show that a considerable number of events from $Z_{b}(10610)$ can be produced on the relevant experiments of EICs and UPCs. Also, one calculates the rates and kinematic distributions for $gamma prightarrow Z_{b}n$ in $ep$ and $pA$ collisions via EICs and UPCs, and the relevant results will provide an important reference for the RHIC, LHC, EIC-US, LHeC, and FCC experiments to search for the bottomonium-like $Z_{b}$ states.
Inspired by the observation of the fully-charm tetraquark $X(6900)$ state at LHCb, the production of $X(6900)$ in $bar{p}prightarrow J/psi J/psi $ reaction is studied within an effective Lagrangian approach and Breit-Wigner formula. The numerical res
ults show that the cross section of $X(6900)$ at the c.m. energy of 6.9 GeV is much larger than that from the background contribution. Moreover, we estimate dozens of signal events can be detected by D0 experiment, which indicates that searching for the $X(6900)$ via antiproton-proton scattering may be a very important and promising way. Therefore, related experiments are suggested to be carried out.
Symmetry principles play a critical role in formulating the fundamental laws of nature, with a large number of symmetry-protected topological states identified in recent studies of quantum materials. As compelling examples, massless Dirac fermions ar
e jointly protected by the space inversion symmetry $P$ and time reversal symmetry $T$ supplemented by additional crystalline symmetry, while evolving into Weyl fermions when either $P$ or $T$ is broken. Here, based on first-principles calculations, we reveal that massless Dirac fermions are present in a layered FeSn crystal containing antiferromagnetically coupled ferromagnetic Fe kagome layers, where each of the $P$ and $T$ symmetries is individually broken but the combined $PT$ symmetry is preserved. These stable Dirac fermions protected by the combined $PT$ symmetry with additional non-symmorphic $S_{rm{2z}}$ symmetry can be transformed to either massless/massive Weyl or massive Dirac fermions by breaking the $PT$ or $S_{rm{2z}}$ symmetry. Our angle-resolved photoemission spectroscopy experiments indeed observed the Dirac states in the bulk and two-dimensional Weyl-like states at the surface. The present study substantially enriches our fundamental understanding of the intricate connections between symmetries and topologies of matter, especially with the spin degree of freedom playing a vital role.
The ability to assemble nanomaterials, such as quantum dots, enables the creation of functional devices that present unique optical and electronic properties. For instance, light-emitting diodes with exceptional color purity can be printed via the ev
aporative-driven assembly of quantum dots. Nevertheless, current studies of the colloidal deposition of quantum dots have been limited to the surfaces of a planar substrate. Here, we investigate the evaporation-driven assembly of quantum dots inside a confined cylindrical geometry. Specifically, we observe distinct deposition patterns, such as banding structures along the length of a capillary tube. Such coating behavior can be influenced by the evaporation speed as well as the concentration of quantum dots. Understanding the factors governing the coating process can provide a means to control the assembly of quantum dots inside a capillary tube, ultimately enabling the creation of novel photonic devices.
Most of existing rendezvous algorithms generate channel-hopping sequences based on the whole channel set. They are inefficient when the set of available channels is a small subset of the whole channel set. We propose a new algorithm called ZOS which
uses three types of elementary sequences (namely, Zero-type, One-type, and S-type) to generate channel-hopping sequences based on the set of available channels. ZOS provides guaranteed rendezvous without any additional requirements. The maximum time-to-rendezvous of ZOS is upper-bounded by O(m1*m2*log2M) where M is the number of all channels and m1 and m2 are the numbers of available channels of two users.
