Papers

Neural networks routinely develop representations in which individual neurons contribute to multiple distinct functions, leading to the widespread view that such networks lack modular organization. We challenge this conclusion by introducing a formally nested hierarchy of modularity: unit-level, which requires distinct computations to occupy disjoint neurons; orthogonal, which asks whether a rotation reveals functionally independent subspaces; and linear, which asks whether a bounded linear map can separate task-relevant computations into independent subspaces. We develop a unified framework for learning decompositions at each level and evaluating them causally through selective ablation. Across vision models, language models and multi-task Recurrent Neural Networks (RNNs), we find that networks which lack unit-level modularity nonetheless often exhibit clear functional modularity at the subspace level. In vision and language models, content versus style and syntax versus semantics respectively dissociate under subspace ablation despite sharing overlapping populations of neurons. These learned subspaces also align differentially with brain data, with content and style subspaces predicting higher and lower visual cortex respectively, and semantic subspaces outperforming syntactic ones in predicting language cortex activity. In RNNs trained on multiple cognitive tasks and previously shown to lack modular structure, our decomposition recovers sharp double dissociations between tasks. The absence of unit-level modularity frequently coexists with clear subspace-level modularity across domains, enabling targeted intervention on specific computations even in networks where no individual neuron is specialized, with broad implications for mechanistic interpretability and for understanding functional organization in both biological and artificial neural systems.

Emotion classifiers are routinely applied beyond their training data, on the assumption that same-named labels such as fear or surprise mean the same thing across resources. We test this assumption for five labels shared by GoEmotions, XED, and BRIGHTER, comparing models on identical inputs and under cross-corpus transfer. On the same texts, one fear classifier flags 71% as fearful and another 8%, yet the two largely agree on how the texts rank. Across 30 resource-label comparisons, training-label prevalence explains about half of this kind of scale mismatch, and a fitted intercept removes most of it. What recalibration cannot repair is larger: transferred models lose 55% of native discrimination in aggregate, agreement in ranking between classifiers is only weakly related to that loss, and for surprise the resources disagree about which texts express the emotion. The choice of training resource can therefore change conclusions: on two external tweet collections, BRIGHTER- and XED-trained classifiers give opposite answers about which shows more surprise, even after rank normalization, with intervals excluding zero in both directions. A shared emotion label name does not establish an interchangeable measurement target.

By performing complete enumeration of all configurations of magnetic moments, we have determined the ground-state phase diagram of the effective model with antiferromagnetic nearest-neighbor interactions $J_1$ and ferromagnetic next-nearest-neighbor interactions $J_2$ on the 1/1 periodic approximant to icosahedral quasicrystal. Depending on the ratio of $J_2/J_1$ and the magnetic anisotropy arising from the crystalline electric field, various non-collinear ferrimagnetic orders and antiferromagnetic orders are shown to be stabilized, whose magnetic space groups are identified as $R\bar{3}$, $C2'/m'$, $Pm'm'2$ and $I_{\rm P}m'\bar{3}'$. We have revealed the topological as well as magnetic property of each ground state with degeneracy, which is useful to understand experimental observations of magnetic structures and domains in the rare-earth-based 1/1 approximants.

We prove comparison results about $\infty$-categories of derived commutative rings with extra algebraic structure. For example, we show that several notions of derived $\delta$-rings agree and obtain similar results for derived $\lambda$-rings, derived binomial rings, and derived Boolean rings. Our comparison results apply more generally to modules over a specific kind of monad on derived rings that we call derived plethory.

In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion $(-\Delta)^s$ $(0<s<1)$, which was considered in \cite{CDN26} as a model for species spreading, where the density function $u(t,x)$ satisfies the fractional Fisher-KPP equation $u_t+(-\Delta)^su=f(u)$ over a one-dimensional spatial domain $(g(t), h(t))$, representing the expanding population range, with $x=g(t)$ and $x=h(t)$ the free boundaries. It was shown in \cite{CDN26} that the model is well-posed, and its long-time dynamics exhibits a spreading-vanishing dichotomy. We show in this paper that, when spreading occurs, \[ \begin{cases} \lim_{t\to\infty}\frac{h(t)}{t^{1/(2s)}} =\lim_{t\to\infty}\frac{-g(t)}{t^{1/(2s)}} =\left(\frac{2^{1-2s}\mu}{1-2s}\right)^{1/(2s)} & \mbox{ if $s\in(0,1/2)$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t\ln t} =\lim_{t\to\infty}\frac{-g(t)}{t\ln t} =\mu & \mbox{ if $s=1/2$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t} =\lim_{t\to\infty}\frac{-g(t)}{t} =c_0(s) & \mbox{ if $s\in(1/2,1)$}, \end{cases} \] where $c_0(s)\in (0,\infty)$ is uniquely determined by the associated semi-wave problem. The level sets of $u(t,x)$ propagate with the same rates as the free boundaries. The proof uses delicate upper and lower solutions constructed from a smooth profile function in the case $s\in (0, 1/2]$, and from the semi-wave profile when $s\in (1/2, 1)$.

Knowledge distillation aims to train a smaller student model to mimic a larger teacher model for resource-constrained environments. However, the process of distillation is typically data-intensive and computationally expensive, particularly for generative tasks. Few-shot distillation offers an alternative by transferring teacher knowledge using limited training samples. However, we observe that few-shot distillation performance is heavily dependent on the choice of these training samples. In fact, we demonstrate that using more uncertain samples near the teacher's decision boundary can significantly deteriorate distillation performance. Motivated by this observation, we propose Counterfactual-guided Few-shot Distillation in Open-book Question Answering (COFFEQA), a novel strategy that aptly uses counterfactuals to provide additional supervision around challenging factual samples. In the generative QA setting, we define a counterfactual as a minimally modified context that induces the teacher to change its response to an alternative answer. We provide geometric insights on how factual-counterfactual pairs can constrain the deviation between the teacher's and student's decision boundaries. We further introduce a new metric, Context Following Difficulty (CFD) that further validates counterfactuals prior to distillation based on agreement with the intended alternative answer and dependence on the modified context. COFFEQA uses CFD-validated factual-counterfactual pairs for few-shot distillation. Experiments across five in-domain and three out-of-domain QA benchmarks using T5 and Qwen2.5 families show that COFFEQA outperforms factual-only distillation baselines under the same few-shot budget. For instance, on TriviaQA, COFFEQA improves over the factual-only baseline by 8.77 average EM points and 9.98 average F1 points under the T5 setting with 32 few-shot pairs.

For every $d\ge4$, every prescribed connectivity $r\ge d$, and every $\varepsilon>0$, we construct a simplicial $d$-polytope whose graph has connectivity exactly $r$, is nontraceable, and has toughness less than $\varepsilon$. Thus the classical theorems of Whitney (1931) and Tutte (1956), which guarantee hamiltonicity under sufficiently high connectivity in dimension three, have no higher-dimensional counterpart. This answers a question posed by Bojan Mohar in 2001 on whether sufficiently high connectivity guarantees hamiltonicity of $4$-polytopal graphs and the connectivity questions raised by Ikegami, Maezawa, and Zamfirescu in 2021. For fixed $d\ge4$, $r\ge d$, and integer $k\ge2$, the examples can be chosen to have no closed spanning walk with at most $k$ visits per vertex, no spanning tree with degrees at most $k$, and nontraceable prisms. For fixed $d$ and $r$, we obtain graphs on $n$ vertices whose longest paths and cycles contain $O(\sqrt n)$ vertices. Consequently, the family of simplicial $d$-polytopes with connectivity exactly $r$ has Grünbaum–Walther shortness coefficient zero and shortness exponent at most $1/2$.

UAV thermal-to-satellite geo-localization localizes UAVs by matching real-time thermal images against georeferenced satellite images, enabling autonomous navigation in GNSS-denied and low-visibility conditions. However, substantial cross-modal disparities in color, brightness and viewpoint hinder conventional appearance-based methods from establishing reliable correspondences. While existing approaches predominantly address global modality shifts, they overlook the invariant scene structures inherently shared between thermal and satellite views. To bridge this gap, we propose SAMF, a Structure-Aware Matching Framework that explicitly exploits stable cross-modal structural cues to improve UAV thermal geolocalization. Specifically, the Directional Structure Enhancement Module (DSEM) strengthens robust structural representations by capturing directional and low-frequency spatial patterns, and the Modality-Aware Affine Remapping Module (MARM) subsequently applies affine correction to these enhanced features, further promoting cross-modal alignment. Additionally, we introduce the Thermal-UAV-Paired benchmark derived from the Thermal-UAV dataset. Extensive experiments on both the Boson-nighttime and Thermal-UAV-Paired datasets show that SAMF achieves state-of-the-art performance. Ablation studies further validate the effectiveness and synergistic complementarity of DSEM and MARM. The code and dataset are released at: https://github.com/YuxinWang879/SAMF.git.

Generative AI is increasingly studied as a tool for supporting lesson authoring; the creation and adaptation of classroom materials, yet little is known about how it should fit into secondary computing teachers' subject-specific practices. We conducted two complementary qualitative studies with eight secondary computing teachers: focus groups examining existing authoring practices and a think-aloud study using LessonBuilder, a structured AI-supported lesson authoring system, as a research probe. We found that teachers' instructional intentions developed through creating, inspecting, and adapting materials, while teachers selectively delegated tasks to AI and retained responsibility for judging classroom readiness. The think-aloud sessions further showed how examining AI-generated materials prompted teachers to clarify and refine their intentions, making authoring an ongoing negotiation between what teachers intended and what the AI produced. In the context of computing lessons, classroom readiness also involved considering how programming materials and supporting guidance matched students' understanding and intended learning goals. We synthesize these findings into a conceptual model explaining how teachers coordinate instructional intent, AI assistance, and judgments of classroom readiness during lesson authoring. We derive design recommendations for supporting this coordination while preserving teacher control and addressing computing-specific authoring needs.

Sobolev gradient flows for the Gross–Pitaevskii ground state problem are Riemannian gradient descent methods on the sphere under a metric induced by a discrete elliptic operator. We analyze their convergence through the condition number of the Riemannian Hessian at the ground state. Bounds of the Rayleigh quotient of the Riemannian Hessian give a uniform analysis of these methods for different metrics. The uniform analysis applies to the classical metrics, such as the shifted Laplacian and the energy-adaptive metrics. Inspired by this analysis, we also consider three new metrics. One of them is a Thomas–Fermi metric, whose condition number is bounded uniformly in the interaction strength for the one-dimensional harmonic trap and whose iteration count does not grow in the Thomas–Fermi regime in the numerical tests. The performance suggested by our analysis is consistent with three-dimensional numerical tests with up to one billion unknowns, for both Riemannian gradient descent and Riemannian conjugate gradient methods, whose practical performance is also related to the Hessian condition number. The Thomas–Fermi flow needs the fewest iterations, while the metrics inverted exactly by tensor-product solvers are the fastest in computational time on one GPU card.

Pretrained vision-language models (VLMs) already exhibit strong visual, spatial and reasoning capabilities relevant to robotic manipulation. The key challenge in turning pretrained VLMs into generalist manipulation policies is to expose these capabilities through an appropriate action interface. Vision-language-action (VLA) models provide a unified trainable policy, but typically introduce learned action tokens, action heads, or action experts whose semantics must be learned from robot data pretraining. In contrast, modular foundation-model-based manipulation methods can exploit pretrained VLM capabilities for zero-shot manipulation, but rely on external planning and control scaffolds that are not end-to-end trainable as a unified continuous-action policy. In this work, we introduce EASG, an Explicit Action-Space Grounding framework that bridges these two paradigms. By projecting a tool center point (TCP)-centered action frame into the wrist camera observation and expressing actions in the form of VLM's native textual generation interface, EASG makes the semantics of a continuous robot action space directly observable in the VLM's perceptual context. This allows pretrained VLMs to directly generate executable robot actions using their existing visual-spatial reasoning capabilities, without robot-action pretraining or additional action modules. Besides, the same action interface supports direct fine-tuning with robot trajectories, allowing the models to continuously improve beyond their manipulation capabilities unleashed by EASG. Overall, EASG enables zero-shot manipulation and generalization across object positions, categories, and robot embodiments, while supporting effective fine-tuning in the same continuous action space. These results demonstrate that, given an appropriate action interface, pretrained VLMs can themselves serve as native action models for robotic manipulation.

Given an integer $k \geq 2$ and a prime $p$ large enough depending on $k$, we show that, if $A \subseteq \mathbb F^n_p$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \\ y_2^2 \\ \vdots \\ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \\ y_2^{k-1} \\ \vdots \\ y_n^{k-1} \end{pmatrix}, \end{align*} where $x \in \mathbb F^n_p$ and $y \in \{0,2, 3, \ldots, M_k\}^n \backslash \{\vec{0} \}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{(\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\geq 4$, the best previously known bound came from the density Hales–Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.

Reasoning post-training draws on offline demonstrations and policy-generated rollouts, yet conventional supervised fine-tuning (SFT) followed by reinforcement learning (RL) separates their objectives and limits continued reuse of collected responses. We propose Cross-Behavior Policy Optimization (CBPO), which makes these sources compatible with one regression loss through a fixed-reference target. Because the generating policy does not enter this target, positive and negative offline completions, reference responses, and current and historical rollouts can share a common reward-labeled record format and replay buffer. As new responses enter the buffer, optimization retains the same regression objective without coordinating separate SFT and RL objectives or requiring stored behavior-policy probabilities, importance weighting, or a learned critic. Restricting the available sources gives offline, on-policy, and off-policy variants, while CBPO-Unified combines them. Across two model scales and four mathematical reasoning benchmarks, CBPO delivers leading avg@32 and pass@32 performance across benchmarks and data regimes. On Qwen3-1.7B, CBPO-Unified improves benchmark-averaged avg@32 and pass@32 over SFT DAPO by 2.33 and 4.44 percentage points, respectively. In the replay ablation, retaining historical rollouts improves these metrics by 1.87 and 3.72 percentage points over the no-history configuration.

Determinantal point processes (DPPs) on a finite set are probability models that exhibit repulsion where inter-item similarities are encoded by a positive definite kernel matrix. Scaling DPP learning to large datasets typically requires imposing structural constraints on the kernel matrix, such as a low-rank-plus-diagonal form. However, structural assumptions introduce a severe trade-off between expressive power and computational scalability. Our study addresses this challenge by (a) modeling the full-rank interactions via a nonlinear kernel function and latent features, (b) designing a surrogate function for the maximum likelihood estimator, and (c) adopting stochastic optimization using random features of the nonlinear kernel. Numerical experiments on real-world datasets demonstrate that our method significantly improves this trade-off, achieving better generalization performance with remarkable parameter efficiency.

Outcome-supervised group-relative policy optimization (GRPO) assigns the same advantage to every token in an LLM agent trajectory, leaving useful and unhelpful decisions undifferentiated. Hindsight credit assignment (HCA) can refine this feedback, but its estimates rely on trajectories containing informative alternatives at consequential histories. We introduce TreeHCA, which couples hindsight-guided exploration with hindsight credit assignment. TreeHCA uses changes in reference-answer likelihood to revisit intermediate histories and sample alternative continuations, forming a tree of reasoning and tool interactions. Because this exploration changes the distribution of sampled continuations, we derive the corresponding inverse-hindsight importance correction, including an exact finite-pool identity. We use this structure in a normalized tree backup that propagates terminal outcomes into turn-level advantages. This shared hindsight structure is TreeHCA's central design principle, linking informative exploration with hindsight-guided credit assignment. Across seven question-answering benchmarks and three Qwen backbones, TreeHCA outperforms all baselines in aggregate exact match, validating the effectiveness of paired exploration and credit assignment for multi-turn LLM agent training. Our code is available at github.com/TheBlackCat22/TreeHCA.

Pseudospectral methods are a powerful framework for spacecraft trajectory optimisation that rely on fixed, problem-independent collocation nodes such as Legendre-Gauss-Lobatto (LGL) nodes. These nodes are located according to polynomial root distributions, which can result in insufficient resolution in regions of high dynamical activity, such as those encountered during collision avoidance manoeuvres. This paper presents an adaptive node distribution strategy for Birkhoff pseudospectral trajectory optimisation, where the collocation nodes are placed according to a probability density function derived from the thrust magnitude profile of the spacecraft. By concentrating nodes in regions of high dynamical variation, the proposed method simultaneously achieves feasibility preservation and improved convergence with fewer nodes in a collision avoidance case study when compared to the universal Birkhoff method, with Model Predictive Control serving as an independent validation comparator. The case study highlights the practical consequences of inter-node constraint violations and validates the adaptive node scheme as a robust and efficient alternative for fuel-optimal spacecraft manoeuvre design.

We presented a program that automates the construction and solution of the cubic spline interpolation problem for a sequence of position-time waypoints, producing trajectories that are continuous in position, velocity, and acceleration and therefore avoid the infinite-acceleration problem inherent to piecewise-linear waypoint following. By assembling the interpolation and continuity conditions into a single linear system and solving directly for the spline coefficients, the program removes the need to manually derive per-segment equations for each new waypoint set. We illustrated the approach on free-form curve drawing and signature digitization, two tasks that use the same underlying machinery required for smooth robot and drone trajectory generation. Future work includes extending the implementation to multi-axis trajectories with explicit velocity and acceleration bounds, adopting the more computationally efficient tridiagonal solution method for larger waypoint sets, and evaluating the approach quantitatively against alternative trajectory generation methods.

We construct a geometric theory on the brane space $S^2M=C^\infty(S^2,M)$ associated to a gerbe with connection on $M$. Normalized logarithmic holonomy around the latitude circles defines an average holonomy line bundle on the brane space, whose connection and averaged flux give an equivariantly flat superconnection. We prove that the completed periodic exotic twisted cohomology defined by this superconnection localizes to the twisted cohomology of $M$. This brane space localization happens in two stages: circle localization lands on the free path space of $M$, and contracting the paths, after a correction of the twisting form, brings us back to $M$. Parallel transport along the meridians lets us average gerbe module connections, after which the circular iterated-integral construction applies. The resulting twisted brane Bismut-Chern forms restrict to the twisted Chern forms on $M$; their variation is described by twisted brane Bismut-Chern-Simons forms. For a T-dual pair, transporting the classical Hori morphisms through localization yields graded genus-0 T-duality on brane cohomology. The composite of each morphism with its dual is minus the Euler operator in the level variable, so these morphisms are isomorphisms at every nonzero level. At level one these morphisms are compatible with the brane character and twisted $K$-theory T-duality.

This paper develops a smoothed generalized method of moments (GMM) framework for panel quantile models with individual-specific fixed-effects and endogenous regressors. By smoothing the indicator in the quantile moment conditions and profiling out the individual effects through smoothed individual quantile restrictions, the proposed estimator can be implemented using standard GMM algorithms. We establish consistency, asymptotic normality, and feasible inference under weak time dependence as both panel dimensions diverge. This dependence structure accommodates dynamic panel specifications, serially dependent disturbances, lagged instruments, and within-individual heteroskedasticity and autocorrelation consistent inference. Monte Carlo simulations show that the estimator is nearly unbiased under endogeneity and yields generally accurate inference while remaining computationally tractable. An application to OECD inflation dynamics suggests asymmetric inflation propagation: under the structural IVQR interpretation, the response of current inflation to lagged inflation is substantially larger at the lower structural quantile than at the upper structural quantile.

Web agents increasingly browse, interpret, and act on web content on behalf of users. In many realistic workflows, however, the visual evidence they consume is not neutral: charts and dashboards can be misleading, shaping judgments without issuing any explicit malicious instruction. Unlike chart question answering, Web Agents treat charts as intermediate evidence whose misinterpretation can propagate into concrete downstream actions — a reliability risk that remains underexplored. We introduce MisVisAgentBench, the first paired executable benchmark for evaluating how misleading visualizations affect Web Agent execution. It comprises 280 task instances in matched clean–misleading pairs that differ only in chart evidence. We further provide a diagnostic framework that isolates misleading-induced failures from baseline task difficulty (Zhu et al., 2025) and captures execution-level effects such as extra browsing steps and decision hesitation. Across thirteen vision-language Web Agents, replacing clean charts with misleading counterparts reduces the average task success rate by more than 34 percentage points. Moreover, these failures predominantly follow the targeted misleading action branch, suggesting that deceptive charts systematically redirect rather than randomly disrupt agent decisions. These results reveal that visualization robustness is a critical yet understudied dimension of Web Agent safety and reliability.

Multi-agent systems are increasingly used to solve coding tasks by coordinating agents that share code and plans. These agents are often optimized with reinforcement learning (RL) using automated evaluators, but bypassing evaluation can earn reward without producing a genuine solution. Prior work has largely treated reward hacking as a single-agent failure. In this work, we show that a reward-hacking strategy elicited in one agent can propagate to others and affect system coordination. Across four open-weight backbones, coding RL elicits and amplifies reward hacking in two settings: explicit mechanism descriptions in the system prompt and mid-training that models upstream exposure to facts about the coding environment. Using post-RL agents from the mid-training setting as sources, we observe propagation through code and shared memory, and across roles through natural-language plans. Even agents based on small open-weight models induce reward hacking in stronger closed-weight targets. On hard tasks, propagation can intensify along a transmission chain after the original source leaves as the code acquires more persuasive justifications. We also theoretically analyze conditions under which reward hacking amplifies, persists, or decays across agent interactions. Beyond propagation, reward hacking can bias task routing toward reward-hacking agents and survive candidate aggregation. As potential mitigations, we evaluate code review by the coder or a separate verifier, both of which reduce delivered hacks. Together, these results establish reward hacking as a transmissible strategy whose effects extend beyond the agent in which it first emerges.

The precorrected-FFT has been used to rapidly evaluate $N$-body sums arising in electromagnetics and acoustics settings. It has, however, been historically limited to kernels that satisfy a Green's identity. In this work, we demonstrate that, using an improved version of the method of proxy points, the precorrected-FFT can be used to efficiently compute $N$-body sums with a wide variety of kernels. This algorithm requires no analytic information beyond pointwise access to the kernel. Our examples include kernels from non-local PDEs, Matern kernels, anisotropic kernels, and the Green's functions of high order PDEs. We also provide a detailed algorithm for choosing the proxy points and spreading grid, as implemented in the new pcfft package. We show these algorithmic choices result in favorable scaling when compared to other kernel-independent fast summation methods. Finally, we use the algorithm to accelerate a wide variety of integral equations solvers involving non-standard kernels, with applications in surface scattering, geophysics, and microbial community formation.

The shock response spectrum (SRS) characterizes the peak responses of single-degree-of-freedom systems to transient acceleration. Because the mapping from an acceleration time series (waveform) to its SRS is many-to-one, inverse synthesis seeks candidate waveforms consistent with a prescribed spectrum. We extend an earlier conditional variational autoencoder (CVAE) framework with a recurrent decoder that predicts multiscale wavelet coefficients and reconstructs waveforms through an inverse discrete wavelet transform. Individual models exhibit limited variability under latent sampling, motivating an ensemble of independently trained models spanning different mother wavelets and decomposition levels. Each member produces a separate candidate waveform for the same target SRS. Evaluation on four held-out benchmark datasets shows lower mean and median spectral errors for the illustrated wavelet model than for the earlier direct time-domain CVAE and a sum-of-decayed-sinusoids (SDS) baseline. A conditioning study examines frequency-grid and encoding choices. Comparisons with published inverse-SRS methods demonstrate superior performance across four holdout benchmark datasets (two operational shock datasets, one earthquake dataset, and one synthetic shock dataset) and competitive performance on synthetic targets reported by Brake. Single-model inference requires approximately 0.1-0.6 ms per waveform during batched inference compared with 5-8 s for SDS, excluding offline training. This low inference cost enables rapid generation of distinct SRS-compatible candidate waveforms across the ensemble, although their physical suitability requires assessment beyond spectral agreement.

Alon, Bishnoi, Das and Neri constructed sequences of strong blocking sets in $(k-1)$-dimension projective space $P({\bf F}_q, k-1)$ with sizes at most $Cqk$, where $k$ goes to the infinity and $C$ is a fixed positive constant not depending on $q$, in their paper published in Transactions AMS, vol. 277, pp. 5389-5410, 2024. This is equivalent to a sequence of minimal $[n_i,k_i]_q$ codes with positive rate $\frac{k_i}{n_i} \geq c$, where $k_i$ goes to the infinity and $c$ is a fixed positive constant only depending on $q$. They solved one of the main open problems on minimal codes. The Alon-Bishnoi-Das-Neri construction was based on asymptotically good sequences of algebraic geometric codes and expander graphs. However, the constant $C$ is large for small prime power $q$. In this paper, based on a simple approach using $(q-1)$-power intersecting algebraic geometric codes via concatenation to construct minimal codes, we construct explicit strong blocking sets in $(k-1)$-dimension projective space with sizes around $2(q^2-1)k$, equivalently, explicit minimal $[n,k]_q$ codes with length around $2(q^2-1)k$. This is an essential improvement on the previous result of Alon-Bishnoi-Das-Neri, when $q \leq 11$. For large $q$, strong blocking sets and minimal codes in our construction are worse than the previous construction of Alon-Bishnoi-Das-Neri. However, our approach is much simpler and does not depend on expander graphs.

We study classical reconstruction of an outgoing scalar-wave component from radial-shell observations. Starting from the noncompact-time Rindler metric, we construct Klein–Gordon evolution by closed quadratic forms and reduce the massless characteristic dynamics to translation in optical distance. For an initial optical interval of length $L$, a shell of width $d$, and recording time $T$, the optimal squared observability bounds are $A(T,L)=\min\{d,(T-L)_+\}$ and $B(T,L)=\min\{d,T\}$. At $T=L$, the Rindler observation map is injective with nonclosed range. Discrete shell observations form a spatial fusion frame precisely when their transported regions cover the initial interval; we compute optimal uniform-grid bounds and characterize robustness under profile erasures. Finite-resolution reconstruction separates localization error, approximation error and noise amplification. For fixed Schwarzschild mass and angular momentum, we compare the scalar-wave observation operator with its Rindler limit. Anchored finite-energy preparation permits arbitrary prescribed outgoing $L^2([-L,0])$ profiles without a zero-integral condition. The operator error is explicitly $O(e^{-2\kappa a})$ as an apparatus of fixed optical size is translated toward the horizon; whenever $T>L$, positive Rindler bounds transfer to Schwarzschild for all sufficiently large $a$. A positive Schwarzschild lower bound is impossible at or below the optical travel time, for every translation. These results concern classical observation and reconstruction, not quantum information or dissipative horizon evolution.

In this paper we introduce a class of non-flat Lorentzian manifolds, called Lorentzian torse-forming structure manifolds (briefly, LTFSMs). This class contains Lorentzian concircular structure manifolds (briefly, LCSMs), LSP-Sasakian manifolds, LP-Sasakian manifolds with coefficient $\alpha$, LP-$\alpha$-Sasakian manifolds, LP-Sasakian manifolds, and LP-$\alpha$-Kenmotsu manifolds. The interrelations among these various classes of manifolds are illustrated in Figures 1 and 2. We investigate several geometric properties of LTFSMs. In particular, we show that the expansion scalar, given by the divergence of $\xi$, is nonzero and nonconstant, and that the flow of $\xi$ is irrotational. In an LCSM, the Ricci operator commutes with the structure tensor $\phi$ [12]. However, it is shown that, in an LTFSM, the Ricci operator does not generally commute with $\phi$. If the gradient of the torse-forming scalar $\alpha$ is collinear with the torse-forming vector field $\xi$, then the Ricci operator commutes with $\phi$, and the manifold reduces to an LCSM. It is known that a conformally flat LCSM is quasi-Einstein and hence is a manifold of quasi-constant curvature. We show that a conformally flat LTFSM is a generalized quasi-Einstein manifold in the sense of Chaki [1] and is also a manifold of generalized quasi-constant curvature. Several curvature-related properties of LTFSMs are established, and their sectional curvature is determined. The existence of LTFSMs is demonstrated through several proper nontrivial examples. We also present examples that are simultaneously LCSMs and LTFSMs, examples of LTFMs characterizing twisted cubic spacetimes, and generalized Robertson-Walker (GRW) spacetimes characterizing LCSMs. Finally, we provide examples of LTFSMs in which the vector field $\xi$ is a unit timelike torse-forming vector field but is not concircular; see Examples 5.6–5.8.

In previous publications it has been shown that it is possible to calculate (in terms of zero and higher order elliptic integrals) the propagation time of a signal, emitted by a satellite, moving along a plane (or space-determined) Kepler orbit. 2. By means of a method of intersecting null cones, the propagation time was found for the case, when the signal is intercepted by a second satellite. The present paper has the purpose to generalize the above approach for satellites, exchanging signals and moving along arbitrary space trajectories. Previously, the important physical notions of space-time distance and geodesic distance (only positive and greater than the Euclidean distance) have been introduced. It will be shown that these notions can be introduced also for the case of general motion of the satellites, and more importantly, they have the same physical meaning. A generalization of the s. c. ranging equation for the case of a curved space-time has been proposed. It has been proved that the geodesic distances of propagation of the signal in the forward and in the backward directions are not equal. The physical reason is hidden in the non-inertial reference systems of motion of the two satellites. Thus, signal propagation in non-inertial systems is an irreversible process, meaning that the propagation times in the two directions is different. A mathematical model has been created for the forward and subsequent backward propagation of the signal between two moving satellites in a gravitational field. Such a general theory of inter-satellite communications is the first step towards constructing a modern theory for optimal propagation of signals in an integrated dense satellite communication network systems, such as the current 6G and future 7G systems.

We prove the conjecture of Prellberg and Stanton (2003) that $\frac{1-q}{(q^n;q)_m}\genfrac{[}{]}{0pt}{}{n}{m}_{q}$ has nonnegative power-series coefficients for $0<m<n$, except when $n$ is even and $m\in\{2,n-2\}$. We obtain a polynomial version by bounding a Möbius correction by first differences of Gaussian coefficients. The proof uses partitions and quadratic and cubic semi-invariants of binary forms. An extension of this bound proves the Reiner–Stanton–White conjecture on positivity of orbit polynomials for flags of subspaces.

We revisit self-interacting dark matter (SIDM), originally proposed to resolve small-scale CDM problems and drive late-time cosmic acceleration. Using latest DESI BAO, Pantheon+/DESY5 type Ia supernova, and cosmic chronometer Hubble data, we constrain three representative SIDM scenarios including creation and annihilation. We focus on the annihilation of dark matter particles in model SIDM1; our findings indicate the effective annihilation cross section is $\log_{10}(\kappa C_1)=-4.32$ (with an upper limit of $\log_{10}(\kappa C_1)=-3.90$). For the SIDM2 model, the SIDM fluid behaves similarly to a dark energy component with a mildly dynamical equation of state ($w_{\rm{SIDM}}=-0.82 \pm 0.04$). We also quantitatively discuss the dark matter self-interactions in a more general SIDM model (SIDM3), which reveals a strong correlation between the annihilation cross section and equation of state of dark matter. For the three SIDM models, the measured Hubble constant agrees with that derived from the recent Planck 2018 results; both are inconsistent with the results of the SH0ES collaboration, who report a significantly higher $H_0$ value. The Bayesian evidence indicates that SIDM2 and SIDM3 are more supported by the available observational data than the concordance standard $\Lambda$CDM. Furthermore, the cosmological constraints on the effective annihilation parameter can be translated into bounds on the velocity-averaged annihilation cross section and the self-interaction cross section per unit mass. While low-redshift data alone imply a relatively large cross section, the inclusion of cosmic microwave background distance priors significantly tightens the constraint at cluster-scale velocities, highlighting the necessity of a velocity-dependent cross section to reconcile cosmological and astrophysical observations.

Balanced Minimum Evolution (BME) reconstructs an unrooted binary phylogenetic tree from a distance matrix $D$ by minimizing the length $\sum_{i<j}D_{ij}w_{ij}$, which is linear in the weights $w_{ij}=2^{-d_T{ij}}$. We study the formulation of BME in these $\binom{n}{2}$ weight variables. Theoretical results. (i) For a fixed topology $T$, the Kraft equalities, the cut equalities $\{W[S]=\tfrac12\}$ of $T$, and the single convex constraint $\sum w_{ij} \log_2 w_{ij} \leq \tfrac{3}{2}-n$ admit exactly one point, the tree's weight vector. (ii) For every tree and every bipartition, the cut value $W_T[S]$ lies in $\{\tfrac{1}{2}\} \cup \{\tfrac{3}{4}\} \cup [\tfrac{7}{8}, \infty)$; the value $\tfrac{3}{4}$ is attained exactly by the $2(n-3)$ bipartitions created by one nearest-neighbour interchange; and the number of bipartitions with $W_T[S]\le1$ is $2n^2-10n+15$ for every binary topology. (iii) The crossing graph of nontrivial bipartitions has an edge clique cover with $O(n^4)$ cliques, and every edge clique cover needs $\Omega(n^2)$. (iv) The necessary conditions of Catanzaro et al. for a path-length matrix, without the four-point condition, are sufficient for $n \leq 11$ and first fail at $n=12$. Computational results. A compact branch-and-bound solver built on these results, which commits and forbids splits and imposes the convex constraint exactly, certifies optimality on benchmark instances with up to $n=20$ taxa within an hour. It is faster than the distance-indexed model of Catanzaro et al. on two of these instances and slower on the others; that model, whose root bound is tighter, certifies 13 of 14 benchmark instances with $n \leq 30$ within an hour in every seed. Although the compact solver does not improve on the computational state of the art overall, we explain why, and we discuss the open problems that our results raise.