Papers

We develop a unified description of convolution on the sphere by lifting to the rotation group. A number of definitions of spherical convolution have been proposed, differing in whether the kernel is axisymmetric or directional, whether the output is on the sphere or the rotation group, whether the signals carry spin, and in their computational cost. No unified treatment exists. Regarding the sphere as the homogeneous space $S^2 \cong \text{SO(3)}/\text{SO(2)}$, we lift a spin $s$ field to the rotation group, convolve on $\text{SO(3)}$, and project back to the sphere through the adjoint of the lift for a potentially different spin $s'$. We recover every rotation-equivariant convolution considered previously through constraints on the kernel, spin choices and inclusion or omission of the lift and projection. For maps between fields of fixed input and output spin, we prove that directional kernels, output on the sphere and equivariance cannot be achieved simultaneously. We make explicit that spin fields are sections of associated line bundles, linking the physics and bundle literatures, and express the spin rotation phase as the character of the twist of the section. We present harmonic-space algorithms for the exact and efficient computation of lifted spherical convolutions of band-limited fields by appealing to sampling theorems on the sphere and rotation group. All operators are implemented in the open-source s2conv code, and numerical experiments confirm the reductions and the equivariance of the operators to machine precision. Finally, we illustrate the operators on fields measured on the Earth and on the celestial sky.

Light Detection and Ranging (LiDAR) is critical for autonomous perception, yet its performance is known to degrade in adverse weather. While existing literature models snow as atmospheric noise that degrades performance, our granular evaluation on the WADS-3D and CADC datasets reveals a counter-intuitive phenomenon: detection accuracy consistently improves as snow intensity increases. Through ablation study and meta-information analysis, we reconcile this by confirming that while falling snow acts as noise, its negative impact is outweighed by a dominant, concurrent factor: significantly increased object point count. We hypothesize that this object point count boost acts as a geometric enhancement, driven by snow accumulation increasing vehicle surface area and a shift from specular to diffuse reflectance. Our findings demonstrate that evaluating perception systems using broad weather-severity bins without accounting for correlated object-level data can lead to misleading conclusions about detector robustness. We conclude that accurate evaluation requires decoupling atmospheric noise from surface accumulation effects and outline future research to validate these physical models. The WADS-3D dataset is available at: https://digitalcommons.mtu.edu/wads-3d/.

Organizations increasingly rely on social networks to drive product adoption and word-of-mouth diffusion, yet a central challenge is that consumers differ not only in whether they adopt but also in how strongly they subsequently advocate. In particular, individuals who adopt under strong social reinforcement often develop greater conviction and become more persuasive. Existing diffusion models largely abstract away from this mechanism by treating influence strength as static. We introduce a conviction-based diffusion framework, the Pressure Threshold (PT) model, that captures how social reinforcement shapes both adoption and subsequent advocacy. In this model, an individual's influence increases with the cumulative social pressure experienced at adoption, providing a tractable representation of conviction formation while preserving the interpretability of classical threshold-based models. We study the managerial implications of this mechanism through marketing targeting and seeding. We show that accounting for reinforcement fundamentally alters optimal targeting strategies, shifting emphasis toward consumers in cohesive network regions where repeated exposure can amplify downstream word-of-mouth. While reinforcement introduces nonstandard features into the optimization problem, we demonstrate that scalable heuristics remain effective and provide data-dependent performance guarantees. Across synthetic and real-world networks, we find that reinforcement often substantially increases the effectiveness of seeding interventions, particularly in dense and clustered environments where opportunities for social validation are abundant. Using time-stamped social media cascades, we further show that the PT model improves predictive accuracy relative to standard diffusion models, highlighting the empirical relevance of conviction-driven amplification.

As AI agents increasingly rely on external tools, accurate retrieval is essential for reliable behavior. Existing methods optimize semantic relevance but largely ignore tool availability during execution. To address this gap, we propose RATED (Reliability-Aware Tool Retrieval via Execution Dynamics), a retrieval framework that combines robust semantic matching with dynamic reliability adaptation. RATED employs a frozen dual-encoder with contrastive fine-tuning for semantic matching, and a lightweight contextual bandit module that re-ranks candidates based on real-time execution feedback. This design enables rapid adaptation to runtime tool failures without expensive model retraining. We introduce EcoAgent, a benchmark of 5,172 real-world tools and over 210,000 query-tool pairs for offline retrieval and dynamic reliability evaluation. Experimental results demonstrate that RATED achieves a 52.2% relative MRR improvement over the ToolLLM API retriever on real-world queries, and reduces fault selection rate from 86.2% to 2.9% under severe API outage scenarios with a 30% failure rate, while incurring negligible latency overhead of less than 1 ms over encode-only inference. The code and dataset are available at https://github.com/Elliotes/EcoAgent.

In this paper, we explore mixed Buchsbaum–Rim multiplicities of finite-colength submodules of free modules over Noetherian local rings. We extend the Hilbert-polynomial construction to submodules in arbitrary homogeneous degrees and prove a product formula for the resulting coefficients. We then examine module analogues of Teissier's coefficient inequalities and the Minkowski inequality. Explicit examples show that the coefficientwise power inequalities, log-convexity, and the product-root inequality can all fail; several failures occur over regular local rings. We also prove that equality of integral closures forces equality of all mixed Buchsbaum–Rim multiplicities, while a one-dimensional example shows that the converse fails for arbitrary modules. A converse is recovered when the module closures are the integral closures of modules of the form $IF$ for $\mathfrak m$-primary ideals $I$. The general converse problem in dimension at least two is stated at the end.

We study the logarithmic Schrödinger equation with an arbitrary positive semidefinite quadratic potential and a periodic potential in the unconfined directions. The main variational difficulty is to turn a constrained minimizer into a critical point when the energy is extended-valued and not of class $C^1$ on the natural Hilbert space. We establish an abstract principle for convex lower semicontinuous energies with a smooth perturbation, allowing a scaling law for the full energy without requiring homogeneity of its convex part. Combined with a profile decomposition under integer translations, this principle yields a positive ground state for every rank of the quadratic form. When the periodic potential vanishes, we prove a sharp anisotropic logarithmic Sobolev inequality, determine its optimal constant and all equality cases, and classify positive ground states as explicit anisotropic Gaussons, unique up to translations along the kernel of the quadratic form. When the periodic component is constant, we also establish existence and classify all normalized ground states at every prescribed positive mass, with explicit formulas for the minimum energy and the associated Lagrange multiplier.

Personal agents increasingly connect to services across the web and applications by using tools. As the effects of their actions accumulate and interact across services, unintended or harmful outcomes can arise when agents fulfill benign user requests. Proactively identifying these compositional failures requires systematically exploring diverse cross-service paths. Existing red-teaming methods often simply prompt LLMs to identify potentially harmful paths from task and environment descriptions or decompose harmful objectives specified by humans or LLMs into seemingly benign action combinations. However, generation is biased toward patterns learned during training, potentially favoring familiar tool combinations and leaving novel cross-service interactions underexplored. To address this, we propose X-AgentRed, a self-evolving red-teaming framework that discovers harmful tool call paths through a graph of tool dependencies and expands and refines them using execution feedback. By analyzing each tool's effects, potential harms, and preconditions for execution, the red-teaming agent constructs a directed graph that captures how one tool's effects can satisfy another's execution preconditions. Recipes sampled from this graph are instantiated as realistic scenarios in which agents can produce harmful outcomes. Execution feedback enables self-improvement by diversifying paths that produce harm through tool substitution and refining preconditions at blocked steps in unsuccessful paths to enable further execution. Across four frontier agentic models, X-AgentRed achieves $1.77\times$ the harmful path rate of STAC and explores a more diverse set of tools, yielding $2.99\times$ the tool coverage on average.

Current AI predicts materials by speeding up energy minimisation and training on large databases. However, scaling data generation prevents manual analysis by domain experts. If novelty is evaluated by chemical compositions or traditional representations in Crystallographic Information Files, or standard structure matchers based on manually chosen thresholds, one can easily generate millions of “new” compositions or files without discovering truly novel materials. This paper uses a hierarchy of crystal invariants to continuously trace and quantify structural novelty based on the new mathematical definition of a crystal structure and asymptotic guarantees for invariant descriptors. The invariant-based analysis through cascade computations on a modest computer exposes the unexpected scale of duplication in the world's largest materials databases, shown here in the high-profile case of GNoME datasets. To prevent current “embarrassment of over-prediction”, we propose necessary conditions for next-generation AI that should be automatically enforced before claiming any new materials.

We introduce a general class of interacting reinforced stochastic systems evolving on a network. The dynamics are governed by two distinct interaction structures: a sampling network, which determines how information is acquired, and a reinforcement network, which determines how the sampled information influences future evolution. Processes generate Bernoulli observations through preferential, de-preferential, or mixed sampling, while observations are transmitted through an edge-dependent affine reinforcement mechanism. We formulate the dynamics as a stochastic approximation scheme and study their asymptotic behaviour. We establish sufficient conditions for almost sure convergence to a deterministic limit and, when the sampling and reinforcement graphs coincide, derive graph-theoretic criteria involving odd cycles, mixed-sampling vertices, and subcritical reinforcement. Under additional structural assumptions, we establish almost sure convergence and partial synchronization in a random-limit regime. In the deterministic regime, we also prove a law of the iterated logarithm and a central limit theorem under suitable spectral and regularity conditions. Finally, we show how the framework encompasses interacting urn models with edge-dependent reinforcement, urns with memory, and random walks with memory.

We study a continuum model of frustrated magnetism that supports particle-like field configurations with nontrivial Hopf index. The energy functional combines competing second and first order exchange interactions with a stabilizing external magnetic field and gives rise to a variational problem for maps from three-dimensional space into the two-sphere. We prove the existence of energy minimizers in infinitely many nontrivial homotopy classes. We further establish the existence of a unit-charge Hopfion in the regime of strong external fields. Finally, we analyze the corresponding large-field limit, identify the natural concentration scale, and prove strong convergence of suitably rescaled minimizers to a minimizer of a limiting variational problem.

Air pollution poses a significant threat to public health, exacerbated by the rapid industrialization and urbanization of modern society. While traditional deep learning offers predictive capabilities, recent advancements in quantum machine learning (QML) present a compelling alternative. Here, we introduce a scalable hybrid framework that integrates variational quantum circuits with quantum annealing. Leveraging a neuromorphic quantum computing platform to transcend the qubit limitations of the NISQ era, we model particulate matter (PM$_{2.5}$) dynamics using high-density urban data from the UK. We demonstrate that our Quantum Neural Networks (QNNs) exhibit an intrinsic resilience to input volatility, significantly outperforming classical baselines (MLP and LSTM) on raw, high-entropy environmental data. Furthermore, we successfully scaled the model to 172 qubits, observing a synergistic enhancement when coupled with wavelet decomposition that achieves state-of-the-art predictive fidelity. These findings establish a critical benchmark for quantum utility, proving that scalable, neuromorphic quantum machine learning can now address complex, non-linear real-world problems.

A memory that reproduces every current prediction may still have forgotten how to learn from the next observation. We characterize this distinction for exact cumulative learning and attention. For smooth convex residual losses with unrestricted real features and labels, a continuous recurrent state of fixed dimension exists exactly when the loss is polynomial. With ordinary, unweighted observations, degree $D$ requires precisely $r=\binom{d+D}{D}-1$ coordinates; even among histories with one fixed current predictor, exactly $r-d$ coordinates can remain necessary for continuation. Nonpolynomial losses have unbounded requirements on such equal-prediction histories. Bounded scalar logistic examples retain this obstruction after fixing any finite derivative jet. For ridge regression, the hidden evidence is the design matrix: the exact state costs $dp+d(d+1)/2$, versus $dp$ for the predictor. On a well-conditioned zero-prediction family, optimal $B$-bit compression has average one-update error of order $d^{-1}2^{-2B/s_d}$, $s_d=d(d+1)/2$, even with arbitrary randomized codes. Kernel attention gives a second exact closure classification. The results identify what future updates reveal, rather than proposing a new regression algorithm or asserting trained-model gains.

Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about $10^8$ two-block codes with up to $320$ qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with $n \le 200$ to reach $kd^2/n=12$, no weight-six two-block code with $n \le 300$ beats $[[288,16,16]]$, and the known $[[64,18,8]]$ is the best two-block code by $kd^2/n$ across weights five to nine with $n \le 94$. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small $n$ and non-abelian groups at large $n$, with the crossover appearing to move out as the check weight grows; and high quality codes become $100$ to $1000$ times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are $[[102,20,10]]$ and $[[120,24,10]]$ at check weight nine, which we simulate in noisy syndrome-extraction circuits, and $[[186,20,18]]$, a weight-nine code with every check in a $5\times5$ window. At check weights seven to nine, our best codes exceed the best published $kd^2/n$ with exact distance at the same check weight in most length bands up to $320$ qubits. All codes are available in a public database.

Quantum low-density parity-check (qLDPC) decoding requires reliable corrections and low latency over long syndrome histories. We extend BP-SF, an oscillation-guided belief-propagation (BP) decoder with independent syndrome-flip trials, by introducing windowed and streaming operations. Our CUDA implementation combines disordered-memory retries, sliding windows, correlation-aware graphs, and layered scheduling. Circuit-level simulations test fixed-basis memory and complete logical Pauli-frame recovery. On the [[144,12,12]] bivariate bicycle code at physical error rate p = 0.002, parallel BP-SF reduces 99.9th-percentile call latency by a factor of 13.15 relative to an eight-chain Relay-BP GPU baseline on 100,000 matched samples for which every tested method's initial BP stage fails within 80 iterations. Windowed BP-SF maintains a flat logical error rate and nearly constant per-round compute time through 512 rounds, reaching a 17.1-fold speedup over monolithic decoding. In 512-round full-frame streaming decoding with 2.97-ms syndrome arrivals, layered BP-SF on a transformed joint graph with 20 retries reduces 99.9th-percentile correction-release age from 166.29 to 36.25 ms relative to the joint graph with flooding updates and 100 retries. These results connect parallel BP post-processing to lower decoding costs and earlier correction release in streaming quantum memory.

We propose a new class of cyclic (noncyclic) operator type condensing mappings and establish conditions that guarantee the existence of best proximity points (pairs) using measure of noncompactness. The results obtained are further applied to investigate the optimum solution of an infinite system of differential equations in $c_0$ and $\ell_1$.

Lazard theory is becoming increasingly important in $p$-adic representation theory, especially when studying mod $p$ representations. For $G$ a connected reductive group defined over a finite extension $K/\mathbb Q_p$, a question going back to Lazard is when a pro-$p$ Iwahori subgroup of $G(K)$ admits a saturated $p$-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated $p$-valuation on $G_{x,\frac{1}{p-1}+}$ for any point $x$ in the semisimple Bruhat-Tits building of $G$, given a mild assumption on the maximal torus of $G$ and that $p>2$. We also compute Lazard Lie algebras defined over $\mathbb Z_p$ and the polynomial ring $\mathbb F_p[\varepsilon]$ associated to the saturated group $G_{x,\frac{1}{p-1}+}$, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over $\mathbb F_p[\varepsilon]$.

Foundation models have recently gained significant attention in fields such as high energy physics (HEP). The tokenizers that convert detector data into discrete inputs for such models, however, are comparatively unexplored. This paper addresses this gap by learning discrete and compressed tokenized representations of HEP datasets. We demonstrate the approach on four modalities covering two different simulated detectors that vary in complexity. Treating tokenization as a rate-distortion problem, a systematic hyperparameter scan yields four design rules for tokenizing HEP data that hold across all set-based modalities studied. The cost of tokenization is evaluated on tasks including jet $p_T$ regression, particle flow reconstruction, and flavor tagging. We find that tokenization preserves much of the information needed for these tasks at compression factors of approximately 4-11$\times$ while keeping the jet $p_T$ resolution below 1%. We also present heptokens, an open-source library for tokenizing HEP data.

We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-$\sigma$ KdV charge inserted with chemical potential $\alpha_\sigma$. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the $c=1/2$ free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-$5$ deformation for the first few charges evaluated on low-lying states.

We investigate the rate of change of equilibrium potential energy for nonlinear elastic solids undergoing evolutions that may involve changes in body geometry, material properties, applied tractions, and prescribed displacements. We represent such an evolution by a one-parameter family of variational problems and identify a class of these families for which the potential-energy release rate is well defined. Particular attention is given to the formulation of the potential energy at each evolution state: each constituent variational problem must yield the appropriate equilibrium state, while its equilibrium value must also correctly represent the physical change in potential energy between states. Within this framework, we derive a closed-form expression for the potential-energy release rate for solids with bounded domains, allowing for spatially nonuniform material evolution, sharp material interfaces, and evolving boundary geometry and prescribed boundary data. The expression is determined entirely by the equilibrium solution at the state under consideration and by the quantities prescribing the local evolution, so neighboring equilibrium problems need not be solved. Finally, we apply the result to the extension of a notch in a two-dimensional elastic body and show that the resulting potential-energy release rate reduces to the classical $J$-integral.

Extending the weak-coupling expansion of the pressure of a cold and dense ultrarelativistic electron gas to next-to-next-to-next-to-leading order (N3LO) in the gauge coupling has remained an open problem since the classic N2LO result of Freedman and McLerran, published in 1977. Here, we present the full N3LO pressure, the first thermodynamic quantity determined to this order in a physical gauge theory. The pressure exhibits good perturbative convergence and weak renormalization-scale dependence even at sizable couplings, while a Bayesian analysis of missing higher-order contributions indicates uncertainties that can exceed those inferred from scale variation alone.

Accurate detection of drivable area is essential for autonomous driving and advanced driver assistance systems. While state-of-the-art methods focus on general traversability estimation, they prove inadequate for snow-covered roads where following existing wheel tracks offers crucial advantages: improved traction, predictable vehicle dynamics, and verified path stability. To address this, we propose a low-cost, self-supervised approach detecting safety-critical wheel tracks using synchronized RGB-I (RGB + Long-Wave Infrared) imagery. Our method automatically generates training labels by projecting future wheel trajectories from recorded GNSS data. A neural network is designed to identify wheel tracks with the 4-channel RGB-I images from challenging, white-out road surfaces. Experimental results demonstrate the effectiveness of our approach in visually monotonous and featureless adverse environment, offering a robust and cost-effective alternative for winter autonomy. The newly introduced ThermalTrack dataset is available at: https://digitalcommons.mtu.edu/thermaltrack/.

Many optimization problems provide direct access to only part of the gradient, while the remaining component must be inferred from function evaluations. This structure arises in performative prediction, performative reinforcement learning, and bilevel optimization with a black-box lower level. We introduce Partially Informed Stochastic Optimization (PISO), which combines the available gradient component with a running estimate of the residual. By centring each zeroth-order observation on this estimate, PISO makes the variance due to random probing depend on the tracking error rather than the residual's magnitude. This yields a randomized sketch iteration with an explicit contraction mechanism and a principled interpretation of momentum. For smooth nonconvex objectives, we establish convergence guarantees that depend on the residual's variation and initial magnitude, without requiring a uniform bound on the unavailable gradient component. With independently sampled function evaluations and Gaussian or spherical probes, PISO attains leading function-evaluation complexities of \(O(d^{2}\varepsilon^{-6})\) under Lipschitz gradients and \(O(d^{2}\varepsilon^{-5})\) under an additional Lipschitz-Hessian assumption. These rates require only one probe direction, two function samples, and one known-component sample per update. The analysis also quantifies how batching trades observations for fewer deployments. Across five settings, PISO variants improve on the tested baselines with comparable oracle access in three, reach a comparable solution faster in bilevel routing, and perform best among methods without response-model derivatives in performative reinforcement learning.

To balance task performance and computational cost, agentic workflows can use a lower-cost model for execution and selectively consult a stronger model. In such executor-advisor systems, the executor decides when to seek advice, making escalation depend on self-assessment, although current language models struggle to detect where their own reasoning fails. The need for stronger reasoning emerges during execution and varies across workflow phases. We introduce ARIA, a meta-controller that learns when to consult by balancing task success against consultation cost. Our approach formulates consultation as a sequential resource-allocation problem. At workflow contact points, ARIA maps execution evidence to a compact state abstraction and selects SILENT or CONSULT, with action values learned by Monte Carlo control from cost-penalized outcomes. Across evaluated tasks of SWE-bench Verified, Terminal-Bench 2.1, and $\tau^2$-bench, ARIA recovers 44-81% of the executor-advisor success gap at less than half the advisor model's cost. ARIA achieves the best observed success-cost trade-off among ten other evaluated conditions, including routing and cascading methods, fixed and prompted consultation, and random placement at matched advice volume. The policies are compact and inspectable, with a single consultation-price parameter governing the success-cost trade-off without retraining either underlying model.

In this paper, we study biholomorphic mappings $F: \mathbb{B} \to X$ of the form $F(z) = f(z)z$ for $z \in \mathbb{B}$, where $f: \mathbb{B} \to \mathbb{C}$ is a holomorphic function satisfying $f(0) = 1$. Our main contributions are as follows: We establish sharp estimates for the coefficients of starlike functions of order $\alpha$ as well as quasi-convex functions of type $B$ and order $\alpha$ by employing the notion of majorization. The results established here directly extend the results due to Robertson \cite{Robertson-1936} to the broader setting of complex Banach spaces.

Nucleosynthesis simulations of the r-process require neutron capture rates for neutron-rich nuclei, many of which lie beyond experimental reach. Predicting these rates relies on theoretical calculations of the low-energy electric dipole (E1) strength near the neutron threshold. We present the first ab initio computation of the E1 strength along the nickel isotopic chain, for $^{58}$Ni and the even-even isotopes $^{68-80}$Ni. The E1 strength is obtained with the Lorentz integral transform coupled-cluster (LIT-CC) method, extended to axially symmetric reference states to reach open-shell nuclei, using two- and three-nucleon interactions from chiral effective field theory. The approach is benchmarked against the measured photoabsorption cross section of $^{58}$Ni and the E1 strength of $^{68}$Ni. Along the chain we find low-lying E1 strength above the neutron threshold that grows with neutron number. Including it in the reaction code TALYS enhances the neutron capture rates forming $^{68-78}$Ni between 3 and 10 GK, with an effect that increases with temperature and neutron number and an uncertainty dominated by the inversion of the integral transform. For $^{80}$Ni, where strong pygmy dipole strength lies just above threshold, the enhancement sets in already around 0.5 GK. Together, these results open the way to ab initio input for neutron capture rates along isotopic chains far from stability.

Contemporary large-scale structure analyses center around the galaxy power spectrum, modeled with one-loop perturbation theory. This description naturally breaks down at large wavenumbers, prohibiting the extraction of cosmological information from small-scale clustering. In this work, we present a complete model for the redshift-space galaxy power spectrum up to two-loop order, derived in the framework of the Effective Field Theory of Large Scale Structure. This accounts for a wide variety of effects, including fifth-order galaxy bias, redshift-space distortions, small-scale non-linearities, spatial non-locality, long-wavelength bulk flows, coordinate distortions, and stochasticity, involving a total of 49 free coefficients. Our model additionally encodes all deterministic contributions to the one-loop bispectrum and trispectrum (as well as the tree-level quadspectrum and pentaspectrum), which depend on many of the same parameters. We implement the model numerically, using the vegas algorithm to precompute the loop integrals; this allows the full likelihood to be evaluated in milliseconds. The approach is validated using power spectra measured from high-resolution simulations, considering both dark matter and realistic galaxy samples. The extension to two-loop order greatly increases the range of scales over which the power spectrum can be modeled, with $k_{\rm max}$ rising from $0.2h\,\mathrm{Mpc}^{-1}$ to $0.34h\,\mathrm{Mpc}^{-1}$ for a luminous red galaxy sample at $z=0.5$. Much of the additional information is absorbed by the numerous fifth-order biases and counterterms; for this reason, constraints on $\sigma_8$ are not significantly enhanced by the inclusion of two-loop terms. Future work will assess the full implications of the two-loop model for cosmological parameter inference, both alone and in combination with higher-order statistics.

In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a $C^{1,\beta}$-manifold. This extends the analysis in \cite{Ca24} to dimensions $\geq 3$ and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones $\beta$-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.

In cosmic ray propagation through turbulent magnetic fields, transport perpendicular to the mean field plays an important role in various astrophysical environments. Numerical simulations have shown that parallel and perpendicular diffusion coefficients exhibit different energy scalings, causing their ratio to increase with energy, but a comprehensive theoretical explanation remains elusive. Here, we investigate perpendicular diffusion using test particle simulations in synthetic isotropic turbulence spanning 6-7 orders of magnitude in wavenumber, together with a theoretical extension of nonlinear guiding center theory that incorporates stochastic drifts. Our approach combines field line random walk and random guiding center drifts to describe perpendicular transport. We find that parallel and perpendicular diffusion coefficients exhibit the same energy scaling only for $r_L \lesssim 10^{-2} L_c$, where $r_L$ is the particle Larmor radius and $L_c$ is the turbulence coherence length. At higher energies, perpendicular diffusion develops a stronger energy dependence, consistently across different turbulence levels. Our theoretical model suggests that this behavior arises from stochastic drifts induced by turbulent magnetic field components parallel to the mean field.

We find the Dirichlet series and convergence bounds for the counting function of sum systems (collections of sets of non-negative integers whose Minkowski sum is an arithmetic progression). This motivates the definition and study of divisor function polynomials, which are generating functions for the non-trivial divisor function and for the balanced divisor function that counts, up to a sign, non-trivial ordered factorisations into squarefree factors. These divisor function polynomials turn out to be intimately connected between each other and with Bernoulli numbers and ordered Bell polynomials.

Two graphs are degree similar if their adjacency matrices and their degree matrices are simultaneously similar, a notion recently introduced by Godsil and Sun. We extend it to signed graphs in two ways, degree similarity and net-degree similarity, according to whether the degree or the net-degree matrix is used. We prove that signed trees are determined by their degree similarity class, but not by their net-degree similarity class. We then develop constructions of (net-)degree similar signed graphs, based on a signed version of Godsil–McKay switching, on partial transposition and on block extensions. As a byproduct, we obtain infinite families both of pairs of net-degree similar unicyclic signed graphs with non-isomorphic underlying graphs, and of unicyclic signed graphs, not necessarily regular, which are not determined by their degree similarity class. This settles the net-degree version of a problem of Godsil and Sun on unicyclic graphs.