Fanaroff-Riley class I (FRI) radio galaxies show centre-brightened emission from disrupted lower power jets, while traditionally more luminous class II (FRIIs), are edge-brightened, with relativistic jets terminating in hotspots. Population studies of radio-loud AGN (RLAGN) with low frequency, deep, wide-field surveys have revealed FRII-like radio structures at lower luminosities. We present the first high-resolution morphological investigation of a representative LOFAR-selected sample of low-luminosity FRIIs, to determine whether this new population is physically distinct from traditional high-luminosity FRIIs. Using new $1.5$-GHz Jansky Very Large Array (VLA) observations for a sample of 19 low-luminosity FRIIs, from the LOFAR Two Metre Sky Survey Data Release 1 (LoTSS DR1), with luminosities up to three orders of magnitude lower than the typical FR break ($L_{150} = 10^{26}$ W Hz$^{-1}$). We examine the compact features and perform spectral index analysis to identify hotspots, cores and signatures of restarting or remnant activity. We find a higher prevalence of cores and a comparable number of hotspots in the low-luminosity FRII sample compared to a randomly-selected sample of luminous ($L_{150}>10^{26}$ W Hz$^{-1}$) FRIIs selected from the same parent LOFAR sample. Approximately 32 per cent of low-luminosity FRIIs show restarting or remnant behaviour, while $\sim 32$ per cent are active FRIIs with compact hotspots. Our results show that FRII source dynamics occur at low radio luminosities, but reinforce earlier conclusions that the population of low-luminosity edge-brightened RLAGN is highly diverse. Binary morphological classifications should be used cautiously as a first step towards more nuanced investigations of the complexity of jet life cycles and evolution.

We study homeomorphisms of the circle that are smooth diffeomorphisms away from a finite set of $n$ points. These "broken diffeomorphisms" do not form a Lie group, but instead naturally assemble into a Lie groupoid. We construct an explicit nontrivial $n$-dimensional central extension of this groupoid, which restricts to the classical Virasoro group when confined to smooth diffeomorphisms. We further describe the associated "broken Virasoro" algebroid, defined as a nontrivial $n$-dimensional central extension of the Lie algebroid of vector fields on the circle that are smooth except at $n$ points. This construction generalizes the Virasoro algebra. As a byproduct, we analyze a related setting on an interval: we construct a nontrivial central extension of the Lie algebra of vector fields vanishing at the endpoints, together with the corresponding central extension of the group of diffeomorphisms fixing the endpoints. We also describe the associated Lie algebroid and groupoid obtained by allowing the endpoints to vary.

AstroPix is a high-voltage CMOS HVCMOS monolithic active pixel sensor MAPS developed for future space-based gamma-ray missions. It is also a candidate technology for the imaging layer of the Barrel Imaging Calorimeter BIC in the ePIC experiment at the future Electron-Ion Collider EIC. We report the first AstroPix test-beam results obtained at the KEK Photon Factory Advanced Ring PF-AR and the CERN Proton Synchrotron PS T10 beam line in 2025, using the third prototype AstroPix-v3. AstroPix-v3 sensors were operated as both standalone tracking layers and imaging layers interleaved with prototype lead/scintillating-fiber Pb/SciFi calorimeter modules, using electron and hadron beams in the few-GeV/c momentum range. Event synchronization between the continuous readout of AstroPix-v3 and the trigger-based readout of the Pb/SciFi calorimeter was achieved using a common timestamp. The AstroPix-v3 sensors exhibit stable performance, reaching a maximum hit efficiency of 68 percent at a bias voltage of -400 V under pion-dominated beam conditions. When combined with the Pb/SciFi calorimeter, the AstroPix layers successfully capture the development of electromagnetic showers. Using Cherenkov-based particle identification, electron-induced events exhibit significantly higher hit multiplicities and broader spatial distributions than pion-induced events, thereby providing clear discrimination between electromagnetic and hadronic showers. These results demonstrate that AstroPix-v3 provides effective, high-granularity imaging of shower development and is well suited as an imaging layer in future calorimeter systems for both collider and space-based experiments.

Long-term \textit{Fermi}-LAT monitoring makes it possible to ask whether a blazar light curve shows signs of an upcoming flare before the flare becomes obvious in the $γ$-ray emission. We present a strictly causal machine-learning framework for forecasting $γ$-ray blazar flares from 3-d binned LAT light curves. Flare intervals are identified with Bayesian Blocks, and each light curve is sampled with 365-d trailing windows from which 42 variability features are measured. We train separate WATCH and TRIGGER models: WATCH predicts whether flare activity will appear within the next 90 d, while TRIGGER predicts whether a new flare onset will occur within the next 45 d. To avoid temporal leakage, all scaling, calibration, threshold selection, and validation use only the pre-cutoff data before MJD 60000. We apply the method to the FSRQ 4FGL\,J1048.4$+$7143, using 13 bright blazars as auxiliary training sources. Among logistic regression, polynomial logistic regression, and random forest classifiers, polynomial logistic regression gives the strongest held-out WATCH performance, with ROC AUC $=0.891$, average precision $=0.396$, and a block-permutation probability $p_{\rm perm}=0.006$. At the selected WATCH threshold, it recovers 18 of the 21 positive windows in the held-out WATCH set, corresponding to a recall of 0.86. The same model also gives the best held-out TRIGGER ranking, with TRIGGER AUC $=0.770$ and TRIGGER AP $=0.123$, although no reliable pre-onset TRIGGER alert is obtained. The WATCH state appears before both held-out flare episodes, with final alerts 4.5 and 2.5 d before onset. The corresponding broader WATCH-active periods begin 88.5 and 72.5 d before flare onset. These results suggest that long-term {\fermi} light curves contain useful predictive information about the build-up to blazar flares.

An imprimitive symmetric indecomposable association scheme of rank $5$ is said to be Higmanian. In the present paper, we prove a necessary and sufficient condition for a Higmanian association scheme with two nontrivial parabolics to be uniform. We also provide examples of uniform Higmanian Cayley schemes.

False-positive bug reports represent a significant yet underexplored challenge in the development and maintenance of the Linux kernel. They occur when correct system behavior is mistakenly flagged as a defect, consuming developer effort without leading to actual code improvements. Such reports can mislead developers, waste debugging resources, and delay the resolution of real bugs. In this paper, we present the first comprehensive empirical study of false-positive bug reports in the Linux kernel. We manually construct a dataset of 2,006 bug reports comprising 1,509 genuine bugs and 497 false positives collected from Bugzilla and Syzkaller. Our analysis indicates that false positives demand effort comparable to real bugs, often requiring extended discussions and non-trivial closure time. They occur in several components, especially File Systems and Drivers, mainly due to external dependencies and semantic misunderstandings. To address this challenge, we evaluate large language models (LLMs) for automated false-positive bug report mitigation. Among various prompting strategies, retrieval-augmented generation (RAG) performs best, achieving 91% recall and an F1 score of 88%. These findings highlight the non-negligible cost of false positive bug reports and show the promise of LLMs for more efficient false positive mitigation in the Linux kernel.

In this paper, we analyze a function space consisting of functions for which both the function and its Fourier transform exhibit Gaussian decay together with exponential growth governed by suitable weight functions. First, we examine logarithmic-type weights, in which case these function spaces are equivalent to Pilipović spaces. In this setting, we establish a decay estimate for the Hermite coefficients of functions. Furthermore, by combining these estimates with the asymptotic behavior of Hermite functions, we prove a decay rate for solutions to the harmonic oscillator Schrödinger equation. Second, we consider a class of weights and prove the exponential decay of the Hermite projection operators on these spaces by analyzing Laguerre expansions and the short-time Fourier transform. Additionally, we revisit the subcritical Hardy uncertainty principle and obtain a partial improvement toward a conjecture posed by Vemuri.

A Tatra association scheme is an association scheme arising from a symmetric bilinear form defined on the equivalence classes of nonzero $2$-dimensional vectors modulo some subgroup of the multiplicative group of a finite field. In the present paper, we prove that every such association scheme is $2$-separable, i.e. it is determined up to isomorphism by the tensor of its $2$-dimensional intersection numbers.

Disorder and defective coordination in the catalytic plane are crucial for enhancing the Hydrogen Evolution Reaction (HER) on two-dimensional catalysts. Amorphous materials are disordered, making them catalytically adaptive for many reactions. In this work, the HER capabilities of Monolayer Amorphous Carbon (MAC) were studied in comparison with crystalline carbon derivatives, such as pristine graphene (GE) and graphyne derivatives. MAC generated from melt-quench simulations revealed a diverse framework of predominantly sp2 and sp3 carbons with numerous 5-, 6-, and 7-membered rings. Density Functional Theory (DFT) calculations investigated free-energy variations in hydrogen adsorption for each material. According to Sabatier's principle, optimum activity is achieved when the Gibbs free energy (Delta GH) change approaches zero. Crystalline carbon materials possess limited active sites, with beta-graphyne showing the best Delta GH value of +0.34 eV. The adsorption study for MAC was conducted in 30 distinct local environments, where core structural properties were analyzed against varying radii. Calculations showed a Delta GH distribution for MAC ranging from -0.02 eV to +1.35 eV. To evaluate activity across the entire MAC surface, a MACE MLIP foundation model was finetuned, achieving optimal energy and force fitting of 1.67 meV/atom and 29.15 meV/A, respectively. The MLIP predicted Delta GH values from -0.91 eV to +1.70 eV, with approximately 15% of sites exhibiting values below +0.25 eV. Feature analysis revealed that 7-membered rings, curvature, and ripple height enhance HER activity. Our findings suggest that, with careful optimization of local features, MAC can be tuned to compete with noble metal catalysts.

We study the Cauchy problem for the improved Boussinesq equation \[ u_{tt}-u_{xx}-u_{xxtt}-(u^2)_{xx}=0 \] on the real line with spatially quasi-periodic initial data. For a non-resonant frequency vector $ω\in\mathbb R^ν$, we prove local existence and uniqueness of classical spatially quasi-periodic solutions with the same frequency vector $ω$ in two Fourier-side classes. First, for exponentially decaying initial Fourier coefficients, we obtain a spatially quasi-periodic solution whose Fourier coefficients remain exponentially decaying on an explicit time interval. Second, for initial Fourier coefficients $c(n)$ and $d(n)$ satisfying the polynomial decay $ |c(n)|+|d(n)|\lesssim (1+|n|)^{-r}, \; r>ν+2, $ we prove that the corresponding spatially quasi-periodic solution preserves the same polynomial decay rate as the initial data. We also extend these results to the nonlinearity $u^p$ with integer $p \geq 3$.

Quantum simulation of complex many-body systems beyond classical computational capabilities provides a promising route toward understanding novel quantum phases and their transitions. In particular, analog quantum simulators with global control fields have attracted considerable attention due to their potential to simulate arbitrary Hamiltonians and perform quantum computing tasks. However, a clear, quantitative measure for the complexity of implementing specific quantum operations in such systems is still lacking. In this Letter, we address this challenge by introducing generalized Krylov complexity, a concept originating from operator growth dynamics, as a direct diagnosis for this synthesis complexity. We construct the block Krylov basis generated by a set of Hamiltonians, which naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators. By analyzing representative systems including Rydberg atom arrays, we demonstrate that the generalized Krylov complexity of a target operation serves as a strong predictor of the minimum time required for its realization. Our results establish Krylov complexity as an intuitive and predictive tool for designing efficient control protocols in analog quantum simulators.

We study weighted Walsh--Carleson maximal operators arising from dyadic martingale transforms associated with Walsh--Fourier partial sums. For weights satisfying a uniform dyadic variation condition and a uniform bound at the top dyadic scale, we prove weak type~$(1,1)$ estimates for the corresponding maximal operators along subsequences. We also give divergence criteria in terms of the behavior of the weights near the top dyadic scale and, under suitable admissibility assumptions, relate these criteria to explicit ratio conditions. As applications, we obtain results on matrix transforms of Walsh--Fourier partial sums, including de la Vallée Poussin means, Cesàro means with varying parameters, Nörlund logarithmic means, and general Nörlund means. In particular, we prove a Walsh--Paley analogue of the Leindler--Tandori theorem and establish everywhere divergence results for several summability methods.

Let $A$ and $B$ be unital complex Banach algebras having no quotients isomorphic to $\mathbb{C}$ or $M_2(\mathbb{C})$. Assume additionally that $B$ is semisimple. If a surjective additive mapping $Φ\colon A\to B$ satisfies $[Φ(x^2),Φ(x)] = 0$ for all $x\in A$, then there exist a surjective direct sum of an additive homomorphism and an additive anti-homomorphism $Ψ\colon A\to B$, an invertible element $λ\in\mathcal{Z}(B)$, and an additive mapping $ζ\colon A\to\mathcal{Z}(B)$ such that $Φ(x)=λΨ(x)+ζ(x)$ for all $x\in A$.

In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence, describes the local behaviour of a sequence of graphs. Here we show that for a Benjamini-Schramm-convergent sequence of trees, the subtree entropy, i.e., the logarithm of the number of subtrees divided by the order, converges to a constant depending only on the limit. The same holds true for the subtree density, i.e., the probability of a uniformly random vertex being contained in a uniformly random subtree, provided that long paths are ruled out in the limit. Related to this, we show that the subtree density and the average subtree entropy are dense in different parts of the unit interval $[0,1]$ for both general trees and series-reduced trees.

In recent work, we initiated a research program aimed at the systematic investigation of quantum superintegrable systems describing the interaction of two non-relativistic spin-$1/2$ particles in three-dimensional Euclidean space. In that study, we classified all such superintegrable systems admitting additional first-order scalar integrals of motion. In the present paper, we continue this program by focusing on systems that admit additional pseudo-scalar integrals of motion. Starting from the most general rotationally invariant Hamiltonian for two interacting spin-$1/2$ particles, we construct the most general first-order pseudo-scalar operator in the form of a matrix polynomial in the momenta. Imposing the commutativity of this operator with the Hamiltonian leads to a system of determining equations. By solving these equations, we obtain a complete classification of such superintegrable systems and determine the corresponding pseudo-scalar integrals of motion. The resulting classification provides new families of superintegrable systems with spin-dependent interactions. These systems enrich the class of integrable models relevant to nucleon--nucleon interactions and contribute to the broader program of classifying superintegrable quantum systems with spin. For selected cases, we further construct the associated polynomial symmetry algebras generated by the integrals of motion, providing additional insight into the algebraic structure of the systems.

The successful completion of collaborative tasks relies on the effective selection of trustworthy collaborators. To accurately evaluate the trustworthiness of potential collaborators, it is necessary to combine insights from their past collaborations with assessments of their resource capabilities under specific task contexts. However, the coexistence of diverse trust perspectives, along with complex spatiotemporal dependencies among devices, makes accurate trust evaluation particularly challenging. To address these challenges, we propose a customized Graph Neural Network (GNN)-Mamba (GM) model for trust evaluation and collaborator selection. In this model, the GNN model performs spatial trust fusion by leveraging inter-device spatial dependencies extracted from historical collaborations, while the Mamba-based temporal model captures both short-term fluctuations and long-term evolution of device trust. In addition, task-specific resource trust is incorporated to reflect the practical capabilities of devices under varying task conditions. Experimental results demonstrate that the proposed GM model outperforms baseline approaches in terms of the accuracy and stability of trust evaluation.

Direct electric drivelines without power-split open new design freedom for frame and suspension design, along with often lower energy losses. This paper focuses on self-propelled agricultural machinery (combine and forage harvest-ers, root crop harvesters), equipment carriers, propelled trailers and field robots. For a typical vehicle with four driven wheels, the electric motors can be packaged as two axle modules or four wheel modules, both defined herein as self-contained mechatronic units with integrated power electronics, distributed control intelligence and steering. Axle module and wheel module concepts are compared in detail against engineering requirements including loads, effi-ciency, steerability, controllability, braking, suspension, structural load support, asymmetric wheel loading and manu-facturing cost. The wheel module offers maximum design freedom, redundancy and controllability, while the axle module provides lower cost, structural rigidity, automatic load sharing through the differential and the ability to be used in existing vehicle structures. Both concepts are defined such that distributed control intelligence and steering are integral to each unit, requiring only a DC power bus and communication interface from the vehicle.

We demonstrate that altermagnetism provides a field-free mechanism for stabilizing finite-momentum superconductivity in two dimensions. Using a non-perturbative static path approximation Monte Carlo approach, we show that a d-wave altermagnet supports a robust pair-density-wave (PDW) phase that persists over a finite temperature window despite strong thermal fluctuations. The underlying mechanism originates from momentum-dependent spin splitting, which effectively enhances pairing instabilities at finite center-of-mass momentum without Zeeman fields. We identify distinct thermal scales associated with phase coherence, gap closing, and pseudogap formation, and establish characteristic spectroscopic and real-space signatures of the PDW state. Our results reveal altermagnetism as a robust route to thermally stable finite-momentum superconductivity and provide experimentally testable signatures for altermagnetic materials.

We present high-resolution UVES+VLT observations of neutral nickel and iron atoms in the coma of the interstellar comet 3I/ATLAS taken after perihelion. Metal emission was strong shortly after perihelion and persisted at large heliocentric distances. At $r_h \sim 2$ au the total metal production rate was found to be at least an order of magnitude larger than that of typical solar-system comets. Post-perihelion production rates exhibit pronounced asymmetry compared to the pre-perihelion behavior: production rates are higher after perihelion and decline more gradually with $r_h$, the difference being stronger for FeI. The NiI/FeI abundance ratio, initially anomalously large before perihelion, evolved toward values comparable to solar-system comets near 2 au, and shows a weaker $r_h$ dependence after perihelion. To interpret these results, we revisited and extended the carbonyl hypothesis in which FeI and NiI are produced by the rapid photodissociation of Fe(CO)$_5$ and Ni(CO)$_4$ vaporized from the nucleus. Fits that include direct sublimation of carbonyls reproduce the observed rates and the high NiI/FeI line ratio, which is determined by the higher volatility of Ni(CO)$_4$. Desorption of carbonyls from sublimating CO$_2$ and H$_2$O ices is found to be negligible. The temperature profiles needed to reproduce the observations were found to be shallower than the equilibrium $T \propto r_h^{-1/2}$ relation, suggesting that the sublimation could occur below the surface of the nucleus. Fits using temperature profiles from thermal models require sublimation from depths of several cm, especially post-perihelion. An additional transient heat source ($T \simeq$ 100-140~K), possibly linked to the amorphous-crystalline ice transition, is proposed to explain the early NiI excess before perihelion.

We investigate the orbit space of the scalar potential of a $Δ(54)$-symmetric three-Higgs-doublet model. We find that, if the potential enjoys $CP$ invariance, then its three-dimensional orbit space is a polytope; if the potential has no $CP$ symmetry, then its four-dimensional orbit space has a boundary that is sometimes slightly concave, but seems never to be convex. Consequently, we conjecture necessary and sufficient conditions for the potential to be bounded from below; brute-force minimization of a large number of potentials affirms the accuracy of our conjecture. We list all possible charge-conserving and charge-breaking minima of the potential.

Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover, any polynomial system is the specialization of such a parametrized system. We prove that the generic number of complex zeros of an augmented vertically parametrized system is the tropical intersection number of a tropical linear space and a classical linear space. In the special case when the matroid of the tropical linear space is cotransversal, we express this number as a mixed volume. We also obtain bounds on the maximal number of positive zeros, which is often the significant number in applications. We derive lower bounds from the number of intersections between positive tropicalizations, and when the positive zeros have toric structure, we provide upper bounds that are simpler and in some cases smaller than the generic root count. The resulting algorithms are implemented in Julia.

Graph coloring is a fundamental problem in computer science. In the semi-streaming model, an input graph $G$ on $n$ vertices and maximum degree $Δ$ is presented as a stream of edges, and the goal is to compute a vertex coloring using a small number of colors while storing only $\tilde{O}(n)$ bits of memory. Recent work has revealed an exponential separation between randomized and deterministic approaches in this setting: while randomized algorithms can achieve a $(Δ+1)$-coloring in a single pass [Assadi, Chen, and Khanna, 2019], any single-pass deterministic algorithm requires $\exp(Δ^{Ω(1)})$ colors [Assadi, Chen, and Sun, 2022]. Consequently, deterministic algorithms that use few colors must necessarily make multiple passes over the stream. Prior to this work, the best known deterministic trade-offs were: an $O(Δ^2)$-coloring in 2 passes, an $O(Δ)$-coloring in $O(\log Δ)$ passes [Assadi, Chen, and Sun, 2022], and a $(Δ+1)$-coloring in $O(\log Δ\cdot \log\log Δ)$ passes [Assadi, Chakrabarti, Ghosh, and Stoeckl, 2023]. It remained open whether better trade-offs -- particularly with sub-logarithmic pass complexity and linear-in-$Δ$ palette size -- were achievable. In this paper, we present a new deterministic semi-streaming algorithm that computes an $O(Δ)$-coloring in $O(\sqrt{\log Δ})$ passes. This is the first deterministic streaming algorithm to achieve a coloring with palette size linear-in-$Δ$ using sublogarithmic-in-$Δ$ passes.

Synthetic antiferromagnetic skyrmions (SAFsk) are nanoscale, topologically protected spin textures with strong potential for spintronic technologies because of their high stability and the absence of the skyrmion Hall effect. However, robust zero field stabilization remains a central challenge. Here, a synthetic antiferromagnetic (SAF) bias system is introduced as a novel strategy to stabilize both ferromagnetic skyrmions (FMsk) and SAFsk at zero field. Ferromagnetic (FM) and SAF multilayers are designed, fabricated and integrated with the SAF bias system to enable controlled skyrmion stabilization and polarity setting via multilayer design and a preparatory field cycle. Combining quantitative and high-sensitivity magnetic force microscopy (MFM) with micromagnetic modeling, reliable zero field skyrmion formation is demonstrated and sub 20nm SAFsk are directly observed, the smallest SAFsk reported to date. Moreover, the SAF bias system concept introduced here offers a robust and scalable route to bias future skyrmion multilayers, as its compensated nature suppresses domain formation and preserves a uniform exchange field.

The recent detection of the ultra-high energy neutrino KM3-230213A by KM3NeT/ARCA marks the first observation of an astrophysical neutrino with energy above 100 PeV, opening a new window to the ultra-high energy Universe. In this contribution, the current global ultra-high energy neutrino landscape in light of this event is reviewed, including tension of this observation with existing limits set by the IceCube and the Pierre Auger Observatories. Different scenarios are discussed to explain its origin. Recent efforts to constrain features of potential source populations using the inferred diffuse ultra-high energy neutrino flux are also presented.

We study smooth morphisms $f \colon X \to S$ that are $\mathbb{A}^1$-contractible in the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}(S)$. For base schemes $S$ of finite Krull dimension, we show that $\mathbb{A}^1$-contractibility is a fiberwise property: such a morphism is $\mathbb{A}^1$-contractible if and only if all its geometric fibers are $\mathbb{A}^1$-contractible. We apply this criterion to $\mathbb{A}^n$-fiber spaces, obtaining a geometric description of their $\mathbb{A}^1$-contractibility in terms of local factorizations as towers of torsors under vector bundles, building on results of Asanuma. In low relative dimensions, we establish rigidity results. In relative dimension $1$, $\mathbb{A}^1$-contractible morphisms over normal bases are precisely Zariski locally trivial $\mathbb{A}^1$-bundles. In relative dimension $2$, we show that over bases with characteristic zero residue fields, $\mathbb{A}^1$-contractible morphisms are $\mathbb{A}^2$-fiber spaces, and we obtain Zariski local triviality under additional hypotheses on the base. We also exhibit counterexamples in positive and mixed characteristic and formulate open problems concerning the existence of exotic $\mathbb{A}^1$-contractible surfaces.

In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in $\mathbb{C}$, $g$ an entire function in $\mathbb{C}^n$, and $h(z)=f(z_1+z_2+\ldots+z_n)$. We study the equations \begin{align*} \frac{\partial h(z)}{\partial z_i}=a G^g_{h}(z)+bh(z)+c\;\;\text{and}\;\;\frac{\partial h(z)}{\partial z_i}=a(z)G^g_{h}(z)+b(z)h(z)+c(z), \end{align*} where $z\in\mathbb{C}^n$, $i\in\{1,2,\ldots,n\}$, $a(\neq 0), b, c\in\mathbb{C}$ or $a(z)(\not\equiv 0), b(z),c(z)$ are polynomials in $\mathbb{C}^n$, and $G^g_h(z)=h(g(z),g(z),\ldots,g(z))$. The results obtained in the paper, extend previous studies on meromorphic solutions of functional-differential equations to the setting of several complex variables, and further illustrate the rigidity imposed by value distribution properties on nonlinear functional equations.

We prove the lossless unit interval Strichartz theorem on asymptotically conic surfaces, assuming that a large enough neighborhood of its trapped set has negative curvature. We also prove the spectral projection theorem on surfaces with Euclidean ends and nonpositive curvature, assuming a large enough neighborhood of its trapped set has negative curvature. We also discuss the spectral projection theorem on asymptotically Euclidean manifolds with dimension greater than or equal to 3, assuming some local smoothing estimates.

Combinatorial optimization problems play a central role in computer science with many real world applications. A number of relevant problems remain computationally difficult to solve as they lie in the NP-hard complexity class. We present a unified framework for solving such optimization problems represented in the quadratic unconstrained binary optimization (QUBO) formalism, namely two-SAT, XOR-SAT, mixed-two-XOR-SAT, set packing, quadratic assignment, binary clustering, and protein folding, by expanding the domain of applications of \textit{PRR, 6(2), 023031}. A direct mapping from the QUBO form of these problems onto the Rydberg quantum platform is demonstrated as our first step. This mapping to the Rydberg system depends on distance-dependent long-range interactions and configurable local detuning, thus reducing resource overhead and improving scalability. Following-up on the encoding, the solution is reached by steering the system toward the ground state of the target Hamiltonian using an optimized quantum annealing protocol that controls the time-dependent detuning and Rabi frequency profiles. The framework can handle a variety of problems, each with different complexity. To quantify the complexity of any problem, a generalized hardness parameter is introduced that compares different problems based on the structure of their optimization landscapes. This is a proceedings contribution to the Athens Workshop in Theoretical Physics: 10th Anniversary, held at the National and Kapodistrian University of Athens on December 17-19 2025.

We develop a global arithmetic framework for studying endomorphism rings inside ordinary elliptic isogeny classes over finite fields. Let p be a prime and let I(t,p) be an ordinary isogeny class over the finite field F_p with Frobenius trace t. The discriminant Delta = t^2 - 4p can be written as Delta = v^2 D_K, where D_K is the fundamental discriminant of an imaginary quadratic field K. In this setting, the possible endomorphism rings are precisely the quadratic orders O_f = Z + f O_K, with f dividing v. Building on Deuring's correspondence, we express the distribution of these orders in terms of weighted class numbers h*(D) = h(D)/w(D), and obtain explicit formulas for global distributions across the entire isogeny class. This approach goes beyond the classical local viewpoint, where the endomorphism ring is constant along each level of an ell-isogeny volcano. In particular, we introduce weighted exact and cumulative distributions of endomorphism rings. These distributions induce canonical laws for the ell-adic valuation of conductors and recover the vertical stratification of ell-volcanoes in an averaged sense. On the global side, by varying the prime p, we relate the existence of curves with a prescribed CM order O_D to splitting conditions in the associated ring class field L_D. Using the Chebotarev density theorem, we obtain the natural density 1/(2h(D)) for primes admitting CM by O_D. This gives a horizontal distribution law complementary to the vertical conductor distribution. These results establish a unified perspective linking Deuring theory, isogeny graph geometry, and class field theory. They also provide a natural framework for quantitative and algorithmic studies of ordinary isogeny classes.

Denoising diffusion models have evolved into a state-of-the-art method for tasks in various fields, such as denoising and generation of images, text generation, or generation of synthetic data for training of other machine learning models. First hitting diffusion models (FHDM) are a particular class of denoising diffusion models with \textit{random} adaptive generation time tailored to generate data on a known manifold. Building on the conditioning framework of Doob's $h$-transform these models leverage the given information on the target data manifold to demonstrate strong performance across tasks while offering distinct features such as time-homogeneous dynamics of the generating process and a reduced average simulation time. Even though the theoretical investigation of standard forward-backward diffusion models has attracted much attention in the recent past, the statistical convergence properties of FHDMs are not yet understood. In this work, we show that, up to logarithmic factors, FHDMs achieve the minimax optimal convergence rate in total variation for spherically supported Sobolev smooth data distributions. In particular, this is the first statistical optimality result for denoising diffusion modelling with random generation time.