Beam theory has traditionally been restricted to small elastic strains and rigid cross-sections. Relaxing these assumptions within closed-form analytical frameworks remains challenging. In contrast, the cross-sectional warping problem provides a computational approach that enables the derivation of general, nonlinear constitutive relations for beam models, thereby overcoming both limitations. In this work, we reinterpret the cross-sectional warping problem for hyperelastic beams and propose a fully material formulation in terms of the Green-Lagrange strain and the second Piola-Kirchhoff stress tensors. Owing to the symmetry of these tensors, the formulation can be expressed efficiently in Voigt notation and is thus particularly well-suited for straightforward numerical implementation. We demonstrate the validity of this alternative formulation in numerical examples, including the computation of the effective beam stiffness, for which we derive the sensitivities of the warping displacement. To promote reproducibility, we accompany this article with an open-access repository containing the isogeometric finite element implementation and all numerical examples presented herein, enabling other researchers to readily reproduce and build upon our results.

The attitude dynamics of aircraft and spacecraft exhibit significantly nonlinear behaviour. In spacecraft, torque is generated through reaction wheels and control moment gyros. In aircraft, torque is generated using lift on control surfaces. In both cases, complex geometries, unique configurations, and internal/environmental changes imply that models must be identified, verified, and updated using in-flight experimental data. However, this data is often noisy, sparsely sampled, and partial in that modeled states may not be directly measurable. In this paper, we propose a method for estimating key parameters in realistic Ordinary Differential Equation (ODE) models of both spacecraft and aircraft dynamics. This method avoids the need to directly measure state derivatives by coupling sampled outputs using the Picard mapping -- an integral constraint on the solution of the parameterized ODE. This constraint is then enforced, and optimal parameter estimates are found using a gradient contraction algorithm. This algorithm is applied to well-studied models of spacecraft and aircraft motion. First, the algorithm is used to estimate the inertia tensor in a 4 control-moment gyro (CMG) model of spacecraft motion. Second, we estimate the 28 higher-order control surface coefficients in a model of the F/A-18 aircraft.

At the fundamental conceptual level, two alternatives have traditionally been considered for how mutations arise and how evolution happens: 1) random mutation and natural selection, and 2) Lamarckism. Recently, the theory of Interaction-based Evolution (IBE) has been proposed, according to which mutations are neither random nor Lamarckian, but are influenced by information accumulating internally in the genome over generations. Based on the estimation-of-distribution algorithms framework, we present a simulation model that demonstrates nonrandom, non-Lamarckian mutation concretely while capturing indirectly several aspects of IBE: selection, recombination, and nonrandom, non-Lamarckian mutation interact in a complementary fashion; evolution is driven by the interaction of parsimony and fit; and random bits do not directly encode improvement but enable generalization by the manner in which they connect with the rest of the evolutionary process. Connections are drawn to Darwin's observations that changed conditions increase the rate of production of heritable variation; to the causes of bell-shaped distributions of traits and how these distributions respond to selection; and to computational learning theory, where analogizing evolution to learning in accord with IBE casts individuals as examples and places the learned hypothesis at the population level. The model highlights the importance of incorporating internal integration of information through heritable change in both evolutionary theory and evolutionary computation.

Let \(H(n)\) denote the Hilbert number, i.e.\ the maximal number of limit cycles of planar polynomial vector fields of degree \(\le n\). A classical lower-bound mechanism for \(H(n)\) is \emph{replication}: one pulls back a vector field by a polynomial map and lifts each existing limit cycle to several disjoint copies while controlling the resulting degree. In this paper we give a fully self-contained replication theorem based on the separable Chebyshev covering \[ Φ(u,v)=(T_m(u),T_m(v)). \] Using the \(m\) monotone full branches of \(T_m\) on \((-1,1)\), we prove that every degree-\(\le n\) polynomial vector field with \(k\) limit cycles gives rise to a degree-\(\le nm+m-1\) polynomial vector field with at least \(m^2k\) limit cycles. Consequently, \[ H(nm+m-1)\ge m^2H(n)\qquad (m\ge 2). \] We then extend the construction to general separable pullbacks \((u,v)\mapsto (p(u),p(v))\), show that Chebyshev attains the maximal possible branch count among degree-\(m\) separable pullbacks, and prove a quadratic ceiling for replication-only schemes: if one iterates separable pullbacks and no additional limit cycles are created beyond those forced by lifting, then the number of resulting limit cycles is at most quadratic in the final degree. This shows that superquadratic lower bounds, such as the known \(n^2\log n\)-type bounds, necessarily require mechanisms beyond pure separable replication. Finally, combining our replication theorem with the strongest currently published seed bounds, we obtain new explicit lower estimates in several degrees, including \begin{gather*} H(14)\ge 252,\qquad H(29)\ge 1080,\\ H(31)\ge 1380,\qquad H(39)\ge 2012. \end{gather*}

Surrogate markers offer the potential to reduce the burden of data collection by replacing costly or invasive primary outcomes with more accessible measurements, provided that they can faithfully indicate the effectiveness of a treatment. However, appropriate evaluation of a surrogate is particularly complex in longitudinal studies, where both outcomes and surrogates can evolve dynamically over time and interest lies not only in the treatment effect at one time, but rather treatment effects that may vary along the entire trajectory. In this paper, we develop a statistical framework for surrogate evaluation when both the surrogate and primary outcome are measured over time. Specifically, within the potential outcomes framework, we propose a formal causal definition of the proportion of the treatment effect on the longitudinal primary outcome that is explained by the treatment effect on the longitudinal surrogate. For estimation, we leverage state-space models, together with the Kalman filter and smoother, enabling efficient estimation of treatment effects under realistic conditions of temporal evolution and patient-level variability. We introduce a nonparametric bootstrap strategy for state-space models, a temporal homogeneity test, and demonstrate the finite-sample performance of our proposed methods via a simulation study and application to a diabetes clinical trial.

Code reasoning tasks are increasingly crucial to evaluating large language models (LLMs). Yet most existing benchmarks rely on simplistic, LLM-generated snippets or human-written solutions to code challenges and often restrict inputs and outputs to primitive types, failing to reflect the structure and dependencies of real-world projects. These simplifications limit their ability to measure practical generalizability. We present R2Eval1, a benchmark of 135 code reasoning problems drawn from ten widely used Python projects. Unlike prior work, R2Eval serializes compound and custom types, preserving real-world data complexity and enabling a more realistic assessment of LLMs.

Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only elementary methods and apply it to several types of Hurwitz numbers: single, double, or with an arbitrary numbers of fixed ramifications; simple and / or including completed cycles type of ramification and / or finitely many blocks of weakly monotone and / or strictly monotone types of ramifications. These, to the best of our knowledge, cover most of the Hurwitz numbers studied, and include for instance correlators of the HCIZ matrix model, Grothendieck dessins d'enfant, weighted Hurwitz numbers, and Gromov-Witten invariants of the Riemann sphere. We also apply our method to b-content Hurwitz numbers. As a specialisation, we recover some previously known about the large genus asymptotics of Hurwitz theory, namely classical results by Hurwitz and recent results of Do-He-Robertson, C. Yang, and results connected to recent work of X. Li.

Digital waveguide physical modeling offers efficient simulation of acoustic wave propagation as compared to general finite-difference schemes commonly used in computational physics. This efficiency has enabled the real-time implementation of physically modeled musical instruments and sound effects, as well as real-time vocal models and artificial reverberation. This paper provides an overview of the historical evolution and applications of digital waveguide modeling and highlights recent advances in the field. Parametric optimization using classical, evolutionary and neural approaches are also discussed and compared. Digital waveguides provide physically accurate simulations with reduced computational cost, and can now be optimized with modern machine learning and differentiable digital signal processing techniques.

We prove a general Fueter Theorem over real alternative *-algebras. We show that a suitable power of the Laplacian maps Dunkl-regular functions to Dunkl monogenic functions with axial symmetries. Using the embedding of hypercomplex function theories in the class of Dunkl monogenic functions, we subsume several Fueter-type results known in the literature and obtain the most general form for the action of the Laplacian on function spaces over hypercomplex subspaces. We show that Fueter Theorems are in a one-to-one correspondence with a class of graphs, the Fueter trees, that describe the interactions between Dunkl-regular function spaces and the relation with the iterated Laplacian. We obtain that the number of distinct Fueter trees on a hypercomplex space of dimension $n+1$ is equal to the number of partitions in odd parts of the integer $n$.

We present an integral formulation of classical Yang-Mills theory coupled to fermionic and scalar matter fields in (1+1)-dimensional Minkowski spacetime. By reformulating the local dynamics in terms of loop-space holonomies, we demonstrate that the path independence of the holonomy eigenvalues constitutes a conservation law, yielding an infinite hierarchy of gauge-invariant, dynamically conserved charges. While a zero-curvature equation is associated with a necessary condition for this path invariance, we note that it is not strictly sufficient on its own. Employing a first-order symplectic formalism, we show that these non-abelian charges generate global symmetry transformations on the fundamental phase-space variables. We rigorously prove that these transformations preserve the physical dynamics, leaving the total Hamiltonian invariant up to first-class constraints. Furthermore, an analysis of the Poisson algebra reveals that these conserved charges are in involution, provided the boundary integration constant lies within the center of the gauge group. This exact, lower-dimensional framework provides a highly tractable setting to investigate the algebraic structures of these hidden symmetries and the meaning of the conserved charges as physical observables, establishing a classical foundation for exploring their role in the quantum regime, such as in strongly coupled lattice gauge theories.

We consider the problem of reconstructing missing data on a smooth manifold from incomplete and nonuniform samples. While classical methods for manifold approximation typically assume quasi-uniform data, their performance deteriorates significantly in the presence of large gaps or holes. We propose a unified framework for manifold data imputation that reduces the problem to function reconstruction on locally defined tangent spaces. The approach combines two complementary strategies. The first is a Fourier-based method that determines missing values by prescribing a decay rate of the discrete Fourier coefficients, thereby enforcing high-order smoothness through a global spectral criterion. The second is a local variational method based on minimizing high-order central differences, leading to sparse least-squares systems with favorable stability and conditioning properties. We establish a discrete inverse estimate linking decay of Fourier coefficients to uniform bounds on high-order divided differences, providing a theoretical foundation for the spectral approach. For the variational method, we analyze existence, uniqueness, and scaling behavior, showing that conditioning depends primarily on the geometry of the missing region. These functional reconstruction techniques are integrated with a moving least-squares projection framework to yield a practical algorithm for manifold completion. Numerical experiments, including reconstruction on surfaces with significant missing regions, demonstrate accurate and stable recovery without requiring a global parameterization. The proposed framework provides a flexible and effective approach to manifold data imputation in challenging settings with incomplete data.

We propose a theory based on dynamical systems to explain and predict the occurrence of extreme events, of which critical transitions form a subset. In fast-slow nonlinear systems, we identify a cascade of events preceding extreme events: (i) a slow regime, in which the fast covariant Lyapunov vectors (CLVs) are both tangent to the fast eigenvectors and remain transversal to the slow subspace; (ii) a transition regime, in which the fast eigenvalues become neutrally stable while the fast CLVs are no longer tangent to the fast eigenvectors; and (iii) a critical regime, in which a strong spectral gap in the eigenvalues causes both fast and slow CLVs to become tangent along the dominant fast direction, breaking the transversality between fast and slow subspaces. Building on this cascade, we propose two precursors to forewarn the occurrence of extreme events. We numerically test the theory and precursors on low- and higher-dimensional systems. The proposed precursors predict extreme events and critical transitions with 100% precision and recall. This work opens opportunities for time-forecasting extreme events using theoretically grounded precursors.

Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union of residue classes. This gives a new inverse theorem for Kneser's sumset inequality in the integers, and provides a partial answer to a long-standing open question of Erdős and Graham.

Online Feedback Optimization leverages properties of optimization algorithms to develop controllers for systems with limited model availability, which is often the case in process control. The interplay between the parameters of the chosen optimization algorithm, as well as lack of direct connection to the characteristics of the underlying process make their tuning challenging. We propose a method for adaptive tuning of Online Feedback Optimization controllers based on scaled projected gradient descent by using sensitivity of the desired objective to the parameters of the algorithm. The proposed adaptive tuning method limits the operator-tunable parameters to scalar values that represent how much the control inputs and the objective can change between iterations without requiring either additional information about the controlled system or repeated experiments. Numerical studies on a gas lift and a continuously-stirred tank reactor processes confirm that our adaptive scheme improves closed-loop performance of Online Feedback optimization compared to standard manual tuning methods.

We develop the interpolatory $\mathcal{H}_2$ optimal model reduction framework for linear control systems posed on infinite dimensional state, input and output spaces. Specifically, we consider linear systems formulated as controlled abstract Cauchy problems on a Banach space and approximate them via Petrov-Galerkin projection onto finite dimensional trial and test subspaces. We show that the resulting reduced order transfer function interpolates the original at prescribed points, and we characterize precisely how the projection subspaces must be constructed to enforce this interpolation. Building on this, we develop a data-driven realization framework -- an infinite dimensional analogue of the Loewner approach -- that recovers the system behavior directly from input-output data without requiring access to the underlying operators. Finally, we derive $\mathcal{H}_2$ optimality conditions for the reduced model and show that the classical interpolatory characterization persists in this infinite dimensional setting: first-order optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles. Taken together, these results establish that the interpolatory $\mathcal{H}_2$ optimal model reduction theory extends naturally and completely to infinite dimensional linear control systems with infinite dimensional input and output spaces.

Gas hydrates are potential candidates in future energy sources while simultaneously providing structures with extensive applications in carbon capture and storage, gas transport, and important separation processes. Prior research in the field considers the dynamics of the water molecule backbone in particular. We investigated the pressure-enthalpy landscape and mechanical stability envelope of sI methane and carbon dioxide hydrates simulated using DFT. We investigated the effect of the revPBE + DFT-D2 and the SCAN + rVV10 and their treatment of the exchange correlation interactions. We examined the zero pressure material properties, finding that revPBE comparatively underbinds the interactions, causing more flexible structures with large equilibrium volumes. Under pressure, the carbon dioxide molecule was found to align itself parallel to the hexagonal faces of the large cage despite the functional used. Additionally, the property differences are caused by the ability of the carbon dioxide molecule to rotate and disperse the changes in the energy landscape in ways that methane molecules cannot. This computational methodology describes the elastic stability of gas hydrate, marginal stability, and critical differences across important molecular interactions, confirming experimentally observed restrictions in guest molecule rotations and novel pressure behaviors under hydrostatic loads

Multivariate mixed-type outcomes are difficult to model jointly, and additional complexity arises when both marginal effects and dependence structures vary with a covariate such as age or time. Existing approaches often impose restrictive dependence assumptions or lack sufficient flexibility to accommodate heterogeneous response types in a unified framework. To address this issue, we propose a Bayesian nonparametric framework for multivariate conditional copula regression with varying coefficients. The proposed model combines adaptive spline-based marginal regressions with an infinite mixture of Gaussian copulas whose weights vary with the covariate through a probit stick-breaking process. This construction provides flexible covariate-dependent dependence modeling while avoiding explicit global constraints on functional correlation matrices. We further establish approximation results for the proposed copula representation and develop a Markov chain Monte Carlo algorithm for posterior inference. Simulation studies show accurate recovery under correct specification and robust performance under copula misspecification. In an analysis of the BRFSS 2023 data, the proposed model reveals age-varying marginal effects and dependence patterns among multiple health outcomes, providing a coherent joint view of multimorbidity beyond separate marginal analyses.

We consider the Schrödinger equation with a multipoint potential of Bethe-Peierls-Thomas-Fermi type. For this singular potential, we develop scattering and inverse scattering at high energies. In particular, in this framework, our results include analogs of the "regular" Born-Faddeev formula for the scattering amplitude and analogs of related "regular" inverse scattering reconstructions at high energies. Related results for scattering solutions at high energies are also presented.

Celestial symmetries of gravity and gauge theory can be enhanced to a $w_{1+\infty}$ algebra and an $S$-algebra respectively, when restricting to a single graviton/gluon helicity sector. Difficulties in combining both sectors in the full theory have been pointed out in the previous literature. In this work, we face this problem from the covariant phase space perspective and analyze in detail the structure of the mixed-helicity bracket of the higher-spin charges for both gravity and Yang--Mills theory. We show that, when restricting one of the two helicities to the wedge sector, a closed algebra can be obtained for all spins in terms of a notion of shadow charge we introduce. Furthermore, when focusing on the lower spin subalgebra sectors, in the case of gravity, we show that a dual mass extension of the BMS algebra can be consistently constructed; in the case of Maxwell theory, inclusion of magnetic charges allows us to recover a non-vanishing expression for the electromagnetic central charge previously obtained through different methods.

Despite their global prevalence, many Large Language Models (LLMs) are aligned to a monolithic, often Western-centric set of values. This paper investigates the more challenging task of fine-grained value alignment: examining whether LLMs can emulate the distinct cultural values of demographic subgroups. Using Singapore as a case study and the World Values Survey (WVS), we examine the value landscape and show that even state-of-the-art models like GPT-4.1 achieve only 57.4% accuracy in predicting subgroup modal preferences. We construct a dataset of over 20,000 samples to train and evaluate a range of models. We demonstrate that simple fine-tuning on structured numerical preferences yields substantial gains, improving accuracy on unseen, out-of-distribution subgroups by an average of 17.4%. These gains partially transfer to open-ended generation. However, we find significant pre-existing performance biases, where models better emulate young, male, Chinese, and Christian personas. Furthermore, while fine-tuning improves average performance, it widens the disparity between subgroups when measured by distance-aware metrics. Our work offers insights into the limits and fairness implications of subgroup-level cultural alignment.

With increasing emphasis on transparency in digital governance, users expect more than silence when their access requests are denied by a system. However, authorization methods are notorious for their inability to provide any form of meaningful feedback under such situations. This paper shows a direction towards how the problem of explainability can be mitigated in the context of Attribute-based Access Control (ABAC), arguably the most researched topic in access control in recent years. We introduce EXTree, which represents ABAC policies optimized for both fast evaluation (Efficiency) and human-centric feedback (Explainability) in the form of a tree. Two strategic dimensions are investigated, namely, Feedback Evaluation Strategies - how to craft actionable explanations when access is denied, and Tree Construction Strategies - how the policy trees should be structured for efficient yet interpretable decisions. Through extensive experiments, we compare entropy-based, changeability-based, and randomly generated trees across multiple configurations. Our results demonstrate that EXTree, built for efficiency and interpretability, can bridge the gap between complex authorization logic and human understanding.

Motivated by generalized geometry (in the sense of Hitchin), the product bundle ${\mathcal Z}\times_{M} {\mathcal Z}$ of the twistor space ${\mathcal Z}$ of a Riemannian manifold $(M,g)$ is considered. The product twistor space admits a natural family of Riemannian metrics and four compatible almost complex structures, analogs of the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space. The Gray-Hervellal classes of these almost Hermitian structures are determined in the case when the dimension of the base manifold $M$ is four.

Trimming is a ubiquitous operation in computer-aided-design whereby parts of a geometry are merged, intersected, or simply discarded. While it grants virtually unlimited flexibility in geometric design, it introduces a plethora of other difficulties when such geometries are used within immersed finite element methods. In particular, small cut elements lead to severely ill-conditioned system matrices requiring dedicated penalization, stabilization, or preconditioning techniques. In this work, we highlight the limitations of existing preconditioning strategies by first carefully examining the condition number of the diagonally scaled matrix and later providing realistic counter-examples for some well-established preconditioning strategies. Building on those insights, we propose a robust deflation-based preconditioning technique tailored to immersed finite element methods.

We study large-scale dynamics in the Ginzburg-Landau equation (GLE) using a reduced description derived from a WKB expansion. Rigorous mathematical results establishing that this reduced equation accurately approximates the full GLE are currently limited to the real GLE (RGLE) and exclude phase-slip dynamics. For the RGLE, we find that the reduced equation has conserved gradient form and show that, upon inclusion of a higher-order regularization, it admits exact stationary solutions. In the reduced dynamics, all nonuniform steady states are linearly unstable and among them, localized hole solutions identified through the reduced description differ from the classical hole solution of the RGLE due to Langer and Ambegaokar. In the Eckhaus-unstable regime, we derive a self-similar description of the approach to finite-time singularities in the reduced equation, with scaling exponents that agree with direct numerical simulations (DNS), and a similarity profile obtained from a nonlinear 4th-order boundary value problem. Extending the reduction to the complex GLE (CGLE) with nearly real coefficients introduces a Burgers nonlinearity that generates traveling shocks connecting two distinct plane-waves. We obtain exact expressions for the shock profile and perform extensive DNS to demonstrate convergence to the predicted profile in the appropriate large-scale, nearly real-coefficient limit of the CGLE. Away from this limit, the wave number profile loses monotonicity, which we explain in the framework of spatial dynamics. We further show that the exact shock solutions found here are qualitatively distinct from the Nozaki-Bekki solutions. Taken together, our results reveal how a single, scalar reduced equation elucidates unstable stationary states, self-similar collapse toward phase slips, and shock formation, providing an understanding large-scale phase dynamics in pattern-forming systems.

Path geometries provide a geometric encoding of systems of second order ODE, which serves as a model for the geometric theory of more general systems of ODE and for cone structures. They are an instance of the family of parabolic geometries, thus they are second order structures that are difficult to study using the usual tools of differential geometry. The general theory of parabolic geometries provides several efficient tools for the study of path geometries, but these use Cartan geometry methods and hence are not easily accessible. In this article, we build a bridge between these general methods and an elementary approach to path geometries. Motivated by the general theory of Weyl structures (but not using it), we first define a family of distinguished connections that is analogous to Webster-Tanaka connections in CR geometry. These are parametrized by (local) non-vanishing sections of a line bundle naturally associated to the geometry, and the dependence of this choice is described explicitly. We also discuss the Schouten tensor associated to such a choice and its dependence on the choice. We explain how these ingredients can be used to obtain an elementary approach to tractor calculus for path geometries and give examples of applications to the construction of invariant operators. A second major result that we prove is that in the case of path geometries, there is a smaller subclass of distinguished Weyl structures which does not seem to have an analog for any other type of parabolic geometries. This has interesting relations to the refinement of the de Rham complex induced by a path geometry via the machinery of BGG sequences. Again, all this is proved using elementary methods without reference to the general theory.

This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional Laplacian and some operators compared to it. Based on the $Γ$-convergence method and compactness arguments, we prove the homogenization theorems for these nonlocal operators with product-type and symmetric coefficient-structured kernels respectively. Furthermore, these results are extended to general nonlinear nonlocal equations.

Charged lepton-flavor violation is a null-test frontier of the Standard Model and a direct probe of physics beyond it. We present a global effective field theory (EFT) analysis across FCC-ee, ILC, CLIC, HL-LHC, HE-LHC, and muon colliders at 3 and 10 TeV, with operator identification as the primary target rather than exclusion reach alone. The analysis combines low-energy constraints, collider differential observables, and Dalitz-level $μ\to 3e$ information in a common profile-likelihood framework. Key hadron-collider and muon-collider signal/background samples are generated at event level and propagated through Delphes detector simulation, while clean $e^+e^-$ benchmark channels are modeled with CDR-calibrated parametric response. We include one-loop renormalization-group (RG) running and operator mixing between UV matching and measurement scales, finding 10--30\% shifts in selected operator-correlation entries when comparing tree-level and RG-evolved coefficient mappings at multi-TeV matching scales. Polarization asymmetries are used to separate $c_{H\ell}$ and $c_{He}$ directions, and UV discrimination is quantified with Bayes factors for benchmark leptoquark and heavy-neutral-lepton hypotheses. The full code chain for event generation, detector response, inference, and figure reproduction is provided.

Let $\mathcal{V}^c(\mathfrak{gl}_N)$ be Etingof--Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $ϕ$-coordinated $\mathcal{V}^c(\mathfrak{gl}_N)$-modules and the representations of quantized enveloping algebra $U_h(\mathfrak{gl}_N)$ and reflection equation algebra $\mathcal{O}_h(Mat_N)$. As an application, we demonstrate how the elements of the center of $\mathcal{V}^c(\mathfrak{gl}_N)$ at the critical level $c=-N$ give rise to the $q$-analogues of quantum immanants for $U_h(\mathfrak{gl}_N)$, which were recently found by Jing, Liu and Molev. Finally, we derive the analogues of these results for the quantum affine vertex algebra associated with the normalized Yang $R$-matrix.

Fast, scalable decoding architectures that operate in a block-wise parallel fashion across space and time are essential for real-time fault-tolerant quantum computing. We introduce a scalable AI-based pre-decoder for the surface code that performs local, parallel error correction with low decoding runtimes, removing the majority of physical errors before passing residual syndromes to a downstream global decoder. This modular architecture is backend-agnostic and composes with arbitrary global decoding algorithms designed for surface codes, and our implementation is completely open source. Integrated with uncorrelated PyMatching, the pipeline achieves end-to-end decoding runtimes of order $\mathcal{O}(1 μ\text{s})$ per round at large code distances on NVIDIA GB300 GPUs while reducing logical error rates (LERs) relative to global decoding alone. In a block-wise parallel decoding scheme with access to multiple GPUs, the decoding runtime can be reduced to well below $\mathcal{O}(1 μ\text{s})$ per round. We observe further LER improvements by training a larger model, outperforming correlated PyMatching up to distance-13. We additionally introduce a noise-learning architecture that infers decoding weights directly from experimentally accessible syndrome statistics without requiring an explicit circuit-level noise model. We show that purely data-driven graph weight estimation can nearly match uncorrelated PyMatching and exceed correlated PyMatching in certain regimes, enabling highly-optimized decoding when hardware noise models are unknown or time-varying, as well as training pre-decoders with realistic noise models. Together, these results establish a practical, modular, and high-throughput decoding framework suitable for large-distance surface-code implementations.

Operation at steady state is often not optimal when optimizing over an economic cost objective. In many cases, periodic operation yields better performance. Therefore, we derive asymptotic stability guarantees of an economic model predictive control scheme with terminal conditions for systems with optimal periodic operation for a more general setup than existing methods can handle. Moreover, we establish a non-averaged closed-loop performance bound by defining the closed-loop cost via a Cesàro summation instead of ordinary summation. Such a non-averaged performance bound provides new insights for systems with periodic optimal operation.