In this work, we investigate the thermodynamic properties of the quantum Blume-Capel model with spin \( S = 5/2 \) in the presence of transverse and random crystalline fields. The system is described by a Hamiltonian that includes ferromagnetic exchange interactions between nearest neighbors, a longitudinal single-ion anisotropy, and a transverse single-ion anisotropy. Using a mean-field approach based on Bogoliubov's inequality for the Gibbs free energy, we derive the fundamental thermodynamic potential and the equation of state for the magnetization. The influence of the longitudinal and transverse anisotropy parameters on the magnetic ordering and phase transitions is analyzed in detail. We present magnetization versus temperature diagrams for various combinations of the anisotropies, exploring both positive and negative values. Our results reveal that the system exhibits standard second-order phase transitions for most parameter ranges, with no evidence of tricritical behavior. However, for certain positive values of the anisotropies, the model displays a first-order phase transition within the ordered phase, characterized by a jump from a higher-spin ordered state to a lower-spin ordered state. The critical temperatures are shown to be sensitive to the magnitude and sign of the anisotropy parameters. In particular, negative transverse anisotropies favor magnetic order, raising the critical temperature, while positive anisotropies promote disorder, lowering the critical temperature. This study provides a comprehensive analysis of the phase diagram of the \( S = 5/2 \) quantum Blume-Capel model and highlights the role of transverse fields in modifying the critical behavior.

The deterministic many-interacting-worlds method proposed in 2014 showed potential among the numerous interpretation of quantum mechanics. The successful application of this method in harmonic oscillator has been promoted for a long time. In this article we continue the idea about using this method to solve some bounded systems different from harmonic oscillator potential and extend to 2 dimension cases. We focus on the potential with singularity like coulomb potential and finite trap potential by some asymptotic smooth method. The numerical simulation mainly based on the dynamical algorithm proposed in many-interacting-worlds method will be used to approach the stationary states of given systems. Our results shows the consistency to the matrix Numerov method in standard quantum mechanics in solving bounded systems and provides the possibility to solve more complex systems.

The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $μ$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter $λ_c(μ)$ for the one-dimensional model. In particular, we prove that $λ_c(μ)<+\infty$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument.

We consider the numerical solution of partial differential equations with coefficients that are strongly heterogeneous in space. We provide an overview of higher-order localized orthogonal decomposition (LOD) methods for the elliptic setting, including recent advancements, and then present a generalization of the strategy to linear hyperbolic multiscale problems. We address the limitations of earlier constructions for the wave equation, which only achieve second-order convergence in space, independent of the chosen polynomial degree. Building on the methodology of enriched corrections recently developed for parabolic multiscale problems, we motivate and propose an enriched higher-order LOD method for the wave equation. The enriched corrections exhibit exponential decay and can be computed on patches. Under minimal assumptions on the coefficient and standard well-preparedness conditions on the data, we derive a priori error estimates that achieve optimal high-order convergence rates, thereby overcoming the previously observed saturation of the convergence rate. With the fifth-order Rosenbrock-Wanner (ROW) time integrator, we conduct a series of numerical examples to verify our theoretical results. We provide examples showing the optimal spatial convergence of the method including the localization errors for different polynomial orders. We also present examples showing the optimal convergence rates of the time discretization.

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $ϕ=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to $\mathcal{H}(b)$, and its norm in $\mathcal{H}(b)$ can be expressed in terms of the Taylor coefficients of $f$ and $ϕ$ via the formula \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2. \] However, the formula can break down for some other $f\in\mathcal{H}(b)$. In this article we extend the scope of the formula to all $f\in H^2$ for which the right-hand side is finite, provided that either $ϕ\in H^2$ or $ϕ$ is rational. If merely $ϕ\in H^p$ for some $p\in(0,2]$, then the formula still holds provided that, in addition, $\sum_{m\ge0}m^{2/p-1}|\hat{f}(m)|^2<\infty$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\in\mathcal{H}(b)$.

A central question in high-dimensional statistics is to understand statistical--computational gaps: regimes in which recovering a hidden signal is information-theoretically possible but conjectured to be computationally intractable. The low-degree framework offers a concrete way to study this gap by restricting attention to estimators that are polynomials of degree at most $D$ in the observed data. In this paper, we study low-degree estimation in planted dense subhypergraph, sparse tensor PCA, and tensor PCA with a general prior. For the planted dense subhypergraph model on $n$ vertices, we identify two regimes depending on whether the planted set is larger or smaller than $\sqrt{n}$. Above this scale, we identify a sharp threshold for low-degree estimation. Below this scale, we establish hardness in the regimes predicted by prior work, thereby resolving a question of Schramm and Wein (2022) and Sohn and Wein (2025). For sparse tensor PCA, we identify an analogous sharp phase transition. For tensor PCA with a general prior, we prove a low-degree estimation lower bound at the critical signal scale, matching the degree--signal tradeoff suggested by prior work. Our lower bounds apply to degree $D=n^δ$, where $n$ is the dimension and $δ>0$ is a constant, and we complement them with corresponding low-degree upper bounds. In addition, for planted dense subhypergraph and sparse tensor PCA above the $\sqrt{n}$ scale, we convert our upper bounds into polynomial-time algorithms that achieve almost exact recovery above the sharp threshold, yielding polynomial-time algorithms succeeding up to this threshold. Our proofs extend the framework of Sohn and Wein (2025) through a conditional variant that yields the correct signal-to-noise ratio in settings where the unconditional approach is insufficient.

In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes. As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022]. We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.

Recent progress in diverse intelligence has shown simple learning capacities below the organism level - single cells and even molecular networks. However, there are still many knowledge gaps around learning capacity above the organism level, and about memory implemented purely by dynamical interactions without explicit memory media. We demonstrate that minimal ecological dynamics (in silico) are sufficient for several kinds of learning, assayed as changes in both, magnitude of response, and of recovery time. Systematic exploration of over 220,000 parameter combinations in a simulated classic predator-prey model revealed that, when perturbed by stimuli, recovery time exhibits habituation, sensitization, and a form of discrete number learning in a scale-invariant manner. Robustness analysis revealed that habituation and sensitization persist under stochastic perturbations, while discrete number learning is disrupted even at low noise levels. Dimensionality reduction revealed that the incidence of learning capacity is primarily determined by ecological interaction strengths. Clear, unique clustering patterns in parameter space allow high prediction accuracy for novel parameter combinations that enable learning. Response magnitude revealed a striking asymmetry: 90.6% of parameter combinations exhibited recovery time sensitization paired with habituation of response magnitude, while the opposite pattern was extremely rare. These findings highlight a set of phenomena at the intersection of ecology, basal cognition, and mathematics with many implications for a wide range of systems describable by similar kinds of equations. These properties provide numerous efforts in biology and engineering with a substrate that has considerable, pre-patterned, propensity for learning, which ultimately arises from mathematics, not depending on the details of physics or biology.

The overhead exponent -- characterizing the scaling of the number of noisy magic states with respect to the target distillation error -- has been a central quantity to benchmark magic state distillation protocols. On the other hand, a related but less investigated quantity motivated by an information-theoretic viewpoint is the asymptotic distillation rate, the largest ratio of output to input magic states such that error vanishes asymptotically. These two quantities are tightly related in the specific case -- the overhead exponent is zero if and only if the asymptotic distillation rate is linear. However, their relationship in other regimes has been unclear. Here, we show that their quantitative relation is generally not robust, by presenting a family of magic state distillation protocols with an overhead exponent not close to zero -- in fact, larger than one -- that still achieves the asymptotic rate arbitrarily close to the linear rate. This implies that the distillation rate is not constrained by the overhead exponent within the sublinear rate regime. Notably, our protocol is based on error checking by measurements of logical Clifford operators, which underlies the recent magic state cultivation protocol, suggesting the potential of this mechanism for asymptotic magic state distillation.

We describe a verification pipeline that takes production Rust cryptographic code and produces machine-checked correctness proofs in Lean 4. The pipeline combines three components: symbolic extraction tools (Charon and Aeneas, or Hax) that lift Rust into Lean 4; formal cryptographic specification libraries (ArkLib and CompPoly, from the Verified zkEVM project) that provide the mathematical targets; and AI provers (Aristotle from Harmonic AI and Aleph from Logical Intelligence) that close the resulting proof obligations. Every proof is checked by the Lean kernel, so AI output cannot compromise soundness. Within the scope of the Ethereum Foundation's zkEVM Verification Project, we applied the pipeline to cryptographic primitives in Plonky3 (FRI folding, Mersenne31 and KoalaBear field arithmetic, Horner polynomial evaluation) and RISC Zero (Merkle inclusion verification). In addition, Aleph authored proofs of two bounds-style theorems in Plonky3's compute_log_arity_for_round that previously stood as sorry. The paper describes the architecture, walks through a running example based on Aleph's two proofs, reports which classes of proof obligations AI closed and which required manual work, and discusses the engineering gaps we encountered: Lean 4 toolchain drift across tools and specific Aeneas/Hax extraction limits. We also document concrete missing lemmas, tactic gaps, and code-generation friction points discovered during proof development. We hope this contribution lowers the barrier to adoption of formal verification and facilitates more effective use of AI in this pipeline. The result is a working pipeline for formal verification of Rust, with kernel-checked proofs and reproducible artefacts.

Large Language Models (LLMs) have shown promise for automated vulnerability repair (AVR), but they still face several limitations, including the lack of intra-vulnerability experience accumulation and the lack of cross-vulnerability experience reuse. As a result, LLMs may repeatedly make similar mistakes during iterative repair and underutilize valuable repair knowledge from historical vulnerabilities. To address these challenges, we propose EvoRepair, the first experience-based self-evolving AVR agent framework that enables LLMs to accumulate, refine, and leverage domain-specific knowledge across long-horizon vulnerability repairs. EvoRepair follows a cyclic learn-and-repair process that retrieves relevant past experiences to guide repair, extracts new experiences from repair trajectories, and updates an experience bank using quality-aware scoring. We evaluate EvoRepair against 12 representative vulnerability repair baselines on PATCHEVAL and SEC-bench using GPT-5-mini. Results show that EvoRepair achieves the best overall performance, reaching 93.47% on PATCHEVAL, 87.00% on SEC-bench, and 90.46% overall. In particular, EvoRepair outperforms latest LLM-based baseline LoopRepair by 39.56% and 33.50% on PATCHEVAL and SEC-bench, respectively, and surpasses IntentFix by 70.86% and 50.50%. Across both benchmarks, EvoRepair also exceeds the recent self-evolving agent Live-SWE-Agent by 6.98% overall. Additional transfer experiments on VUL4J further demonstrate the robustness of EvoRepair across models, programming languages, and datasets. These findings demonstrate that experience-based self-evolution substantially strengthens agentic AVR and goes beyond existing self-evolving techniques.

We prove a list recovery guarantee for random low-rate linear codes over sufficiently large prime fields. For fixed dimension $d$, error fraction $α$, and accuracy parameter $\varepsilon$, a random $d$-dimensional linear code $C \subseteq \mathbb{F}_p^n$ is, with high probability, $(α,\ell,\frac{1+\varepsilon}{1-α}\ell)$-list recoverable simultaneously for all input list sizes $\ell\le 2^{O_{α, \varepsilon, d}(n/\log n)}$. The proof is inspired by work of Matoušek, Př\'ıvětivý, and Škovroň on reconstructing point sets from their projections. It combines a deterministic graph-theoretic certificate, a nonvanishing determinant criterion, and the Schwartz--Zippel lemma. We also give a lower bound showing that any linear code $C \subseteq \mathbb{F}_p^n$ of dimension at least two cannot be $(α,\ell,\frac{1+\varepsilon}{1-α}\ell)$-list recoverable for feasible list sizes $\ell \geq 2^{Ω_{α, \varepsilon}(n)}$. In this sense, our result is nearly optimal.

We know that the charged Reissner Nordstr$ö$m black hole metric is obtained from the Einstein Hilbert gravitational action. This action has the kinetic term $F^2 = (da)^2$. Motivated by the higher-form symmetry structure of the EFT of Force Free Electrodynamics, we replace the Maxwell field-strength contribution in the Einstein Hilbert action by the gauge-invariant combination $(b-da)^2$, where $a_μ$ is the worldsheet gauge field and $b_{μν}$ is a background two-form field. This ensures that the new action has a higher form symmetry $b \rightarrow b+dΛ, a\rightarrow a+Λ$. Here, unlike in qed, $Λ$ may be any one form (not necessarily a differential one form $\partial_μϕ$). The higher form symmetry here is one with the conserved current being a two form and the charge integrated on surfaces. Intuitively, it is the number/current of vector field lines that is conserved here, not the current of particles. Thus, integrating over a surface through which the field lines pierce is sufficient to find the number of these lines that are passing through; so the charge is integrated on surfaces, rather than on the volume. After fixing a particular gauge for the fields $a_μ$ and $b_{μν}$, we obtain a generalized black-hole metric. We find that on hypersurfaces satisfying $(r-t)=$ constant, this metric reduces locally to a Reissner Nordstr$ö$m geometry with an effective charge parameter depending on the constant $(r-t)$.

In this paper, we answer negatively to a question posed in the context of the 2025 Oberwolfach Mini-Workshop ``The Yang-Baxter Equation and Representations of Braid Groups'' regarding the existence of split extensions classifiers in the category of skew braces. To this end, we show that the variety of skew braces is not action accessible by investigating the categorical notion of centraliser in the sense of Huq. In light of their intrinsic relationship with skew braces, an analogous result is proven for the variety of post-Lie algebras over a fixed field.

We study estimation in the low signal-to-noise ratio (SNR) regime for a broad class of Gaussian latent-variable models, including Gaussian mixtures and orbit recovery problems. We show that, in this regime, the generalized method-of-moments (GMoM) matches the first-order asymptotic efficiency of maximum likelihood. In particular, if the moment features are chosen up to the minimal local order required for identification and are weighted optimally, then the resulting GMoM estimator has the same leading asymptotic covariance as the maximum-likelihood estimator. Our analysis shows that, in low SNR, this equivalence is governed by a layered local geometry: different directions become informative at different moment orders, partitioning the space into layers with distinct SNR scalings. We prove that the observed Fisher information and the GMoM information operator admit matching layerwise expansions across these layers. As a consequence, in the low-SNR regime, GMoM provides a statistically efficient alternative to maximum likelihood, while preserving the computational advantages of moment-based estimation.

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$-systems. The main result is a proof of the Aljohani--Bamberg--Cameron conjecture which claims that if $n > n_0(k)$ and there are an $(n,k,L)$-system and an $(n,k,\{0,\dots,k-1\}\setminus L)$-system whose sizes have product $\binom{n}{k}$, then they are a $t$-intersecting family and a Steiner system $S(t,k,n)$ for some $t$.

We derive closed form expressions for the lower expectations that correspond to total variation distance and chi-squared divergence balls around a probability mass function over a finite set.

In this work, we study several inequalities related to a Dirichlet problem on Riemannian manifolds whose Ricci curvature is bounded from below. First, we establish inequalities involving the torsional rigidity and discuss rigidity results characterizing metric balls in this setting. Next, we derive an integral identity associated with a Dirichlet problem, which measures the spherical deficit arising in this context. In particular, we apply this identity to the setting of Einstein manifolds.

We construct canonical extensions of $p$-adic shtukas on integral models of toroidal compactifications of abelian-type Shimura varieties with quasi-parahoric levels at any prime number $p$. More precisely, we define the notion of a log diamond as a $v$-sheaf associated with a log scheme over $\mathbb{Z}_p$ and construct a $p$-adic log shtuka over the log diamond of an integral toroidal compactification of an abelian-type Shimura variety by studying the ``degeneration'' of the shtuka at the boundary. Moreover, we provide a definition of canonical integral models of toroidal and minimal compactifications in the sense of Pappas and Rapoport, and verify it in the same generality as above. Applications include the canonicity and functoriality of integral toroidal compactifications, as well as an axiomatic proof of the well-positionedness of all well-known stratifications on the special fiber.

A model combining Enskog's collision integral for dense fluids with a Vlasov-style description of the van der Waals force is applied to supercooling. First, the spinodal temperature $T_{s}$ is calculated, at which a liquid becomes unstable to small perturbations and transitions to solid. In particular, it turns out that isochoric cooling allows one to reach a lower temperature than isobaric cooling. Second, the surface tension of a supercooled liquid-vapor interface is shown to diverge at $T_{s}$. The singularity is caused by an oscillatory region emerging on the liquid side of the interface as $T\rightarrow T_{s}$; it develops because the liquid approaches instability, and the interface starts radiating (so far, evanescent) waves. At $T=T_{s}$, the waves cease to be evanescent and the oscillatory region extends to infinity -- hence, the singularity of the surface tension. Since this effect has a clear physical interpretation, it should occur regardless of the model and approximations under which it was obtained. This and the other results of the paper are illustrated using argon and several other fluids.

Many experimental studies have reported variations in interfacial tension. Isolating all the geometric and fluid material parameters and varying the interfacial tension can be useful to check their influence. Numerical investigations using Free Surface 3D (FS3D), have been conducted to compare varying values of interfacial tension and evaluate the sensitivity. A grid independence study compared the compound crown height of a splash to determine the required resolution for validation. A qualitative validation showed FS3D could correctly capture the impact morphology while varying the viscosity ratio of the drop and film liquid when compared to the experimental results. A quantitative validation for a water drop impacting onto an oil film shows a good match for the crown heights of the numerical and experimental data. The same setup was then extended to study the variation of interfacial tension, where the deviation of the overall compound crown height and spreading diameter of the internal crowns was compared. Results revealed minor changes in the compound crown height and spreading diameter of the drop liquid, but the internal crown composition showed significant differences. In order to run FS3D efficiently on the new supercomputer Hunter, which has a new APU architecture-based system, extensive work had to be done. To adapt to the new hardware architecture, large parts of FS3D have been ported to utilise the AMD Instinct MI300A accelerated processing units (APUs) at HLRS using OpenMP. Implementation of Umpire memory pools improved performance for larger workloads per APU. The GPU-accelerated code achieves a 4 times speedup compared to CPU-only execution on the same hardware. Strong and weak scaling tests have been conducted, showing good strong scaling for up to 4 APUs, and linear weak scaling for up to 512 APUs, resulting in a total of 4096**3 cells for the first time.

Less-salient taxes can ease the classic equality-efficiency trade-off by making people respond less to taxation. But deliberately obscuring taxes may be viewed as dishonest. This creates a three-way trade-off between equality, efficiency, and honesty. We analyze this trade-off in a simple setting with a linear income tax. We define and characterize the morally efficient frontier, trading off utilitarian welfare against honesty or transparency. Complete honesty is Pareto inefficient but not morally inefficient. More generally, any increase in honesty reduces utilitarian welfare. When utilitarian welfare is decomposed into equality and efficiency, the cost of honesty falls most robustly on equality: higher salience always reduces equality, while the effect on efficiency is ambiguous. This asymmetry is explained by the fact that salience increases the price of equality, which is the efficiency cost of a marginal increase in equality. Our approach could be applied to other settings in which utilitarian and procedural or deontological values conflict.

This paper presents an extensible ns-3-based simulation framework for evaluating intent-based, semantics-aware control in Open RAN architectures. The framework integrates external Radio Access Network (RAN) Intelligent Controller (RIC) components and supports fine-grained control via internal distributed applications (dApps), enabling intent-based RAN orchestration across different timescales while maintaining standardized network behavior. As an illustrative use case, we implement an intent-based dApp for radio resource management (RRM) under realistic observability constraints. The scheduling problem is formulated using realistic key performance measurements (KPMs) available to dApps, together with a newly introduced Intent Satisfaction Score (ISS), which quantifies the delivery of intent-relevant information by combining distortion- and perception-oriented measures. Simulation results show that intent-based RRM can improve ISS while significantly reducing radio resource usage and computational overhead, at the cost of a moderate reduction in packet delivery ratio and throughput.

Sub-Neptunes and Neptunes are often modeled with distinct, fully convective layers. Yet, there are several arguments for compositions gradients that can inhibit convection. In these regions, energy transport depends on the thermal conductivity and radiative opacity. We compare three thermal conductivity models and investigate their impact on planetary evolution accounting for the possibility of convective mixing eroding composition gradients. Using a modified version of MESA, we model the evolution of planets with masses of Mp=5, 10, 15 Mearth and three initial entropies. We implement thermal conductivities for: pure water, fully ionized matter, and constant electron conductivity. Convective mixing complicates the relation between conductivity, evolution, and radius. For hot forming planets with a large composition gradient, where the heavy-element mass fraction changes gradually from the core to the envelope, convective mixing has a large impact on the radius evolution. In this case, the thermal conductivity is less relevant and the radii converge to similar values after billions of years. For cold forming planets or narrow composition gradients, convective mixing is less efficient. If the composition profile is not altered significantly, the thermal conductivity becomes critical. It determines how much energy can be trapped beneath stable composition gradients. For intermediate initial entropies, high thermal conductivity inhibits convection. Further work is required to determine the thermal conductivity for various mixtures expected in sub-Neptune and Neptunes at high densities and temperatures. In addition, further constraints on the entropy and composition profile after formation can reduce the degeneracy of the planetary evolution, in particular, the dependence of the radius with time.

This paper develops a reduced-order framework for modelling the two-way coupling between gravity waves and turbulent wakes in large-scale wind farms. Linearising the non-hydrostatic Boussinesq equations and introducing simplifications appropriate to the boundary layer and the overlying stratified free atmosphere yield separate governing equations for the two regions. These are coupled through a dynamic boundary condition at the capping inversion, which directly captures the feedback of gravity waves on the boundary-layer flow. A mixed spectral-finite-difference discretisation yields a computationally efficient model while retaining vertical boundary-layer structure. Comparisons with large-eddy simulations (LES) confirm the model successfully reproduces both internal wind-farm flow and large-scale gravity-wave effects. It captures the upstream blockage induced by adverse pressure gradients, as well as the accelerated wake recovery within and downwind of the farm, driven by favourable pressure gradients.

The origin of quasi-one-dimensional (q1D) ferromagnetic (FM) excitations in the candidate spin-triplet superconductor CeSb$_2$ has remained unclear. Here we report an electronic mechanism for emergent q1D magnetism in the quasi-two-dimensional lattice of CeSb$_2$, revealed by angle-resolved photoemission spectroscopy (ARPES). High-resolution ARPES resolves no spin-density-wave gap on the dispersive Fermi pockets, disfavoring a nesting-driven mechanism for the q1D FM excitations. Instead, resonant ARPES reveals a pronounced selective enhancement of Ce 4$f$ spectral weight on the $C_2$-distributed Fermi pockets aligned with the Ce ladder. This observation signifies band-selective Kondo coupling that generates strongly anisotropic magnetic exchange interactions, which can naturally account for both the q1D ferromagnetic excitations and the competing magnetic orders. Our results identify a band-selective Kondo coupling mechanism for emergent low-dimensional magnetism in correlated $f$-electron systems.

We propose a Bayesian framework for uncertainty quantification and comparison in brain connectivity graph analysis. Standard graph-based approaches typically rely on point estimates of correlation matrices, overlooking the uncertainty induced by high-dimensional estimation from limited data. Our methodology constructs and compares credible hyperrectangles derived from posterior distributions, providing interpretable tools for subject-level inference and longitudinal monitoring. We develop scalable algorithms for estimating these regions in high dimensions and establish theoretical guarantees in the inverse-Wishart model for resting-state fMRI data, including a Bernstein--von Mises theorem for correlation matrices and control of a Bayesian family-wise error rate. The proposed framework enables principled detection of significant connectivity differences both globally and locally while preserving joint dependency structures. While demonstrating competitive performance against multiple-testing procedures on synthetic datasets, our approach also facilitates the direct comparison of two distinct scans from a single patient, a capability currently absent from the literature. We leverage this novelty on real datasets to improve interpretability. Beyond fMRI data, the approach provides a general framework for comparison problems in high-dimensional dependent settings.

Hjort and Glad (1995) present a method for semiparametric density estimation. Relative to the ordinary kernel density estimator, this technique performs much better when a parametric vehicle distribution fits the data, and otherwise performs at broadly the same level. Jones, Linton, and Nielsen (1995) present a somewhat similar method for density estimation which has higher order bias for all sufficiently smooth densities. In this paper, we combine the two methods. We show that, theoretically, the desired properties of general higher order bias allied with even better performance for an appropriate vehicle model are achieved. Simulations suggest that the new estimator realises only a little of its theoretical potential in practice for small to moderately large sample sizes.

Here we present an updated shape model of (486958) Arrokoth, the bilobate Kuiper Belt Object (KBO) which the NASA New Horizons spacecraft flew past in 2019. This updated shape model uses all of the resolved images of Arrokoth obtained by the New Horizons LOng Range Reconnaissance Imager (LORRI). We developed an updated shape modeling algorithm which allowed the shape and rotational pole of Arrokoth to be fit to much better quality with an efficient use of GPU-accelerated features. The resulting model of Arrokoth's contact binary shape is significantly thicker and of larger volume than the one previously published immediately after the flyby by Spencer et al (2020). We show that Arrokoth's smaller lobe Weeyo is roughly spherical in shape, while the larger lobe Wenu is more flattened, with the volume ratio between the lobes being roughly 2:1. Owing to Wenu's oblate shape, Arrokoth's rotational lightcurve would have significantly lower mean reflectance when viewed from subobserver latitudes that would have shown lightcurve variation. We discuss the impact this may have on estimates of the frequency of contact binaries in the Kuiper Belt. We also discuss the implications of this shape for the formation of Arrokoth, particularly in the context of the Streaming Instability.

We investigate integration by parts (IBP) formulae for stochastic Volterra equations and we establish the smoothing effect of the expectation. Due to the inherent path-dependent dynamics of this class of processes, standard Bismut--Elworthy--Li (BEL) formulae and lifting procedures fail to produce representations for directional derivatives with respect to the initial curve. We exhibit a new type of fractional IBP for these derivatives which, by means of the Riemann--Liouville fractional derivative, interpolates between the standard chain rule and a pure BEL formula with Cameron--Martin path directions. Our assumptions describe precisely the trade-off between the direction's and the test function's regularities. Crucially, we reveal that more roughness leads to more smoothing: for a power-law kernel with Hurst parameter $H\in(0,1/2)$, we show that the expectation is differentiable along constant directions provided that the test function has Hölder continuity $β>2H$. The proof of the formula relies on a careful analysis of the conditional expectation's temporal regularity and on the well-posedness of its Riemann--Liouville derivative. We complement these results with a BEL formula along all square integrable directions whenever the noise is additive, a second order BEL formula and an application to forward and rough volatility models. In the latter case, the derivative is interpreted as the sensitivity with respect to the whole initial forward variance curve.