Phantom divide line transitions are not possible in the context of single scalar field scalar-tensor theories. In this article we study a combined framework of a tachyonic minimally coupled single scalar field theory in the presence of an $\mathcal{R}^2$ correction term and with a rescaled Einstein-Hilbert term of the form $\sim λ\frac{\mathcal{R}}{16πG}$. Such terms can be part of an $f(\mathcal{R})$ gravity which in the large curvature regime yields such correction terms effectively. Alternatively, such terms can simply be quantum corrections to the scalar field action. We aim to answer two questions, firstly if this framework can lead to phantom divide line transitions and secondly whether the resulting model can be compatible with the ACT data. The model we studied is an inverse square power-law model, well known from tachyon inflation models. As we show, the field equations can be cast in terms of the scalar field solely, however the resulting theory is distinct from a single scalar field theory, because the phantom divide line is crossed during inflation. Thus initially the tachyonic nature of the scalar field generates a phantom equation of state parameter, and during inflation the phantom divide line is crossed, with the effective equation of state parameter at the end of inflation being $w=-1/3$ which corresponds to the non-accelerating state of the Universe. The model is proved to be compatible with the ACT data, only when the gravity during inflation is stronger than Einstein-Hilbert gravity, with the effective gravitational constant during inflation being $\frac{G}λ$. The effective theory is valid only during inflation, thus Big-Bang nucleosynthesis is not affected by the rescaling of the Einstein-Hilbert gravity. The feature of a phantom crossing in $f(\mathcal{R},ϕ)$ frameworks is new in the literature.

This work describes the formalism for estimating thermonuclear reaction rates for astrophysical applications, emphasizing modern statistical approaches such as Monte-Carlo sampling and Bayesian models. We discuss related topics including the calculation of resonance energies from nuclear Q values, indirect estimates of particle partial widths, and matching of reaction rates at elevated temperatures to statistical-model results. We have evaluated available experimental data on cross sections, resonance energies and strengths, partial widths, life-times, spin-parities, and spectroscopic factors. Based on these results, we have estimated numerical values of 78 experimental charged-particle thermonuclear reaction rates for target nuclei in the A = 2 to 40 mass region, for temperatures ranging from 1 MK to 10 GK. For each reaction, three rate values are provided: low, median, and high, corresponding to the 16th, 50th, and 84th percentiles, respectively, of the cumulative reaction rate probability density distribution. Additionally, we present the factor uncertainty of each rate at each temperature grid point. These results enable users to sample the reaction rate probability density in nucleosynthesis calculations, facilitating uncertainty estimates of nuclidic abundances. The rates presented here refer to their laboratory values. For use in stellar model simulations, these values need to be corrected for the effects of thermal excitations of the interacting nuclei. For each reaction, we include graphs that illustrate the fractional contributions to the overall reaction rate along with the associated uncertainty. These visuals are designed to assist both stellar modelers and nuclear experimentalists by identifying the primary sources of rate uncertainty at specific stellar temperatures. A graphical comparison with earlier Monte-Carlo rates is also provided.

The integration of Large Language Models (LLMs) into wireless networks presents significant potential for automating system design. However, unlike conventional throughput maximization, Covert Communication (CC) requires optimizing transmission utility under strict detection-theoretic constraints, such as Kullback-Leibler divergence limits. Existing benchmarks primarily focus on general reasoning or standard communication tasks and do not adequately evaluate the ability of LLMs to satisfy these rigorous security constraints. To address this limitation, we introduce CovertComBench, a unified benchmark designed to assess LLM capabilities across the CC pipeline, encompassing conceptual understanding (MCQs), optimization derivation (ODQs), and code generation (CGQs). Furthermore, we analyze the reliability of automated scoring within a detection-theoretic ``LLM-as-Judge'' framework. Extensive evaluations across state-of-the-art models reveal a significant performance discrepancy. While LLMs achieve high accuracy in conceptual identification (81%) and code implementation (83%), their performance in the higher-order mathematical derivations necessary for security guarantees ranges between 18% and 55%. This limitation indicates that current LLMs serve better as implementation assistants rather than autonomous solvers for security-constrained optimization. These findings suggest that future research should focus on external tool augmentation to build trustworthy wireless AI systems.

Let $\mathbf{G}=\{G_1,\dots,G_{s}\}$ be a collection of $s$ bipartite graphs with the same bipartition $V=(X,Y)$. For a path $P$ with $V(P)=V$ and $|E(P)|=s$, if there exists an injection $ϕ$: $E(P)\rightarrow [s]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$, then we say that the Hamiltonian path $P$ is a $\mathbf{G}$-transversal. A bipartite graph collection $\mathbf{G}$ is called Hamiltonian connected if for any two vertices $x\in X$ and $y\in Y$, there exists a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path between $x$ and $y$. In this paper, we give the minimum degree conditions that ensure the existence of a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path and the Hamiltonian connectivity of a balanced bipartite graph collection $\mathbf{G}$, which improve the results of [Hu, Li, Li and Xu, Discrete Math., 2024]. Moreover, we also provide a minimum degree condition that guarantees a nearly balanced bipartite graph collection $\mathbf{G}$ contains a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path.

The spatial and kinematic structure of the circumgalactic medium (CGM) remains poorly constrained observationally. In this article we compute the clustering of CIV absorption systems at cosmic noon using quasar pairs. We analyze VLT/UVES and Keck/HIRES high-resolution spectra (R = 45000) of a sample of 8 projected and 4 lensed quasar pairs that probe transverse separations, $Δr$, from sub-kpc to a few Mpc, over the redshift range 1.6 < z < 3.3. We detect and fit Voigt profiles to a total of 141 CIV systems, corresponding to 620 velocity components across all quasar lines-of-sight. We compute the two-point correlation function of CIV, $ξ(Δv, Δr)$, where $Δv$ is the velocity difference between components across all available scales. We find a strong dependence of $ξ(Δr)$ with $Δr$ at all velocities. $ξ(Δr)$ reaches a sharp peak at the smallest scales analyzed here, $Δr\approx 0.1$ kpc, decreases steadily up to $Δr\approx 5$ kpc and remains flat up to $Δr\approx 500$ kpc, where it begins to decrease again. By fitting power-laws to the projected transverse correlation function $Ξ(Δr)$, we infer two coherence lengths: $r_1 = 654^{+100}_{-87}$ kpc, which we interpret as a representative size for the CIV enriched regions at $z\approx 2$, and $r_2 = 4.70^{+1.60}_{-1.19}$ kpc for the individual CIV-bearing "clouds". Projecting instead in $Δr$, we find consistent amplitudes of $ξ(Δv)$ with previous work using quasars and extended background sources. Our results suggest that CIV may be a good tracer of not only the small, internal structure of the circumgalactic medium, but also of the way in which galaxies cluster at cosmic noon.

Recent dielectric and magnetic studies of (Ca0.5Mn1.5)MnWO6 ceramics [A.A. Belik, Chem. Mater. 36, 7604 (2024)] have classified this material as a rare hybrid multiferroic, with both antiferromagnetic and (anti)ferroelectric ordering occurring at the same temperature of 22 K. The pronounced dielectric anomaly observed at this temperature indicated that the structural change is primarily induced by a phonon soft mode and not by a spin arrangement, as is usually the case in type II multiferroics. However, our comprehensive investigation involving new ceramic samples as well as the sample from the above-mentioned reference does not support this conclusion. Low-temperature polarization measurements revealed no evidence of either ferroelectric or antiferroelectric order in both sample series. The dielectric permittivity exhibits only a slight change at the antiferromagnetic transition, and phonon modes observed in IR and Raman spectra show no indication of a symmetry change at low temperatures. In the new samples the Neel temperature is shifted to TN = 18 K. XRD, SEM, EDS and WDS analyses confirmed the composition (Ca0.5Mn1.5)MnWO6 of both ceramics, but also indicated a small amount (percentage points) of MnO and CaO impurities in the sample from the previous publication and Mn3O4, CaWO4 secondary phases (<4%) in the new ceramics. The differences in dielectric and magnetic properties of the two samples can therefore be explained by their different chemical purity. The small dielectric anomaly of the new sample at the antiferromagnetic transition temperature is explained by a spin-phonon coupling. We conclude that (Ca0.5Mn1.5)MnWO6 is not a multiferroic, but a paraelectric antiferromagnet.

Implementing privacy by design (PbD) according to the General Data Protection Regulation (GDPR) is met with a growing number of requirements engineering (RE) approaches. However, the question of which RE method for PbD fits best the goals of organisations remains a challenge. We report our endeavor to close this gap by synthesizing a goal-centric approach for PbD methods assessment. We used literature review, interviews, and validation with practitioners to achieve the goal of our study. As practitioners do not approach PbD systematically, we suggest that RE methods for PbD should be assessed against organisational goals, rather than process characteristics only. We hope that, when further developed, the goal-centric approach could support the development, selection, and tailoring of RE practices for PbD.

Statistical data depth plays an important role in the analysis of multivariate data sets. The main outcome is a center-outward ordering of the observations that can be used both to highlight features of the underlying distribution of the data and as input to further statistical analysis. An important property of data depth is related to symmetric distributions as the point with the highest depth value, the center, coincides with the point of symmetry. However, there are applications in which it is more natural to consider symmetry with respect to a subspace of a certain dimension rather than to a point, i.e. a subspace of dimension zero. We provide a general framework to construct statistical data depths which attain maximum value in a subspace, providing a center-outward ordering from that subspace. We refer to these data depths as central subspace data depths. Moreover, if the distribution is symmetric with respect to a subspace, then the depth is maximized at that subspace. We introduce general notions of symmetry about a subspace for distributions, study the properties of central subspace data depths and provide asymptotic convergence for the corresponding sample versions. Additionally, we discuss connections with projection pursuit and dimension reduction. An application based on custom data fraud detection shows the importance of the proposed approach and strengthens its potential.

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, the First Isomorphism Theorem for graded polyadic algebras and concrete examples including ternary superalgebras and polynomial algebras over $n$-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility.

Natural interfaces in ferroic oxides have developed into versatile playgrounds for studying electronic correlation effects in 2D systems. The microscopic origin of the emergent local electronic properties is often debated, however, as quantitative atomic-scale characterization remains challenging. A prime example is enhanced conductivity at ferroelectric domain walls, attributed to mechanisms ranging from local band gap reduction to point defect accumulations. Here, we resolve the microscopic mechanisms for domain wall conduction in the ferroelectric model system BiFeO3, by combining transport measurements with atom probe tomography to quantify the local chemical composition and correlate it with the electrical properties. Significant chemical variations along the walls are observed, demonstrating an outstanding chemical flexibility at domain walls, which manifest in spatially varying physical properties. The results give a unifying explanation for the diverse electronic behavior observed and establish the fundamental notion that multiple conduction mechanisms can coexist within individual domain walls.

In this paper we extend the notion of $φ$-mixing to set-valued random sequences that take values in the family of closed subsets of a Banach space. Several strong laws of large numbers for such $φ$-mixing sequences are stated and proved. Illustrative examples show that the hypotheses of the theorems are both natural and sharp.

Gamma-ray bursts (GRBs) offer a powerful probe of the cosmic expansion history far beyond the redshift range accessible to Type Ia supernovae. However, the study of cosmological models using GRBs is hindered by the circularity problem, which arises from assuming a fiducial cosmological model during GRB luminosity distance calibration. In this work, we perform a model-independent calibration of GRB luminosity relations using observational measurements of the Hubble parameter from the A220 and J220 compilations, thereby avoiding explicit cosmological assumptions. We employ an Artificial Neural Network to reconstruct the calibration relation directly from the data. In addition, we implement a Bayesian Neural Network framework as an alternative approach, enabling a data-driven treatment of both statistical and systematic uncertainties. The calibrated GRB sample is used to constrain the Amati relation, and we systematically compare the outcomes obtained from different calibration techniques and datasets. We find that the Amati relation slopes derived from the two neural network approaches are consistent with each other and with previous low-redshift calibrations obtained using model-independent methods. The Bayesian Neural Network approach provides a more robust framework for propagating uncertainties in the calibration procedure.

For any continuous self-map of a compact metric space, we provide sufficient conditions under which the infinite direct product of the map is $ω$-chaotic. We also apply the result to obtain some examples of unusual $ω$-chaotic maps.

We propose generalized Fermat's conjecture in the framework of arithmetic dynamics, and give evidences. The multi-indexed version is added.

We study the mass spectra and various mass differences of heavy hadrons containing one or more bottom quarks using MILC's $N_f = 2+1+1$ HISQ gauge ensembles at three lattice spacings. For the valence quarks, we employ a combination of lattice actions: the NRQCD action is used for bottom quarks, the anisotropic Clover action for charm quarks, and the $O(a)$-improved Wilson--Clover action for strange and lighter (up/down) quarks. Heavy hadron operators with at least one bottom quark are constructed by considering all possible combinations with charm, strange, and light quarks corresponding to various quantum numbers.

In 6G mobile communications, acquiring accurate and timely channel state information (CSI) becomes increasingly challenging due to the growing antenna array size and bandwidth. To alleviate the CSI feedback burden, the channel knowledge map (CKM) has emerged as a promising approach by leveraging environment-aware techniques to predict CSI based solely on user locations. However, how to effectively construct a CKM remains an open issue. In this paper, we propose F$^4$-CKM, a novel CKM construction framework characterized by four distinctive features: radiance Field rendering, spatial-Frequency-awareness, location-Free usage, and Fast learning. Central to our design is the adaptation of radiance field rendering techniques from computer vision to the radio frequency (RF) domain, enabled by a novel Wireless Radiator Representation (WiRARE) network that captures the spatial-frequency characteristics of wireless channels. Additionally, a novel shaping filter module and an angular sampling strategy are introduced to facilitate CKM construction. Extensive experiments demonstrate that F$^4$-CKM significantly outperforms existing baselines in terms of wireless channel prediction accuracy and efficiency.

Soft factorization theorems can be reinterpreted as Ward identities for (asymptotic) symmetries of scattering amplitudes in asymptotically flat space-time. In this paper we study the symmetries implied by the all loop soft photon theorems when all the charged particles are highly energetic and the relation $ω<< m << E$ holds where $E$ is the typical energy of a charged particle, $m$ is the typical mass and $ω$ is the soft photon energy. Loop level soft theorems are qualitatively different from the tree level soft theorems because loop level soft factors contain multi-particle sums. If we want to interpret them as Ward identities or define celestial OPE between between soft and hard operators then we need to introduce additional fields which live on the celestial sphere but do not appear as asymptotic states in any scattering experiment. For example, if we want to interpret the one-loop exact $\mathcal{O}(\lnω)$ soft theorem for a positive helicity soft photon (with energy $ω$) as a Ward identity then we need to introduce a pair of antiholomorphic currents on the celestial sphere which transform as a doublet under the $SL(2,\mathbb{R})_{R}$. We call them dipole currents because the corresponding charges measure the monopole and the dipole moment of an electrically charged particle on the celestial sphere. More generally, the soft photon theorem at $\mathcal{O}(ω^{2j-1}(\lnω)^{2j})$ for every $j\in \frac{1}{2}\mathbb{Z}_+$ gives rise to $(2j+1)$ antiholomorphic currents which transform in the spin-$j$ representation of the $SL(2,\mathbb{R})_{R}$. These currents exist in the quantum theory because they follow from loop level soft theorems. We argue that under certain circumstances the (classical) algebra of the higher spin currents is the wedge subalgebra of the $w_{1+\infty}$.

It has long been thought that black hole accretion flows are driven by magnetohydrodynamic (MHD) turbulence, and there are now many general relativistic global simulations illustrating the dynamics of this process. However, many challenges must be overcome in order to predict observed spectra from luminous systems. Ensuring energy conservation, local thermal balance, and local ionization equilibrium, our post-processing method incorporates all the most relevant radiation mechanisms: relativistic Compton scattering, bremsstrahlung, and lines and edges for 30 elements and all their ions. Previous work with this method was restricted to black holes of $10 M_\odot$; here, for the first time, we extend it to $10^8 M_\odot$ and present results for two sub-Eddington accretion rates and black hole spin parameter 0.9. The spectral shape predicted for stellar-mass black holes matches the low-hard state for the lower accretion rate and the steep power law state for the higher accretion rate. For high black hole mass, both accretion rates yield power-law continua from $\sim 0.5 - 50$~keV whose X-ray slopes agree well with observations. For intermediate mass black holes, we find a soft X-ray excess created by inverse Compton scattering of low-energy photons produced in the thermal part of the disk; this mechanism may be relevant to the soft X-ray excess commonly seen in massive black holes. Thus, our results show that standard radiation physics applied to GRMHD simulation data can yield spectra reproducing a number of the observed properties of accreting black holes across the mass spectrum.

We investigate the Rice-Mele model with on-cell Kerr-type nonlinearities, where the interaction depends on the total particle number within each unit cell rather than on individual sites. This interaction enables a nontrivial interplay between topology and nonlinear dynamics in soliton pumping. In the weakly interacting regime, the ground-state soliton undergoes quantized Thouless pumping. At intermediate interaction strengths, soliton creation and annihilation break adiabaticity and disrupt quantized transport. In the strong-coupling regime, the coexistence of ground- and excited-state solitons leads to negligible coupling at energy crossings, giving rise to discrete time-translation symmetry breaking (DTTSB) in the soliton dynamics. Comparison of mean-field results with exact diagonalization along closed circular pumping paths confirms both the validity of the mean-field description and the robustness of DTTSB across different pumping trajectories. Our findings reveal how interaction-induced effects can fundamentally modify topological transport and suggest that these phenomena may be explored in cold-atom, photonic, and superconducting-circuit platforms.

We define a grid graph $G$ as a Cartesian product of path-graphs $P_n$ or cycle-graphs $C_n$ as shown in Figure 1, and we ask, when can the edge set of a complete graph be expressed as a disjoint union of graphs isomorphic to $G$? That is, we are asking for which grid graphs a $G$-design exists, where a $G$-design is defined as a decomposition of a complete graph into edge-disjoint subgraphs isomorphic to $G$. We show that when $n$ is an odd prime or the square of an odd prime, the toroidal grid-graph $G = C_n \square C_n$ admits a $G$-design. In the less symmetrical case of products of path-graphs, we prove that $G = P_3 \square P_3$ does not admit a $G$-design but that $G = P_4 \square P_4$ does. This last result is the special case that motivated the present paper: a $P_4 \square P_4$-design corresponds to a way of successively scrambling a Connections puzzle so that each pair of words occurs adjacently exactly once. Our constructions use the arithmetic of finite fields.

Vertical federated learning (VFL) enables multi-laboratory collaboration on distributed multi-omics datasets without sharing raw data, but exhibits severe instability under extreme data scarcity (P >> N) when applied generically. Here, we investigate how domain-aware design choices; specifically gradient saliency-guided feature selection with biologically motivated priors; affect the stability, interpretability, and failure modes of VFL architectures in small-sample coral stress classification (N = 13 samples, P = 90,579 features across transcriptomics, proteomics, metabolomics, and microbiome data). We benchmark REEF (Robust Expert Encoder Federation), a domain-aware VFL framework, against two baselines on the Montipora capitata thermal stress dataset: (i) a standard NVFlare-based VFL and (ii) LASER, a state-of-the-art label-aware VFL method. REEF achieves an AUROC of 0.776 +/- 0.039 after reducing dimensionality by 98.6% (90,579 to 1,300 features), substantially outperforming NVFlare VFL at chance level (AUROC 0.500 +/- 0.125, p = 0.0106, Cohen's d = 2.265) and numerically exceeding LASER (AUROC 0.557 +/- 0.191, p = 0.0995, Cohen's d = 1.068), with 3-5-fold variance reduction. An equal-weights ablation confirms that biological priors specifically contribute stability: removing priors yields statistically indistinguishable mean AUROC (p = 0.405) but 2.3x higher variance (CV 0.110 vs 0.050). Negative control experiments using permuted labels produce AUROC near or below chance (0.357 for REEF, 0.238 for NVFlare), consistent with the absence of gross data leakage. These results motivate design principles for VFL in extreme P >> N regimes, emphasizing domain-informed dimensionality reduction, stability-focused evaluation, and interpretable feature selection for scarce biological data.

We study the continuum limit of branched polymers (BPs) with loops coupled to Ising spins at the zero-temperature critical point. It is known that the continuum partition function can be represented by a Hermitian two-matrix model, and we propose a string field theory whose Dyson-Schwinger equation coincides with the loop equation of this continuum matrix model. By setting the matrix size to one, we analyze a convergent non-perturbative partition function expressed as a two-dimensional integral, and show that it satisfies a third-order linear differential equation. In contrast, in the absence of coupling to the critical Ising model, the continuum partition function of pure BPs with loops is known to satisfy the Airy equation. From the viewpoint of two-dimensional quantum gravity, we introduce a non-perturbative loop amplitude that serves as a solution to the Wheeler-DeWitt equation incorporating contributions from all genera. Furthermore, we demonstrate that the same Wheeler-DeWitt equation can also be derived through the stochastic quantization.

Fully automated verification of large-scale software and hardware systems is arguably the holy grail of formal methods. Large language models (LLMs) have recently demonstrated their potential for enhancing the degree of automation in formal verification by, e.g., generating formal specifications as essential to deductive verification, yet exhibit poor scalability due to long-context reasoning limitations and, more importantly, the difficulty of inferring complex, interprocedural specifications. This paper presents Preguss -- a modular, fine-grained framework for automating the generation and refinement of formal specifications. Preguss synergizes between static analysis and deductive verification by steering two components in a divide-and-conquer fashion: (i) potential runtime error-guided construction and prioritization of verification units, and (ii) LLM-aided synthesis of interprocedural specifications at the unit level. We show that Preguss substantially outperforms state-of-the-art LLM-based approaches and, in particular, it enables highly automated RTE-freeness verification for real-world programs with over a thousand LoC, with a reduction of 80.6%~88.9% human verification effort.

Rate-Splitting Multiple Access (RSMA) is a key enabling technique for sixth-generation (6G) wireless systems due to its powerful interference management, and Reconfigurable Intelligent Surface (RIS) improves communication performance by shaping wireless propagation. However, conventional RSMA--RIS architectures employ fixed antennas, limiting spatial degrees of freedom and system performance. To address this, we propose a movable-antenna (MA) assisted RSMA--RIS framework and formulate a sum-rate maximization problem that jointly optimizes the transmit beamforming matrix, RIS reflection matrix, common-rate partition, and MA positions. After yielding a closed-form solution for common rate splitting, the problem is transformed via fractional programming (FP). Using Karush--Kuhn--Tucker (KKT) conditions, we give iterative updates for Lagrange multipliers and beamforming matrix, obtain the RIS reflection matrix via the dual problem, and determine optimal antenna positions via gradient ascent. Numerical results show that with the existence of RIS, integrating MA yields additional gains of approximately 33.3\% for SDMA and 35.6\% for RSMA.

This paper shares some experience in advanced mathematical education. We show how a high school student can be naturally and gradually introduced to basic steps of scientific research: developing intuition by finding and correcting mistakes through discussions and writing a paper, (transparent) anonymous peer review, recognition and award. We show that most of this can be done in research projects not aiming at scientific novelty. We share the experience (both principles and examples) of the Moscow Mathematical Conference of High School Students.

We study the holographic construction of timelike entanglement entropy (tEE) in black hole backgrounds in Lorentzian geometries. The holographic tEE is realized through extremal surfaces consisting of spacelike and timelike branches that encode its real and imaginary components, respectively. In the BTZ black hole, these surfaces extend into the interior of the black hole and reproduce the field-theoretic results. The analysis is further generalized to higher-dimensional AdS-Schwarzschild black holes, where the characteristics of tEE are obtained with increasing size of the boundary subsystem. Besides, we also show that the boundary subsystem length diverges at a dimension-dependent critical turning point. Notably, this critical point moves closer to the black hole horizon as the dimensionality of the bulk increases. For large subsystem lengths, the finite part of the tEE displays a characteristic volume-plus-area structure, with a real volume term and a complex coefficient of the area term approaching constant values at large dimensions. Besides, we also study the monotonicity of a new quantity, timelike entanglement density, which offers insights into a timelike area theorem in specific limits. Subsequently, we investigate the near-horizon dynamics in various black hole backgrounds, where the spacelike and timelike surfaces exhibit exponential growth of the form $e^{\frac{2π}β Δt}$ with inverse black hole temperature $β$.

Lorentzian and completely log-concave polynomials have recently emerged as a unifying framework for negative dependence, log-concavity, and convexity in combinatorics and probability. We extend this theory to variational analysis and cone-constrained dynamics by studying $K$-Lorentzian and $K$-completely log-concave polynomials over a proper convex cone $K\subset\mathbb{R}^n$. For a $K$-Lorentzian form $f$ and $v\in\operatorname{int}K$, we define an open cone $K^\circ(f,v)$ and a closed cone $K(f,v)$ via directional derivatives along $v$, recovering the usual hyperbolicity cone when $f$ is hyperbolic. We prove that $K^\circ(f,v)$ is a proper cone and equals $\operatorname{int}K(f,v)$. If $f$ is $K(f,v)$-Lorentzian, then $K(f,v)$ is convex and maximal among convex cones on which $f$ is Lorentzian. Using the Rayleigh matrix $M_f(x)=\nabla f(x)\nabla f(x)^T - f(x)\nabla^2 f(x)$, we obtain cone-restricted Rayleigh inequalities and show that two-direction Rayleigh inequalities on $K$ are equivalent to an acuteness condition for the bilinear form $v^T M_f(x) w$. This yields a cone-restricted negative-dependence interpretation linking the curvature of $\log f$ to covariance properties of associated Gibbs measures. For determinantal generating polynomials, we identify the intersection of the hyperbolicity cone with the nonnegative orthant as the classical semipositive cone, and we extend this construction to general proper cones via $K$-semipositive cones. Finally, for linear evolution variational inequality (LEVI) systems, we show that if $q(x)=x^T A x$ is (strictly) $K$-Lorentzian, then $A$ is (strictly) $K$-copositive and yields Lyapunov (semi-)stability on $K$, giving new Lyapunov criteria for cone-constrained dynamics.

Circular photon orbits have become an attractive topic in recent years. They play extremely important roles in black hole shadows, gravitational lensings, quasi-normal modes, and spacetime topological properties. In our recent work, \href{https://doi.org/10.1103/PhysRevD.106.L021501}{Phys. Rev. D \textbf{106}, L021501 (2022)}, a geometric approach to circular photon orbits was proposed for spherically symmetric spacetimes. In the present study, we extend this geometric approach from spherically symmetric spacetimes to axially symmetric spacetimes. In this geometric approach, light rings in the equatorial plane are determined by the intrinsic curvatures in the optical geometry of Lorentz spacetime, which gives rise to a Randers-Finsler geometry in axially symmetric cases. Specifically, light rings can be precisely determined by the vanishing of geodesic curvature, and the stability of light rings is classified using the intrinsic flag curvature in Randers-Finsler optical geometry. This geometric approach presented in this work is generally applicable to any stationary and axially symmetric spacetime, without imposing any restriction on the spacetime metric forms. Furthermore, we provide a rigorous demonstration to show that our geometric approach yields results that are completely equivalent to those derived from the conventional approach (based on the effective potential of photons).

Gate-based quantum image processing is constrained by qubit scarcity and the high overhead of quantum state preparation, limiting its applicability to realistic geometric data. We introduce a quantum-native framework for image matching on neutral-atom analog quantum computers that advances our earlier Sparse-Dots Representation (SDR) approach. A classical pre-processing pipeline -- Sobel edge extraction followed by the Ramer--Douglas--Peucker (RDP) algorithm -- converts an input image into a geometrically faithful Sparse-Dots point cloud of substantially fewer atoms. This atom layout is virtually embedded into the programmable tweezer array of QuEra's Aquila device via its Bloqade SDK, where the image geometry is encoded physically in the distance-dependent van der Waals interaction term of the Rydberg Hamiltonian. After time-evolution, we extract the many-body fingerprint of each image using two observables -- the Pearson-normalized two-site correlation matrix which encodes the blockade-induced correlation structure of the quantum state, and the two-dimensional static structure factor evaluated on a fixed wavevector grid, yielding a fingerprint vector of constant length regardless of atom count. In Stage~1, image matching is performed by cosine similarity on the fingerprint vectors, a scale-invariant metric appropriate for Fourier-domain descriptors. In Stage~2, this approach is extended to quantum reservoir computing~(QRC) to enable machine learning via dramatically reduced training data and training cycles, as a preliminary proof-of-concept. Simulations using the Bloqade software stack confirm successful matching of industrial objects, often with fewer than 24 atoms. To our knowledge, this constitutes the first application of the static structure factor -- a condensed-matter quantum observable -- as an image retrieval descriptor in an analog quantum computing context.

We present a lattice-renormalized formalism for configurational tunneling two-level systems (TLS) that overcomes limitations of minimum-energy-path and light-particle models. Derived from the nuclear Hamiltonian, our formulation introduces composite phonon coordinates to capture lattice distortions between degenerate potential wells. This approach resolves deficiencies in prior models and enables accurate computation of tunnel splittings and excitation spectra for hydrogen-based TLS in bcc Nb. Our results bound experimental tunnel splittings and reveal strong anharmonic couplings between tunneling atoms and lattice phonons, establishing a direct link between TLS dynamics and phonon-mediated strain interactions. The formalism further generalizes to multi-level systems (MLS), providing insight into defect-induced decoherence in superconducting qubits and guiding strategies for materials design to suppress TLS-related loss.