Diffusion-weighted magnetic resonance imaging (D-MRI) is a noninvasive in vivo technique for probing the microstructural architecture of biological tissues. At each voxel, the fiber orientation distribution (FOD) characterizes local fiber configurations and orientations and is therefore a central object of estimation in D-MRI analysis. We propose the Nearest-Neighbor Adaptive Regression Model (NARM), a spatially adaptive framework for FOD estimation that performs weighted local likelihood estimation over nested spatial neighborhoods, where the weights jointly encode spatial proximity and similarity among neighboring FODs, measured by either the optimal transport or Hellinger distance. To prevent over-smoothing while preserving structural heterogeneity, we introduce a voxel-wise rescaling scheme and a data-driven stopping rule based on minimum nearest-neighbor dissimilarity. We further develop a configuration-aware strategy for selecting the similarity-smoothing parameter, allowing the smoothing strength to adapt to local fiber complexity. Simulation studies demonstrate that NARM improves FOD estimation accuracy relative to voxel-wise methods and the existing spatial smoothing approach PMARM. Application to test-retest data from the Human Connectome Project additionally shows that NARM yields more reproducible FOD estimates. Implementation details and scripts for the simulation and real data analyses are available at https://github.com/jie108/NARM

Heart failure (HF) treatment relies heavily on pharmacotherapy, particularly combining multiple therapies as recommended by clinical guidelines. However, non-adherence to prescribed regimens remains a significant challenge, contributing to increased hospitalizations and poorer patient outcomes. This study introduces a novel methodological pipeline that integrates Latent Markov Models (LMM) with dynamic adherence modeling to evaluate adherence behaviors and their impact on HF rehospitalization. Using administrative healthcare data from Lombardy, Italy, we analyzed 6,818 patients hospitalized for HF between July and December 2020. Adherence was assessed monthly over a six-month observation period, and adherence profiles were linked to clinical outcomes using Cox regression. Seven latent behavioral profiles were identified, reflecting varying levels and trajectories of adherence. The findings revealed that higher adherence levels significantly reduced the risk of rehospitalization. Patients with consistently high adherence exhibited a 56% lower risk of HF rehospitalization compared to those with low adherence. Importantly, improving adherence during the observation period was associated with better survival probabilities, highlighting the potential benefits of timely interventions. Additionally, adherence behaviors were influenced by factors such as age, comorbidity burden, and hospitalization during the observation period. This study underscores the importance of dynamic and personalized strategies to monitor and enhance adherence to polytherapy. By linking adherence patterns to clinical outcomes, the proposed approach offers actionable insights for improving patient management and reducing the burden of HF on healthcare systems.

Let $π$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$-function of $π$ in the $t$-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.

We introduce two models of consensus following a majority rule on time-evolving stochastic block models (SBM), in which the network evolution is Markovian or non-Markovian. Under the majority rule, in each round, each agent simultaneously updates their opinion according to the majority of their neighbors. Our network has a community structure and randomly evolves with time. In contrast to the classic setting, the dynamics is not purely deterministic, and reflects the structure of SBM by resampling the connections at each step, making agents with the same opinion more likely to connect than those with different opinions. In the Markovian model, connections between agents are resampled at each step according to the SBM law and each agent updates their opinion via the majority rule. We prove a power-of-one type result, i.e., any initial bias leads to a non-trivial advantage of winning in the end, uniformly in the size of the network. In the non-Markovian model, a connection between two agents is resampled according to the SBM law only when at least one of them changes opinion and is otherwise kept the same. We identify the phase-transition threshold, up to the second-order leading term, between halting and fast convergence to consensus. We also give sufficient initial-lead conditions for consensus to occur within one, two, or three rounds.

We prove homogenization properties of random Hamilton-Jacobi-Bellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is non-elliptic and its law is non-stationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finite-range dependence (i.i.d.) assumption on the percolation models and the effective Hamiltonian admits a variational formula which reflects some key properties of percolation. The proof is inspired by a method of Kosygina-Rezakhanlou-Varadhan [KRV06] developed for the case of HJB equations with constant viscosity and uniformly coercive Hamiltonian in a stationary, ergodic and elliptic random environment. In the non-stationary and non-elliptic set up, we leverage the coercivity property of the underlying Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB, in any framework) and make use of the random geometry of continuum percolation.

We consider the empirical neighborhood distribution of marked sparse Erdős-Rényi random graphs, obtained by decorating edges and vertices of a sparse Erdős-Rényi random graph with i.i.d. random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam~(2019) which is inspired on the previous work by Bordenave and Caputo~(2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erdős-Rényi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.

A preference profile (i.e., a collection of linear preference orders of the voters over a set of alternatives) with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed into a $d$-dimensional space such that between each pair of alternatives, every voter prefers the one which has a shorter Manhattan (resp. Euclidean) distance to the voter. We study how $d$-Manhattan preference profiles depend on the values $m$ and $n$. First, we provide explicit constructions to show that each preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan whenever $d \ge \min(n, m - 1)$. We further extend this positive result for other $p$-norms with $p \in R_{\ge 1} \cup \{\infty\}$. Second, for $d = 2$, we develop forbidden substructures-preference patterns among small sets of voters that constrain any 2-Manhattan embedding -- and use them to show that the smallest non-2-Manhattan preference profile has either 3 voters and 6 alternatives, or 4 voters and 5 alternatives, or 5 voters and 4 alternatives. This is more complex than the case with $d$-Euclidean preferences (see (Bogomolnaia and Laslier, 2007) and (Bulteau and Chen, 2022)). We also show that $d$-Manhattan preferences imply $(2d-1)$-dimensional single-peakedness, while 2-Manhattanness is incomparable with single-peakedness and single-crossingness.

In this article we study the continuity and sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. For both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different percolation models based on spin-flip dynamics. We also discuss some of the problems that can be tackled in a similar fashion.

Let ${\cal M}(X,G)$ be the moduli space of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group with centre $Z$. We describe the fixed points of the action of a finite group $Γ$ on ${\cal M}(X,G)$, induced by holomorphic actions of $Γ$ on $X$ and $G$, a character of $Γ$ and a homomorphism from $Γ$ to the group of $Z$-bundles over $X$. Two important ingredients in this study are provided by the theory of twisted $Γ$-equivariant bundles developed by Barajas--García-Prada--Gothen--Mundet i Riera, and the Prym--Narasimhan--Ramanan construction given by Barajas--García-Prada. Via the non-abelian Hodge correspondence, our results provide a description of the fixed-point subvarieties of certain finite group actions on the $G$-character variety of the fundamental group of $X$.

We propose a quantum bosonic qubit model on a fcc lattice that realizes the canonical source structure of mimetic dark matter as a defect of a rank-two lattice Gauss law. The standard contribution from general relativity is implemented similarly to previous works in the literature, while the mimetic sector modifies the constraint equations through additional source terms. Different theories such as mimetic dark matter, vector mimetic dark matter, and tensor-vector-scalar models are implemented on the lattice. In all these cases, a generalized Gauss law incorporates an additional Gauss-law (topological) defect depending on the type of generalization, but always fitting into the structure of the defects from the general relativity contribution. The mimetic constraint is treated in its full ADM form, retaining the normal derivative of the scalar field, and the known ghost and gradient instabilities of the minimal continuum mimetic theory are summarized. The lattice construction is therefore presented as a formal realization of the canonical source structure rather than as a complete cosmological model.

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.

We establish the existence of abelian envelopes for interpolation categories of wreath product groups $G\wr S_n$, for a fixed finite group $G$ with the symmetric groups $S_n$, for $n\ge0$. Our approach consists of showing directly via essentially combinatorial methods that certain generalized restriction functors admit adjoints.

We construct an operadic model for the higher-genus Teichmüller tower. More precisely, we define a modular operad $\mathbf{S}$ in groupoids built from mapping class groups, with compositions and contractions encoding gluing operations on surfaces. We prove a presentation theorem for maps out of $\mathbf{S}$, showing that they are determined by a small number of genus-zero and genus-one generators and relations. Using this presentation and the work of Nakamura--Schneps, we construct a faithful action of the Nakamura--Schneps subgroup $\widehatΓ\subseteq\widehat{\mathsf{GT}}$ on the profinite completion $\widehat{\mathbf{S}}$, and hence an action of $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$. The genus-zero truncation of $\mathbf{S}$ recovers the cyclic operad of parenthesized ribbon braids, and its group of object-fixing profinite automorphisms recovers $\widehat{\mathsf{GT}}$. Finally, the profinite completion of the classifying spaces of $\mathbf{S}$ assemble into a modular $\infty$-operad in profinite spaces whose values identify with the étale homotopy types of moduli stacks of curves with marked tangent vectors, and the $\widehatΓ$-action extends to this homotopy-coherent Teichmüller tower.

Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings. In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. This variety is typically still not smooth, but it is the closure of a smooth subvariety of a torus. The latter then a smooth, tropical compactification, using work of Tevelev. We construct explicitly such a compactification and a morphism to the normalized Nash blow-up for every configuration, described in terms of bipermutohedral matroid combinatorics introduced by Ardila, Denham and Huh. Along the way, we find that the normalized Nash blow-up of the configuration hypersurface has strongly $F$-regular singularities in positive characteristic. We deduce this by certifying $F$-rationality of its biprojective cone, and infer from it that the normalized Nash blow-up has rational singularities over the complex numbers.

We reassess prospects for direct detection of solar chameleons, in light of recent progress in modeling their production, and the availability of new XENONnT data. We show that the contribution from Primakoff production in the electric fields of electrons and ions dominates the electron recoil event rate, which is enhanced compared to earlier estimates based on magnetic conversion in the tachocline alone. We argue that the signal is governed by the effective coupling $β_{\text{eff}} \equiv β_γM_e^{-4}$, which encodes the combined effects of production and detection, where $β_γ$ and $M_e$ are the chameleon-photon (conformal) coupling and chameleon-electron disformal coupling scale, respectively. Setting the height of the chameleon potential to the dark energy (DE) scale $Λ\simeq 2.4\,{\text{meV}}$, we show that XENONnT electron recoil data set the upper limit $\log_{10}β_{\text{eff}}<-6.9$. This limit is independent of the conformal matter coupling $β_m$ and index $n$, and applies to the whole class of inverse power-law chameleons, well beyond the $n=1$ case usually studied. We comment on how future multi-target experiments and lower-threshold analyses could distinguish solar chameleons from other light (pseudo)scalar particles such as axions. Our work demonstrates that existing dark matter direct detection experiments can probe regions of parameter space relevant to screened DE models, providing complementary tests to astrophysical and fifth-force searches at no additional experimental cost.

We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization (MBL). In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy (CSE) as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.

The orbital motion of chargeless bosons, unlike that of electrons, does not generate a magnetic moment and thus cannot directly interact with magnetic fields. To formulate the orbital angular momentum (OAM) of magnons, we first identify its proper conjugate variable by considering the Aharonov-Casher effect, which gives rise to a virtual perturbation to the equilibrium state, allowing us to calculate the magnon OAM as a virtual response to an infinitesimal electric field divergence. At finite temperatures, both self-rotation and topological contributions to the magnon OAM are explicitly derived, analogous to their electronic counterpart but with the correct bosonic statistics. In a two-dimensional honeycomb lattice, we show that the Dzyaloshinskii-Moriya interaction induces a large magnon OAM in both the ferromagnetic and antiferromagnetic phases. Our formalism can be generalized to other chargeless bosons with intrinsic spin.

Introduction. The high prevalence of students not achieving basic learning competencies in Latin America (LAC) is concerning, even more so considering the region's deep structural inequalities and the larger post-pandemic learning losses. Within this scenario, the paper aims to contribute to the identification of the determinants of bottom and low performers (below level 2). Methodology. Based on 2022 data from the Programme for International Student Assessment (PISA) for 10 LAC countries, and using a stacking model integrating binary classification models as well as by applying Shapley Additive Explanations (SHAP) analysis for interpretability, we identify critical factors impacting on the student performance across low performers groups. Results. We find that a student with the highest probability of being a not achiever speaks a minority language and had repeated, has no digital devices at home, comes from a poor family and works for payment half of the week, and the school the student attends has wide disadvantages such as bad school climate, weak Information and Communication Technology (ICT) infrastructure and poor teaching quality (only a third of teachers being certified). For countries' estimates, we find quite homogeneous patterns regarding the contribution of top ranked factors, with repetition at primary, household wealth, and educational ICT inputs being top ten ranked covariates in at least 8 out of the 10 total countries. Discussions. The paper findings contribute to the broad literature on strategies to identify and to target those most left behind in Latin American education systems.

Randomized benchmarking (RB) is a widely adopted protocol for estimating the average gate fidelity in quantum hardware. However, its standard formulation relies on the assumption of temporally uncorrelated noise, an assumption often violated in current devices. In this work, we derive analytic expressions for the average sequence fidelity (ASF) in the presence of temporally correlated (non-Markovian) noise with classical memory, including cases where such correlations originate from interactions with a quantum environment. We show how the ASF can be interpreted to extract meaningful benchmarking parameters under such noise and identify classes of interaction Hamiltonians that render temporal correlations completely invisible to RB. We further provide operational criteria for witnessing temporal correlations due to quantum memory through RB experiments. Importantly, while classical correlations may remain undetectable in the ASF data, they can nonetheless significantly affect worst-case errors quantified by the diamond norm, a metric central to fault tolerant quantum computing. In particular, we demonstrate that temporal correlations may suppress worst-case errors highlighting that temporal correlations may not always have detrimental effects on gate performance.

We investigate the fast ion effects on the threshold conditions of ion temperature gradient (ITG) mode and electron temperature gradient (ETG) mode both analytically and numerically using gyrokinetic equation. The onset condition for ITG mode shows a strong and monotonic favorable dependence on the fraction of fast ions, and mostly favorable but non-monotonic dependence on the fast ions' normalized temperature $T_f/T_i$ ($T_f$ is the effective temperature of fast ions, $T_i$ is the temperature of thermal ions). Overall favorable parametric trends are consistent with those for the linear growth rate reported in previous papers, as they are largely determined by kinetic wave-particle resonance effects. While general analytic expressions for the critical normalized thermal ion temperature gradient scale length $(R/L_{T_i})_c$ are quite complicated, an explicit compact expression $\left(\frac{R}{L_{T_i}}\right)_c=\left(\frac{4}{3}+\frac{3}{2}\sqrt{\fracπ{2}}\frac{|\hat{s}|}{q}\right)\left(1+\frac{T_i}{Z_i(1-f_h)T_e}\right)$ has been derived for the mode with its perpendicular scale larger than thermal ion gyroradius, but much smaller than the fast ion gyroradius so that finite Larmor radius effects are manifested in opposite asymptotic limits depending on ion species when $T_f\gg T_i$, and weak density gradient. Here, $q$ is safety factor, $\hat{s}$ is magnetic shear, $Z_i$ is thermal ions' charge, and $f_h$ is fast ion charge density fraction. In this limit, only the fast-ion-induced thermal ion dilution effects persist as fast ion density response becomes unmagnetized and negligible. On the other hand, the fast ion effects on ETG-threshold are found to be unfavorable.

Modified divacancies in the 4H polytype of silicon carbide (SiC) exhibit enhanced charge stability and spin addressability at room temperature, making them attractive for quantum applications. However, their low formation yield and lack of direct structural identification have hindered progress. Here, we demonstrate a controllable method for high-yield engineering and identification of oxygen-related modified divacancy color centers in 4H-SiC via oxygen-ion implantation. Based on their distinct optical and spin-resonance characteristics, we experimentally resolve four types of modified divacancies. Furthermore, by measuring isotope-resolved 17O hyperfine interactions, we identify them as the four crystallographic configurations of oxygen-vacancy (OV) complexes. Remarkably, single OV centers account for over 90% of the total defect population and exhibit superior optical properties and spin coherence compared with defects created by conventional carbon or nitrogen implantation. We characterize the zero-phonon lines of these OV centers and reveal distinct temperature-dependent behavior in spin-readout contrast. By optimizing implantation dose and annealing temperature, we achieve high-density ensembles and observe Rabi-oscillation beating patterns associated with different orientations of basal-type defects. These results establish a high-yield route for scalable engineering of these four oxygen-related modified divacancies in 4H-SiC and clarify their atomic structure, opening new opportunities for solid-state quantum technologies.

Cygnus X-3 presents significant challenges to standard accretion models. Recent polarimetric observations by IXPE reveal high polarization degrees (PD) in the hard state ($\sim 23\%$) and unexpectedly significant polarization in the soft state ($\sim 12\%$), which are difficult to reconcile with static scattering models at low inclination ($i \approx 30^\circ$). We present a relativistic scattering model within a funnel-shaped geometry that resolves this discrepancy. We show that a single funnel-outflow configuration with variable bulk velocity $β$ can reproduce both polarization states, with lower velocities ($β\approx 0$) yielding $\sim 12\%$ polarization (soft state) and mildly relativistic velocities ($β\lesssim 0.4$) producing $\sim 23\%$ polarization (hard state) at $i \approx 30^\circ$ for half funnel opening angles of $\sim 13^\circ$-$16^\circ$. Relativistic aberration modifies the effective scattering angle in the comoving frame, enhancing polarization in the hard state while recovering the static limit in the soft state. The model also yields a consistent estimate of the intrinsic luminosity, of order $\sim 10^{40}$ erg s$^{-1}$, supporting a super-Eddington interpretation. This framework provides a unified explanation of the observed polarization properties of Cygnus X-3.

We investigated the effects of Ni doping on carrier density and anomalous electrical transport properties in CeCo$_{1-x}$Ni$_x$In$_5$ ($x \leq 0.3$) by performing Hall resistivity measurements. The carrier density, estimated from the Hall coefficient $R_{\rm H}$ at a temperature of 0.5 K in high magnetic fields, increases linearly with $x$, indicating that the doped Ni ions act as electron dopants. In CeCoIn$_5$, the magnitude of $-R_{\rm H}$ is strongly enhanced at magnetic fields near the superconducting upper critical field $H_{c2}$ and in the low-field region above the superconducting transition temperature $T_c$. However, these anomalies are found to be significantly suppressed by Ni doping. Possible origins of this suppression in $-R_{\rm H}$ are discussed.

Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common generalization of these two structures: $B_3$-generalized complex structures. We prove that any closed orientable 3-manifold admits such a structure, which can be chosen to be stable, that is, generically cosymplectic up to generalized diffeomorphism.

Convection in Earth's outer core is driven by the release of heat and light elements at the inner core boundary. A key question is whether these buoyancy sources drive convection throughout the core, or whether a stable layer exists just below the core-mantle boundary (CMB). Recent simulations incorporating CMB heat flux heterogeneities propose locally stable ``regional inversion lenses'' (RILs) rather than a global layer, allowing stable and unstable regions to coexist. However, these simulations combine thermal and compositional anomalies, ignoring differences in diffusivities and boundary conditions. Here we simulate thermal, chemical, and thermochemical convection at Ekman number $E=10^{-5}$, with thermal and chemical flux Rayleigh numbers $\widetilde{Ra}_T=30-4000$ and $\widetilde{Ra}_ξ=30-100000$, and Prandtl numbers $Pr_T=1$ and $Pr_ξ=10$. Purely chemical simulations accumulate light elements below the CMB, forming locally stable regions near the poles or global layers, depending on $\widetilde{Ra}_ξ$. These chemically stratified regions persist in thermochemical simulations even when thermal forcing is destabilising. Introducing heterogeneous CMB heat flux produces thermally stratified RILs even with strongly destabilising compositional buoyancy. Our simulations reveal a diverse range of locations, properties, and morphologies of stable regions depending on $\widetilde{Ra}_T$ and $\widetilde{Ra}_ξ$, they can have a seismically detectable thickness and strength and might also have a signature in geomagnetic observations.

We investigate the propagation of gravitational waves in the presence of Lorentz- and diffeomorphism-violating operators within the linearized gravitational sector of the Standard Model Extension. Focusing on isotropic contributions, we analyze the combined effects of the CPT-even dimension-four coefficient $\mathring{k}^{(4)}_{(I)}$ and the CPT-odd dimension-five coefficient $\mathring{k}^{(5)}_{(V)}$ on tensorial gravitational radiation. The modified dispersion relation induces both a rescaling of the propagation speed and helicity-dependent corrections, leading to birefringence and polarization mixing without introducing additional propagating degrees of freedom. We derive the retarded Green function associated with the modified wave operator and obtain explicit expressions for the gravitational waveform generated by matter sources. As an application, we examine a binary black hole system and show how Lorentz violation alters the observed strain through shifted retarded times, amplitude rescaling, and higher derivative corrections to the quadrupole formula. Using GW170817/GRB 170817A, published GWTC-3 propagation tests, and conservative polarization consistency arguments, we translate existing observational constraints into bounds on $\mathring{k}^{(4)}_{(I)}$ and $\mathring{k}^{(5)}_{(V)}$.

Multispecies ecosystems modelled by generalized Lotka-Volterra equations exhibit stationary population abundances, where large number of species often coexist. Understanding the precise conditions under which this is at all feasible and what triggers species extinctions is a key, outstanding problem in theoretical ecology. Using standard methods of random matrix theory, I show that distributions of species abundances are Gaussian at equilibrium, in the weakly interacting regime. One consequence is that feasibility is generically broken before stability, for large enough number of species. I further derive an analytical expression for the probability that $n=0,1,2,...$ species go extinct and conjecture that a single-parameter scaling law governs species extinctions. These results are corroborated by numerical simulations in a wide range of system parameters.

The development of eccentric waveform models enables us to explore the growing catalog of gravitational-wave events with measurable eccentricity. This opens new opportunities to gain insight into the formation channels and evolutionary pathways of compact binary systems using eccentricity. However, most recent population analyses have been limited to quasi-circular binaries, primarily due to constraints in waveform modeling and sensitivity estimates. We are now entering an era where both of these limitations are being addressed, allowing for a more comprehensive investigation of eccentric binary populations. In this work, we perform a very first population analysis that simultaneously fits the mass, spin, redshift, and eccentricity distribution. Specifically, we use source-parameter estimation on 153 binary black holes in GWTC-4 catalog provided by the Rapid Iterative FiTting (RIFT) framework using the SEOBNRv5EHM waveform model. We extend the default O4a population model to include orbital eccentricity. We find that inferred population properties are broadly consistent with conclusions obtained in previous analyses assuming quasi-circular binaries. To assess sensitivity of our results to the most eccentric sources, we repeat our analysis excluding GW200129_065458. Consistent with our conclusions about each event and using Nonoverlapping Mixture eccentricity model, we bound the branching ratio for eccentric events to be below $0.051890$ and $0.022011$ at $90\%$ confidence with and without GW200129_065458 respectively. Using four different parametric population models for eccentricity, we argue that the rate of eccentric events is weakly constrained by observations and highly model-dependent.

We show how to use divisors on the projectivized Hodge bundle to construct special vector-valued modular forms and then apply invariant theory to construct all vector-valued Siegel modular forms of level two and degree two. Thus we construct all modular forms in terms of certain basic modular forms that are intimately connected to the moduli of curves of genus two.

Certain Feynman integrals can be expressed as periods of differential forms on Calabi--Yau manifolds. We provide a mathematical proof of a result of Duhr and Maggio on the modularity of the two-dimensional conformal traintrack integral. Our approach is based on a factorization of the associated Picard-Fuchs system into a tensor product of Gauss hypergeometric systems via a gauge transformation due to Clingher, Doran and Malmendier.