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Published work

25 published item(s)

preprint2026arXiv

Attention Hijacking: Response Manipulation Across Queries in Vision-Language Models

Existing adversarial attacks on vision-language models (VLMs) can steer model outputs toward attacker-specified target responses, but their effectiveness often degrades when the same perturbed input is paired with different textual queries. This paper studies cross-query response manipulation, where a single adversarial example is expected to remain effective across diverse user queries. We first analyze the limitations of existing attacks and find that successful transfer is closely associated with preserving an image-dominant attention pattern during response generation. Motivated by the observation, we propose \textbf{Attention Hijacking}, a novel adversarial attack that explicitly steers internal attention distributions toward a persistent image-dominant pattern. By amplifying the influence of visual tokens on target response tokens while suppressing the competing influence of textual tokens, our method reduces the dependence of the manipulated output on the specific wording of the query. Extensive experiments on widely used VLMs show that Attention Hijacking substantially improves cross-query transferability across diverse target responses and unseen queries. The method also extends effectively to multiple attack scenarios, offering new insights into the role of attention stability in transferable response manipulation for VLMs.

preprint2026arXiv

AutoVulnPHP: LLM-Powered Two-Stage PHP Vulnerability Detection and Automated Localization

PHP's dominance in web development is undermined by security challenges: static analysis lacks semantic depth, causing high false positives; dynamic analysis is computationally expensive; and automated vulnerability localization suffers from coarse granularity and imprecise context. Additionally, the absence of large-scale PHP vulnerability datasets and fragmented toolchains hinder real-world deployment. We present AutoVulnPHP, an end-to-end framework coupling two-stage vulnerability detection with fine-grained automated localization. SIFT-VulMiner (Structural Inference for Flaw Triage Vulnerability Miner) generates vulnerability hypotheses using AST structures enhanced with data flow. SAFE-VulMiner (Semantic Analysis for Flaw Evaluation Vulnerability Miner) verifies candidates through pretrained code encoder embeddings, eliminating false positives. ISAL (Incremental Sequence Analysis for Localization) pinpoints root causes via syntax-guided tracing, chain-of-thought LLM inference, and causal consistency checks to ensure precision. We contribute PHPVD, the first large-scale PHP vulnerability dataset with 26,614 files (5.2M LOC) across seven vulnerability types. On public benchmarks and PHPVD, AutoVulnPHP achieves 99.7% detection accuracy, 99.5% F1 score, and 81.0% localization rate. Deployed on real-world repositories, it discovered 429 previously unknown vulnerabilities, 351 assigned CVE identifiers, validating its practical effectiveness.

preprint2026arXiv

MCP-ITP: An Automated Framework for Implicit Tool Poisoning in MCP

To standardize interactions between LLM-based agents and their environments, the Model Context Protocol (MCP) was proposed and has since been widely adopted. However, integrating external tools expands the attack surface, exposing agents to tool poisoning attacks. In such attacks, malicious instructions embedded in tool metadata are injected into the agent context during MCP registration phase, thereby manipulating agent behavior. Prior work primarily focuses on explicit tool poisoning or relied on manually crafted poisoned tools. In contrast, we focus on a particularly stealthy variant: implicit tool poisoning, where the poisoned tool itself remains uninvoked. Instead, the instructions embedded in the tool metadata induce the agent to invoke a legitimate but high-privilege tool to perform malicious operations. We propose MCP-ITP, the first automated and adaptive framework for implicit tool poisoning within the MCP ecosystem. MCP-ITP formulates poisoned tool generation as a black-box optimization problem and employs an iterative optimization strategy that leverages feedback from both an evaluation LLM and a detection LLM to maximize Attack Success Rate (ASR) while evading current detection mechanisms. Experimental results on the MCPTox dataset across 12 LLM agents demonstrate that MCP-ITP consistently outperforms the manually crafted baseline, achieving up to 84.2% ASR while suppressing the Malicious Tool Detection Rate (MDR) to as low as 0.3%.

preprint2026arXiv

MindGuard: Intrinsic Decision Inspection for Securing LLM Agents Against Metadata Poisoning

The Model Context Protocol (MCP) is increasingly adopted to standardize the interaction between LLM agents and external tools. However, this trend introduces a new threat: Tool Poisoning Attacks (TPA), where tool metadata is poisoned to induce the agent to perform unauthorized operations. Existing defenses that primarily focus on behavior-level analysis are fundamentally ineffective against TPA, as poisoned tools need not be executed, leaving no behavioral trace to monitor. Thus, we propose MindGuard, a decision-level guardrail for LLM agents, providing provenance tracking of call decisions, policy-agnostic detection, and poisoning source attribution against TPA. While fully explaining LLM decision remains challenging, our empirical findings uncover a strong correlation between LLM attention mechanisms and tool invocation decisions. Therefore, we choose attention as an empirical signal for decision tracking and formalize this as the Decision Dependence Graph (DDG), which models the LLM's reasoning process as a weighted, directed graph where vertices represent logical concepts and edges quantify the attention-based dependencies. We further design robust DDG construction and graph-based anomaly analysis mechanisms that efficiently detect and attribute TPA attacks. Extensive experiments on real-world datasets demonstrate that MindGuard achieves 94\%-99\% average precision in detecting poisoned invocations, 95\%-100\% attribution accuracy, with processing times under one second and no additional token cost. Moreover, DDG can be viewed as an adaptation of the classical Program Dependence Graph (PDG), providing a solid foundation for applying traditional security policies at the decision level.

preprint2026arXiv

Transferability of Adversarial Attacks in Video-based MLLMs: A Cross-modal Image-to-Video Approach

Video-based multimodal large language models (V-MLLMs) have shown vulnerability to adversarial examples in video-text multimodal tasks. However, the transferability of adversarial videos to unseen models - a common and practical real-world scenario - remains unexplored. In this paper, we pioneer an investigation into the transferability of adversarial video samples across V-MLLMs. We find that existing adversarial attack methods face significant limitations when applied in black-box settings for V-MLLMs, which we attribute to the following shortcomings: (1) lacking generalization in perturbing video features, (2) focusing only on sparse key-frames, and (3) failing to integrate multimodal information. To address these limitations and deepen the understanding of V-MLLM vulnerabilities in black-box scenarios, we introduce the Image-to-Video MLLM (I2V-MLLM) attack. In I2V-MLLM, we utilize an image-based multimodal large language model (I-MLLM) as a surrogate model to craft adversarial video samples. Multimodal interactions and spatiotemporal information are integrated to disrupt video representations within the latent space, improving adversarial transferability. Additionally, a perturbation propagation technique is introduced to handle different unknown frame sampling strategies. Experimental results demonstrate that our method can generate adversarial examples that exhibit strong transferability across different V-MLLMs on multiple video-text multimodal tasks. Compared to white-box attacks on these models, our black-box attacks (using BLIP-2 as a surrogate model) achieve competitive performance, with average attack success rate (AASR) of 57.98% on MSVD-QA and 58.26% on MSRVTT-QA for Zero-Shot VideoQA tasks, respectively.

preprint2024arXiv

Spectrality of random convolutions generated by finitely many Hadamard triples

Let $\{(N_j, B_j, L_j): 1 \le j \le m\}$ be finitely many Hadamard triples in $\mathbb{R}$. Given a sequence of positive integers $\{n_k\}_{k=1}^\infty$ and $ω=(ω_k)_{k=1}^\infty \in \{1,2,\cdots, m\}^\mathbb{N}$, let $μ_{ω,\{n_k\}}$ be the infinite convolution given by $$μ_{ω,\{n_k\}} = δ_{N_{ω_1}^{-n_1} B_{ω_1}} * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} B_{ω_2}} * \cdots * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} \cdots N_{ω_k}^{-n_k} B_{ω_k} }* \cdots. $$ In order to study the spectrality of $μ_{ω,\{ n_k\}}$, we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if $\mathrm{gcd}(B_j - B_j)=1$ for $1 \le j \le m$, then all infinite convolutions $μ_{ω,\{n_k\}}$ are spectral measures. This implies that we may find a subset $Λ_{ω,\{n_k\}}\subseteq \mathbb{R}$ such that $\big\{ e_λ(x) = e^{2πi λx}: λ\in Λ_{ω,\{n_k\}} \big\}$ forms an orthonormal basis for $L^2(μ_{ω,\{ n_k\}})$.

preprint2022arXiv

120-fs single-pulse generation from stretched-pulse fiber Kerr resonators

Fiber Kerr resonators are simple driven resonators with desirable wavelength and repetition rate flexibility for generating ultrashort pulses for applications including telecommunications, biomedicine, and materials processing. However, fiber Kerr resonators to date often generate longer pulses and require more complicated techniques for generating single pulses than would be desirable for applications. Here we address these limits by demonstrating robust single-pulse performance with 120-fs pulse durations in fiber Kerr resonators based on stretched-pulse solitons. Through matching numerical and experimental studies, stretched-pulse soliton performance is found to strongly depend on the total cavity length, and the optimum length is found to depend on the drive, Raman scattering, and the total pulse stretching. By designing the cavity for this optimum with the described setup, stable stretched-pulse solitons with 120-fs duration are experimentally observed. In addition, soliton trapping is demonstrated with a pulsed drive source despite large intracavity breathing and single-pulse performance is observed. Robust with high performance single-pulse generation is a critical step toward useful femtosecond pulse generation.

preprint2022arXiv

Benchmarks for Corruption Invariant Person Re-identification

When deploying person re-identification (ReID) model in safety-critical applications, it is pivotal to understanding the robustness of the model against a diverse array of image corruptions. However, current evaluations of person ReID only consider the performance on clean datasets and ignore images in various corrupted scenarios. In this work, we comprehensively establish six ReID benchmarks for learning corruption invariant representation. In the field of ReID, we are the first to conduct an exhaustive study on corruption invariant learning in single- and cross-modality datasets, including Market-1501, CUHK03, MSMT17, RegDB, SYSU-MM01. After reproducing and examining the robustness performance of 21 recent ReID methods, we have some observations: 1) transformer-based models are more robust towards corrupted images, compared with CNN-based models, 2) increasing the probability of random erasing (a commonly used augmentation method) hurts model corruption robustness, 3) cross-dataset generalization improves with corruption robustness increases. By analyzing the above observations, we propose a strong baseline on both single- and cross-modality ReID datasets which achieves improved robustness against diverse corruptions. Our codes are available on https://github.com/MinghuiChen43/CIL-ReID.

preprint2022arXiv

GLSD: The Global Large-Scale Ship Database and Baseline Evaluations

In this paper, we introduce a challenging global large-scale ship database (called GLSD), designed specifically for ship detection tasks. The designed GLSD database includes a total of 212,357 annotated instances from 152,576 images. Based on the collected images, we propose 13 ship categories that widely exist in international routes. These categories include Sailing boat, Fishing boat, Passenger ship, Warship, General cargo ship, Container ship, Bulk cargo carrier, Barge, Ore carrier, Speed boat, Canoe, Oil carrier, and Tug. The motivations of developing GLSD include the following: 1) providing a refine and extensive ship detection database that benefits the object detection community, 2) establishing a database with exhaustive labels (bounding boxes and ship class categories) in a uniform classification scheme, and 3) providing a large-scale ship database with geographic information (covering more than 3000 ports and 33 routes) that benefits multi-modal analysis. In addition, we discuss the evaluation protocols corresponding to image characteristics in GLSD and analyze the performance of selected state-of-the-art object detection algorithms on GSLD, aiming to establish baselines for future studies. More information regarding the designed GLSD can be found at https://github.com/jiaming-wang/GLSD.

preprint2022arXiv

Heat-bath approach to anomalous thermal transport: effects of inelastic scattering

We present results for the entire set of anomalous charge and heat transport coefficients for metallic systems in the presence of a finite-temperature heat bath. In realistic physical systems this necessitates the inclusion of inelastic dissipation mechanisms; relatively little is known theoretically about their effects on anomalous transport. Here we demonstrate how these dissipative processes are strongly intertwined with Berry-curvature physics. Our calculations are made possible by the introduction of a Caldeira-Leggett reservoir which allows us to avoid the sometimes-problematic device of the pseudogravitational potential. Using our formulas, we focus on the finite-temperature behavior of the important anomalous Wiedemann-Franz ratio. Despite previous expectations, this ratio is found to be non-universal as it can exhibit either an upturn or a downturn as temperature increases away from zero. We emphasize that this derives from a \textit{competition} between Berry curvatures having different signs in different regions of the Brillouin zone. We point to experimental support for these observations and for the behavior of an alternative ratio involving a thermoelectric response which, by contrast, appears to be more universal at low temperatures. Our work paves the way for future theory and experiment, demonstrating how inelastic scattering at non-zero temperature affects the behavior of all anomalous transport coefficients.

preprint2022arXiv

Spectral pulsations of dissipative solitons in ultrafast fiber lasers: period doubling and beyond

Period doubling is a universal bifurcation of central importance in all disciplines of nonlinear science, which generally signals the existence of chaotic dynamics in the vicinity of the system parameters. Although observed in diverse ultrafast laser configurations, there is still no consensus on its physical origin. The observations also include other types of pulsating dissipative solitons, with either short or long periods. Real time spectral characterization allows to investigate optical spectral oscillations, whose features reveal the intracavity dynamics leading to instabilities. Following a contextual review, this article presents a variety of period doubling dynamics manifesting in the spectral domain of dissipative solitons. These dynamics are obtained with ultrafast fiber lasers featuring either anomalous or normal dispersion. It reveals a sequence of period doubling bifurcations and instabilities within transient dynamics, unveiling intertwined bifurcations and the entrainment of new pulsating frequencies. The oscillating frequencies tend to lock to the integral of roundtrip numbers as well as coexist with period doubling, demonstrating new combinations of the period doubling bifurcation with other bifurcations. These experimental findings are confirmed by numerical simulations, emphasizing both the universality of the period doubling bifurcations and their potentially highly complicated manifestations within ultrafast laser systems.

preprint2022arXiv

Symmetry and Asymmetry in the 1+N Coorbital Problem

The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an emphasis on the relation between symmetries of the configurations and symmetries in the masses, or lack thereof. We prove that in the $N=4$, $N=6$, and $N=8$ Newtonian coorbital problems there exist symmetric relative equilibria with asymmetric positive masses. This result can be generalized to other homogeneous potentials, and we conjecture similar results hold for larger even numbers of infinitesimal masses. We prove that some equalities of the masses in the $1+4$ and $1+5$ coorbital problems imply symmetry of a class of convex relative equilibria. We also prove there is at most one convex central configuration of the symmetric $1+5$ problem.

preprint2022arXiv

Weak Convergence and Spectrality of Infinite Convolutions

Let $\{ A_k\}_{k=1}^\infty$ be a sequence of finite subsets of $\mathbb{R}^d$ satisfying that $\# A_k \ge 2$ for all integers $k \ge 1$. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution $$ν=δ_{A_1}*δ_{A_2} * \cdots *δ_{A_n}*\cdots, $$ where all sets $A_k \subseteq \mathbb{R}_+^d$ and $δ_A = \frac{1}{\# A} \sum_{a \in A} δ_a$. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in $\mathbb{R}$ and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.

preprint2021arXiv

Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials

We prove the reducibility of 1-D quantum harmonic oscillators in $\mathbb R$ perturbed by a quasi-periodic in time potential $V(x,ωt)$ under the following conditions, namely there is a $C>0$ such that \begin{equation*} |V(x,θ)|\le C,\quad|x\partial_xV(x,θ)|\le C,\quad\forall~(x,θ)\in\mathbb R\times\mathbb T_σ^n. \end{equation*} The corresponding perturbation matrix $(P_i^j(θ))$ is proved to satisfy $(1+|i-j|)| P_i^j(θ)|\le C$ and $\sqrt{ij}|P_{i+1}^{j+1}(θ)-P_i^j(θ)|\le C$ for any $θ\in\mathbb T_σ^n$ and $i,j\geq 1$. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element $P_{i}^j(θ)$ as well as the discrete difference matrix element $P_{i+1}^{j+1}(θ)-P_i^j(θ)$. For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.

preprint2021arXiv

SSCAN: A Spatial-spectral Cross Attention Network for Hyperspectral Image Denoising

Hyperspectral images (HSIs) have been widely used in a variety of applications thanks to the rich spectral information they are able to provide. Among all HSI processing tasks, HSI denoising is a crucial step. Recently, deep learning-based image denoising methods have made great progress and achieved great performance. However, existing methods tend to ignore the correlations between adjacent spectral bands, leading to problems such as spectral distortion and blurred edges in denoised results. In this study, we propose a novel HSI denoising network, termed SSCAN, that combines group convolutions and attention modules. Specifically, we use a group convolution with a spatial attention module to facilitate feature extraction by directing models' attention to band-wise important features. We propose a spectral-spatial attention block (SSAB) to exploit the spatial and spectral information in hyperspectral images in an effective manner. In addition, we adopt residual learning operations with skip connections to ensure training stability. The experimental results indicate that the proposed SSCAN outperforms several state-of-the-art HSI denoising algorithms.

preprint2020arXiv

Boundary feedback stabilization of quasilinear hyperbolic systems with partially dissipative structure

In this paper, we study the boundary feedback stabilization of a quasilinear hyperbolic system with partially dissipative structure. Thanks to this structure, we construct a suitable Lyapunov function which leads to the exponential stability to the equilibrium of the $H^2$ solution. As an application, we also obtain the feedback stabilization for the Saint-Venant-Exner model under physical boundary conditions.

preprint2020arXiv

Correct Normalization Matters: Understanding the Effect of Normalization On Deep Neural Network Models For Click-Through Rate Prediction

Normalization has become one of the most fundamental components in many deep neural networks for machine learning tasks while deep neural network has also been widely used in CTR estimation field. Among most of the proposed deep neural network models, few model utilize normalization approaches. Though some works such as Deep & Cross Network (DCN) and Neural Factorization Machine (NFM) use Batch Normalization in MLP part of the structure, there isn't work to thoroughly explore the effect of the normalization on the DNN ranking systems. In this paper, we conduct a systematic study on the effect of widely used normalization schemas by applying the various normalization approaches to both feature embedding and MLP part in DNN model. Extensive experiments are conduct on three real-world datasets and the experiment results demonstrate that the correct normalization significantly enhances model's performance. We also propose a new and effective normalization approaches based on LayerNorm named variance only LayerNorm(VO-LN) in this work. A normalization enhanced DNN model named NormDNN is also proposed based on the above-mentioned observation. As for the reason why normalization works for DNN models in CTR estimation, we find that the variance of normalization plays the main role and give an explanation in this work.

preprint2020arXiv

GateNet: Gating-Enhanced Deep Network for Click-Through Rate Prediction

Advertising and feed ranking are essential to many Internet companies such as Facebook. Among many real-world advertising and feed ranking systems, click through rate (CTR) prediction plays a central role. In recent years, many neural network based CTR models have been proposed and achieved success such as Factorization-Machine Supported Neural Networks, DeepFM and xDeepFM. Many of them contain two commonly used components: embedding layer and MLP hidden layers. On the other side, gating mechanism is also widely applied in many research fields such as computer vision(CV) and natural language processing(NLP). Some research has proved that gating mechanism improves the trainability of non-convex deep neural networks. Inspired by these observations, we propose a novel model named GateNet which introduces either the feature embedding gate or the hidden gate to the embedding layer or hidden layers of DNN CTR models, respectively. The feature embedding gate provides a learnable feature gating module to select salient latent information from the feature-level. The hidden gate helps the model to implicitly capture the high-order interaction more effectively. Extensive experiments conducted on three real-world datasets demonstrate its effectiveness to boost the performance of various state-of-the-art models such as FM, DeepFM and xDeepFM on all datasets.

preprint2020arXiv

On the Observability Inequality of Coupled Wave Equations: the Case without Boundary

In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau's paper \cite{DehmanLebeau09} to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds.

preprint2020arXiv

Order parameter dynamics of the non-linear sigma model in the large $N$ limit

We study non-equilibrium order parameter dynamics of the non-linear sigma model in the large $N$ limit, using Keldysh formalism. We provide a scheme for obtaining stable numerical solutions of the Keldysh saddle point equations, and use them to study the order parameter dynamics of the model either following a ramp, or in the presence of a periodic drive. We find that the transient dynamics of the order parameter in the presence of a periodic drive is controlled by the drive frequency displaying the phenomenon of synchronization. We also study the approach of the order parameter to its steady state value following a ramp and find out the effective temperature of the steady state. We chart out the steady state temperature of the ordered phase as a function of ramp time and amplitude, and discuss the relation of our results to experimentally realizable spin models.

preprint2020arXiv

Robot Navigation with Map-Based Deep Reinforcement Learning

This paper proposes an end-to-end deep reinforcement learning approach for mobile robot navigation with dynamic obstacles avoidance. Using experience collected in a simulation environment, a convolutional neural network (CNN) is trained to predict proper steering actions of a robot from its egocentric local occupancy maps, which accommodate various sensors and fusion algorithms. The trained neural network is then transferred and executed on a real-world mobile robot to guide its local path planning. The new approach is evaluated both qualitatively and quantitatively in simulation and real-world robot experiments. The results show that the map-based end-to-end navigation model is easy to be deployed to a robotic platform, robust to sensor noise and outperforms other existing DRL-based models in many indicators.

preprint2020arXiv

Security of Medical Cyber-physical Systems: An Empirical Study on Imaging Devices

Recent years have witnessed a boom of connected medical devices, which brings security issues in the meantime. Medical imaging devices, an essential part of medical cyber-physical systems, play a vital role in modern hospitals and are often life-critical. However, security and privacy issues in these medical cyber-physical systems are sometimes ignored. In this paper, we perform an empirical study on imaging devices to analyse the security of medical cyber-physical systems. To be precise, we design a threat model and propose prospective attack techniques for medical imaging devices. To tackle potential cyber threats, we introduce protection mechanisms, evaluate the effectiveness and efficiency of protection mechanisms as well as its interplay with attack techniques. To scoring security, we design a hierarchical system that provides actionable suggestions for imaging devices in different scenarios. We investigate 15 devices from 9 manufacturers to demonstrate empirical comprehension and real-world security issues.

preprint2020arXiv

Spin-orbit coupling and spin-triplet pairing symmetry in $\mathrm{Sr_2 Ru O_4}$

Spin-orbit coupling (SOC) plays a crucial role in determining the spin structure of an odd parity psedospin-triplet Cooper pairing state. Here, we present a thorough study of how SOC lifts the degeneracy among different p-wave pseudospin-triplet pairing states in a widely used microscopic model for $\mathrm{Sr_2 Ru O_4}$, combining a Ginzburg-Landau (GL) free energy expansion, a symmetry analysis of the model, and numerical weak-coupling renormalization group (RG) and random phase approximation (RPA) calculations. These analyses are then used to critically re-examine previous numerical results on the stability of chiral p-wave pairing. The symmetry analysis can serve as a guide for future studies, especially numerical calculations, on the pairing instability in $\mathrm{Sr_2 Ru O_4}$ and can be useful for studying other multi-band spin-triplet superconductors where SOC plays an important role.

preprint2020arXiv

Univoque bases of real numbers: local dimension, Devil's staircase and isolated points

Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal U(x)$ such that when $x$ passes the first critical value the set $\mathcal U(x)$ changes from a set with positive Hausdorff dimension to a countable set, and when $x$ passes the second critical value the set $\mathcal U(x)$ changes from an infinite set to a singleton. Denote by $\mathbf U(x)$ the set of all unique $q$-expansions of $x$ for $q\in\mathcal U(x)$. We give the Hausdorff dimension of $\mathbf U(x)$ and show that the dimensional function $x\mapsto\dim_H\mathbf U(x)$ is a non-increasing Devil's staircase. Finally, we investigate the topological structure of $\mathcal U(x)$. In contrast with $x=1$ that $\mathcal U(1)$ has no isolated points, we prove that for typical $x>0$ the set $\mathcal U(x)$ contains isolated points.