AI for Science

As a natural sequel to the study of A-motivic cohomology initiated in "On the integral part of A-motivic cohomology", we develop a notion of regulator for rigid analytically trivial Anderson A-motives. In accordance with the conjectural picture over number fields, we define it as the morphism at the level of extension modules induced by the exactness of the Hodge-Pink realization functor. The purpose of this article is twofold: first, we prove a finiteness result for A-motivic cohomology; second, under a weight assumption, we show that the source and the target of the regulator have the same dimension. It came as a surprise to the author that the image of this regulator may fail to have full rank, thereby preventing an analogue of Beilinson's celebrated conjecture from holding in our setting.

We give an example of a unital C*-algebra $\mathbf{A}$ with a computable presentation and for which neither $K_0(\mathbf{A})$ nor $K_1(\mathbf{A})$ has a computable presentation.

Ultrafast online learning is essential for high-frequency systems, such as controls for quantum computing and nuclear fusion, where adaptation must occur on sub-microsecond timescales. Meeting these requirements demands low-latency, fixed-precision computation under strict memory constraints, a regime in which conventional Multi-Layer Perceptrons (MLPs) are both inefficient and numerically unstable. We identify key properties of Kolmogorov-Arnold Networks (KANs) that align with these constraints. Specifically, we show that: (i) KAN updates exploiting B-spline locality are sparse, enabling superior on-chip resource scaling, and (ii) KANs are inherently robust to fixed-point quantization. By implementing fixed-point online training on Field-Programmable Gate Arrays (FPGAs), a representative platform for on-chip computation, we demonstrate that KAN-based online learners are significantly more efficient and expressive than MLPs across a range of low-latency and resource-constrained tasks. To our knowledge, this work is the first to demonstrate model-free online learning at sub-microsecond latencies.

Engineered systems typically separate mechanical function from information processing, whereas biological systems can exploit physical structure as a medium for information processing and computation. Motivated by this contrast, recent work in mechanics has explored embedding information-processing capabilities directly into mechanical structures. However, quantitative frameworks for evaluating such capabilities remain limited. Here we address a foundational question: how does information propagate through a solid body? Using elastic bodies as a model system, we apply information-theoretic tools to treat an elastic domain as an information encoder and quantify how information transmits from applied loads to discrete sensor locations. We further connect these measures to familiar mechanical phenomena, including Saint-Venant's effect and principal stress lines. Moving toward design, we show how geometry and architected materials can tune transmission, enabling elastic domains to either transmit or block information. Overall, this work advances quantifiable metrics and benchmark tasks for mechanical intelligence, supporting comparable designs of mechanically embodied information processing.

In Evans and Hendel's recent proof of an outstanding conjecture on the resistance distances of a family of linear 3-trees, a key technique in the proof was calculating the recursion satisfied by a family of determinants. The underlying algorithm employed to prove the conjecture converged (i.e., terminated) in the particular case studied, and the paper presented an open question on when such a procedure converges in general. This paper proves the convergence of a Laplace expansion procedure for an arbitrary family of determinants of banded, square, Toeplitz matrices. A comparison of the procedure presented in this paper, the paper by Evans and Hendel, and a paper by Jia, Yang, and Li is presented.

We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the family of monitorable sets is $Π^0_3$ and determine the exact complexities it can have. In contrast, we show that if $X$ is not second countable then the family of monitorable sets can be much more complex, giving an example where it is $ Π^1_1$-complete.

We modify the proof of the disc embedding theorem for $4$-manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to construct geometrically dual spheres. These were claimed in the statement but not constructed in the proof. We also prove Proposition 1.6 from the Freedman-Quinn book regarding generic homotopies of discs or spheres in a 4-manifolds, which was not proven there.

We find a layered permutation $w\in S_n$ whose Schubert polynomial $\mathfrak S_w(x_1, \dots, x_n)$ has support of size asymptotically at least $n!/4^n$. This gives precise asymptotics for the growth rate of $β(n):= \max_{w\in S_n}|\mathrm{supp}(\mathfrak S_w)|$. We find a different layered permutation $w\in S_n$ whose Grothendieck polynomial has support of size asymptotically at least $n!/e^{\sqrt{2n} \cdot \ln(n)}$ and obtain more precise asymptotics for the growth rate of $β^{\mathfrak G}(n):=\max_{w\in S_n}|\mathrm{supp}(\mathfrak G_w)|$.

We give a short and relatively elementary proof of the Hilton-Milner Theorem.

This article focuses on establishing a positive weak solution to a class of semipositone problems over the Heisenberg group $\mathbb{H}^N$. In particular, we are interested in the positive weak solution to the following problem: \begin{equation}\label{p1} -Δ_{\mathbb{H}}u= g(ξ)f_a(u) \text{ in } \mathbb{H}^N \tag{$P_a$}, \end{equation} where $a>0$ is a real parameter and $g$ is a positive function. The function $f_a: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and of semipositone type which means it becomes negative on some parts of the domain. Due to this sign-changing nonlinearity, we can not directly apply the maximum principle to obtain the positivity of the solution to \eqref{p1}. For that purpose, we need some regularity results for our solutions. In this direction, we first prove the existence of weak solutions to \eqref{p1} via the mountain pass technique. Further, we establish some regularity properties of our solutions and using that we prove the $L^\infty$-norm convergence of the sequence of solutions $\{u_a\}$ to a positive function $u$ as $a \rightarrow 0$, which yields $u_a \geq 0$ for $a$ sufficiently small. Finally, we use the Riesz-representation formula to obtain the positivity of solutions under some extra hypothesis on $f_0$ and $g$. To the best of our knowledge, there is no article dealing with semipositone problems in Heisenberg group set up.

We consider a bandit problem where the buget is smaller than the number of arms, which may be infinite. In this regime, the usual objective in the literature is to minimize simple regret. To analyze broad classes of distributions with potentially unbounded support, where simple regret may not be well-defined, we take a slightly different approach and seek to maximize the expected simple reward of the recommended arm, providing anytime guarantees. To that end, we introduce a distribution-free algorithm, OSE, that adapts to the distribution of arm means and achieves near-optimal rates for several distribution classes. We characterize the sample complexity through the rank-corrected inverse squared gap function. In particular, we recover known upper bounds and transition regimes for $α$ less or greater than $1/2$ when the quantile function is $λ_η= 1-η^α$. We additionally identify new transition regimes depending on the noise level relative to $α$, which we conjecture to be nearly optimal. Additionally, we introduce an enhanced practical version, PROSE, that achieves state-of-the-art empirical performance for the main distribution classes considered in the literature.

We continue the study of Adin, Alon and Roichman [arXiv:2502.14398, 2025] on the number of steps required to sort $n$ labelled points on a circle by transpositions. Imagine that the vertices of a cycle of length $n$ are labelled by the elements $1,\dots,n$. We are allowed to change this labelling by swapping the labels of any two vertices on the cycle. How many swaps are needed to obtain a labelling that has the elements $1,\dots,n$ in clockwise order? We provide evidence for their conjecture that at most $n-3$ transpositions are needed to sort a circular permutation when $n$ is not prime. We prove this conjecture when $2\mid n$ or $3\mid n$ and when restricting to permutations given by a polynomial over $\mathbb{Z}_n$. We also provide various algebraic constructions of circular permutations that take many transpositions to sort, most notably providing one that matches our upper bound when $n=3p$ for $p$ an odd prime, and disproving their second conjecture by providing non-affine circular permutations that require $n-2$ transpositions (for $n$ prime). We also improve the lower bounds for some sequences of composite numbers. Finally, we improve the bounds for small $n$ computationally. In particular, we prove a tight upper bound for $n=25$ via an exhaustive computer search using a new connection between this problem and strong complete mappings.

We prove that in dimension 3, Anosov flows which are $\mathbb{R}$-covered and skewed are orbit equivalent to Reeb-Anosov flows. We characterize the existence of an invariant contact form or of a Birkhoff section with a given boundary, in terms of linking numbers between two invariant signed measures. Furthermore, we prove the existence of open book decompositions with one boundary component for Reeb-Anosov flows.

We connect the existence of a ternary classical universal quadratic form over a totally real number field $K$ with the property that all totally positive multiples of 2 are sums of squares (if $K$ does not contain $\sqrt 2$ or contains a nonsquare totally positive unit). In particular, we get that Kitaoka's Conjecture holds for all fields of odd discriminant.

We define and study a cohomology theory for the space of Jacobi $n$-point functions generated by a vertex operator (super)algebra, using precise analogues of Zhu's reduction formulas. A cochain complex $(C^{\bullet}(W), δ^{\bullet})$ is constructed whose coboundary operators are given by Zhu-type reduction maps, and whose cohomology groups $H^{n}_{J}(W)$ we call the {reduction cohomology of Jacobi forms}. We prove that the $n$-th reduction cohomology of a $V$-module $W$ is isomorphic to the space of analytic continuations of solutions to a vertex-operator-algebraic analogue of the Knizhnik-Zamolodchikov equations. We further show that Jacobi $n$-point reduction formulas are $n$-point connections on the vertex operator algebra bundle over the torus, yielding a Bott-Segal-type theorem: $H^{n}_{J}(W)$ is isomorphic to the cohomology of the space of deformed sections of the VOA bundle.

Many algorithms in scientific computing and data science take advantage of low-rank approximation of matrices and kernels, and understanding why nearly-low-rank structure occurs is essential for their analysis and further development. This paper provides a framework for bounding the best low-rank approximation error of matrices arising from samples of a kernel that is analytically continuable in one of its variables to an open region of the complex plane. Elegantly, the low-rank approximations used in the proof are computable by rational interpolation using the roots and poles of Zolotarev rational functions, leading to a fast algorithm for their construction.

In this paper, we would like to study the critical exponent for semilinear damped wave equations with the nonlinearity terms of Coulomb-type singularities $|x|^{-α} |u(t,x)|^p$ and the initial data belonging to Sobolev spaces of negative order $\dot{H}^{-β}$. Precisely, we obtain a critical exponent $$p_{\rm c}(α,β,n): = 1 + \frac{4-2α}{n+2β} $$ for $1 \leq n \leq 4$ and $ 0 \leq α, β< n/2,$ by proving the global (in time) existence of small data solutions when $p \geq p_{\rm c}(α,β,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(α,β,n)$. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.

Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in this realm. By explicitly formulating a fundamental theorem for Hopf modules via the concept of Hopf-Galois comodules, we prove that the category of Hopf bimodules can be endowed with the structure of a (pre-)braided monoidal category in two different ways, which, in turn, are shown to be both braided monoidally equivalent to the category of Yetter-Drinfel'd modules, that is, to the monoidal centre of the category of left bialgebroid modules or comodules. As an illustration, we discuss relative Hopf bimodules associated to Ehresmann-Schauenburg bialgebroids.

Pinching-antenna systems (PASS) improve wireless links by configuring the locations of activated pinching antennas along dielectric waveguides, namely pinching beamforming. In this paper, a novel adjustable power radiation model is proposed for PASS, where power radiation ratios of pinching antennas can be flexibly controlled by tuning the spacing between pinching antennas and waveguides. A closed-form pinching antenna spacing arrangement strategy is derived to achieve the commonly assumed equal-power radiation. Based on this, a practical PASS framework relying on discrete activation is considered, where pinching antennas can only be activated among a set of predefined locations. A transmit power minimization problem is formulated, which jointly optimizes the transmit beamforming, pinching beamforming, and the numbers of activated pinching antennas, subject to each user's minimum rate requirement. (1) To solve the resulting highly coupled mixed-integer nonlinear programming (MINLP) problem, branch-and-bound (BnB)-based algorithms are proposed for both single-user and multi-user scenarios, which is guaranteed to converge to globally optimal solutions. (2) A low-complexity many-to-many matching algorithm is further developed. Combined with the Karush-Kuhn-Tucker (KKT) theory, locally optimal and pairwise-stable solutions are obtained within polynomial-time complexity. Simulation results demonstrate that: (i) PASS significantly outperforms conventional multi-antenna architectures, particularly when the number of users and the spatial range increase; and (ii) The proposed matching-based algorithm achieves near-optimal performance, resulting in only a slight performance loss while significantly reducing computational overheads. Code is available at https://github.com/xiaoxiaxusummer/PASS_Discrete

Understanding how sentiment toward COVID-19 mitigation measures evolves on social media can inform both epidemiological models and public health policy. We analyzed 655,642 posts by 28,559 users on r/Belgium from January 2020 to June 2022, classifying posts into three mitigation topics (lockdowns, masks, vaccinations) using a BERT-based topic model and scoring sentiment with a RoBERTa-based classifier. Post volume tracked external events such as policy announcements, but we found no evidence of within-Reddit social contagion in topic initiation, suggesting topics are seeded by external information rather than platform-internal spread. Sentiment, however, exhibited significant homophily: comment sentiment correlated with that of the parent post. To capture the underlying dynamics, we developed the Smooth Latent-Expressed Bounded Confidence (SLEBC) model, which distinguishes a latent sentiment trajectory from noisy expressed sentiment and uses bounded confidence rather than linear update rules. Evaluated against two alternatives by WAIC, SLEBC fit best across all three topics. The model indicates that expressed sentiment adapts more strongly to the immediate parent comment than the user's latent state updates from interaction history, suggesting that expressed sentiment is a poor proxy for underlying opinion. These findings imply that infodemic models for Reddit-like platforms should seed topics from external sources and model sentiment spread via bounded confidence mechanisms.

Extending Bayesian optimization to batch evaluation can enable the designer to make the most use of parallel computing technology. However, most of current batch approaches do not scale well with the batch size. That is, their optimization efficiencies often deteriorate as the batch size increases. To address this issue, we propose a simple and efficient approach to extend Bayesian optimization to large-scale batch evaluation in this work. Different from existing batch approaches, the idea of the new approach is to draw a batch of axis-aligned subspaces of the original problem and select one point from each subspace using existing acquisition functions. Numerical experiments show that our proposed approach speedups the convergence significantly when compared with the sequential Bayesian optimization algorithm, and performs very competitively when compared with ten batch Bayesian optimization algorithms. The implementation of our proposed approach is available at https://github.com/zhandawei/SubSpace_Acquisition_Functions.

The deepest arithmetic invariants attached to an algebraic variety defined over a number field $F$ are conjecturally captured by the integral part of its motivic cohomology. There are essentially two ways of defining it when $X$ is a smooth projective variety: one is via the $K$-theory of a regular model, the other is through its $\ell$-adic realization. Both approaches are conjectured to coincide. This paper initiates the study of motivic cohomology for global fields of positive characteristic, hereafter named $A$-motivic cohomology, where classical mixed motives are replaced by mixed Anderson $A$-motives. Our main objective is to set the definitions of the model version and the $\ell$-adic version of the integral part of $A$-motivic cohomology, using Gardeyn's notion of maximal models of $A$-motives as the analogue of regular models of varieties. Our main result states that the model version is contained in the $\ell$-adic version. As opposed to what is expected in the number field setting, we show that the two approaches do not match in general. We conclude this work by introducing the submodule of regulated extensions of mixed Anderson $A$-motives, for which we expect the two approaches to match, and solve some particular cases of this expectation.

OD matrix estimation is a critical problem in the transportation domain. The principle method uses the traffic sensor measured information such as traffic counts to estimate the traffic demand represented by the OD matrix. The problem is divided into two categories: static OD matrix estimation and dynamic OD matrices sequence(OD sequence for short) estimation. The above two face the underdetermination problem caused by abundant estimated parameters and insufficient constraint information. In addition, OD sequence estimation also faces the lag challenge: due to different traffic conditions such as congestion, identical vehicle will appear on different road sections during the same observation period, resulting in identical OD demands correspond to different trips. To this end, this paper proposes an integrated method, which uses deep learning methods to infer the structure of OD sequence and uses structural constraints to guide traditional numerical optimization. Our experiments show that the neural network(NN) can effectively infer the structure of the OD sequence and provide practical constraints for numerical optimization to obtain better results. Moreover, the experiments show that provided structural information contains not only constraints on the spatial structure of OD matrices but also provides constraints on the temporal structure of OD sequence, which solve the effect of the lagging problem well.

Crowdsourced smartphone-based earthquake early warning systems recently emerged as reliable alternatives to the more expensive solutions based on scientific-grade instruments. For instance, during the 2023 Turkish-Syrian deadly event, the system implemented by the Earthquake Network citizen science initiative provided a forewarning up to 25 seconds. We develop a statistical methodology based on a survival mixture cure model which provides full Bayesian inference on epicentre, depth and origin time, and we design an efficient tempering MCMC algorithm to address multi-modality of the posterior distribution. The methodology is applied to data collected by the Earthquake Network, including the 2023 Turkish-Syrian and 2019 Ridgecrest events.

Time series anomaly detection plays a crucial role in a wide range of real-world applications. Reconstruction-based methods have become the mainstream paradigm, but they suffer from over-generalization and under-generalization problems, which are challenging to balance. To address this, we introduce multi-scale clustering to enhance reconstruction-based methods. At the representation level, we integrate the cluster center representations of normal patterns to constrain the model to target representative normal patterns for reconstruction, preventing dominance of powerful capacity and representation capability. At the anomaly criterion level, we derive anomaly confidence score based on cluster membership probability and combine it with reconstruction error, providing dual criteria for detection. Furthermore, the effectiveness of the cluster center representations and anomaly confidence score depends on the clustering performance. Accordingly, we extract neighborhood-centered representations for multi-view clustering to improve clustering performance. Extensive experiments on multiple real-world datasets from diverse application domains demonstrate the state-of-the-art performance of SCAN.

We consider the equal sum partition problem, motivated by distance magic graph labeling: Given $n,k \in \N$ such that $k\, | \sum_{i=1}^ni$ and a partition $p_1+\cdots+p_k=n$, when is it possible to find a partition of the set $\{1,2,\ldots,n\}$ into $k$ subsets of sizes $p_1,\dots,p_k$, such that the element sum in each subset is the same? A known necessary condition is the \emph{slack condition}, requiring that for all $j$, placing the largest possible elements in the $j$ smallest sets yields a total sum that is at least what is needed. However, this condition is not sufficient, and known counterexamples exist. This work clarifies the boundary between solvable and unsolvable instances of the problem. We extend the list of unsolvable problem instances satisfying the slack condition by exhibiting infinite families where the $n/k$ ratio is any rational number in the interval $(2,\frac{24}{7})$, and a new criterion for unsolvability. Furthermore, we show that the slack condition is natural, as it is both necessary and sufficient for the fractional relaxation of the problem. Based on this result, we prove that the problem is solvable for the class of linear partitions, where $k$ is fixed, $p_1,\ldots,p_k$ grow linearly with $n$, and where the slack condition holds in a strong sense. We do this by applying a randomized rounding algorithm to a solution of the fractional relaxation of the problem and proving that the algorithm has an exponentially small failure probability.

The study of random walks has increasingly been popular across diverse disciplines such as statistics, mathematics, quantum physics, where they are used to model paths consisting of successive random steps in a mathematical space. A fundamental quantity of interest is the probability that a simple symmetric random walk returns to the origin after 2n steps. In this paper, we develop a unified probabilistic approach that connects the return probabilities in arbitrary dimensions with moment representations. Using this framework, we provide probabilistic proofs of several combinatorial identities involving beta and gamma functions, and derive new combinatorial identities in general dimensions.

This paper aims to develop a non-commutative geometrical version of the theory of Yang--Mills--Scalar--Matter fields. To accomplish this purpose, we will dualize the geometrical formulation of this theory, in which principal $G$--bundles, principal connections, and linear representations play the most important role. In addition, we will present the non-commutative geometrical Lagrangian of the system as well as non-commutative geometrical associated field equations. At the end of this work, we show some examples

In this paper, we prove a decomposition lemma for symmetric matrix fields on bounded domains: $D+\mathrm{Sym}\nablaΦ=\sum_i a_i^2ξ_i\otimesξ_i$ with uniform control on $Φ$ and $a_i^2$, using fewer than the usual $n(n+1)/2$ rank-one symmetric terms. Except possibly in dimensions $n=8,16$, the decomposition is shown to be optimal through algebraic arguments. This reduces the number of steps in convex integration for a nonlinear PDE system, improving Hölder regularity of flexible solutions in dimension $n\ge3$. This PDE is a partial linearization of the codimension-one local isometric embedding equation in the Nash--Kuiper theorem, and also yields improved regularity for very weak solutions of related 2D Monge--Ampére and $2$-Hessian systems. The improved Hölder exponent is any $α<(n^2+1)^{-1}$ for $n=2,4,8,16$ and any $α<(n^2+n-2ρ(n/2)-1)^{-1}$ otherwise, where $ρ$ is the Radon--Hurwitz number, related to Bott periodicity. The proof involves novel applications of algebraic geometry and topology that yield the optimality of decomposition, including Adams' theorem on vector fields on spheres, intersections of projective varieties, and projective duality, combined with an elliptic method that avoids loss of differentiability.

A step-reinforced random walk is a discrete-time stochastic process with long-range dependence. At each step, with a fixed probability $α$, the so-called positively step-reinforced random walk repeats one of its previous steps, chosen randomly and uniformly from its entire history. Alternatively, with probability $1-α$, it makes an independent move. For the so-called negatively step-reinforced random walk, the process is similar, but any repeated step is taken with its direction reversed. These random walks have been introduced respectively by Simon (1955) and Bertoin (2024) and are sometimes refered to the self-confident step-reinforced random walk and the counterbalanced step-reinforced random walk respectively. In this work, we introduce a new class of unbalanced step-reinforced random walks for which we prove the strong law of large numbers and the central limit theorem. In particular, our work provides a unified treatment of the elephant random walk introduced by Schutz and Trimper (2004) and the positively and negatively step-reinforced random walks.