Let $A\subset N$, and define its ordered representation function $r(n)=\#\{(a,b)\in A\times A:a+b=n\}.$ The Erdos--Turan conjecture asserts that, if $r(n) > 0$ for all sufficiently large $n$, then $r(n)$ is unbounded. Konstantoulas proved a density-theoretic version: if the upper density of $E=N\setminus(A+A)$ is less than $1/10$, then $\limsup_{n\to\infty} r(n)> 5$. In this paper, we improve Konstantoulas' constant to $7/32$. We also prove that $D(E)< 1/2$ implies $\limsup_{n\to\infty} r(n) > 3$, and give a conditional criterion forcing $\limsup_{n\to\infty} r(n)>7$.

Altermagnets, featuring momentum-dependent spin splitting without net magnetization, provide a promising platform for spintronic functionalities beyond conventional ferromagnets and antiferromagnets. Here, we propose an altermagnetic spin photovoltaic battery consisting of a nonmagnetic semiconducting layer sandwiched between two altermagnetic electrodes. Using first-principles quantum-transport simulations, we show that a V2Te2O/ZnSe/V2Te2O junction supports a pure spin photocurrent for opposite Néel vectors in the two altermagnetic electrodes, with spin-up and spin-down photocurrents equal in magnitude and opposite in sign. The effect persists under both linearly and circularly polarized light and remains tunable with photon energy and polarization angle. Our results establish a realistic route toward light-driven pure spin-current generation in altermagnetic junctions.

We derive theoretical bounds on the Kronecker rank of block-structured matrices that possess both inner and outer structure. Building on the matrix-to-tensor and tensor-to-matrix framework of Kilmer and Saibaba ($\textit{SIAM J. Matrix Anal. Appl.}$, 2022), we show that the Kronecker rank of a matrix $\mathbf{A}$ equals the dimension of the span of its distinct blocks (the $\textit{inner blockspan}$), and equals as well the dimension of the corresponding span for a permuted matrix $\mathbf{B}$ whose inner and outer structures are interchanged. We give two proofs of this equality: one via a direct dimension argument, and one via an explicit isomorphism between the outer blockspan of $\mathbf{A}$ and the column space of the second-mode unfolding of the associated tensor. These results yield nested containments that translate structural information, such as Toeplitz, Hankel, banded, or sparse patterns, into computable upper bounds on the Kronecker rank. We also establish a duality between element-level sparsity in $\mathbf{A}$ and block-level sparsity in $\mathbf{B}$. Numerical experiments confirm the theory across several classes of structured matrices and, for two sparse matrices drawn from the SuiteSparse collection, provide a structural explanation for observed singular value decay.

Virtualization obfuscation is a more powerful obfuscation technique compared to other obfuscation methods, and as it is increasingly being applied to malware, it demands significant effort and time from analysts. This study analyzes virtualization obfuscation and proposes a tool called VMPredator that automatically extracts semantic units. The proposed tool performs various analyses including memory analysis and trace analysis, while minimizing dependency on the specific internal structure of virtual machines in order to handle diverse forms of virtualization obfuscation that existing tools are unable to process. Experimental results demonstrate that the length of obfuscated programs was reduced by approximately 85%, and it was verified through validation that small-scale programs were fully restored to semantics identical to the original.

Optimal finite-time control is essential for energy-efficient operation of nanoscale devices. While existing work has largely focused on transitions between equilibrium states in overdamped systems, many settings of practical interest -- including nanomechanical resonators, biomolecular conformational dynamics, and quantum Brownian motion -- are governed by underdamped dynamics where both particle inertia and frequency-dependent friction (memory) play a non-negligible role. In this study, we analytically and computationally investigate optimal transitions between nonequilibrium steady states (NESS) for an underdamped particle in a moving harmonic trap with general memory kernels. We find that inertia qualitatively alters optimal control in the presence of memory. Compared to the overdamped case, underdamped dynamics break the time-reversal symmetry, making the forward and backward optimal protocols fundamentally distinct. Across the memory-kernel types examined, the asymmetry, rather than the detailed form of the kernel, governs the structure of the optimal strategy. These results offer a unified framework for optimal control in underdamped systems with memory.

We develop a systematic framework for constructing parameter-free algorithms for smooth nonconvex optimization. The framework is based on higher-order regularization: each step is computed from a regularized local model whose regularization exponent exceeds the order of the model error. This design makes the resulting method robust to misspecification of the regularization parameter and yields complexity bounds without backtracking or other acceptance tests. We apply the framework to gradient descent, Newton's method, the Gauss--Newton method, stochastic gradient descent, and PAGE. Without prior knowledge of problem-dependent parameters, the resulting algorithms achieve complexity bounds with optimal or best-known dependence on the target accuracy. When the problem-dependent parameters are known up to constant factors, suitable tuning also recovers the optimal or best-known dependence on those parameters.

The reduction of complex labour to simple labour is an unresolved difficulty in Marx's labour theory of value, and a key obstacle that has prevented the transformation problem from being settled definitively. This paper proposes a two-step solution framework. First, we prove that as long as the macroeconomy generates a physical surplus, the reduction coefficients that respect the floor of labour-power reproduction form a bounded ``value feasible region''; within this region the two macro aggregate equalities can hold simultaneously for a reasonable range of the profit rate. Second, we propose a linear mapping method that exploits the observable structure of nominal wages and the reproduction floor constraint to systematically construct the implicit reduction coefficients from the value feasible region. We show that this mapping is a homeomorphism between the price feasible region and the value feasible region, and that it preserves the boundary structure. An empirical calibration based on China's 2017 inter-provincial input--output table with 1272 sectors shows that the reduction coefficients obtained by the mapping method substantially outperform the homogeneous labour method and the wage-proxy method in matching the macro profit share.

Fluctuations can drive otherwise continuous phase transitions to first order through the Brazovskii mechanism. We study how these fluctuation-induced transitions are modified in active systems by introducing nonequilibrium spatiotemporally correlated noise. We show that, while the transition remains fluctuation-induced first order, activity systematically suppresses these fluctuation effects, shifting the transition to higher temperatures and rendering it increasingly weakly first order. As a result, ordering is enhanced without inducing a spinodal instability of the isotropic phase, as confirmed by direct numerical simulations. In the strong-activity limit, fluctuation effects disappear and mean-field behavior is recovered. Our results identify activity as a generic control parameter for tuning the strength of fluctuation-induced first-order transitions.

Deriving analytic solutions for optimal mixed strategies in zero-sum linear-quadratic differential games (ZSLQDGs) remains an open problem. In this paper, we analytically synthesize near-optimal mixed strategies for ZSLQDGs and establish rigorous performance certifications. Specifically, we construct a surrogate pure-strategy stochastic differential game (SDG) by matching the first two moments of the mixed strategies. This method achieves an $\mathcal{O}(\barπ^2)$ weak approximation of state distributions and expected costs with respect to the maximum commitment delay $\barπ$. By analytically resolving the surrogate SDG, we derive closed-form optimal control laws for the matched moments. Crucially, we reveal that the surrogate game is governed by a Generalized Riccati Differential Equation (GRDE), which explicitly dictates a dynamic energy allocation law for variance injection. Building on these solutions, we propose a robust dual-routing architecture to execute the near-optimal mixed strategies. Furthermore, we certify that both the global value approximation error and the strategy suboptimality gaps are bounded by $\mathcal{O}(\barπ^{\frac{1}{2}})$. Finally, numerical experiments on a double-integrator pursuit-evasion game illustrate the induced physical behaviors and validate the theoretical bounds.

This paper introduces a novel framework for polar codes, designed for flexible Incremental Redundancy Hybrid Automatic Repeat Request (IR-HARQ). By generalizing the decoding order beyond the standard 1$\to$N sequence, we enable a capacity-aware scheduling strategy that prioritizes the decoding of reliable subblocks. The framework integrates nested parity-check polar construction and reverse bit-mapping to support continuous and arbitrary transmission lengths $E \in [N_{\min}, N_{\max}]$. Simulation results show that the proposed rateless codes match the coding gain of independently optimized fixed-rate codes across the entire range of rates and lengths. With a validated hardware implementation, this work provides a practical solution for next-generation wireless data channels.

The rise of LLM agents introduces a new threat by enabling planning, coding, and even end-to-end execution of expert-level attack workflows. However, this threat remains underexplored and underestimated since (i) safety alignment prevents LLMs from directly generating harmful instructions, and (ii) most existing jailbreak methods cannot consistently induce agents to execute malicious operations. In this paper, we propose TRACE, a practical agentic jailbreaking framework to further reveal the risks of this threat surface. To conceal the malicious intent, TRACE decomposes a malicious task into multiple subtask sequences under different schemes and selects the sequence with the fewest explicitly harmful subtasks. TRACE then disguises the remaining harmful subtasks as benign-looking instructions by embedding them in task-aware scenarios with related roles, environments, directives, and heuristics. The scenarios are iteratively evolved through well-defined transformation actions, which are sampled by a Q-learning-inspired mechanism, for inducing the agent to execute on the harmful subtasks. Extensive evaluations on AgentHarm and AdvCUA show that TRACE consistently outperforms existing jailbreak baselines across multiple advanced LLM agents, achieving up to 100% bypass rate and 0.73 average success score. We also demonstrate the effectiveness of TRACE in controlled cyberattack instances. Our code and demos are available at https://github.com/ZJU-LLM-Safety/TRACE.git.

The sensory cortices of the brain perform perceptual inference efficiently through their complex networks of neurons. One of the theoretical accounts of this process is the free-energy principle (FEP), which postulates that the brain performs variational Bayesian inference. Pioneering studies have shown that FEP can correspond to the predictive coding (PC) hypothesis under the Gaussian assumption and Laplace approximation. However, PC-based implementations of FEP within such a limited Gaussian regime have failed to capture several properties of biological neural networks, such as nonlinearity and heterogeneity of input--output properties within a network, and the biological implausibility of negative firing rates. This study shows that, when a broader class of probability distributions, namely the exponential family of distributions (EFD), is assumed for the variational posterior and prior, these missing characteristics are exhibited within the network, maintaining the FEP--PC correspondence up to the second cumulant of the posterior. We also show that the proposed model can be trained by biologically plausible local plasticity rules. Our results enrich the explanatory power of FEP regarding neural dynamics involved in perception as variational inference.

We study the spin Hall effect in two-dimensional topological insulators with "Mexican hat" dispersion and a ring-shaped Fermi surface which are formed due to the band inversion. Electron transitions between different isoenergetic contours and the quantum metric of band states play an important role in the transport properties of such materials, since they largely determine the spatial distribution of the electron charges screening the impurity potential and the scattering probability [Phys.B, 719, 417942 (2025)]. Here we study a spin-dependent skew scattering, which is enabled by the second-order scattering processes, and show that the extrinsic spin-Hall current (SHC) can significantly exceed the intrinsic SHC arising from the Berry curvature. Furthermore, due to Mexican-hat dispersion, the SHC exhibits a very unusual dependence on the Fermi energy ($E_F$). The extrinsic SHC reaches a maximum at some $E_F$, then decreases with increasing $E_F$ and can even change a sign. This complicated behavior reflects an interplay of energy dependencies of such important factors as probabilities of inter- and intra-contour transitions, as well as different electron velocities in two contours.

We study many-to-one matching with transfers and peer effects, such as matching workers to firms, students to schools, residents to neighborhoods, or consumers to status goods. With flexible prices (as in the labor market), competitive equilibrium exists and is efficient under general conditions. We characterize when workforces are segregated by skill and matched to firms in a positively assortative manner. In general, equilibrium features alternating intervals of workforce segregation and compression (mixing). Comparative statics characterize when workforces are more segregated or more compressed, and when profits and wages are more or less unequal. With uniform prices (as in school or neighborhood choice), the value generated by peer effects accrues to schools rather than students, and equilibrium can be excessively segregated. Our model generalizes both assignment models (optimal transport) and Bayesian persuasion.

We introduce a wetting boundary condition for curved solid boundaries within a diffuse interface framework for lattice Boltzmann method. The boundary condition relies on updating the order parameter (color/phase-field) values on ghost nodes inside the solid phase. The ghost node color modification rule, in turn, extends the equilibrium color profile into the solid phase. Numerical simulations performed on an NVIDIA A100 GPU demonstrate that the wetting scheme retains the model's ability to handle large density and viscosity contrasts while producing relatively small spurious currents. The present scheme agrees well with analytical solutions/other numerical works for both static and dynamic contact lines on curved solid boundaries.

We prove that any Fano threefold of genus 9 and 10 contains a cylinder, i.e. an open subset isomorphic to the product of a quasiprojective variety and the affine line. Moreover, we show that any Fano threefold of genus 10 has a point such that the Hilbert scheme of lines through the point has length three.

Using the operator method and properties of $O(2,1)$ algebra, the integral representation for the two-dimensional Green's function of an electron in the field of a magnetic monopole is found. This representation is valid in all complex plane of the electron energy.

Roaming reactions occur when a molecule enters a near-dissociation region, avoids immediate separation, and later forms products by a pathway not controlled by the conventional tight transition-state bottleneck. Classical studies have shown that roaming is best understood in phase space: inner and outer transition-state structures, together with their invariant manifolds, organize trapping, return, and dissociation. The corresponding quantum question is less settled. Can a single quantum resonance carry a recognizable signature of the classical roaming region? We address this question in Chesnavich's two-degree-of-freedom model for the ion--molecule reaction $\mathrm{CH}_4^+\rightarrow\mathrm{CH}_3^+ + \mathrm{H}$. Resonance states are computed with a complex absorbing potential and analyzed using diagnostics designed to mirror the classical phase-space picture: radial probability weights derived from the tight and outer transition-state structures, radial Husimi projections, angular-momentum channel weights, and coherent-state probes of the classical periodic orbits. One resonance is distinguished from the rest of the computed resonance ensemble. Its wavefunction is concentrated in the projected region between the inner and outer transition-state structures, its radial phase-space distribution is centered at intermediate radius with nearly zero radial momentum, and its angular structure is consistent with a standing rather than a directed rotating component. We interpret this state as a phase-space-localized quantum analogue of classical roaming. The result provides a controlled example in which quantum roaming is identified directly from a resonance wavefunction and its phase-space diagnostics, rather than only from product-state or scattering signatures.

The James Webb Space Telescope (JWST) has revolutionized the characterization of exoplanetary atmospheres, offering unprecedented sensitivity to probe their chemical and physical properties. Recently, a growing trend has emerged to obtain atmospheric information directly from pixel-level planetary spectra. In this work, we re-analyzed the WASP-18b NIRISS/SOSS dataset by employing a direct extraction method. This new method preserves the spectral information at the native instrumental resolution, thereby enabling the application of cross-correlation techniques and providing atmospheric retrievals with enhanced precision and richer information content. With this methodology, we report detections of CO at $4.4σ$ significance, H$_2$O at $3.4σ$, and OH at $7.8σ$, where CO and OH were previously unseen. Building on these unambiguous detections, our subsequent retrieval analysis significantly improves the constraints on atmospheric abundances. Our results demonstrate that the cross-correlation technique effectively extracts molecular signals from medium-resolution JWST data, enhancing detection sensitivity. By revisiting JWST archival data with cross-correlation and retrieval analysis, we can achieve a more comprehensive survey of planetary atmospheric chemistry, thereby placing precise constraints on key parameters such as planetary metallicity and C/O ratio.

We present a narrowband integrated Mach-Zehnder interferometer (MZI) capable of precise transmission control within a targeted wavelength band while maintaining out-of-band transparency. This functionality enables its use as a fundamental building block for fully programmable on-chip spectral shaping. The device is implemented on a novel dual-mode (TE0/ TE1) transmission platform, where anti-symmetric multimode waveguide Bragg gratings (AM-WBGs) and asymmetric Y-branches are combined to function as an equivalent narrowband 1*2 or 2*2 coupler. Experimentally, the MZI achieves wide extinction ratio tuning 0 dB to 30 dB across a 2.5 nm bandwidth, with independent and simultaneous control of both wavelength and extinction ratio. Cascaded multiple narrowband MZIs are experimentally characterized, demonstrating independent intensity control at individual wavelengths without cross-interference. Furthermore, the device's application as a tunable, channel-selective optical blocker/passer in high-speed communication systems is experimentally validated. Compared to prior approaches relying on dual-grating-assisted contra-directional couplers, the AM-WBG-based design overcomes fundamental bandwidth limitations caused by unintended intra-waveguide coupling bands. In addition, their single-waveguide-grating structures enhance the reliability of both fabrication and spectral control, while enabling compact spiral configurations for significant miniaturization. These advantages position the proposed MZI a promising, scalable candidate for advanced spectral shaping applications.

We show that Young translation has dense orbits in the space of rough paths: for any two geometric rough paths, one can translate the first by a sequence of smooth paths so that it converges to the second in rough path topology. As applications, we obtain full-support criteria for rough paths arising from random series and Gaussian processes, including non-centered fractional Brownian rough paths.

Helical segments in polymer chains are often transient, finite, and dynamically evolving, yet their origin and stability remain incompletely understood. Here we develop a minimal coarse-grained statistical-mechanical theory that explains how such living helices emerge in fluctuating polymer systems. Using a three-state model with cooperative interactions, we show that helix formation proceeds through a multistep nucleation mechanism. An initial constrained pre-nucleus forms first, followed by cooperative stabilization that promotes the growth of finite helical segments. The resulting free-energy landscape naturally favors marginally stable helices whose size is determined by a competition between cooperative gains and nonlinear penalties arising from stiffness, torsional strain, and solvent fluctuations. By formulating the dynamics as a stochastic process in segment size, we derive analytical expressions for both formation times and lifetimes within a mean first-passage framework. For representative parameters relevant to flexible polymers and peptide segments, the theory predicts characteristic timescales in the nanosecond to sub-microsecond range. The present analysis supports a view of living helices as finite, mobile excitations whose stability is controlled by cooperativity, boundary motion, and solvent-induced fluctuations.

People often seek out ways to watch others perform complex action sequences (e.g., sports). What makes some sequences more enjoyable to watch than others? We generated 24 video clips of gameplay from a Flappy Bird-style video game. Clips varied in difficulty (how often players succeeded on average) and in moment-to-moment uncertainty (how likely the player was to crash at any given step). Participants (N=864) rated each video on one of three dimensions: how much they enjoyed it, how difficult the level appeared, or how dangerous the player's trajectory appeared. We found that participants preferred videos where the player seemed to be completing more difficult obstacle courses, but dangerousness did not predict enjoyment ratings. These findings show how procedurally generated stimuli can isolate the factors that affect how enjoyable an action sequence is to watch.

Bloch equations are a powerful tool in describing the dynamics of nuclear magnetization in magnetic resonance phenomena. The fractional generalization of the Bloch equation effectively captures the anomalous relaxation and diffusion in porous, heterogeneous, and complex media. These equations describe how nuclear magnetization evolves under the influence of magnetic fields and relaxation processes. This work effectively employs a hybrid approach, the Laplace residual power series method, to investigate and analyze the fractional Bloch equation. A series solution is derived as the approximate solution for magnetization components. The influence of fractional order on each magnetization component in magnetization dynamics is analyzed and illustrated graphically. We conduct an error analysis to demonstrate the reliability and effectiveness of the proposed approach. The superiority of the suggested approach is shown using a comparative study with existing methods. The findings indicate the potential of the suggested approach as a reliable tool in understanding fractional magnetic resonance systems arising in applications such as NMR spectroscopy, MRI, MRF, and other complex heterogeneous materials.

We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy networks. We analyze how such diffusive uncertainty reshapes the computational landscape of robust graph optimization. For convex network problems, such as minimum-cost flow and maximum flow, the resulting formulations remain convex and admit polynomial-time solution methods across all diffusion regimes considered. For combinatorial problems, the effect is more delicate. We focus on two canonical combinatorial graph problems, shortest path and the traveling salesman problem (TSP), which provide complementary benchmarks: shortest path is polynomial-time solvable in the nominal setting, whereas TSP is already NP-hard. We show that, for shortest path, propagation depth induces a sharp transition between tractable and intractable robust counterparts. For the traveling salesman problem, robustness often adds no computational complexity beyond ordinary TSP, because the structure of Hamiltonian cycles makes the fixed-tour adversarial problem collapse to explicit formulas. Together, these results show that topology-aware uncertainty can fundamentally change robust combinatorial optimization, with tractability governed by the interaction between propagation, budget geometry, and the structure of feasible solutions.

Parallel decoding improves generation efficiency by processing multiple decode positions within a single decode forward, but reported speedups conflate algorithmic token utilization with the system cost of executing multiple positions. We isolate the system side by introducing Near-Free Parallelism (NFP), the maximum number of positions executable at near-free latency. Analyzing Dense FFNs, MoE FFNs, and Attention against an idle-compute baseline, we find that NFP is shaped not by memory-bound resource slack alone, but also by implementation-induced kernel-granularity slack. Based on these mechanisms, we establish a Near-Free Parallelism principle that predicts the NFP boundary from hardware balance and implementation granularity. Validation on representative Dense and MoE models -- spanning both diffusion and autoregressive decoding -- shows that the principle accurately predicts practical NFP boundaries, revealing that the standard idle-compute intuition can over-predict by up to 23x -- offering a system-side budget for parallelism selection and model-system co-design.

Disordered superconductors represent an important area in modern condensed matter physics, where superconductivity survives even in the presence of strong electron scattering and localization effects. Understanding how disorder modifies the high-frequency electrodynamic response is not only important from physics point of view, but is also essential for developing next-generation quantum detectors and superconducting devices. In this work, we investigate the terahertz electrodynamics of disordered Ti40V60 alloy thin films using terahertz time-domain spectroscopy (THz-TDS) to understand the relationship between disorder, quasiparticle dynamics, and kinetic inductance. By analysing the complex conductivity, penetration depth and superfluid response, we show that structural disorder can be systematically used to tune the inductive response while maintaining a robust superconducting phase. Unlike conventional nitride superconductors that require tightly controlled reactive growth conditions, Ti40V60 alloys provide a simpler and more adaptable route for tuning the superconducting energy scales directly through the deposition conditions. These findings establish Ti40V60 alloys as a promising material for kinetic inductance detectors and provide useful insights into the electrodynamics of strongly disordered superconductors.

A group $G$ is called a Howson group if the intersection of any two finitely generated subgroups of $G$ is again finitely generated. It is called strongly Howson if, in addition, the rank of such an intersection is bounded only in terms of the ranks of the two subgroups. Strongly Howson groups are indeed Howson. Recently, Zhang and Zhao showed that the converse fails, by constructing the first examples of infinitely generated Howson groups which are not strongly Howson. In this note, we provide finitely generated examples with and without torsion elements, thereby answering a question posed by Zhang and Zhao.

In this paper, we propose and analyze a novel two-field symmetric formulation with solid displacement and fluid pressure as main unknowns for the Biot's consolidation model in poroelasticity. Firstly, we prove the well-posedness of the new formulation and then show the existence and uniqueness of optimal control where the fluid sources in the model act as a control variable. We prove a priori error estimates for the fully discrete scheme with backward Euler time discretization and a variational approximation of the control variable. A numerical example is presented to validate the performance of the proposed novel scheme.

The group $\E(X)$ of homotopy self-equivalences of a topological space $X$ is a well-known group in homotopy theory and has been studied by many people since it was first introduced in the late 1950s. $\E$ is not a functor in the usual sense. In this paper we show that $\E$ is a Lax functor from the category $\mathscr Top$ of topological spaces to a strict $2$-category $\op{Corr}_{\mathscr Gr}$ of \emph{correspondences} of groups.