A note on some critical thresholds of Bernoulli percolation

Consider Bernoulli bond percolation a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons' conjecture of $p_{\mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahn's modified question. The key observation is that in Kahn's modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (\textit{Comm. Math. Phys. Volume 343, 725-745.} (2016)). We also study some applications for Bernoulli percolation on periodic trees

A subperiodic tree whose intermediate branching number is strictly less than the intermediate growth rate

We construct an example of a subperiodic tree whose intermediate branching number is strictly less than the intermediate growth rate. This answers a question of Amir and Yang (2022) in the negative.Comment: 8 page

The wired minimal spanning forest on the Poisson-weighted infinite tree

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.Comment: 35 page

Contextualized Non-local Neural Networks for Sequence Learning

Recently, a large number of neural mechanisms and models have been proposed for sequence learning, of which self-attention, as exemplified by the Transformer model, and graph neural networks (GNNs) have attracted much attention. In this paper, we propose an approach that combines and draws on the complementary strengths of these two methods. Specifically, we propose contextualized non-local neural networks (CN$^{\textbf{3}}$), which can both dynamically construct a task-specific structure of a sentence and leverage rich local dependencies within a particular neighborhood. Experimental results on ten NLP tasks in text classification, semantic matching, and sequence labeling show that our proposed model outperforms competitive baselines and discovers task-specific dependency structures, thus providing better interpretability to users.Comment: Accepted by AAAI201

Time Reversal Method for Arch Bridge Cables Inspection using Longitudinal Guided Waves

Arch bridge cables consist of anchor heads, steel wires parallel arranged in an equilateral hexagon and hot-extruding PE sheathing layers outside the wires. The complex structure and contact force between wires aggravates the dispersion and attenuation of guided waves in steel wires. In order to reduce the attenuation of acoustical energy, below 80kHz low-frequency longitudinal guided waves is usually adopted. Low-frequency guided waves attenuate more slowly than high-frequency waves, but the received signal packets are wider and less recognizable. In this paper, the process of the time reversal method[1] is presented and the related parameters are calculated. Over a wide frequency range, using narrow-band pulse signals with different center-frequencies to drive comb-like magnetostrictive transducer array round the cable, extract the echo signals, which contains some feature information such as flaws, anchor heads, structural noise caused by contact force between wires. By taking advantage of the time-space compression characteristics of the method, the identification of anchor heads and flaws can be improved effectively and noise can also be decreased by driving the transducers again with the time reversed signal. Verification experiments show that the acoustical energy of guided waves can be focused on the position of flaws and the amplitude of flaws echo waves can be increased. At severe dispersion frequency, time reversal focusing process can improve the signal-noise ratio and suppress dispersion phenomenon caused by structural contact force