arXiv subjects

Zhipeng Yang

Publications and source records attributed to Zhipeng Yang.

At least 37 records · Page 2Linked to original sources

arXiv2026

On the existence of normalized solutions to a class of fractional Choquard equation with potentials

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(x) u=λu+f(x)\left(I_α*\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_α*\left(g|u|^p\right)\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N $$ subject to the mass constraint $$ \int_{\mathbb{R}^N}|u|^2 d x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N)$, and $\frac{N+α}{N} \leq q<p \leq \frac{N+α+2 s}{N}$. Here, the parameter $λ\in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x), f(x)$, and $g(x)$, we establish several existence results for normalized solutions.

math.AP↗

arXiv2026

Fractional Sobolev-type embedding on CR sphere and Heisenberg group

This paper studies critical fractional Sobolev inequalities with lower-order terms on the standard CR sphere $\mathbb S^{2n+1}$. Let $Q=2n+2$, let $s\in(0,1)$, let $1<p<Q$, and let $p_s^*=\frac{Qp}{Q-sp}$. For the inequality $\|u\|_{L^{p_s^*}(\mathbb S^{2n+1})}\le A[u]_{s,p}+B\|u\|_{L^p(\mathbb S^{2n+1})}$, we prove that the admissible lower-order coefficients are exactly $\left[|\mathbb S^{2n+1}|^{-s/Q},\infty\right)$. For the power-type inequality $\|u\|_{L^{p_s^*}(\mathbb S^{2n+1})}^p\le A[u]_{s,p}^p+B\|u\|_{L^p(\mathbb S^{2n+1})}^p$, we show that the admissible set is $\left[|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $1<p\le 2$, and $\left(|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $2<p<Q$. Via the Cayley transform, we derive the exact weighted counterpart on the Heisenberg group and prove that the corresponding admissible sets coincide with those on the sphere. We also show that nonlinear first-moment constraints do not improve the optimal lower-order coefficient, whereas finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.

math.AP↗

arXiv2026

Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation

In this paper, we study the mass-constrained fractional Choquard equation $ (-Δ)^s u = λu + α(I_μ* |u|^{\frac{2N-μ}{N}})|u|^{\frac{2N-μ}{N}-2}u + (I_μ* |u|^p)|u|^{p-2}u $ in $ \mathbb{R}^N $, under the constraint $ \int_{\mathbb{R}^N} |u|^2 \, dx = c^2 > 0 $, where $ N > 2s $, $ s \in (0,1) $, $ μ\in (0,N) $, $ α> 0 $, and $ 2 + \frac{2s-μ}{N} \le p < \frac{2N-μ}{N-2s} $. We first establish a nonexistence result in the $ L^2 $-critical case $ p = 2 + \frac{2s-μ}{N} $. Then, in the $ L^2 $-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity $ \mathcal{M}_1(c) $. Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps.

math.AP↗

arXiv2026

Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

We study a logarithmic fractional Schrödinger--Poisson system in $\R^{3}$: \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+ϕu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αϕ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here $α\in\bigl(\frac34,1\bigr)$, $4 0$ and all sufficiently small $\varepsilon>0$, the system admits at least $\operatorname{cat}_{M_δ}(M)$ distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of $V$ as $\varepsilon\to0$.

math.AP↗

arXiv2026

Improved Fractional Sobolev Embeddings on Closed Riemannian Manifolds under Isometric Group Actions

In this paper, we study symmetry-improved fractional Sobolev embeddings on closed Riemannian manifolds under the action of compact isometry groups. We prove that $G$-invariant fractional Sobolev spaces embed into higher $L^p$ spaces, with corresponding compactness results depending on the minimal orbit dimension. We also investigate the associated optimal constants in the improved critical inequality and in the standard critical inequality under finite-orbit symmetry.

math.AP↗

arXiv2026

Improved Sobolev Inequalities on the Quaternionic Sphere

In this paper we establish improved Sobolev inequalities on the quaternionic sphere under higher-order moment vanishing conditions with respect to the measure $|u|^{p^*}\,dξ$. As an application, we give a new proof of the existence of extremals for the sharp Sobolev embedding \[ S^{1,2}(S^{4n+3}) \hookrightarrow L^{2^*}(S^{4n+3}). \]

math.AP↗

arXiv2026

Spectral properties and bound states of the Dirac equation on periodic quantum graphs

We investigate nonlinear Dirac equations on a periodic quantum graph $G$ and develop a variational approach to the existence and multiplicity of bound states. After introducing the Dirac operator on $G$ with a $\mathbb Z^{d}$-periodic potential, we describe its spectral decomposition and work in the natural energy space. Under asymptotically linear or superquadratic assumptions on the nonlinearity, we establish the required linking geometry and a Cerami-type compactness property modulo $\mathbb Z^{d}$-translations. As a consequence, we prove the existence of at least one bound state and, when the nonlinearity is even, infinitely many geometrically distinct bound states.

math.AP↗

arXiv2026

Bound states of nonlinear Dirac equations on periodic quantum graphs

We study nonlinear Dirac equations (NLDE) on periodic quantum graphs endowed with Kirchhoff-type vertex conditions. Our main goal is to establish existence and multiplicity of bound states, which arise as critical points of the associated NLDE action functional. The underlying Dirac operator has a spectral gap around the origin, so the corresponding functional is strongly indefinite, and in addition the Palais--Smale condition fails due to the noncompactness and the periodic structure of the graph. To overcome these difficulties, we combine the spectral properties of the periodic Dirac operator with critical point theorems for strongly indefinite functionals and a concentration--compactness analysis adapted to periodic quantum graphs, and derive existence and multiplicity results for bound states with frequencies lying in the spectral gap.

math.AP↗

arXiv2026

Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}Δu+\varepsilon^{2s}(-Δ)^{s}u+V(x)u=\varepsilon^{μ-2}\left(\frac{1}{|x|^μ}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where $\varepsilon>0$, $s\in(0,1)$, $0<μ<2$, $f$ has Trudinger--Moser critical exponential growth, and $F(t)=\int_{0}^{t}f(τ)\,dτ$. By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as $\varepsilon\to0^{+}$.

math.AP↗

arXiv2026

NextFlow: Unified Sequential Modeling Activates Multimodal Understanding and Generation

We present NextFlow, a unified decoder-only autoregressive transformer trained on 6 trillion interleaved text-image discrete tokens. By leveraging a unified vision representation within a unified autoregressive architecture, NextFlow natively activates multimodal understanding and generation capabilities, unlocking abilities of image editing, interleaved content and video generation. Motivated by the distinct nature of modalities - where text is strictly sequential and images are inherently hierarchical - we retain next-token prediction for text but adopt next-scale prediction for visual generation. This departs from traditional raster-scan methods, enabling the generation of 1024x1024 images in just 5 seconds - orders of magnitude faster than comparable AR models. We address the instabilities of multi-scale generation through a robust training recipe. Furthermore, we introduce a prefix-tuning strategy for reinforcement learning. Experiments demonstrate that NextFlow achieves state-of-the-art performance among unified models and rivals specialized diffusion baselines in visual quality.

cs.CV↗

arXiv2025

Sharp Fractional Sobolev Embeddings on Closed Manifolds

We develop an intrinsic, heat-kernel based fractional Sobolev framework on closed Riemannian manifolds and study the critical fractional Sobolev embedding. We determine the optimal coefficient of the lower-order $L^{p}$ term and prove that the fully sharp $p$-power inequality cannot hold globally in the superquadratic range. We further establish an almost sharp inequality whose leading constant is arbitrarily close to the Euclidean best constant, and we derive improved inequalities under finitely many orthogonality constraints with respect to sign-changing test families.

math.AP↗

arXiv2025

Convexification Numerical Method for Imaging of Moving Targets

The problem of imaging of a moving target is formulated as a Coefficient Inverse Problem for a hyperbolic equation with its coefficient depending on all three spatial variables and time. As the initial condition, the point source running along a straight line is used. Lateral Cauchy data are known for each position of the point source. A truncated Fourier series with respect to a special orthonormal basis is used. First, Lipschitz stability estimate is obtained. Next, a globally convergent numerical method, the so-called convexification method, is developed and its convergence analysis is carried out. The convexification method is based on a Carleman estimate. Results of numerical experiments are presented.

math.NA↗

arXiv2025

Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

In this paper we study the following fractional Choquard equation with mixed nonlinearities: \[ \left\{ \begin{array}{l} (-Δ)^s u = λu + α\left( I_μ* |u|^q \right) |u|^{q-2} u + \left( I_μ* |u|^p \right) |u|^{p-2} u, \quad x \in \mathbb{R}^N, \\[4pt] \displaystyle \int_{\mathbb{R}^N} |u|^2 \,\mathrm{d}x = c^2 > 0. \end{array} \right. \] Here $N > 2s$, $s \in (0,1)$, $μ\in (0, N)$, and the exponents satisfy \[ \frac{2N - μ}{N} < q < p < \frac{2N - μ}{N - 2s}, \] while $α> 0$ is a sufficiently small parameter, $λ\in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_μ$ denotes the Riesz potential. We establish existence and multiplicity results for normalized solutions and, in addition, prove the existence of ground state normalized solutions for $α$ in a suitable range.

math.AP↗

arXiv2025

Towards an Oracle for Binary Decomposition Under Compilation Variance

Third-Party Library (TPL) detection, which identifies reused libraries in binary code, is critical for software security analysis. At its core, TPL detection depends on binary decomposition-the process of partitioning a monolithic binary into cohesive modules. Existing decomposition methods, whether anchor-based or clustering-based, fundamentally rely on the assumption that reused code exhibits similar function call relationships. However, this assumption is severely undermined by Function Call Graph (FCG) variations introduced by diverse compilation settings, particularly function inlining decisions that drastically alter FCG structures. In this work, we conduct the first systematic empirical study to establish the oracle for optimal binary decomposition under compilation variance. We first develop a labeling method to create precise FCG mappings on a comprehensive dataset compiled with 17 compilers, 6 optimizations, and 4 architectures; then, we identify the minimum semantic-equivalent function regions between FCG variants to derive the ground-truth decomposition. This oracle provides the first rigorous evaluation framework that quantitatively assesses decomposition algorithms under compilation variance. Using this oracle, we evaluate existing methods and expose their critical limitations: they either suffer from under-aggregation failure or over-aggregation failure. Our findings reveal that current decomposition techniques are inadequate for robust TPL detection, highlighting the urgent need for compilation-aware approaches.

cs.SE↗

arXiv2025

Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

This paper has two main purposes. In the first part, combining the nondegeneracy of the ground state with the Lyapunov--Schmidt reduction method, we prove the existence of multi-peak positive solutions to the singularly perturbed problem \[\Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b\int_{\mathbb{R}^{N}}|(-Δ)^{\frac{s}{2}}u|^2\,dx\Big)(-Δ)^s u+V(x)u=u^p\quad \text{in }\mathbb{R}^{N},\] for all sufficiently small $\varepsilon> 0$, under the assumptions $2s<N<4s$, $1<p<2^*_s-1$, and some mild conditions on the potential $V$. The main difficulty comes from the interplay between the nonlocal operator $(-Δ)^s$ and the nonlocal Kirchhoff term, which makes the corresponding limiting problem a coupled system of partial differential equations rather than a single fractional Kirchhoff equation. In the second part, under additional assumptions on $V$, we establish the local uniqueness of positive multi-peak solutions by means of a local Pohozǎev identity.

math.AP↗

arXiv2025

Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,σ_1\,\partial_x u + m c^2 σ_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P} \] where $V:\mathcal{G}\to\mathbb{R}$ and $f:\mathbb{R}\to\mathbb{R}$ are continuous, $\varepsilon>0$ is a semiclassical parameter, $m>0$ denotes the mass, and $c>0$ the speed of light. Here $σ_1,σ_3$ are Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We prove that when $\varepsilon$ is sufficiently small, the number of solutions to $(P)$ is at least the number of global minima of $V$. Moreover, these solutions exhibit semiclassical concentration: as $\varepsilon\to0$, their concentration points approach the set of global minima of $V$.

math.AP↗

arXiv2025

Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: \[ -i c \frac{d}{d x} σ_1 ψ+m c^2 σ_3 ψ-ωψ=g(|ψ|) ψ, \quad \text { in } \mathcal{G} \] where $ g : \mathbb{R}\rightarrow\mathbb{R} $ is a continuous nonlinear function, $ c>0 $ represents the speed of light, $ m>0 $ is the particle's mass, $ ω\in\mathbb{R} $ is related to the frequency, $ σ_1 $ and $ σ_3 $ denote the Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on $\mathcal{G}$, and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schrödinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light $ c \to \infty $) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in $ c $, thereby offering insight into their asymptotic behavior.

math.AP↗

arXiv2025

Internal Chain-of-Thought: Empirical Evidence for Layer-wise Subtask Scheduling in LLMs

We show that large language models (LLMs) exhibit an $\textit{internal chain-of-thought}$: they sequentially decompose and execute composite tasks layer-by-layer. Two claims ground our study: (i) distinct subtasks are learned at different network depths, and (ii) these subtasks are executed sequentially across layers. On a benchmark of 15 two-step composite tasks, we employ layer-from context-masking and propose a novel cross-task patching method, confirming (i). To examine claim (ii), we apply LogitLens to decode hidden states, revealing a consistent layerwise execution pattern. We further replicate our analysis on the real-world $\text{TRACE}$ benchmark, observing the same stepwise dynamics. Together, our results enhance LLMs transparency by showing their capacity to internally plan and execute subtasks (or instructions), opening avenues for fine-grained, instruction-level activation steering.

cs.CL↗