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Li Cai

Publications and source records attributed to Li Cai.

At least 37 records · Page 2Linked to original sources
arXiv2023

On the graph products of simplicial groups and connected Hopf algebras

In this paper we study the classifying spaces of graph products of simplicial groups and connected Hopf algebras over a field, and show that they can be uniformly treated under the framework of polyhedral products. It turns out that these graph products are models of the loop spaces of polyhedral products over a flag complex and their homology, respectively. Certain morphisms between graph products are also considered. In the end we prove the structure theorems of such graph products in the form we need.

math.AT↗
arXiv2023

Ichino period for CM forms

In both local and global settings, we establish explicit relations between Ichino triple product period and Waldspurger toric periods for CM forms via the theta lifting and the see-saw principle.

math.NT↗
arXiv2023

An Effective and Efficient Time-aware Entity Alignment Framework via Two-aspect Three-view Label Propagation

Entity alignment (EA) aims to find the equivalent entity pairs between different knowledge graphs (KGs), which is crucial to promote knowledge fusion. With the wide use of temporal knowledge graphs (TKGs), time-aware EA (TEA) methods appear to enhance EA. Existing TEA models are based on Graph Neural Networks (GNN) and achieve state-of-the-art (SOTA) performance, but it is difficult to transfer them to large-scale TKGs due to the scalability issue of GNN. In this paper, we propose an effective and efficient non-neural EA framework between TKGs, namely LightTEA, which consists of four essential components: (1) Two-aspect Three-view Label Propagation, (2) Sparse Similarity with Temporal Constraints, (3) Sinkhorn Operator, and (4) Temporal Iterative Learning. All of these modules work together to improve the performance of EA while reducing the time consumption of the model. Extensive experiments on public datasets indicate that our proposed model significantly outperforms the SOTA methods for EA between TKGs, and the time consumed by LightTEA is only dozens of seconds at most, no more than 10% of the most efficient TEA method.

cs.AI↗
arXiv2023

Families of Canonical Local Periods on Spherical Varieties

We consider the variation of canonical local periods on spherical varieties proposed by Sakellaridis-Venkatesh in families. We formulate conjectures for the rationality and meromorphic property of canonical local periods and establish these conjectures for strongly tempered spherical G-varieties without type N-roots when G is split.

math.NT↗
arXiv2023

Higher Ext-groups in the triple product case

In this short note, we compute higher extension groups for all irreducible representations and deduce the multiplicity formula for finite length representations in triple product case.

math.NT↗
arXiv2022

Anomalous Nernst effect in compensated ferrimagnetic CoxGd1-x films

The anomalous Nernst effect (ANE) is one of the most intriguing thermoelectric phenomena which has attracted growing interest both for its underlying physics and potential applications. Typically, a large ANE response is observed in magnets with pronounced magnetizations or nontrivial Berry curvature. Here, we report a significant ANE signal in compensated ferrimagnetic CoxGd1-x alloy films, which exhibit vanishingly small magnetization. In particular, we found that the polarity of ANE signal is dominated by the magnetization orientation of the transition metal Co sublattices, rather than the net magnetization of CoxGd1-x films. This observation is not expected from the conventional understanding of ANE but is analogous to the anomalous Hall effect in compensated ferrimagnets. We attribute the origin of ANE and its Co-dominant property to the Co-dominant Berry curvature. Our work could trigger a more comprehensive understanding of ANE and may be useful for building energy-harvesting devices by employing ANE in compensated ferrimagnets.

cond-mat.mtrl-sci↗
arXiv2022

Infinite time bubble towers in the fractional heat equation with critical exponent

In this paper, we consider the fractional heat equation with critical exponent in $\mathbb{R}^n$ for $n>6s,s\in(0,1),$ \begin{equation*} u_t=-(-Δ)^su+|u|^{\frac{4s}{n-2s}}u,\quad (x,t)\in \mathbb{R}^n\times\mathbb{R}. \end{equation*} We construct a bubble tower type solution both for the forward and backward problem by establishing the existence of the sign-changing solution with multiple blow-up at a single point with the form \begin{equation*} u(x,t)=(1+o(1))\sum_{j=1}^{k}(-1)^{j-1}μ_j(t)^{-\frac{n-2s}{2}}U\left(\frac{x}{μ_j(t)}\right) \quad\mbox{as}\quad t\to+\infty, \end{equation*} and the positive solution with multiple blow-up at a single point with the form \begin{equation*} u(x,t)=(1+o(1))\sum_{j=1}^{k}μ_j(t)^{-\frac{n-2s}{2}}U\left(\frac{x}{μ_j(t)}\right) \quad\mbox{as}\quad t\to-\infty, \end{equation*} respectively. Here $k\ge2$ is a positive integer, $$U(y)=α_{n,s}\left(\frac{1}{1+|y|^2}\right)^{\frac{n-2s}{2}},$$ and \begin{equation*} μ_j(t)=β_j |t|^{-α_j}(1+o(1))~\mathrm{as}~t\to\pm\infty, \quad α_j=\frac{1}{2s}\left(\frac{n-2s}{n-6s}\right)^{j-1}-\frac{1}{2s}, \end{equation*} for some certain positive numbers $β_j,j=1,\cdots,k.$

math.AP↗
arXiv2022

A Simple Temporal Information Matching Mechanism for Entity Alignment Between Temporal Knowledge Graphs

Entity alignment (EA) aims to find entities in different knowledge graphs (KGs) that refer to the same object in the real world. Recent studies incorporate temporal information to augment the representations of KGs. The existing methods for EA between temporal KGs (TKGs) utilize a time-aware attention mechanism to incorporate relational and temporal information into entity embeddings. The approaches outperform the previous methods by using temporal information. However, we believe that it is not necessary to learn the embeddings of temporal information in KGs since most TKGs have uniform temporal representations. Therefore, we propose a simple graph neural network (GNN) model combined with a temporal information matching mechanism, which achieves better performance with less time and fewer parameters. Furthermore, since alignment seeds are difficult to label in real-world applications, we also propose a method to generate unsupervised alignment seeds via the temporal information of TKG. Extensive experiments on public datasets indicate that our supervised method significantly outperforms the previous methods and the unsupervised one has competitive performance.

cs.AI↗
arXiv2021

Isolation of the Cuspidal Spectrum: the Function Field Case

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case.

math.NT↗
arXiv2021

A Kernel-free Boundary Integral Method for the Bidomain Equations

The bidomain equations have been widely used to mathematically model the electrical activity of the cardiac tissue. In this work, we present a potential theory-based Cartesian grid method which is referred as the kernel-free boundary integral (KFBI) method which works well on complex domains to efficiently simulate the linear diffusion part of the bidomain equation. After a proper temporal discretization, the KFBI method is applied to solve the resulting homogeneous Neumann boundary value problems with a second-order accuracy. According to the potential theory, the boundary integral equations reformulated from the boundary value problems can be solved iteratively with the simple Richardson iteration or the Krylov subspace iteration method. During the iteration, the boundary and volume integrals are evaluated by limiting the structured grid-based discrete solutions of the equivalent interface problems at quasi-uniform interface nodes without the need to know the analytical expression of Green's functions. In particular, the discrete linear system of the equivalent interface problem obtained from the standard finite difference schemes or the finite element schemes can be efficiently solved by fast elliptic solvers such as the fast Fourier transform based solvers or those based on geometric multigrid iterations after an appropriate modification at the irregular grid nodes. Numerical results for solving the FitzHugh-Nagumo bidomain equations in both two- and three-dimensional spaces are presented to demonstrate the numerical performance of the KFBI method such as the second-order accuracy and the propagation and scroll wave of the voltage simulated on the real human left ventricle model.

math.NA↗
arXiv2021

How Medical Crowdfunding Helps People? A Large-scale Case Study on Waterdrop Fundraising

While online medical crowdfunding achieved tremendous success, quantitative study about whether and how medical crowdfunding helps people remains little explored. In this paper, we empirically study how online medical crowdfunding helps people using more than 27, 000 fundraising cases in Waterdrop Fundraising, one of the most popular online medical crowdfunding platforms in China. We find that the amount of money obtained by fundraisers is broadly distributed, i.e., a majority of lowly donated cases coexist with a handful of very successful cases. We further investigate the factors that potentially correlate with the success of medical fundraising cases. Profile information of fundraising cases, e.g., geographic information of fundraisers, affects the donated amounts, since detailed description may increase the credibility of a fundraising case. One prominent finding lies in the effect of social network on the success of fundraising cases: the spread of fundraising information along social network is a key factor of fundraising success, and the social capital of fundraisers play an important role in fundraising. Finally, we conduct prediction of donations using machine learning models, verifying the effect of potential factors to the success of medical crowdfunding. Altogether, this work presents a data-driven view of medical fundraising on the web and opens a door to understanding medical crowdfunding.

cs.SI↗
arXiv2020

Simultaneous confidence bands for nonparametric regression with partially missing covariates

In this paper, we consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation studies which support our asymptotic theory. The proposed method is demonstrated via an analysis of a data set from the Canada 2010/2011 Youth Student Survey.

stat.ME↗
arXiv2020

On the KO-groups of toric manifolds

In this paper we consider the real topological K-groups of a toric manifold M, which turns out to be closely related to the topology of the small cover MR, the fixed points under the canonical conjugation on M . Following the work of Bahri and Bendersky [2], we give an explicit formula for the KO-groups of toric manifolds, and then we characterize the two extreme classes of toric manifolds according to the two A(1) modules shown in [2].

math.AT↗
arXiv2019

On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves

In the present paper, we prove, for a large class of elliptic curves defined over $\mathbb{Q}$, the existence of an explicit infinite family of quadratic twists with analytic rank $0$. In addition, we establish the $2$-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch--Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over $\mathbb{Q}$ without complex multiplication.

math.NT↗
arXiv2019

Integral cohomology groups of real toric manifolds and small covers

For a simplicial complex $K$ with $m$ vertices, there is a canonical $\mathbb Z_2^m$-space known as a real moment angle complex $\mathbb R \mathcal Z_K$. In this paper, we consider the quotient spaces $Y=\mathbb R \mathcal Z_K / \mathbb Z_2^{k}$, where $K$ is a pure shellable complex and $\mathbb Z_2^k \subset \mathbb Z_2^m$ is a maximal free action on $\mathbb R \mathcal Z_K$. A typical example of such spaces is a small cover, where a small cover is known as a topological analog of a real toric manifold. We compute the integral cohomology group of $Y$ by using the PL cell decomposition obtained from a shelling of $K$. In addition, we compute the Bockstein spectral sequence of $Y$ explicitly.

math.AT↗
arXiv2016

Heegner Points on Modular Curves

In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold.

math.NT↗
arXiv2015

On products in a real moment-angle manifold

In this paper we give a necessary and sufficient condition for a (real) moment-angle complex to be a topological manifold. The cup and cap products in a real moment-angle manifold are studied: the Poincaré duality via cap products is equivalent to the Alexander duality of the defining complex $K$. Consequently, the cohomology ring (with coefficients integers) of a polyhedral product by pairs of disks and their bounding spheres is isomorphic to that of a differential graded algebra associated to $K$, and the dimensions of the disks.

math.AT↗