This paper addresses the stability analysis and state estimation of generalized Persidskii systems subject to time-varying delays and external disturbances. The generalized Persidskii class, which couples linear dynamics with sector-bounded nonlinear feedback loops, offers a tractable yet expressive framework for modeling electromechanical and neural network systems. We develop delay-dependent conditions for input-to-state stability (ISS) via Lyapunov--Krasovskii functionals incorporating Persidskii-type integral terms, and cast these conditions as linear matrix inequalities (LMIs). A structured robust observer is proposed for systems with partial state measurement, and its convergence is guaranteed through an $H_\infty$ synchronization criterion. To handle plant uncertainty, the system matrices are identified from trajectory data using a stability-preserving Koopman lifting procedure, in which the ISS-LMI constraint is embedded as a convex side condition during parameter regression. The identified model populates the prediction horizon of an ICODE-MPPI (Input-dependent Control-oriented Dynamical Estimation -- Model Predictive Path Integral) controller. The complete framework is validated on a 1.5 kW Permanent Magnet Synchronous Motor (PMSM) drive equipped with a programmable load brake. Experimental results confirm a 35\% reduction in velocity estimation RMSE relative to an Extended Kalman Filter and a 67\% improvement in speed-tracking accuracy relative to standard Field-Oriented Control, corroborating the theoretical ISS bounds established herein.

We show that the operational quantum formalism -- the Born rule, Lüders state updating, quantum interference, non-Markovian effective dynamics, and Bell inequality violation at the Tsirelson bound $2\sqrt{2}$ -- arises within Accessibility Theory (AT) from the Aperture construction together with explicit coherence and locality assumptions stated in the paper. AT is a framework built on real graded spectral triples and a single algebraic selection principle. The Principle of Universal Accessibility Balance requires three independent measures of the complexity of a spectral triple -- its algebraic, gauge-theoretic, and geometric content -- to be exactly equal and minimized, uniquely selecting the algebra $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ and with it the Standard Model gauge group, particle content, four-dimensional Lorentzian spacetime, three generations, and gravitational dynamics. Restriction to a codimension-one geometric boundary reduces this algebra to its commutative center $\mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C}$ -- the Aperture -- which defines a permanent information bottleneck for any embedded observer. Coherence conditions on inference through this bottleneck, together with Gleason's theorem on the 48-dimensional internal Hilbert space, uniquely determine the Born rule; the remaining operational features follow from the same observer-level framework under the stated assumptions. At the ontological level the theory is deterministic and state-realist, while the operational quantum formalism appears at the observer level as a consequence of structurally limited access to the underlying algebraic reality.

Limited connectivity makes nonlocal quantum gates expensive on near-neighbor hardware, where compilation typically relies on SWAP transport, inheriting both depth overhead and path congestion. We present a swap-free routing framework in which higher levels of a qudit act as orthogonal spectral buses that transport control information without moving the computational state. We show that exact congestion relief in nearest-neighbor architectures requires local Hilbert-space expansion. In this model, a nonlocal operation over a path of length $L$ requires $2L+1$ logical routing primitives, compared to the $3L$ baseline. Overlapping routes remain distinguishable through bus labels encoded in the same physical qudits. This routing algebra extends to Boolean fan-in at a common target: multiple controls arriving on distinct buses trigger a local unitary based on an arbitrary Boolean function of bus digits, yielding multi-control operations of depth $2L + D_g + O(1)$ for fan-in size $K$ and target-synthesis cost $D_g$. We prove decodability, reversibility, and correctness for CNOT and Boolean fan-in, along with a state-count lower bound $d \geq 2^{K+1}$ for exact overlap routing. Cirq simulations confirm single-control correctness and zero crosstalk. Compiler-level benchmarks on QFT, QAOA, and mirror-interaction circuits verify the predicted congestion law and transport reduction. Noisy QuTiP simulations show that the architectural advantage depends on higher-level coherence and speed. These results identify spectral qudit routing as a congestion-relief architecture that separates nonlocal control delivery from local target-side aggregation, providing a minimal mechanism for overcoming qubit routing limitations.

\emph{Literate programming}, introduced by Knurth, interleaves code and prose so that a program can be read as both executable and explanatory text. We propose \emph{literate execution}, which inverts this relationship: rather than embedding code within a static narrative, we treat documentation -- and other expository elements such as visualisations -- as first-class artefacts that can be computed alongside a running program and then integrated into a view of its execution. We explore this idea through Fluid, a programming language with a provenance-tracking runtime that records fine-grained dependencies between inputs and outputs. These provenance relationships can be surfaced as interactions that allow readers to explore how intermediate values contribute to a result. By integrating visualisation, provenance, and exposition, literate execution aims to make programs more explorable and self-explanatory, and explorable explanations easier to program.

This paper studies exact fixed-cardinality Solow--Polasky diversity subset selection on ordered finite $\ell_1$ point sets, with monotone biobjective Pareto fronts and their higher-dimensional staircase analogues as central applications. Solow--Polasky diversity was introduced in biodiversity conservation, whereas the same inverse-matrix expression appears in metric geometry as magnitude: for a finite metric space $(X,d)$ with exponential similarity matrix $Z_{ij}=e^{-q d(x_i,x_j)}$, the quantity $\1^\top Z^{-1}\1$ is the magnitude of the scaled finite metric space $(X,qd)$ whenever the weighting is defined by the inverse matrix. Thus, in this finite exponential-kernel setting, Solow--Polasky diversity and magnitude are mathematically the same object viewed through different motivations. Building on the linear-chain magnitude formula of Leinster and Willerton, the paper gives a detailed proof of the scaled consecutive-gap identity $ \SP(X)=1+\sum_r \tanh(qg_r/2),$ where the $g_r$ are the gaps between consecutive selected points. It then proves an exact Bellman-recursion theorem for maximizing this value over all subsets of a prescribed cardinality, yielding an $O(kn^2)$ dynamic program for an ordered $n$-point candidate set and subset size $k$. Finally, the paper proves ordered $\ell_1$ reductions showing that the same algorithm applies to monotone biobjective Pareto-front approximations and, more generally, to finite coordinatewise monotone $\ell_1$ staircases in $\R^d$. These are precisely the ordered $\ell_1$ chains for which the $\ell_1$-distance becomes a line metric along the chosen order, so the one-dimensional dynamic program applies without modification. Keywords: Solow--Polasky diversity; magnitude; metric geometry; dynamic programming; ordered points; $\ell_1$ geometry; Pareto-front approximation.

In pedestrian dynamics, the internal drive that propels individuals toward their goals is typically captured by a single, fixed parameter, the desired walking speed. This simplification overlooks that motivation fluctuates in response to changing spatial and social conditions within a crowd. This paper proposes a dynamic motivation model grounded in expectancy-value theory from psychology, in which each agent's motivation evolves over time depending on proximity to the goal, relative position among other pedestrians, and individual goal importance. The resulting motivation modulates multiple movement parameters simultaneously, including walking speed, gap-closing behavior, and interpersonal spacing. The model is evaluated in simulated pre-bottleneck waiting scenarios using paired statistical comparisons across multiple random seeds and population sizes, and compared with trajectory data from the CROMA concert-entry bottleneck experiments under low- and high-motivation framings. Simulations show that the dynamic model produces structured heterogeneity in the crowd: agents self-organize into differentiated positions near the bottleneck, with those closer to the front occupying less space, a pattern absent in the static baseline but clearly present in the experimental data. These findings suggest that motivation in crowds should be understood not as a uniform increase in urgency, but as a mechanism that reorganizes competitive positioning along spatial and social axes. Future work should extend the framework to open-door throughput scenarios, larger populations, and richer social interactions such as group cohesion and cooperative strategies.

Lutwak's affine quermassintegral theory is a foundational component of modern affine Brunn--Minkowski theory. Developed in the 1980s, it provides affine analogues of the classical quermassintegrals and has led to a rich family of sharp affine isoperimetric inequalities. A central question in this program, going back to Lutwak's 1988 work, is an Alexandrov--Fenchel-type monotonicity principle for the normalized $L^{-n}$-moment quermassintegrals $I_{k,-n}$. In one form, this principle predicts that \[ I_{m,-n}(K)^{1/m}\ge I_{k,-n}(K)^{1/k}, \qquad 1\le m<k\le n . \] The question was recorded in Gardner's 2006 book Geometric Tomography as part of its problem list, and the comparison with the top dimension, $k=n$, was established by Milman and Yehudayoff in their 2023 JAMS paper. We show that the proposed monotonicity does not persist in the full range. More precisely, for every triple of integers $m,k,n$ satisfying $1\le m<k\le n-1$ and $n>(m+2)(k+2)-2$, there exists an origin-symmetric $C^2_+$ convex body $K\subset\mathbb R^n$ such that \[ I_{m,-n}(K)^{1/m} < I_{k,-n}(K)^{1/k}. \] The example is obtained from the Euclidean ball by an arbitrarily small degree-four spherical harmonic perturbation. On the positive side, we prove that the endpoint chain is true in dimension three: for every convex body $K\subset\mathbb R^3$, \[ I_{1,-3}(K)\ge I_{2,-3}(K)^{1/2}\ge I_{3,-3}(K)^{1/3}=1. \] The equality cases in both non-trivial inequalities are exactly ellipsoids, up to translation and nonsingular affine transformations.

We study a projection-type gradient flow for equality-constrained maximisation of a smooth bilinear control objective on $\mathcal{H}=L^2(0,T;\mathbb{R})$, eliminating Lagrange multipliers through an $(M{+}1)\times(M{+}1)$ moving Gram matrix $Γ(s)_{\ell\ell'}=\int_0^T S(t)\,c_\ell(s,t)\,c_{\ell'}(s,t)\,\mathrm{d}t$. The flow generates monotonic ascent in continuous time but becomes unstable on discretisation; existing implementations rely on heuristic step-size safeguards lacking rigorous justification. We close this gap by replacing $Γ$ with $Γ_{\varepsilon}:=Γ+\varepsilon^{2}I$ and prove: (i) an exact spectral identity giving $κ(Γ_{\varepsilon})=(σ_{\max}^{2}+\varepsilon^{2})/(σ_{\min}^{2}+\varepsilon^{2})$; (ii) objective monotonicity $\mathrm{d}J/\mathrm{d}s\ge 0$ for all $\varepsilon\ge 0$; (iii) constraint drift $|h_{m}-C_{m}|=\mathcal{O}(\varepsilon^{2})$ with a computable prefactor; (iv) convergence of the regularised trajectory to the unregularised one in $L^{2}(0,T)$ at rate $\mathcal{O}(\varepsilon^{2})$ under uniform invertibility of $Γ$; and (v) a discrete CFL criterion $Δs\,G\,\|Γ_{\varepsilon}^{-1}\|\leα<2$ guaranteeing objective monotonicity of the forward-Euler scheme up to $\mathcal{O}(Δs^{2})$ local truncation error. The theory is validated on a three-level bilinear benchmark for all-optical Bell-state preparation, where $κ(Γ)\in[10^{9},10^{11}]$, the predicted $\varepsilon^{2}$ rate is confirmed over eight decades, and moderate regularisation eliminates step rejections and reduces constraint drift by more than an order of magnitude at unchanged final fidelity.

This paper studies the Nash equilibrium seeking problem for stochastic games under heavy-tailed noise. The gradient noise is considered to have a finite $δ$-th moment ($1<δ\le 2$), which generalizes the Gaussian noise and covers cases with infinite variance. In this work, we employ the classic method Median-of-Means (MoM) in robust estimation. MoM works by dividing samples into blocks, taking the average of each block, and then taking the median of these block averages, achieving a breakdown point of up to $1/2$. This makes the final estimate reliable even when some samples are very noisy or wrong, and thus is effective to handle the heavy-tailed noise. The method also naturally defends against malicious gradient attacks. Compared with gradient clipping, which is the most popular method to deal with the heavy-tailed noise, MoM requires no preset clipping threshold and is insensitive to the tail behavior of the noise. Under standard assumptions, we prove the almost sure convergence of the algorithm and derive its almost sure convergence rate. To address the systematic bias caused by asymmetric noise, we further design an online bias correction strategy. Simulation results show the effectiveness and efficiency of the proposed algorithms.

This note deals with the question: If T is a linear mapping between Banach spaces X and Y, and x belongs to X and has small norm, is x close to the kernel of T? It draws on notions of Z-stability and provides an affirmative constructive answer when T is onto Y, sequentially continuous, and has located kernel.

ISAC is currently being standardized within the 3GPP New Radio (NR) to enable cellular infrastructure to perform sensing using existing communication waveforms. While standardization is progressing, practical deployment may be limited by scenario-dependent observability constraints. For example, in UMa-AV scenarios, sensing with a single TRP can be affected by restricted angular coverage, partial blockage, and limited field of view, which may degrade detection reliability in three-dimensional UAV environments. For this reason, multi-TRP solutions have been suggested to improve spatial diversity and sensing robustness. In this paper, we present a system-level investigation of multi-TRP assisted monostatic sensing for UAV detection under standardized 3GPP UMa-AV channel assumptions and Release 19 evaluation parameters. We propose a spatial diversity fusion framework and evaluate the achievable performance of a 3GPP network by combining the measurements obtained independently at different TRP. Extensive evaluations demonstrate that multi-TRP assistance improves target observability, reduces spurious detections, and tightens localization error distributions at the cost of additional sensing overhead due to the need for multiple TRPs to periodically allocate radio resources for sensing measurements. In the evaluated scenario, results show that a voting threshold of two assisting TRPs achieves an optimal trade-off between miss detection probability and false alarm suppression, meeting 3GPP performance objectives. Furthermore, we quantify the sensing overhead and show that proper system design, tuned to the application requirements, can substantially reduce its impact: for example, extending the sensing refresh interval beyond the 128 ms coherent processing interval to 1 s reduces the effective overhead from 29 % to approximately 3.7 %, enabling more scalable network deployment.

We develop the theory of nilpotency and the Frattini theory for transposed Poisson algebras. The lower central series is shown to admit a simplified form, and an analogue of Engel's theorem is established: a finite-dimensional transposed Poisson algebra is nilpotent precisely when the left multiplication operators in both the associative and the Lie structures are nilpotent. Constructions of nilpotent and solvable algebras via tensor products and derivations are given. For a finite-dimensional Lie-nilpotent transposed Poisson algebra, we prove that the derived Lie subalgebra is a nilpotent ideal, which implies that the nilpotent radical coincides with the associative radical. In the framework of Frattini theory, we show that the Frattini subalgebra is always contained in the derived algebra and the Frattini ideal is associative nilpotent. When the algebra is nilpotent, all maximal subalgebras are ideals and the Frattini subalgebra equals the derived algebra. Conversely, for a Lie-nilpotent transposed Poisson algebra, if all maximal subalgebras are ideals, the algebra either is nilpotent or decomposes as a direct sum of a one-dimensional algebra generated by an idempotent and the nilpotent radical; if the Frattini subalgebra equals the derived algebra, the algebra is necessarily nilpotent. We also prove that the zero socle coincides with the nilpotent radical, and when the Frattini ideal is zero, the algebra splits into a subalgebra and its zero socle; in the Lie-nilpotent case this subalgebra is abelian as a Lie algebra.

We show that the faithful highest weight hearts in an algebraic triangulated category are the serially faithful glued hearts, equivalently the hearts containing a dual pair of full exceptional collections in the sense of Bodzenta--Bondal (arXiv:2601.22004). We then characterise faithful highest weight categories of perverse sheaves on topologically stratified spaces algebraically, in terms of the exactness of certain functors, and topologically, in terms of the vanishing of certain cohomology groups of pairwise links. We prove that the global dimension of a faithful category of perverse sheaves on a topologically stratified space $X$ with finitely many strata is bounded by the dimension of $X$. Finally, we show that in this setting the hypercohomology of a perverse sheaf can be computed from a projective resolution of the constant sheaf, and conversely that the multiplicities of the terms in a minimal projective resolution of the constant sheaf can be computed as intersection cohomology groups.

This article presents a partial differential equation (PDE) of Keller-Segel (KS) type that reproduces patterns commonly observed during the growth of brain microvasculature. We provide mathematical insights into the mechanisms underlying the emergence of these patterns. In addition, we derive a data-driven equation that ensures a consistent temporal evolution of the chemoattractant associated with the observed microvascular dynamics. Beyond numerical simulations, the aim of this study is to advance a comprehensive mathematical modeling framework, spanning blood flow in cerebral arterial networks to biochemical processes, in order to better understand how vascular impairments may contribute to neurodegenerative diseases.

We propose a parallel protocol for implementing distributed nonlocal quantum gates between spatially separated stationary qubits encoded in dual-species quantum emitters (i.e., color-center and superconducting qubits). By utilizing entangled photon pairs with distinct frequencies as a quantum data bus, our approach connects spatially separated devices without requiring quantum frequency conversion or preshared entanglement, while maintaining an always-ready and resource-efficient property for distributed quantum computing and networks. Furthermore, we demonstrate the feasibility of implementing parallel distributed nonlocal quantum gates on multiple pairs of spatially separated qubits using a single high-dimensional entangled photon pair, which directly benefits from the enhanced quantum capacity provided by optical qudit encoding. Our protocol establishes a scalable and practically implementable framework for distributed quantum networks, potentially enabling the development of future large-scale quantum computing architectures.

We propose and analyze a new method of detecting center vortices and monopoles in lattice Yang-Mills theory. This procedure is sensitive to the intrinsic degeneracy of the center charges, which play a crucial role in how these topological objects interact and correlate with one another. Our approach is based on fixing the Maximal Abelian gauge (MAG) and decomposing the link configuration in a suitable way to look for so-called Cartan fluxes. Our discussion is general for $SU(N)$ gauge theory, but we focus our applications on the $SU(3)$ case. For the $SU(2)$ case, our proposed parametrization is equivalent to the usual one.

Vehicle-to-Grid (V2G) technology allows bidirectional power flow for real-time grid support, making electric vehicles (EVs) well-suited for ancillary services such as frequency regulation. However, existing methods for flexibility estimation and coordinating aggregated EVs often rely on individual EV traveling information (e.g., arrival/departure time) and/or characteristic parameters (e.g., charging efficiency, battery capacity) as well as real-time state-of-charge (SOC), which raises privacy concerns and faces data quality issues. To address these challenges, this paper proposes a data-driven, privacy-preserving modeling and control framework for frequency regulation using aggregated EVs. The proposed method can provide accurate estimation for power outputs and flexibility of aggregated EVs and carry out effective frequency regulation without any individual EV information. Simulation results validate the accuracy and effectiveness of the proposed method, which also outperforms the model-based and federated learning-based method under SOC data inaccuracies.

Gatemon qubits are based on a superconductor-quantum dot-superconductor (S-QD-S) junction which enables in situ electrostatic tuning via a gate electrode. For a single-channel QD this structure gives rise to two subgap Andreev bound states (ABSs), and generally leads to a richer quantum phase dynamics as compared to conventional transmons. In a recent work [Phys. Rev. B 111, 214503 (2025)] we derived the quantum phase dynamics from a many-body treatment which leads to an effective gate voltage-dependent Hamiltonian that self-consistently incorporates the phase quantization. It predicts (i) a renormalization of the junction's effective capacitance and (ii) the presence of gate voltage and occupation-dependent charge offsets in junctions with tunneling asymmetry. Here, we quantify the observable impact of these effects on the qubit's energy spectrum and anharmonicity, by studying the interplay of the two Andreev branches as a function of dot-gate voltages and junction transparencies. We show the relation of these predictions to simplified gatemon models and propose a protocol to experimentally detect the predicted charge offsets.

Numerical analysis to determine the form of the scalar power spectrum in Warm Inflationary paradigm is inevitable. One further needs numerical techniques to analyse any Warm Inflation model with the current observational data through the MCMC codes that are available publicly, like COSMOMC or Cobaya. We present SWIM (Stochastic Warm Inflation Module) written in C++ and Python, that not only helps generate the Warm Inflationary scalar power spectrum, either semi-analytically or fully numerically, but also is integrated with Cobaya enabling the user to constrain the model parameters with current CMB data and thus to put any Warm Inflation model to test. SWIM numerically solves the standard stochastic perturbation equations of Warm Inflation without any approximations, uses machine learning techniques to speed up the MCMC analysis while analysing the fully numerical power spectrum that significantly reduces the computational cost, and is able to accommodate any Warm Inflation model with any form of inflationary potential and dissipative coefficient for numerical analysis. We show that SWIM, in most of the cases, outperforms other numerical codes on Warm Inflation that are designed to yield only the semi-analytical power spectrum as far as the runtimes are concerned. We further point out that there can be situations where the semi-analytical way of determining the scalar power spectrum in Warm Inflation can fall short, and one needs the full numerical power spectrum for parameter estimation given the observational data. In such cases, SWIM is the only code available so far that is designed to perform the task. Hence, SWIM offers a complete numerical platform for thorough analysis of Warm Inflation models against the current cosmological data. SWIM has been made publicly available at https://github.com/umg-kmr/SWIM.

Accurate estimation of thermospheric mass density is a prerequisite for orbit prediction and space situational awareness, where the upper atmosphere responds nonlinearly to solar and geomagnetic forcing across several orders of magnitude. Physics-based general circulation models resolve this response but are computationally expensive, while empirical models run cheaply but lack a time-evolving atmospheric state. This work couples a data-driven reduced-order thermospheric model with a Kalman filter that assimilates in situ density observations. An autoregressive Sparse Identification of Nonlinear Dynamics with control (SINDy$_c$-AR) reduced-order model derived from the Thermosphere-Ionosphere-Electrodynamics General Circulation Model (TIE-GCM) captures the dominant modes of variability and their dependence on solar and geomagnetic drivers at a fraction of the parent model's cost. Density observations from CHAMP, GRACE, GRACE-FO, GOCE, and Swarm are assimilated across a range of orbital configurations and geomagnetic conditions, with a linear DMDc model evaluated as a reference. Assimilation reduces density estimation error relative to open-loop predictions, most visibly during geomagnetic storms and under single-satellite coverage. SINDy$_c$-AR and DMDc perform comparably on assimilated orbits; on withheld orbits, SINDy$_c$-AR is more accurate in the in-training scenarios while DMDc is better in the out-of-training 2024 Swarm-C case. Benchmarks against NRLMSIS~2.1 and HASDM (2000--2019, where available) show that empirical references can outperform the assimilated model far from the assimilated track, so results are framed as improvements over the open-loop forecast.

The nonlinear synergy between global warming and urbanization is amplifying extreme climate risks in cities worldwide. While observations and simulations confirm these compounding effects, two fundamental bottlenecks impede predictive understanding: (1) fragmented, case-specific perspectives that hinder the discovery of universal mechanisms, and (2) a methodological divide between computationally prohibitive high-resolution models and AI-based tools that lack physical interpretability at urban scales. This article advocates for a paradigm shift toward the deep integration of physical principles with data intelligence. To this end, we propose a transformative "Classification-Mechanism-Inference" (CMI) framework. Classification involves establishing a global urban "climate-morphology-development" typology to enable systematic comparison beyond isolated case studies. Mechanism advocates for physics-informed machine learning (PIML) as the core engine to develop efficient, physics-constrained surrogate models for uncovering nonlinear interactions. Inference leverages these models for high-throughput, tailored risk projection to directly inform context-specific adaptation planning. The CMI framework aims to bridge the cognitive and methodological gaps, thereby advancing urban climate science from phenomenological description towards mechanistic, predictive, and decision-relevant science, which is crucial for building climate-resilient cities globally.

Cluster randomized trials are widely used when individual randomization is logistically infeasible or when correlations between observations cannot be ignored, especially in fields such as ophthalmology, infectious disease, vaccine research, and sociology. The desirability of outcome ranking (DOOR) framework evaluates patient-centric benefit-risk using an ordinal outcome and a Wilcoxon-Mann-Whitney statistic-based approach to compare outcome distributions between interventions. We propose a suite of new methods to extend DOOR to cluster trials based on properties of U-statistics and influence functions to estimate within-cluster and between-cluster treatment effects. These approaches can be applied in different scenarios, including mixtures of clusters with two treatment groups and clusters with only one group, and both small and large numbers of clusters. Simulations demonstrate that the proposed methods perform well under various scenarios regarding the number of clusters and cluster sizes. As an illustration, we apply the proposed methods to a cluster randomized crossover trial comparing delayed cord clamping and umbilical cord milking for newborns.

In many astrophysical transients, outflows drive shocks into the ambient medium, accelerating electrons to non-thermal energy distributions that produce broadband synchrotron emission. At late times, even initially collimated relativistic jets evolve into quasi-spherical Newtonian blastwaves. As the shock decelerates, the post-shock internal energy per particle decreases; below a critical velocity $β_{\rm DN} \approx 0.2$, only a fraction $ξ_e < 1$ of electrons are accelerated to relativistic energies, defining the deep Newtonian (DN) regime. We develop a unified analytic framework for synchrotron emission in this phase, applicable to both single-velocity and stratified ejecta. For gamma-ray burst afterglows in a uniform medium, the DN transition occurs at $t_{\rm DN} \approx 3.7\,E_{51}^{1/3} n_0^{-1/3}$~yr, yielding a shallower decay by $δα= 6(p-2)/5$ relative to standard Newtonian predictions. For kilonova remnants ($E_0 = 10^{50.5}$~erg, $M_{\rm ej} = 0.1\,M_\odot$), the DN phase begins prior to deceleration; neglecting it underestimates radio flux by factors of $\sim 3$--$5$ during coasting and even more thereafter. Magnetar-boosted remnants ($E \sim10^{52}$~erg) should reach $\sim$\,10\,--\,100\,$μ$Jy at 3~GHz at $\sim$\,40\;Mpc, though limits on GW170817 already disfavor a long-lived millisecond magnetar. In core-collapse supernovae in a wind medium ($ρ\!\propto\!r^{-k}$), the peak luminosity remains constant during coasting, while $ν_{\rm pk} \propto t^{-1}$; for SN~2023ixf, we find $k = 1.29 \pm 0.14$. The DN spectral energy distribution typically satisfies $ν_m\!<\!ν_{\rm sa}\!<\!ν_c$, peaking at sub-GHz frequencies where LOFAR and SKA-low are most sensitive. Even non-detections place robust constraints on ambient density and outflow energetics.

Automated failure diagnosis requires correlating browser-visible symptoms with backend observability signals, yet existing benchmarks do not evaluate this cross-modal reasoning task. Constructing one is non-trivial: multi-modal failure scenarios are costly to annotate, and live-environment capture introduces stochasticity that makes cross-run agent comparison unreliable. We present CUJBench, to our knowledge, the first benchmark to combine browser-visible failure evidence with backend observability in a diagnostic framing. CUJBench addresses annotation cost through an LLM-assisted generation pipeline with a multi-agent review loop and a three-layer annotation scheme, producing 87 labeled scenarios across five fault families, and ensures reproducibility by packaging each failure as a deterministic multi-modal snapshot with a fixed tool interface. Evaluating six frontier models under retrieval, browser-only, and full-toolset baselines, the benchmark yields an overall accuracy of 19.7% with a ceiling of 52%, well below saturation. Contrary to expectation, browser-only agents outperform full-toolset agents in aggregate, with expanded evidence access inducing unfocused exploration rather than improved synthesis. Trajectory analysis identifies cross-modal synthesis as the primary bottleneck: agents retrieve the decisive evidence but fail to attribute it correctly - a structural limitation uniform across all six models that model scale and richer tool access alone cannot resolve.

For every nonzero integer $m$ and every integer $n \ge 1$, the $n$\textsuperscript{th} harmonic number $H_n = 1 + \tfrac12 + \dots + \tfrac1n$ satisfies the identity \[ H_n \;=\; \frac{1}{m}\,\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k}\, \binom{m k}{k}\binom{n + (m-1)k}{n - k}. \] The cases $m = 1$ and $m = 2$ are classical; for general nonzero integer $m$ the identity was conjectured by P.~Bala in the OEIS entry A001008 in 2022 and remained open. We prove it here, working throughout in $\QQ[[x]]$. The proof reduces, via a substitution $u = x/(1-x)^m$, to two formal-power-series identities: a Lagrange--Bürmann evaluation of $\sum_{k\ge1} \binom{mk}{k} u^k / k$, and the fixed-point fact that under that substitution the unique solution $v(u)$ of $v = u(1-v)^{m}$ is $v = x$. The argument extends verbatim to arbitrary complex $m \ne 0$.

We study the exact counting problem for all lattice rectangles contained in the square $[0,n)\times[0,n)$, including non-axis-parallel ones. Starting from the standard parametrization by a primitive direction $(u,v)$ and two side lengths, we derive several exact algorithms: the classical $O(n^2)$ sweep, decompositions of complexity $O(n^{3/2}\log n)$ and $O(n^{4/3}\log n)$, a ten-moment weighted-floor-sum reduction of complexity $O(n\log^3 n)$, and a divisor-layer algorithm with the complexity $O(n\log^2 n)$. We also give an all-values algorithm that computes $F(1),\ldots,F(N)$ in $O(N^{3/2})$ arithmetic operations. The main idea behind the near-linear one-value algorithms is to reduce the geometric summation to constant-size families of weighted floor sums closed under Euclidean-style affine and reciprocal transformations. Besides the exact algorithmic results, we derive a two-term asymptotic expansion, $F(n)=\frac{4\log 2-1}{π^2}n^4\log n+B\,n^4+o(n^4)$ with the explicit formula for $B$, which provides an independent consistency check for the large-$n$ numerical data produced by the algorithms.

We study the integrability conditions of the conformal Killing equations for the Eisenhart lift of a scalar field in a flat Friedmann-Lema\^ıtre-Robertson-Walker universe. We show that the potential found in our earlier work is already the most general local potential that admits a non-trivial conformal Killing vector in the sector independent of the cyclic Eisenhart coordinate. The determinant condition of the prolonged conformal Killing equations reduces to a nonlinear second-order differential equation for $h=V'/V$. We solve this equation locally and find two branches. The regular branch reproduces exactly the family of potentials obtained before, while the singular branch lies on the locus where the determinant equation cannot be written locally in normal form with respect to $h''$ and is incompatible with the full conformal Killing equations. Hence the ansatz used in our earlier work is exhaustive.

Novel target materials with anisotropic response will play a key role in detecting low-mass dark matter in upcoming experiments. Bilayer graphene is one such material that has been proposed for the detection of sub-MeV mass dark matter particles via electronic excitations. In this work, we calculate scattering rate via a massive mediator in bilayer graphene. With an exposure as small as $\sim$ 0.5 mg-year, bilayer graphene can probe new regions of the parameter space. The anisotropic response function of bilayer graphene leads to a sidereal-day modulation in the scattering rate, depending on its orientation with respect to the Galactic dark matter wind. We find significant modulation in the scattering rate for sub-MeV mass dark matter, demonstrating bilayer graphene's promise for a future experiment. We hope that our work will motivate the community to investigate bilayer graphene as a novel target material, and that it may lead us to discover the particle nature of dark matter.

There is no microscopic quantum approach based on the many-body time-dependent Schrödinger equation which capable to describe the formation and the evolution of a compound nucleus. The most advanced microscopic approach developed so far to describe multi-nucleon transfer (MNT) reactions in complex nuclear systems (with total number of nucleons $\gg 100$) is the time-dependent Hartree Fock (TDHF) mean field theory. In any mean field approach, however, the mean field is an expectation value of a quantum operator, thus classical in nature and unable to describe its quantum fluctuations, which are often expected to be crucial. Quantum fluctuations can be in principle be included in a configuration interaction (CI) framework, which in the case of reactions has to be implemented in the continuum. Here we describe the first such implementation within a novel extension of the well known Generator Coordinate Method (GCM), dubbed the enhanced GCM (eGCM), applied to the MNT reaction $^{48}$Ca+$^{208}$Pb near the Coulomb barrier, which demonstrates major qualitative differences with either TDHF or GCM previous approaches.

We study properties of $A^p_α$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that \[ \int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^α \, dA(z) < \infty, \] when $0<α< 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_α$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_α$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_α$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_α$ is contained in the weak product of a Dirichlet-type space.