This constant is the maximum of the sum $|c_0|+|c_1|+|c_2|+|c_3|$ of the moduli of the coefficients of a trigonometric polynomial $c_0+c_1e^{it}+c_2e^{2it}+c_3e^{3it}$ bounded by 1. Its value is still unknown, but I will present some ideas on how to compute it and describe a distinguished torus of extremal functions.

We give a sufficient condition for a countable group $G$ to possess a probability measure $μ$ that admits a non-trivial $μ$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $μ$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $μ$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $μ\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $μ$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $μ$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries.

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is Lamba(p) for all p but is not a Rosenthal set. This holds also for the sequence of primes.

Let $(\mathcal F_n)_{n\ge 1}$ be a filtration and let $f\ge0$ belong to $L^1(\mathcal F_\infty)$. For the martingale $f_n=\mathbb E[f\mid \mathcal F_n]$ and each $λ>0$ we prove a Gundy--Stein decomposition \[ f=g+h+k \] with explicit numerical constants. In the positive closed case the three parts satisfy explicit bounds, and the bounded part is bounded above by $λ$. We also prove a one-parameter form for the bounded part and two-point sharpness results, including a joint sharpness statement for arbitrary decompositions under the condition $0\le k\le λ$. We also obtain an exact four-term refinement of the decomposition, separating the bounded term into a stopped part and a conditional expectation term. As applications we obtain an explicit weak-type $(1,1)$ estimate for truncated martingale multipliers and a John--Nirenberg inequality for martingale $\mathrm{BMO}$ on atomic $α$-regular filtrations.

For any $1 < p < q < \infty$, we investigate fixed-time hypercontractive bounds from $L^p$ to $L^q$ of Poisson semigroups associated with the Ornstein--Uhlenbeck, Laguerre and Jacobi operators. We prove that, in the Ornstein--Uhlenbeck and Laguerre cases, the Poisson semigroups fail to be $L^p \to L^q$ bounded for any fixed $t > 0$. In contrast, for Jacobi operators with $α, β\ge -1/2$, the associated Poisson semigroups are ultracontractive, namely bounded from $L^1$ to $L^\infty$. More generally, we study Bernstein subordinations of these semigroups and show that fixed-time hypercontractivity is not stable under subordination. The analysis relies on quantitative $L^q$-estimates for the corresponding orthogonal polynomial eigenfunctions, together with a bilinear test with the exponential family.

Wildfire spread is strongly influenced by the transport and ignition of embers. While long-range spotting driven by plume lofting has received significant attention, embers transported near the surface by turbulent winds can also influence fire propagation. We develop a stochastic model for near-surface ember transport, referred to as ember wash. The model represents ember motion as a sequence of short displacements analogous to saltation-like transport and incorporates a probabilistic ignition process that depends on ember survival during transport. This formulation leads to an exponential distribution of ember flight times. The model is implemented within a simplified fire spread model to examine burn patterns and growth dynamics. Simulations demonstrate that ember wash produces spread behavior that differs fundamentally from classical plume-driven spotting. These results suggest that ember wash provides a plausible mechanism for wildfire spread regimes that differ from those predicted by geometric or plume-driven spread models.

Modeling the Scape-off layer (SOL) of stellarator fusion devices is challenging due to the complicated magnetic topology, requiring numerical tools to solve transport equations for realistic geometries. Previously the flux coordinate independent (FCI) method has been successfully applied to model the SOL in simplified geometries. The current work presents some of the recent improvements for the BOUT++ modeling implemented to simulate the SOL in realistic geometries with the example of Wendelstein 7-X. The changes include improvements for the grid generation tool, the physics model as well as the BOUT++ library itself. A short outlook is given on current modeling work using the new features.

Decentralized primal-dual methods are widely used for solving decentralized optimization problems, but their updates often rely on the potentially crude first-order Taylor approximations of the objective functions, which can limit convergence speed. To overcome this, we replace the first-order Taylor approximation in the primal update of EXTRA, which can be interpreted as a primal-dual method, with a more accurate multi-cut bundle model, resulting in a fully decentralized bundle EXTRA method. The bundle model incorporates historical information to improve the approximation accuracy, potentially leading to faster convergence. Under mild assumptions, we show that a KKT residual converges to zero. Numerical experiments on decentralized least-squares problems demonstrate that, compared to EXTRA, the bundle EXTRA method converges faster and is more robust to step-size choices.

In this article, we focus on determining a minimum-cost treatment program aimed at maintaining the size of a cancerous tumor at a level that allows the patient to live comfortably. At each predetermined point in a treatment horizon, the patient either receives drug treatment or does not. In the first case, the tumor shrinks and its size is multiplied by a constant factor lower than 1; in the second, it grows following an exponential or Gompertz growth law. We first demonstrate that a simple heuristic solution provides an optimal treatment program. We then show that the Gompertz function can be described, like the exponential function, by a simple recurrence relation that does not explicitly depend on time. Thanks to the characteristics of the logarithmic function, this property allows us to formulate the problem as a mixed-integer linear program. This result makes it possible to solve the problem very efficiently using one of the many solvers available to handle this type of program, and above all to consider and solve several extensions to the problem. In particular, we show how to determine which of the optimal equivalent solutions to the initial problem are the most relevant. We also show how to measure the effect of a marginal increase in the treatment budget on patient quality of life. Numerous computational experiments are presented to illustrate these issues.

The study of Ramsey-type problems for linear equations originated with Schur's theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting, one fixes a pair of distinct linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and asks for the least integer $N$ such that every red--blue coloring of $\{1, 2, \dots, N\}$ must yield either a red solution to $\mathcal{E}_1$ or a blue solution to $\mathcal{E}_2$. This threshold integer is referred to as the off-diagonal Rado number of the system $(\mathcal{E}_1, \mathcal{E}_2)$. In this work, we study the discrete and continuous off-diagonal Rado number for non-homogeneous linear system of equations $x+y+c=z$ and $x+y+k=z$ where $c\le k$. We determine the exact two-color discrete and continuous off-diagonal Rado number $R_2(c,k)$ associated with this system of equations.

We give criteria for when finitely generated local modules over a commutative algebra $A$ in the ind-completion $\widehat{\mathcal{C}}$ of a braided tensor category $\mathcal{C}$ inherit the structure of a (rigid, braided, ribbon) tensor category. We then apply this to simple current algebras $A = \bigoplus_{g \in Γ} E_g$, where $Γ$ is a subgroup of invertible objects in $\mathcal{C}$. Using a description of simple $A$-modules, we verify the required hypotheses for this class of algebras and deduce rigidity, braided, ribbon, and non-degeneracy properties for their finitely generated local modules. As applications, we construct examples of ribbon tensor categories from quantum supergroup categories for unrolled $\mathfrak{gl}(1|1)$.

Free-electron lasers (FELs) provide a revolutionary tool for capturing the structure and dynamics of matter in real time at the atomic scale. The size and cost of FELs can be substantially reduced by using laser wakefield acceleration (LWFA), which offers acceleration gradients orders of magnitude beyond radiofrequency technology, producing multi-GeV electron beams within tens of centimeters. This compactness opens the possibility of integrating multiple operating modes - from the EUV to X-rays including broadband operation - into one facility. Realizing this vision, however, faces key challenges: current LWFA bunches are too short to sustain sufficient radiation slippage, limiting FEL pulse energy at EUV wavelengths, while the large energy spread and emittance make X-ray lasing even more demanding. Here we present a LWFA-driven FEL scheme that addresses these challenges, enabling multi-mode operation spanning different wavelengths and bandwidths within a single facility. The scheme employs staged acceleration to reach multi-GeV energies while preserving beam quality, combined with a dual-chicane beamline that stretches the bunch to mitigate the radiation slippage for EUV FEL and tailors the energy chirp for diverse FEL bandwidth modes. Simulations demonstrate that the scheme can generate high-quality electron beams with energies up to 7 GeV and tunable energy chirp, enabling both FEL saturation from the EUV to X-ray wavelengths and large bandwidth operation with a bandwidth of up to 11%. This work provides a roadmap for compact, multi-mode FELs based on plasma acceleration, and the high-energy, high-quality beams achieved also point toward compact injectors for next-generation storage-ring light sources.

This paper investigates the Fréchet mean of the Erdős-Rényi random graph $G_{n,p}$ with respect to the Frobenius distance on graph Laplacians, a metric that captures global structural information beyond local edge flips. We first characterize the Fréchet mean set as consisting of quasi-regular graphs (i.e., graphs where all vertex degrees differ by at most one). We then analyze the asymptotic behavior of the Frobenius distance $F_n=d_{\mathrm{F}}(G_{n,p},R)$ as $n\to\infty$, where $R$ is any Fréchet mean. Closed-form expressions for the mean and variance of $F_n^2$ are derived, which are invariant to the choice of $R$. Leveraging these results, we establish several weak convergence laws for the Frobenius distance over all regimes of $p \in (0,1)$ as $n \to \infty$. Finally, under the scaling condition $n^2 p(1-p) \to \infty$ we prove the asymptotic normality of this distance, which exhibits a phase transition governed by the growth rate of $np(1-p)$. Our results reveal how metric selection fundamentally shapes Fréchet mean geometry in random graphs.

This study investigates the role of inertia in moving contact lines using experiments, theoretical analysis, and numerical simulations. Experiments are conducted using a plate immersion configuration over a wide range of Reynolds numbers from $O(10^{-3})$ to $O(10)$. Flow configurations and quantitative measurements are obtained using high-speed imaging and particle image velocimetry. The streamfunction contours reconstructed from the experimental velocity fields are compared with the viscous modulated wedge solution (viscous-MWS) and inertial-MWS theory. Experimental observations show that the streamfunction contours agree well with viscous predictions at low Reynolds numbers; however, systematic deviations emerge as the Reynolds number increases. The inertial-MWS theory, an inertial extension of the Huh and Scriven framework, accounts for these deviations, but only within a narrow range of Reynolds numbers $10^{-1} < Re < 1$. At higher Reynolds numbers, inertial theory fails to accurately capture the deviations in the streamfunction contours observed in the experiments. Moreover, simulations conducted using the volume of fluid method support our findings, exhibiting deviations in streamfunction contours consistent with experimental observations. We demonstrate that inertia does not fundamentally alter the underlying flow configuration but instead induces a systematic deviation in the streamfunction contours. At finite $Re$, the interfacial speed transitions from a nearly constant value in the viscous regime to a monotonic decay along the interface. These findings expose the need for more sophisticated models of moving contact lines.

Context. Via scaling relations, it is well-known that active galactic nuclei (AGN) and bulges are linked. This link was thought to be driven by mergers, but recent studies show that secular processes are the dominant mechanism of supermassive black hole growth. One such secular mechanism is gas inflow driven by large-scale bars. Since bulges can also grow via these bars, there is likely some common process between these three features. Aims. We investigate whether the observed correlation between AGN and bars is real or arises as a result of correlations between bars and bulges. Methods. Using a catalogue of AGN identifications and galaxy morphologies in the DESI Legacy Survey at $z\leq0.1$, we control for mass and colour and investigate the AGN fraction variation with bulge prominence and bar strength. Results. We first show that the variation in AGN fraction between strongly barred, weakly barred and unbarred galaxies does not qualitatively change if we additionally control for bulge prominence. Second, we find that in fixed bins of bulge prominence, the AGN fraction increases with increasing bar strength. In subsamples split by bar strength, the AGN fraction increases with bulge prominence, indicating that AGN presence correlates with both bar strength and bulge prominence simultaneously.

Transition-region loops are a type of critical magnetic structure in the solar atmosphere, yet their physical properties and evolutionary characteristics remain statistically poorly constrained. We aim to statistically characterize the physical properties of propagating brightening events in transition-region loops and to explore the underlying heating mechanism responsible for these brightenings.Using coordinated observations from the Extreme Ultraviolet Imager onboard the Solar Orbiter and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory, we analyze 42 propagating brightening events in loops that are unambiguously detected in both instrument data. Each of these events evolve simultaneously in the AIA 94, 131, 171, 193, 211, 304, and 335 passband images, suggesting that they are in the transition-region or low-coronal temperature range. Our analyses show that these brightenings are impulsive, with an average brightening time of 118.4 s and a mean intensity decreasing time of 159.4 s. The propagating brightenings are predominantly subsonic, with velocities in the range of 0-90 km/s and an average of 51.3 km/s. The lengths of brightenings range from 3 to 11 Mm, with an average and standard deviation of 6.3 Mm, which are closely related to the propagation velocity and the lifetime. The initial brightening sites are predominantly located near the footpoints of these loops, and the number of brightening events decreases systematically with increasing of loop height. Our results are consistent with an energizing mechanism regulated by enthalpy flows and radiative cooling.

This article studies wave equations and their solutions on plane wave spacetimes of arbitrary dimension, developing the interplay among three structural layers: the Ward progressing-wave representation of solutions to the scalar wave equation, the Fourier analysis of the Heisenberg group naturally associated to the plane wave, and the Schrödinger propagator governing the evolution of initial data. The central geometric object is a positive curve in the Lagrangian Grassmannian determined by the plane wave metric, previously studied in the authors' series. The conformal tensor $H(u)$ that parametrises this curve plays a dual role: it encodes the null-cone geometry of the spacetime and simultaneously appears as the time-dependent parameter in the Schrödinger representation of the Heisenberg group acting by isometries on the plane wave. Parallel to the classical Fourier inversion theorem, convolution by Lagrangian delta distributions on the Heisenberg group furnishes an intrinsic description of the Schrödinger propagator, and the intertwining of different polarisations by this propagator is captured by a diagram that commutes up to a Maslov phase. The theta functions and Bargmann transforms that arise from imaginary polarisations complete the analytic picture, connecting the present work to the theory of the Weil representation as developed by Lion--Vergne and to Mumford's systematic treatment of theta functions.

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $δ_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $δ_2(H) \geq 4n/5 + o(n)$ is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of $3$-graphs with large minimum codegree.

The second solar spectrum is the solar spectrum of Stokes parameter Q observed in linear polarization close to the solar limb. It differs significantly of the usual intensity spectrum (Stokes parameter I). The second solar spectrum contains in the visible range a few polarized lines with Q/I > 1% (such as CaI, SrI, SrII, BaII), but most lines exhibit weak or very faint polarization rates (Q/I < 0.3%). This paper presents unpublished observations made in 2004-2006 of weak polarized lines performed with the Pic du Midi Turret Dome spectropolarimeter, such as atomic lines of Na, Al, H, He, Ti, Li, Sc as well as C2 and MgH molecules.

The stringent link budget, caused by long propagation distances and payload constraints, poses a fundamental bottleneck for single-satellite transmission. Although LEO mega-constellations make multi-satellite cooperative transmission (MSCT), such as distributed precoding (DP), increasingly feasible, its cooperative gains critically rely on stringent time-frequency-phase synchronization (TFP-Sync), which is difficult to maintain under rapid channel variation and feedback latency. To address this issue, this paper proposes a joint CSI acquisition, feedback, and phase-level synchronization (JCAFPS) framework for MSCT. Specifically, to enable reliable, overhead-efficient CSI acquisition, we design a beam-domain adjustable phase-shift tracking reference signal (TRS) transmission scheme, along with criteria for the TRS and CSI-feedback periods. Then, exploiting deterministic orbital motion and dominant LoS propagation, we establish a polynomial model for the temporal evolution of delay and Doppler shift, and derive an OFDM-based multi-satellite signal model under non-ideal synchronization. The analysis reveals that, unlike the single-satellite case, the composite multi-satellite channel exhibits nonlinear time-frequency-varying phase behavior, necessitating symbol- and subcarrier-wise phase precompensation for coherent transmission. Based on these results, we develop a practical closed-loop realization integrating single-TRS-based channel parameter estimation, multi-TRS-based channel prediction, predictive CSI feedback, and user-specific TFP precompensation. Numerical results demonstrate that the proposed framework achieves accurate CSI acquisition and precise TFP-Sync, enabling DP-based dual-satellite cooperative transmission to approach the theoretical 6 dB power gain over single-satellite transmission, while remaining robust under extended prediction durations and enlarged TRS periods.

In this paper we study a two-component coagulation equation that models the aggregation of rouleaux in blood. We consider product kernels that have homogeneity $2$ and we characterize the initial data that lead to gelation. We prove that, when gelation occurs, the solution to the two-component coagulation equation localizes along a direction of the space of cluster as $ t $ approaches the gelation time $0 < T_* < \infty $. The localization direction is determined by the initial datum. We also prove that the solution converges to a self-similar solution along the direction of localization.

We show that inclusions of $p$-metric spaces always produce genuine linear embeddings at the level of Lipschitz-free $p$-spaces. More precisely, for every $0<p<1$ and every inclusion $ \mathit{N}\subset \mathit{M}$ of $p$-metric spaces, the canonical map from $ \mathit{F}_p(\mathit{N})$ into $ \mathit{F}_p( \mathit{M})$ is always an isomorphic embedding, as it plainly happens for $p=1$. Our proof relies on a versatile extension procedure for $p$-Banach-valued Lipschitz maps, allowing us to control the geometry of canonical molecules and uncover a rigidity principle governing the structure of Lipschitz free $p$-spaces. As an application, we prove that, given $0<p<q\le 1$, the natural envelope map from the Lipschitz-free $p$-space $ \mathit{F}_p( \mathit{M})$ to its $q$-Banach envelope $ \mathit{F}_q( \mathit{M})$ is one-to-one. These results give positive answers to two foundational questions that were originally raised by Kalton in [Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335], and provide tools for furthering the understanding of subspace structures, hereditary properties, and geometric invariants in Lipschitz-free $p$-spaces.

In this letter, we prove that under mild conditions, the scaled graph of a reset control system is bounded by the scaled graph of its underlying base linear system, i.e., the system without resets. Building on this new insight, we establish that the negative feedback interconnection of a linear time-invariant plant and a reset controller is stable, if the scaled graphs of the underlying base linear components are strictly separated. This result simplifies reset system analysis, as stability conditions reduce to verifying properties of linear time-invariant systems. We exploit this result to develop a systematic approach for reset control system design. Our framework also accommodates reset systems with time-regularization, which were not addressed in the context of scaled graphs before.

Algorithmic content targeting homogenizes information, with implications for strategic interactions. For example, this increased homogenization was arguably responsible for the run on the Silicon Valley Bank. We argue that existing measures of similarity are inappropriate for studying games -- especially coordination games -- because they do not discipline agents' conditional beliefs. We propose a class of stochastic orders, Concentration Along the Diagonal (CAD), built on agents' conditional beliefs. In canonical binary-action coordination games, greater CAD-similarity is both necessary and sufficient for strategic similarity -- agents adopt the same strategy. We further demonstrate CAD's applicability in congestion games, collective action, and second-price auctions.

We performed a comprehensive study of thermonuclear bursts from the millisecond X-ray pulsar SAX J1808.4$-$3658 with XMM-Newton and NICER. We report the results from the analysis of an intense burst with NICER using a self-consistent and physically motivated disc reflection modeling approach and investigate the burst-disc interaction. The dynamic evolution of the spectral parameters suggested evidence of photospheric radius expansion (PRE) of the neutron star using the disc reflection modeling approach, which indicates a maximum expansion of the photosphere up to 14.8$\pm$0.7 km. The corresponding blackbody temperature drops to a minimum of 1.9 keV. In addition, an emission line at 1 keV is observed, possibly originating from the Ne or Fe L-band transition as a result of the reprocessing of burst photons by cold gas in the accretion disc. The 1 keV emission line flux is found to be strongly correlated with the flux of the reflection component. We also investigated a thermonuclear burst observed with XMM-Newton EPIC-PN from SAX J1808.4$-$3658 using the variable persistent emission method and the disc reflection modeling approach. The X-ray reflection feature is also investigated in persistent emission using a NuSTAR observation. The best-fitting results provide an inner disc radius of $14_{-5.9}^{+9.7}$ $R_g$ and an inclination of $ 38^\circ-60^\circ$ during the NuSTAR observation. The magnetic field is estimated to be $\simeq$3.7 $\times$10$^8$ G at the poles of the neutron star.

The rapid advancement of digital technologies necessitates significant progress in functional materials, which are often derived from scarce elements and involve complex manufacturing processes. Additionally, the trend towards miniaturization in high-tech devices has heightened the demand for extremely small components with tailored functionalities. In the domains of ferromagnetic materials, the market is mostly dominated by rare-earth elements-based structures, which are also limited in abundance. In this work, we focus on the microstructure and properties of MnSi. It is a ferromagnetic material with a relatively low Curie temperature (TC) of 30 K. However, our study demonstrates that Tc can be increased by a factor of 4 through careful control of the crystal size. MnSi thin films were synthesized by combining two non-equilibrium techniques: magnetron sputtering and laser annealing. Laser annealing provoked the crystallinity evolution, by heat accumulation, of the barely crystallized films deposited by magnetron sputtering. The laser beam scanning parameters were adjusted to achieve different fluence values, pulse numbers, and pulse frequencies at each point of the film. Films with a crystal size of around 20 nm exhibited a TC of up to 120 K. These properties were obtained under conditions of low fluence and a high number pulse. Local laser impacts were applied to as-deposited samples, enabling spatially controlled crystallization. The interface between the poorly and well-crystallized regions was showcased using high-resolution transmission electron microscopy (HR-TEM). A spatial resolution of approximately 100 μm was achieved. These results demonstrate the strong potential of laser annealing as a versatile and promising approach for the fabrication of miniaturized devices.

In one-dimensional (1D) electron systems, the Fermi liquid state breaks down due either to electron interactions, which results in a Tomonaga-Luttinger liquid (TLL) state, or to Peierls instability, which leads to an insulating charge-density-wave (CDW) phase. In general, these two phenomena are mutually exclusive, and their coexistence remains elusive in real materials. Here, we report the discovery of a new quasi-1D material, Cs$_{1-δ}$Cr$_3$S$_3$, which unexpectedly exhibits coexistence of the antithetical CDW and TLL states. The CDW state is evidenced by the intra-unit-cell dimerization, and the opening of an optical band gap of $\sim$250 meV. Meanwhile, TLL behaviour is unambiguously demonstrated by the measurements of electrical transport and angle-resolved photoemission spectroscopy, which reveal a power-law scaling with temperature, bias voltage and electron energy. Band structure calculations reveal isolated, linearly dispersive, 1D bands around the Fermi level. For the dimerized CDW phase, the 1D Fermi-surface sheets located at the boundary of the Brillouin zone are gapped from intra-unit-cell bond symmetry breaking. Experimentally, subtle Cs vacancies shift the Fermi level into the linearly dispersive valence band, enabling the observation of TLL behaviour without interrupting the CDW order. This work establishes Cs$_{1-δ}$Cr$_3$S$_3$ as a rare material platform in which the antagonistic Fermi-liquid instabilities coexist and intertwine, opening new avenues for studying emergent quantum phenomena in 1D systems.

Given a finitely generated, torsion-free nilpotent group, we find the maximum possible (critical) regularity for its faithful actions by diffeomorphisms of the closed or half-open interval and of the circle. Our result gives an expression for its value in purely algebraic terms (using the relative growth of appropriate subgroups), generalizing many preceding works. As an intermediate step, we generalize the Bass-Guivarc'h formula, obtaining a formula for the relative growth of subgroups of nilpotent groups, as well as for the growth of the corresponding Schreier graphs.

Adders are fundamental building blocks in modern digital systems, and their performance, power, and area (PPA) directly impact system efficiency. Contemporary adders typically use parallel-prefix architectures with established PPA trade-offs, but these often fail to deliver globally optimal PPA for specific design goals. Prior work lacks netlist-/cell-level awareness, and general synthesis heuristics are not adder-specific, resulting in suboptimal PPA. To address this, we propose AXON, an automated netlist optimization framework for adders. It performs design space exploration from architectural to netlist level, integrating prefix topology search with standard-cell-aware mapping via a hierarchical approach to quickly converge to near-optimal PPA solutions. We also introduce a hybrid ultra-high-speed adder combining parallel-prefix and Ling architectures to shorten the critical path. Experiments on TSMC 28nm library show AXON improves delay, area-delay product, and energy-delay product by up to 10.3%, 12.6%, and 32.1% respectively, compared to commercial synthesis tools.

We investigate the dynamics of spinning test particles in the vicinity of Einstein--geometric Proca (EGP) Anti-de Sitter (AdS) compact objects, which arise from metric-Palatini gravity extended by the antisymmetric part of the affine curvature. Using the Mathisson-Papapetrou-Dixon (MPD) equations with the Tulczyjew spin supplementary condition, we derive the effective potential and analyze the equatorial motion of spinning particles. The influence of the model parameters $q_{1}$, $q_{2}$, and the Proca mass parameter $σ$ on the innermost stable circular orbits (ISCO), superluminal spin bounds, and orbital stability is systematically explored. Our results show that increasing $q_{1}$ and $q_{2}$ reduces the ISCO radius, angular momentum, and energy, while spin orientation introduces significant modifications to orbital behavior. We further examine head-on collisions of spinning particles near the horizon and demonstrate how the center-of-mass energy depends on spin and the EGP theory parameters. The study reveals that Einstein-geometric Proca AdS black holes may act as efficient particle accelerators, with distinctive features absent in Schwarzschild or standard AdS backgrounds. These findings provide new insights into the interplay between spin dynamics, modified gravity, and strong-field compact object physics.