We study generic $d$-dimensional rigidity in sparse random graphs. Our main result is that for every $d\ge 2$, the Erdős--Rényi random graph $G\sim G(n,c/n)$ undergoes a $d$-rigidity phase transition at the known, explicit, $d$-orientability threshold $c_d$: If $c<c_d$, then $G$ is asymptotically almost surely (a.a.s.) independent in the generic $d$-rigidity matroid. Moreover, in this regime $G$ has no linear-size rigidity components: it contains no induced $d$-rigid subgraphs with more than $3$ vertices, and the largest clique in its $d$-rigidity closure has size at most $o(\sqrt n)$. If $c>c_d$, then $G$ is a.a.s. not independent in the generic $d$-rigidity matroid, and we give a sharp asymptotic estimate for its rank. In addition, the $d$-rigidity closure of $G$ has a giant clique of linear size, which contains all but at most $o(n)$ vertices of the $((d+1)+d)$-core of the graph. More generally, we compute, up to a $1+o(1)$ factor, the generic $d$-rigidity rank of random graphs with a given degree distribution. For example, we show that the uniform $n$-vertex $k$-regular graph a.a.s. has rank $\min(k/2,d)n+o(n).$ Our approach is to estimate the rigidity rank of a random graph from its Galton--Watson local weak limit, using a parameter that we call {\em local flexibility}.

Recently Watanabe has given an algorithm to compute a bijection, that he calls (quantum) Littlewood-Richardson (LR) map (or quantum LR rule of type AII), between semi-standard Young tableaux of shape a partition with at most $2n$ parts and pairs of tableaux consisting of a symplectic tableau with shape a partition with at most $n$ parts, and a recording tableau of skew-shape given by the two previous shapes. The recording tableaux in that algorithm are shown to be equinumerous to Littlewood-Richardson-Sundaram tableaux whose injectivity is shown combinatorially while the surjectivity is concluded via representation theory of a quantum symmetric pair of type AII$_{2n-1}$. Henceforth, the algorithm to compute the quantum LR map provides a new branching model for the branching multiplicities from $GL_{2n}(\C)$ to $Sp_{2n}(\C)$. Here, as morally suggested by Watanabe, one provides a combinatorial proof for the surjectivity of the quantum LR map which in turn exhibits the restriction of the LR orthogonal transpose symmetry map to LR-Sundaram tableaux. The surjectivity is exhibited via the reverse Schensted insertion on the quantum recording tableaux, ruled by the slack data, followed with the inverse of the reduction map on the bumped entries that we explicitly compute for certain families of symplectic columns. As an application of the inverse of the quantum LR map, we compute and characterize by certain linear inequalities a family of $\mathfrak{k}$- highest weight semi-standard tableaux in the recent proof of the Naito-Sagaki conjecture using the Watanabe'sbranching rule based on the crystal basis theory for $\imath$quantumgroups of type AII$_{2n-1}$.

Escobar, Harada, and Manon introduced polyptych lattices as a piecewise-linear extension of the lattice-polytope formalism of toric geometry. In this paper we study the first genuinely non-toric case: rank-two polyptych lattices with a single shear. A detropicalization is given by a polynomial \(f(y)\), and the corresponding affine surface is $U_f=\operatorname{Spec} K[x_1,x_2,y^{\pm 1}]/\langle x_1x_2-f(y)\rangle.$ We classify these detropicalizations, compute the complexity of their projective compactifications, and show that the resulting log Calabi--Yau surface pairs are of cluster type. Conversely, we prove that every normal projective \(\mathbb Q\)-factorial index-one log Calabi--Yau surface pair with reduced boundary, ample boundary support, and a nontrivial \(\mathbb G_m\)-action arises from this single-shear construction. We also construct a global family interpolating between the two toric degenerations associated with the two charts, and compute the Cox rings of the resulting tropical mutation surfaces.

Governments increasingly operate centralized, algorithm-run admission clearinghouses that implement affirmative action through reserve systems. To sustain public trust, many such clearinghouses disclose category-specific cutoffs, but cutoffs need not allow participants to verify whether reserved and open seats are correctly assigned. We formulate cutoff-based verifiability as a governance constraint on the clearinghouse: each participant must be able to verify her assigned school and seat type using only her own score and the public cutoffs, under two intuitive verification protocols. In a controlled school choice model with multiple reserve categories, we characterize mechanisms that are individually rational, strategy-proof, and verifiable. The characterization identifies deferred acceptance mechanisms induced by two choice rules. We recommend one rule that assigns reserved seats only when a student cannot secure an open seat on merit, so that every reserved-seat assignment reflects genuine affirmative action. The results explain mechanism choices across China's high school admission systems and provide design guidance for affirmative action systems in Brazil and India.

This paper proposes estimation and inference procedures for quantiles of the heterogeneous individual-specific coefficients in panel data. Unlike conventional panel quantile regression, which focuses on outcome heterogeneity, our approach targets the $τ$-quantile of the cross-sectional distribution of individual-specific slopes. We establish the asymptotic theory under both stochastic and deterministic designs, with convergence rates $\sqrt{N}$ and $\sqrt{N\sqrt{T}}$, respectively. We also develop two corresponding bootstrap procedures for practical inference, and formally establish their validity. The suggested methods are of practical interest since they require weaker sample size growth conditions than standard fixed-effect quantile regression, and accommodate large $N$ settings. Numerical simulations and an empirical application illustrate the empirical effectiveness of the methods under both designs.

We report the discovery of a spectroscopically confirmed z = 1.824 minor merger with a mass ratio of ~35:1 in which the secondary (smaller) galaxy hosts a luminous AGN. The system is identified in the 3D-HST survey and exhibits clear tidal features in James Webb Space Telescope imaging, confirming an ongoing interaction. Using archival Chandra X-ray observations, we detect 121 +/- 11 X-ray counts associated with the secondary galaxy, corresponding to a rest-frame 2-10 keV luminosity of L_X ~ (9 +/- 0.1) x 10^43 erg/s and a photon index of Gamma ~ 2.0-2.3. Analysis of the HST/WFC3 G141 grism spectrum yields an [O III] lambda5007 luminosity of (2 +/- 0.5) x 10^42 erg/s. Independent bolometric luminosity estimates from X-ray and [O III] emission are consistent, implying L_bol ~ (3-7) x 10^45 erg/s. Assuming standard black hole-galaxy scaling relations, the expected black hole mass is ~2 x 10^6 M_sun, which would require extreme super-Eddington accretion to explain the observed luminosity. On the other hand, assuming Eddington-limited or moderately sub-Eddington accretion implies a black hole mass more than an order of magnitude above expectations. The observed X-ray spectral slope disfavors low accretion rates, restricting the allowed parameter space to high lambda_Edd and elevated black hole masses. We conclude that the secondary AGN must be powered by an overmassive, highly accreting black hole, providing direct observational support for theoretical predictions that minor mergers can drive rapid black hole growth in secondary, smaller companions.

Reachability analysis plays a central role in low-thrust spacecraft trajectory optimization by identifying which target states can be achieved under constraints on time, thrust, and propellant. Classical approaches construct reachable sets by solving many optimal control problems over grids of terminal states, requiring extensive forward simulations with fixed initial conditions. While effective, this approach is computationally expensive and becomes impractical for high-dimensional systems or strongly nonlinear dynamics, such as those encountered in cislunar environments or solar sail missions. This work introduces a dual formulation of the reachability problem. Instead of computing reachable sets directly, we determine, for fixed transfer time and boundary conditions, the maximum allowable initial mass (or, for solar sails, a scalar sail-strength parameter) that permits a successful transfer. A target is reachable if the spacecraft's initial mass does not exceed this threshold. This reformulation reduces reachability assessment to a scalar optimization problem for each target, producing a smooth scalar field that encodes equivalent feasibility information to classical reachable sets. We develop indirect maximum-initial-mass (MIM) formulations for both electric low-thrust and solar-sail dynamics and show how they can serve as efficient reachability oracles. Building on this formulation, we construct data-driven surrogate models to approximate the MIM-based reachability indicator. We investigate fully connected neural networks and demonstrate that residual networks provide the best trade-off between accuracy, training stability, and model complexity. The resulting surrogates enable rapid reachability evaluation while preserving the numerical advantages of the dual formulation, offering a practical tool for preliminary mission design and feasibility assessment.

It is shown that the lowest (second) order truncation of the Carleman linearization of the fluid equations (C2) recovers the late stage of the evolution, namely the steady-state solution, although to a decreasing degree of accuracy at increasing Reynolds number. This asymptotic property is first proved analytically for the decaying logistic with external forcing and then shown to hold to a significant degree of accuracy also for the more complex case of two-dimensional Kolmogorov-like fluid flow at low Reynolds numbers, below $Re \sim 10$. This time-asymptotic property may open interesting prospects for the quantum simulation of low-Reynolds steady-state fluid flows.

AI coding assistants are now central to professional software development, yet their impact on how developers think about and practice security remains poorly understood. While prior work has documented vulnerability rates in AI-generated code, a more fundamental question persists: how do these tools transform security awareness in authentic, ongoing development practice? We conducted semi-structured interviews with 15 professional software engineers and observed them completing security-relevant coding tasks with AI assistance, spanning 3 experience cohorts defined by their relationship to AI tools during professional formation. We find that AI coding assistants reorganize rather than eliminate security thinking, shifting it from the act of writing code to the act of reviewing it. This transition from preventive to reactive security is structurally encouraged by interaction models that frame code generation as a functional task, leaving security as an afterthought. Notably, none of our coding session participants specified security requirements in their initial prompts, even when they possessed the relevant knowledge, revealing a decoupling of security awareness from security behavior. We further document informal coping strategies developers had independently invented to manage AI security risk, none of which are supported by current tools or organizations, and find that the experience cohort did not reliably predict security performance. This paper contributes a practice-grounded account of how AI-assisted development reshapes the human side of secure coding, offering empirical foundations for the design of more security-aware tools, training programs, and organizational policies.

The structure of the ground $(2^+)$ and excited $(1^+)$ bound states of the $^8$B and $^8$Li nuclei is studied within the framework of the $α+^3$He($^3$H)+$p(n)$ three-body potential cluster model based on the hyperspherical Lagrange-mesh method. The two-body realistic potentials have been applied from the literature. Convergent theoretical estimates for the three-body binding energy and matter radius have been obtained with the maximal hypermomentum $K_{max}=22$ and 28 for the ground and excited $1^+$ states, respectively. The ANC value of the virtual transition of the $^8$B nucleus is estimated self-consistently by matching the overlap integral of the $^8$B three-body and the $^7$Be two-body wave functions with it's asymptotics. The obtained values are $0.211$~fm$^{-1/2}$ and $0.739$~fm$^{-1/2}$ in the spin 1 and spin 2 channels, respectively. For the ANC values of the $^8$Li nucleus the estimates $0.220$~fm$^{-1/2}$ and $0.774$~fm$^{-1/2}$ are extracted. The ratio $C^2(^8 {\rm B})/C^2(^8 {\rm Li})=0.912$ implies a breaking of the mirror symmetry of the strong nuclear forces of order 27\% due to the Coulomb interaction and the dynamical three-body effects. For the $S_{17}(0)$ -factor an estimate $22.492\pm0.014$ eV b was obtained based on the asymptotic theory developed by D. Baye [Phys. Rev. C {\bf 62},065803 (2000)]. The spin 2 channel contributes with $S^{(2)}_{17}(0)=20.838 \pm 0.014$ eV b, while the spin 1 channel yields $S^{(1)}_{17}(0)=1.654 \pm 0.003$ eV b. These results for $S_{17}(0)$ are in a good agreement with the estimate $20.8\pm0.7{\rm(th)}\pm1.4{\rm(exp)}$ eV b of the SF II, but larger than the recommended value $20.5\pm0.70$ eV b of the SF III. At the same time, our estimate is very close to the value 22.4 eV b used in the most successful Solar Model BAR2M [W.~Yang and Z.~Tian, AJ {\bf 970} (2024), 38].

We propose a deep learning framework based on an encoder-decoder architecture for the design and evaluation of cloaking devices, demonstrated in this work for two-dimensional wave propagation governed by the Helmholtz equation. The cloaks under consideration are concentric layered media surrounding the object, whose geometry and material parameters determine the scattering response. We consider circular and object-fitted layer configurations and parameterize all designs by the layer thicknesses, enabling a unified representation for direct comparison of different cloaks for the same object. Training data are generated using a boundary element formulation suitable for geometries where analytic solutions are not available, and neural networks are trained with standard hyperparameters on geometry-specific datasets. The proposed approach is applied to circular, star-shaped, and kite-shaped objects. Results show that object-fitted configurations consistently outperform simpler circular-layer designs in scattering reduction, highlighting the importance of geometry in cloaking performance. Overall, we present a flexible, data-driven approach for systematic comparison of cloaking strategies, with potential extension to more complex geometries and broadband settings.

Generalized partially linear single-index additive models (GPLSIAMs) have been increasingly applied across diverse areas due to their versatility in integrating functional flexibility with parametric dimension reduction while maintaining interpretability. However, the estimation presents severe computational challenges. This paper introduces a novel stable method that uses the model matrix for each single-index effect, defined by its single-index coefficients, and the penalized complete Fisher information matrix to dynamically update the boundaries of the single-index covariates within a unified iterative framework. The derived model matrices enable the fast computation of the estimated effective degrees of freedom and pointwise confidence bands for the single-index effects. The smoothing parameter updates are integrated into the iterative process via the generalized Fellner-Schall method, which recycles the derived matrix decompositions, thereby providing an efficient approximation to the global penalized optimization problem. Simulation studies with moderate sample sizes under non-Gaussian distributions confirm the empirical consistency of the estimation across multiple scenarios. Notably, the proposed approach remains stable where state-of-the-art competitive methods fail to recover true single-index coefficients and nonlinear functions, and is 80.13 times faster than the usual two-step method in the most computationally intensive scenario. The modeling advantage is illustrated through an application to Capital Bike Sharing data, where we deal with a single-index interaction effect for each year, with distinct single-index coefficients, a complex structure that makes competitive methods inapplicable. The proposed method is implemented in R, with functions available for reproducibility and transparency in comparisons.

We introduce the concept of \emph{orbital altermagnetism}, a symmetry-protected magnetic order of pure orbital degrees of freedom. It is characterized with ordered anti-parallel orbital magnetic moments in real space but momentum-dependent orbital band splittings, analogous to spin altermagnetism. Using a minimal tight-binding model with complex hoppings in a square-kagome lattice, we show that such order inherently arises from staggered loop currents, producing a $d$-wave-like orbital-momentum locking. First-principles calculations show that orbital altermagnetism emerges independent of spin ordering in in-plane ferromagnets of CuBr$_2$ and VS$_2$, so that it can be unambiguously identified experimentally. On the other hand, it may also coexist with spin altermagnetism, such as in monolayer MoO and CrO. The orbital altermagnetism offers an alternative platform for symmetry-driven magnetotransport and orbital-based spintronics, as exemplified by large nonlinear current-induced orbital magnetization.

Giant pulses are rare bright radio bursts that occur in restricted ranges of pulsar rotational phase. Here we report giant pulses from PSR B0355+54, a pulsar with spin period ~0.156 s. Using 7.97 hours of observations centred at 1.46 GHz, with 128 MHz bandwidth in each circular polarizations, we identify 432 pulse periods containing bright pulses. The giant pulses recur in two compact longitude regions inside the radio emission window. They are narrow compared with the mean profile, with median W50=290.3 us, and reach relative peak flux density ratios up to 149.7. The early longitude group has a timing scatter of 139.7 us, or 0.00089 of a rotation. The first longitude group favours right, while the second favours left circular polarization.

Agentic large-language-model systems can coordinate scientific tools, but many implementations remain difficult for domain scientists to extend without modifying the source orchestration code or relying on unconstrained code generation. DynaMate2 is a LangGraph-based multi-agent framework for converting expert-defined Python functions into persistent AI-callable tools. The architecture separates domain execution from LLM supervision: registered tools perform scientific operations, while a supervisor LLM decomposes goals, selects specialist agents, routes inputs, and propagates outputs across steps. DynaMate2 supports: runtime tool registration from inline code, source files, and explicitly requested natural-language specifications; persistent storage of tools, agents, and conversation state; and a web interface for interactive workflow assembly. We demonstrate the framework on a molecular simulation workflow in which a single instruction retrieves a MACE foundation model, builds a NaCl-water configuration, runs an ASE molecular dynamics trajectory, and generates energy and temperature diagnostics. The demonstration illustrates how validated workflow components can be composed into a supervised agentic pipeline without rewriting the framework. DynaMate2 therefore provides a reusable template for extending LLM-based automation to research groups with existing Python workflows, while preserving the need for explicit tool validation, reproducibility logs, and deployment-specific safeguards.

We determine the basepoint-freeness threshold of a very general polarized abelian surface over the field of complex numbers. We also give the first example of a polarized abelian surface whose basepoint-freeness threshold is irrational.

We prove the weak cosmic censorship conjecture in $2+1$ spacetime dimensions for the circularly symmetric Einstein-scalar field system in the presence of a negative cosmological constant $Λ<0$. More precisely, we show that for any integer $k\geq2$, the maximal development of generic $C^k$ initial data does not contain naked singularities. An essential step of the proof is establishing the presence of a mass gap. In particular, this implies that all naked singularities have infinite blueshift, which represents the main instability mechanism.

We present a holographic dual description for the O(N) vector model in $d$-dimensional Euclidean space within the functional renormalization group (FRG) framework. By iterating Wilsonian renormalization group transformations, the extra-dimensional scale coordinate is identified as the radial direction of an emergent $(d+1)$-dimensional bulk spacetime. We construct a bidirectional holographic dictionary that maps non-perturbative fluctuations to the emergent bulk metric warping factors. Under the massless critical configuration, the emergent gravitational vacuum reduces to an Anti-de Sitter ($\text{AdS}_{d+1}$) geometry, satisfying the local energy conditions.

Constructing explicit Runge--Kutta (ERK) methods with as few stages as possible for a given order is a classical problem in numerical analysis. In this work, we introduce a $Q$/$D$-space framework of sufficient order conditions for ERK methods. This framework generalizes Butcher's classical simplifying assumptions by reformulating them in terms of simplified $Q$- and $D$-spaces defined through their residual vectors. It yields sufficient conditions which, together with $B(p)$, ensure order $p$. It also leads to a recursive construction procedure for ERK methods of arbitrary even order, in which the Butcher coefficients are obtained from two structured linear systems. For every even order $p\ge 4$, the construction produces ERK methods with stage number $s(p)=(p^2-2p+8)/4$. This stage count has the same leading term as that of the classical Gragg families, while improving the linear term. The free parameters retained by the construction further provide a systematic framework for designing methods with enhanced stability and short-time accuracy.

While energy scale competition dictates equilibrium phase behavior, kinetic barriers often drive soft-matter mixtures into dynamically arrested states, rendering conventional phase diagrams incomplete. We extend the classification of binary systems into regions of thermodynamic instability to explore the interplay between phase separation and dynamical arrest. Strong cross-attractions kinetically suppress demixing, whereas competitive energy scales yield either condensation- or demixing-driven amorphous states. We show that this morphological crossover is parameterized by a structural order parameter, $χ$, providing a unified non-equilibrium framework that reconciles theoretical predictions with experimentally observed arrested mixtures.

Identifying topological invariants are endowed with profound physical connotations in the fields of condensed matter physics. However, they are often limited by the failure of standard theorems and high computational costs. Traditional machine learning methods typically treat this problem as a black-box regression, which fails to learn the underlying mathematical structures of lattices. To this end, we propose a straightforward yet powerful multimodal model that fundamentally improves topological invariants discovery through two key methodological innovations. First, instead of feeding raw data into a standard network, our model introduces a dual-track alignment mechanism. This mechanism treats eigenvalues as context sequences and eigenvectors as matrix structures, enabling the model to naturally capture the interrelated algebraic and geometric properties of topological states. Second, to resolve the common problem where deep learning models make numerical mistakes in exact mathematical calculations, we use a tool-integrated reasoning paradigm. Within this paradigm, the neural network acts as a logical controller to calculate discrete topological indices. This design eliminates the uncertainty of deep learning, ensuring zero-error calculations from continuous physical data. We demonstrate that our model can accurately reconstruct complex three-dimensional generalized Brillouin zones and achieve 97% accuracy in calculating non-Bloch Chern numbers in long-range transition systems. By combining the flexible logic of multimodal artificial intelligence with the rigorous precision of symbolic computation, this work provides a reliable and highly versatile tool for exploring exotic topological phases.

We introduce a non-equilibrium version of the Caldeira-Leggett model in which a quantum particle is strongly coupled to a set of engineered reservoirs. The reservoirs are composed by collections of squeezed and displaced thermal modes, in contrast to the standard case in which the modes are assumed to be at equilibrium. The model proves to be very versatile. Strongly displaced/squeezed reservoirs can be used to generate an effective time dependence in the system Hamiltonian and can be identified as sources of pure work. In the case of squeezing, the time dependence is stochastic and breaks the fluctuation-dissipation relation, this can be reconciled with the second law of thermodynamics by correctly accounting for the energy used to generate the initial non-equilibrium conditions. To go beyond the average description and compute the full heat statistics, we treat squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour. As an application of this technique, we show the quantum-classical correspondence between the heat statistics in the non-equilibrium Caldeira-Leggett model and the statistics of a classical Langevin particle under the action of squeezed and displaced colored noises. Finally, we discuss thermodynamic symmetries of the heat generating function, proving a fluctuation theorem for the energy balance and showing that the conservation of energy at the trajectory level emerges in the classical limit.

We show analytically that the convergence of the QNM expansion of the retarded Green's function of the Schwarzschild spacetime is set by a singularity in the complex time plane. The singularity has a simple geometric origin: it is an example of a `bouncing singularity' in the language of AdS/CFT literature, caused by a null geodesic which bounces from the black hole singularity. Our work explains why the QNM convergence region at real times is bounded by null ray which scatters from the gravitational potential at a seemingly unremarkable point ($r_* = 0$ in the conventions of previous work) -- this ray is the same distance from the origin as the bouncing singularity in the relevant complex plane. The same set of singularities are responsible for an annular region of convergence for the Matsubara mode sum which describes the early time behaviour of the Schwarzschild Green's function for perturbations close to the horizon.

Understanding when linear immersions of nonlinear dynamical systems exist is important since such immersions allow us to leverage the rich tools of linear system theory to analyze nonlinear dynamics. Recently, Liu et al. (2023) showed that continuous-time dynamical systems that admit countably many but more than one omega-limit sets cannot be immersed into finite dimensional linear systems with a one-to-one and continuous mapping. In this paper, we extend these results to discrete-time dynamics and show that similar obstructions exist also in discrete time. We further consider a generalization involving alpha-limit sets. Several examples are provided to demonstrate the results.

In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends, generalizing the result of Li-Wang [LW04] on manifolds with nonnegative sectional curvature.

Computer-use agents are increasingly capable of operating on real operating systems, but this capability has also increased the risks posed by prompt injection, indirect instructions, and visual attacks. Existing defenses typically rely on analyzing the prompt or each potentially malicious input with a second large model at inference time, which can limit coverage or increase deployment cost. We propose ProjGuard, an alternative based on behavioral trajectory monitoring. At each step, we derive a lightweight scalar risk signal from the agent's accumulated interaction history and evaluate, online, whether execution is beginning to drift toward an unsafe region. This enables early warnings before the trajectory reaches a potentially harmful action. When an alert is raised, we selectively activate an auxiliary vision-language model to propose a corrected next step and steer execution back toward task completion. Experiments on OS-Harm show that monitoring with on-demand correction reduces the unsafe rate from 16 percent to 3 percent while improving task completion from 59 percent to 65 percent. We further evaluate transfer to RiosWorld, where the method remains competitive, reaching 4 percent unsafe and 64 percent completion. Overall, these results support a hierarchical safety strategy in which always-on monitoring anticipates deviations and activates correction only when needed.

Given rationals $α$ and $β$, the sure-almost-sure problem for a threshold Boolean objective $φ$ in a Markov decision process (MDP) asks if one can simultaneously ensure that all outcomes of the MDP have $φ$-value at least $α$ (i.e. sure $α$ satisfaction) and with probability $1$ the outcome has $φ$-value at least $β$ (i.e. almost-sure $β$ satisfaction). The sure-limit-sure problem asks if for all $\varepsilon > 0$ one can simultaneously ensure that all outcomes have $φ$-value at least $α$ and with probability at least $1 - \varepsilon$ the outcome has $φ$-value at least $β$. Moreover, if simultaneous satisfaction of objectives is possible, then one would also like to construct a strategy (for sure-almost-sure) or a family of strategies (for sure-limit-sure) that achieves this. In this paper, we solve the sure-almost-sure and sure-limit-sure problems for window mean-payoff objectives. The window mean-payoff objective strengthens the standard mean-payoff objective by requiring that eventually, from every point in the infinite run, the average payoff becomes greater than a given threshold within a finite window length. We study two variants of window mean payoff: in the fixed variant, the window length $\ell$ is given, while in the bounded variant, the length is not given but is required to be bounded throughout the run. We show that the sure-almost-sure problem and the sure-limit-sure problem are both in P for the fixed variant (if $\ell$ is given in unary) and are both in NP $\cap$ coNP for the bounded variant, matching the computational complexity of sure satisfaction and almost-sure satisfaction when considered separately for these objectives. We also give bounds for the memory requirement of winning strategies for all considered problems.

We study semileptonic $B$ meson decays $B^+ \to D_1^{(\prime)}\ell^+ν_\ell$ and $B_s^0 \to D_{s1}^{-(\prime)}\ell^+ν_\ell$, where $\ell=μ,τ$. The final state axial vector mesons are treated as mixtures of the heavy quark basis states with light degree of freedom angular momenta $j_\ell=1/2$ and $j_\ell=3/2$, parametrized by the mixing angle $θ_{D_1}$. Using form factors obtained in the covariant light front quark model, we analyze the dependence of various observables on $θ_{D_1}$ such as polarized and unpolarized branching ratios, the lepton forward-backward asymmetry, the longitudinal polarization fraction, and the lepton flavor universality ratios. In addition, we also discuss correlations among different observables. We study these observables in the experimentally motivated mixing angle regions as well as over a wider range of $θ_{D_1}$. Our results show that branching ratios and other observables are sensitive to the axial-vector mixing structure. These predictions provide useful Standard Model benchmarks for future measurements of semileptonic $B_{(s)}$ decays into orbitally excited mesons and may help to clarify the long standing $1/2$ vs. $3/2$ puzzle through semileptonic $B$ decays.

Many real-world networks exhibit hierarchical, tree-like structure and heavy-tailed degree distributions, phenomena not readily captured by standard statistical models for network data. Extensions of the popular continuous latent space modeling framework have been proposed to accommodate such networks. Drawing on insights from statistical physics, continuous latent space models with underlying hyperbolic geometry have been proposed as a natural framework, probabilistically embedding nodes in a latent Riemannian manifold with constant negative curvature. Most statistical implementations, however, simplify the original physics-based model by omitting the ``temperature parameter," which controls the sharpness of the latent distance-to-probability mapping. We argue this omission is critical. We demonstrate that temperature is the fundamental parameter governing a network's tree-like topology, and that failing to infer it weakens model expressiveness. We formalize a Bayesian hyperbolic continuous latent space model with an unknown, learnable temperature parameter. We then develop two inferential procedures: a Hamiltonian Monte Carlo approach for rigorous posterior characterization and a scalable auto-encoding variational Bayes algorithm for large-scale networks. Through simulation and real data examples, we show that our model outperforms models with fixed temperature and misspecified Euclidean geometries in graph reconstruction tasks in most settings, confirming temperature is a crucial and inferable feature of complex networks.

We develop a theoretical framework for describing superposed coherent states in graphene quantum systems using the concept of catability as a phase-sensitive metric functional measure. In this case, the formalism quantifies interference stability and coherence structure via phase-dependent contributions of quantum superposition states. Catability is defined as a functional measure sensitive to relative phase variations within coherent state combinations, serving as a diagnostic tool for quantum interference effects in graphene-based systems. Also, the formulation is extended using Lie algebra techniques, where the underlying symmetry structure of graphene quantum states is represented through operator algebras governing state transformations in quantum space. In this context, to describe nonlocal propagation and phase-resolved dynamics, a Green function approach is incorporated, enabling systematic treatment of quantum correlations in a spatially extended structures framework. A unified framework is constructed by combining Lie algebraic symmetry analysis with Green function propagation theory, yielding a consistent description of phase-sensitive catability in complex graphene quantum configurations within the framework approach. Results provide a structured route for testing coherence, interference stability, and quantum state control in low-dimensional quantum materials systems.