arXiv subjects

Zhijie Chen

Publications and source records attributed to Zhijie Chen.

At least 55 records · Page 3Linked to original sources

arXiv2023

Asymptotic behavior of least energy nodal solutions for biharmonic Lane-Emden problems in dimension four

In this paper, we study the asymptotic behavior of least energy nodal solutions $u_p(x)$ to the following fourth-order elliptic problem \[ \begin{cases} Δ^2 u =|u|^{p-1}u \quad &\hbox{in}\;Ω, \\ u=\frac{\partial u}{\partial ν}=0 \ \ &\hbox{on}\;\partialΩ, \end{cases} \] where $Ω$ is a bounded $C^{4,α}$ domain in $\mathbb{R}^4$ and $p>1$. Among other things, we show that up to a subsequence of $p\to+\infty$, $pu_p(x)\to 64π^2\sqrt{e}(G(x,x^+)-G(x,x^-))$, where $x^+\neq x^-\in Ω$ and $G(x,y)$ is the corresponding Green function of $Δ^2$. This generalize those results for $-Δu=|u|^{p-1}u$ in dimension two by (Grossi-Grumiau-Pacella, Ann.I.H.Poincaré-AN, 30 (2013), 121-140) to the biharmonic case, and also gives an alternative proof of Grossi-Grumiau-Pacella's results without assuming their comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$.

math.AP↗

arXiv2022

3D Interconnected Magnetic Nanowire Networks as Potential Integrated Multistate Memristors

Interconnected magnetic nanowire (NW) networks offer a promising platform for 3-dimensional (3D) information storage and integrated neuromorphic computing. Here we report discrete propagation of magnetic states in interconnected Co nanowire networks driven by magnetic field and current, manifested in distinct magnetoresistance (MR) features. In these networks, when only a few interconnected NWs were measured, multiple MR kinks and local minima were observed, including a significant minimum at a positive field during the descending field sweep. Micromagnetic simulations showed that this unusual feature was due to domain wall (DW) pinning at the NW intersections, which was confirmed by off-axis electron holography imaging. In a complex network with many intersections, sequential switching of nanowire sections separated by interconnects was observed, along with stochastic characteristics. The pinning/depinning of the DWs can be further controlled by the driving current density. These results illustrate the promise of such interconnected networks as integrated multistate memristors.

physics.app-ph↗

arXiv2022

Sharp estimates, uniqueness and nondegeneracy of positive solutions of the Lane-Emden system in planar domains

We study the Lane-Emden system $$\begin{cases} -Δu=v^p,\quad u>0,\quad\text{in}~Ω, -Δv=u^q,\quad v>0,\quad\text{in}~Ω, u=v=0,\quad\text{on}~\partialΩ, \end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as $p,q\to+\infty$ and $|q-p|\leq Λ$. In this paper, we obtain sharp estimates of such multi-bubble solutions, including sharp convergence rates of local maxima and scaling parameters, and accurate approximations of solutions. As an application of these sharp estimates, we show that when $Ω$ is convex, then the solution of this system is unique and nondegenerate for large $p, q$.

math.AP↗

arXiv2022

Above-room-temperature ferromagnetism in ultrathin van der Waals magnet

Two-dimensional (2D) magnetic van der Waals materials provide a powerful platform for studying fundamental physics of low-dimensional magnetism, engineering novel magnetic phases, and enabling ultrathin and highly tunable spintronic devices. To realize high quality and practical devices for such applications, there is a critical need for robust 2D magnets with ordering temperatures above room temperature that can be created via exfoliation. Here the study of exfoliated flakes of cobalt substituted Fe5GeTe2 (CFGT) exhibiting magnetism above room temperature is reported. Via quantum magnetic imaging with nitrogen-vacancy centers in diamond, ferromagnetism at room temperature was observed in CFGT flakes as thin as 16 nm. This corresponds to one of the thinnest room-temperature 2D magnet flakes exfoliated from robust single crystals, reaching a thickness relevant to practical spintronic applications. The Curie temperature Tc of CFGT ranges from 310 K in the thinnest flake studied to 328 K in the bulk. To investigate the prospect of high-temperature monolayer ferromagnetism, Monte Carlo calculations were performed which predicted a high value of Tc ~270 K in CFGT monolayers. Pathways towards further enhancing monolayer Tc are discussed. These results support CFGT as a promising platform to realize high-quality room-temperature 2D magnet devices.

cond-mat.mtrl-sci↗

arXiv2022

Learning Neural Hamiltonian Dynamics: A Methodological Overview

The past few years have witnessed an increased interest in learning Hamiltonian dynamics in deep learning frameworks. As an inductive bias based on physical laws, Hamiltonian dynamics endow neural networks with accurate long-term prediction, interpretability, and data-efficient learning. However, Hamiltonian dynamics also bring energy conservation or dissipation assumptions on the input data and additional computational overhead. In this paper, we systematically survey recently proposed Hamiltonian neural network models, with a special emphasis on methodologies. In general, we discuss the major contributions of these models, and compare them in four overlapping directions: 1) generalized Hamiltonian system; 2) symplectic integration, 3) generalized input form, and 4) extended problem settings. We also provide an outlook of the fundamental challenges and emerging opportunities in this area.

cs.LG↗

arXiv2022

Modular Ordinary Differential Equations on ${\rm SL}(2,\mathbb{Z})$ of Third Order and Applications

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form $y'''+Q_2(z)y'+Q_3(z)y=0$, $z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,\operatorname{Im}z>0 \}$, where $Q_2(z)$ and $Q_3(z)-\frac12 Q_2'(z)$ are meromorphic modular forms on ${\rm SL}(2,\mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on ${\rm SL}(2,\mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $\hatρ\colon {\rm SL}(2,\mathbb{Z})\to{\rm SL}(3,\mathbb{C})$ for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain ${\rm SU}(3)$ Toda systems. Note that the ${\rm SU}(N+1)$ Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to $\mathbb{CP}^N$.

math.NT↗

arXiv2021

On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

We study the $SU(3)$ Toda system with singular sources \[ \begin{cases} Δu+2e^{u}-e^v=4π\sum_{k=0}^m n_{1,k}δ_{p_k}\quad\text{ on }\; E_τ,\\ Δv+2e^{v}-e^u=4π\sum_{k=0}^m n_{2,k}δ_{p_k}\quad\text{ on }\; E_τ, \end{cases} \] where $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with $\operatorname{Im}τ>0$ is a flat torus, $δ_{p_k}$ is the Dirac measure at $p_k$, and $n_{i,k}\in\mathbb{Z}_{\geq 0}$ satisfy $\sum_{k}n_{1,k}\not\equiv \sum_k n_{2,k} \mod 3$. This is known as the non-critical case and it follows from a general existence result of \cite{BJMR} that solutions always exist. In this paper we prove that (i) The system has at most \[\frac{1}{3\times 2^{m+1}}\prod_{k=0}^m(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2)\in\mathbb{N}\] solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry. (ii) For $m=0$ and $p_0=0$, the system has even solutions if and only if at least one of $\{n_{1,0}, n_{2,0}\}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}<n_{2,0}$, then except for finitely many $τ$'s modulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$ even solutions. Differently from \cite{BJMR}, our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.

math.AP↗

arXiv2021

Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy

In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lamé equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_{N}$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)Ψ(N)}{24}-\left( a_{n}% ϕ(N)+b_{n}ϕ(\tfrac{N}{2}) \right) \right) +\frac{2}% {3}\varepsilon_{n}(N).\] Our main tool is the new pre-modular form $Z_{r,s}^{(n)}(τ)$ of weight $n(n+1)/2$ introduced by Lin and Wang \cite{LW2} and the associated modular form $M_{n,N}(τ):=\prod_{(r,s)}Z_{r,s}^{(n)}(τ)$ of weight $Ψ(N)n(n+1)/{2}$, where the product runs over all $N$-torsion points $(r,s)$ of exact order $N$. We show that this conjecture is equivalent to the precise formula of the vanishing order of $M_{n,N}(τ)$ at infinity: \[v_{\infty}(M_{n,N}(τ))=a_{n}ϕ(N)+b_{n}ϕ( N/2).\] This formula is extremely hard to prove because the explicit expression of $Z_{r,s}^{(n)}(τ)$ is not known for general $n$. Here we succeed to prove it by using certain Painlevé VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlevé VI equations. The main results of this paper has been announced in \cite{Lin-CDM}.

math.NT↗

arXiv2020

The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence

In this paper, the third in a series, we continue to study the generalized Lamé equation H$(n_0,n_1,n_2,n_3;B)$ with the Darboux-Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2}|τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} and a related linear ODE with additional singularities $\pm p$ from the monodromy aspect.We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+2,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations $ρ:π_1(E_τ)\to SL(2,\mathbb{C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $τ=\frac12+i\frac{\sqrt{3}}{2}$, there are $B_1,B_2$ such that the monodromy representations of H$(1,0,0,0;B_1)$ and H$(4,0,0,0;B_2)$ are {\bf the same}, namely the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+3,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations is {\bf not} necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on $\mathbb{CP}^1$ with finite singularities.

math.CA↗

arXiv2020

Optimizing AD Pruning of Sponsored Search with Reinforcement Learning

Industrial sponsored search system (SSS) can be logically divided into three modules: keywords matching, ad retrieving, and ranking. During ad retrieving, the ad candidates grow exponentially. A query with high commercial value might retrieve a great deal of ad candidates such that the ranking module could not afford. Due to limited latency and computing resources, the candidates have to be pruned earlier. Suppose we set a pruning line to cut SSS into two parts: upstream and downstream. The problem we are going to address is: how to pick out the best $K$ items from $N$ candidates provided by the upstream to maximize the total system's revenue. Since the industrial downstream is very complicated and updated quickly, a crucial restriction in this problem is that the selection scheme should get adapted to the downstream. In this paper, we propose a novel model-free reinforcement learning approach to fixing this problem. Our approach considers downstream as a black-box environment, and the agent sequentially selects items and finally feeds into the downstream, where revenue would be estimated and used as a reward to improve the selection policy. To the best of our knowledge, this is first time to consider the system optimization from a downstream adaption view. It is also the first time to use reinforcement learning techniques to tackle this problem. The idea has been successfully realized in Baidu's sponsored search system, and online long time A/B test shows remarkable improvements on revenue.

cs.LG↗

arXiv2020

A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$

In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;τ)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential \[q(x;τ):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{ω_{k}}{2};τ\right),\] where $n_k\in\mathbb{Z}_{\geq 0}$ with $\max n_k\geq 1$ and $z_0\in\mathbb{C}$ is chosen such that $q(x;τ)$ has no singularities on $\mathbb{R}$. For any fixed $τ\in i\mathbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $σ(L)$ is \[σ(L)=(-\infty, E_{2g}]\cup[E_{2g-1}, E_{2g-2}]\cup \cdots \cup[E_{1}, E_{0}],\quad E_j\in \mathbb{R},\] and hence generalizes Ince's remarkable result in 1940 for the Lamé potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1}$, $E_{2j-2})$, which generalizes the recent result in \cite{HHV} where the Lamé case $n_1=n_2=n_3=0$ was studied.

math.CA↗

arXiv2019

Current Distribution on Capacitive Electrode-Electrolyte Interfaces

The distribution of electric current on an electrode surface in electrolyte varies with time due to charge accumulation at a capacitive interface, as well as due to electrode kinetics and concentration polarization in the medium. Initially, the potential at the electrode-electrolyte interface is uniform, resulting in a non-uniform current distribution due to the uneven ohmic drop of the potential in the medium. Over time, however, the non-uniform current density causes spatially varying rate of the charge accumulation at the interface, breaking down its equipotentiality. We developed an analytical model to describe such transition at a capacitive interface when the current is below the mass-transfer limitation, and demonstrated that the steady distribution of the current is achieved when the current density is proportional to the capacitance per unit area, which leads to linear voltage ramp at the electrode. More specific results regarding the dynamics of this transition are provided for a disk electrode, along with an experimental validation of the theoretical result. These findings are important for many electrochemical applications, and in particular, for proper design of the electro-neural interfaces.

physics.app-ph↗

arXiv2019

Decoding Spiking Mechanism with Dynamic Learning on Neuron Population

A main concern in cognitive neuroscience is to decode the overt neural spike train observations and infer latent representations under neural circuits. However, traditional methods entail strong prior on network structure and hardly meet the demand for real spike data. Here we propose a novel neural network approach called Neuron Activation Network that extracts neural information explicitly from single trial neuron population spike trains. Our proposed method consists of a spatiotemporal learning procedure on sensory environment and a message passing mechanism on population graph, followed by a neuron activation process in a recursive fashion. Our model is aimed to reconstruct neuron information while inferring representations of neuron spiking states. We apply our model to retinal ganglion cells and the experimental results suggest that our model holds a more potent capability in generating neural spike sequences with high fidelity than the state-of-the-art methods, as well as being more expressive and having potential to disclose latent spiking mechanism. The source code will be released with the final paper.

q-bio.NC↗

arXiv2019

A Pre-Allocation Design for Cost Minimization and Delay Constraint in Vehicular Offloading System

To accommodate exponentially increasing traffic demands of vehicle-based applications, operators are utilizing offloading as a promising technique to improve quality of service (QoS), which gives rise to the application of Mobile Edge Computing (MEC). While the conventional offloading paradigms focus on delay and energy tradeoff, they either fail to find efficient models to represent delay, especially the queueing delay, or underestimate the role of MEC Server. In this paper, we propose a novel \textbf{P}re-\textbf{A}llocation \textbf{D}esign for vehicular \textbf{O}ffloading (\textbf{PADO}). A task delay queue is constructed based on an allocate-execute separate (AES) mechanism. Due to the dynamics of vehicular network, we are inspired to utilize Lyapunov optimization to minimize the execution cost of each vehicle and guarantee task delay. The MEC Server with energy harvesting devices is also taken into consideration of the system. The transaction between vehicles and server is decided by a Stackelberg Game framework. We conduct extensive experiments to show the property and superiority of our proposed framework.

eess.SY↗

arXiv2019

An end-to-end Generative Retrieval Method for Sponsored Search Engine --Decoding Efficiently into a Closed Target Domain

In this paper, we present a generative retrieval method for sponsored search engine, which uses neural machine translation (NMT) to generate keywords directly from query. This method is completely end-to-end, which skips query rewriting and relevance judging phases in traditional retrieval systems. Different from standard machine translation, the target space in the retrieval setting is a constrained closed set, where only committed keywords should be generated. We present a Trie-based pruning technique in beam search to address this problem. The biggest challenge in deploying this method into a real industrial environment is the latency impact of running the decoder. Self-normalized training coupled with Trie-based dynamic pruning dramatically reduces the inference time, yielding a speedup of more than 20 times. We also devise an mixed online-offline serving architecture to reduce the latency and CPU consumption. To encourage the NMT to generate new keywords uncovered by the existing system, training data is carefully selected. This model has been successfully applied in Baidu's commercial search engine as a supplementary retrieval branch, which has brought a remarkable revenue improvement of more than 10 percents.

cs.IR↗

arXiv2018

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2}|τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} from the monodromy aspect. We prove the existence of a pre-modular form $Z_{r,s}^{\mathbf{n}}(τ)$ of weight $\frac{1}{2}\sum n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by $Z_{r,s}^{\mathbf{n}}(τ)=0$. This generalizes the result in \cite{LW2}, where the Lamé case (i.e. $n_1=n_2=n_3=0$) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations \[Δu+e^u=16πδ_{0}\quad\text{and}\quad Δu+e^u=8π\sum_{k=1}^3δ_{\frac{ω_k}{2}}\] on a flat torus $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ has the same number of even solutions. This result is quite surprising from the PDE point of view.

math.CA↗

arXiv2017

Proactive Caching for Energy-Efficiency in Wireless Networks: A Markov Decision Process Approach

Content caching in wireless networks provides a substantial opportunity to trade off low cost memory storage with energy consumption, yet finding the optimal causal policy with low computational complexity remains a challenge. This paper models the Joint Pushing and Caching (JPC) problem as a Markov Decision Process (MDP) and provides a solution to determine the optimal randomized policy. A novel approach to decouple the influence from buffer occupancy and user requests is proposed to turn the high-dimensional optimization problem into three low-dimensional ones. Furthermore, a non-iterative algorithm to solve one of the sub-problems is presented, exploiting a structural property we found as \textit{generalized monotonicity}, and hence significantly reduces the computational complexity. The result attains close performance in comparison with theoretical bounds from non-practical policies, while benefiting from higher time efficiency than the unadapted MDP solution.

eess.SP↗

arXiv2017

Local Core Members Aided Community Structure Detection

The relationship of friends in social networks can be strong or weak. Some research works have shown that a close relationship between friends conducts good community structure. Based on this result, we propose an effective method in detecting community structure in social networks based on the closeness of relations among neighbors. This method calculates the gravity between each neighbor node to core nodes, then makes judgement if the node should be classified in the community or not, and finally form the process of community detection. The experimental results show that the proposed method can mine the social structure efficiently with a low computational complexity

cs.SI↗