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Zhipeng Yang

Publications and source records attributed to Zhipeng Yang.

At least 19 recordsLinked to original sources
arXiv2026

Strict ground-level gaps on long-pendant generalized star graphs for nonlinear Dirac equations

We study strict ground-level gaps for autonomous nonlinear Dirac equations on noncompact metric graphs. The relevant level at infinity is the autonomous full-line level \(d_λ^\infty=d_{λ,\mathbb R}\), since a concentrating sequence escaping along a half-line sees the real line after recentering far from all vertices. We prove that, uniformly for \(λ\) in a compact subinterval \(I\Subset(-mc^2,mc^2)\), the presence of a sufficiently long pendant path in the essential reduction of the graph implies \[ d_λ(\mathcal G)<d_λ^\infty. \] The proof uses a half-line restriction of a symmetric full-line ground state, uniform exponential decay, resolvent localization for massive Dirac-Kirchhoff spectral projections, and a compactness argument for generalized Nehari fibres. We also identify elementary graph classes for which no such gap can hold: pure transmission graphs whose essential reduction is isometric to the full line satisfy \(d_λ(\mathcal G)=d_λ^\infty\).

math.AP↗
arXiv2026

A Sharp Mass Threshold for Fractional Choquard Equations with Lower Hardy-Littlewood-Sobolev and \(L^2\)-Critical Terms

For \(\frac12\le s<1\), we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and \(L^2\)-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass \(a_*\). For \(0 a_*\), the energy is unbounded from below on the mass sphere, while the Pohozaev set is nonempty.

math.AP↗
arXiv2026

Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants

We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants $C_{G,L}$ and $β_{G,L}$. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carathéodory distance; the leading coefficient $\mathfrak F_{G,K}(Z)$ is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of $\mathrm{SU}(N)$, recovering the classical $\mathrm{SU}(2)$ and CR sphere spectra.

math.CA↗
arXiv2026

Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\leq N \leq5$

Let \(Ω\Subset\mathbb H^N\), \(N\in\{3,4,5\}\), be a bounded connected \(C^2\) domain. We prove that the pure critical Dirichlet problem \[ -Δ_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }Ω, \qquad u=0 \quad\text{on }\partialΩ \] admits a positive solution whenever \(H_d(Ω;\mathbb F_2)\neq0\) for some \(1\le d\le N-1\). This gives a hyperbolic Bahri-Coron theorem for \(3\le N\le5\) under \(C^2\) boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

math.AP↗
arXiv2026

N2NMatcher: Towards Inlining-Resilient Binary Decomposition and Module Matching

Program-level Binary Code Similarity Analysis (BCSA) aims to identify semantically similar code regions across binary programs, serving as a fundamental technique for software plagiarism detection, vulnerability search, and malware analysis. Existing approaches often decompose binaries into modules following the structure of function call graphs (FCGs) and then match these modules by their contained functions. However, function inlining changes both FCG structures and binary function semantics, making existing decomposition and module matching methods less effective. In this work, we propose N2NMatcher, an inlining-resilient framework for binary decomposition and module matching. We first conduct an empirical study to examine whether binaries still contain functions that provide stable module boundaries across compilation settings. N2NMatcher learns to predict such functions as anchors using a hierarchical graph neural network that encodes binary ACFG-FCG representations built from opcode sequences, control-flow structures, and FCG calling context. It then performs anchor-bounded decomposition and matches the generated modules using learned module graph embeddings. Experimental results show that N2NMatcher improves both the decomposition quality and module matching accuracy compared to existing works, enabling more effective program-level BCSA.

cs.SE↗
arXiv2026

Word Recovery in Large Language Models Enables Character-Level Tokenization Robustness

Large language models (LLMs) trained with canonical tokenization exhibit surprising robustness to non-canonical inputs such as character-level tokenization, yet the mechanisms underlying this robustness remain unclear. We study this phenomenon through mechanistic interpretability and identify a core process we term word recovery. We first introduce a decoding-based method to detect word recovery, showing that hidden states reconstruct canonical word-level token identities from character-level inputs. We then provide causal evidence by removing the corresponding subspace from hidden states, which consistently degrades downstream task performance. Finally, we conduct a fine-grained attention analysis and show that in-group attention among characters belonging to the same canonical token is critical for word recovery: masking such attention in early layers substantially reduces both recovery scores and task performance. Together, our findings provide a mechanistic explanation for tokenization robustness and identify word recovery as a key mechanism enabling LLMs to process character-level inputs.

cs.CL↗
arXiv2026

Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations

We study a fractional Choquard equation with an upper-critical Hartree term, an $L^2$-supercritical Hartree perturbation, and a semiclassical potential under a prescribed $L^2$-mass constraint. The potential is bounded and nonnegative, has a nonempty zero set, and has a positive lower limit at infinity. For every prescribed mass and all sufficiently small semiclassical parameters, we prove the existence of a pair of normalized solutions $\pm u_\varepsilon$ with a negative Lagrange multiplier. The proof combines a strict energy bound below the critical one-bubble level, compactness modulo translations for the autonomous ground-state set, a simultaneous cutoff of both Hartree terms, and a localized constrained mountain-pass argument. Moreover, suitable translates of $u_\varepsilon$ converge strongly in $H^s(\mathbb{R}^N)$ to a positive autonomous ground state, and the corresponding concentration points approach the zero set of the potential as $\varepsilon\to0$.

math.AP↗
arXiv2026

Isolated Singularities for Fractional Hartree Equations

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-Δ)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^μ}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental-solution term may occur at the singular point. For non-removable blow-up singularities satisfying a fundamental-order upper bound and a weighted source condition, we derive the corresponding Riesz decomposition and retain its nonnegative singular term in an off-center Kelvin moving-spheres argument. In the range determined by two nonnegative Kelvin weights, this yields radial symmetry and strict radial monotonicity. We also construct and classify positive radial homogeneous singular solutions in the corresponding convergence regime. If a nonzero positive radial homogeneous scaling limit is independently known to exist and to solve the limiting equation, then its coefficient is uniquely determined.

math.AP↗
arXiv2026

Local well-posedness for nonlinear Dirac equation on $N$-star metric graphs

We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t ψ= Dψ- |ψ|^{p-2}ψ, \qquad ψ(0)=ψ_0, \] where $p\ge3$, $ψ:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. Using Bourgain-type spaces defined through the spectral resolution of $D$, together with elementary $L^\infty$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data \[ ψ_0\in H_D^s(G)\cap L^\infty(G;\mathbb C^2), \qquad 0\le s<\frac12 . \] The corresponding solution belongs to \[ C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^\infty([0,T]\times G). \] Moreover, $\|ψ(t)\|_{L^2(G;\mathbb{C}^2)}$ is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined $H_D^s$ and space-time $L^\infty$ control norm.

math.AP↗
arXiv2026

A positive ground state for a planar Choquard equation with mixed diffusion and critical exponential growth

We study a two-dimensional Choquard equation driven by the mixed local and nonlocal operator $L:=-Δ+(-Δ)^s$, where the nonlinearity has critical exponential growth of Trudinger--Moser type. Under a coercive assumption on the potential and suitable one-sided assumptions on the nonlinearity, we prove the existence of a least energy positive solution. The proof combines Nehari manifold minimization, compactness below the critical Trudinger--Moser threshold, local regularity, and a strong maximum principle.

math.AP↗
arXiv2026

Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

We study the Choquard equation involving mixed local and nonlocal operators \[ -Δu + (-Δ)^{s}u + V(x)u = \left(\frac{1}{|x|^μ} * F(u)\right) f(u) \quad \text{in } \mathbb{R}^{2}, \] where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.

math.AP↗
arXiv2026

Infinitely many sign-changing solutions for logarithmic Schrödinger equations via an \(L^p\)-perturbation approach

We study the logarithmic Schrödinger equation \[ -Δu+V(x)u=u\log u^2,\qquad x\in\mathbb R^N,\ N\ge3. \] Since the logarithmic energy is not \(C^1\) on the natural space \(H_V^1(\mathbb R^N)\), direct invariant-set minimax arguments for sign-changing solutions are not available. We introduce an \(L^p\)-regularization perturbation, which restores a \(C^1\) variational structure while preserving the logarithmic nonlinearity, and prove via a limiting argument that the original equation admits infinitely many sign-changing weak solutions.

math.AP↗
arXiv2026

Low-regularity well-posedness for a mixed-sign quadratic Dirac equation on $N$-star metric graphs

We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t ψ= Dψ- \mathcal N(ψ), \qquad ψ(0)=ψ_0, \] where $ψ=(ψ_1,ψ_2)^{\mathsf T}:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. The nonlinearity acts edgewise and is given by a bilinear interaction between the positive and negative spectral parts, \[ \mathcal N(ψ)=\mathcal B\bigl(Π_+ψ,Π_-ψ\bigr), \] where $Π_\pm$ are the spectral projections of $D$ and $\mathcal B$ is a fixed bilinear map on $\mathbb{C}^2$ applied componentwise on each edge. This is a model quadratic interaction tailored to the mixed-sign Bourgain-space mechanism, rather than a general nonlinear Dirac equation on graphs. Using Bourgain-type spaces associated with the spectral resolution of $D$ and a mixed-sign bilinear estimate on $N$-star graphs, we prove local well-posedness in the operator Sobolev space $H_D^s(G)$ for \(s>-\frac18\). We also establish a blow-up alternative in $H_D^s(G)$ for the maximal forward lifespan.

math.AP↗
arXiv2026

Higher-Order Heat Estimates and Semilinear Heat Equations on the Noncommutative Torus

We establish sharp higher-order heat estimates with complete bound on the noncommutative tori \(\mathbb{T}_θ^{n}\) and show the optimality in the small-time order. As an application in polynomial semilinear heat equations on \(\mathbb{T}_θ^{n}\), we give local well-posedness, the blow-up alternative, persistence of higher regularity, and instantaneous smoothing in the Sobolev algebra scale \(H^k_θ\), \(k>n/2\).

math.AP↗
arXiv2026

On a class of logarithmic Schrödinger equations via perturbation method

In this paper, we consider the following logarithmic Schrödinger equation \[ -Δu + V(x)u = u \log u^{2},\quad x\in\mathbb{R}^{N}. \] Assuming that \(V\in C(\mathbb{R}^{N},\mathbb R)\), \(V\) is bounded away from zero, and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the lack of \(C^{1}\)-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions.

math.AP↗
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↗