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Jialun Li

Publications and source records attributed to Jialun Li.

29 records · Page 2Linked to original sources
arXiv2022

Equidistribution and counting of periodic flat tori

Let $G$ be a semisimple Lie group without compact factor and $Γ< G$ a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on maximal flat periodic tori of the space of Weyl chambers. We prove that these flat periodic tori equidistribute exponentially fast towards the quotient of the Haar measure. From the equidistribution formula, we deduce a higher rank prime geodesic theorem. These counting and equidistribution results also hold in the non cocompact, finite covolume case for $G=\mathrm{SL}(d,\mathbb{R})$ and $Γ<\mathrm{SL}(d,\mathbb{Z})$ a finite index subgroup.

math.DS↗
arXiv2021

Speed Planning Using Bezier Polynomials with Trapezoidal Corridors

To generate safe and real-time trajectories for an autonomous vehicle in dynamic environments, path and speed decoupled planning methods are often considered. This paper studies speed planning, which mainly deals with dynamic obstacle avoidance given the planning path. The main challenges lie in the decisions in non-convex space and the trade-off between safety, comfort and efficiency performances. This work uses dynamic programming to search heuristic waypoints on the S-T graph and to construct convex feasible spaces. Further, a piecewise Bezier polynomials optimization approach with trapezoidal corridors is presented, which theoretically guarantees the safety and optimality of the trajectory. The simulations verify the effectiveness of the proposed approach.

cs.RO↗
arXiv2021

Trigonometric series and self-similar sets

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $μ$ on $F$ is a Rajchman measure: the Fourier transform $\widehatμ(ξ) \to 0$ as $|ξ| \to \infty$. The rate of $\widehatμ(ξ) \to 0$ is also shown to be logarithmic if $\log r_i / \log r_j$ is diophantine for some $i \neq j$. The proof is based on quantitative renewal theorems for stopping times of random walks on $\mathbb{R}$.

math.CA↗
arXiv2020

Fourier transform of self-affine measures

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $Γ$ generated by the matrices $A_j$, $j \in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $μ= \sum p_j f_j μ$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehatμ(ξ) \to 0$ as $|ξ| \to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $Γ$ is connected real split Lie group in the Zariski topology, then $\widehatμ(ξ)$ has a power decay at infinity. Hence $μ$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $Γ$ and non-compactness of the image of $Γ$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehatμ$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.

math.DS↗
arXiv2019

Multi-qubit Quantum Rabi Model and Multi-partite Entangled States in a Circuit QED System

Multi-qubit quantum Rabi model, which is a fundamental model describing light-matter interaction, plays an important role in various physical systems. In this paper, we propose a theoretical method to simulate multi-qubit quantum Rabi model in a circuit quantum electrodynamics system. By means of external transversal and longitudinal driving fields, an effective Hamiltonian describing the multi-qubit quantum Rabi model is derived. The effective frequency of the resonator and the effective splitting of the qubits depend on the external driving fields. By adjusting the frequencies and the amplitudes of the driving fields, the stronger coupling regimes could be reached. The numerical simulation shows that our proposal works well in a wide range of parameter space. Moreover, our scheme can be utilized to generate two-qubit gate, Schrödinger states, and multi-qubit GHZ states. The maximum displacement of the Schrödinger cat states can be enhanced by increasing the number of the qubits and the relative coupling strength. It should be mention that we can obtain high fidelity Schrödinger cat states and multi-qubit GHZ states even the system suffering dissipation. The presented proposal may open a way to study the stronger coupling regimes whose coupling strength is far away from ultrastrong coupling regimes.

quant-ph↗
arXiv2019

Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay

Let $Γ$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let $Λ(Γ)$ be its limit set, endowed with a Patterson-Sullivan measure $μ$ supported on $Λ(Γ)$. We show that the Fourier transform $\widehatμ(ξ)$ enjoys polynomial decay as $\vert ξ\vert$ goes to infinity. This is a PSL2(C) version of the result of Bourgain-Dyatlov [8], and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on C. These bounds on exponential sums require a delicate non-concentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.

math.DS↗
arXiv2018

Decrease of Fourier coefficients of stationary measures

Let $μ$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a finite exponential moment, and assume that the subgroup $Γ_μ$ generated by the support of $μ$ is Zariski dense. Let $ν$ be the unique $μ-$stationary measure on $\mathbb P^1_{\mathbb R}$. We prove that the Fourier coefficients $\widehatν(k)$ of $ν$ converge to $0$ as $|k|$ tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.

math.DS↗