AI for Science

Accurate epidemic forecasting requires models that account for the layered and heterogeneous nature of real social interactions. The basic reproduction number $\mathcal R_0$, as calculated from models that assume homogeneous mixing or single-layer contact structures, has limited applicability to complex social systems. Here, we derive an expression for $\mathcal R_0$ in the context of multiplex networks, enabling the analysis of disease transmission across multiple social layers. We adapt the Degree-Based Mean-Field (DBMF) SIR model for single-layer complex networks to the multiplex setting, where each layer is characterized by its own degree distribution and infection rate. Using the Next Generation Matrix method, we derive an analytical expression for the basic reproduction number $\mathcal R_0$. Numerical integration of the multiplex DBMF equations shows that $\mathcal R_0=1$ marks the epidemic threshold and governs the behavior of key outbreak indicators as expected. In addition to the exact expression for $\mathcal R_0$, we introduce an approximation, denoted by $τ$, which is simpler to compute and admits a more transparent interpretation in terms of the epidemiological and topological parameters of the system. Stochastic agent-based simulations support these findings, demonstrating a direct correspondence between $τ$ and the average number of secondary infections generated during the early stages of an outbreak, consistent with the epidemiological interpretation of $\mathcal R_0$. This work provides a robust generalization of $\mathcal R_0$ for layered contact structures, offering a more realistic basis for epidemic forecasting and the design of intervention strategies.

AI systems can increasingly automate scientific workflows, but the reasoning that links prior evidence, generated ideas, experiments and final claims often remains implicit inside model inference. Here we introduce Xcientist, a research harness that externalizes research synthesis and experimental validation into inspectable, contract-governed processes. Xcientist organizes literature evidence, idea states, implementation plans, ablation records and repair traces as persistent research artifacts, so that generated mechanisms can be grounded, executed, tested and revised without losing their evidential basis. We identify claim drift as a failure mode of automated research, where runnable artifacts no longer support the mechanism originally claimed. Across training-free memory systems, graph-structured traffic forecasting and multi-scale physics-informed neural networks, Xcientist preserves traceable trajectories from problem formulation to mechanism design, validation and bounded revision. These results suggest that AI scientists should be evaluated not only by their final artifacts, but by whether their synthesis and validation processes remain attributable, inspectable and scientifically accountable.

This paper studies the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

We study a diffuse--interface model for a binary incompressible mixture in a periodically perforated porous medium, described by a time-dependent Navier--Stokes--Cahn--Hilliard (NSCH) system posed on the pore domain $Ω_p^\varepsilon\subset\mathbb{R}^3$. The microscopic model involves a variable viscosity tensor, a non-conservative source term in the Cahn--Hilliard equation, and mixed boundary conditions: no-slip on the outer boundary and Navier slip with zero tangential stress on the surfaces of the solid inclusions. The capillarity strength $λ^\varepsilon>0$ depends on the microscopic scale $\varepsilon>0$. The analysis consists of two main parts. First, for each fixed $\varepsilon>0$ we prove existence of a weak solution on a finite time interval $(0,T)$ and derive a priori estimates that are uniform with respect to $\varepsilon$ (and $λ^\varepsilon$). Second, we perform the periodic homogenization for the perforated setting in the limit $\varepsilon\to0$. Depending on the limit value $λ$ of the capillarity strength $λ^\varepsilon$, we obtain two distinct effective models: (i) in the vanishing capillarity regime $λ=0$, the limit system decouples completely into a standalone linear Stokes system for the velocity--pressure pair and a standalone Cahn--Hilliard system with source term $G$ for the phase field and chemical potential, with no macroscopic convection, advection, or capillary coupling between the two; (ii) in the balanced regime $λ\in(0,+\infty)$, we derive a Navier--Stokes--Cahn--Hilliard system with nonlinear convection and advective transport of the phase field at the macroscopic scale, coupled through a capillary forcing term. Finally, we establish the convergence of the microscopic free energy to a homogenized energy functional satisfying an analogous dissipation law.

Accurately estimating treatment effects in time series is essential for evaluating interventions in real-world applications, especially when treatment assignment is biased by unobserved factors. In many practical settings, interventions are adopted at different times across individuals, leading to staggered treatment exposure and heterogeneous pre-treatment histories. In such cases, aggregating outcome trajectories across treated units is ill-defined, making individual treatment effect (ITE) estimation a prerequisite for reliable causal inference. We therefore study the problem of estimating the average treatment effect for the treated (ATT) by first recovering individual-level counterfactuals. We introduce a neural framework that learns simultaneously low-dimensional latent representations of individual time series and propensity scores. These estimates are then used to approximate the individual treatment effects through a flexible matching procedure that avoids classical convexity constraints commonly used in synthetic control methods. By operating at the individual level, our approach naturally accommodates staggered interventions and improves counterfactual estimation under latent bias, without relying on explicit temporal modeling assumptions. We illustrate our approach on both real-world energy consumption data and clinical time series, including high-frequency electricity demand-response programs and semi-synthetic data for individuals in intensive care unit (ICU), where hidden confounding, staggered treatment adoption, and non-stationary dynamics are prevalent.

This work presents an extension to Pareto Front Guided Sampling (PFGS), a Human-in-the-Loop (HitL) Bayesian Optimization (BO) framework in which Gaussian process (GP) surrogate-derived quantities are reformulated as objectives of a multi-objective optimization problem, and the resulting Pareto front is exposed to a domain expert for interactive candidate selection rather than returning a single automated recommendation. The framework is extended in two directions: constrained optimization is addressed by incorporating the posterior probability of satisfying output specification limits as an explicit Pareto objective, computed analytically from the GP posterior distribution; robust optimization is addressed by a Monte Carlo sampling strategy that estimates expected lower-confidence performance over a user-defined variability of input perturbations, capturing performance degradation under likely implementation deviations. The resulting multi-dimensional Pareto representation renders trade-offs between predicted performance, model uncertainty, probabilistic constraint satisfaction, and input robustness simultaneously visible through pairwise two-dimensional projections on an interactive dashboard, enabling selection criteria to be iteratively refined as the surrogate model improves and development objectives evolve. The framework is showcased on an eight-dimensional fed-batch Chinese Hamster Ovary (CHO) cell culture simulator demonstrating systematic identification of high-performing, feasibility-compliant, and perturbation-resilient operating conditions, and illustrating how expert-defined requirements provide a principled stopping criterion and support informed allocation of experimental resources.

Multivariate time series anomaly detection (MTSAD) is critical for a wide range of application areas, such as industrial monitoring, cybersecurity, or healthcare. Real-world data is often sparse, irregularly sampled or partially observed, yet existing methods assume uniformly sampled time series. We propose a generative approach based on Latent SDEs that projects the observed time series on a continuous-time stochastic dynamical system, directly being able to handle missing observations and irregular sampling, while also naturally capturing possible cyclic behavior that many real-world use cases inherently possess. Experiments on six anomaly benchmark datasets show that our proposed method ranks first among state-of-the-art baselines. We further demonstrate that our method remains robust under severe data sparsity, while performance significantly degrades for the tested baseline methods. These results highlight latent SDEs as a natural inductive bias for anomaly detection in multivariate time series, especially in presence of real-world irregularities.

Sparse matrix kernels form the computational backbone of scientific computing, traditionally relying on C/C++ and Fortran implementations that prioritize performance over memory safety. This work evaluates Rust as a systems-level alternative for sparse linear algebra by implementing and benchmarking three core workloads: sparse matrix-vector multiplication (SpMV), Lanczos-based Krylov methods, and matrix-exponential evaluation. We compare native Rust code against established baselines (Intel oneMKL, Eigen, PETSc, and PSBLAS) across a suite of representative matrices. Our results show that Rust's sparse kernels achieve performance comparable to Eigen and PSBLAS, tracking the state-of-the-art for CSC formats, while trailing PETSc's advanced blocked CSR optimizations. By analyzing compile-time monomorphization, SIMD vectorization, and FFI boundaries, we assess the practical impact of Rust's safety model and ecosystem readiness. The study provides concrete, evidence-based guidance for modernizing high-performance numerical software stacks.

We study the $I^G(J^{PC})=0^+(1^{++})$ $K\bar Kπ$ system in an infinite-volume three-body unitary framework, focusing on the pole content of the region of the $f_1(1285)$ and $f_1(1420)$ resonances. The coupled $πa_0$-$K\bar K^*$ amplitude is constructed in the spectator-isobar representation, where the one-particle-exchange interaction required by three-body unitarity automatically incorporates the triangle-singularity mechanism. The short-range three-body interaction is constrained by fitting the $0^+(1^{++})$ component of the BESIII $K^0_SK^0_Sπ^0$ invariant-mass distribution in the $J/ψ\toγ(K^0_SK^0_Sπ^0)$ decay. Analytically continuing the fitted amplitude to the relevant unphysical Riemann sheets, we find two robust poles: \begin{align} \sqrt{s_{f_1(1285)}}&= \left(1277\pm2\pm1\right) -i\left(12\pm1\pm0\right)\text{MeV}\,,\notag\\ \sqrt{s_{f_1(1420)}}&= \left(1435\pm2\pm7\right) -i\left(40\pm2\pm1\right)\text{MeV}\,.\notag \end{align} The pole trajectories indicate that the $f_1(1285)$ originates from dressing a bare state introduced in the potential. In contrast, the $f_1(1420)$ is predominantly dynamically generated, and a single-channel analysis traces it to an $S$-wave $K\bar K^*$ quasi-bound state mixed with the nearby bare state, supporting its hadronic-molecule interpretation. We also find an additional pole deeper in the complex plane in the best-fit amplitude on the same Riemann sheet as the $f_1(1285)$. This additional pole is generated by the $P$-wave $πa_0$ contact interaction alone. It has a sizable cutoff and two-body-input dependence, and leaves little visible imprint on the physical lineshape. Finally, we provide a detailed and pedagogical appendix on how three-body cuts affect the solution of the integral equation.

We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $ω=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $ω=3$ analogue. Counterintuitively the problem hardens as $ω$ decreases -- the induction's feed thins -- so $ω=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\le\min\{β(N),β'(N)\}$, proved for all $N\ge10$ by Olson's addition theorem and elementary Chebyshev bounds above a finite, machine-checked base range. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\min\{B_{N_b}/6,\,1/5\}$ is $ω=2$ representable, unconditionally. As an application we give the first complete proof of the rational $ω=3$ statement -- every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions -- that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $ω\ge3$. What remains open is the $ω=2$ regime below this threshold, which we reduce to a single explicit conjecture -- that the gap-free floor of a semiprime subset-sum set tends to zero. This work is a human-AI collaboration: AI tools (notably Anthropic's Claude, used through Claude Code) contributed substantially to the Lean formalisation, the experiments, and the writing; correspondingly, every result is machine-checked in Lean 4 / Mathlib (no sorry; two cited classical axioms, plus the native_decide compiler-trust base for the finite computations), so its correctness is independent of the tools used.

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \(G\) is a finitely generated hyperbolic group and \(A\subseteq G\) is infinite with \[ A^2\subseteq AX \] for some finite \(X\subseteq G\), then either \(\langle A\rangle\) is virtually cyclic, or \(A\) has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \(G\) is hyperbolic with finite generating set \(S\), then there is a constant \(c_{G,S}>0\) such that \[ |UV| \geq c_{G,S}\,\frac{|U||V|}{n+k+1}, \qquad U\subseteq B_n,\; V\subseteq B_k. \] The linear loss is optimal in order whenever \(G\) contains an element of infinite order. In the free group with its standard generating set one may take \(c_{G,S}=1/4\). We also prove that, in a free group, if \(U\subseteq S_n\) and \(V\subseteq S_k\), then \[ |UV|\geq \left(\frac{2}{3}+\frac{1}{3\cdot 4^{\min\{n,k\}}}\right)|U||V|, \] and this constant is sharp for all \(n,k\).

We extend the application of the Guillemin--Abreu formalism to holomorphic sectional and bisectional curvature on $\mathbb C^n$, $\mathcal O(-\ell)$, and Hirzebruch manifolds $M_{n,\ell}$, and further apply it to the total spaces of certain higher-rank vector bundles. The resulting formulas recover known positivity criteria and we show that, when the slope is sufficiently close to $1$, the extremal metrics on $M_{n,\ell}$ have positive holomorphic sectional curvature. We construct complete scalar-flat Kähler metrics on $ \operatorname{Tot}\bigl(\mathcal O(-k)\oplus\mathcal O(-k)\to\mathbb{CP}^n\bigr), $ and identify the Ricci-flat case, which occurs precisely when $2k=n+1$.

We develop a framework for conformal prediction in dyadic regression problems under complex missingness mechanisms. At the theoretical level, we develop general technical tools for establishing finite-sample validity of conformal prediction under distributional invariance conditions weaker than exchangeability. A key result handles the case where the sample itself is a random subset of the index set, a setting not covered by existing theory, via a novel bijection argument that constructs an explicit measure-preserving correspondence between events. In addition, we propose conformal prediction procedures for jointly exchangeable arrays, including full conformal, split conformal, a row-column approach exploiting similarities within rows and columns, and a selective conformal procedure achieving mask-conditional validity. For missing elements, we establish asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism. We further establish conditional validity results for both continuous and discrete responses; to the best of our knowledge, this is the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption. The proposed methods are illustrated on synthetic and real network data.

The deep neural network is a widely used framework in machine learning that has been widely applied in various fields. However, deep neural networks often involve a large number of parameters and inputs, many of which may be irrelevant to the goal or true output. These parameters and input variables not only increase computational complexity, but also contribute to additional computational cost. One solution to this problem is knockoff methods, which have proven successful in controlling false discovery rates in high-dimensional regression. Building on the knockoff methods and using the regularised neural network, this paper proposes three variable screening methods under the condition of controlling false discovery rates: one layer filter, multiple layers filter, and variable weight aggregation filter. In comparison with existing algorithms, we find that our algorithms show satisfactory performance.

Diffusion-based neural solvers have shown strong promise for combinatorial optimization (CO), but existing methods typically rely on supervised training with large collections of near-optimal solutions. In this work, we extend adjoint-based trajectory optimization methods to discrete combinatorial domains. We formulate diffusion-based CO as a stochastic control problem over Continuous-Time Markov Chains and introduce discrete adjoint dynamics for propagating optimization signals through discrete generative trajectories. Building on this formulation, we propose Combinatorial Adjoint Matching (CAM), an unsupervised training framework for discrete diffusion solvers with structured and low-variance trajectory-level optimization signals. Empirically, CAM consistently outperforms existing unsupervised diffusion baselines and achieves performance competitive with strong supervised diffusion solvers and even traditional solvers across diverse combinatorial optimization problems. Our code is available at https://github.com/Shengyu-Feng/CAM.

Cryptocurrencies are increasingly adopted as investment assets, making their interactions with traditional financial markets central to cross-asset diversification and systemic risk. This paper studies the integration of cryptocurrencies, fiat currencies, and S&P500 equities using a balanced panel of 381 assets from October 2017 to February 2024. We combine rolling correlation networks, community structure, market-specific and system-wide Turbulence Indices, and VAR-based connectedness analysis to examine how market stress, network structure, and shock transmission vary across financial regimes. The results show that cross-asset integration is episodic. In calm periods, the three asset classes remain relatively segmented, whereas under stress, local clustering increases, modular separation weakens, and communities become more compositionally mixed across asset classes. Connectedness analysis further shows that regime shifts alter the structure of transmission rather than simply increasing spillover magnitudes. In high-turbulence states, fiat-market turbulence becomes the dominant propagation channel, while network clustering and modularity play a greater role in transmitting forecast uncertainty. These findings support the interpretation of network structure as an emergent, state-dependent transmission layer rather than a persistent exogenous driver of turbulence. The results highlight the need for regime-aware risk monitoring, since full-sample connectedness estimates can understate the cross-asset coupling that emerges precisely when diversification benefits are most fragile.

The 2024-2025 Nobel and Turing awards recognised AI and quantum science simultaneously. Yet no deployed system has brought these streams together for the public. This paper presents QSignAI, a production-deployed platform demonstrating a bidirectional AI-quantum relationship in a real-time event participation system. We address three questions: can quantum-randomness generation via a two-source extractor be embedded in an AI-driven social platform with acceptable latency; can an AI bot make quantum phenomena perceptually legible to general audiences; and does the combined system work in practice? A conversational bot routes each participant's first message through a quantum pipeline comprising a Toeplitz two-source extractor over independent single-qubit Hadamard measurements on SV1 and DM1 simulators, plus a 2-qubit Bell state, producing a unique quantum-randomness-seeded identity signature per participant. The first two questions are answered through system architecture and qualitative deployment evidence from live events; the third through successful production deployment. The current deployment uses cloud quantum simulators; physical QPU randomness is the near-term extension. Measurable benchmarks are identified as priority future work.

Schrödinger bridges for time series (SBTS) generate synthetic paths by projecting, in relative entropy, a Brownian reference onto the path laws that match the joint distribution of the data on the observation grid. The Brownian reference, however, fixes the quadratic variation of the generated paths, which is restrictive when stochastic volatility, correlated noise, or rank-deficient covariance structures must be reproduced. We introduce "Triangular-Reference Schrödinger Bridges for Time Series" (TR-SBTS), which keeps the entropy-projection backbone of SBTS but replaces the Brownian reference by a triangular, volatility-informed, intervalwise frozen reference on a state augmented with latent covariance descriptors. The construction remains a single entropy projection on the augmented state: the minimiser is the \(h\)-transform of the reference, and on each frozen interval the optimal drift has the logarithmic-gradient form \(b^\star(t,x)=A\,\nabla\log H(t,x)\), intrinsic to the active covariance directions when the frozen covariance \(A\) is degenerate. We prove stability of the frozen approximation and consistency of the associated regularised kernel estimators, describe a reference-aware Nadaraya--Watson implementation of the conditional next-increment law, and evaluate the construction on numerical experiments.

We introduce the \emph{Morse ensemble polynomial} $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete distribution of Morse vectors across all discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes. Our main results are the following. \textbf{(I) The Laplacian Formula}: for any connected graph $G$, $\ME_G = z_1^{m-n}\det(z_0z_1\,I_n + L_G)$, identifying $\ME_G$ as a complete Laplacian spectral invariant and showing $\ME_G$ to be incomparable with the Tutte polynomial. \textbf{(II) The Top-Face Recursion}: adding a $d$-simplex $σ$ (with $\partialσ\subset K$) to a complex $K$ gives a recursion $\ME_{K\cup\{σ\}} = z_d\cdot\ME_K + \sum_{τ\precσ}(\ME_{P(K')\setminus\{σ,τ\}}-F(K,σ,τ))$. The correction term is controlled by the top incidence graph: an incidence-separation criterion detects exactly when $F=0$, and the incidence distance gives the leading obstruction term. As a topological application, this recursion gives exact coefficient recursions for perfect and optimal discrete Morse vectors. \textbf{(III) The independence ME polynomial} $Φ(G) := \ME_{\mathrm{Ind}(G)}$ is a fine graph invariant which strictly refines the graph-level Morse ensemble $\ME_G$, separates examples not distinguished by $T_G$ and $I(G;t)$, and records collapse-level information of $\mathrm{Ind}(G)$ through coefficients such as $[z_0]Φ(G)$.

We define a sequence of integer-valued invariants $γ^k(L)$ for a $3$-component link $L$. We prove that the resulting $γ$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish this, we introduce an invariant $h(L)$, a $3$-component analogue of the Kojima--Yamasaki $η$-invariant, and show that it recovers the $γ$-invariants. As applications, we obtain a weak-cobordism classification when the distinguished component has trivial Alexander polynomial and characterize knots that bound continuously embedded disks in $B^4$ whose complements have fundamental group $\mathbb{Z}$.

We develop a first and second fundamental theorem for $n$--tuples of bosonic and fermionic matrices, by developing graded analogues of the classical case.

The exodromy correspondence of Barwick, Glasman, and Haine computes constructible sheaves of spaces on a scheme $X$ as an $\infty$-category of continuous functors from the profinite category $\operatorname{Gal}(X)$. Viewing $\operatorname{Gal}(X)$ instead as a condensed category, this was extended by Wolf to an exodromy correspondence for pro-étale sheaves. Using the condensed perspective from the outset, we give a quick and self-contained proof of the pro-étale exodromy theorem. This is used to extract an exodromy theorem for (Postnikov complete) étale sheaves that does not yet appear in the literature, which is closely related to Lurie's work on ultracategories. Finally, we use this to give a new proof of the constructible étale exodromy correspondence of Barwick, Glasman, and Haine. Without additional effort, our method removes the qcqs hypotheses on the schemes, and gives versions for sheaves with coefficients in more general $\infty$-categories. Finally, we refine the methods to obtain a $κ$-condensed statement for any uncountable cardinal $κ$ such that $κ> \lvert \mathcal O_X(U) \rvert$ for every affine open $U \subseteq X$.

Already in $\bf{R}^4$, there are many minimal hypersurfaces, yet few structural results. We show that minimal submanifolds, of any dimension and codimension, that are confined in space are very restricted. It is well-known that the half-space theorem fails already for hypersurfaces in $\bf{R}^4$, where there are many examples contained in a slab. In $\bf{R}^3$ the height of the catenoid grows at a logarithmic rate, whereas in higher dimensions the height of the catenoid remains bounded. We will see that even in high dimensions, minimal submanifolds that are confined in space must satisfy strong structural restrictions. We show that any proper minimal immersion whose height grows sublinearly must have Euclidean volume growth. A consequence is an optimal Bernstein theorem in any dimension for stable hypersurfaces with sublinearly growing height that generalizes results of Moser, Bombieri-De Giorgi-Miranda, Trudinger, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. Euclidean volume growth is a powerful property and there are many other consequences.

Small area estimation produces estimates of population parameters for geographic and demographic subgroups with limited sample sizes. Such estimates are critical for policy decisions, yet principled validation of these models remains a challenge. Unlike conventional predictive settings, validation data are rarely available. Data thinning splits a single observation into independent training and test components. It enables out-of-sample validation using only the area-level summary statistics routinely available, requiring only their Gaussianity and known sampling variances. However, the properties of thinning-based model comparison have not been formally studied. In this paper, we develop these properties. We construct an unbiased estimator of thinned-data mean squared error and show that it differs systematically from its full-data counterpart; for the standard Fay-Herriot model, the gap admits a closed-form expression that depends on the candidate model's shrinkage behavior. We further show that the estimator variance increases sharply as the training fraction approaches one, producing a bias-variance tradeoff with no universally optimal thinning parameter. Practical recommendations balancing these forces are informed by theory and verified empirically. Design-based simulations using American Community Survey microdata show that the recommended data thinning approach is competitive with information-criterion and simulation-based methods, and substantially more stable across heterogeneous sampling designs.

We take two approaches to classifying the complexity of Presburger models: Scott analysis and degree spectra. In particular, we investigate the possible Scott sentence complexities and possible degree spectra of models of Presburger arithmetic. Many of our results will be achieved by showing how given a linear order $\mathcal{L}$, we can construct a Presburger group $P_\mathcal{L}$ that maintains much of the structure of $\mathcal{L}$.

A triangular structural panel data model with additive separable individual-specific effects is used to model the causal effect of a covariate on an outcome variable when there are unobservable confounders with some of them time-invariant. In this setup, a linear specification for the reduced-form equation might be problematic when the conditional mean of the endogenous covariate and the instrumental variables is nonlinear in the population. The reason is that ignoring the nonlinearity could lead to weak instruments (instruments are weakly correlated with the endogenous covariate) due to misspecification as shown using a generalized concentration parameter for panel data. As a solution, we propose a triangular simultaneous equation model for panel data with additive separable individual-specific fixed effects composed of a linear structural equation with a nonlinear reduced form equation. The parameter of interest is the structural parameter of the endogenous variable. The identification of this parameter is obtained under the assumption of available exclusion restrictions and using a control function approach. We provide an estimator that we call Super Learner Control Function estimator (SLCFE). The estimation procedure is composed of two main steps and cross-fitting. First, we estimate the control function using a super learner. In the following step, we use the estimated control function to control for endogeneity in the structural equation. Cross-fitting is done across the individual dimension. The estimator is consistent and asymptotically normal achieving a parametric rate of convergence. We show that the SLCF estimator differs from both the plug-in IV estimator and a naive plug-in 2SLS estimator, with the former not being consistent without cross-fitting, and the latter not being consistent even with cross-fitting.

We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are $N$ sites (with $N$ finite or infinite), each may contain at most $c$ particles, $1 \le c < \infty$. New particles enter the system at the left-most site $1$ as a Poisson process of rate $α\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $β\le 1$. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site $n$ to $n+1$ at epochs of a rate $1$ Poisson process, as long as the former site has strictly more particles than the latter. When $c=1$, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as $N\to\infty$. In particular, we prove that interesting non-trivial phase transitions take place in a system with $c>1$. For example, if $c>1$ and $1/2 \le β\le 1$, the maximum limiting flux $1/4$ is achieved as long as $α\ge α_c^*$, where $α_c^* < 1/2$ is some non-trivial threshold. (For the standard TASEP the threshold is $1/2$.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.

Low-latency, resource-efficient neural network inference on FPGAs is essential for applications demanding real-time capability and low power. Lookup table (LUT)-based neural networks are a common solution, combining strong representational power with efficient FPGA implementation. In this work, we introduce KANELÉ, a framework that exploits the unique properties of Kolmogorov-Arnold Networks (KANs) for FPGA deployment. Unlike traditional multilayer perceptrons (MLPs), KANs employ learnable one-dimensional splines with fixed domains as edge activations, a structure naturally suited to discretization and efficient LUT mapping. We present the first systematic design flow for implementing KANs on FPGAs, co-optimizing training with quantization and pruning to enable compact, high-throughput, and low-latency KAN architectures. Our results demonstrate up to a 2700x speedup and orders of magnitude resource savings compared to prior KAN-on-FPGA approaches. Moreover, KANELÉ matches or surpasses other LUT-based architectures on widely used benchmarks, particularly for tasks involving symbolic or physical formulas, while balancing resource usage across FPGA hardware. Finally, we showcase the versatility of the framework by extending it to real-time, power-efficient control systems.

Error-mitigation methods for Ising machines are reexamined not merely as noise-suppression techniques but as a structural design problem of replica-coupled Ising models. Using simulated annealing as a hardware-noise-free testbed, we systematically compare the penalty-spin (PS) model, which couples replicas through a centralized auxiliary layer, with the stacked model, which couples adjacent replicas directly. Numerical experiments on the quadratic assignment problem reveal that the ferromagnetically coupled stacked model stably maintains constraint satisfaction and improves solution quality over a broad parameter range, exhibiting favorable scalability with both the number of replicas and problem size. In contrast, the PS model suffers from cooperation collapse at large parallelism: many-replica averaging in the PS layer washes out sparse solution information, preventing effective inter-replica coordination. These findings demonstrate that the topology of inter-replica couplings decisively influences search robustness, and provide practical guidelines for model selection and parameter tuning in constrained optimization.

We construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the orthogonal and symplectic groups. We also prove some results on good moduli spaces of fixed point stacks and mapping stacks from finite groupoids.