AI for Science

Sampling a quantum system's underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that generalize existing 1D tensor networks sampling algorithms. Our first proposed algorithm performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify $K$ high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.

We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and the two-sheet hyperboloid. We construct their energy spectra and normalized wave functions and find that they depend on two quantum numbers, which suggests that the systems are minimally superintegrable.

Power-free integers and related lattice subsets give rise to interesting dynamical systems. They are revisited from a spectral perspective, in the setting of the Halmos--von Neumann theorem. With respect to the natural patch frequency measure, also known as the Mirsky measure, many of these systems have pure-point dynamical spectrum, but trivial topological point spectrum. We calculate the spectra explicitly, in additive notation, and derive their group structure, both for a large class of $\cB$-free lattice systems in $\RR^d$ and for power-free integers in quadratic number fields. Further, in all cases, the eigenfunctions can be given in closed form, via the Fourier--Bohr coefficients of generic elements and their translates, which form a subset of full Mirsky measure. Based on a simple argument via Kolmogorov's strong law of large numbers, we show how the Fourier--Bohr coefficients also provide the eigenfunctions for the unique measure of maximal entropy, and that we get phase consistency for both measures.

Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian. Prior work has identified critical initialization schemes that ensure Jacobian stability, but these analyses are typically restricted to fully connected networks with i.i.d. weights. In this work, we go significantly beyond these limitations: we establish a general stability theorem for deep neural networks that accommodates sparsity (such as that introduced by pruning) and non-i.i.d., weakly correlated weights (e.g. induced by training). Our results rely on recent advances in random matrix theory, and provide rigorous guarantees for spectral stability in a much broader class of network models. This extends the theoretical foundation for initialization schemes in modern neural networks with structured and dependent randomness.

We propose a semismooth Newton-based augmented Lagrangian framework for reconstructing sparse sources in inverse acoustic scattering problems. Rather than working in the unknown source space, our semismooth Newton updates operate in the measurement (adjoint) space, which is especially efficient when the number of measurements is much smaller than the discretized source dimension. The source is then recovered via Fenchel-Rockafellar duality. Our approach substantially accelerates computation and reduces costs. Numerical experiments in two and three dimensions demonstrate the high efficiency of the proposed method.

We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with {the results of} Nam, Nguyen and Sly (Trans.\ Am.\ Math.\ Soc.\ 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.

We analyse the metastable behaviour of the disordered Curie--Weiss--Potts (DCWP) model subject to a Glauber dynamics. The model is a randomly disordered version of the mean-field $q$-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. These random variables are chosen to have fixed mean (for simplicity taken to be $1$) and well defined cumulant generating function, with a fixed distribution not depending on the number of particles. The system evolves as a discrete-time Markov chain with single spin flip Metropolis dynamics at finite inverse temperature $β$. We provide a comparison of the metastable behaviour of the CWP and DCWP models, when $N \to \infty$. First, we establish the metastability of the CWP model and, using this result, prove metastability for the DCWP model (with high probability). We then determine the ratio between the metastable transition time for the DCWP model and the corresponding time for the CWP model. Specifically, we derive the asymptotic tail behavior and moments of this ratio. Our proof combines the potential-theoretic approach to metastability with concentration of measure techniques, the latter adapted to our specific context.

Trajectory mutual information is frequently used to quantify information transfer in biochemical systems. Tractable solutions of the trajectory mutual information can be obtained via the widely used Linear-Noise Approximation (LNA) using Gaussian channel theory. This approach is expected to be accurate for sufficiently large systems. However, recent observations show that there are cases, where the mutual information obtained this way differs qualitatively from results derived using an exact Markov jump process formalism, and that the differences remain even in the large copy number regime. In this letter, we show that these differences can be explained by introducing the notion of reaction- versus state-based descriptions of trajectories. In chemical systems, the information is encoded in the sequence of reaction events, and the reaction-based trajectories of Markov jump processes capture this information. We show that within the Gaussian formalism, trajectories can be defined either based on individual reaction channels, or on a state-based level, where different reaction channels are summarised into a single noise term. While both definitions agree in terms of copy number fluctuations, state-based trajectories contain in general less information than reaction-based trajectories. The commonly used Gaussian mutual information via the Linear-Noise Approximation is consistent with a state-based trajectory notion, which causes a systematic loss of information independent of system size. We show that an alternative, reaction-based variant of the Gaussian mutual information prevents this loss of information. We illustrate the consequences of different trajectory descriptions for two common cellular reaction motifs and discuss their connection with Berg-Purcell and Maximum-Likelihood sensing.

We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group $G$ with a unitary on-site action, our formalism produces a $G$-crossed braided tensor category $G\mathsf{Sec}$. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component $G\mathsf{Sec}_1$. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute $G\mathsf{Sec}$ explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.

This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in earlier work and the results of Part I, this study makes contributions in two main directions. In Part A, we focus on bifurcations of generalized periodic solutions in autonomous Lagrangian systems. By employing Morse index and nullity techniques within the normal space to the $\mathbb{R}$-orbits of solutions, we derive necessary and sufficient conditions for bifurcation, encompassing scenarios of both Fadell--Rabinowitz and Rabinowitz type. In Part B, we extend these results to the geometric setting of geodesic bifurcations in Finsler and Riemannian manifolds. A principal achievement is the significant refinement of the classical Gauss lemma and its generalizations by Morse-Littauer and Savage, providing a precise description of geodesic behavior near critical points of the exponential map. The sharpness of these theoretical results is rigorously tested and confirmed through explicit counterexamples, such as the round sphere. The work is technically rigorous, leveraging a specialized technique developed by the author to establish novel bifurcation theorems. These findings have profound theoretical implications and potential applications in related fields such as the Zermelo navigation problem and the study of stationary spacetimes.

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.

Optically pumped atomic magnetometers (OPAMs) offer high sensitivity at room temperature and are increasingly considered for portable magnetic sensing in geomagnetic-field environments. Here we report a handheld-scale, single-beam scalar $^{87}$Rb OPAM with a sensor-head volume of approximately 110~mL. The device operates in an all-optical Bell-Bloom configuration and uses digital lock-in, dispersive tracking of the $^{87}$Rb Larmor resonance, implemented with a hybrid electronics stack that combines in-house control hardware with commercial modules. A single frequency-modulated laser beam performs both pumping and probing without RF coils. All signal processing is realized in Python on a single-board computer paired with a commercial off-the-shelf (COTS) data-acquisition module, enabling immediate deployment without dedicated signal-processing hardware. The magnetometer has an intrinsic in-band field sensitivity of approximately 21~pT/$\sqrt{\mathrm{Hz}}$, estimated from the lock-in dispersion slope, over a 0.1--30~Hz closed-loop in-band region with a digital-output rate of 200~samples/s. In an unshielded Earth-field deployment, we detect repeatable transient magnetic signatures from a controlled elevator motion sequence and quantify standoff observability over sensor-elevator distances from 1.25~m to 10~m. These results show that compact scalar OPAMs can provide bandwidth and range-resolved event sensitivity suitable for field-deployable magnetic anomaly detection and infrastructure monitoring in realistic geomagnetic environments.

We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.

If AI is a foundational general-purpose technology, we should anticipate that demand for AI compute -- and energy -- will continue to grow. The Sun is by far the largest energy source in our solar system, and thus it warrants consideration how future AI infrastructure could most efficiently tap into that power. This work explores a scalable compute system for machine learning in space, using fleets of satellites equipped with solar arrays, inter-satellite links using free-space optics, and Google tensor processing unit (TPU) accelerator chips. To facilitate high-bandwidth, low-latency inter-satellite communication, the satellites would be flown in close proximity. We illustrate the basic approach to formation flight via an 81-satellite cluster of 1 km radius, and describe an approach for using high-precision ML-based models to control large-scale constellations. Trillium TPUs are radiation tested. They survive a total ionizing dose equivalent to a 5 year mission life without permanent failures, and are characterized for bit-flip errors. Launch costs are a critical part of overall system cost; a learning curve analysis suggests launch to low-Earth orbit (LEO) may reach $\lesssim$\$200/kg by the mid-2030s.

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this dynamic, we introduce several methods to generate topological feature vectors from unreduced boundary matrices and investigate their theoretical and computational properties. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. We also benchmarked the computational performance of an algorithm for computing unreduced diagrams, which was implemented as a heavily modified version of Ripser. These computations are parallelizable and required an order of magnitude less memory on average compared to computing full persistence diagrams. Our results suggest that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.

Gene regulatory networks are routinely translated from Boolean update rules into large continuous ODE systems integrated numerically for attractor identification, sensitivity analysis, and control design. The reliability of that integration depends critically on the sigmoidal kernel representing regulation. This simulation study shows that the Hill function -- the near-universal choice -- is a generically unreliable kernel, while the logistic function is a robust replacement. Two failure modes are demonstrated. First, because the Hill function vanishes at zero input, bistable circuits acquire an absorbing off-state: with experimentally grounded \textit{E. coli} galactose-operon autoregulation parameters, a Hill model stays trapped below the unstable separatrix, whereas the logistic model -- whose basal rate is strictly positive by construction -- escapes in about $44$~minutes through basal production alone, matching an analytical estimate of ${\approx}58$~min. A saddle-node analysis characterises the bistable window via an explicit transcendental equation and identifies the threshold $λθ=2$ separating monostable from bistable regimes. Second, when the Hill exponent is non-integer -- as in dose-response fits -- the power law $x^n=e^{n\ln x}$ turns complex-valued whenever a solver overshoots into negative concentrations. On an $80$-gene Boolean-derived benchmark with $n\approx3.509$, the Hill solver is silently contaminated by complex values from $t\approx52.64$, yielding smooth but spurious trajectories, whereas the logistic formulation completes $t\in[0,200]$ without a single warning. Because the logistic vector field is globally Lipschitz with explicit constant, we further prove an a priori global-error bound of classical order -- a guarantee structurally unavailable to the Hill formulation.

In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every $n$-vertex plane graph $G$ has (under some natural restrictions) a vertex-partition into two sets $V_1$ and $V_2$ such that each $V_i$ is \emph{dominating} (every vertex of $G$ contains a vertex of $V_i$ in its closed neighbourhood) and \emph{face-hitting} (every face of $G$ is incident to a vertex of $V_i$). Their proof works by considering a supergraph $G'$ of $G$ that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement. In this paper, we give a new proof that every $n$-vertex plane graph $G$ has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.

In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for $BV$ functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global $BV$ regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a $BV$ assumption seems to be necessary to get uniqueness.

Optical satellite uplinks rely on high-power optical amplifiers (HPOAs) to overcome free-space attenuation and enable long-distance transmission. However, at high power levels, fiber Kerr nonlinearity becomes significant and degrades system performance. In this work, we develop a realistic model for optical uplinks that accounts for nonlinear effects and analyze their impact, highlighting key differences from conventional longhaul fiber systems. We then introduce low-complexity digital signal processing techniques for nonlinearity compensation, based on constellation shaping via a look-up table (LUT) and a simple nonlinear phase rotation applied at the transmitter and/or receiver. The LUT also enables adaptive rate tuning according to channel conditions, enhancing robustness against link variations. Simulation results show that the proposed techniques increase the maximum acceptable link loss by up to 6 dB with negligible complexity. Finally, we show that, at the system level, propagation in the HPOA can be modeled as a simple nonlinear phase rotation, equivalent to propagation in a zero-dispersion noiseless fiber link, and fully characterized by a single parameter - the characteristic nonlinear power.

In a recent work, Sharma and Panda~\cite{sharma} showed that every bipartite graph with a perfect matching has property (P). In this paper, we investigate the converse direction, i.e., when property (P) forces the existence of a perfect matching in bipartite graphs. We show that such graphs are balanced and establish that property (P) is equivalent to the existence of a perfect matching for several families of bipartite graphs.

We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional "doubling events" when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size of this tree at large times, its degree distribution, and its height profile. We also prove a lower bound for its height. Because of the doubling events that affect the tree globally, the proofs are all much more intricate than in the case of the random recursive tree in which the growing operation is always local.

Evaluating visualization systems in niche domains such as genomics is challenging due to scarcity of domain experts and difficulty recruiting a representative user base. While LLM-based synthetic personas are increasingly used to ease evaluation bottlenecks, they face well-founded skepticism. Rather than weighing synthetic personas as substitutes for real users, we ask a fundamental open question: when synthetic personas evaluate a real visualization system, what do they actually produce, and how does that output change when grounded in documented human contexts? We present Sycamore, an exploratory three-condition probe design using Geranium, a search engine for multimodal genomics visualization, as a case study. Sycamore evaluates Geranium using: (1) ungrounded synthetic personas from generic LLM priors; (2) grounded synthetic personas constrained by voice-of-customer artifacts from a prior interview study; and (3) a published baseline study of real domain experts. We observe that grounding shifts synthetic feedback toward the language and concerns of documented users, while ungrounded evaluators drift toward operational specifics that real participants did not raise; both synthetic conditions, however, converge on a find-and-adapt frame and miss the image-modality preference observed in the expert study. We discuss what these observations imply for where synthetic personas might fit alongside expert studies in domain-specific visualization evaluation. All supplemental materials are available at https://osf.io/kdfr3/.

We revisit the optimization problem solved in Løkka & Zervos (2008), i.e., the maximization of dividends, in a Brownian risk model, with the possibility (not the obligation) of making capital injections. Following the approach introduced in Alvarez & Shepp (1998), Renaud & Simard (2021), Renaud et al. (2023), we consider instead absolutely continuous (AC) dividend strategies with an affine bound on the payment rates, while singular capital injections are still allowed. In addition, we incorporate a parameter for the cost of ruin or, said differently, a penalty at ruin in the performance function. We show that the solution is a so-called Løkka-Zervos dichotomy: the surplus is never ruined by making bail-out payments, or no capital is injected and bankruptcy can occur; in either case, dividends are paid at full rate when the surplus is above a threshold. Our framework allows us to provide explicit conditions to express the dichotomy, either using the cost of capital injections or the cost of ruin as a criterion, which also exposes the underlying structure of the solution. In particular, for some values of the parameters, we show that it is optimal to liquidate. Moreover, we perform a numerical analysis highlighting the range of values generated under this AC affine-bound structure.

Commercial fusion energy requires frameworks to assess both the scientific and economic viability of a wide variety of fusion concepts. Inspired by the Lawson criterion's ability to universally describe fusion energy gain, a generalized framework is developed to determine the economic gain of fusion power plants. The model exploits temporal equilibrium, and engineering and cost parameters normalized to the energy capture surface. The derived criteria for economic gain are therefore independent of the power plant's absolute power, impartial to the particulars of its fusion technology, and can be applied to any fusion confinement concept. The derivation of the economic gain factor, $Q_{econ}$, results in nonlinear equations with ten controlling normalized design parameters ranging from fusion power density and surface component lifetime to energy fluence, price of energy, and component efficiency and cost. These ten controlling parameters are varied over a wide range to provide high-level insights in design, finance and operational tradeoffs that improve the prospects for economically viable fusion energy.

We present a framework to compute non-Gaussian likelihoods for two-point correlation functions. The non-Gaussianity is most pronounced on large scales that will be well-measured by stage-IV weak-lensing surveys. We show how such a multivariate likelihood can be constructed and efficiently evaluated using a copula approach by incorporating exact one-dimensional marginals and a dependence structure derived from the exact multivariate likelihood. The copula likelihood is found to be in better agreement with simulated sampling distributions of correlation functions than Gaussian likelihoods, particularly on large scales. We furthermore investigate the effect of the non-Gaussian copula likelihood on posterior inference, including sampling the full parameter space of contemporary weak-lensing analyses. We find potential parameter shifts in $S_8$ on the order of one standard deviation for $1 \ 000 \ \mathrm{deg}^2$ surveys but negligible shifts for areas of $10 \ 000 \ \mathrm{deg}^2$, suggesting Gaussian likelihoods are sufficient for stage-IV surveys, though results depend on the detailed mask geometry and data-vector structure.

Duality is a foundational tool in robust and distributionally robust optimization (RO/DRO), underpinning both analytical insights and tractable reformulations. While most RO/DRO duality results are derived through saddle-point, Lagrangian, or conic arguments, this paper leverages perturbation duality. We show that this perspective provides a natural and unifying framework for deriving RO/DRO dual formulations, proving the associated duality results, and diagnosing the regularity assumptions on which they depend. First, guided by perturbation duality, we establish new duality theorems for a recent DRO framework that unifies several canonical models, including $ϕ$-divergence and Wasserstein models, through optimal transport subject to conditional moment constraints. Our results resolve an open conjecture on this DRO duality by clarifying the role of compactness: compactness itself is not necessary, but can be replaced by perturbation-based regularity conditions. Second, we revisit \emph{robust duality}, commonly described as \emph{primal-worst equals dual-best.} Using bifunctions, we unify dual-best formulations appearing in the literature and derive concise perturbation-based proofs that streamline recent results. Overall, the paper positions perturbation duality as a versatile and underutilized tool for RO and DRO, offering both conceptual unification and technical generality across a broad class of models.

We prove that for any real number $ξ\neq 0$ and any coprime integers $p>q\ge1$ such that $ξ$ is irrational or $q>1$, the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-p/q)^n)_{n\ge 0}$ is not contained in any interval of length less than $(1+q/p-q^2/p^2)/p$.

Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study the quartic forms GF = (|x|^4 + F(x))/2 associated with isoparametric polynomials F of OT-FKM type with g = 4. We characterize the sos property of GF in terms of the feasibility of an explicit SDP determined by the underlying Clifford system, and in the sos cases we obtain quantitative rank bounds for sos representations, with rigidity when m >= 3.

We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key topology and its Borel sigma-algebra we show that recurrence implies periodicity. Moreover, we establish that if every continuous potential with finite pressure possesses some equilibrium state then we have either finiteness of cycles or infinitely many cycles sharing the same period. The existence of some continuous potential with no equilibrium state is equivalent to the unboundedness of periods of cycles. The uniqueness of periodic orbits is equivalent to the uniqueness of equilibrium state for every bounded and continuous potential. We also prove that we have either infinitely many cycles or no divergent orbits. Finally, we apply our technique to the Baker and Syracuse maps, obtaining a similar result for this general class of important maps.

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation and treating the spectral density as a probability distribution suitable for Monte Carlo approximation. Although this probabilistic interpretation is valid for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. We propose regular Fourier features for harmonizable processes to avoid this limitation. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Under a finite-spectral-support assumption, this yields an efficient low-rank approximation that is consistent and positive semi-definite by construction. When the spectral density is unknown, the framework extends naturally to kernel learning from data. We demonstrate the method on locally stationary and harmonizable mixture kernels, the latter with a complex-valued spectral density, and apply the kernel-learning extension to real and synthetic data.