Differential light shifts (DLS) induced by optical trapping fields fundamentally limit the stability and accuracy of trapped-atom microwave clocks. We demonstrate an in-situ method to cancel DLS by simultaneously interrogating multiple spatially separated atomic ensembles at different trap intensities generated from a common light source. By operating the ensembles at set intensity ratios and performing Ramsey spectroscopy, the intensity-dependent frequency shifts are measured within each experimental cycle and extrapolated to the zero-intensity limit. This approach effectively enables shot-to-shot determination of a DLS-free frequency without requiring magic wavelengths or species-specific cancellation schemes. We validate the method for Rb atoms trapped in time-averaged potentials by introducing controlled variations of the total trap power and show that the extrapolated frequency remains insensitive to these fluctuations. The technique is general and can be extended to other systematic shifts, providing a scalable route toward improved stability and accuracy in compact trapped-atom clocks and related quantum sensors relying on optical dipole traps

We give approximation schemes for Subset TSP and Steiner Tree on unit disk graphs, and more generally, on intersection graphs of similarly sized connected fat (not necessarily convex) polygons in the plane. As a first step towards this goal, we prove spanner-type results: finding an induced subgraph of bounded size that is $(1+\varepsilon)$-equivalent to the original instance in the sense that the optimum value increases only by a factor of at most $(1+\varepsilon)$ when the solution can use only the edges in this subgraph. - For Subset TSP, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $\mathrm{poly}(1/\varepsilon)\cdot\mathrm{OPT}$ in polynomial time, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. - For Steiner Tree, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $2^{\mathrm{poly}(1/\varepsilon)}\cdot\mathrm{OPT}$ in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{2^{\mathrm{poly}(1/\varepsilon)}}\cdot n^{O(1)}$. - An improved algorithm finds a $(1+\varepsilon)$-approximate solution for Steiner Tree in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. An easy reduction shows that approximation schemes for unit disks imply approximation schemes for planar graphs. Thus our results are far-reaching generalizations of analogous results of Klein [STOC'06] and Borradaile, Klein, and Mathieu [ACM TALG'09] for Subset TSP and Steiner Tree in planar graphs. We show that our results are best possible in the sense that dropping any of (i) similarly sized, (ii) connected, or (iii) fat makes both problems APX-hard.

Neural networks (NNs) have gained significant attention across various engineering disciplines, particularly in design optimization, where they are used to build surrogate models for high-dimensional regression problems. Despite their power as global approximators, NNs often fail to accurately capture local nonlinearities without relying on a large number of training parameters. To address these limitations, in this paper we propose domain decomposition methods (DDM), which divide the input feature space into multiple local subdomains, each modeled by a simpler NN, trained in parallel. To recover the accuracy of a global approximation, interface constraints are introduced in the local loss functions to enforce continuity between subdomains. The interface constraints are enforced with two different approaches, by utilizing Lagrange multiplier or augmented Lagrange multiplier methods. Both approaches are validated using synthetic data from 2D and 3D linear compression problems, numerically solved using the finite element method. The study investigates computational time and accuracy across varying numbers of subdomains to identify optimal partitioning strategies. Compared to unconstrained approximations, both methods significantly improve continuity across subdomain interfaces. Also, the use of DDMs improves approximation accuracy in nonlinear regions when compared to standard global NN training. The augmented Lagrange method outperforms the standard Lagrange formulation by converging faster due to lower convergence requirements, albeit with a slightly lower accuracy. Its scalability makes it the preferred choice for large-scale problems, as the faster convergence outweighs the minor loss in accuracy. Overall, these results highlight the augmented Lagrange method as a promising DDM approach for training efficient and scalable NN surrogate models.

Convex polyominoes can be refined according to the number of direction changes in monotone paths connecting pairs of cells, leading to the notion of $k$-convexity. In particular, the cases $k=1$ and $k=2$ correspond to $L$-convex and $Z$-convex polyominoes, two well-studied subclasses of convex polyominoes, with intermediate families such as centered and $4$-stack polyominoes. These families exhibit remarkably different combinatorial behaviours, suggesting that geometric constraints have a strong impact on the nature of the generating function: $L$-convex and centered polyominoes possess rational generating functions and growth of order $(2+\sqrt{2})^n$ and $4^n$, respectively, while $Z$-convex, 4-stack, and convex polyominoes have algebraic functions and asymptotics of order $n4^n$, $\sqrt{n}\,4^n$, and $n4^n$ respectively. In this paper we investigate the structure of $Z$-convex polyominoes by introducing a refinement based on the NW- and NE-convexity degrees, which yields a decomposition into three disjoint subclasses $C(1,2)$, $C(2,1)$, and $C(2,2)$. To enumerate these families we introduce ascending polyominoes, admitting a simple geometric characterization, and construct a generating tree that leads to functional equations for the corresponding generating functions. By solving these equations we obtain explicit algebraic generating functions and the asymptotic growth for all the subclasses.

In this paper, we establish the weak*-convergence of a sequence of the complex Hessian measures $H_m(u_j)$ to the complex Hessian measure $H_m(u)$ in the class $\mathcal{E}(X,ω,m)$ under hypotheses that $u_j$ is convergent to $u$ in $(ω,m)$-capacity.

The goal of this paper is to compute the motivic local density of an isolated algebraic surface singularity, in order to explain its link with algebraic multiplicity. In this context, we can use an additional data: the inner rate related to the bilipschitz geoemtry of the singularity, as studied by A. Belotto da Silva, L. Fantini and A. Pichon.

Models in Algebraic Quantum Field Theory (AQFT) may be generalized including Lie groups of symmetries whose Lie algebras admit an Euler element $h$, characterized by the property that $ad h$ is diagonalizable with eigenvalues in $\{-1, 0, 1\}$. These elements becomes fundamental to the formal description of wedge localization. In this paper, we extend the geometric analysis of Euler wedges and investigate their applications within the AQFT framework. We call a pair of Euler elements $(h, k)$ orthogonal if $e^{i π\operatorname{ad} h}(k) = -k.$ Using the geometric framework established in our previous work, we derive both a Bisognano--Wichmann Theorem and a Spin--Statistics Theorem for nets of standard subspaces and von Neumann algebras. Our results {show} how this generalized approach recovers classical results in the AQFT literature while providing a deeper structural understanding of the underlying geometry in established models.

Rotatable antenna (RA) provides additional spatial degrees of freedom (DoFs) for communication systems by enabling per-antenna dynamic boresight adjustment, which is attractive for fairness-oriented multicast transmission. This letter investigates an RA-enhanced downlink multi-group multicast system. Specifically, we aim to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among all users by jointly optimizing the multicast beamforming vectors and the RA boresight directions under transmit power and rotation constraints. To solve this non-convex problem, we first reformulate the max-min SINR objective via quadratic transform. Then, we develop an alternating optimization (AO) algorithm that iteratively updates the multicast beamforming and RA boresight directions. The beamforming vectors are obtained from a convex subproblem, while the boresight directions are refined using a successive convex approximation (SCA) procedure. Simulation results verify that the proposed RA-based scheme substantially enhances the fairness performance compared with fixed antenna-based and random-orientation benchmarks.

Generalization under manipulation and dataset shift remains a core challenge in forged media detection for AI-driven edge sensing systems. Frozen vision foundation models with linear probes are strong baselines, but most pipelines use default backbone outputs without testing conditioning at the frozen feature interface. We present the first controlled probing study on DINOv3 ConvNeXt and show that, without task-specific fine-tuning, linear probing alone yields competitive forged-media detection performance, indicating that ViT-7B self-supervised distillation transfers to security-critical vision workloads at edge-compatible inference cost. Backbone, head, data, and optimization are fixed while conditioning is varied; LN-Affine, the default ConvNeXt head output, is the natural baseline. On FaceForensics++ c23, five conditioning variants are evaluated under in-distribution testing, leave-one-manipulation-out (LOMO), and cross-dataset transfer to Celeb-DF v2 and DeepFakeDetection. In ConvNeXt-Tiny, conditioning alone changes LOMO mean AUC by 6.1 points and reverses ID-vs-OOD ranking: LN-Affine is strongest on external datasets, while LayerNorm is strongest in-distribution. In ConvNeXt-Base replication, the OOD winner becomes protocol-dependent, and ID-optimal selection still fails as a robust deployment rule. Results show that feature conditioning is a first-order design variable and should be selected with robustness-oriented validation, not ID accuracy alone.

Monolayer transition metal dichalcogenides are a suitable platform for studying excitonic coherence in the light-matter coupling regime. We present an ab initio time-dependent GW-Bethe-Salpeter equation (GW-BSE) investigation of coherent excitonic dynamics in monolayer WS$_2$. By solving the coherent coupling between the A, A$^{*}$, and B excitons under linearly polarized pump fields, we identify the microscopic origin of the resulting oscillatory dynamics and rationalize it using an effective theoretical model. Our results provide the interpretation of recently reported coherent excitonic phenomena in monolayer WS$_2$ (Nano Lett. 24, 8117 (2024)). Building on this first-principles time-resolved framework, we propose a tailored pump-probe scheme that enables the controlled generation and regeneration of coherent oscillations between excitonic states. These findings establish a predictive route for controlling excitonic coherence in two-dimensional materials, with direct relevance for ultrafast optoelectronic switches and solid-state quantum logic devices.

Channel estimation is a central bottleneck in BD-RIS-assisted MIMO systems. The richer inter-element coupling that enables large performance gains also makes training and hardware control substantially harder than in diagonal RIS architectures. Existing estimators either target only cascaded channels or require block-by-block reconfiguration of the BD-RIS interconnections, which is costly and difficult to implement in practice. To overcome this limitation, we propose a pilot-assisted tensor framework for group-connected BD-RIS under a two-timescale protocol, where the scattering structure is designed as a low-rank PARAFAC model with fixed factor matrices. This design keeps the interconnection topology constant across blocks and updates only phase shifts, enabling practical operation without sacrificing estimation quality. Building on this structure, we develop a PARAFAC-based alternating least-squares (PALS) receiver that recovers the individual channels. Numerical results confirm that PALS delivers markedly lower composite-channel NMSE than conventional LS, matches the accuracy of state-of-the-art tensor receivers, and sharply reduces BD-RIS design complexity

Relativistic, polarized pair-photon fireballs are central to understand the microscopic energy transfer of high-energy astrophysical outflows, yet generating an overdense fireball in the laboratory, especially via an ultraintense laser, remains a formidable challenge. Here, we propose a novel method of laser-driven nonlinear-linear quantum electrodynamics (NL-QED) plasma, that dramatically lowers the laser intensity threshold for dense pair-photon fireball creation. By coupling polarization-resolved linear Breit-Wheeler and Compton processes with strong-field nonlinear radiation, we find that a self-organized NL-QED cascade is ignited in the laser-driven hole boring at currently accessible 10-petawatt intensities ($\sim 10^{22}\ \mathrm{W/cm^2}$). Consequently, we demonstrate the generation of an overdense (pair density $n_{\rm}\simeq 4.1\times 10^{16}\ \mathrm{cm^{-3}}$, gamma-photon density $n_γ\simeq 9.6\times 10^{21}\ \mathrm{cm^{-3}}$), highly polarized, and quasi-neutral fireball. Our method provides a comprehensive framework for studying laser-driven QED plasma and its application in laboratory astrophysics, probing multi-process QED physics.

Turbulence in wall-bounded shear flows is increasingly understood through exact coherent structures (ECS) -- invariant solutions of the Navier-Stokes equations that act as organising centres in the high-dimensional state space. Here we report five new ECS of plane Poiseuille flow: two relative periodic orbits (RPOs) and three travelling waves (TWs), that are computed in four distinct symmetry-invariant subspaces using a Newton-Krylov-hookstep solver initialised from direct numerical simulations. We trace each state through one-parameter continuations in both the Reynolds number $Re$ and the spanwise period $L_z$. All five states are organised around counter-rotating rolls sustaining streamwise velocity streaks, yet they exhibit qualitatively different stability properties: the two RPOs are linearly stable within their symmetry subspace, while the three TWs are saddle-type solutions whose instabilities are, respectively, mixed oscillatory-and-monotone, purely monotone and purely oscillatory. The continuation diagrams reveal their bifurcation geometry, from simple saddle-node folds for the RPOs to a pronounced S-shaped multi-branch structure with three coexisting dissipation levels for one of the travelling waves. Along each branch, the roll-streak topology is preserved across the folds, with the different branches distinguished by their amplitude and gradient intensity rather than by a change in spatial organisation. Floquet spectra evaluated at multiple points along each branch shows that folds generically act as sites of reduced instability for the travelling waves, while upper branches develop stronger and sometimes qualitatively different unstable modes. These include the emergence of linearly stable segments on the intermediate and upper branches of the multi-branch travelling wave, despite the solution being unstable at its reference parameters.

The Kaczmarz method is widely recognized as an efficient iterative algorithm for solving large-scale linear systems, owing to its simplicity and low memory requirements. However, the development of its nonlinear extensions for solving large-scale nonlinear systems has seen limited progress. In this work, we introduce a new family of momentum-accelerated averaging block nonlinear Kaczmarz methods tailored for large-scale nonlinear systems and ill-posed problems. Our contributions are twofold: (1) We develop an adaptive strategy for selecting step sizes and momentum coefficients, leading to a new average block nonlinear Kaczmarz method with adaptive momentum (ABNKAm). This algorithm achieves high computational efficiency by requiring only minimal inner-product computations per iteration, which significantly reduces both arithmetic complexity and memory usage. (2) We establish rigorous convergence of the ABNKAm under mild assumptions, proving that the method converges exponentially to the unique solution nearest to the initial point. Moreover, under suitable conditions, we provide a theoretical justification of acceleration of the proposed ABNKAm with momentum. Extensive numerical experiments demonstrate that ABNKAm outperforms existing nonlinear Kaczmarz variants in terms of both iteration count and computational time, with particularly notable gains in large-scale problems.

Age-specific probabilities of death provide a snapshot of population mortality at the country level at a given point in time. Due to the high dimensionality of the data, summarising mortality information is essential for various analyses, such as visualisation and clustering. We propose the use of beta latent variable (BLV) models to summarise mortality information without data transformation. A time-dependent version of the BLV model is developed by incorporating an autoregressive prior for the latent effects. This model aims to represent mortality data with a small set of $K$ latent effects while accounting for time dependence between these effects. Inference is performed using Bayesian methods, with posterior samples generated via Hamiltonian Monte Carlo. The BLV model is applied to probabilities of death from the Human Mortality Database, covering 41 countries and 23 age-specific probabilities of death over several periods. The time-dependent BLV model with $K=6$ latent effects accurately reconstructs observed mortality data, and the model parameters have intuitive and insightful interpretations. The time-dependent BLV outperforms the standard Gaussian factor analysis model applied to logit probability of death, and demonstrates that BLV models can effectively summarise mortality data.

The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. We explore experimentally relevant parameter regimes and find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and how this leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.

We investigate the sensitivity of current and future hadron-collider experiments to axion-like particles (ALPs) through associated diboson production, focusing on a linear effective field theory framework with bosonic ALP couplings. We analyze the dominant production mechanisms and relevant backgrounds, considering the impact of jet misidentification rates on the diboson background. We present our results using conservative jet-misidentification rates, and derive four dimensional constraints on the ALP couplings to gluons, weak bosons, and photons. Our findings highlight the potential of the high-luminosity phase of the CERN Large Hadron Collider to probe the ALP parameter space in the sub-GeV mass range, as well as the codependencies of the various ALP couplings.

We study the following focusing intercritical nonlinear Schrödinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+Δ_{z}u-y^2 u =- |u|^αu,\quad t\in \mathbb{R},\newline u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases} \end{equation*} where $d \geq 1 $ is an integer and the exponent $α$ satisfies \begin{equation}\label{assumption} \frac{4}{d}< α<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline + \infty,\,\,\, \text{if} ~~ d=1. \end{cases} \end{equation} For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions $d \leq 4$ via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold.

Biological systems are governed by coupled interactions between intracellular metabolism and bioreactor operation that span multiple time scales. Constraint-based metabolic models are widely used to describe intracellular metabolism, but repeatedly solving the optimization problem at each time step in dynamic models introduces numerical challenges related to infeasibility and computational efficiency. This work presents a multi-scale modeling framework that integrates genome-scale, constraint-based metabolic models with dynamic bioreactor simulations. Intracellular metabolism is described using positive flux variables in a parsimonious flux balance analysis, and the resulting embedded optimization problem is replaced by a neural network surrogate. The surrogate provides a smooth approximation of the embedded optimization mapping and eliminates repeated linear program solves during simulation. The approach is demonstrated for fed-batch fermentation of Escherichia coli, in which the surrogate model yields intracellular fluxes under substrate-limited conditions, whereas the underlying linear program would otherwise be infeasible. The framework provides a continuous representation of intracellular metabolism suitable for dynamic simulation of genome-scale models in bioreactor configurations.

A crucial assumption to reduce computational complexity in spatial-temporal data analysis is separability, which factors the covariance structure into a purely spatial and a purely temporal component. In this paper, we develop statistical inference tools for validating this assumption for a second-order stationary process under both domain-expanding-infill asymptotics and domain-expanding asymptotics. In contrast to previous work on this subject, the methodology neither requires the assumption of normally distributed data, nor uses spectral methods. Our approach is based on nonparametric estimates of measures for the deviation between the covariance matrix and separable approximations, which vanish if and only if the assumption of separability is satisfied. We derive the asymptotic distributions of appropriate estimators for these measures with non-standard limiting distributions and use these results to develop inference tools for validating the assumption of separability. More specifically, we derive confidence intervals for the deviation measures, tests for the hypothesis of exact separability, and for the hypothesis that the deviation from separability is smaller than a prespecified threshold.

Functor morphing provides a method to translate complex representations of automorphism groups of finite modules over finite rings to representations of automorphism groups of functors in some abelian category. In this paper we give an explicit criterion for a representation to be in the image of functor morphing using the action of parabolic subgroups. We then demonstrate this criterion on Borel groups of finite fields.

Foundation models learn transferable representations, motivating growing interest in their application to wireless systems. Existing wireless foundation models are predominantly based on transformer architectures, whose quadratic computational and memory complexity can hinder practical deployment for large-scale channels. In this work, we introduce WiMamba, a wireless foundation model built upon the recently proposed Mamba architecture, which replaces attention mechanisms with selective state-space models and enables linear-time sequence modeling. Leveraging this architectural advantage combined with adaptive preprocessing, WiMamba achieves scalable and low-latency inference while maintaining strong representational expressivity. We further develop a dedicated task-agnostic, self-supervised pre-training framework tailored to wireless channels, resulting in a genuine foundation model that learns transferable channel representations. Evaluations across four downstream tasks demonstrate that WiMamba matches or outperforms transformer-based wireless foundation models, while offering dramatic latency and memory reductions.

Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an equivalence relation called cut-concordance, which encompasses the topological notion of concordance for classical links. Our main result is that the nilpotent peripheral system of 1--dimensional cut-diagrams is an invariant of cut-concordance, giving along the way a combinatorial version of a theorem of Stallings. We also investigate the relationship with several other equivalence relations in diagrammatic knot theory, in particular in connection with link-homotopy.

Blockchain technology introduces asset types and custody mechanisms that fundamentally break traditional financial auditing paradigms. This paper presents an autoethnographic analysis of cryptoasset auditing challenges, build on top of prior research on a comprehensive framework addressing existence, ownership, valuation, and internal control verification. Drawing from lived experience implementing blockchain systems as an engineer, smart contract auditor, and CTO of a publicly traded cryptoasset firm, we demonstrate how autoethnographic methodology becomes necessary for understanding technical complexities that external analysis cannot capture. Through detailed examination of token airdrops, multi-signature smart contracts, and real-time on-chain reporting, we provide experimental approaches and common scenarios that auditing firms can analyze to address blockchain innovations currently considered technically insurmountable.

Accurate real-time monitoring of disease transmission is crucial for epidemic control, which has conventionally relied on reported cases or hospital admissions. Such metrics are frequently susceptible to delays in reporting, various forms of bias, and under-ascertainment. Cycle threshold values obtained from reverse transcription quantitative polymerase chain reaction offer a promising alternative, serving as a proxy for viral load. In this paper, we aim to jointly model the viral load and the number of deaths (mortality), which involves a continuous bounded and a count time series, and therefore, a proper mixed-type model is needed. This is the motivation to introduce a new mixed-valued time series quasi-likelihood (MixTSQL) model capable of analyzing multivariate time series of different types, like continuous, discrete, bounded, and continuous positive. The MixTSQL model only requires a mean-variance specification with no distributional assumptions needed, and allows for testing Granger causality. Statistical guarantees are provided to ensure consistency and asymptotic normality of the proposed quasi-maximum likelihood estimators. We analyze weekly viral load and Covid-19 death counts in São Paulo, Brazil, using our MixTSQL model, which not only establishes the temporal order in which viral load Granger-causes mortality but also offers a comprehensive joint statistical analysis.

High-visibility Franson interference at telecom C-band wavelengths is achieved using a cascaded periodically poled lithium niobate (PPLN) waveguide photon-pair source combined with fully passive, path-imbalanced Mach-Zehnder interferometers implemented on photonic integrated circuits (PICs). The interferometers require neither on-chip phase shifters nor active stabilization; instead, the phase is scanned via thermal tuning of the chip. By employing a narrow-linewidth continuous-wave (CW) pump and dense wavelength-division multiplexing (DWDM) filtering, energy-time entangled photon pairs with high spectral indistinguishability are generated. We achieve a 4.8% heralding efficiency and a two-photon interference visibility of 97.1% from sinusoidal fringe fitting (raw visibility 95.2% and background-corrected visibility 95.6%), alongside a coincidence-to-accidental ratio (CAR) exceeding 1000 at only 1.7 mW of pump power. These results represent one of the highest Franson-interference visibilities reported for a PIC-based analyzer within a compact, fiber-integrated platform.

Let $a$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In this paper, we show that for arbitrary positive integers $a$ and $b$ with $a|b$, the $b$th power GCD matrix $(S^b)$ and the $b$th power LCM matrix $[S^b]$ are both divisible by the $a$th power GCD matrix $(S^a)$ if $S$ is a gcd-closed (i.e. $\gcd(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i,j\le n$) set satisfying the condition $\mathcal G$ (i.e., for any element $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that $[y_1,y_2]=x$ as well as $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$). This confirms a conjecture of Hong proposed in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.} {\bf 113} (2026), 231-243].

We consider circular particle motion under the action of an unspecified force in a static spherically symmetric spacetime. We derive the machinery that allows one to find the force acting on a circular particle and deduce whether its position is stable or not. This also allows one to extend the definition of ISCO to the case of a non-zero external force. By conducting the near-horizon expansion, we obtain that for any non-extremal black holes, the acceleration for extremal ones is finite, and for ultraextremal (multiple) horizons it tends to zero. Applying the derived machinery to the case of the Schwarzschild metric assuming that a force is constant, we scrutiny how the number of orbits for a given force depends on its value. In particular, if a force is big enough, an additional branch of solutions appears that was absent in the case of geodesic motion. Then, for various circular orbits, we numerically investigate their stability. A similar problem is solved for the Reissner-Nordstrom (RN) metric and uncharged particles. It appears that for the near-extremal and extremal RN black holes, there exist near-horizon circle trajectories (in contrast to the nonextremal case). For the ISCO, the dependence of the orbit radius on $κ$ (the surface gravity) is similar to that in the case of neutral particles moving in the background of rotating black holes. In addition, two scenarios of high-energy particle collisions near such orbits are considered, and it is found that dependence on $κ$ is also similar to that for rotating black holes.

Giant atoms are quantum emitters coupled to waveguides at multiple, spatially separated points, enabling interference effects that fundamentally change their light-matter interactions. A notable consequence of the interference is the emergence of decoherence-free interaction (DFI), which allows coherent excitation exchange between giant atoms via the waveguide without radiative loss. Leveraging DFI offers a promising route to implementing two-qubit quantum gates without the need for additional resources, positioning giant atoms as a versatile platform for scalable universal quantum simulators. However, existing work has focused primarily on continuous, Markovian waveguides; in structured waveguides, where non-Markovian effects become significant, only iSWAP gates have been explored. To address this gap, we introduce and analyze a protocol for implementing controlled-Z (CZ) gates with giant atoms in structured waveguides. We first show that while a minimal two-point coupling scheme supports DFI, it also exhibits strong non-Markovian effects that substantially degrade gate fidelity. To overcome this limitation, we propose an extended design featuring a third coupling point. This configuration suppresses non-Markovian effects and enables CZ gates with fidelities up to $97.7\%$ (assuming typical values for experimental imperfections). Our results broaden the accessible gate set for giant atoms in structured waveguides to include both iSWAP and CZ gates, advancing these systems as a pathway toward universal quantum simulators operating in non-Markovian environments.

The application of zero-knowledge proofs (ZKPs) in autonomous systems is an emerging area of research, motivated by the growing need for regulatory compliance, transparent auditing, and trustworthy operation in decentralized environments. zk-SNARK is a powerful cryptographic tool that allows a party (the prover) to prove to another party (the verifier) that a statement about its own internal state is true, without revealing sensitive or proprietary data about that state. This paper proposes Hermes Seal: a zk-SNARK-based ZKP framework for enabling privacy-preserving, verifiable communication in vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) networks. The framework allows autonomous systems to generate cryptographic proofs of perception and decision-related computations without revealing proprietary models, sensor data, or internal system states, thereby supporting interoperability across heterogeneous autonomous systems. We present two real-world case studies implemented and empirically evaluated within our framework, demonstrating a step toward verifiable autonomous system information exchanges. The first demonstrates real-time proof generation and verification, achieving 8 ms proof generation and 1 ms verification on a GPU, while the second evaluates the performance of an autonomous vehicle perception stack, enabling proof of computation without exposing proprietary or confidential data. Furthermore, the framework can be integrated into AV perception stacks to facilitate verifiable interoperability and privacy-preserving cooperative perception. The demonstration code for this project is open source, available on Github.