Papers

We study density-dependent birth–death processes with a strong Allee effect and absorbing extinction. The deterministic system is bistable: extinction and a positive equilibrium are locally asymptotically stable, separated by an unstable Allee threshold. Let $K$ be the population-size scaling parameter. As $K\to\infty$, we derive a sharp Eyring–Kramers-type asymptotic formula, including the leading prefactor, for the principal eigenvalue of the killed generator, and a uniform approximation of the corresponding positive eigenvector. Consequently, we obtain a sharp asymptotic formula for the mean extinction time under the quasi-stationary distribution (QSD). In the bistable regime, the initial population determines the competition between early extinction and positive metastability. After initial relaxation, the law is approximated in total variation by an initial-state-dependent convex combination of the QSD and the Dirac mass at extinction. The mixing coefficient is identified as the probability of reaching the positive stable region before absorption. We identify a probabilistic transition layer of width $O(\sqrt K)$ around the Allee threshold, where the mixing coefficient converges to an explicit Gaussian profile. Notably, a population starting at the threshold reaches the positive metastable region or becomes extinct with asymptotically equal probabilities. We establish existence and uniqueness of the QSD, quantitative convergence to the quasi-stationary regime, and a discrete Gaussian approximation of the QSD centered at the positive stable level. Our analysis combines a self-adjoint spectral realization of the killed generator, matched asymptotic analysis of the associated second-order recurrence, and discrete Laplace estimates for reversible weights.

Accurate medium-term, from a few months to a few years, electricity load forecasts are crucial for informed decision-making in power plant maintenance scheduling, load dispatch and price settlement. Being comprised between Long-Term Load Forecasting (LTLF) which uses mostly economic projections and appliances development scenarios, and Short-Term Load Forecasting (STLF) driven by weather, calendar and autoregressive patterns, Medium-Term Load Forecasting (MTLF) requires both extrapolation capabilities and variability modeling. Yet, it remains unclear if MTLF can benefit from economic indicators, and especially at which forecast horizon and resolution. To address these challenges we investigated the impact of socioeconomic data on predictions issued 1 month and up to 48 months in advance for France at monthly and daily resolution using a tabular Foundation Model (FM). A dataset covering 20 years of observations of electricity load, weather variables and economic features such as consumer price and production indices, electric vehicle counts or employment is created for the study. To avoid noisy data, we used a new feature selection pipeline, creating ensemble of expert models with diverse feature subsets, to demonstrate that selected economic covariates improve forecast skill by 20% over 2015-2025. This enhancement is steady across lead times and resolutions limiting the Mean Absolute Percentage Error to 4% for monthly granularity and 5% for daily granularity. Explainability of the models is investigated through feature and context importance. Results showed that the FM is limited in the context it leverages pointing towards potential computational savings with a reduced context, while feature importance of economic predictors grows with the forecast horizon. This suggests that including economic data in MTLF could bridge the gap with LTLF leading to seamless forecasts.

We investigate orthogonality under the shift-invariant resolvent delta operator $\mathscr L=D(I-D)^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\mathscr L$-Appell if and only if it is a translated Laguerre sequence. We then move beyond the Appell setting and show that the diagonal family associated with $$\mathcal S_\beta=\frac{\beta}{\beta+1}\mathcal L^0+\frac{1}{\beta+1}\delta$$ satisfies the exact Hahn relation $$\mathscr L B_{n+1}(x;\beta)=(n+1)B_n(x;\beta+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=\theta\,\mathscr L P_{n+1}(c)$ already forces the orthogonality functional to be a translated Laguerre functional or a translated member of the diagonal family. Connection and structure relations, a semiclassical Pearson equation, a differential characterization, and a generating function complete the description. Thus the resolvent operator provides an explicit setting in which delta-operator lowering and semiclassical orthogonality are linked by exact structural identities.

Space weather disturbances driven by the solar wind can degrade satellite navigation, disrupt radio communications, and threaten power infrastructure, making accurate forecasting of the high-latitude ionosphere's response a critical operational need. Existing approaches, empirical climatological models and physics-based magnetohydrodynamic simulations, either smooth out the ionosphere's time-dependent, non-linear response or are too computationally expensive for real-time use, while prior machine learning efforts have mostly targeted scalar indices rather than the full spatial structure of ionospheric convection. Here we train and compare three deep learning architectures, a probabilistic diffusion model conditioned on multi-variate solar wind and interplanetary magnetic field measurements at the L1 Lagrange point, an unconditioned diffusion ablation, and a deterministic U-Net baseline, to forecast Southern Hemisphere high-latitude electric potential maps derived from SuperDARN radar observations. Using five years (2020–2025) of SuperDARN data synchronised with DSCOVR L1 measurements, we evaluate the models under a single-pass regime, an extended autoregressive rollout of up to 350 frames, and an out-of-distribution case study on the intense March 2015 St.\ Patrick's Day storm, unseen during training. We find that the relative advantage of the deterministic and probabilistic approaches is not fixed: the deterministic model is competitive over short horizons and calm conditions, while the diffusion model's advantage grows and eventually dominates as the forecast horizon lengthens and the event becomes more dynamic, evidence that probabilistic, sample-based generative models are the more promising direction for operational, long-horizon space weather forecasting.

Autonomous navigation in dynamic environments is a critical challenge, particularly when spaces are shared with other mobile agents whose future trajectories are known. While traditional grid-based planners efficiently find collision-free paths, their reliance on stop-and-turn mechanics over $2^k$-connected grids produces piecewise-linear trajectories that are kinodynamically highly sub-optimal for differentially constrained robots. In this paper, we present an adaptation of Safe Interval Path Planning (SIPP) that operates on state lattices, utilizing precomputed, kinodynamically smooth motion primitives. To efficiently handle dynamic environments, we rasterize the spatiotemporal swept volumes of moving obstacles directly onto the grid, treating grid cells as atomic units of space, whose resolution is typically dictated by inherent localization noise. We perform a comprehensive comparative analysis between our lattice-based approach and $2^k$-connected grid planners across diverse topological environments. Our evaluation considers a broad spectrum of performance metrics, including planning time, path angularity, cumulative heading change (angle-over-length), and bending energy. The results demonstrate that while the expanded state space of lattice-based search increases computational overhead, it yields trajectories with significantly superior kinodynamic properties. Specifically, our method achieves a reachability comparable to highly connected grids while ensuring smooth, continuous, and physically executable paths ready for real-world deployment.

Let $Z_\alpha$ be a positive $\alpha$-stable random variable, where $0<\alpha<1$. We prove that $Z_\alpha^p$ is infinitely divisible for every $p>0$. Combining this result with Jedidi and Simon's characterization of infinite divisibility of negative powers of $Z_\alpha$, we obtain, for $p\ne0$, $$Z_\alpha^p\ \text{is infinitely divisible} \quad\Longleftrightarrow\quad p\in\left(-\infty,-\frac{\alpha}{1-\alpha}\right]\cup(0,\infty).$$ The proof relies on Kanter's factorization and a general product property: if $X$ is a mixture of exponential distributions, $W$ is a generalized gamma convolution, and $X$ and $W$ are independent, then $(X+c)W$ is infinitely divisible for every $c\ge 0$.

Complex systems such as vehicles, patients, or genomes emit discrete event sequences whose operative question is causal, not predictive: which events cause which other events, and which cause higher-level outcomes such as failures or diseases? This question decomposes along two axes – dependency type (event $\to$ event vs.\ event $\to$ outcome) and causal scope (single sequence vs.\ population) – yielding four structurally distinct regimes with different identifiability conditions. No existing method addresses more than one, because all assume multi-stream structure with low vocabulary, and none scales beyond a few hundred event types. We present Seq2Cause, a unified framework that resolves all four regimes through a single shared primitive: a pretrained autoregressive model repurposed as an amortized conditional independence testing engine requiring no task-specific retraining. We establish a prediction–causality duality: the model's excess cross-entropy simultaneously bounds causal identification error across all four regimes, so that every improvement in next-token prediction tightens causal guarantees for free. On nonlinear SCMs (vocabularies up to $8{,}000$ types) and real-world vehicle diagnostic logs ($29$K event types, $474$ failure outcomes), Seq2Cause is the first method to populate all four regimes at scale with a single frozen backbone. Existing methods are either inapplicable, inaccurate, or computationally intractable in this setting.

We investigate the cosmological dynamics and observational viability of a logarithmic Herglotz-type $f(R,T)$ gravity model described by $f(R,T)=R+\beta\ln(-T/T_0)$. Assuming a spatially flat Friedmann–Lemaître–Robertson–Walker background and pressureless matter, the modified field equations are formulated as a coupled dynamical system involving the Herglotz field and the normalized matter density. A phenomenological closure relation is adopted for the effective Herglotz sector, allowing the background evolution to be determined numerically. The model is constrained using Cosmic Chronometer measurements, the DESI DR2 baryon acoustic oscillation compilation, and the Union3 supernova sample, considering both individual and combined datasets within a Bayesian Markov-chain Monte Carlo framework. The combined analysis provides a statistically consistent description of the background expansion. The reconstructed expansion history yields a present-day effective equation of state $w_{\rm eff,0}\simeq-0.702$ and exhibits the expected transition from accelerated expansion at late times toward decelerated expansion at higher redshifts. We further investigate the $Om(z)$ diagnostic and statefinder parameter. These results indicate that the logarithmic Herglotz model provides a consistent phenomenological description of the homogeneous and isotropic late-time expansion history, while allowing departures from the standard $\Lambda$CDM background.

In his thesis work, the first author studies neighborhoods of singular elliptic curves of type $I_0^*$ (Kodaira's classification) and provides the analytic classification. This is related via a double covering construction to the analytic invariants found by Touzet, Voronin and the second author for the analytic classification of neighborhoods of smooth elliptic curves. After recalling these constructions, we explain in this note how this construction is conjecturally related to the Riemann-Hilbert correspondence at infinity for Painlevé VI case when considering neighborhood of Okamoto divisor.

For an integer $k\ge 3$, let $w(3,k)$ be the least $n$ such that every red-blue coloring of $[n]$ contains either a nontrivial three-term arithmetic progression in blue or a nontrivial $k$-term arithmetic progression in red. A recent breakthrough of Green proved that $w(3,k)\ge k^{\Omega\left(\left(\frac{\log k}{\log\log k}\right)^{1/3}\right)}$ when $k$ is large, and Hunter later improved the bound to $w(3,k)\ge k^{\Omega\left(\frac{\log k}{\log\log k}\right)}$. On the other hand, Green remarked that it is reasonable to believe that $w(3,k)\le k^{O(\log k)}$. We prove that $w(3,k)\ge k^{\Omega(\log k)}$, which perhaps gives some evidence that $k^{\Theta(\log k)}$ is the correct order of magnitude.

We prove dimension-independent stability for the Courtade-Kumar inequality: a Boolean function $f:\{-1,1\}^n\to\{-1,1\}$ whose information is close to the dictator value is close in probability to a signed dictator. The correlation dependence is sharp in order near zero and, for increasing functions, also at the noiseless endpoint.

Activation probes are increasingly used to monitor LLMs in deployment. A probe is typically trained under one inference configuration, then used under whatever batch size and numerical precision the serving stack uses. Because common GPU kernels are not batch-invariant and floating-point formats round differently, the activations seen at deployment are not the ones the probe was trained on. We measure what that costs for Llama-3.1-8B, Qwen3-8B and Gemma-3-4B across batch sizes 4, 8 and 16 and float32, bfloat16 and float16, training 768 probes on one configuration, evaluating each on every other, and comparing verdicts example by example. Probes are stable, but aggregate accuracy is the wrong instrument for showing it: it understates how many verdicts change by a factor of two to nine. At the prompt, accuracy never moves by more than 0.47 percentage points across 1,392 transfers and only 0.076% of verdicts change; under float32 with only the batch size varied, none of 201,960 verdicts change. During decoding the flip rate rises to 2.8%, but rows whose realised tokens matched flip in only 0.12-0.15% of cases, while rows whose tokens diverged flip in 12.9%: the cause is the text, not the arithmetic. A bfloat16 batch-size change flips the first generated token for 2.1% of rows and leaves 25% on different tokens by token 20. Flips are symmetric, Cohen's kappa stays above 0.94, and AUROC moves by at most 0.05 points. Underneath, activations move about as much as the format's rounding: a bfloat16 batch-size change perturbs them by a median relative L2 of 1e-2, roughly 8x the float16 figure. Probes absorb this; the model's own next-token argmax does not. Robustness evaluations of activation monitors should report per-example agreement rather than aggregate accuracy, separate representational noise from input change, and state the serving configuration.

We describe our entry to the LoViF 2026 AIGC Image Compression Challenge, a benchmark for ultra-low-bitrate coding of AI-generated images under a strict global rate budget of 0.025 bits per pixel (BPP). Generated imagery poses a distinct challenge for compression: it frequently contains rendered typography, synthetic edges, repeated motifs, UI-like layout, and stylized micro-texture that conventional distortion-oriented codecs erase at this rate, while unconstrained generative decoders can restore plausible-looking detail that no longer matches the source geometry or symbols. We treat this as a rate-perception allocation problem. Our system fine-tunes four rate-specialized checkpoints of the AEIC one-step diffusion codec, generates per-image candidates from all four, including one latent refined through encoder-side test-time optimization (TTO) with periodic entropy-conditioning refresh, entropy-codes every candidate with practical rANS coding, and selects exactly one bitstream per image with an exact multiple-choice knapsack solved over true coded file sizes. A fixed, zero-additional-bit residual restoration network is applied at decode time. Every submitted bitstream is independently decodable by the shipped decoder, which uses no source image or external side information. We report the full pipeline, an ablation history spanning 77 logged experiments, and a set of negative results, including why PSNR could not be pushed to parity with rate-distortion-oriented competitors under this architecture, useful to future participants. This is a challenge report: our entry scored 31.527739 (PSNR 27.02 dB, MS-SSIM 0.9176, LPIPS 0.0778, DISTS 0.0390 at 0.02495 BPP), the second-best DISTS on the leaderboard, ranking 5th on the final test-phase leaderboard announced August 4, 2026.

We prove the Wang transform inequality in every order: if $n$ is a positive integer, $A$ is a nonnegative $n\times n$ matrix whose row and column sums equal one, and $J_n$ has every entry $1/n$, then \[ \operatorname{per}A\ge \operatorname{per}\!\left(\frac{nJ_n+A}{n+1}\right). \] Equality holds exactly when $A=J_n$. We also obtain an explicit positive quadratic gap for every $n\ge2$.

Artificial Intelligence (AI), particularly GenAI, is becoming an increasingly important part of modern work. In industrial settings, AI can support decision-making, automate routine activities, assist humans, and improve productivity. However, successful AI adoption depends on more than what the technology can do. It also depends on how people experience and work with it. This raises an important question: how should human-AI collaboration be designed so that it works well for both people and organisations? This white paper addresses that question by presenting a practical framework for designing human-AI collaboration. The framework considers the human, the AI system, the task, the organisation, and the wider societal environment. It explains what effective collaboration looks like, what conditions influence it, what requirements should be met, and what design decisions organisations should consider. The report also includes a human-AI collaborative assembly system with cobot use case to demonstrate how the framework can be applied in practice. The use case shows how design requirements can be translated into specific collaboration features and evaluated through a case study. The aim of this white paper is to provide a clear and practical guide for designing human-AI collaboration that is effective, human-centred, and responsible.

While Vision-Language-Action (VLA) models perform strongly on manipulation tasks, their responses to invalid task premises remain underexplored. Existing evaluations of premise conflicts often focus on terminal task outcomes, yet task failure alone cannot distinguish behavioral disengagement from continued pursuit followed by an execution error. We call the latter pattern Failed Persistence. To study this phenomenon, we introduce ConflictVLA-Bench, which pairs conflict rollouts with premise-consistent reference rollouts and evaluates both outcomes and execution processes. Built on LIBERO, the benchmark contains 2,826 prompt-conditioned conflict tasks spanning four conflict families, four structural configurations, and two prompt conditions. Across all eight VLAs, invalid premises reduce original goal completion by at least 17.3 percentage points, with the reduction reaching 56.2 percentage points for OpenVLA. Crucially, even when models succeed on premise-consistent tasks and fail on their matched conflict tasks, they often continue to approach the original targets, retain early trajectory structure, and show limited action magnitude suppression. Failed Persistence therefore recurs across the evaluated models. Explicit premise checking does not consistently produce selective and coordinated behavioral changes. These findings show that terminal failure alone establishes neither behavioral disengagement nor refusal and that outcomes alone are insufficient for VLA evaluation. Experimental data and additional details are available on the project page: https://github.com/EmbodiedAISurvey/ConflictVLA-Bench

In this work, we investigate the unstable circular orbits of spinning test particles and the associated classical orbital chaos bound in a hairy black hole spacetime admitting a double photon sphere structure. This background supports two distinct branches of unstable circular orbits, referred to as the inner and outer branches. We find that, within the parameter ranges considered, the inner branch always satisfies the chaos bound, whereas violations can occur on the outer branch. This behavior contrasts with many previous studies focusing primarily on near-horizon violations of the chaos bound and reveals a clear dependence of the bound on the orbital branch. Our analysis further shows that the total angular momentum of the particle has a pronounced effect on the local orbital instability and can enlarge the bound-violating region on the outer branch. Meanwhile, the deformation parameter of the background geometry regulates the validity of the bound by modifying both the orbital Lyapunov exponent and the black hole surface gravity. By contrast, the effect of the particle spin is comparatively weak and mainly shifts the existing unstable circular orbits and the corresponding critical boundaries through the spin-curvature coupling.

Crystal plasticity (CP) simulations predict the mechanical behavior of polycrystalline metals, yet their routine use is hindered by the manual effort of configuring heterogeneous tools, orchestrating multi-step data pipelines, and calibrating constitutive parameters against experiments. These bottlenecks impede productivity in systematic parameter studies, motivating interest in automated workflows. This study presents CP-Agent, a harness-engineered LLM-based agent that autonomously executes complete CP modeling workflows from natural-language tasks. Operating under the ReAct paradigm, the agent reasons about tool selection and sequencing while delegating numerical search to established optimizers. The harness comprises a minimal system prompt, typed tool definitions, a dispatcher, and a safety-bounded iteration loop, encoding domain knowledge through tool schemas rather than hard-coded logic. CP-Agent is demonstrated on four case studies: calibrating four slip parameters of additively manufactured stainless steel 316L against tensile data; validating the workflow against published copper benchmarks, reproducing stress-strain and texture evolution; recovering the initial crystallographic texture of copper, where the agent correctly identifies a diffuse initial texture; and reproducing the multi-pass rolling texture evolution of a Mg-Zn-Ca alloy, where the agent chains five deformation passes and recovers the experimentally observed weakened, split basal texture. In all cases, the agent inferred the correct execution sequence from the task statement, robustly across repeated runs, and delivered physically interpretable results. This work establishes harness engineering as a systematic approach to automating CP modeling workflows while maintaining physical interpretability and auditability through visible reasoning traces.

In this work, we explore MammoClaw, a training-free agent framework that leverages frozen MLLMs for mammography analysis. To support agentic investigation, we equip the agent with lightweight mammography-specific tools for targeted image analysis, including ROI, paired-view, and contralateral-breast examination. MammoClaw iteratively gathers evidence through these tools, while skill evolution enables non-parametric adaptation by transforming failed trajectories into reusable guidance for later runs. We evaluate the framework on BI-RADS assessment and breast density estimation tasks. In our experiments, we find that tools alone do not reliably improve performance, whereas evolved skills can improve tool-use behavior and performance in some settings. Beyond these results, MammoClaw enables transparent inspection of evidence acquisition, tool interactions, and failure modes, facilitating the analysis and auditing of agent behavior. We view this work as an exploratory study of training-free, self-evolving agentic approaches for mammography and hope it provides a concrete starting point for future work on mammography-specific tools and self-evolution mechanisms. We release our code at https://krishnakanthnakka.github.io/mammoclaw.

Progress in 3D CT report generation is usually sought in increasingly sophisticated architectures and larger pools of training data. We find instead that a 3D CT report generator already holds what its report leaves out, and loses it when the hidden state becomes tokens. Over the 18 CT-RATE abnormalities, this hidden-to-report surfacing gap is reflected by a drop in macro AUROC from 0.848 in the hidden states to 0.739 in the generated report. We propose SelfCue based on contrastive decoding. It promotes what the hidden state already supports and suppresses what it does not. It raises clinical efficacy F1 to 0.481 and the LLM-judged GREEN score to 0.510. Distilling that behaviour into the weights gives SelfCue-KD, a student that keeps most of the gain, needs nothing extra at inference, and drops into any pipeline already serving the baseline. Code is available at https://github.com/renjie-liang/SelfCue-CT.

Representation autoencoders (RAEs) reuse features from a pretrained visual encoder as reconstruction and diffusion latents, integrating strong visual representations into image generation. However, RAEs still need to decide which encoder layers form the shared latent space for the generator and pixel decoder. This choice involves a trade-off: shallower layers tend to preserve fine pixel details better, while deeper layers tend to yield better generation metrics. A fixed heuristic layer fusion therefore couples two stages that benefit from different information. We introduce FuseReg, which replaces heuristic feature selection with training over random subsets of encoder layers. We theoretically analyze the underlying mechanism: subset sampling preserves the full-layer latent mean in expectation while explicitly penalizing sensitivity to cross-layer disagreement. On ImageNet-256 with DINOv3-L, a single FuseReg decoder reconstructs from full, sparse, and single-layer fusions without retraining, achieving higher PSNR than decoders specialized to fixed fusions. This flexibility also benefits generation: decoder replacement alone reduces unguided gFID by 27% with an unchanged RAEv2 DiT-XL generator. Applying FuseReg to both stages also reduces unguided gFID by 29% on DiT-Base. The reconstruction and generation benefits also extend to other encoder families. FuseReg narrows the reconstruction-generation gap without additional training cost or architectural changes.

Reasoning models often generate very long reasoning traces, making inference computationally expensive. Existing approaches typically improve efficiency either through inference-time early-stopping mechanisms or by explicitly encouraging shorter reasoning during training, for example through reinforcement learning with length penalties. We show that substantial efficiency gains can instead emerge from a different kind of supervision: confidence. Using a self-supervised procedure, we fine-tune reasoning models to predict their confidence in the answer at intermediate points along their own reasoning trajectories using only 600 training problems. Confidence is used only as a training target: the loss contains no objective for reasoning length, efficiency, or stopping. At inference, the fine-tuned models use the standard generation procedure, with no confidence elicitation or early-stopping mechanism. Despite this, self-supervised confidence fine-tuning makes reasoning more efficient, reducing generated tokens by up to 25% at matched accuracy across Gemma, Qwen, Nemotron, and GPT-OSS models on mathematical, scientific, and coding reasoning benchmarks, with efficiency gains comparable to methods that explicitly optimize for shorter reasoning. Analysis of reasoning episodes further shows that confidence supervision largely preserves the base models' high-level reasoning composition rather than selectively suppressing particular behaviors. Our results suggest that efficient reasoning may emerge as a downstream consequence of learning metacognitive signals, without being directly optimized.

Skyrmions experience a sideways gyroscopic force when they move. Bound molecules can cancel this force; we call them gyroscopically neutral. We show that the quantum states available on each site of a magnet select the composition needed for cancellation. In two models of $SU(3)$ magnetism, this composition changes from a pair to a trimer. Both models assign exactly the same energy to every configuration of their common classical order parameter, isolating the effect of the local quantum states. Neutral molecules can also unwind when additional local states are allowed. Mean field calculations nevertheless find both compositions bound and locally stable, with the neutral trimer surviving sampled interaction changes. Comparing their motion reveals the changed cancellation rule, and prepared neutral molecules translate over short distances with little deformation.

We develop a general construction of homogeneous solutions to the Bernoulli free boundary problem, as well as general extremal domains on the sphere, from isoparametric foliations of the sphere. Our construction produces rich families of infinitely many new examples with sophisticated topologies connected to minimal surfaces of the sphere by smooth families of interpolating capillary surfaces, which include novel singularity models in low dimensions and recover most known homogeneous one-phase solutions. We also introduce a geometric representation for every such homogeneous map and establish a geometric rigidity theorem: for every prescribed isoparametric foliation, and in particular for cohomogeneity-one subgroups of $O(n)$, its radial geometry uniquely determines the corresponding homogeneous solution. Finally, we study the density of the constructed solutions and conjecture that the lowest density among non-flat cones in a given dimension $n \geq 5$ is attained by an $O(k) \times O(n-k)$-invariant cone.

Isoelectronic compounds are generally expected to exhibit related electronic, structural, and magnetic behavior. Chromium monopnictides CrX (X= Sb, As, P), however, display a striking evolution from high-temperature altermagnetism in NiAs-type CrSb, through double-helical antiferromagnetism in MnP-type CrAs, to the absence of resolved long-range magnetic order in MnP-type CrP. Here, combining first-principles calculations in the non-collinear Curie-paramagnetic state, structural analysis, magnetic phase-diagram calculations, and exchange-parameter extraction, we establish a unified microscopic framework for this evolution. All three compounds share a formal high-spin $d^5$ configuration, nominally Cr$^{1+}$, coupled to itinerant states of predominantly pnictogen character. From Sb to P, chemical pressure enhances ligand itinerancy and drives the structural evolution from the high-symmetry NiAs lattice to increasingly distorted MnP-type networks. The accompanying reconstruction of competing Cr-Cr exchanges favors A-type antiferromagnetism in CrSb and double-helical order in CrAs, while placing CrP near a frustrated regime with closely competing magnetic tendencies. Our results establish chemical-pressure-driven exchange reconstruction as a general mechanism through which a common spin-fermion electronic structure can generate contrasting magnetic phases in strongly correlated materials.

We verify that the stable minimal hypersurfaces with prescribed singular sets constructed by Leon Simon are area-minimizing. Thus, Simon's work settles the existence of area-minimizing hypersurfaces with fractal singular sets.

We give a gap-free $(\epsilon,\delta)$-differentially private algorithm for the principal component analysis (PCA) problem with Gaussian data. The algorithm is based on a private variant of the power iteration method, and it is computationally efficient.

We study the Galois group $G_f$ of a random polynomial $f$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$. We use a Euclidean height adapted to this relation. Our main result is an asymptotic theorem of van der Waerden–Bhargava type: for fixed $b \neq 0$ and $n \geq 4$, the number of polynomials of height at most $H$ whose Galois group is not the full hyperoctahedral group $S_2 \wr S_n$ is an explicit constant times $H^n\log H$, with an error of order $H^n$. We determine the dependence of the leading constant on $b$; the leading-order group $G_1$ is of index $2$. This paper is a sequel to a recent paper by Anderson, Bertelli, and the author addressing reciprocal polynomials (i.e. the case $b = 1$).

Recent literature has shown a strong connection between optimization and sampling. We develop the corresponding first-order theory for diffusion models. First, the SDE-based reverse-time flows of overdamped and underdamped Langevin diffusions contract relative Fisher divergences at explicit exponential rates whenever the stationary potential of the forward process is strongly convex—a condition on the noising process one chooses, not on the data. This is a unique advantage of SDE-based reverse diffusion, absent in the reverse process based on ODEs. Second, we incorporate discretization and establish averaged first-order stationarity bounds—the sampling analog of averaged gradient-norm guarantees in nonconvex optimization—for samplers of both overdamped and underdamped diffusion models. As in nonconvex optimization, the convexity-free certificate is local: it guarantees score consistency, not global mode weights.

We present the implementation of a multidimensional general-relativistic Boltzmann solver for neutrino transport using the finite volume method. We extend the general-relativistic magnetohydrodynamics solver Gmunu to discretize the full $6$D phase space. We discretize the momentum space in spherical coordinates in the comoving frame, and the position space in the lab frame in Cartesian, cylindrical or spherical coordinates. As in the M1 scheme, we expand the interaction kernels up to first order in a Legendre series. We present a discretization scheme that ensures consistency between the number-conservative and non-conservative formulations in empty flat spacetime, multiple dimensions and coordinate systems, conserves energy to $\sim 1\%$ with $20$ energy bins in 1D tests. We discuss optimizations of the implicit solver for stiff source terms to minimize its computational cost. In particular, we introduce a method that leverages the Legendre expansion of the kernels to reduce the dimensionality of the problem. In a 1D core-collapse supernova snapshot, our method yields a speed-up factor of $300$ compared to a full-matrix LU method for $14$ angular bins when including energy- and species-coupling interactions. Finally, we validate our implementation on standard test cases and report quadratic convergence in space, energy and propagation angles. We compare our solver to the M1 scheme in simplified 1D configurations and find differences of about $10\%$ in the free-streaming luminosities in relaxation test cases, which we attribute primarily to the M1 closure relation. The average energies, on the other hand, agree between the two methods. We compare both methods on a core-collapse supernova test case and find an overall good agreement in the fluid variables' profiles at the time of core bounce for $20$ energy bins.