Search

Consider a one dimensional critical branching L\'{e}vy process $((Z_t)_{t\geq 0}, \mathbb {P}_x)$. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction to some $\alpha$-stable distribution with $\alpha\in (1, 2)$, and that the underlying L\'{e}vy process $(\xi_t)_{t\geq 0}$ is non-lattice and has finite $2+\delta^*$ moment for some $\delta^*>0$. We first prove that $$t^{\frac{1}{\alpha-1}}\left(1- \mathbb{E}_{\sqrt{t}y}\left(\exp\left\{-\frac{1}{t^{\frac{1}{\alpha-1}-\frac{1}{2}}}\int h(x) Z_t(\mathrm{d}x) -\frac{1}{t^{\frac{1}{\alpha-1}}} \int g\left(\frac{x}{\sqrt{t}}\right)Z_t(\mathrm{d}x)\right\}\right)\right)$$ converges as $t\to\infty$ for any non-negative bounded Lipschtitz function $g$ and any non-negative directly Riemann integrable function $h$ of compact support. Then for any $y\in \R$ and bounded Borel set of positive Lebesgue measure with its boundary having zero Lebesgue measure, under a higher moment condition on $\xi$, we find the decay rate of the probability $\mathbb {P}_{\sqrt{t}y}(Z_t(A)>0)$. As an application, we prove some convergence results for $Z_t$ under the conditional law $\mathbb {P}_{\sqrt{t}y}(\cdot| Z_t(A)>0).$, Comment: 36 pages

Aspect-Based Sentiment Analysis (ABSA) in tourism plays a significant role in understanding tourists' evaluations of specific aspects of attractions, which is crucial for driving innovation and development in the tourism industry. However, traditional pipeline models are afflicted by issues such as error propagation and incomplete extraction of sentiment elements. To alleviate this issue, this paper proposes an aspect-based sentiment analysis model, ACOS_LLM, for Aspect-Category-Opinion-Sentiment Quadruple Extraction (ACOSQE). The model comprises two key stages: auxiliary knowledge generation and ACOSQE. Firstly, Adalora is used to fine-tune large language models for generating high-quality auxiliary knowledge. To enhance model efficiency, Sparsegpt is utilized to compress the fine-tuned model to 50% sparsity. Subsequently, Positional information and sequence modeling are employed to achieve the ACOSQE task, with auxiliary knowledge and the original text as inputs. Experiments are conducted on both self-created tourism datasets and publicly available datasets, Rest15 and Rest16. Results demonstrate the model's superior performance, with an F1 improvement of 7.49% compared to other models on the tourism dataset. Additionally, there is an F1 improvement of 0.05% and 1.06% on the Rest15 and Rest16 datasets, respectively., Comment: 19 pages, 17 figures

In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift $-\rho$ and offspring distribution $\{p_k:k\ge 0\}$. Let $\widetilde{\zeta}^{-\rho}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-\rho}$ the maximal position of all the particles alive at time $t$ and $\widetilde{M}^{-\rho}:=\max_{t\ge 0}\widetilde{M}_t^{-\rho}$ the all time maximal position. Let $\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at $x\in (0,\infty)$. Under the assumption $\sum_{k=1}^\infty k (\log k) p_k <\infty$, we establish the decay rates of $\mathbb{P}_x(\widetilde{\zeta}^{-\rho}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-\rho}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-\rho}>z(t,\rho))$ as $t\to\infty$, where $z(t,\rho)=\sqrt{t}z-\rho t$ for $\rho\leq 0$ and $z(t,\rho)=z$ for $\rho>0$. As a consequence, we obtain a Yaglom-type limit theorem.

In this paper, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed L\'{e}vy process $(Z_t^{(0,\infty)})_{t\ge 0}$ in $\mathbb{R}$, in which all particles (and their descendants) are killed upon exiting $(0, \infty)$. Let $\zeta^{(0,\infty)}$ and $M_t^{(0,\infty)}$ be the extinction time and maximal position of all the particles alive at time $t$ of this branching killed L\'{e}vy process and define $M^{(0,\infty)}: = \sup_{t\geq 0} M_t^{(0,\infty)}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (1, 2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities $$ \mathbb{P}_{y}(\zeta^{(0,\infty)}>t), \quad \mathbb{P}_{\sqrt{t}y}(\zeta^{(0,\infty)}>t) $$ and the tail probabilities $$ \mathbb{P}_{y}(M^{(0,\infty)}\geq x), \quad \mathbb{P}_{xy}(M^{(0,\infty)}\geq x). $$ We also study the scaling limits of $M_t^{(0,\infty)}$ and the point process $Z_t^{(0,\infty)}$ under $\mathbb{P}_{\sqrt{t}y}(\cdot |\zeta^{(0,\infty)}>t)$ and $\mathbb{P}_y(\cdot |\zeta^{(0,\infty)}>t)$. The scaling limits under $\mathbb{P}_{\sqrt{t}y}(\cdot |\zeta^{(0,\infty)}>t)$ are represented in terms of super killed Brownian motion., Comment: 54 pages, 0 figures

The logarithmic correction for the order of the maximum of a two-type reducible branching Brownian motion on the real line exhibits a double jump when the parameters (the ratio of the diffusion coefficients of the two types of particles, and the ratio of the branching rates the two types of particles) cross the boundary of the anomalous spreading region identified by Biggins. In this paper, we further examine this double jump phenomenon by studying a two-type reducible branching Brownian motion on the real line with its parameters depend on the time horizon t. We show that when the parameters approach the boundaries of the anomalous spreading region in an appropriate way, the order of the maximum can interpolate smoothly between different surrounding regimes. We also determine the asymptotic law of the maximum and characterize the extremal process., Comment: 48 pages, 2 figures

Waveguides are fundamental components in communication systems. However, they suffer from reflection and scattering losses at sharp routes or defects. The breakthrough in developing topological photonic crystals (PhCs) provides promising solutions to robust signal transmission. In this work, we propose a new mechanism for protecting wave-guiding modes by decorating the boundaries of a conventional waveguide with valley-Hall PhCs. This special layout enables the robust propagation of conventional transverse electric waves against defects and bends. Moreover, the proposed waveguide is compatible with the substrate integrated waveguide (SIW). High efficient mode conversion from the SIW to the proposed waveguide is achievable. By leveraging the idea of topology to conventional waveguides, we provide a powerful and practical tool that can largely improve the performance of microwave and millimeter-wave integrated circuits while reserving the features of wave-guiding modes., Comment: 5 figures

Let $N(t)$ be the collection of particles alive at time $t$ in a branching Brownian motion in $\mathbb{R}^d$, and for $u\in N(t)$, let $\mathbf{X}_u(t)$ be the position of particle $u$ at time $t$. For $\theta\in \mathbb{R}^d$, we define the additive measures of the branching Brownian motion by$$\mu_t^\theta (\mathrm{d}\mathbf{x}):= e^{-(1+\frac{\Vert\theta\Vert^2}{2})t}\sum_{u\in N(t)} e^{-\theta \cdot \mathbf{X}_u(t)} \delta_{\left(\mathbf{X}_u(t)+\theta t\right)}(\mathrm{d}\mathbf{x}).$$ In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for $\mu_t^\theta ((\mathbf{a}, \mathbf{b}])$ and $\mu_t^\theta ((-\infty, \mathbf{a}])$ for $\theta\in \mathbb{R}^d$ with $\Vert \theta \Vert <\sqrt{2}$. These expansions sharpen the asymptotic results of Asmussen and Kaplan (1976) and Kang (1999), and are analogs of the expansions in Gao and Liu (2021) and R\'{e}v\'{e}sz, Rosen and Shi (2005) for branching Wiener processes (a particular class of branching random walks) corresponding to $\theta=\mathbf{0}$.

Consider a critical branching L\'{e}vy process $\{X_t, t\ge 0\}$ with branching rate $\beta>0, $ offspring distribution $\{p_k:k\geq 0\}$ and spatial motion $\{\xi_t, \Pi_x\}$. For any $t\ge 0$, let $N_t$ be the collection of particles alive at time $t$, and, for any $u\in N_t$, let $X_u(t)$ be the position of $u$ at time $t$. We study the tail probability of the maximal displacement $M:=\sup_{t>0}\sup_{u\in N_t} X_u(t)$ under the assumption $\lim_{n\to\infty} n^\alpha \sum_{k=n}^\infty p_k =\kappa\in(0,\infty)$ for some $\alpha\in (1,2)$, $\Pi_0(\xi_1)=0$ and $\Pi_0 (|\xi_1|^r)\in (0,\infty)$ for some $r> 2\alpha/(\alpha-1)$. Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of which are proved under the assumption $\sum_{k=0}^\infty k^3 p_k<\infty$.

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related L\'{e}vy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom's distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

Let $Z_t^{(0,\infty)}$ be the point process formed by the positions of all particles alive at time $t$ in a branching Brownian motion with drift and killed upon reaching 0. We study the asymptotic expansions of $Z_t^{(0,\infty)}(A)$ for $A= (a,b)$ and $A=(a,\infty)$ under the assumption that $\sum_{k=1}^\infty k(\log k)^{1+\lambda} p_k <\infty$ for large $\lambda$ in the regime of $\theta \in [0,\sqrt{2})$. These results extend and sharpen the results of Louidor and Saglietti [J. Stat. Phys, 2020] and Kesten [Stochastic Process. Appl., 1978].

Let $X^I_n$ be the coalescence time of two particles picked at random from the $n$th generation of a critical Galton-Watson process with immigration, and let $A^I_n$ be the coalescence time of the whole population in the $n$th generation. In this paper, we study the limiting behaviors of $X^I_n$ and $A^I_n$ as $n\to\infty$.

Photonic topological insulators (PTIs) have been widely studied due to the robustness of energy transport via supported edge modes immune to structural disorder. In this work, a topological gap waveguide is constructed by introducing line defect into a topological photonic crystal structure and combining it with a gap waveguide structure, which design therefore combines the advantages of both topological and gap waveguides. Not only does it give high transmission efficiency, but also enables high robustness for energy transmission under structural defects and sharp bends. Our proposed topological waveguide design can be implemented with conventional semiconductor technology and integrated into optical circuits for communication systems., Comment: 16 pages, 7 figs

In this paper we consider a large class of super-Brownian motions in $\mathbb{R}$ with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval $(-\delta t,\delta t)$ for $\delta>0$. The growth rate is given in terms of the principal eigenvalue $\lambda_{1}$ of the Sch\"{o}dinger type operator associated with the branching mechanism. From this result we see the existence of phase transition for the growth order at $\delta=\sqrt{\lambda_{1}/2}$. We further show that the super-Brownian motion shifted by $\sqrt{\lambda_{1}/2}\,t$ converges in distribution to a random measure with random density mixed by a martingale limit.