Let $Γ$ be an additive abelian group equipped with a commutative factor $ω$. We describe the simple Lie colour algebras and the associated untwisted affine Lie colour algebras graded by $Γ$, which fulfil the Cartan-Weyl paradigm. The quantised universal enveloping algebras of these (affine) Lie colour algebras are constructed, which are colour analogues of the Drinfeld-Jimbo quantum groups including the latter as the special case of trivial $Γ$. We develop the quasi-triangular Hopf colour algebraic structure of these ``colour quantum groups'', which has immediate applications in areas such as knot theory and statistical mechanics.

We study two families of quandles arising from Coxeter quandles. One is the quandle defined by Andruskiewitsch-Graña, which is the set of roots with binary operation defined by using the negatives of reflections. We observe that this is realized as a conjugation quandle in a Pin group. The other, which we call a rotational $D_n$ quandle, is the set of some right angle rotations in the Coxeter group of type $D_n$ with binary operation given by conjugation. We determine their inner automorphism groups, and observe that they are quite similar.

We introduce a family of constants \[ C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac1k\right) - \log n \right), \] which may be regarded as polylogarithmic analogues of Euler's constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants \(C_m\) to special values of certain Dirichlet series involving the Riemann zeta function.

In the design of computational methods for Bayesian inverse problems, costly forward model evaluations make it difficult to sample from or compute the posterior. This motivates the need for approximate forward models that are cheaper to evaluate. We consider reduced-order forward models which exploit the lower-dimensional structure in the Bayesian inverse problem by projecting to the "likelihood-informed subspace" of the parameter space where the prior-to-posterior update is significant. However, the theoretical properties of these reduced-order forward models and their impact on the solution of the Baysian inverse problem are not always well-understood. In this work we consider linear Gaussian inverse problems with a possibly singular prior covariance matrix. We analyse a recently proposed reduced-order model which uses a Petrov-Galerkin projection to likelihood-informed subspaces that arise in optimal low-rank approximations of the posterior covariance matrix. We bound the error in the resulting approximation of the root prior-preconditioned Hessian of the data misfit. Based on this we also bound the errors of the approximate posterior covariance and mean. Our analysis shows that this reduced-order model recovers the exact posterior when the rank of the reduced-order model is equal to the "intrinsic dimension" of the inverse problem, i.e. the rank of the prior-preconditioned Hessian. Two numerical experiments from structural engineering illustrate the performance of our bounds.

Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle $θ$, requiring $O(\log 1/δ)$ T gates. We show that it is possible to do much better for small angles, reducing the T cost to $\tilde{O}(θ^2/δ)$, and returning to existing $O(\log1/δ)$ results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new $θ$-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of $θ$, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources.

Let $B(Λ_a) \, (a=0,1)$ be the crystal of the level 1 integrable irreducible highest weight representation of the affine quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. We consider the crystal graphs of degree $n$ associated with the irreducible $(2r+1)$-dimensional (resp. $(2r+2)$-dimensional) $U_q(\mathfrak{sl}_2)$ module in $B(Λ_0)$ (resp. $B(Λ_1)$). In this paper, we construct an explicit combinatorial procedure providing a bijection between the set of highest weight paths in these graphs with respect to the action of the Kashiwara operator $\tilde{f}_{1}$, and the set of integer partitions of $n$ with sqrank (resp. rerank) $r$, which is a recently introduced partition statistic. As a byproduct, we also obtain a precise interpretation of the motif description of spinons suggested by Bernard-Pasquier-Serban in the spinon picture for Wess-Zumino-Witten conformal field theory models.

We construct a new family of $K_s$-free graphs that leads to improved lower bounds for Ramsey numbers across a wide range of parameters. For any fixed $s \ge 4$, we show that the off-diagonal Ramsey numbers satisfy $r(s, k) \ge k^{s-2 + o(1)}.$ For $s \ge 6,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen logarithmic improvements. This nearly matches the best known upper bound which is of the form $r(s, k) \le k^{s-1 + o(1)}$ and which is widely believed to give the correct exponent. More generally, we show that if $s, k/s \rightarrow \infty$, then $r(s, k) = \left(\frac{k}{s}\right)^{(1+o(1)) s},$ where the upper follows from the seminal work of Erdős and Szekeres in 1935. We also obtain improved lower bounds for Ramsey numbers extremely close to the diagonal as well as for diagonal multicolor Ramsey numbers.

Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

Engineered infrastructure systems pose inverse problems in which hidden states, unknown parameters, and subsystem couplings must be inferred from sparse and noisy measurements. These problems are difficult because physical subsystems are heterogeneous, sensing is partial, uncertainty is distributed across subsystem interfaces, and computational cost grows rapidly with system size. We address this challenge with probabilistic compositional inference, a graph-based architecture that represents a coupled system as interacting subsystems, each retaining its own local model, estimator, and uncertainty representation, while coupling is handled through physically meaningful stochastic messages exchanged across subsystem interfaces. This formulation allows mechanistic, learned, and deterministic components to coexist within a single inference framework and propagates calibrated uncertainty without assembling a global augmented state or covariance. We validate the framework in three increasingly demanding settings: a sparse-sensing canonical inverse problem, where interface couplings can also be learned from data; infrastructure-scale power networks, where the method matches centralized joint state-and-parameter inference while reducing computational scaling from approximately cubic to approximately linear; and a multi-physics turbine embedded in a power-grid network, where heterogeneous subsystems compose hierarchically without degrading local inference or collapsing local posteriors into a global estimate. Together, these results show that subsystem structure can be exploited as the organizing principle for uncertainty-aware inverse inference in coupled engineered systems.

Given the canonical seesaw mechanism as a most natural extension of the standard model in its neutrino sector, we find out a special but brand new solution to the exact seesaw equation: $m^{}_i/M^{}_i = - R^2_{αi}/U^2_{αi}$ for the masses and flavor mixing matrix elements of light and heavy Majorana neutrinos of the $i$-th family (for $i = 1, 2, 3$ and $α= e, μ, τ$). This family-separated seesaw scenario allows us to establish simple relations between the original seesaw parameters and the active degrees of freedom, and thus offers a number of testable predictions in neutrino phenomenology.

The human brain represents objects in a way that is both invariant across instances and flexible enough to support different contexts and tasks. Yet it remains unknown how object representations are dynamically remapped as the same object shifts across contextual roles. Using fMRI during naturalistic movie viewing we investigated how the same objects are represented when they are passive scene elements versus targets of goal-directed actions. Action targets engaged a parietal action network centered in the supramarginal and postcentral gyri, while passive objects recruited a distributed occipito-temporal network involved in visual object recognition. Within context-selective networks, representational geometry showed a double dissociation: target objects were organized by action affordance and hand posture affordance dimensions, while passive objects aligned with semantic dimensions. Visual representational structure was invariant to context. Outside these networks, representational content retained invariance, indicating that flexibility and invariance operate at different levels of the same representational system. These findings demonstrate neural remapping of object representations depending on moment-to-moment changes in contextual roles during a naturalistic scene.

We show that every stable UCT Kirchberg algebra has a principal étale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our methods. In particular, we obtain a principal étale groupoid model for the Cuntz algebra $\mathcal{O}_\infty$.

Simulations of chaotic systems can only produce high-fidelity trajectories if the initial and boundary conditions are well specified. When these conditions are unknown but measurements are available, variational state estimation can reconstruct a trajectory that is consistent with both the data and the governing equations. A key open question is how many measurements are required for accurate reconstruction, making the full system trajectory observable from sparse data. We establish observability criteria for variational state estimation applied to the Kuramoto-Sivashinsky equation by linking its observability to embedding theory for dissipative dynamical systems. For a system whose attractor lies on an inertial manifold of dimension $d_M$, we show that $m \geq d_M$ measurements ensures local observability from an arbitrarily good initial guess, and $m \geq 2d_M + 1$ implies global observability using a gradient-based smoother since the only critical point on $M$ is the global minimum. We also analyze optimization-dynamic limitations that persist even when these topological conditions are met, including drift off the manifold, degeneracy of the Hessian, negative curvature, and vanishing gradients. To address these issues, we introduce a robust reconstruction strategy that combines non-convex Newton updates with a novel pseudo-projection step. Numerical simulations of the Kuramoto-Sivashinsky equation validate our analysis and show practical limits of observability for chaotic systems with low-dimensional inertial manifolds.

Cross-market factor research studies whether firm-level signals from one or more markets can predict returns in a target market, but existing public benchmarks do not support cross-market disclosure-to-return evaluation. Building such a benchmark is challenging because filings differ across languages and regulatory systems, disclosure-derived similarity can be biased by common reporting components, and cross-market signals must be evaluated under feasible trading-time alignment. We introduce \textbf{CrossAlpha}, a public annual-report benchmark for cross-market factor research. CrossAlpha addresses these challenges through three corresponding components: \emph{Disclosure Distillation}, which standardises heterogeneous filings into ten-category English business descriptions; \emph{Residual Schema Graph Construction}, which builds PCA-whitened cross-market firm-pair scores from schema-level disclosures; and \emph{Timing-Aligned Evaluation}, which pairs the graph with 11 years of daily OHLCV data to construct forward-return labels under feasible cross-market execution protocols. CrossAlpha covers about 3,600 firms and 10,700 firm-year reports from the United States, Japan, Taiwan, South Korea, and Hong Kong, and releases about 19M directed firm-pair scores. In experiments, disclosure-derived cross-market peers outperform domestic text, industry-code, and return-correlation peers in the US-to-Japan setting (ICIR 0.39 versus 0.07--0.18), and cross-market sources beat the domestic text baseline in most target markets. CrossAlpha offers an open-sourced, reusable, return-grounded benchmark for cross-market financial NLP.

This paper explores the sensitivity gains afforded by spin-squeezed states in atom interferometry, in particular using Bragg diffraction. We introduce a generalised input-output formalism that accurately describes realistic, non-unitary interferometers, including losses due to velocity selectivity and scattering into undesired momentum states. This formalism is applied to evaluate the performance of one-axis twisted spin-squeezed states in improving phase sensitivity. Our results show that by carefully optimising the parameters of the Bragg beam splitters and controlling the degree of squeezing, it is possible to improve the sensitivity of the interferometer by several dB with respect to the standard quantum limit despite realistic levels of losses in light pulse operations. However, the analysis also highlights the challenges associated with achieving these improvements in practice, most notably the impact of finite temperature on the benefits of entanglement. The results suggest ways of optimising interferometric setups to exploit quantum entanglement under realistic conditions, thereby contributing to advances in precision metrology with atom interferometers.

In sequential CTR prediction, a central design question is at what granularity the target should interact with the user behaviour sequence. Existing models mainly follow two routes. Raw-item architectures such as DIN let the target score each item in the sequence directly. This relies on well-trained item embeddings and becomes brittle for sparse items. Latent-query architectures such as HyFormer, MixFormer, and OneTrans build query representations by combining the target with other information. This is more robust across item-density regimes but blunter: target-specific control is diluted. We propose LENS to restore target-specific control within these coarser bottlenecks. LENS has two modules: a Target-Conditioned Query Gate (TCQG) for query activation and a Target-Conditioned Position Bias (TCPB) for history retrieval. We further introduce Query-Specific Position Bias (QueryPos), a simple static position-aware reference for latent-query backbones. Across three representative latent-query backbones and four datasets, the combined QueryPos+LENS design achieves positive total-gain point estimates in all twelve evaluated backbone--dataset cells. We also identify a density-dependent conditioning rule: as item density decreases, the optimal condition source shifts from item-only to item-plus-sequence.

Let X be an affine variety. The local finiteness of a Lie subalgebra h of Lie(Aut(X)) is equivalent to the existence of an algebraic subgroup G of Aut(X) such that h is contained in Lie(G). Let h be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether h is itself locally finite. After presenting some criteria for the local finiteness of h, we answer this question in the affirmative in the particular case where X is the affine plane.

The geometrically defined wide class of time-like surfaces in $\mathbb R^3$, admitting real asymptotic lines is considered. A fundamental theorem of Bonnet-type is obtained for these surfaces. It states that a surface in this class is determined (up to a motion) by four invariant functions, satisfying some natural PDEs. Then canonical parameters are defined for these surfaces and it is proved that such a surface is determined (up to a motion) in canonical parameters with only two invariant functions (which in particular can be the Gauss and the mean curvature), satisfying a partial differential equation, equivalent to the Gauss equation.

This paper develops a unified explicit solution theory for optimal execution through sequential limit-order placement in a limit order book. Rather than controlling only the trading speed of a metaorder, we determine how individual limit orders should be quoted over time. The model incorporates signal-dependent drift, price impact, inventory risk, and execution risk, with fills modeled by point processes whose intensities depend on the submitted quotes. We formulate four execution criteria: expected terminal wealth, expected terminal wealth with running inventory penalty, CARA utility of terminal wealth, and CARA utility with running inventory penalty. For general price-impact and inventory-penalty functions, we derive the corresponding HJB equations and show that all four problems reduce to a triangular finite-dimensional structure which can be solved explicitly, leading to fully explicit value functions and optimal quotes across all cases. We also prove well-posedness, admissibility, and verification results. The explicit formulas reveal connections between quoting strategies under different criteria, support long-horizon asymptotic analysis, and show numerically that signal-dependent drift can substantially affect optimal execution.

We investigate whether the Lagrangian density for an interacting vector field or an interacting massless spin-2 field can be determined by imposing Poincaré invariance and the conservation of energy-momentum for the entire system. We adopt the Belinfante-Rosenfeld energy-momentum tensor for systems involving either a vector field or a spin-2 field. For the vector field coupled to a system of point masses, it is not possible to determine the Lagrangian density of the vector field. On the other hand, we show that for the spin-2 field coupled to a material system such as a system of point particles, its Lagrangian density is uniquely given by the Einstein Lagrangian density. Furthermore, the Belinfante-Rosenfeld tensor for the spin-2 field becomes Papapetrou's gravitational energy-momentum pseudotensor.

We study risk-neutral density extraction from short-dated option chains. As expiry approaches, option premia decline and bid--ask spreads can be large relative to prices, making mid quotes particularly uninformative. Stale or asynchronous quotes may also generate potential static arbitrages, rendering standard procedures infeasible or unstable. We develop a model-free pipeline that treats bid-ask quotes as the primitive market constraint. The pipeline consists of two steps. First, a procedure called ``Arbitrage Removal Iterative Executable Strategy'' (ARIES) filters executable static arbitrage at quoted bid and ask prices under market-depth constraints. Second, the ``Smooth Entropic Density EXtraction'' (SEDEx) then recovers the density through a criterion leveraging smoothness and entropy under bid-ask constraints. We test the pipeline on synthetic Heston panels and short-dated SPX option data, sampled from a few hours to one week before expiry. Computation is fast and returns robust densities across various market conditions, including scheduled macroeconomic announcements. As an empirical application, we use the recovered densities to construct short dated implied-volatility smiles.

Network configurations are prone to errors, which can lead to catastrophic service outages. A tool that can achieve automatic configuration repair (ACR) is highly desired by operators. Existing tools for ACR follow a \textit{semantics-driven approach}: they model network semantics as a set of SMT constraints, and solve them for a location or fix of the error. Due to the complex semantics of networks, constructing and solving these constraints can be prohibitively expensive, making these tools neither general nor scalable. Inspired by automatic program repair (APR), we explore another direction, i.e., a \textit{syntax-driven approach}, which generates and validates syntactically-valid candidate updates without modeling program semantics, often drawing on existing code in the same repository. Following this direction, we propose Astragalus, a syntax-driven method for ACR. It uses multiple iterations of a "localize-fix-validate" pipeline to search for repairs, and proves quite effective on configurations of our production network. Specifically, we show that Astragalus can repair every incident in multiple sizes of a synthesized network, and 97.5% of the incidents on a real network, both with 15 types of errors injected, within an average time of 6.93 seconds. It has also provided valid repairs in under 6 minutes for 7 recent network incidents or undesired changes, in a real production network with O(1,000)~O(10,000) devices.

We present a Renormalization Group (RG) framework based on a nonlocal effective action ansatz to analyze the strong coupling dynamics of the three-dimensional relativistic $ϕ^{4}$ theory. By implementing a Hubbard-Stratonovich transformation, we decouple the quartic interaction into the primary field $ϕ$ and an auxiliary field $φ\sim ϕ^2$, allowing both exponents $Δ_ϕ$ and $Δ_φ$ to act as independent, unconstrained variables rather than fixed scaling dimensions. Within this nonlocal propagator framework, both the field self-energies and vertex corrections are evaluated at the one-loop order. The resulting one-loop logarithmic derivatives determine the renormalization group flows of the couplings and the scaling exponents. For $d = 3$ and $ε= 0.5$, the self-consistent equations yield a physical fixed point at $Δ_ϕ \approx 0.81479$ and $Δ_{ϕ^2} \approx 1.37042$. These exponents result in a kinematic anomalous dimension $η_ϕ \approx 0.37042$, an energy operator dimension $Δ_{ϕ^2} \approx 1.37042$, and a thermal correlation length exponent $ν\approx 0.61366$. For $ε\approx-0.28996$, the self-consistent equations yield another fixed point at $Δ_ϕ\approx0.87284$, $Δ_φ\approx-0.53564$, and $Δ_{ϕ^2}\approx1.25432$, corresponding to $η_ϕ\approx0.25432$ and $ν\approx0.57284$. Although these leading-order results show deviations from high-precision Quantum Monte Carlo (QMC) and conformal bootstrap benchmarks, they characterize the baseline scaling behavior of the Wilson-Fisher universality class.

Barycentric algebras are an abstraction of the notion of convex sets, defined by a set of equations. We study semitopological and topological barycentric algebras, in the spirit of a previous study by Klaus Keimel on semitopological and topological cones (2008), which are special cases of semitopological and topological barycentric algebras. For example, the space of all continuous valuations (a very close cousin of measures) over a topological space is a topological cone, while probability valuations form a topological barycentric algebra, and subprobability valuations form a pointed topological barycentric algebra. Among other results, we show the existence of free semitopological cones over semitopological barycentric algebras and over pointed semitopological algebras, we investigate which semitopological barycentric algebras embed into semitopological cones and which pointed semitopological barycentric algebras embed strictly into semitopological cones. We study notions of local convexity, which split into weak local convexity, local convexity, local affineness and local linearity. We show that the weakly locally convex topological barycentric algebras are exactly the affine retracts of locally affine topological barycentric algebras. On locally convex barycentric algebras, we show sandwich theorems, extending theorems by Roth and Keimel on cones. A running theme of this paper is the notion of barycenters, which we progressively generalize until we reach a general notion of barycenters of continuous (resp., subprobability, probability) valuations, inspired by a definition of Choquet. We conclude with a general barycenter existence theorem, whose proof relies on the study of the Smyth poweralgebra, namely the topological barycentric algebra of all non-empty convex compact saturated subsets of a topological barycentric algebra.

We demonstrate that novel limits on prompt axion-like particles (ALPs) in the hard-to-probe mass range near the neutral pion - the so-called pion chimney - may be obtained from recasting $K_L \to 3π^0 \to 6γ$ data taken by the J-PARC KOTO experiment, to search for $K_L \to 2π^0a \to 6γ$. We also explore the power of KOTO $6γ$ data to probe $K_L \to 2π^0a$ for a broader range of ALP masses, incorporating displaced decays.

Recent advances in spoken dialogue language models have shifted from turn-based to full-duplex designs, where the model continuously listens to the user while generating responses. However, existing duplex backbones still lack a native channel for in-conversation planning and tool calling, leaving real-time agentic behaviour either tied to turn boundaries or relegated to an external cascade. We propose DuplexSLA, a native full-duplex Speech-Language-Action foundation model that decodes assistant audio together with a structured action stream on a shared 160 ms chunk timeline. DuplexSLA is built on a dual-stream three-channel formulation: a continuous user audio channel, a discrete assistant audio channel, and a rate-limited textual action channel, all decoded jointly by a single backbone, so that listening, speaking, planning, and tool calling unfold on one shared clock. Two capabilities define the model: (1) semantic-driven turn-taking control, where interruption, pause, and backchannel are handled inside the same backbone instead of by an external semantic VAD; and (2) in-conversation planning and tool calling, where planning text and structured tool calls are emitted on the action channel without halting assistant audio, so that multi-action and backchannel-triggered tool use are interleaved with ongoing speech. To evaluate these capabilities together, we further construct DuplexSLA-Bench, a duplex benchmark covering pause, interrupt, and backchannel turn-taking together with three styles of in-conversation tool calling. Our project page, interactive demos, and the DuplexSLA-Bench evaluation suite are publicly available at https://github.com/hyzhang24/DuplexSLA.

Transition-metal halides V$X_3$ ($X$ = Br and I) have emerged as promising candidates for two dimensional spintronic and quantum applications due to their layer-dependent magnetism and tunable electronic states. However, experimental insights into their ground state electronic structures remain limited. Here, we present a comprehensive investigation of V$X_3$ using high resolution resonant inelastic x-ray scattering (RIXS) combined with ligand field multiplet calculations. The RIXS spectra reveal distinct $dd$ and charge-transfer excitations, allowing precise determination of electronic structure parameters, including the crystal field splitting, trigonal distortion, and Racah parameters. The determined parameters showed clear variation, indicating an increase in covalency from Br to I. The trigonal distortion parameters $Δ_{D_{3d}}$ were determined to be -0.096 eV in VBr$_3$ and 0.07 eV in VI$_3$, indicating a sign opposition between the two compounds, reflecting good agreement with experimental RIXS data. Cluster model calculations yield a high-spin V$^{3+}$ $(S = 1)$ configuration, with an $e'^2_g$ ground state in VBr$_3$ and an $e'^1_ga^1_{1g}$ ground state in VI$_3$, consistent with trigonal elongation and compression, respectively. Our findings provide the most complete experimental determination of the low energy electronic structure in V$X_3$, offering valuable insights for designing 2D magnetic and spintronic materials based on vanadium halides.

In this paper, we present a novel numerical framework for solving a specific biological reaction-diffusion-advection system of cancer growth in three dimensions (3D) using particles of variable mass. We adopt empirical particle measures to represent cell density and dynamically construct the concentration fields of multiple related chemical species throughout the 3D domain. Efficient interaction between the particles and the spatial grid is achieved through a Particle-in-Cell (PIC) algorithm, while diffusion in space is solved rapidly using a spectral method. We demonstrate that for this particular system, the rate of change of particle mass remains bounded over finite time intervals. Furthermore, in addition to the inherent positivity preservation of cell density guaranteed by the empirical particle measures, the concentrations constructed by the algorithm are also unconditionally positivity-preserving on the spatial grid. Moreover, we present a rigorous error analysis for the proposed method, and numerical experiments confirm the theoretical convergence rates. To the best of our knowledge, this is the first numerical work to solve this system in three dimensions, wherein a rapid spread of cells driven by haptotactic flux is observed, similar to the behavior documented in the two-dimensional case.

Polarimetric interferometry is a powerful tool for probing both black hole accretion physics and the background spacetime. Current models aimed at explaining the observed multiwavelength flares in Sgr A* often assume hotspots moving on geodesic, Keplerian orbits. In many scenarios, though, a hotspot may instead follow an inspiraling trajectory, potentially transitioning into a plunge toward the black hole. In this work, we present a general framework to simulate the polarized emission from generic equatorial inspiraling hotspots in Kerr spacetime using a parametric four-velocity profile. This parametrization defines a continuous family of flows, ranging from Cunningham's disk model (fixed radius orbits outside the innermost stable circular orbit and plunging motion within the innermost stable circular orbit) to purely radial motion, thereby extending the standard assumptions. Within this framework, we show that inspiral motion produces a distinctive observational signature: a precessing, unwinding evolution of the polarimetric Stokes Q-U looping pattern, in sharp contrast with the closed Q-U loops associated with stable orbits at a fixed radius. We then explore how the morphology of these signatures depends on black hole spin, observer inclination, and magnetic-field configuration. The presented model can be applied to current and near-future interferometric observations of linear polarization, offering a new avenue to probe the physics of matter spiraling inward and the relativistic velocities of plunging plasma.

Causal mediation analysis decomposes the total treatment effect into a portion operating through a hypothesized mediator and a residual direct portion. Identification of natural direct and indirect effects typically rests on the mediator stage of sequential ignorability, which cannot be empirically verified and requires explicit sensitivity analysis. We formulate the \emph{bridge score}, a mediator-stage balancing score, as a low-dimensional vector formed from the two treatment-specific mediator densities at a common mediator value, and show that it balances baseline covariates for the mediator stage relevant to natural effect identification. Conditional on the bridge score, we derive a sharp pointwise variance envelope on the unidentified mediator-outcome confounding function in terms of latent outcome relevance and residual selection. To make the bound operational for sensitivity analysis, we further introduce a residual budget calibration approach based on local residual outcome variation and record a complementary range bound for support-based restrictions. Finally, we show how the pointwise bound can be operationalized for inference through a scalar functional reduction and a Bayesian g-computation algorithm that combines observed-data posterior uncertainty with user-specified sensitivity uncertainty, rather than treating the unidentified sensitivity corrections as learned from the likelihood.