In this short note we observe that the gamma factor defined by Gelfand and Kazhdan coincides with the Rankin-Selberg root number defined by Jacquet, Piatetskii-Shapiro and Shalika.

In particle physics, experiments generate substantial amounts of data that can be difficult to process without preliminary scaling. To avoid losing potentially crucial data, experimental collaborations are studying novel techniques for real-time data processing to extract features for further physics analysis. A common approach, especially in neutrino physics, is to use FPGAs for data acquisition and pre-processing. This paper presents an advanced Real-Time Wiener deconvolution algorithm designed to leverage the processing capabilities of the FPGA integrated into the readout boards of the Jiangmen Underground Neutrino Observatory (JUNO). The goal is to enable real-time reconstruction of the signal generated by photomultiplier tubes (PMTs) when neutrino interactions are detected. By exploiting online reconstruction of the signal generated by PMTs, we expect to improve the detection of low-energy depositions, such as those produced by transient astrophysical phenomena. These depositions are usually not saved because of the significant background that affects the low end of the energy spectrum, which would result in a large trigger rate, hence a large amount of data required for storage. This paper presents the features of the algorithm, including its ability to manage high-throughput data streams with minimal latency, adaptability, and resilience in discerning the characteristics of input data. Performance is evaluated on a JUNO electronic board. This study further demonstrates the potential of FPGA-based solutions for neutrino physics.

By applying the nonlinear Legendre transform to the continuity equation, this paper derives exact solutions to the Schrödinger equation and the equations of continuum mechanics. A generalized Maxwell distribution has been used as the momentum density function. Explicit expressions for the vector fields of time independent flows, density distributions, quantum and classical potentials have been found, and a detailed mathematical and physical analysis of the results obtained has been carried out.

Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$.

The interaction between phase oscillators is conservative if the phase volume is conserved throughout the dynamics. We derive a general condition, based on the notion of a pair-Hamiltonian, for the pairwise couplings to be conservative. The conservative networks with Winfree-type and Kuramoto-Daido-type couplings are also discussed. It is demonstrated that although, in contradistinction to genuine Hamiltonian dynamics, there is no exact pairwise symmetry of the Lyapunov exponents, the Lyapunov spectrum for a large network is nearly symmetric. The concept is also generalized to triplet and quadruplet couplings.

Accurately measuring the magnetocaloric effect is necessary to foster the development of magnetic refrigeration devices. However, current methods are inconvenient, requiring different instruments to measure each individual property or a custom-made setup. By measuring the time-varying magnetization in a commercially available VersaLab\textsuperscript{\textregistered} PPMS\textsuperscript{\textregistered} from Quantum Design, we have determined the adiabatic temperature change ($Δ$T$_{\textrm{ad}}$) of the first-order phase transition material Gd$_5$Si$_2$Ge$_2$, for a magnetic field change of 0 to 1 T, under high vacuum ($<$ 0.1 mTorr). For each temperature and magnetic field, the equilibrium magnetization is used as the magnetization-to-temperature conversion curve, allowing us to extend the validity of a previously proposed technique to the first-order phase transition material Gd$_5$Si$_2$Ge$_2$, which exhibits significant hysteresis. Our method thus enables full characterization (magnetic entropy change, adiabatic temperature change, and heat capacity) of any magnetocaloric material, whether it has a first-order or a second-order phase transition, using a single instrument. Comparing to a directly measured $Δ$T$_{\textrm{ad}}$, our method resulted in a peak $Δ$T$_{\textrm{ad}}$ value of 4.47 K, within 1\% of the directly measured value for a sample of the same composition.

A pair $\{u,v\}$ of vertices is said to be globally linked in a $d$-dimensional framework $(G,p)$ if there exists no other framework $(G,q)$ with the same edge lengths, in which the distance between the points corresponding to $u$ and $v$ is different from that in $(G,p)$. We say that $\{u,v\}$ is globally linked in $G$ in $\R^d$ if $\{u,v\}$ is globally linked in every generic $d$-dimensional framework $(G,p)$. We give a complete combinatorial characterization of globally linked vertex pairs in graphs in $\R^2$, solving a conjecture of Jackson, Jordán and Szabadka from 2006 in the affirmative. Our result provides a refinement of the characterization of globally rigid graphs in $\R^2$ as well as an efficient algorithm for finding the globally linked pairs in a graph. We can also deduce that globally linked pairs in $\R^2$, globally linked pairs in ${\mathbb C}^2$, and stress-linked pairs in ${\mathbb R}^2$ are all the same, settling conjectures of Jackson and Owen, and Garamvölgyi, respectively. In higher dimensions we determine the globally linked pairs in body-bar graphs in $\R^d$, for all $d\geq 1$, verifying a conjecture of Connelly, Jordán and Whiteley.

In this paper, we obtain new lower bounds for the evolution of the radius of analyticity of solutions to two initial value problems (IVPs) with initial data belonging to the class of analytic functions $H^{σ,s}(\mathbb{R})$ defined via a hyperbolic cosine weight. First, we consider the IVP for the modified Korteweg-de Vries (mKdV) equation. For this problem, we prove that the evolution of the radius of analyticity $σ(T)$ of the solution admits an algebraic lower bound $cT^{-\frac 12}$ for some $c>0$ and given arbitrarily large $T>0$. Next, we analyze the IVP for the mKdV equation with generalized dispersion (mKdVm) and a damping term. For this problem, we guarantee the local well-posedness in $H^{σ,s}(\mathbb{R})$ and demonstrate that the local solution can be extended globally in time and admits constant lower bounds for the radius of analyticity $σ(t)$ as time goes to infinity. The outcome of this paper concerning the mKdV equation represents an improvement on that achieved by the authors' previous work in [R. O. Figueira and M. Panthee, New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations, J. Evol. Equ. 24 No. 42 (2024)]. As far as we know, the results for the mKdVm with damping are new.

Two-dimensional metallic altermagnets are rare, and no correlated 2D material has been established to host large nonrelativistic spin splitting. Here we show that spontaneous orbital order, driven by electronic correlations and Fermi surface nesting, provides a general microscopic route to two-dimensional metallic altermagnetism. Antiferro-orbital ordering between the d$_{xz}$ and d$_{yz}$ orbitals breaks the equivalence of magnetic sublattices with opposite spins and generates a symmetry-enforced altermagnetic spin texture. As a concrete realization, we identify monolayer YbMn$_2$Ge$_2$ as a stable correlated metallic altermagnet exhibiting giant nonrelativistic spin splitting of order 1 eV. The resulting phase supports an exceptionally large and gate-tunable transverse spin conductivity. These results establish correlation-driven orbital order as a robust and general mechanism for designing correlated altermagnets with large spin splitting.

Within the framework of the left-right symmetric model (LRSM) and under the assumption of a diagonal Dirac neutrino mass matrix $M_{\rm D}$, this paper systematically investigates 20 types of three-zero textures in the Majorana neutrino mass matrix $M_{\rm R}$. The study reveals that only five three-zero textures of $M_{\rm R}$ satisfy the constraints of the latest NuFit 6.0 global fit results. Furthermore, we phenomenologically explore the correlations between Majorana phases $ρ$ and $σ$, as well as the relationships between the heavy neutrino mass spectrum $M_{I}~(I=1,2,3)$, ratio of Dirac matrix elements $y_{2}$, $y_{3}$ and the scale factor $r$. The results indicate strong correlations among the model parameters. In particular, the allowed regions for the Majorana CP phases are significantly restricted and depend on the specific texture of $M_{\rm R}$. On this basis, leptogenesis originating from heavy right-handed neutrino decays is investigated. Numerical results demonstrate that the $M_{\nu3}$ pattern can achieve successful leptogenesis within specific $r$ intervals for both the normal ordering (NO) and the inverted ordering (IO) of the light neutrino masses, while the $M_{\nu4}$ and $M_{\nu5}$ patterns possess viable parameter space for successful leptogenesis only in the NO case.

Angular broadening observations of compact radio sources provide a powerful method for probing the solar corona and solar wind. Such observations enable studies of the phase structure function, turbulence amplitude, intermediate-scale density fluctuations, solar-wind heating rates, and dissipation scales. When a compact radio source is observed through coronal or solar-wind plasma, several observable effects can arise: (1) the apparent source size increases because of scattering by turbulent plasma, producing angular broadening; (2) the source flux density decreases because of scattering and absorption; (3) the observed angular broadening can be anisotropic, reflecting anisotropic turbulence in the corona and solar wind; and (4) the position angle of the anisotropy, measured from north through east, can help infer the orientation of the coronal magnetic field. These effects provide insights into the physical processes governing the solar wind and its interaction with electromagnetic waves, and they offer constraints on coronal turbulence and magnetic-field structure. At present, angular broadening studies remain limited and have mostly focused on very bright radio sources such as Tau A. The unprecedented sensitivity and angular resolution of the Square Kilometre Array are expected to greatly expand the number of suitable background sources, opening a new window on the solar corona, solar wind, and heliosphere.

Homogeneous Boltzmann-type equations are an established tool for modelling interacting multi-agent systems in sociophysics by means of the principles of statistical mechanics and kinetic theory. A customary implicit assumption is that interactions are "all-to-all", meaning that every pair of randomly sampled agents may potentially interact. However, this legacy of classical kinetic theory, developed for collisions among gas molecules, may not be equally applicable to social interactions, which are often influenced by preferential connections between agents. In this paper, we discuss ongoing research on incorporating graph structures into homogeneous Boltzmann-type equations, thereby accounting for the "some-to-some" nature of social interactions.

The constancy of the gravitational constant $G$ is a cornerstone of the strong equivalence principle and of general relativity, yet its possible temporal variation remains a key target in tests of fundamental physics. Gravitational-wave (GW) astronomy, especially when combined with electromagnetic observations, provides an unprecedented new opportunity to probe this principle in the strong-field and dynamical regime. In this work, we develop a GW waveform model with a slowly varying gravitational constant, incorporating its effects both on compact binary dynamics and GW propagation in an expanding universe. Applying this framework to the binary neutron star merger GW170817, together with independent electromagnetic constraints on the luminosity distance, sky localization and binary inclination from GRB 170817A, we perform a joint Bayesian analysis that disentangles varying-$G$ effects from astrophysical degeneracies. We find no evidence for a temporal variation of the gravitational constant, and constrain its fractional time derivative to $\dot{G}/G \in [-3.36 \times 10^{-9}, 5.34\times10^{-10}]~{\rm yr^{-1}}$, representing the most stringent bounds obtained to date from real GW observations. Our results demonstrate the power of multi-messenger astronomy as a precision probe of the strong equivalence principle in the relativistic regime.

The spin alignment conjecture was originally formulated in connection with the additivity of coherent information for a class of quantum channels known as platypus channels. Recently, a stronger majorization-based version was proposed by M. A. Alhejji and E. Knill [Commun. Math. Phys. 405, 119, 2024], asserting that the spectrum of the alignment operator is always majorized by that of the perfectly aligned configuration. In this letter, we show that this strong spin alignment conjecture is false in general by constructing an explicit counterexample in the smallest unresolved case, namely three qubits. The example uses two-body states that are not jointly compatible with any single three-qubit global state, which naturally leads to a compatibility-constrained variant of the conjecture.

Quantum theory brings into question the compatibility of the twin desiderata of exact knowability of the present state of the physical world and perfect predictability of its future states. Bohr's coordination-causality complementarity principle transforms this tension into one between properties (as ordinarily understood in classical physics) and deterministic causality. Here, we develop an explicit model of quantum properties which accommodates this essential tension. Our approach integrates operational, reconstructive, and metaphysical standpoints. In particular, we make use of an operational framework employed in a recent operational reconstruction of Feynman's formulation of quantum theory; base our property model on an analysis of property types; and use the notions of actuality and potentiality to frame the model. We show that this quantum property model provides a natural resolution of Zeno's paradox of motion, and provides reliable intuitions about phenomena such as electron diffraction and the non-local behaviour of entangled states of non-identical particles.

We focus on the automated classification of eclipsing binary stars using deep learning methods to handle the vast data generated by large-scale photometric sky surveys. These surveys produce extensive datasets that are impractical for manual analysis. By using machine learning to classify eclipsing binary stars based on light curve morphology, this study aims to contribute to the efforts to efficiently process and accurately interpret massive data from projects Kepler, TESS and Gaia missions.

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose $G$ is a group and $M$ and $N$ are normal subgroups so that $N < M$, and suppose that there is an element $g \in G$ so that the conjugacy class of $g$ is $M \setminus N$. We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class. If we add the condition that $M/N$ is the unique minimal normal subgroup of $G/N$, then we obtain a generalization of a result by S.M. Gagola.

This paper proposes a physics-informed neural operator (PINO) framework for solving inverse scattering problems, enabling rapid and accurate reconstructions under diverse measurement conditions. In the proposed approach, the dielectric property is represented as a learnable tensor, while a neural operator is employed to predict the induced current distribution. A hybrid loss function, consisting of the state loss, data loss and total-variation (TV) regularization, is constructed to establish a fully differentiable formulation for a joint optimization of network parameters and dielectric property. To demonstrate the framework's generality and flexibility, PINO is implemented using three representative neural operators, i.e., the Fourier Neural Operator (FNO), the enhanced Fourier Neural Operator (U-FNO) and the Factorized Fourier Neural Operator (F-FNO). Compared with conventional approaches, the proposed framework offers a simpler formulation and universal modeling capability, making it readily applicable to various measurement scenarios, including multi-frequency and phaseless inversion. Numerical simulations demonstrate that the proposed PINO achieves high accuracy and robust reconstruction across samples with and without phase information, under single-frequency and multi-frequency settings in the presence of noise. The results demonstrate that PINO consistently outperforms conventional contrast-source inversion (CSI) methods and provides an efficient, unified solution to complex electromagnetic inverse-scattering problems.

An A-type coefficient polynomial introduced by Kawauchi recovers the HOMFLY-PT polynomial as a formal power series within skein theory. A notable feature of this construction is that each coefficient defines a link invariant, yielding an infinite sequence of invariants, while the low-degree coefficients are relatively easy to compute. In this paper, we extend this viewpoint to the B-type setting. Unlike the A-type case, the B-type setting requires a genuinely new inductive scheme due to the four-term skein relation. More precisely, we introduce coefficient polynomials associated with the B-type skein relation and show that their generating series recovers the Kauffman polynomial. We further prove that these coefficient polynomials are well-defined and that the resulting generating series is invariant under the corresponding Reidemeister moves.

We study nonsmooth convex minimization through a continuous-time dynamical system that can be seen as a high-resolution ODE of Nesterov Accelerated Gradient (NAG) adapted to the nonsmooth case. We apply a time-varying Moreau envelope smoothing to a proper convex lower semicontinuous objective function and introduce a controlled time-rescaling of the gradient, coupled with a Hessian-driven damping term, leading to our proposed inertial dynamic. We provide a well-posedness result for this dynamical system, and construct a Lyapunov energy function capturing the combined effects of inertia, damping, and smoothing. For an appropriate scaling, the energy dissipates and yields fast decay of the objective function and gradient, stabilization of velocities, and weak convergence of trajectories to minimizers under mild assumptions. Conceptually, the system is a nonsmooth high-resolution model of Nesterov's method that clarifies how time-varying smoothing and time rescaling jointly govern acceleration and stability. We further extend the framework to the setting of maximally monotone operators, for which we propose and analyze a corresponding dynamical system and establish analogous convergence results. We also present numerical experiments illustrating the effect of the main parameters and comparing the proposed system with several benchmark dynamics.

We study the one-arm probability in the level-set percolation of the discrete and metric-graph Gaussian free field (GFF) defined on a box with Dirichlet boundary conditions. For the metric-graph case, we establish asymptotic estimates on two one-arm probabilities of interest. For the discrete case, we show up-to-constants bounds on the point-to-bulk probability and demonstrate its difference from the metric-graph case.

Serverless computing is increasingly adopted for AI-driven workloads due to its automatic scaling and pay-as-you-go model. However, its function-based architecture creates significant security risks, including excessive privilege allocation and poor permission management. In this paper, we present ALPS, an automated framework for enforcing least privilege in serverless environments. Our system employs serverless-tailored static analysis to extract precise permission requirements from function code and a fine-tuned Large Language Model (LLM) to generate language- and vendor-specific security policies. It also performs real-time monitoring to block unauthorized access and adapt to policy or code changes, supporting heterogeneous cloud providers and programming languages. In an evaluation of 8,322 real-world functions across AWS, Google Cloud, and Azure, ALPS achieved 94.8\% coverage for least-privilege extraction, improved security logic generation quality by 220\% (BLEU), 124\% (ChrF++) and 100\% (ROUGE-2), and added minimum performance overhead. These results demonstrate that ALPS provides an effective, practical, and vendor-agnostic solution for securing serverless workloads.

In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this function essentially by design, it is natural to ask whether they arise as special values of more intrinsic zeta-type objects. In this article, we show that a shifted log-sine integral provides such an example. Its analytically continued values at negative integers are given by anti-diagonal sums of poly-Bernoulli numbers with negative index.

Every software organization faces the onboarding challenge: helping newcomers navigate complex codebases, compensate for insufficient documentation, and comprehend code they did not author. Expert walkthroughs are among the most effective forms of support, yet they are expensive, repetitive, and do not scale. We present Lacy, a hybrid human-AI onboarding system that captures expert mentoring in reusable code tours-to our knowledge, the first hybrid approach combining AI-generated content with expert curation in code tours. Our design is grounded in requirements derived from 20+ meetings, surveys, and interviews across a year-long industry partnership with Beko. Supporting features include Voice-to-Tour capture, comprehension quizzes, podcasts, and a dashboard. We deployed Lacy on Beko's production environment and conducted a controlled study on a legacy finance system (30K+ LOC). Learners using expert-guided tours achieved 83% quiz scores versus 57% for AI-only tours, preferred tours over traditional self-study, and reported they would need fewer expert consultations. Experts found tour creation less burdensome than live walkthroughs. Beko has since adopted Lacy for organizational onboarding, and we release our code and study instruments as a replication package.

Local Operator Entanglement (LOE) has emerged an indicator of quantum chaos in many-body systems. Numerical studies have shown that, in chaotic systems, LOE grows linearly in time and displays a volume-law behavior at late times, scaling proportionally with the number of local degrees of freedom. Despite extensive numerical evidence, complemented by analytical studies in integrable systems, a fully analytical understanding of the emergence of the volume law remains incomplete. In this paper, we contribute toward this goal by deriving a late-time expression for LOE in chaotic systems that exhibits volume-law scaling. Our derivation proceeds by expressing the late-time LOE in the Liouville eigenstate basis and relies on three main assumptions: a higher-order non-resonance condition for the Hamiltonian eigenenergies, the Eigenstate Thermalization Hypothesis (ETH) ansatz for the matrix elements of the initial local operator, and the replacement of Hamiltonian eigenstates with random states in the final expression for LOE. Under these assumptions, we obtain an explicit formula displaying volume-law scaling. Finally, we complement our analytical derivation with numerical simulations of the 1D mixed-field Ising model, testing the resulting formula and exploring the regime of validity of our assumptions.

We present a detailed analysis of the neutral atomic gas (HI) in the dwarf galaxy UGCA 320, observed with the MeerKAT telescope as part of the MHONGOOSE (MeerKAT HI Observations of Nearby Galactic Objects: Observing Southern Emitters) programme. In a small group consisting of three dwarf galaxies, all of which contain HI, it is the most massive. Detailed kinematic modelling shows that UGCA 320 contains a substantial amount of (kinematically) anomalous gas (>=20%), at least ~30% of which is likely the result of a tidal interaction with its neighbour UGCA 319. It also reveals that UGCA 320 likely harbours a star-formation driven outflow, and that ~10% of its HI is extra-planar and has a filamentary structure. Although UGCA 320 aligns with established scaling relations from the literature, its neutral hydrogen content is notably complex - shaped by its immediate environment. This underscores the importance of deep, resolved observations and detailed kinematic analyses to capture the nuances of galaxy evolution.

As most optical satellites remotely acquire multispectral images (MSIs) with limited spatial resolution, multispectral unmixing (MU) becomes a critical signal processing technology for analyzing the pure material spectra for high-precision classification and identification. Unlike the widely investigated hyperspectral unmixing (HU) problem, MU is much more challenging as it corresponds to the underdetermined blind source separation (BSS) problem, where the number of sources is larger than the number of available multispectral bands. In this article, we transform MU into its overdetermined counterpart (i.e., HU) by inventing a radically new quantum deep image prior (QDIP), which relies on the virtual band-splitting task conducted on the observed MSI for generating the virtual hyperspectral image (HSI). Then, we perform HU on the virtual HSI to obtain the virtual hyperspectral sources. Though HU is overdetermined, it still suffers from the ill-posed issue, for which we employ the convex geometry structure of the HSI pixels to customize a weighted simplex shrinkage (WSS) regularizer to mitigate the ill-posedness. Finally, the virtual hyperspectral sources are spectrally downsampled to obtain the desired multispectral sources. The proposed geometry/quantum-empowered MU (GQ-$μ$) algorithm can also effectively obtain the spatial abundance distribution map for each source, where the geometric WSS regularization is adaptively and automatically controlled based on the sparsity pattern of the abundance tensor. Simulation and real-world data experiments demonstrate the practicality of our unsupervised GQ-$μ$ algorithm for the challenging MU task. Ablation study demonstrates the strength of QDIP, not achieved by classical DIP, and validates the mechanics-inspired WSS geometry regularizer.

We extend the theory of complex cells introduced by Binyamini and Novikov to the sharply o-minimal setting, obtaining cellular preparation and parameterization theorems which are polynomially effective in the degrees of the relevant sets. Our constructions are definable, and so applying them to sets in a given reduct of R_an yields cells and cellular maps definable in the same reduct.

We report the results of the second iteration of an experiment searching for neutron-hidden-neutron oscillations in a beam of ultracold neutrons, conducted at the PF2 facility of the Institut Laue Langevin (ILL). Oscillations were tested via neutron disappearance as a function of an applied magnetic field, in the context of a phenomenological two-parameter model assuming zero hidden potentials. The magnetic field was varied in a step-wise manner in order to resonantly enhance the oscillation probability at different mass splittings ($δm$) across a 60--1550 peV range. No evidence for neutron disappearance is observed and conservative limits on the neutron-hidden-neutron oscillation period ($τ_{nn'}$) have been set at 95 % confidence level: $τ_{nn'} > 200$ms for $|δm| \in [60, 400]$ peV and $τ_{nn'} > 100$ ms for $|δm| \in [400, 1550]$ peV

We present a unified analytical and numerical study of the one-dimensional Feshbach--Villars (FV) equation for spin-0 particles in the presence of several representative external potentials. Starting from the FV formulation of the Klein--Gordon equation, we derive the corresponding one-dimensional master equation and analyse its solutions for Coulomb, power-exponential, Cornell, Pöschl--Teller, and Woods--Saxon interactions. For the singular Coulomb and Cornell cases, a Loudon-type cutoff regularisation is implemented on the full line, allowing a mathematically controlled treatment of the origin and an explicit classification of the states by parity. The Coulomb problem exhibits the expected near-degenerate even--odd structure in the cutoff limit, while the Cornell potential combines short-distance Coulomb behaviour with long-distance confinement and produces a finite set of bound states for fixed parameters. The power-exponential potential with $p=1$ is reduced to a Whittaker-type equation and yields an intrinsically relativistic spectrum with no standard Schrödinger bound-state limit in the parameter regime considered. For the smooth short-range Pöschl--Teller and Woods-Saxon potentials, the FV formalism reveals, respectively, the effects of definite parity and spatial asymmetry on the spectrum, wave functions, and particle--antiparticle mixing. In all cases, we reconstruct the full FV spinor, analyse the associated charge density, and compare the relativistic behaviour with the corresponding non-relativistic expectations whenever such a limit exists. The results provide a coherent set of analytical and numerical benchmarks for relativistic scalar bound states in one dimension.