We compute numerically the time evolution of simple semiclassical states describing homogeneous and isotropic spatial geometries in quantum-reduced loop gravity under a deparametrized formulation of the dynamics, in which a reference matter field is taken as a relational time variable for the dynamics of quantum states of the gravitational field. The states which we consider are defined on the Hilbert space of a spin network graph formed by a single six-valent vertex. We find that the quantum dynamics is generally in close agreement with the semiclassical effective dynamics of a homogeneous and isotropic universe throughout the range of validity of the numerical approximation. In particular, an initial state describing a contracting geometry undergoes a dynamical "bounce", where the contraction is halted and turned into an expansion by the quantum dynamics.

The analysis of extremal dependence in high dimensions is a key challenge in modern extreme-value statistics. Existing methodology primarily focuses on modeling and estimation of extremal dependence structures, often supported by concentration bounds for empirical tail quantities. However, comparatively little is known about general inferential procedures in high-dimensional extremes. In this paper, we develop foundational results that enable inference for rank-based empirical tail dependence coefficients, stable tail dependence functions, and functionals derived from them. We start by establishing finite-sample probability bounds that quantify the linearization error for such estimators uniformly over collections of coordinates. Moreover, we derive high-dimensional central limit theorems and establish the validity of multiplier bootstrap procedures for collections of empirical tail dependence statistics. Within an asymptotic framework, our results allow the dimension to grow exponentially with the effective sample size. We illustrate the usefulness of the results through two applications: uniform expansions and normal approximations for M-estimators of tail dependence parameters and inference for spatial isotropy based on collections of tail dependence functions.

We present a multi-epoch study of the extreme X-ray variability of the type~1 quasar SDSS~J000532.84+200717.4 using archival observations from \textit{XMM-Newton}, \textit{Swift}/XRT, \textit{EP-FXT}, and \textit{ROSAT}, together with new optical spectroscopy and multi-wavelength photometry. The 0.2--10~keV X-ray flux exhibits a transition from a high state to a subsequent low state, declining by more than an order of magnitude and placing the source in the X-ray--weak regime ($Δα_{\rm ox} \lesssim -0.3$). Significant variability on timescales of days to weeks persists within the low state. In contrast, the optical and mid-infrared emission remain stable over decade-long timescales, while the UV continuum varies only mildly and broadly tracks the X-ray evolution. Multi-epoch optical spectroscopy shows no significant long-term changes in either the continuum shape or the broad emission-line profiles. The \ion{Mg}{2} emission is relatively weak compared with typical quasars, suggesting similarities to weak-line quasars. The pronounced wavelength-dependent variability indicates that the accretion disk remains largely intact while the X-ray emission undergoes dramatic changes. The spectral hardening in the low state and the viability of ionized partial-covering models are consistent with variable, largely dust-free absorbing gas, possibly associated with clumpy inner disk winds, although intrinsic coronal variations cannot be excluded. SDSS~J0005+200717.4 therefore provides evidence that extreme X-ray weakness can arise as a transient phase in otherwise normal quasars.

The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $ι_{\rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $ι_{\rm g}'(G)$. In this paper, $ι_{\rm g}(C_n)$, $ι_{\rm g}(P_n)$, $ι_{\rm g}'(C_n)$, and $ι_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $ι_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $ι_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $ι_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $ι_{\rm g}(G) = ι_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds.

A binary fluid mixture in contact with lateral particle reservoirs is considered. By imposing different particle concentrations in these reservoirs, the system can be maintained under controlled non-equilibrium conditions. Previous stochastic approaches have revealed an unexpected property of the system's state trajectory, namely that it remains time-reversible even when the system is driven out of equilibrium. In the absence of relevant experimental evidence, we employ microscopic molecular dynamics simulations to assess the validity of this surprising result. Remarkably, the simulation results unambiguously confirm the prediction of the stochastic analysis.

The empirical Bayes $g$-modeling approach based on the nonparametric maximum likelihood estimator (NPMLE) has been central to large-scale estimation and inference in the normal means problem. However, theoretical guarantees for uncertainty quantification remain scarce. A key obstacle is that the NPMLE is necessarily discrete, which yields discrete posterior credible sets and a slow logarithmic deconvolution rate. We address both limitations by introducing a hierarchical Gaussian smoothing layer that restricts the mixing distribution to a Gaussian location mixture. Our smooth NPMLE inherits the favorable properties of the classical NPMLE: it is computable via convex optimization and achieves nearly parametric denoising performance. Moreover, it achieves a polynomial deconvolution rate that is asymptotically minimax over the corresponding class. Our procedure also leads to estimated smooth posteriors that converge to the true posteriors at a polynomial rate. Further, we characterize marginal coverage sets that are optimal in expected length, construct plug-in estimators of these sets, and establish theoretical guarantees for the estimated sets in terms of both coverage probability and expected length. We also extend the theory to settings with model misspecification and heteroscedastic Gaussian observations, and study identifiability of the proposed hierarchical model.

The Prisoner's Dilemma, zero-sum games, LQR team problems, and differential games have shaped game theory in controls for decades, but the field's most pressing adversarial challenges demand a richer framework, and its name is Colonel Blotto. Strategic adversarial constraints represent a fundamental consideration in control systems, from cybersecurity defense to infrastructure protection. Colonel Blotto games, despite their direct relevance to such applications, remain underutilized in the controls community relative to other game-theoretic approaches. This article aims to close that gap for the controls community. Indeed, theoretical advances within the last two decades have spurred a resurgence of interest and enabled their applications across several domains. In this article, we introduce the Colonel Blotto framework, survey key analytical and computational results, and demonstrate how problems spanning cybersecurity, network defense, and multi-agent systems fit naturally within this structure. Three research directions are examined in depth: interdependent contest objectives that capture networked vulnerabilities, alternate winning rules that model partial rewards and structural asymmetries, and multi-agent competitive environments involving coalition formation and strategic concessions. Taken together, these directions reveal a framework that is both practically deployable and rich enough to capture the strategic complexity inherent in adversarial resource allocation.

We consider the hypothetical possibility of neutral gauge bosons ($Z^\prime$) with flavor-specific leptophilic couplings. For such New Physics (NP) interactions, the current experimental constraints are much relaxed in the heavy mass regime, particularly for masses $\geq \mathcal{O}(1)$ TeV. However, in the presence of a leptophilic $Z^\prime$, leptonic decay modes of the electroweak gauge bosons and Higgs can be corrected at the loop level. Using the existing upper bounds on the corresponding decay widths, we find that one can impose stronger exclusion limits on the interactions of a heavy $Z^\prime$. Future updates on the aforesaid decay channels can be used in complementarity with the proposed lepton colliders to probe even weaker leptophilic NP interactions at the TeV scale and beyond.

It is a study note detailing the connection between Boolean inverse monoids and ample groupoids.

A system of two cubic reaction-diffusion equations for two independent gene frequencies arising in population dynamics is studied. Depending on values of coefficients, all possible Lie and $Q$-conditional (nonclassical) symmetries are identified. A wide range of new exact solutions is constructed, including those expressible in terms of a Lambert function and not obtainable by Lie symmetries. An example of a new real-world application of the system is discussed. A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.

For graphs $F$ and $G$, $F$-multicolor Turán number of $G$, denoted by $\mathrm{ex}_F(n,G)$, is the maximum number of edge-disjoint copies of $F$ in an $n$-vertex graph such that there is no copy of $G$ whose edges come from distinct copies of $F$. We study this parameter mainly for cycle pairs and determine, up to asymptotic order, when $\mathrm{ex}_{C_k}(n,C_\ell)$ attains the three natural thresholds: the upper bound, the lower bound, and the $n^{2-o(1)}$ regime. In particular, for every odd $k\ge 5$ and every $t\ge 1$, where $C_k(t)$ denotes the $t$-blow-up of $C_k$, we prove $\mathrm{ex}_{C_k(t)}(n,C_{k-2})=n^2/(kt)^2+o(n^2),$ and establish a corresponding stability theorem. We further show that if $F$ and $G$ have the same odd girth $k$ and there exist homomorphisms from both $F$ and $G$ to $C_k$, then $\mathrm{ex}_F(n,G)=n^{2-o(1)}$; in particular, $\mathrm{ex}_{C_k}(n,C_k)=n^{2-o(1)}$ for odd $k$. In addition, we prove $\mathrm{ex}_{C_{2k+1}}(n,C_{2\ell+1})=O\!\left(n^{1+1/\lceil \ell/k\rceil}\right)$ for $\ell>k$ and $\mathrm{ex}_F(n,G)=O(\mathrm{ex}(n,G))$ for bipartite $G$. We particularly establish $\mathrm{ex}_{C_4}(n,C_4)=\frac{\sqrt{2}}{8}n^{3/2}+O(n)$, and give a sufficient condition under which the lower bound cannot be attained.

Measurements of the shear induced by weak gravitational lensing around galaxy cluster lines of sight are the gold standard for calibrating cluster observable-mass relations, thereby enabling a robust and precise inference of cosmological parameters. The weak-lensing mass bias is the systematic offset between the true halo mass and the mass that is inferred from the lensing data using an imperfect model for the halo mass distribution. We study the impact of cosmology on the lensing mass bias to inform future cosmological analyses of galaxy clusters. We create synthetic lensing shear maps for 115,920 projections of clusters with $M_{200\mathrm c}>1.56\times10^{14}\,h^{-1}M_\odot$ in a suite of Magneticum simulations. The simulation boxes are $896\,h^{-1}$Mpc on a side and are set up with 15 different combinations of the cosmological parameters $Ω_\mathrm{m}$, $Ω_\mathrm{b}$, $σ_8$, and $H_0$. Assuming a Navarro-Frenk-White profile, we extract weak-lensing mass measurements and quantify their bias $b_\mathrm{WL}$ with respect to the true halo mass. To investigate the impact of baryonic effects, we perform the analysis on gravity-only simulations and on their full-physics hydrodynamical counterparts. We confirm that assuming a fixed halo concentration or a fixed concentration-mass relation leads to cosmology-dependent changes of the mass bias. We report changes of up to $Δ\ln b_\mathrm{WL}=0.030$ with respect to the bias obtained at the fiducial WMAP7 cosmology. Adopting a model for the concentration that also depends on cosmology absorbs the changes in halo profiles and we recover essentially constant values for the mass bias. Our analysis of hydrodynamical simulations suggests that future, more accurate models will also need to explicitly account for the strength of baryonic effects.

The global transition towards renewable energy has accelerated the deployment of utility-scale wind farms, increasing the need for accurate performance and economic assessments. Although wind energy offers substantial potential for carbon emission reduction, investment decisions are highly sensitive to predicted annual energy production and economic profitability. Conventionally wind farm analyses often estimate turbine power output based solely on incoming wind conditions, neglecting wake interactions between turbines. These wake effects can significantly reduce downstream turbine performance, leading to overestimation of energy yield and financial returns. This study proposes WAKE-NET, a 3D wake-aware optimization framework that integrates turbine layout optimization, turbine capacity selection, cable routing, and hub height diversification within a unified profit-driven formulation. Unlike traditional approaches that assume a uniform hub height and turbine capacities or ignore wake dynamics, the proposed framework accounts for wake-induced power losses during optimization. A benchmark wake-ignorant model is also evaluated to quantify the impact of neglecting wake interactions. Results indicate that the wake-ignorant optimization can significantly overestimate annual profits, while the use of multiple hub heights and capacities reduce wake overlap and improve spatial utilization. Overall, the findings demonstrate that wake-aware optimization coupled with hub height and capacity diversification provides more reliable energy yield prediction and economic assessment, offering valuable guidance for large-scale wind farm planning and investment.

The Cauchy combination test (CCT) is widely used because it yields a closed-form combined $p$-value and is known to be asymptotically valid as the nominal level $α\downarrow0$ under broad dependence structures. We study a different asymptotic question: whether the usual Cauchy cutoff remains accurate at an ordinary fixed level when the number $K$ of combined $p$-values grows under dependence. Under a canonical one-factor equicorrelated Gaussian copula model, we show that the raw CCT is generally not asymptotically exact at fixed $α$. With fixed positive correlation, the statistic converges to a random latent-factor limit, so there is no universal fixed-level reference law. When the common correlation $ρ_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\sqrt{ρ_K}(\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $ρ_K(\log K)^3\to0$. Because the size distortion arises entirely from the reference law and not from the statistic, it can be corrected without modifying the test statistic itself. We propose the boundary-layer calibrated CCT (BL-CCT), which replaces the standard Cauchy reference by a one-parameter Gaussian-smoothed Cauchy family. Unlike recent variants that modify the test statistic, BL-CCT leaves the statistic unchanged and corrects only the reference law. BL-CCT is asymptotically exact under the weaker condition $ρ_K\log K\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. We also conduct several power analyses: although BL-CCT only raises the cutoff, it incurs no first-order power loss relative to the raw CCT on the exactness scale, under local dense, sparse, and dense Gaussian alternatives. Numerical experiments support the calibration theory.

We apply a finite-time geometric optimization framework to investigate thermal fluctuations and (non)equilibrium optimal processes in the $(2+1)$-dimensional BTZ black hole. Employing Hessian thermodynamic information metrics, we construct geodesic trajectories that define optimal protocols connecting distinct thermodynamic configurations. Finite-time state transitions are described by paths that extremize entropy production or energy dissipation, depending on the chosen thermodynamic representation. { We compare our optimization framework with a non-optimal blackbody Hawking evaporation model, revealing substantial differences between the two descriptions. Finally, we quantify the intrinsic efficiency of both types of processes in terms of the extractable rotational energy stored in the black hole configurations.} This work presents the first formulation of a geometric optimal control theory for the BTZ black hole.

We determine a set of permutation patterns $q$ so that the number of permutations with $r$ occurrences of $q$ is asymptotically $n^r$ times the number of permutations avoiding $q$, partially settling a conjecture of Conway and Guttman. We also use these asymptotics to prove nonrationality and nonalgebraicity for certain ordinary generating functions for permutations with $r$ copies of a pattern.

We propose a mathematical model for the growth and treatment of Undifferentiated Pleomorphic Sarcoma (UPS) using a system of nonlinear differential equations. The model combines Gompertz-type tumor growth with surface-dependent necrotic loss, surgical resection with residual disease, postoperative recovery, tumor--immune interaction, and radiation treatment scheduling. We study the mathematical properties of the model and obtain several results. The growth equation shows the existence of a threshold below which the tumor cannot survive and may disappear. The postoperative phase exhibits an early inflammatory stage followed by proliferative recovery. For the tumor-immune subsystem, equilibrium states and local stability conditions are identified. The radiation treatment problem is formulated as an optimal control problem, and the optimal strategy is shown to be of bang-bang type. The model suggests that tumor recurrence depends not only on tumor growth itself but also on residual disease, postoperative dynamics, immune response, and treatment timing.

RR Lyrae stars (RRLs) are classical tracers of old stellar populations, yet growing evidence suggests the presence of a metal-rich ([Fe/H]>-0.5), intermediate-age (2-7 Gyr) sub-population in the Milky Way disc. Binary evolution, particularly stable mass transfer, has been proposed as a viable formation channel, predicting that most metal-rich, intermediate-age (<9 Gyr) RRLs should reside in binaries with orbital periods of ~900-2000 days. However, no genuine RRL binaries have been robustly identified, including in the Gaia DR3 astrometric binary catalogues, despite Gaia being sensitive to the predicted orbital-period range. We investigate whether the lack of detections in Gaia DR3 reflects an intrinsically low binary fraction or instead arises from observational biases. We analyse a carefully selected sample of 100 Gaia DR3 RRLs designed to trace the metal-rich population with thin-disc kinematics and compare them with predictions from binary evolution models. We generate realistic Gaia observation mocks, including variability-induced astrometric biases, and assess the detectability of binaries and the posterior constraints on the hidden binary fraction using astrometric quality indicators, such as RUWE, and a robust Bayesian inference. While current uncertainties prevent a definitive rejection of a high fraction of hidden binaries, our results reveal tensions between existing binary evolution predictions and the Gaia DR3 non-detections. This suggests either the presence of unaccounted systematics in the modelling of Gaia observations or the need to revise assumptions in binary evolution models. We predict that Gaia DR4 will significantly improve the binary detectability and provide powerful new constraints on the post-interaction binary populations.

GUTs unify quarks and leptons into same representations and predict correlations between their masses and mixing. We perform numerical scans in SO(10) GUTs to explore the flavor space with new data of JUNO taken into account. The quark-lepton correlation shows the preference of normal ordering for light neutrino masses, predicts favored region of the CP-violating phase in neutrino oscillations, and classifies GUT models based on their testability in neutrinoless double beta decay experiments. The quark-lepton correlation predicts mass spectrum of right-handed neutrinos, pointing to the energy scale of baryon and lepton number violation and providing sources for baryogenesis. We emphasize that, as high precision measurements of neutrino physics are coming, the quark-lepton correlation will provide increasingly important role in the testability of GUTs, complementary to proton decay measurements.

Code smoothing is a phenomenon in which an error distribution makes a code statistically close to the uniform distribution over the ambient space. This closeness is measured by total variation distance. Recently, Debris-Alazard et al.\ introduced a smoothing bound, which is an upper bound on this total variation distance. Although the smoothing bound evaluates how the error distribution smooths a code, this bound applies only to linear codes. In this paper, we generalize this bound to not only linear codes but also specific non-linear codes. While the smoothing bound in previous work was obtained by Fourier analysis over finite abelian groups, we derive this bound using a graph-theoretic approach. To derive the smoothing bound, we consider code smoothing as the mixing of random walks on a specific graph, and use the concept of equitable partitions, which is well-studied in graph theory.

Photoplethysmography (PPG) is widely used as a non-invasive and accessible modality for continuous health monitoring. However, despite being a peripheral hemodynamic signal intrinsically coupled with systemic circulation, existing research has largely confined its scope to a narrow range of cardiovascular tasks, leaving a fundamental question underexplored: to what extent can PPG support holistic health profiling beyond traditional cardiovascular applications? To answer this question, we present AnyPPG, a foundation model-based framework designed to reveal the broader health-profiling potential of PPG. To ensure reliable performance for this investigation, AnyPPG is pretrained with ECG guidance on the most diverse PPG corpus with synchronized ECG to date, comprising over 100,000 hours of recordings from six large-scale data sources. This pretraining yields robust and physiologically grounded PPG representations that provide a reliable basis for subsequent analysis. Building upon this pretrained model, we conduct a systematic investigation into the association between PPG and holistic health through, to our knowledge, the first PPG-based phenome-wide disease detection study, spanning 1,468 disease phenotypes in more than 15,000 subjects. Our evaluation demonstrates the effectiveness of AnyPPG: across eight clinical and wearable datasets covering 15 downstream tasks, it achieves the best performance in 13 tasks. More importantly, in the phenome-wide analysis, AnyPPG exhibits meaningful discriminative capability (AUC $\ge$ 0.70) for 307 phenotypes across 16 distinct phecode chapters, including 230 non-circulatory conditions such as dementia and chronic kidney disease, many of which have rarely been explored using PPG. Collectively, these findings indicate that easily acquired PPG signals encode rich health-related information extending well beyond conventional cardiovascular assessment.

We prove that groups of the form $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$, where $m,n \in \mathbb N$, are regularly bi-interpretable with $\mathbb Z$ and therefore are first-order rigid: every finitely generated group elementarily equivalent to $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$ is isomorphic to $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$. On the other hand, we show that $\mathbb Z^2 {\,\rm wr\,} \mathbb Z$ admits $2^{\aleph_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.

Testing general relativity in the strong-field and highly dynamical regime is now possible through current gravitational-wave observations, where even a single high-quality detection can place competitive constraints on deviations from Einstein's theory. The parametrized post-Einsteinian framework provides a theory-agnostic approach to search for such deviations, but it typically assumes that systematic uncertainties in the base waveform model, particularly those arising from calibration to numerical relativity, are negligible. In this work, we investigate how calibration errors in the late-inspiral fitting coefficients of the IMRPhenomD waveform model can lead to spurious detections of departures from general relativity in parametrized tests. We use an uncertainty-aware version of IMRPhenomD, recalibrated to a set of numerical relativity surrogate waveforms and equipped with a probabilistic description of its fitting coefficients, to simulate general-relativity-consistent signals. We inject these signals into an O5 ground-based detector network and recover them with the original IMRPhenomD model augmented with a parametrized post-Einsteinian phase deformation. We find that false violations of general relativity using this model arise for network signal-to-noise ratios as low as 60. When the uncertainty-aware model is used instead, the inferred parametrized post-Einsteinian phase deformation remains consistent with zero even for signals with a signal-to-noise ratio up to 330. These results demonstrate the need to account for numerical relativity calibration uncertainty in order to perform reliable inspiral tests of general relativity. They also illustrate that explicitly incorporating numerical relativity calibration uncertainty into the waveform model preserves our ability to robustly test general relativity.

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main features are a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, and a ``surpassing relation,'' taking the place of equality, which permit generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' $(\mcA,\mcA_0)$ is studied along the lines of universal algebra.

We study a dispatching and pricing problem in two-sided spatial queues with fixed supply, motivated by ride-hailing and robotaxi platforms. Idle drivers queue on one side, waiting to pick up riders, while riders queue on the other, waiting to be matched with available drivers. The platform seeks to maximize net profit, penalized by rider waiting penalties, by jointly optimizing state-dependent dispatching and pricing decisions. We formulate this problem as a Markov decision process with state-dependent service times that capture key features of spatial matching. We show that, under mild assumptions, the optimal dispatching policy admits a closed-form expression with a zigzag structure. This policy significantly improves the tractability of pricing optimization due to the resulting closed-form stationary distribution and a substantially reduced state space. Building on this insight, we propose an efficient and scalable dynamic programming heuristic to approximate the optimal zigzag policy in more general settings. Extensive numerical experiments with both the analytical model and ride-hailing simulations demonstrate that our algorithm is both near-optimal and highly scalable.

Accurate mode tracking is essential for elastic waveguide dispersion analysis in ultrasonic nondestructive evaluation and structural health monitoring. Its reliability, however, deteriorates near mode veering and closely spaced eigenvalues, where rapid eigenvector exchange causes mode misidentification. Although widely observed, the quantitative relationship between veering, eigenvector evolution, and tracking robustness has not been systematically established. By specializing classical perturbation theory to the single-parametric Hermitian SAFE eigenproblem--exemplified by conservative elastic waveguides--we obtain explicit expressions for eigenvector derivatives and modal coupling strength. This yields a unified, quantitative interpretation of mode veering, symmetry-protected crossings, and degeneracies, and clarifies their distinct tracking implications: eigenvector sensitivity scales inversely with the eigengap, explaining modal repulsion and the degradation of correlation-based tracking near avoided crossings, whereas symmetry-protected crossings remain benign because symmetry-induced decoupling preserves smooth eigenvector evolution, and symmetry-protected degeneracies require rotation-invariant subspace tracking. A numerical consistency condition and an existence result for a critical step size are then derived, motivating a two-level adaptive strategy with an a posteriori error indicator that separates numerical tracking consistency from symmetry-based physical correctness. Numerical examples validate the theoretical predictions and demonstrate improved robustness in regions of strong modal interaction, providing practical guidance for reliable dispersion calculations and ultrasonic inspection.

The spectral function of the charged pion form factor was extracted from two data sets. The difference between the two sets is based on the presence or absence of the recent measurement of the $π^+ π^-$ production by CMD-3 . Although the CMD-3 data are largely incompatible with other recent measurements, no excess was found when the data were used for analytical continuation to the spacelike region and compared to JLaB-$π$ experiment. Instead, both data sets provide a spacelike form factor that differs from the expected behavior known from perturbative analyses of quantum chromodynamics (QCD). A precise fit of the spectral functions is provided, either based on he bare form factor and the full $e^+e^-\rightarrow π^+π^-$ as well . Furthermore, the extracted pionic spectral functions were used to obtain the QED running charge. This was then compared with the KLOE-2 measurements.

Nonlocal quantum gates, coupling quantum systems located at a distance, are crucial for distributed quantum computing. To this aim, high-capacity optical noiseless connections between different processing units are essential for transmitting large amounts of information per mode. Simultaneously, optical quantum computing offers future high-speed multimode quantum processors. We propose a library of feasible protocols to implement a necessary nonlocal continuous-variable (CV) quantum nondemolition (QND) gate between two distant users sharing a quantum channel and exploiting classical communication. The users are endowed with a newly achieved high-fidelity and large-bandwith element - single-pass phase-sensitive optical parametric amplifier (OPA), that allows for both online squeezing and channel-loss compensation. The use of OPAs enhances quality of the resulting gate in terms of both excess noise and entangling capability. The proposed schemes are also applicable to CV cluster state fusion, providing a first step towards development of distributed CV measurement-based quantum computation.

The tautological $\mathbb{Q}$-subalgebra $\mathsf{R}^*(\mathcal{A}_g) \subset \mathsf{CH}^*(\mathcal{A}_g)$ of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical $\mathbb{Q}$-linear projection operator $\mathsf{taut}: \mathsf{CH}^*(\mathcal{A}_g) \rightarrow \mathsf{R}^*(\mathcal{A}_g).$ We present here new calculations of intersection products of the Torelli locus in $\mathcal{A}_g$ with the product loci $\mathcal{A}_{r}\times \mathcal{A}_{g-r} \rightarrow \mathcal{A}_g$ for $r\leq 3$. The results suggest that $\mathsf{taut}$ is a $\mathbb{Q}$-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property. Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps. We define tautological projections of cycles on the fiber products $\mathcal X_g^s \to \mathcal A_g$ of the universal family. We compute these projections for a class of product cycles on $\mathcal X_g^s$ in terms of a determinant involving the universal theta divisors and Poincaré classes. Using Abel-Jacobi pullbacks of product cycles on $\mathcal X_g^s$ and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings $\mathsf{R}^*(\mathcal M^{\mathrm{ct}}_{g,n})$. In particular, we construct nontrivial elements of the Gorenstein kernels of $\mathsf{R}^5(\mathcal{M}_{5,2}^{\mathrm{ct}})$ and $\mathsf{R}^5(\mathcal{M}_{4,4}^{\mathrm{ct}})$.

We study the parabolic fractional $p-$Laplace equation $\partial_t u+(-Δ_p)^su = 0$ in the degenerate range $2 < p < 2/(1-s)$. We show that weak solutions are Lipschitz continuous in space and, if $p > 1/(1-s)$, also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.