Virtually every biological function emerges through the organization of molecules in time and space. Consequently, a major challenge in statistical physics is to uncover the universal principles governing macromolecular self-organization within the crowded, non-equilibrium environment of the cell. Here, we investigate a class of models where molecules maintain a conserved total concentration but can switch "identities", thereby modulating their intermolecular interactions. By enforcing thermodynamic consistency via the local detailed balance condition, we derive the steady-state criteria determining coexisting concentrations in a binary mixture. For non-constant transition rates and using a sharp-interface approximation, we obtain jump conditions that generalize Gibbs' coexistence criteria of equal pressure and chemical potential. We demonstrate that these jumps balance the chemical potential differences of individual species against their currents, which are confined to the interfacial region.

We present results from examining the silicate cloud modeling of four JWST-observed hot Jupiters in the context of brown dwarf theory to further explore signatures of formation in present-day atmospheres. We contextualize our understanding of protoplanetary disk refractory chemistry with empirical evidence from chondritic meteorites to show that giant planets forming and accreting in the outer disk adopt their stellar Mg/Si value. We show that current silicate cloud species determinations of WASP-17 b, WASP-107 b, WASP-39 b and HD 189733 b are in line with predictions laid out in Calamari et al. 2024 based on each system's host star Mg/Si ratio, further supporting this hypothesis. We discuss physical motivations for potential atmospheric scenarios where apparent silicate cloud species is not in agreement with that predicted by its host star chemistry. Additionally, we compare current transit spectroscopy for three of these four exoplanets against brown dwarf spectra to examine molecular absorption trends across the substellar mass temperature regime.

This paper presents a thorough study of the local and nonlocal amplitudes in $B^+ \to K^+μ^+μ^-$ transitions through an amplitude analysis of the dimuon mass spectrum of the decay. The analysis is based on $pp$ collision data corresponding to an integrated luminosity of 8.4fb$^{-1}$ collected by the LHCb experiment. This measurement employs a model that describes both one-particle and two-particle nonlocal amplitudes across the entirety of the dimuon mass spectrum, enabling the determination of both short- and long-distance contributions to the decay. The compatibility of the Wilson coefficient combinations $C_9+C_9'$ and $C_{10}+C_{10}'$ with the Standard Model prediction is found to vary between $1.6\,σ$ and $4\,σ$, depending on the choice of local form factors.

Property graphs often contain tree-shaped substructures, yet they are not captured by existing proposals for graph schemas; likewise, query languages and query engines offer little-to-no native support for managing them systematically. As a first contribution, we report on a micro experiment that demonstrates the optimization potential of treating tree-shaped substructures as first class citizens in graph database systems. In particular, we show that in systems backed by relational engines, we can achieve substantial speedups by leveraging structural indexes, as originally developed for XML databases, to accelerate path queries. Based on our findings, we put forward a vision in which tree-shaped substructures are systematically managed throughout the graph query lifecycle, from modeling and schema design to indexing and query processing, and outline arising research questions.

Many physical systems involve two types of orientational order, which are coupled together. For example, ferroelectric nematic liquid crystals have coupled polar and nematic order, and tilted hexatic phases have coupled polar and hexatic order. In these systems, defect structures can be quite complex. Here, we investigate phases with two types of two-dimensional orientational order, $m$-atic and $n$-atic, where $m$ and $n$ are two distinct integers. We simulate these phases in a flat disk with strong radial anchoring, and on a spherical surface, because both of these geometries require the presence of defects. If the coupling between the two types of order is weak, then the defects are connected by a network of diffuse walls, and the system forms a stable domain structure. As the coupling increases, the domain walls become sharper and shorter. For very strong coupling, the higher-order defects merge into the lower-order defects, forming stretched defect cores.

We employ optical pump-THz probe spectroscopy to chart the ultrafast superconductivity suppression in NbN over a broad range of excitation fluences. Our measurements uncover a pronounced lengthening of the superconductivity quenching time when the absorbed optical energy is close to the condensation energy of the superconductor, which constitutes a non-equilibrium analog of critical slowing-down on a timescale comparable to that of superconducting fluctuations. Time-dependent Ginzburg-Landau simulations reproduce this behavior and ascribe it to the flattening of the free energy landscape at the dynamical phase transition boundary. Our findings represent a direct observation in real time of slowed-down superconductivity dynamics in proximity to a critical point and open a pathway for investigating out-of-equilibrium critical phenomena with time-resolved THz spectroscopy.

We study the residual Eisenstein cohomology of semisimple groups in the context of maximal parabolic subgroups which remain maximal over $\mathbb{R}$. Under certain general hypotheses, we show that these residual representations are cohomological one degree below middle, and one above; however, the classes above middle vanish in the full automorphic cohomology. The proof of this vanishing finds an explicit cochain which provides a primitive to the image of a nontrivial class from the cohomology of such a residual representation. This cochain is valued in regular Eisenstein series. Along the way, we study in detail the archimedean component of the relevant induced representation. In particular, we prove that it has a subrepresentation which is the sum of two discrete series, whose Harish-Chandra parameters we describe, and that the intertwining operator vanishes to order exactly $1$ on that subrepresentation.

Dirac points (DPs) are topological singularities that determine the extraordinary properties of two-dimensional materials. They are generally classified by discrete topological invariants, which determine the possibility of DPs' annihilation upon their collision. Here, we study the behaviors of DPs within a photonic Kagome lattice created in atomic vapor. With optically engineering the potential difference among three sites constituting the Kagome unit cell while preserving time-reversal symmetry and the stability of an isolated DP, the DPs move in reciprocal space. By employing conical diffraction to measure their position and the topological invariant (Euler number), we demonstrate an obstruction to DPs' annihilation during collision and a transition to a case where the Euler number changes and annihilation occurs. Such topological transition is induced by a non-Abelian frame rotation of the eigenstates around the Brillouin zone torus. The associated conversion of the DP quaternionic charges during their motion explains the change of Euler number.

A long standing problem in high energy astrophysics is the nature of galactic accelerators of particles with energies above PeV. Such objects are sources of galactic cosmic rays and can produce PeV-regime photons observed by ground-based observatories. Among very likely accelerators are astrospheres of pulsars in gamma-ray binaries. These binaries have long been observed as bright sources of TeV gamma-rays. Recently, 2D relativistic magnetohydrodynamic (rMHD) simulations have shown that the astrospheres can accelerate particles to energies well above PeV, provided that they harbor a Gauss-range magnetic field. Such a strong field is necessary in the region of two colliding winds: the relativistic outflow of the pulsar or accreting black hole and the wind of its stellar companion, a massive early-type star. Here, the wind collision region is explored as the site of PeV protons acceleration. The local structure of colliding flows is illustrated using rMHD simulations of a powerful pulsar wind in 2D and 3D models. The relativistic outflow of a pulsar or black hole, evolving inside the strongly magnetized stellar wind, have an elongated shape and surrounded by a kind of magnetic cocoon providing favorable conditions for acceleration of ultra high energy ions. The simulated spectra of particles, accelerated by intermittent relativistic turbulence in these systems, have piece-wise power-law shape and extend well above PeV energies for powerful outflows. The model indicated that gamma-ray binaries harboring a powerful relativistic outflow, produced either by a pulsar or accreting black hole, can be bright sources of synchrotron MeV-regime photons and multi-PeV regime gamma-rays, as recently detected from galactic microquasars like Cyg X-3. The Gauss-range magnetic field of a massive star wind strongly influences the non-thermal emission of gamma-ray binaries with relativistic companions.

With digital learning environments becoming more prevalent, the ease with which generative AI enables the scalable production of real-time, automated feedback holds the potential to reshape learning and teaching experiences. This meeting report synthesizes the interdisciplinary perspectives of 50 scholars from educational psychology, computer science, science education, and the learning sciences on the use of generative AI for feedback and its promises and risks in educational practice. We highlight points of convergence in the scholarship, identify areas of debate and unresolved challenges, and outline open questions and future directions for research and educational practice that emerged from structured small-group activities designed to bridge disciplinary barriers.

We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in $1$-variational inequalities for all connected graphs on at most five vertices and pose a conjecture on the corresponding sharp constants for path graphs. Finally, we construct finite graphs with arbitrarily large $p$-variational constants.

3D DRAM has emerged as a promising approach for continued density scaling, but its viability is limited by routing and hybrid bonding constraints to periphery, which may degrade sensing margin, latency, and array efficiency. With device characteristics and array parasitics extracted from TCAD, SPICE simulations are performed with peri logic in a CMOS-Bonded-Array (CBA). The analysis shows that the bitline strap architecture with amorphous oxide semiconductor (AOS) selectors is essential to manage routing congestion and parasitics. The optimized design achieves a bit density of 2.6 Gb/mm^2 (137 layers with Si access transistors or 87 layers with AOS), representing ~6x density scaling over D1b 2D DRAM. The design further demonstrates a nominal row cycle time (tRC) of 10.5 ns, compared to 21.3 ns in D1b, and a 60% reduction in read/write energy.

Disordered solids distort and fail as particle contacts become unstable and rearrange under sufficiently large shear strains. Such instabilities can occur at different locations and, because of their proximity, can interact with one another. We develop a tool for simulations with periodic boundary conditions that allows strains to be applied at a continuously variable angle, $θ$. We show that instabilities can persist over a broad angular ranges of applied shear to form instability lines in phase space. By applying strain at different $θ$, we examine the correlations between the instabilities encountered at different angles and different positions in the sample. We find instabilities that pass through one another, others that change continuously as the angle is varied, and yet others that end by smoothly decreasing their magnitudes to zero as the instability fades away. Examining hysterons, i.e., instabilities that undo themselves upon reversing the direction of shear, we find that as $θ$ is varied towards the point where the instability disappears, the separation between the forward and backward instabilities shrinks to zero so as to produce an enhanced number of very small hysterons.

The most accurate approach to determine the compressibility of nuclear matter remains the one based on microscopic Energy Density Functionals (EDFs). Recent analyses yield a value for nuclear incompressibility modulus $K_\sat=240\pm 20$~MeV, defined in nuclear matter as the second derivative of the energy per particle at saturation density. However, we demonstrate that the compressibility modulus can be reduced to values shifted by four times the suggested uncertainty, i.e., $K_\sat\approx 160$~MeV, by providing examples based on models where the second derivative ($K_\sat$) and third derivative ($Q_\sat$) of the energy per particle at saturation density can be independently varied, while the experimental binding energies, charge radii, and ISGMR data in $^{120}$Sn and $^{208}$Pb are enforced. The present work suggests a new methodology to access the compressibility of nuclear matter from nuclear experiments, still based on microscopic models, but using EDFs containing more flexibility than the ones employed up to now. Consequences of our results for nuclear matter at supra-saturation density are also discussed by exploring the quarkyonic cross-over. We predict that, for our models with low values for $K_\sat$, the quark onset density has to be low for neutron stars to exist.

We develop an iterative framework for Bayesian inference problems where the posterior distribution may involve computationally intensive models, intractable gradients, significant posterior concentration, and pronounced non-Gaussianity. Our approach integrates: (i) a generalized annealing scheme that combines geometric tempering with multi-fidelity modeling; (ii) expressive measure transport surrogates for the intermediate annealed and final target distributions, learned variationally without evaluating gradients of the target density; and, (iii) an importance-weighting scheme to combine multiple quadrature rules, which recycles and reweighs expensive model evaluations as successive posterior approximations are built. Our scheme produces both a quadrature rule for computing posterior expectations and a transport-based approximation of the posterior from which we can easily generate independent Monte Carlo samples. We demonstrate the efficiency and accuracy of our approach on low-dimensional but strongly non-Gaussian Bayesian inverse problems involving partial differential equations.

Multiple-input multiple-output (MIMO) transceiver design and probabilistic shaping (PS) are key enablers for high spectral efficiency in 6G wireless networks. This work proposes a distribution-aware MIMO transceiver optimized for PS constellation symbols, including a Bayesian geometric-mean decomposition (BGMD) precoder and a maximum a posteriori-VBLAST (MAP-VBLAST) detector. BGMD precoder incorporates PS priors into the derivation and equalizes layer gains to facilitate a single modulation and coding scheme for low-complexity transmissions while preserving channel capacity. MAP-VBLAST leverages these PS priors for optimal MAP detection within a successive interference cancellation (SIC) framework. Furthermore, a new codeword-to-layer mapping scheme, termed layer-contained MIMO (LC-MIMO), is proposed. By containing each codeblock (CB) within a single layer, LC-MIMO enables SIC at CB level, allowing the receiver to exploit the error-correction capability of channel coding to mitigate error propagation. Numerical results show that the BGMD transceiver with LC-MIMO achieves notable performance gains over state-of-the-art methods.

This work implements a hybrid device based on a semiconductor quantum dot embedded within a nanowire to bridge a non-continuous curved waveguide structure. The geometry takes advantage of evanescent coupling between the photonic structures to recover single photons emitted from both outputs of the device. Auto- and cross-correlation measurements were performed on different output facets of the device. We demonstrate single-photon emission from both ends of the nanowire for both neutral, X and XX, and charged X-, excitonic complexes. We further demonstrate the cascaded XX-X emission by collecting each complex from a different facet. This work lays the foundation for on-chip architectures which utilize multi-directional integration of quantum emitters.

We determine properties of two-dimensional normal affine semigroup rings, and in particular of weighted Veronese rings, including determinantal presentation, Gröbner basis, graded Hilbert series and graded Betti numbers, the structure of their associated graded rings, and their Koszul property. We give examples in higher dimensions illustrating that the first and last properties may fail. Our approach leverages convex monomial ideals as introduced in Herzog-Qureshi-Saem(2019), which give rise to convex semigroups.

This paper presents a Lyapunov-Halanay method to study global asymptotic stabilization (GAS) of nonlinear retarded systems subject to large constant delays in input/output - a challenging problem due to their inherent destabilizing effects. Under the conditions of global Lipschitz continuity (GLC) and global exponential stabilizability (GES) of the retarded system without input delay, a state feedback controller is designed based on sequential predictors to make the closed-loop retarded system GAS. Moreover, if the retarded system with no output delay permits a global exponential observer, a dynamic output compensator is also constructed based on sequential predictors, achieving GAS of the corresponding closed-loop retarded system with input/output delays. The predictor based state and output feedback stabilization results are then extended to a broader class of nonlinear retarded systems with input/output delays, which may not be GES but satisfy global asymptotic stabilizability/observability and suitable ISS conditions. As an application, a pendulum system with delays in the state, input and output is used to illustrate the effectiveness of the proposed state and output feedback control strategies based on sequential predictors.

Superconductivity has been a vigorously researched topic since its discovery in 1911. Raising the superconducting transition temperature (Tc) has been the main driving force behind such long-sustained efforts due to its potential for impacting humanity and the fundamental knowledge gained from understanding this macroscopic coherent quantum state at high temperatures. The successful development of high-Tc superconductivity will make possible extraordinarily efficient generation, delivery, and utilization of energy, and could also enable the development of controlled fusion while impacting other burgeoning fields like quantum computation and quantum electronics. However, progress has been hindered by a longstanding plateau in the record ambient-pressure Tc, unchanged since 1993. Subsequent significant advancements in Tc have been achieved only under high pressures, preventing the realization of superconductivity's full potential. To directly address this challenge, we developed a pressure-quench protocol (PQP) to stabilize pressure-induced/-enhanced superconducting states at ambient pressure. Here we achieve a record ambient-pressure Tc of 151 K in the cuprate HgBa2Ca2Cu3O8+δ via PQP. The experimental results are further supported by synchrotron X-ray diffraction measurements and phonon and electronic structure calculations. This breakthrough opens new avenues for stabilizing and exploring ambient-pressure high-Tc superconducting states and other quantum states that have been previously only accessible under pressure, paving the way for deeper understanding and practical applications of high-Tc superconductivity and beyond.

State-of-the-art DRAM read disturbance mitigations rely on the read disturbance threshold (RDT) (e.g., the number of aggressor row activations needed to induce the first read disturbance bitflip) to securely and performance- and energy-efficiently prevent read disturbance bitflips. However, accurately and exhaustively characterizing the RDT of every DRAM row in a chip is time intensive. Rapidly determining RDT is important for enabling secure, performance- and energy-efficient systems. Our goal is to develop and evaluate a reliable and rapid read disturbance testing methodology. To that end, we develop DiscoRD building on the key results of an extensive experimental characterization study using 212 real DDR4 chips whereby we measure the RDT of hundreds of thousands of DRAM rows millions of times. We develop an empirical model for read disturbance bitflips and evaluate the probability of read-disturbance-induced uncorrectable errors when a read disturbance mechanism is configured using a single $RDT_{min}$ measurement. Using this model we demonstrate that 1) relying on a lightweight error-correcting code (ECC) alone yields relatively high uncorrectable error probability and 2) combining ECC, infrequent memory scrubbing, and configurable read disturbance mitigation mechanisms can greatly reduce the error probability. Building on our observations and analyses, we discuss the RDT of each individual row can be identified more precisely. Our results show that error tolerance, memory scrubbing, online profiling, and run-time configurable read disturbance mitigation techniques are important to enable secure and energy-efficient spatial-variation aware read disturbance mitigations. We hope that DiscoRD drives research that enables us to quantitatively navigate the performance/cost - reliability tradeoff space for read disturbance mitigation techniques.

I'll begin with some well-deserved acknowledgements: I am grateful to Daniel Harlow for discussions of time-reversal holonomies. I have also benefited from a long ongoing correspondence with Edward Witten, but frankly in both cases I can't tell whether they agree with me or not. I have often been accused of imprecision, especially toward the later parts of a paper, where I expect that my readers have ``caught on." That does eventually happen -- the readers catching on and I thank them -- but I'm now almost 86 and I can't wait. So I've tried to maintain a level of conceptual if not mathematical rigor throughout. Mathematical rigor(mortis) can sometimes be the enemy of conceptual clarity. I thank my friend Richard Feynman for reminding me of that lesson. Finally I thank the chatbot who gave me the definition of scaffold in section \ref{Scaff}. It was better than anything I was able to do. Symmetries of a Holographic theory; whether continuous or discrete, local or global, are gauge symmetries of the bulk. This includes discrete space-time symmetries such as C and P. But time-reversal is sufficiently different from other symmetries that we may question the standard wisdom and ask whether symmetries involving T should be gauged in the bulk. Harlow and Numasawa \cite{Harlow:2023hjb} say yes; time-reversal is a gauge symmetry. Witten \cite{Witten:2025ayw} says no: time reversal is different and does not manifest as a gauge symmetry of the bulk. My view is -- yes -- but with a twist: Time-reversal is indeed a gauge symmetry; but it is hidden by spontaneous symmetry breaking. In this paper I will review the case for spontaneous symmetry breaking of time-reversal and explain the ``smoking gun" -- a closed curve and a holonomy which flips forward-going clocks to backward going clocks, and vice versa.

The propagation of a spherical wave through a two-dimensional random Lorentz gas composed of small fixed scatterers is studied. Inspired by the Mott problem (how an initially isotropic quantum wave can give rise to a single particle-like track), we investigate, on a schematic model, whether such a directional behavior can emerge purely from the multiscattering process, without any explicit measurement or decoherence mechanism. Using the Foldy-Lax formalism, we derive the far-field angular behavior of the wavefunction, and introduce a directionality vector to quantify its anisotropy and identify its preferred direction. Numerical simulations reveal the existence of a strongly directional regime within a specific wavenumber range, which emerges from multiscattering with more than $100$ scatterers and which can be related to Anderson localization.

Interacting dark sector (IDS) models provide a commonly explored extension of the standard $Λ$CDM cosmology, allowing for non-gravitational energy--momentum exchange between cold dark matter (CDM) and dark energy (DE). Although such models can be constructed to reproduce the same background expansion history as $Λ$CDM, their impact on the growth of cosmic structures is fundamentally different and requires a careful treatment of cosmological perturbations. In this work, we develop the one-loop Standard Perturbation Theory (SPT) formalism for IDS cosmologies without invoking the Einstein--de~Sitter (EdS) approximation. We show that even weak dark sector interactions induce a non-trivial time dependence in the perturbative kernels, leading to a breakdown of the EdS approximation commonly assumed in $Λ$CDM analyses. By deriving and numerically solving the evolution equations for the second- and third-order kernels, we compute the corresponding one-loop corrections to the matter power spectrum and find that the resulting deviations can significantly exceed the percent level, even for small interaction strengths. Our results demonstrate that nonlinear corrections are systematically enhanced in IDS models and that neglecting the full time dependence of the kernels can lead to biased predictions on mildly nonlinear scales. These findings establish the necessity of a time-dependent perturbative treatment for IDS scenarios and provide a robust framework for precision tests using nonlinear large-scale structure (LSS) observables.

Collisional ring galaxies are produced by the collision of a disk galaxy with a compact galaxy plunging through the disk, forming a ring-shaped expanding density wave, triggering star formation at its wake. The wave expansion is expected to produce negative stellar age gradients in radial profiles of post-collision stellar populations. Integral field spectroscopy combined with stellar population synthesis allows us to spatially resolve the stellar populations, to separate the post-collision and pre-collision components, and to produce the radial profiles. We analyse three candidate galaxies: Arp~143, NGC~2793, and VII~Zw~466. Observations were performed with the Calar Alto 3.5~m~telescope using the PMAS/PPak spectrophotometer. NGC 2793 presents a positive stellar age gradient, dismissing the collision hypothesis. For Arp~143 and VII~Zw~466, we found negative stellar age gradients for the youngest stellar populations, up to the ring radii, consistent with the collision hypothesis. We estimated that the collisions occurred $\sim$300~Myr and $\sim$100~Myr (expansion velocities of 33~$\pm$~10 km s$^{-1}$ and 108~$\pm$~26 km s$^{-1}$), respectively, before the density waves reached the observed ring radii. A spatially resolved analysis of the specific star formation histories (sSFH), reveals an expected star formation enhancement following the collision. The sSFH also allowed to identify the most probable intruder galaxy for VII~Zw~466. We report new redshifts for its group members. Finally, radial profiles of light contributions from pre-collisional and post-collisional stars show that the density wave dragged old pre-collisional stars along, as predicted by simulations.

We show that every finite-dimensional pointed Hopf algebra over a finite simple Chevalley group, different from $PSL_2(q)$ with q= 3 mod 4 (and from $PSL_3(2)\simeq PSL_2(7)$), is isomorphic to the corresponding group algebra. To do this, we complete the analysis of the Nichols algebras of Yetter-Drinfeld modules over such groups whose support is a semisimple orbit, begun in arXiv:1506.06794, arXiv:2301.03361. In addition to the techniques used in loc. cit., we introduce a general procedure to determine when a semisimple conjugacy class in a Chevalley or Steinberg group is of type C and a new criterion based on the results of arXiv:2411.02304 that applies to arbitrary racks. Throughout the process, we obtain results on Nichols algebras over racks beyond the framework of Chevalley groups.

The magnetic fields in the solar polar region are important to our understanding of the internal dynamo process, the global coronal structure, and the origin of the solar wind. The inference of polar fields based on spectropolarimetric observation is highly model-dependent and can suffer from various systematic effects. Here we analyze a raster map of the southern polar region taken by the Hinode Spectro-Polarimeter, utilizing the Stokes Inversion based on Response functions code. The inversions provide height-dependent vector magnetic field maps between optical depths $\log_{10}τ= -2$ and $0$. We examine the impact on the total magnetic flux estimate from adopting (1) 1- vs 2-component atmospheric models via a "filling factor" parameter and (2) different analysis schemes. At $\log_{10}τ= -1.5$, the polar magnetic flux is estimated to be $(1.84 \pm 0.03) \times 10^{21}$ Mx and $(1.38 \pm 0.02) \times 10^{21}$ Mx under the 1- and 2-component atmosphere assumption, respectively. The magnetic flux is approximately constant or increases slightly with height, respectively. We find that the 2-component (1-component) configuration is preferred for 58.3% (32.3%) of the pixels. Different initial guesses, including the input atmosphere model and the filling factor, as well as different inversion settings, can significantly affect the results, especially for locations with weaker polarization signals. Our work highlights the importance of including unresolved magnetic structures or stray light into consideration. Model degeneracy and the convergence to local minima limit the precision of the polar magnetic flux inference (no better than several tens of percent in this case). Higher-resolution observations and advanced inversion and disambiguation algorithms may alleviate these limitations.

For regular continued fractions (CFs), points with finite expansions are exactly the rationals and, by Lagrange's theorem, points with eventually-periodic expansions are exactly the roots of non-degenerate quadratic equations with integer coefficients. We extend both results to proper and discrete Iwasawa CFs, including real, complex, 3D, quaternionic, octonionic, and Heisenberg CFs. Namely, the following three conditions are equivalent for a point $p$: $p$ has a finite expansion, $p\in \mathcal M(\infty)$ for the appropriate modular group $\mathcal M$, and $p$ is a fixed point of a parabolic transformation in $\mathcal M$. Eventually-periodic points correspond exactly to fixed points of loxodromic elements of $\mathcal M$, which can be interpreted as roots of non-degenerate quadratics using the Clifford Algebra formalism of Ahlfors. In particular, this provides a new geometric proof of Lagrange's theorem for nearest-integer real CFs and Hurwitz complex CFs. Lastly, we comment on generalizations of the identity $i+1/i=0$.

Unravelling the origins of single-atom catalyst reactivity is a central challenge in heterogeneous catalysis research. A key question is whether the activity arises solely from atomic isolation or from distinct structural and electronic configurations of the single atoms. Here, we use precisely defined Fe-N$_3$ and Fe-N$_4$ model catalyst sites synthesized on an inert support to quantify the effect of coordination geometry on chemical reactivity. Both the Fe-N$_3$ and Fe-N$_4$ models have the same electronic configuration (high-spin Fe$^{2+}$ with S=2), and even their d-orbital occupancies and positions with respect to Fermi level are almost identical. Despite this electronic similarity, the adsorption energy of CO differs by more than 0.6 eV between the Fe-N$_3$ and Fe-N$_4$ sites, as indicated by density functional theory computations and confirmed by atomically-resolved scanning tunneling microscopy experiments. We trace this reactivity difference to the structural flexibility of the Fe-N$_3$ sites, which allows strengthening of the Fe 3d$_{xz/yz}$-CO 2$π$* back-bonding by lifting the Fe atom from the -N$_3$ plane. These results demonstrate that coordination geometry plays a crucial role in defining the reactivity of single-atom catalysts, and that such effects cannot be predicted by analysis of the sites' electronic structures alone.

Human mobility studies how people move among meaningful places over time and how these movements aggregate into population-level patterns that shape accessibility, congestion, emissions, and public health. Large language models (LLMs) are increasingly used in this domain because many human mobility problems require reasoning about place and activity semantics, travelers' intentions and preferences, and diverse real-world constraints that are difficult to capture using coordinates and other purely numerical attributes. Despite rapid growth, the literature is still scattered, and there is no clear overview that connects human mobility tasks, challenges, and LLM designs in a consistent way. This survey therefore provides a comprehensive synthesis of LLM-based research on human mobility across five tasks, including travel itinerary planning, trajectory generation, mobility simulation, mobility prediction, and mobility semantics and understanding. For each task, we review representative work, connect core challenges to the specific roles of LLMs, and summarize typical LLM-based solution designs. We conclude with open challenges and research directions toward reliable, grounded and privacy-aware LLM-based approaches for human mobility.