Let $X=\sum_{n\geq1}ξ_n2^{-n} $ be a random number where we model the digits $ξ_n$ as independent Bernoulli random variables with possibly non-identical parameters $p_n=\mathbb{P}(ξ_n=1)$. For any polynomial $P\in\mathbb{R}[X]$ with degree $d\geq2$, we prove almost sure absolute normality of $P(X)$ under the condition $p_n(1-p_n)\geq (\log n)^{Γ(d)} n^{-(d-1)/d}$ for a suitable constant $Γ(d)$ depending only on the degree $d$. Our analysis reveals the sharp power law $n^{-(d-1)/d}$, which is suggested by an elementary heuristics regarding carrier interactions. Our results establish a transition as we further show that the pure critical power law is insufficient, but the precise critical window remains an interesting open problem. As far as we know, this is the first sharp result on digit mixing. We complement our main results by structurally convenient summability criteria, which turns out to be sharp at least for $X^2$, and we formulate a more general conjecture for higher degrees. Our proofs rely on Fourier decay estimates which we obtain by probabilistic argument involving conditioning and non-resonancy estimates combined with a subtle triangularization argument.

In this manuscript we develop the direct and inverse scattering problem for the cubic focusing nonlinear Schrödinger equation and for initial data that are asymptotic to an elliptic travelling wave with distinct phase at $\pm \infty$. We consider the case in which the spectral bands intersect the real axis. We then show that this class of initial data has non zero intersection with the full soliton gas initial data.

A novel concept for wide-angle scanning is proposed based on multi-mode multi-port antennas. The theory of multi-mode multi-port antennas based on aperture radiators is developed and applied towards the design of an antenna array consisting of multi-mode aperture radiators. An advanced beamforming algorithm is developed and implemented, making use of the higher degrees of freedom available to multi-mode multi-port antennas. The manufactured antenna array is measured and compared to the expected performance. Wide-angle steering up to $\pm77^\circ$ from broadside with respect to a scan loss of $3\,\mathrm{dB}$ is achieved in both the horizontal and vertical plane with no visible grating lobe.

Non-volatile phase-change material (PCM)-integrated metasurfaces offer a promising pathway toward next-generation solid-state reconfigurable free-space optics. However, their practical operation is currently bottlenecked by the highly non-uniform thermal profiles generated by the external heaters used to switch the PCM between its amorphous and crystalline states. The non-uniform heat profile in turn severely restricts the active switching area of the PCM integrated devices. In this work, we present an inverse-designed doped silicon resistive heater on a silicon-on-sapphire platform, featuring a spatially varying doping profile explicitly tailored to generate uniform heat. The new heater significantly improves spatial temperature uniformity, reducing the thermal gradient from 110 K in the case of typical conventional heater to merely 25 K in the case of the inverse-designed heater at a target temperature of ~1000 K. By meticulously optimizing the doping and annealing processes to suppress lateral dopant diffusion, we achieve a near-perfect spatial doping resolution capable of patterning two distinct doped Silicon filaments just 100 nm apart. We experimentally fabricate these devices and demonstrate their efficacy by successfully switching a large area (~18 x 14 micrometer square) of the wide-bandgap phase-change material (PCM) Sb2S3 using a compact heater geometry of size 26 x 26 micrometer square. We show that our inverse-designed heater achieves a 10-fold increase in active PCM switching area despite a reduction in the total heater footprint when compared to a conventional heater. Ultimately, this work provides a crucial steppingstone toward the development of non-volatile PCM-based reconfigurable free-space optics.

The rise of polyglot data management and AI-ready database architectures has created a complex design space across diverse database paradigms. However, architecture selection in modern enterprise environments continues to rely heavily on ad-hoc engineering intuition, with limited systematic frameworks to guide decision-making across heterogeneous database systems. This paper introduces a unified cross-paradigm evaluation and selection framework for database architecture design in AI-ready data platforms. The framework is based on nine architectural dimensions and incorporates a structured multi-stage selection process involving workload characterization, constraint filtering, and compatibility scoring to enable systematic comparison and decision-making. To ground the framework, we conduct a structured comparative analysis across thirteen major database paradigms spanning transactional, analytical, and AI-oriented systems. This analysis reveals three recurring patterns in database evolution: decoupling of storage and compute, workload-driven specialization, and convergence toward integrated AI-ready platforms. The proposed framework is demonstrated through a representative enterprise case study in financial fraud detection, illustrating how hybrid, polyglot architectures emerge as optimal solutions for multidimensional workload requirements. The cross-paradigm analysis culminates in an AI-ready reference architecture that integrates lakehouse storage, feature processing, and semantic retrieval layers as the unified substrate for modern analytics, machine learning, and Retrieval-Augmented Generation applications.

Inverse problems play a central role in current areas of research in particle phenomenology. In these lectures we focus on two examples, the extraction of Parton Distribution Functions (PDFs) from experimental data (or, equivalently, from pseudo- and quasi-PDFs computed in lattice QCD), and the extraction of spectral functions from lattice Euclidean time correlators. We investigate in detail three different approaches, namely Backus-Gilbert, Gaussian Processes and fits based on Neural Network parametrizations.

This paper benchmarks sequential feedback optimization (SFO) for wind farm power maximization using a medium-fidelity dynamic flow model. We compare SFO with two well-established approaches, adjoint-based economic model predictive control (AMPC) and extremum seeking control (ESC), under a common nine-turbine layout and identical operating constraints. The comparison focuses on steady-state power production and computational efficiency, both relevant for real-time implementation. The simulation results illustrate that SFO achieves higher steady-state power while preserving real-time feasibility, AMPC provides a better transient performance at a higher online computational cost and without guarantees of convergence to the steady-state optimum, and ESC offers a computationally inexpensive model-free baseline that may converge to locally optimal solutions. These results provide a practical reference for selecting wind farm control strategies and for designing scalable, real-time optimization methods.

This paper presents a feedforward nonlinear equalizer (FFNE) framework for short- to medium-reach wireline links that removes the feedback-timing bottleneck of decision-feedback equalizers (DFEs) while approaching the noise-margin advantage within a characterized operating region. The proposed FFNE reduces short-window maximum-likelihood sequence estimation to a compact binary decision rule, enabling a low-complexity feedforward realization without transmitter-side encoding. For the single-postcursor NRZ case, the mathematical foundation, hardware implementation, tap adaptation, statistical analysis, and equalization limit relative to an ideal 1-tap DFE are established. A window-length-3 FFNE quantifies the performance-complexity tradeoff of longer sequence windows. The framework is further extended to PAM-4 modulation and simultaneous precursor/postcursor equalization through a pattern-detection-based FFNE (PD-FFNE), which outperforms conventional FFE+DFE baselines under representative channel conditions.

Despite the growing integration of Deep Research tools into academic workflows, empirical evidence on the operation, stability, and potential biases of their citation systems remains scarce. This study addresses this gap by evaluating the intensity, consistency, and bibliographic characteristics of references cited in the literature reports generated by Ai2 Asta, with the aim of understanding how its citation system operates and assessing its implications for scholarly communication. To this end, ten domain-specific queries were submitted to Asta's Summarise Literature feature, and two independent rounds of data collection were conducted. From each report, in-text citations, cited references, as well as other metrics related to the response process were extracted and examined. The results reveal high citation intensity, with reports integrating numerous in-text citations grounded in retrieved evidence and a diverse yet concentrated set of venues. However, notable instability is observed in the composition of cited references across identical queries, alongside a lack of concordance between retrieved documents and those ultimately cited, suggesting additional opaque selection mechanisms during report generation. These findings indicate that, while Ai2 Asta produces well-structured and quality reports, its instability and opacity in the citation process pose challenges in quantitative science studies due to their lack of reproducibility and transparency. Despite the restricted number of queries and disciplinary scope, the results offer valuable insights for researchers, bibliometricians, developers, and research evaluators seeking to understand, use or regulate AI-based scholarly assistants responsibly.

We study a canonical $G_2$-equivariant operator $h:\mathbb{O}\otimes_{\mathbb{R}}V\to \mathbb{O}\otimes_{\mathbb{R}}V$ defined using only octonion multiplication, where $V$ is the standard $7$-dimensional $G_2$-module. We first compute its ordinary real spectrum using the $G_2$-decomposition of $\mathbb{O}\otimes_{\mathbb{R}}V$. We then analyze the octonionic right-eigenvalue problem $$ h(\widehat w)=\widehat wλ, \qquad λ\in\mathbb{O}. $$ After fixing a complex slice $\mathbb{R}\oplus\mathbb{R}\widehat u\subset\mathbb{O}$, the problem becomes a real spectral problem for $H_{u,Q}=h-Q R_{\widehat u}$, whose residual symmetry is $\mathrm{SU}(3)$. The resulting $\mathrm{SU}(3)$-block decomposition yields two explicit spectral loci in each slice: a quartic curve and a circle. The equations defining these loci are independent of the slice, and the full right spectrum is obtained by allowing $\widehat u$ to vary over the unit sphere in $\operatorname{Im}\mathbb{O}$.

When we observe 3D objects, such as nanocrystals using a transmission electron microscope (TEM), the significant depth of the sample causes the contrast transfer function (CTF) to mix, resulting in complex and confused image. This paper predicts that a high-resolution projected image can be obtained by tilting the illumination and filtering it with a ring slit. This method produces images similar to those obtained using scanning transmission electron microscopy (STEM), which is not affected by the CTF problem. Potential applications are such as MOFs (metal-organic frameworks), perovskites and Solid-State Electrolyte (SSE) for Li-ion battery, where small ions or molecules play important role.

Online hate speech is a global challenge amplified by engagement8-driven social media algorithms. This paper develops an epidemiological model of hate speech propagation capturing the strategic interaction between a profit-maximizing platform and a welfare-maximizing government. The platform's profit depends on the prevalence of hate speech and on its own algorithmic reactivity, creating a feedback loop between the epidemic and economic incentives. The government sets an optimal tax on amplification to internalize the social costs, balancing the benefit of tax revenue against the deadweight loss of taxation. The Stackelberg equilibrium is characterised analytically and solved numerically. The optimal tax reduces hate speech prevalence, eliminates bistability and lowers victim harm.

In this paper, we propose a second-order-in-time, structure-preserving, and mesh-robust parametric finite element method for surface diffusion and volume-preserving mean curvature flow. We first reformulate the original evolution equations into new systems in which the tangential motion is governed by a harmonic map heat flow. This heat flow maps a fixed reference surface onto the unknown evolving surface and drives points on the evolving surface to move in their tangent spaces so as to reduce the associated harmonic energy. As a result, in the discrete setting, the mesh quality can be maintained at a level comparable to that of the reference surface, unless singularities occur. The volume-preserving property is theoretically guaranteed by the careful design of the scheme, while energy dissipation is enforced through a Lagrange multiplier. We present several numerical experiments to demonstrate second-order convergence in time and the advantage of the proposed method in preserving mesh quality. The structure-preserving properties are further confirmed by the numerical results. Finally, the proposed framework can be readily extended to other geometric flows.

We show that the standard power-series method described in many textbooks of quantum-mechanics and quantum-chemistry is simpler and more powerful than the wavefunction approach proposed by Ikhdair and Sever [Cent. Eur. J. Phys. 6, 697 (2008)]. As illustrative examples we choose the pseudoharmonic and Kratzer-Fues potentials already treated by those authors.

National statistical offices (NSOs) produce their estimates under a single weighting system (uni-weight approach): one set of weights, independent of the variable of interest, is used to estimate multiple parameters and multiple subpopulations (domains). In this paper we study, within the family of model-assisted estimators and from a design-based perspective of direct estimation, the use of regression trees as the assisting model for estimating totals in unplanned domains. We distinguish two strategies: (i) fitting a single tree at the population level and deriving from it uni-weight weights applicable to any domain, and fitting a domain-specific tree. We show that both estimators can be written as weighted sums with weights that do not depend on $y$, preserving the uni-weight property and additivity benchmarking with respect to the population total. Extending to trees the classical result, we argue why the estimator built from a population-level model tends to behave like the Horvitz-Thompson estimator within domains, whereas the domain-specific model can achieve substantial variance reductions. A simulation study based on microdata from the Uruguayan Continuous Household Survey (ECH) illustrates the behavior of the estimators at the population level and by department

We test whether large language models (LLMs) add value in commodity portfolio construction when the information set and implementation rules are held fixed across strategies. A Hawkish Agent (inflation-tightening prior), a Dovish Agent (growth-easing prior), a Debate Agent, and a deterministic z-score Rule Agent each receive identical FRED macro z-scores and route their tilt signals through the same portfolio engine. Across 124 weekly rebalancing dates spanning the 2023 U.S. rate peak and the 2024-2025 soft landing, all three LLM strategies outperform the Rule Agent in Sharpe terms; the Hawkish and Debate Agents record the largest gains (ΔSharpe = +0.044 and +0.040, both p < 0.10 under a block bootstrap) and preserve a net-of-cost advantage over the passive inverse-volatility benchmark at one-way trading costs up to 30 basis points, while the Rule Agent's thin margin over passive disappears at approximately 5 basis points.The Debate Agent does not outperform the best single agent (ΔSharpe = -0.004, p = 0.769); its contribution is bias correction -- averaging out the Dovish Agent's miscalibrated prior -- rather than deliberation-generated return. The performance advantage is concentrated in the soft-landing sub-period, the evaluation window spans a single rate cycle, and the reported $p$-values are unadjusted for multiple comparisons. Within these limits, the results suggest that an LLM acting as a constrained macro-interpretation function can add modest but economically meaningful value over a transparent rule layer, though the margin is small and its persistence beyond this sample is unknown.

We study the parametric sensitivity of two-dimensional massive Dirac fermions subject to Aharonov-Bohm flux insertion, using the Bures metric (fidelity susceptibility) as the central statistical-mechanical response function. Near integer values of the reduced flux, the low-energy spectrum undergoes a flux-induced avoided crossing whose structure is controlled by the Dirac mass. Through a controlled low-energy projection of the full Dirac=Aharonov-Bohm operator onto an effective two-level subspace, valid near integer flux, we derive an exact closed-form expression for the ground-state Bures metric, which takes a universal Lorentzian profile centered at integer flux with width set by the mass parameter. The peak value scales as $g^{\max}_{λλ}\sim m^{-2}$, diverging in the chiral limit in direct analogy with the divergence of thermodynamic susceptibilities near critical points. We introduce an integrated geometric susceptibility $χ(m) = π/(8m)$, whose inverse-mass scaling is the information-geometric counterpart of power-law critical behavior, with the Dirac mass playing the role of a relevant coupling controlling the distance from the chiral fixed point. The Lorentzian profile is shown to arise from the curvature of the ground-state manifold on the Bloch sphere, requiring no dynamical input beyond the spectral structure. Importantly, this geometric response is independent of Berry curvature and topological invariants, emerging instead from a universal local spectral mechanism. Through its spectral representation, the Bures metric is identified as the geometric (paramagnetic) contribution to the persistent current susceptibility, encoding the sensitivity of persistent currents and orbital magnetization to flux variations and connecting information geometry to measurable response functions in mesoscopic Dirac systems.

We show that every probability-measure-preserving equivalence relation generated by a locally-finite Borel graph with planar connected components is sofic in the sense of Elek--Lippner. In particular, every unimodular random planar graph is sofic. This removes the additional assumptions in the works of Angel--Hutchcroft--Nachmias--Ray and Timár on the soficity of unimodular random planar maps and graphs. To prove this, we investigate Borel simplicial complexes with planar components and approximate them by treeable covering spaces. To construct these coverings, we use a canonical family of tracks on planar simplicial complexes introduced by Dunwoody.

The emergence of large language model (LLM)-based agents and multi-agent systems has enabled a shift from narrow task automation to more autonomous decision-making. Despite progress in language generation, planning, tool use, and coordination, most agents still treat intelligence as prediction, optimization, and task completion. Human environments are social and normative, where people reason under bounded rationality, communicate in culturally situated language, and make decisions guided by values, beliefs, trust, and social norms. This survey argues that future AI agents, especially those acting on behalf of humans, must move beyond task competence toward human-centered capabilities. We review research across six areas: (1) evolution of intelligent agents, (2) human cognition and decision-making, (3) language, culture, and social context, (4) human values and belief systems, (5) human-agent collaboration, and (6) multi-agent coordination and modeling of human characteristics. We synthesize work from cognitive science, sociolinguistics, computational social science, and AI alignment, along with recent advances in LLM agents, cultural alignment benchmarks, preference learning, explainability, and agent societies. We identify a key gap: existing systems do not provide a unified framework integrating cognition, culture, values, and social behavior into autonomous agents. We conclude with directions for building culturally aware, value-aligned, cognitively grounded, and cooperative multi-agent systems.

The recent Belle II measurement of $\mathcal{B}(B^+\to K^+ν\barν)$, which deviates from the Standard Model prediction, provides a strong motivation to search for New Physics in $b\to sν\barν$ transitions. We perform a model-independent analysis within the low-energy effective theory, focusing on dimension-six vector operators involving right-handed neutrinos. Using the Belle II measurement of $\mathcal{B}({B\to Kν\barν})$ together with the upper limit on $\mathcal{B}({B\to K^{*}ν\barν})$, we constrain the new physics interactions. We study the correlation of $\mathcal{B}({B\to Kν\barν})$ with $\mathcal{B}({B\to K^{*}ν\barν})$, $\mathcal{B}({B_s\toϕν\barν})$, $\mathcal{B}({Λ_b\toΛν\barν})$, and the longitudinal polarization fraction $F_L^{K^*}$. We find sizeable enhancements in all branching fractions studied, while $F_L^{K^*}$ offers complementary sensitivity to the underlying RHN interaction structure. These results provide testable signatures of right-handed neutrino interactions at Belle~II and LHCb, and at the future FCC-ee experiment.

We prove the Ashbaugh--Benguria conjecture for bounded domains with smooth boundary in $\mathbb R^m$. More precisely, among all smooth bounded domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

This paper asks when MR-subset selection is a real mutant-level requirement for minimum complete evidence in metamorphic testing rather than a coarse fault-class counting artifact. We define a layer-relative completeness criterion over an admitted mutant--draw coverage universe. The central result is a support-set domination boundary: it states when class-level abstraction is safe and when mutant-level MR minimization is necessary. The boundary is governed by kill-signature heterogeneity, which yields a scoped fault-signature kernel and separates the MR-specific question from ordinary fault-class counting. The resulting Min-MR-Complete problem is Set-Cover-equivalent over the selected coverage universe, giving NP-hardness, the classical logarithmic approximation boundary, a greedy approximation, an exact ILP formulation, and an SMS-rank upper bound that is not a lower bound or tight predictor. Artifact lanes provide lane-local minimization and audit evidence; separately, route witnesses instantiate both collapse and non-collapse regimes for the boundary theorem and are not pooled as population-level experiments. Other MR-class-proxy rows remain intermediate signals rather than route-admitted witness evidence.

We study high-energy charged-lepton-flavor-violating (cLFV) channels in $e^- μ^+$ scattering to probe heavy sterile neutrinos, which arise naturally in minimal extensions of the Standard Model. For $\sqrt{s} \le 2M_W$, we consider the process $e^- μ^+ \to e^+ μ^-$, which is dominated by one-loop box diagrams. We numerically evaluate these diagrams, involving a high-energy extension of the Inami-Lim functions, and find that the amplitudes are strongly suppressed because of their quartic dependence on light-heavy mixing. Using current bounds on active-sterile neutrino mixing, we determine the maximal rates allowed by existing constraints. For $\sqrt{s} > 2M_W$, we analyze the process $e^- μ^+ \to W^+ W^-$ and compute the corresponding cross sections in both single-sterile and minimal type-I seesaw scenarios. We find this latter process to be significantly more promising for revealing the presence of heavy sterile neutrinos at $e-μ$ colliders.

James Dixon introduced the Data Lake in 2010. The pitch was simple: store data raw, postpone schema, cut up-front transformation. It promised flexibility and easier analytics. Fifteen years on, that promise has mostly gone unmet: survey after survey reports high failure rates, whether a big data program, a Data Lake, or a data science effort. This paper asks why. Reading 64 sources across academic work, analyst reports, and practitioner accounts, we found seven recurring anti-patterns, the Seven Deadly Sins of Data Lakes, and offer an explanation for them: Governance Debt, the compounding cost of governance decisions organizations keep deferring. A second pattern surfaced on its own: when governance gets hard, organizations drift back toward structured, warehouse-style approaches, a pull we name governance gravity. The term Data Swamp is used loosely in the literature, so we give it a working definition with measurable indicators, plus a qualitative rubric, the Governance Debt Assessment Model, for catching decay early. The root causes are organizational far more than technical. We also asked whether the newer paradigms, Data Lakehouse and Data Mesh, absorbed the lesson; the technology advanced, the organizational record barely moved. For practitioners we provide two tools, a Reality Check Framework and a Stage-Based Intervention Matrix. The paper rests on more than the analyst literature: it draws on a primary catalogue of close to five hundred field reality checks recorded over fifteen years of building and rescuing enterprise Data Lakes in financial services and telecommunications across Morocco and West Africa. Assembled independently of that literature, the catalogue lands on the same anti-patterns, surfaces two dimensions the literature under-reports, operational debt and engineering-discipline debt, and reads the problem from an emerging-market vantage.

We consider bounded positive semi-definite Hankel operators $H$, realised as integral operators on the positive semi-axis. For each value of $E$, not necessarily in the spectrum of $H$, we analyse solutions $f$ of the eigenvalue equation $Hf=Ef$, understood as an integral equation on the semi-axis. Our analysis reveals strong analogies with properties of solutions of the one-dimensional Schrödinger equation.

We introduce a generalization of generalized Weyl Poisson algebras. This is a Poisson analogue of the twisted generalized Weyl algebras defined by Mazorchuk and Turowska. We prove existence of these algebras in two ways, using Ore extensions and by using skew Laurent Poisson algebras. It is shown that this structure is preserved under tensor products, Poisson twists, and by taking invariant rings. Finally, we prove a simplicity criterion for these Poisson algebras.

Accurately monitoring mental fatigue is critical for improving workplace safety and productivity. A recent study examined unobtrusively collected smartphone typing speed as a potential ambulatory proxy assessment of mental fatigue using data from the Intern Health Study (IHS). While population-level average typing speed patterns were found to be consistent with validated measures of mental fatigue, how these trajectories vary across participants and days may inform opportune moments for just-in-time interventions and remains an open question. Treating typing speed trajectories as sparsely observed functional data, we propose a novel sparse longitudinal functional principal component analysis (sparse LFPCA) method for decomposing variability and predicting individual curves. Specifically, sparse data are accommodated by casting covariance estimation as a structured penalized spline regression problem, enabling simultaneous estimation and smoothing of multiple covariance components while borrowing information across locations in the functional domain. Simulations show that sparse LFPCA (1) accurately estimates eigenfunctions and generates reasonable predictions for underlying curves, and (2) achieves similar or superior performance compared to existing alternatives. Our analysis of typing speed data collected from IHS reveals new and interpretable participant- and day-level patterns not captured by previous analyses and can be used to tailor behavioral interventions.

It has been a long standing conjecture that the $ \infty $-harmonic functions in the plane have a 1/3-Hölder continuous gradient. It \emph{is} known that solutions are $ C^1 $ and that the gradient is locally $ α$-Hölder, but $ α$ comes without any positive lower bound. Aronsson's solution $ x^{4/3} - y^{4/3} $ shows that no better general regularity is possible. In the plane there is also a connection between the $ \infty $-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-Hölder continuity. Of course, an injective gradient is by no means a \emph{necessary} condition, as seen by smooth solutions such as the planes and cones.

For control-affine systems, standard and high-order control barrier function conditions are affine in the control input and are commonly enforced through quadratic-program-based safety filters. Although convex, these optimization problems may be undesirable in embedded, high-rate, or resource-limited implementations. This letter studies when the corresponding Euclidean projection can be computed exactly without solving a quadratic program. Given a nominal control input, we form the set of affine inequalities violated by that input and compute the minimum-norm correction that enforces those inequalities with equality. This correction need not equal the exact Euclidean projection onto the full feasible set. The main result gives structural conditions under which it coincides with the Euclidean projection onto the feasible set. These conditions are interpreted through interactions between affine-inequality normals and are expressed using a Gram matrix. Finally, an online certification procedure is given for determining whether the optimization-free update is exact.

Federated learning aims to protect data privacy by collaboratively learning a model without sharing private data among clients. Unlike traditional parameter-based FL methods that exchange model weights or gradients during training, emerging logit-based FL approaches share model outputs (logits) on public data. This strategy promotes model heterogeneity, reduces communication overhead, and enhances clients' privacy. However, the potential privacy risks associated with these logit-based methods have been largely overlooked. This research presents the first theoretical and empirical analysis of a hidden privacy risk in logit-based FL methods - the risk that a semi-honest server (adversary) may learn clients' private models from logits. To quantify and address this threat, we develop the Adaptive Model Stealing Attack (AdaMSA) by leveraging historical logits during training. Notably, we observe that this inherent privacy risk persists even when public data is unrelated to private data, emphasizing the urgency to address privacy vulnerabilities in logit-based FL methods. Moreover, our theoretical analysis establishes the bounds of this privacy risk. We then propose a simple but effective defense strategy that perturbs the transmitted logits in the direction that minimizes the privacy risk while maximally preserving the training performance. The experimental results validate our analysis and demonstrate the effectiveness of AdaMSA and our defense strategy.