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Shenghao Li

Publications and source records attributed to Shenghao Li.

18 recordsLinked to original sources
arXiv2026

Bump-Friedberg type periods beyond the cuspidal spectrum

In this article, we study several Bump--Friedberg type periods beyond the cuspidal spectrum. We first consider the twisted Bump--Friedberg period on $\textnormal{GL}_{2n}$, as well as a variant on $\textnormal{GL}_1\times \textnormal{GL}_{2n}$. Under suitable regularity conditions on the cuspidal datum, these periods extend continuously to automorphic functions of uniform moderate growth. Such extensions are characterized by entire Whittaker-type zeta integrals. We then introduce a Bump--Friedberg type period on $\textnormal{GL}_{2n+1}$, integrating over the subgroup $\textnormal{SL}_{n+1}\times \textnormal{GL}_n$. For certain Eisenstein series, we evaluate this period as a finite sum of products of special values of $L$-functions and normalized local zeta integrals. Assuming the expected global Langlands correspondence, the sum is indexed by the fixed points of the extended $L$-parameter on the conjectural dual variety, and the resulting $L$-factors agree with the tangent space prediction of the global numerical conjecture of Ben-Zvi-Sakellaridis-Venkatesh.

math.RT↗
arXiv2026

Base change fundamental lemma for Bernstein centers of principal series blocks

Let $G$ be an unramified group over a $p$-adic field $F$. This article introduces a base change homomorphism for the Bernstein center of a principal series block, and proves that two functions related by this base change homomorphism are associated. This result provides new evidence for the conjecture on twisted endoscopic transfer of the stable Bernstein center proposed by T. Haines, which will be applied to a general conjecture on test functions for Shimura varieties due to R. Kottwitz and T. Haines.

math.RT↗
arXiv2026

Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transport

Diffusion Models (DMs) have achieved remarkable progress in generative modeling. However, the mismatch between the forward terminal distribution and reverse initial distribution introduces prior error, leading to deviations of sampling trajectories from the true distribution and severely limiting model performance. This issue further triggers cascading problems, including non-zero Signal-to-Noise Ratio, accumulated denoising errors, degraded generation quality, and constrained sampling efficiency. To address this issue, this paper proposes a prior error elimination framework based on Optimal Transport (OT). Specifically, an OT map from the reverse initial distribution to the forward terminal distribution is constructed to achieve precise matching of the two distributions. Meanwhile, the upper bound of the prior error is quantified using the Wasserstein distance, proving that the prior error can be effectively eliminated via the OT map. Additionally, by deriving the asymptotic consistency between dynamic OT and probability flow, this method is revealed to be highly compatible with the intrinsic mechanism of the diffusion process. Experimental results demonstrate that the proposed method completely eliminates the prior error both theoretically and practically, providing a universal and rigorous solution for optimizing the performance of DMs.

cs.LG↗
arXiv2025

LFC-DA: Logical Formula-Controlled Data Augmentation for Enhanced Logical Reasoning

For complex logical data augmentation, heavy reliance on human annotation is costly, whereas direct generation with large language models yields uninterpretable and logically homogeneous examples. To address this, we present LFC-DA, a symbolic-logic-controlled pipeline: logical text is first mapped to propositional expressions, a compact rule library is compiled, and a bounded state-space search systematically discovers valid formulas that are then verbalized back into natural-language questions, ensuring both diversity and logical rigor under propositional logic. Experiments on ReClor and LogiQA show significant improvements in the logical-reasoning accuracy of pretrained models, confirming the effectiveness of LFC-DA for LLM-guided logical data augmentation.

cs.CL↗
arXiv2025

Standing wave solutions of $abcd$-systems for water waves

We continue the study for standing wave solutions of $abcd$-systems which was started by Chen and Iooss \cite{chen2005standing} for the BBM system via the Lyapunov-Schmidt method. In this paper, we will first discuss the feasibility of the Lyapunov-Schmidt method for bifurcating standing wave solutions of $abcd$-systems. These systems will be characterized into three categories: feasible, infeasible and uncertain feasible ones. In particular, we prove the existence of nontrivial bifurcating standing waves for the Bona-Smith system.

math.AP↗
arXiv2024

In-Place Panoptic Radiance Field Segmentation with Perceptual Prior for 3D Scene Understanding

Accurate 3D scene representation and panoptic understanding are essential for applications such as virtual reality, robotics, and autonomous driving. However, challenges persist with existing methods, including precise 2D-to-3D mapping, handling complex scene characteristics like boundary ambiguity and varying scales, and mitigating noise in panoptic pseudo-labels. This paper introduces a novel perceptual-prior-guided 3D scene representation and panoptic understanding method, which reformulates panoptic understanding within neural radiance fields as a linear assignment problem involving 2D semantics and instance recognition. Perceptual information from pre-trained 2D panoptic segmentation models is incorporated as prior guidance, thereby synchronizing the learning processes of appearance, geometry, and panoptic understanding within neural radiance fields. An implicit scene representation and understanding model is developed to enhance generalization across indoor and outdoor scenes by extending the scale-encoded cascaded grids within a reparameterized domain distillation framework. This model effectively manages complex scene attributes and generates 3D-consistent scene representations and panoptic understanding outcomes for various scenes. Experiments and ablation studies under challenging conditions, including synthetic and real-world scenes, demonstrate the proposed method's effectiveness in enhancing 3D scene representation and panoptic segmentation accuracy.

cs.CV↗
arXiv2024

Multicam-SLAM: Non-overlapping Multi-camera SLAM for Indirect Visual Localization and Navigation

This paper presents a novel approach to visual simultaneous localization and mapping (SLAM) using multiple RGB-D cameras. The proposed method, Multicam-SLAM, significantly enhances the robustness and accuracy of SLAM systems by capturing more comprehensive spatial information from various perspectives. This method enables the accurate determination of pose relationships among multiple cameras without the need for overlapping fields of view. The proposed Muticam-SLAM includes a unique multi-camera model, a multi-keyframes structure, and several parallel SLAM threads. The multi-camera model allows for the integration of data from multiple cameras, while the multi-keyframes and parallel SLAM threads ensure efficient and accurate pose estimation and mapping. Extensive experiments in various environments demonstrate the superior accuracy and robustness of the proposed method compared to conventional single-camera SLAM systems. The results highlight the potential of the proposed Multicam-SLAM for more complex and challenging applications. Code is available at \url{https://github.com/AlterPang/Multi_ORB_SLAM}.

cs.RO↗
arXiv2024

Local well-posedness for a generalized sixth-order Boussinesq equation

A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived, $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} - (u^2)_{xxxx} - (uu_{xx})_{xx} - (u^3)_{xx} = 0.$$ Such equation is analogous to original Boussinesq equation but with higher order approximation which may ensure a more accuracy description on a long time scale. Moreover, through a rigorous derivation from Boussiensq systems, it has redeemed all the non-linear terms neglected in the sixth order Boussinesq equation (SOBE), $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} = 0.$$ The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, $X^{s,b}$, framework. The multi-linear estimates for $(u^2)_{xx}$, $(u^2)_{xxxx}$, $(uu_{xx})_{xx}$ and $(u^3)_{xx}$ are given, the local wellposedness of the gSOBE is established for $s>\frac{1}{2}$.

math.AP↗
arXiv2023

Effect of lower order terms on the well-posedness of Majda-Biello systems

This paper investigates a noteworthy phenomenon within the framework of Majda-Biello systems, wherein the inclusion of lower-order terms can enhance the well-posedness of the system. Specifically, we investigate the initial value problem (IVP) of the following system: \[ \left\{ \begin{array}{l} u_{t} + u_{xxx} = - v v_x, v_{t} + αv_{xxx} + βv_x = - (uv)_{x}, (u,v)|_{t=0} = (u_0,v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}), \end{array} \right. \quad x \in \mathbb{R}, \, t \in \mathbb{R}, \] where $α\in \mathbb{R}\setminus \{0\}$ and $β\in \mathbb{R}$. Let $s^{*}(α, β)$ be the smallest value for which the IVP is locally analytically well-posed in $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R}) $ when $s > s^{}(α, β)$. Two interesting facts have already been known in literature: $s^{*}(α, 0) = 0$ for $α\in (0,4)\setminus\{1\}$ and $s^*(4,0) = \frac34$. Our key findings include the following: For $s^{*}(4,β)$, a significant reduction is observed, reaching $\frac12$ for $β> 0$ and $\frac14$ for $β< 0$. Conversely, when $α\neq 4$, we demonstrate that the value of $β$ exerts no influence on $s^*(α, β)$. These results shed light on the intriguing behavior of Majda-Biello systems when lower-order terms are introduced and provide valuable insights into the role of $α$ and $β$ in the well-posedness of the system.

math.AP↗
arXiv2023

OT-Net: A Reusable Neural Optimal Transport Solver

With the widespread application of optimal transport (OT), its calculation becomes essential, and various algorithms have emerged. However, the existing methods either have low efficiency or cannot represent discontinuous maps. A novel reusable neural OT solver OT-Net is thus presented, which first learns Brenier's height representation via the neural network to obtain its potential, and then gained the OT map by computing the gradient of the potential. The algorithm has two merits, 1) it can easily represent discontinuous maps, which allows it to match any target distribution with discontinuous supports and achieve sharp boundaries. This can well eliminate mode collapse in the generated models. 2) The OT map can be calculated straightly by the proposed algorithm when new target samples are added, which greatly improves the efficiency and reusability of the map. Moreover, the theoretical error bound of the algorithm is analyzed, and we have demonstrated the empirical success of our approach in image generation, color transfer, and domain adaptation.

cs.CV↗
arXiv2023

who is snoring? snore based user recognition

Snoring is one of the most prominent symptoms of Obstructive Sleep Apnea-Hypopnea Syndrome (OSAH), a highly prevalent disease that causes repetitive collapse and cessation of the upper airway. Thus, accurate snore sound monitoring and analysis is crucial. However, the traditional monitoring method polysomnography (PSG) requires the patients to stay at a sleep clinic for the whole night and be connected to many pieces of equipment. An alternative and less invasive way is passive monitoring using a smartphone at home or in the clinical settings. But, there is a challenge: the environment may be shared by people such that the raw audio may contain the snore activities of the bed partner or other person. False capturing of the snoring activity could lead to critical false alarms and misdiagnosis of the patients. To address this limitation, we propose a hypothesis that snore sound contains unique identity information which can be used for user recognition. We analyzed various machine learning models: Gaussian Mixture Model (GMM), GMM-UBM (Universial Background Model), and a Deep Neural Network (DNN) on MPSSC - an open source snoring dataset to evaluate the validity of our hypothesis. Our results are promising as we achieved around 90% accuracy in identification and verification tasks. This work marks the first step towards understanding the practicality of snore based user monitoring to enable multiple healthcare applicaitons.

cs.SD↗
arXiv2023

Non-homogeneous boundary value problems for coupled KdV-KdV systems posed on the half line

In this article, we study an initial-boundary-value problem of a coupled KdV-KdV system on the half line $ \mathbb{R}^+ $ with non-homogeneous boundary conditions: \begin{equation*} \left\{ \begin{array}{l} u_t+v_x+u u_x+v_{xxx}=0, \quad v_t+u_x+(vu)_x+u_{xxx}=0, \quad u(x,0)=ϕ(x),\quad v(x,0)=ψ(x), \quad u(0,t)=h_1(t),\quad v(0,t)=h_2(t),\quad v_x(0,t)=h_3(t), \end{array} \right. \qquad x,\,t>0. \end{equation*} It is shown that the problem is locally unconditionally well-posed in $H^s(\mathbb{R}^+)\times H^s(\mathbb{R}^+)$ for $s> -\frac34 $ with initial data $(ϕ,ψ)$ in $H^s(\mathbb{R}^+)\times H^{s}(\mathbb{R}^+)$ and boundary data $(h_1,h_2,h_3) $ in $H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s}{3}}(\mathbb{R}^+)$. The approach developed in this paper can also be applied to study more general KdV-KdV systems posed on the half line.

math.AP↗
arXiv2021

Lower Regularity Solutions of the Non-homogeneous Boundary-Value Problem for a Higher Order Boussinesq Equation in a Quarter Plane

We continue to study the initial-boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: \begin{equation*} \begin{cases} u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x,t\in \mathbb{R}^+,\\ u(x,0)=φ(x), u_t(x,0)=ψ''(x), \\ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t), \end{cases} \end{equation*} where $β=\pm1$. We show that the problem is locally analytically well-posed in the space $H^s(\mathbb{R}^+)$ for any $ s> -\frac34 $ with the initial-value data $$(φ,ψ)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$$ and the boundary-value data $$(h_1,h_2,h_3) \in H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s-1}{3}}(\mathbb{R}^+)\times H^{\frac{s-3}{3}}(\mathbb{R}^+).$$

math.AP↗
arXiv2019

An Introduction to Mordell Weil Theorem

This article is an introduction to Mordell-Weil theorem. In this article, I introduced some basic properties about ellptic curves and proved the theorem in two different ways.

math.HO↗
arXiv2019

Lower regularity solutions of non-homogeneous boundary value problems of the sixth order Boussinesq equation in a quarter plane

In this article, we study an initial-boundary-value problem of the sixth order Boussinesq equation on a half line with nonhomogeneous boundary conditions: \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x>0\mbox{, }t>0,\] \[u(x,0)=φ(x), u_t(x,0)=ψ''(x),\] \[ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t),\] where $β=\pm1$. It is shown that the problem is locally well-posed in $H^s(\mathbb{R}^+)$ for $-\frac12<s\leq 0$ with initial condition $(φ,ψ)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$ and boundary condition $(h_1,h_2,h_3) $ in the product space $H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s-1}{3}}(\mathbb{R}^+)\times H^{\frac{s-3}{3}}(\mathbb{R}^+)$.

math.AP↗
arXiv2018

Exact controllability and stability of the Sixth Order Boussinesq equation

The article studies the exact controllability and the stability of the sixth order Boussinesq equation \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f, \quad β=\pm1, \] on the interval $S:=[0,2π]$ with periodic boundary conditions. It is shown that the system is locally exactly controllable in the classic Sobolev space, $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$, for "small" initial and terminal states. It is also shown that if $f$ is assigned as an internal linear feedback, the solution of the system is uniformly exponential decay to a constant state in $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$ with "small" initial data assumption.

math.AP↗
arXiv2018

Hexagonal standing wave patterns of a two-dimensional Boussinesq system

We prove the existence of a large family of two-dimensional standing waves, that are triple periodic solutions, for a Boussinesq system which describes two-way propagation of water waves in a channel. Our proof uses the Lyapunov-Schmidt method to find the bifurcation standing waves.

math.AP↗