According to activity-rotation relations, rapid rotators are expected to show high levels of magnetic activity. However, recent studies with TESS have found Ultra Fast Rotating (UFR) M dwarfs with periods $<1$ d displaying low levels of flaring activity. There have been efforts to explore their magnetic field strengths through spectropolarimetric measurements and to assess the potential for binarity. However, neither could fully explain the lack of observed flaring activity despite their rapid rotation. Another avenue for investigation is to measure their coronal emission for signs of supersaturation: an underluminosity in X-rays observed for some rapidly rotating FGK stars. Therefore, in this study, we utilise X-ray observations from Swift and XMM-Newton of ten M dwarf UFRs with P$_{\rm{rot}}$<1 d to determine their X-ray luminosities. Overall, we do not find evidence for supersaturation amongst our UFR M dwarf stars, instead determining them to be at the saturated level, or perhaps even enhanced. Therefore, supersaturation seems not to be the main driver behind the reduced level of flaring activity observed in these stars, and the mystery behind the magnetic activity of UFR low-mass stars remains. Additionally, we provide an updated analysis on the long term variability within our sample using TESS light curves taken during Cycles 5 and 7. We identify 352 optical flares from our sample with energies between $1.2\times10^{31}$ and $8.7\times10^{34}$ erg. We determine flare rates for each TESS cycle, compare them, identifying variations across a 7 year timespan and attribute this to potential activity cycles.

In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the Lagrange equilateral triangle solutions of the three-body problem in $\mathbb R^4$. We then initiate a new approach to stability based on electromagnetic curvature. In a two-dimensional model, we relate linear stability to both the Mañé critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold. This criterion is then applied to the planar $n$-body problem: in the three-body case, we recover Routh's classical criterion, and, more generally, we obtain an instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes. These results suggest a new geometric perspective on linear stability and on questions related to Moeckel's conjecture.

The existence of an upper limit to the human lifespan has been widely debated, with studies offering both supporting and opposing evidence. Using unique individual-level death and population records for individuals aged 90 and older in Belgium and the Netherlands between 1995 and 2022, we provide statistical evidence supporting the existence of an upper limit. A related yet unexplored question is whether this life span limit differs across socio-demographic groups. Our microdata include information on the sex, origin, civil status, type of household, and education level of each individual. Using tools from extreme value theory, we quantify and compare the upper tail of human lifespan distributions across these socio-demographic characteristics. We find that men have a statistically lower maximum lifespan than women and that individuals who are widowed or live in institutional households have a clearly lower maximum lifespan. Finally, individuals of non-Western European origin and those with higher educational attainment exhibit longer maximum lifespans.

The Polyak-Łojasiewicz (PŁ) condition is often invoked in nonconvex optimization because it allows fast convergence of algorithms beyond strong convexity. A function $f \colon \mathcal{M} \to \mathbb{R}$ on a Riemannian manifold $\mathcal{M}$ is globally PŁ if $\|\nabla f(x)\|^2 \geq 2μ(f(x) - f^*)$ for all $x$, where $f^* = \inf f$ and $μ> 0$. How much does this pointwise, first-order inequality constrain $f$ and its set of minimizers $S$? We show that if $f$ is also smooth ($C^\infty$) and $\mathcal{M}$ is contractible (e.g., if $\mathcal{M} = \mathbb{R}^n$), then the PŁ condition imposes a firm global structure: such a function is necessarily of the form $f(x) = f^* + \|φ(x)\|^2$ (a nonlinear sum of squares) where $φ\colon \mathcal{M} \to \mathbb{R}^k$ is a submersion, and $k$ is the codimension of $S$ in $\mathcal{M}$. The proof hinges on showing that the end-point map of negative gradient flow on $f$ is a trivial smooth fiber bundle over $S$. This rigidity leads to a striking dichotomy. Either $S$ is diffeomorphic to a Euclidean space, in which case $f$ can be transformed into a convex quadratic by a smooth change of coordinates. Or $S$ must display genuinely exotic geometry; for example, it can be diffeomorphic to the Whitehead manifold. As a further consequence, we show that there exists a complete Riemannian metric on $\mathcal{M}$ under which $f$ remains PŁ and becomes geodesically convex.

Ground-state cooling is a prerequisite for exploring macroscopic quantum effects in mechanical motion of massive objects. Here we construct a polarization-angle-controllable coupled cavity-levitated-nanoparticle system in which two nanoparticles trapped by individual tweezers are coupled to a single-mode field in a cavity. We also study the simultaneous ground-state cooling of six mechanical displacement modes of the two levitated nanoparticles through the coherent scattering mechanism. By deriving the Hamiltonian of the system and performing the linearization, we obtain a linearized seven-mode Hamiltonian, which can exhibit the coupling structure and cooling mechanism. We confirm the physical condition for the appearance of dark modes, which will suppress the simultaneous ground-state cooling of these mechanical modes. We also find that, by properly tuning the polarization angle $θ$ between the cavity field and the optical tweezer fields, the coupling channels can be controlled on demand and simultaneous ground-state cooling of these six motional modes of the two nanoparticles can be realized. Our work paves the way for generation and manipulation of collective macroscopic quantum effects in multiple levitated nanoparticles.

The yields of prompt and non-prompt J/$ψ$ and the fraction of non-prompt J/$ψ$ are measured at midrapidity ($|y| < 0.9$) via the dielectron decay channel as a function of the midrapidity charged-particle multiplicity ($|η| < 0.9$) in pp collisions at $\sqrt{s} = 13$ TeV. The J/$ψ$ yields and the multiplicity are normalized by their average value in inelastic collisions. The multiplicity-dependent yield ratio between prompt J/$ψ$ and D$^0$ is reported. The multiplicity is further divided into three azimuthal regions with respect to the J/$ψ$ momentum: toward the J/$ψ$ emission direction, transverse, or opposite to it. A stronger-than-linear increase of the self-normalized yields is observed for both prompt and non-prompt J/$ψ$ production, with similar trends. This behaviour is also observed in the toward region, while a weaker increase is observed in the transverse and away regions.

The ALICE collaboration presents the first measurement of the dijet invariant mass spectra of charged-particle jets in pp and p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV. Charged particles in the mid-pseudorapidity region, $|η| < 0.9$, are clustered into jets using the anti-$k_{\rm T}$ algorithm with a resolution parameter $R = 0.4$. The leading and subleading jets are required to have a transverse momentum $p_{\rm T} > 20$ GeV$/c$ and to be contained within $\left|η_{\rm jet}\right| < 0.5$. The dijet invariant mass spectrum and the nuclear modification factor $R_{\rm pA}$ are presented in the low-mass region of 75 to 150 GeV/$c^2$. The nuclear modification factor for charged-particle dijet invariant mass is consistent with unity. This is in line with previous small-system jet studies. Comparisons with Monte Carlo simulations suggest that the low-mass region is sensitive to anti-shadowing effects on parton densities in the nucleus, however, the expected signal is subtle and below the present experimental sensitivity.

We study quantum property testing for directed graphs with maximum in-degree and out-degree bounded by some universal constant $d$. For a proximity parameter $\varepsilon$, we show that any property that can be tested with $O_{\varepsilon,d}(1)$ queries in the classical bidirectional model, where both incoming and outgoing edges are accessible, can also be tested in the quantum unidirectional model, where only outgoing edges are accessible, using $n^{1/2 - Ω_{\varepsilon,d}(1)}$ queries. This yields an almost quadratic quantum speedup over the best known classical algorithms in the unidirectional model. Moreover, we prove that our transformation is almost tight by giving an explicit property $P_\varepsilon$ that is $\varepsilon$-testable within $O_\varepsilon(1)$ classical queries in the bidirectional model, but requires $\widetildeΩ(n^{1/2-f'(\varepsilon)})$ quantum queries in the unidirectional model, where $f'(\varepsilon)$ is a function that approaches $0$ as $\varepsilon$ approaches $0$. As a byproduct, we show that in the unidirectional model, the number of occurrences of any constant-size subgraph $H$ can be approximated up to additive error $δn$ using $o(\sqrt{n})$ quantum queries.

The interaction between nucleons and hyperons - baryons containing a strange quark - is key to understanding the properties of dense nuclear matter, such as that expected in the interior of neutron stars. Direct scattering experiments are hindered by the short lifetime of hyperons, prompting the study of hypernuclei - bound states of nucleons and hyperons - as an alternative approach. The lightest known hypernucleus, the hypertriton ($^3_Λ$H), is a weakly bound state composed of a proton, a neutron and a $Λ$ hyperon, and is believed to exhibit a halo-like structure with the $Λ$ being loosely bound to a deuteron core. Based on the first measurement of hypertriton production in proton-proton collisions at the CERN Large Hadron Collider (LHC), its halo structure is confirmed. A successful description of the hypertriton production yield within the nuclear coalescence framework enables an estimation of the $Λ$ separation from the deuteron core as $9.54^{+2.67}_{-1.11}$ fm.

Existing numerical methods for modeling magnetization in thin type-II superconducting films have mostly been developed for flat films. This work introduces an efficient spectral method for axisymmetric magnetization problems involving non-flat films. The method is based on the integral thin-shell current-density formulation of the problem, employs Chebyshev polynomial expansions for spatial discretization, and uses the method of lines for time integration. It applies to both open and closed axisymmetric shells and is so accurate that the solutions obtained can serve as benchmarks for numerical methods for general, not necessarily axisymmetric, thin-shell magnetization problems. As one of the examples, we consider magnetic shielding by a superconducting sphere.

Under ambient humidity, water spontaneously condenses in pores only a few nanometers in size, making nanoscale capillarity central to numerous natural phenomena and technological applications. At these dimensions, water may no longer be treated as a continuous fluid, yet the consequences of molecular discreteness for capillary condensation and filling remain poorly understood. Here we study nanocapillaries fabricated by van der Waals assembly and, using atomic force microscopy, monitor their wall deformations during humidity-driven water uptake. We observe two distinct regimes: layer-by-layer filling of flexible capillaries and abrupt filling of rigid ones. Flexible walls deform in steps of ~3 Å, corresponding to the sequential entry of individual water molecular layers. The different filling regimes are explained by the competition between deformation energy and oscillatory wall-water interactions. Our findings show that the molecular discreteness of water can profoundly affect ubiquitous capillary phenomena, with wall compliance selecting between discrete and abrupt filling.

Let $(M,\mathsf{g})$ be a connected and compact Riemannian manifold admitting an isometric action by a compact Lie group $G$ whose principal orbits have codimension one. We show that any $G$-invariant, smooth, and divergence-free vector field $u_0$ on $(M,\mathsf{g})$ initiates a $G$-invariant time-varying velocity-pressure pair $(u,p)$ which has time interval $\mathbb{R}$, is smooth, and solves the incompressible Euler fluid equations.

Bayesian methods for the detection of continuous gravitational waves (CGWs) in Pulsar Timing Array (PTA) data incur substantial computational costs that grow rapidly due to the number of noise and signal parameters characterizing the fitted model being proportional to the size of the PTA. This computational burden limits the scalability of these methods for large-scale PTAs comprising hundreds of pulsars anticipated from next-generation radio astronomy facilities. In this work, we introduce a computationally efficient frequentist method designed to circumvent this challenge. This is achieved by combining an adaptive spline fitting algorithm that non-parametrically suppresses red noise, thereby eliminating the need for complex noise modeling inherent to Bayesian methods, with a novel scheme for optimizing the subsets of pulsars included in the search. We quantify the performance of our method on a simulated dataset based on the NANOGrav 15-year data release and find that it achieves a performance comparable to that of Bayesian analysis: for a CGW signal with a signal-to-noise ratio of $\approx 10$, our method yields a relative characteristic strain error of 1.0\% and a frequency error of 0.072\% from the injected values by using the optimal pulsar selections, while the same errors are 1.7\% and 0.16\%, respectively, for the standard Bayesian analysis. At the same time, our analysis completes in less than 5 hours, in contrast to the 1-2 days required by Bayesian methods. This allows us to perform a rigorous study of our method using multiple data realizations and signal parameters, establishing it as an efficient and scalable tool for CGW searches with large-scale PTAs.

The rotation-activity connection is the standard paradigm for interpreting chromospheric and coronal activity in late-type stars, namely, stars with outer convection zones. This paradigm states that activity increases with decreasing rotation period until a saturation limit is reached. By scaling rotation periods with the convective turnover time via the Rossby number, $\text{Ro}$, saturation is expected to occur at a universal value across all spectral types. In our paper, we systematically investigate the relationship between rotation and activity as measured though X-ray emission for a large sample of late-type stars to test the universal applicability of this paradigm. To this end, we utilized TESS short-cadence space photometry to determine the rotation periods for late-type stars identified in the eROSITA all-sky survey. This combined dataset provides rotation and X-ray measurements for 14004 stars, representing a sample size increase of more than an order of magnitude compared to previous studies. We find that the convective turnover times derived from this sample closely agree with theoretical computations, supporting the idea that Rossby number-activity relations hold for all late-type main sequence stars. The lower level of activity in earlier spectral types (e.g., F-type and late A-type stars) is a physical consequence of their short convective turnover times, which prevent them from rotating rapidly enough to ever reach the saturation regime. We demonstrate that a simple model incorporating our derived turnover times versus color can successfully reproduce the observed characteristics of the eROSITA X-ray activity distribution, as measured by the L$_X$/L$_{\text{bol}}$ ratio and {\it Gaia} BP-RP color.

The Hardware Lottery posits that research directions are dictated by available silicon compute platforms. We identify a derivative phenomenon, the Hyperscale Lottery, where model architectures are optimized for cloud throughput at the expense of algorithmic efficiency. While State-Space Models (SSMs) such as Mamba were lauded for their linear complexity, ideal for edge intelligence, their evolution from Mamba-1 to Mamba-3 reveals a systematic divergence from edge-native efficiency. We demonstrate that Mamba-3's architectural changes, designed to saturate hyperscale GPUs, impose a significant edge penalty: a 28% latency increase at 880M parameters, worsening to 48% for 15M-parameter models. We argue for decoupling cloud-scale saturation strategies from core architectural design to preserve the viability of single-user, real-time edge intelligence.

In moiré superlattices, the band flatness governs the degree of wave localization, which is central to harnessing emergent phenomena and designing functional meta-devices. While research has focused on the magic conditions such as magic angle and magic distance for optimal flatness, a fundamental understanding of how flatness changes with the supercell size has remained elusive. Here, we establish a universal scaling between band flatness and supercell size. Theoretically, by recognizing the statistical equivalence between structural perturbations in moiré superlattices and disordered systems, we introduce the Thouless number to evaluate the strength of moiré localization. This approach allows us to establish a scaling theory for the evolution of band flatness with the supercell size, from which an analytical expression is derived. Our full-wave simulations with one-dimensional and two-dimensional moiré superlattices show excellent agreement with the theoretical prediction. Our work reveals a general scaling law for moiré band flatness, offering a new perspective for understanding and designing moiré-based resonant systems.

We present detailed light-curve analyses of ten binary-lens microlensing events observed during the 2023--2025 seasons and selected as candidates for hosting brown-dwarf companions. The sample includes OGLE-2023-BLG-0249, KMT-2023-BLG-1246, OGLE-2023-BLG-0079, KMT-2024-BLG-0072, KMT-2024-BLG-0897, KMT-2024-BLG-1876, KMT-2024-BLG-2379, KMT-2025-BLG-0922, KMT-2025-BLG-1056, and KMT-2025-BLG-2427. For each event, we carry out modeling of the light curve, explore relevant degeneracies, and, when finite-source effects are present, determine the angular Einstein radius. For OGLE-2023-BLG-0249, we additionally measure the microlens parallax, which allows a direct determination of the lens masses and distance. For the remaining events, we estimate the physical lens properties via Bayesian analyses incorporating Galactic priors. The resulting posteriors show that the lens companions in all systems have median masses in the brown-dwarf regime, and the lenses of two events (KMT-2025-BLG-0922 and KMT-2025-BLG-1056) are consistent with binaries in which both lens components fall within the brown-dwarf mass range. Spanning a wide range of projected separations and distances, these detections illustrate the power of high-cadence microlensing surveys to build a census of brown-dwarf companions, including faint and distant systems beyond the reach of flux-limited methods.

This paper studies how complicated and irregular behavior, known as chaos, can arise in a simple mathematical model that includes time delays. The model is a delay differential equation in which the present rate of change depends not only on the current state but also on past states at two different delay times. The system is described by \begin{equation} \dot{x}(t) = -γx(t) + g\big(x(t - τ_1)\big) - e^{-γτ_2}, g\big(x(t - τ_1 - τ_2)\big), \qquad 0 < α\le 1, \end{equation} where $g(x)=k \sin{x}, k\in\mathbf{R}$. Here, the delays $τ_1$ and $τ_2$ represent memory effects in the system, while the sine terms introduce strong nonlinearity. Numerical simulations are used to study the system behavior for different parameter values. Chaotic motion is identified using Lyapunov exponents and phase portraits, which show irregular and unpredictable dynamics. For certain parameter ranges, the system exhibits multi-scroll chaotic attractors, in which the motion alternates among several complex patterns. Finally, chaos is controlled by adding a simple linear feedback term, which suppresses irregular oscillations and stabilizes the system. In addition, synchronization between master and slave systems is investigated using linear state feedback control, and a delay-independent sufficient condition for synchronization is derived and verified numerically. The results show that even complex delayed systems can be effectively controlled and synchronized using simple feedback techniques. The study is further extended to a fractional-order version of the system to examine the influence of memory effects, where it is observed that chaotic behavior can persist even for lower fractional orders.

We propose a Physics Informed Learning framework for reconstructing traffic density from sparse trajectory data. The approach combines a second-order Aw-Rascle and Zhang model with a first-order training stage to estimate the equilibrium velocity. The method is evaluated in both equilibrium and transient traffic regimes using SUMO simulations. Results show that while learning the equilibrium velocity improves reconstruction under steady state conditions, it becomes unstable in transient regimes due to the breakdown of the equilibrium assumption. In contrast, the second-order model consistently provides more accurate and robust reconstructions than first-order approaches, particularly in nonequilibrium conditions.

We introduce a general semiparametric clusterwise elliptical distribution to assess how latent cluster structure shapes continuous outcomes. Using a subjectwise representation, we first estimate cluster-specific mean vectors and a cluster-invariant scatter matrix by minimizing a weighted sum of squares criterion augmented with a separation penalty; we provide an initialization scheme and a computational algorithm with guaranteed convergence. This initial estimator consistently recovers the true clusters and seeds a second phase that alternates pseudo-maximum likelihood (or pseudo-maximum marginal likelihood) estimation with cluster reassignment, yielding asymptotic semiparametric efficiency and an optimal clustering that asymptotically maximizes the probability of correct membership. We also propose a semiparametric information criterion for selecting the number of clusters. Monte Carlo simulations and empirical applications demonstrate strong finite-sample performance and practical value.

We investigate quantum field theory in Rindler space with a UV cutoff by considering a free scalar field on a lattice in Rindler coordinates. We find that the Minkowski vacuum is not exactly thermal with respect to the local lattice Rindler Hamiltonian. Nevertheless, for observables sufficiently far from the horizon, the Wightman function and the Unruh--DeWitt detector response reproduce the expected thermal behavior in the continuum limit. Thus, the Unruh effect survives operationally, even though exact thermality is lost at the state level. We also show that the Rindler vacuum energy density reproduces the standard continuum behavior away from the horizon, while the UV singularity at the horizon is replaced by a stretched-horizon contribution. Furthermore, the retarded Green function exhibits a component reflected at the stretched horizon, implying that an ingoing wave packet is reflected at a proper distance of order the cutoff. This provides an effective brick-wall picture in the UV-regulated theory. Our analysis suggests that, once a cutoff is introduced, the global Minkowski description and the wedge description based on a local Rindler Hamiltonian are no longer equivalent at the operator level.

Contextual learning seeks to learn a decision policy that maps an individual's characteristics to an action through data collection. In operations management, such data may come from various sources, and a central question is when data collection can stop while still guaranteeing that the learned policy is sufficiently accurate. We study this question under two precision criteria: a context-wise criterion and an aggregate policy-value criterion. We develop unified stopping rules for contextual learning with unknown sampling variances in both unstructured and structured linear settings. Our approach is based on generalized likelihood ratio (GLR) statistics for pairwise action comparisons. To calibrate the corresponding sequential boundaries, we derive new time-uniform deviation inequalities that directly control the self-normalized GLR evidence and thus avoid the conservativeness caused by decoupling mean and variance uncertainty. Under the Gaussian sampling model, we establish finite-sample precision guarantees for both criteria. Numerical experiments on synthetic instances and two case studies demonstrate that the proposed stopping rules achieve the target precision with substantially fewer samples than benchmark methods. The proposed framework provides a practical way to determine when enough information has been collected in personalized decision problems. It applies across multiple data-collection environments, including historical datasets, simulation models, and real systems, enabling practitioners to reduce unnecessary sampling while maintaining a desired level of decision quality.

We investigate incentives for reducing the carbon emissions of video streaming that depend on the energy consumption of segments in the end-to-end video delivery path, the carbon intensity, and the user type, i.e., quality-sensitive and green or environmentally conscious users. The incentives can be offered through a practical 2-tier subscription model with a discount and carbon rewards, which gives providers the flexibility to reduce the quality for up to a maximum percentage of videos within a time period, such as one month. The key features of our approach are i) it is preferable to offer subscriptions where the reduced-quality tier is set one resolution level below the resolution required for maximum user satisfaction; ii) when a video is streamed from a local data center, the maximum percentage of videos streamed at a lower quality depends solely on the carbon intensity and the average intensity cap, whereas the incentives also depend on the users' level of environmental consciousness; iii) when a video can be streamed from a local or a remote data center with different carbon intensities, the maximum percentage of videos streamed at lower quality and the incentives depend on the relative carbon intensity and energy consumption at the data centers, and the additional network energy costs from the remote data center.

Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment.

In the first part of this two-part paper a game-theoretic decentralized real-time control is proposed in the context of Electric Vehicle (EV) Charging Station (CS). This method, relying on a Stackelberg Game-based Alternating Direction of Multipliers (SG-ADMM), intends to steer the EVs' individual objectives towards the CS optimum by means of an incentive design mechanism, while controlling the EV power dispatch in a distributed manner. We integrate SG-ADMM in a hierachical multi-layered Energy Management System (EMS) as the real-time control algorithm, formulating the two-layer approach so that the SG leader (i.e., the CS), holding commitment power, trades off the available power with the incentives to the EVs, and the SG followers (i.e., the EVs) optimizes their charging curve in response to the leader decision. In this second part, we demonstrate the applicability of SG-ADMM as a incentive design mechanism inside an EVCS EMS, testing it in a large-scale EVCS. We benchmark this method with a decentralized (ADMM-based), a centralized and a uncontrolled approach, showing that our method exploits EV-level flexibility in a cost-effective, fair and computationally efficient manner.

We investigate not only the associated curves of regular plane curves, but also those of Legendre curves. As associated curves, we consider Bertrand regular plane curves and Bertrand Legendre curves. These curves contain parallel, evolute and involute curves, as well as evolutoid and involutoid curves. Since associated curves may have singular points even if the original curve is regular, Legendre curves provide a suitable framework for investigating the properties of such curves. We give existence conditions of Bertrand regular plane curves and Bertrand Legendre curves. Moreover, we give an inverse operation for Bertrand Legendre curves. Furthermore, we define a mapping between sets of Legendre curves using Bertrand Legendre curves and prove that this mapping is bijective up to equivalence relations.

We prove that sparse string graphs in a fixed surface have linear expansion. We extend this result to the more general setting of sparse region intersection graphs over any proper minor-closed class. The proofs are combinatorial and self-contained, and provide bounds that are within a constant factor of optimal. Applications of our results to graph colouring are presented.

Granlund and Montgomery proposed an optimization method for unsigned integer division by constants [3]. Their method (called the GM method in this paper) was further improved in part by works such as [1] and [7], and is now adopted by major compilers including GCC, Clang, Microsoft Compiler, and Apple Clang. However, for example, for x/7, the generated code is designed for 32-bit CPUs and therefore does not fully exploit 64-bit capabilities. This paper proposes an optimization method for 32-bit unsigned division by constants targeting 64-bit CPUs. We implemented patches for LLVM/GCC and achieved speedups of 1.67x on Intel Xeon w9-3495X (Sapphire Rapids) and 1.98x on Apple M4 (Apple M-series SoC) in the microbenchmark described later. The LLVM patch has already been merged into llvm:main [6], demonstrating the practical applicability of the proposed method.

Large-scale Electric Vehicle (EV) Charging Station (CS) may be too large to be dispatched in real-time via a centralized approach. While a decentralized approach may be a viable solution, the lack of incentives could impair the alignment of EVs' individual objectives with the controller's optimum. In this work, we integrate a decentralized algorithm into a hierarchical three-layer Energy Management System (EMS), where it operates as the real-time control layer and incorporates an incentive design mechanism. A centralized approach is proposed for the dispatch plan definition and for the intra-day refinement, while a decentralized game-theoretic approach is proposed for the real time control. We employ a Stackelberg Game-based Alternating Direction Method of Multipliers (SG-ADMM) to simultaneously design an incentive mechanism while managing the EV control in a distributed manner, while framing the leadership-followership relation between the EVCS and the EVs as a non-cooperative game where the leader has commitment power. Part I of this two-part paper deals with the SG-ADMM approach description, literature review and integration in the abovementioned hierarchical EMS, focusing on the modifications needed for the proposed application.

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this paper, we introduce an equivalence relation for Legendre curves called the curvature equivalence. We investigate properties of the curvature equivalence and give local and global classifications of Legendre curves under the curvature equivalence.