What is the meaning of physical causal closure? Jaegwon Kim explicitly adopts a conception of causation according to which physical causation is effectively identified with deterministic physical lawfulness, and equates it with physical determinism. While this conception is internally coherent, it differs from currently dominant theories of causation. Physics and the theory of causation serve different descriptive purposes. In this study, we refer to them, respectively, as the Physical Stance and the Causal Stance. Within this framework, physical determinism belongs to the Physical Stance, and physical causal closure is defined only within the Causal Stance. Consequently, the two should not be equated. On this basis, this study reconstructs Davidson's anomalous monism as a materialist position that acknowledges mental causation without contradicting physical determinism. Furthermore, we propose a linguistic framework in which physical causal closure does not hold in the Causal Stance while physical determinism remains intact in the Physical Stance.

Well-known conjectures of Charney--Davis, Gal, and Nevo--Petersen predict increasingly strong positivity phenomena for the h-vectors of flag simplicial spheres. In this paper, we formulate and prove matroid analogues of these conjectures in the setting of Chow polynomials of matroids with building sets. Our proofs rely on toric geometry and make crucial use of tropical intersection theory. We begin by introducing complete building sets, a class encompassing all maximal building sets and other important families such as minimal building sets of braid matroids. For matroids with complete building sets, we analyze the Chow rings of the associated toric varieties, and prove that their Hilbert--Poincaré polynomials are gamma-positive. From this analysis, we derive a combinatorial formula for the coefficients of the gamma-expansion, and use it to explicitly construct a simplicial complex $Γ$, whose f-vector coincides with the gamma-vector. This establishes a matroid analogue of the Nevo--Petersen conjecture. When the building set is maximal, we further prove that $Γ$ is balanced, confirming the strongest such analogue in this case. As an application, we obtain a new combinatorial formula for the gamma-expansion of the Poincaré polynomial of the Deligne--Mumford--Knudsen compactification $\overline{\mathcal{M}}_{0,n}$, and derive several novel numerical inequalities for its coefficients. We also study the toric varieties of matroids with flag building sets, another class containing maximal building sets as well as several other prominent families. We prove that the Hilbert--Poincaré polynomials of these toric varieties are gamma-positive. This result establishes matroid analogues of the Charney--Davis and Gal conjectures, and simultaneously extends several recent gamma-positivity results for Chow polynomials.

We present, in detail and with a modern rigor, the two title proofs. The Weak Jordan Theorem (WJT) states that the complement of any topological circuit in the plane is disconnected.

Small area estimation (SAE) produces estimates of population parameters for geographic and demographic subgroups with limited sample sizes. Such estimates are critical for informing policy decisions, ranging from poverty mapping to social program funding. Despite its widespread use, principled validation of SAE models remains challenging and general guidelines are far from well-established. Unlike conventional predictive modeling settings, validation data are rarely available in the SAE context. External validation surveys or censuses often do not exist, and access to individual-level microdata is often restricted, making standard cross-validation infeasible. In this paper, we propose a novel model validation scheme using only area-level direct survey estimates under the widely used Fay-Herriot model. Our approach is based on data thinning, which splits area-level observations into independent training and test components to enable out-of-sample validation. Our theoretical analysis reveals a fundamental tension inherent in thinning-based validation: performance metrics measured on the thinned training component target a different quantity than those based on the full data, with the gap varying by model complexity. Increasing the information allocated for training reduces this gap but inflates the variance of the estimator. We formally characterize this bias-variance tradeoff and provide practical recommendations for the thinning parameters that balance these competing considerations for model comparison. We show that data thinning with these settings provides consistent and stable performance across heterogeneous sampling designs in design-based simulations using American Community Survey microdata.

Safe operation of autonomous systems requires robustness to both model uncertainty and uncertainty in the environment. We propose DRP-$\mathcal{L}_1$AC, a hierarchical framework for stochastic nonlinear systems that integrates distributionally robust model predictive control (DR-MPC) with $\mathcal{L}_1$-adaptive control. The key idea is to use the $\mathcal{L}_1$-adaptive controller's online distributional certificates that bound the Wasserstein distance between nominal and true state distributions, thereby certifying the ambiguity sets used for planning without requiring distribution samples. Environmental uncertainty is captured via data-driven ambiguity sets constructed from finite samples. These are incorporated into a DR-MPC planner enforcing distributionally robust chance constraints over a receding horizon. Using Wasserstein duality, the resulting problem admits tractable reformulations and a sample-based implementation. We show theoretically and via numerical experimentation that our framework ensures certifiable safety in the presence of simultaneous system and environmental uncertainties.

This paper presents a dissipativity-based distributed droop-free control and communication topology co-design framework for voltage regulation and current sharing in DC microgrids (MGs), where constant-power loads (CPLs) and voltage-source converter (VSC) input saturation introduce significant nonlinearities. In particular, CPLs introduce an inherently destabilizing nonlinearity, while VSC input saturation imposes hard amplitude constraints on applicable control input at each distributed generator (DG), collectively making the DC MG control system design extremely challenging. To this end, the DC MG is modeled as a networked system of DGs, transmission lines, and loads coupled through a static interconnection matrix. Each DG is equipped with a local PI-based controller with an anti-windup compensator and a distributed consensus-based global controller, from which a nonlinear networked error dynamics model is derived. The CPL nonlinearity is characterized via sector-boundedness with the S-procedure applied directly to yield tight LMI conditions, while the VSC input saturation is handled via a dead-zone decomposition and sector-boundedness, with both nonlinearities simultaneously absorbed into the dissipativity analysis. Both nonlinearities are simultaneously absorbed into the dissipativity analysis using the S-procedure. Subsequently, local controller gains and passivity indices, and distributed controller gains and the communication topology are co-designed by solving a sequence of local and global Linear Matrix Inequality (LMI) problems, enabling a one-shot co-design process that avoids iterative procedures. The effectiveness of the proposed framework is validated through simulation of an islanded DC MG under multiple operating scenarios, demonstrating robust performance superior to conventional control approaches.

A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation \(E \propto |p|^{j}\), where \(j = N - 1\). Starting from the generalized Minkowski distance in \(L^j\)-normed spaces, the conventional quadratic kinetic structure of three-dimensional geometry is extended to higher-order spatial derivatives, yielding a consistent \(j\)-th order Schrödinger equation. The formalism is applied to free particles and to particles confined within a one-dimensional infinite potential well for 2G, 3G, 4G, and 5G geometries. While plane-wave solutions and translational invariance are preserved, the spectral structure is modified, with bound-state energies scaling as \((2n+1)^{j}\), leading to cubic and quartic growth in higher geometries. The corresponding eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms determined by the roots of negative unity. A generalized probability framework based on \(j\)-fold conjugation is introduced, ensuring a real-valued probability density and consistent expectation values. Despite these generalizations, the Heisenberg uncertainty principle is preserved. The formulation presents quantum mechanics as a geometry-dependent theory in which dispersion relations, spectral properties, and probabilistic structure emerge from the underlying spatial metric.

In recent years, layered kagome magnets have emerged as promising platforms for Berry-curvature engineering and unconventional transport phenomena. Here, we present the single-crystal growth, magnetization, and electrical transport characterizations of the van der Waals-like layered antiferromagnet GdTi3Bi4. The system exhibits pronounced field-induced first-order phase transitions. Comprehensive frequency, temperature, and field-dependent ac susceptibility measurements, and Hall analysis, reveals the formation of a spin-cluster-like glassy magnetic phase attributed to noncollinear spin textures. Additionally, the system demonstrates a colossal anomalous Hall conductivity σ_xy^{A}~ 8.6(7)10^{3} Ohm-1 cm-1 at 2 K). Detailed scaling analyses reveal the coexistence of skew scattering and intrinsic Berry-curvature contributions to the anomalous Hall effect. First-principles calculations highlight flat-band near the Fermi level, with f-electrons of the Gd ion contributing large intrinsic Hall response. Thus, GdTi3Bi4 emerges as a rare layered kagome magnet, exhibiting Berry curvature-induced giant anomalous and spin texture-driven Hall responses, providing a versatile platform for exploring spin-texture physics and advancing low-dimensional spintronic functionalities.

Negative avoiding sequences of span $n$ are periodic sequences of elements from $\mathbb{Z}_k$ for some $k$ with the property that no $n$-tuple occurs more than once in a period and if an $n$-tuple does occur then its negative does not. They are a special type of cut-down de Bruijn sequence with potential position-location applications. We establish a simple upper bound on the period of such a sequence, and refer to sequences meeting this bound as maximal negative avoiding sequences. We then go on to demonstrate the existence of maximal negative avoiding sequences for every $k\geq3$ and every $n\geq2$.

We study the influence of the loop quantum gravity (LQG) holonomy-correction parameter $ξ$ on black hole horizon structure, timelike geodesic motion, and gravitational wave emission in two rotating LQG-inspired black hole spacetimes, constructed via Newman-Janis algorithm from two distinct spherically symmetric seed metrics (type BH-I and BH-II). The physically admissible range of $ξ$ is determined by requiring the existence of event horizons, marginally bound orbits, and innermost stable circular orbits simultaneously, and is found to shrink monotonically with increasing spin parameter $a$. For equatorial periodic orbits, increasing $ξ$ at fixed angular momentum enlarges the bound energy range, while for off-equatorial orbits, it suppresses the allowed range of the Carter constant, effectively confining trajectories closer to the equatorial plane. The effects of $ξ$ and $a$ on orbital dynamics are systematically antagonistic. Gravitational waveforms computed within a leading-order post-Newtonian extreme-mass-ratio inspiral (EMRI) model show that larger $ξ$ produces enhanced deviations from the Kerr waveform, more prominently so for type BH-II than type BH-I. The resulting characteristic strains occupy the $(10^{-3}, 0.1)$ Hz frequency band but fall below the sensitivity curves of current and near-future space-based detectors for the EMRI parameters considered ($M=10^7 M_\odot$, $m=10 M_\odot$, $D_L = 200$ Mpc). Adiabatic inspiral calculations confirm that $ξ$ and $a$ drive orbital evolution in opposite directions, with their relative magnitude determining whether quantum corrections accelerate or retard the inspiral. These results establish systematic observational signatures of holonomy corrections in rotating LQG black holes and motivate higher-fidelity waveform modeling for future space-based gravitational wave detectors.

Data unfolding -- the removal of noise or artifacts from measurements -- is a fundamental task across the experimental sciences. Of particular interest are applications in physics, where the dominant approach is Richardson-Lucy (RL) deconvolution. The classical RL approach aims to find denoised data that, once passed through the noise model, is as close as possible to the measured data in terms of Kullback-Leibler (KL) divergence. This requires that the support of the measured data overlaps with the output of the noise model, a hypothesis typically enforced by binning, which introduces numerical error. As a counterpoint, the present work studies an alternative formulation using a Wasserstein loss. We establish sharp conditions for existence and uniqueness of optimizers, answering open questions of Li, et al., regarding necessary conditions for uniqueness in the case of transport map noise models. We then develop a provably convergent generalized Sinkhorn algorithm to compute approximate optimizers. Our algorithm requires only empirical observations of the noise model and measured data and scales with the size of the data, rather than the ambient dimension. Numerical experiments on one- and two-dimensional problems inspired by jet mass unfolding in particle physics demonstrate that the optimal transport approach offers robust, accurate performance compared to classical RL deconvolution, particularly when binning artifacts are significant.

In solid-state batteries, ceramic solid electrolytes are penetrated by dendrites when plating above a critical current density $J_\mathrm{crit}$. A dendrite will propagate by metal deposition at a pre-existing dendrite tip if the mechanical energy required to crack the ceramic open is less than the electrical energy (Joule heating) wasted by forcing the current to detour around the dendrite to the flat electrode surface. Based on this principle of minimal power dissipation, a dependence of $J_\mathrm{crit}\propto c_\mathrm{max}^{3/2}$ is derived. $c_\mathrm{max}$ is the length of the longest preexisting, sufficiently thin interfacial defect. Consequentially, scattering of $J_\mathrm{crit}$ between samples must follow a Weibull-distribution, similar to the tensile strength of ceramic components. Furthermore, a theory is presented by which residual electron conduction of the solid electrolyte leads to electrochemical stress corrosion cracking at dendrite tips, thus expanding the concept of the critical current density to a subcritical crack growth regime.

The origin of Saturn's rings has been debated for decades. Measurements from Voyager and Cassini have suggested that the rings could be as young as ~100 Myr and composed of nearly pure water ice. Several scenarios have been proposed to explain these properties. One hypothesis (Wisdom et al 2022) is that the rings formed through the recent tidal disruption of a pre-existing moon, Chrysalis, which experienced a close encounter with Saturn following its highly eccentric orbit. However, the mechanism by which this hypothesis would have formed the rings remains largely unexplored, in particular, whether Chrysalis could supply ring material of the desired mass and composition. To address these questions, we perform smoothed particle hydrodynamics simulations to investigate the tidal response of Chrysalis during close encounters with Saturn. Our results demonstrate that preferential tidal stripping of the ice mantle from a differentiated Chrysalis can produce rings with both mass and composition resembling the present rings -- provided that the closest encounter occurs between the parabolic Roche limits for ice ~1.53Rs and rock ~1.07Rs -- consistent with Wisdom et al 2022. Moreover, multiple close encounters can extend the effective disruption limit by spinning up the body, enhancing the tidal stripping efficiency. Following close encounters, the rocky remnant of Chrysalis would have been removed in less than few kyr, either by collision with Saturn or ejection onto a hyperbolic orbit. These findings support the hypothesis that Saturn's rings could originate from a recent lost moon, and imply a highly dynamical evolution of the Saturnian system over the past few hundred million years.

Recent work has highlighted the deep connection between quantum information and spacetime geometry. Bousso and Shahbazi-Moghaddam (Phys. Rev. Lett. 128, 231301 (2022)) proved that ``hyperentropic'' regions -- where entropy exceeds the area bound -- inevitably lead to singularity formation. In this work, we explore the converse implication: does the thermodynamic consistency of such singularities require them to be hidden? We answer in the affirmative, establishing a Quantum Weak Cosmic Censorship principle governed by Generalized Entropy. This provides a semiclassical mechanism for censorship which forbids naked singularities. Since Quantum Weak Cosmic Censorship is a semiclassical statement, it is more robust than the classical Weak Cosmic Censorship showing naked singularities are forbidden in nature even if quantum effects are taken into account.

Rosenmann and Ventura asked "What is the right definition of dependence of subgroups for general groups?". Here we aim to answer this question. We consider a definition of subgroup independence which is a special case of a category-theoretic one. It is that: Two subgroups of a group are independent if and only if any two endomorphisms, one acting on each subgroup, can be extended to an endomorphism of the group generated by these subgroups. This definition helps to illuminate that the usual condition of almost disjointness of subgroups (two subgroups $A$ and $B$ are almost disjoint if and only if $A \cap B = \{e\}$, where $e$ is the identity element) is not enough to force independence and here we find necessary and (different) sufficient conditions for subgroup independence. The aim of this note is to introduce this general notion of subgroup independence to the group theory community and to pose the open question of its characterisation. We present the partial results known up to this point. Moreover, we use the progress made so far to give a heuristic algorithm that decides subgroup independence for many cases.

Two independent electron temperature diagnostics applied to the quiet solar corona yield systematically different results. Radio brightness temperatures from the Nancay Radioheliograph indicate T_e ~ 0.6 MK, while hydrostatic scale-height modeling requires T_e ~ 1.5 MK. Both probe electrons; they disagree by a factor of R = 2.4 +/- 0.3. This discrepancy persists across eight years spanning solar minimum and is confirmed by LOFAR at lower frequencies. We consider turbulent scattering, which suppresses brightness temperature, but comparison with the FORWARD/PSIMAS Maxwellian model shows the standard thermal structure predicts ~1.6 MK; scattering accounts for the reduction toward observed MWA values but not the gap to 620 kK. The ratio R is also cycle-invariant despite measured variations in turbulence. We propose the residual discrepancy reflects non-Maxwellian electron velocity distributions. Radio bremsstrahlung samples the distribution core; ionization and scale heights are dominated by the suprathermal tail. For kappa distributions, the predicted ratio kappa/(kappa - 3/2) matches R = 2.4 at kappa ~ 2-3, consistent with spectroscopic measurements in active regions but in tension with perturbative predictions of kappa ~ 10-25. We predict Active Region cores should show a collapsed ratio (R <= 1.5) as collisionality restores thermal equilibrium.

We show that a topologically generating set $S$ of a connected compact Lie group $G$ of size larger than a fixed polynomial in the rank of $G$ must be redundant (i.e., some proper subset of $S$ still topologically generates $G$). Similar results are obtained for amenable Lie groups and for reductive algebraic groups with the Zariski topology. The quantitative bounds produced by our method are controlled by corresponding bounds for finite simple groups of Lie type. We also treat redundancy up to Nielsen transformations, thereby partially answering a few conjectures of Gelander. We show that these conjectures are implied by the Wiegold conjecture.

We investigate the evolution of matter density perturbations within a fractional cosmological framework inspired by fractal space-time constructions in field theory, where a deformation of the integration measure induces non-locality and memory effects in the dynamics. Working in the matter-dominated era and adopting a covariant fluid-flow approach, we derive the modified growth equation for the density contrast and obtain exact analytical solutions. The resulting dynamics depends explicitly on the fractional parameter $α$ and smoothly reduces to the corresponding standard case in the limit $α=1$. We show that the model admits both growing and decaying modes, and we identify the parameter range in which structure formation is physically viable. Focusing on the growing mode, we compute the evolution of density fluctuations from recombination to the present epoch. By confronting theoretical predictions with observational constraints from large-scale structure, in particular the $σ_8$ normalization and the Sachs-Wolfe effect, we derive a stringent upper bound on the fractional parameter, $α\lesssim1.07$, which significantly improves upon previous constraints obtained at the background level. Our results show that the growth of density perturbations exhibits distinct fractional signatures, providing a sensitive observational probe of the underlying framework.

We provide asymptotics for the number of faces of a certain family of Hanner polytopes. As a corollary, we come close to saturating the FLM inequality for a certain family of parameters.

We performed $^{85}$Rb nuclear magnetic resonance (NMR) measurements on the $S$ = 1 bilayer triangular-lattice antiferromagnet Rb$_2$Ni$_2$(SeO$_3$)$_3$ in magnetic fields up to 26 T. In the field range from 3 T to 26 T, the NMR spectral lines split and their respective spectral weight ratios reveal the existence of the magnetic up-up-down (UUD) phase, although the 1/3-plateau phase is only reached at fields above 16 T. Two distinct gapless regimes are further identified: one at low fields and low temperatures, and the other at high fields and high temperatures, consistent with the spin supersolid Y and V phases. Notably, the UUD-V phase boundary exhibits a negative slope in $dT/dH$, where the supersolid phase is located at temperatures above the solid phase due to strong low-energy spin fluctuations.

Proton decay searches in the next generation of water Cherenkov detectors, such as Hyper-Kamiokande, are expected to probe the $10^{35}$-year lifetime regime where atmospheric neutrino backgrounds and systematic uncertainties begin to play an increasingly important role. In this study, we employ the GiBUU framework and reevaluate the proton decay search sensitivity for the $\textrm{p}\rightarrow\textrm{e}^{+}π^{0}$ channel by incorporating a typical event reconstruction performance in water Cherenkov detectors. Using sophisticated models implemented in GiBUU -- most notably the mean-field potential and Boltzmann transport -- which have been benchmarked against accelerator neutrino scattering data, in particular pion production, we find that the resulting proton decay signal detection efficiency and atmospheric neutrino background rate are comparable to those previously evaluated for the current and near future water Cherenkov experiments using $\textit{ad hoc}$ nuclear models. In addition to pion final-state interactions, we evaluate the impact of differences in the Fermi momentum distribution of nucleons in the nucleus, as a source of systematic uncertainty, on the signal detection efficiency and the expected background event rate. We find that the uncertainty associated with pion final-state interactions is moderate, whereas the choice of Fermi momentum distribution can significantly affect the estimated atmospheric neutrino background rate and constitutes the dominant contribution. Our study provides an independent and complementary characterisation of nuclear effects on proton decay searches and helps to refine sensitivity estimates in the regime where systematic uncertainties become more relevant.

In this paper, we provide explicit recursive constructions of infinitely many non-equivalent wild knots contained in the Menger sponge, in such a way that we can control their set of wild points that lies in a usual Cantor set contained in the Menger sponge. Furthermore, we show that wild knots of dynamically defined type arising from Kleinian group actions can be isotoped into the sponge. We want to emphasize that our approach is constructive and geometric.

A group $G$ is said to have restricted centralizers if for every $x\in G$ the centralizer $C_G(x)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups $G$ for which there is an integer $n$ such that $C_G(x^n)$ is either finite or open whenever $x\in G$. It is shown that such a group $G$ has an open normal subgroup $T$ with the property that $G/Z(T)$ has finite exponent.

We present balnet, an R package for scalable pathwise estimation of covariate balancing propensity scores via logistic covariate balancing loss functions. Regularization paths are computed with Yang and Hastie (2024)'s generic elastic net solver, supporting convex losses with non-smooth penalties, as well as group penalties and feature-specific penalty factors. For lasso penalization, balnet computes a regularization path of balancing weights from the largest observed covariate imbalance to a user-specified fraction of this maximum. We illustrate the method with an application to spatial pixel-level balancing for constructing synthetic control weights for the average treatment effect on the treated, using satellite data on wildfires.

Artificial Intelligence (AI) increasingly automates various parts of the software development tasks. Although AI has enhanced the productivity of development tasks, it remains unstudied whether AI essentially outperforms humans in type-related programming tasks, such as employing type constructs properly for type safety, during its tasks. Moreover, there is no systematic study that evaluates whether AI agents overuse or misuse the type constructs under the complicated type systems to the same extent as humans. In this study, we present the first empirical analysis to answer these questions in the domain of TypeScript projects. Our findings show that, in contrast to humans, AI agents are 9x more prone to use the 'any' keyword. In addition, we observed that AI agents use advanced type constructs, including those that ignore type checks, more often compared to humans. Surprisingly, even with all these issues, Agentic pull requests (PRs) have 1.8x higher acceptance rates compared to humans for TypeScript. We encourage software developers to carefully confirm the type safety of their codebases whenever they coordinate with AI agents in the development process.

We study the fair allocation of indivisible goods among agents with additive valuations. The fair division literature has traditionally focused on two broad classes of fairness notions: envy-based notions and share-based notions. Within the share-based framework, most attention has been devoted to the maximin share (MMS) guarantee and its relaxations, while envy-based fairness has primarily centered on EFX and its relaxations. Recent work has shown the existence of allocations that simultaneously satisfy multiplicative approximate MMS and envy-based guarantees such as EF1 or EFX. Motivated by this line of research, we study for the first time the compatibility between ordinal approximations of MMS and envy-based fairness notions. In particular, we establish the existence of allocations satisfying the following combined guarantees: (i) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EFX for ordered instances; (ii) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EF1 for top-$n$ instances; and (iii) simultaneous $1$-out-of-$4\lceil n/3 \rceil$ MMS and EF1 for ordered instances.

We review three well known inconsistencies in the standard mathematical formulation of semiclassical gravity: the factorization problem, the information problem, and the closed universe problem. Building upon recent work, we explore how modifying the semiclassical holographic dictionary may provide the necessary freedom to resolve these three problems in a unified manner while maintaining more well established aspects of the standard correspondence. Using the modified semiclassical holographic dictionary as a scaffolding, we propose a program for constructing an `extended' semiclassical gravitational path integral which (i) is manifestly factorizing, (ii) computes a von Neumann entropy which satisfies the Page curve, and (iii) incorporates new operators that create closed baby universe states. Our construction may be interpreted as imposing a semiclassical version of background independence/a no global symmetry condition, defining a modified large N limit, preparing an ensemble of dual theories, or enforcing observer rules using gravitational degrees of freedom.

The dynamical nature and formation mechanism(s) of galactic spiral arms remain long-standing problems in astrophysics. Most theoretical work is based on analytic calculations or idealised simulations, which has yielded several theories of spiral structure. The radial profile of the spiral arm rotation speed - the pattern speed - is a key observable prediction of these theories. However, observations that infer spiral pattern speeds reveal a mixed picture with no clear consensus. Here, we expand on theoretical efforts by examining the pattern speed profiles in the Auriga Superstars set of high-resolution cosmological magnetohydrodynamic simulatons of Milky Way-mass spiral disc galaxies. These simulations combine galaxy formation in a cosmological environment with the high dynamical fidelity afforded by an $\sim 800$ $\rm M_{\odot}$ star particle resolution, giving $\sim 100$ million star particles in the disc. We show that several different spiral arm theories are realised among our simulations, including large-scale kinematic density waves, manifold spirals, dynamic (co-rotating) spirals, and overlapping modes. In particular, we demonstrate that a strong tidal interaction leads to clear kinematic density waves, and that manifold spirals are present in a strongly-barred galaxy. Interestingly, we find that the same galaxy may show qualitative evolution of their spiral pattern speed profiles, indicating that the nature of spiral arms can evolve on potentially sub-Gigayear timescales. Our results demonstrate that in the absence of a strong external encounter or a strong bar, galactic spiral structure is highly transitional and complex with no clear long-lived underlying wave.

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $λ\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_λ(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\mathcal{S}_λ:=\mathcal{P}+\mathcal{A}_λ(X)$. Fix $\varepsilon>0$, choose $θ$ with $2/15+\varepsilon<θ<0.99$, and set $h=X^θ$. We prove that for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains $\asymp_{\varepsilon} h$ integers of the form $p+a$, where $p$ is prime and $a\in\mathcal{A}_λ(X)$.

The systematic variation of solar active region (AR) properties with their magnetic flux has been the subject of numerous studies but the proposed scaling laws still vary rather widely. A correct representation of these laws and the deviations from them is important for modelling the source term in surface flux transport and dynamo models of space climate variation, and it may also help constrain the subsurface origin of active regions. Here we determine active region scaling laws based on the recently constructed ARISE active region data base listing bipolar ARs for cycle 23, 24 and 25. For the area $A$, pole separation $d$ and tilt angle $γ$ we find scalings against magnetic flux $Φ$ and heliographic latitude $λ$. Residuals from these relations are also modelled. These scaling relations are recommended for use in space climate research for the modelling of future data or missing past data, as well as for the identification of candidate rogue ARs.